STEMscopes Georgia Math Teacher Guide Grade 6 by acceleratelearning

6 Georgia Math Teacher Guide

Grade 6 Teacher Guide

STEMscopes.com ISBN: 979-8-88826-716-5

ISBN: 979-8-88826-665-6

6 GEORGIA

MATH G6

GEORGIA

Teacher Guide: Grade 6 ISBN: 979-8-88826-665-6 Published by Accelerate Learning Inc., 5177 Richmond Ave, Suite 800, Houston, TX 77056. Copyright © 2023, by Accelerate Learning Inc. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior written consent of Accelerate Learning Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. To learn more, visit us at www. www.stemscopes stemscopes.com.

USING THE TEACHER GUIDE

Using the Teacher Guide Plan and Strategize In the Teacher Guide,, you’ll ﬁnd details about each element in our curriculum. Use these summaries to guide what you’ll integrate into your lessons based on the needs of your students and your teaching style. Throughout each scope, facilitation focuses primarily on understanding Vertical Alignment along with core Engage and Explore elements. Please note that all other elements are still available online.

Discover and Facilitate As you move through each scope, find STEMscopes Tips that explain how to use and where to find many of the aligned resources that are included throughout the curriculum. In each Explore lesson, you’ll also find Facilitation Tips to assist you in this critical part of the learning process.

Journal and Record The Teacher Guide includes areas throughout its pages for you to write notes about lessons, your students, and more. There are also areas to sketch out long-range plans, make observations, and coordinate smallgroup sessions.

Reﬂect and Enhance Trying to remember what you did last year when teaching a lesson? Use the notes and plans you write here to remind you. Find out what works, what doesn’t, and how to do it better from year to year with our product to help you along the way. When it’s time for a new year, it’s also time for a new Teacher Guide. Guide. Keep them to reference or share them with a colleague.

Using STEMscopes ............................................................................................... 4 SCOPE 1

Add and Subtract Fractions ................................................................................. 18

SCOPE 2

Multiplication and Division Problem Solving Using Fractions ................................ 32

SCOPE 3

Add and Subtract Decimals ................................................................................. 54

SCOPE 4

Multiply and Divide Decimals .............................................................................. 68

SCOPE 5

Integers ............................................................................................................. 96

SCOPE 6

Rational Numbers ............................................................................................. 116

SCOPE 7

Equivalent Numerical Expressions ..................................................................... 136

SCOPE 8

Algebraic Expressions ...................................................................................... 166

SCOPE 9

Equations and Inequalities ................................................................................ 190

TABLE OF CONTENTS

Table of Contents

SCOPE 10 Ratios, Rates, and Unit Rates ............................................................................ 214 SCOPE 11 Percents .......................................................................................................... 242 SCOPE 12 Measurement Conversions................................................................................ 262 SCOPE 13 Coordinate Planes ............................................................................................ 278 SCOPE 14 Coordinate Plane Problem Solving .................................................................... 292 SCOPE 15 Area and Volume .............................................................................................. 308 SCOPE 16 Surface Area .................................................................................................... 334 SCOPE 17 Represent and Interpret Data............................................................................. 350 SCOPE 18 Summarize Numerical Data ............................................................................... 374

YEAR AT A GLANCE 2

Year at a Glance JULY

AUGUST

SEPTEMBER

OCTOBER

NOVEMBER

DECEMBER

JANUARY

FEBRUARY

MARCH

APRIL

MAY

JUNE

YEAR AT A GLANCE

Year at a Glance

USING STEMSCOPES

Using STEMscopes Utilizing the Home Section In the Home section, build your own content knowledge, analyze the standards, and gain an understanding of everything the scope has to offer. This is where you will ﬁnd all your lesson-planning materials so you can facilitate fun, purposeful experiences for your students. CONTENT SUPPORT • The standard(s) being addressed in the scope • The mathematical thinking and reasoning standards addressed in the scope • Student misconceptions and obstacles teachers may face • Detailed description of the content • Extensive list of terms and deﬁnitions students should know • Sample student responses to example questions • An overview of related concepts students will learn in future grades

Use Content Support to gain background knowledge to fully support the students’ understanding. • Includes the reasons a concept is being taught a certain way, examples that can be used to help teach the concepts, and sample student questions and answers • Explains what the students have already learned and gives insight to the concepts students will learn next • Provides known misconceptions students have about the content and obstacles teachers may face when teaching the content • Includes vocabulary and deﬁnitions students should learn throughout the scope Ideas for using this element: • Use it as a resource to understand why math concepts are taught a certain way and how the concepts should be taught. • Use it to understand what students should know before you teach the content, what they should learn, and what they will need to know to be successful in future grades.

STANDARDS EXPLAINED • The standard(s) being addressed in the scope • The verbs used in the standard that highlight what students should be doing • Concrete words and deﬁnitions students should know • A brief summary of the implications for instruction, including what students should understand by the end of the scope

Use the Standards Explained to fully understand the standard(s) that are being addressed in the scope. • Includes what students should be doing and what words they should know • Explains what the students must know to meet the standard • Shows the vertical alignment of relevant standards throughout the grade levels Ideas for using this element: • Use it to become familiar with the standard(s) being addressed and fully understand the concepts students need to know.

• A vertical alignment of related standards 4

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

MATERIALS LIST • The ability to generate the total quantity of materials needed based on your class • A list of all the supplies that are needed for the Engage, Explore, Explain, Elaborate, Evaluate, Intervention, and Acceleration sections • A breakdown of each material, including the quantity per use; the item; whether the item is used per student, per pair, per group, or per class; whether the item is printed, reusable, or consumable; and the total quantity needed

Use the Materials List to plan for the materials that will be needed throughout the scope.

USING STEMSCOPES

Home

• Includes the ability to individualize the number of materials needed based on the total number of students, number of groups and stations, maximum class size, and total number of classes • Lists the materials needed for all the activities throughout the scope Ideas for using this element: • Use it to plan the materials you will need throughout the scope.

SCOPE OVERVIEW Use the Scope Overview to see every component of the scope. • Provides an easy-to-read, color-coded graphic showing the activities included in each element • Shows the sequential path students will take as they move through the scope • Includes the standard(s) and suggestions of how to use certain elements Ideas for using this element:

• The standard(s) addressed in the scope • Each element in the scope • The title of each part of an element • The order in which the scope should be taught

• Use it to quickly see the parts of the scope and how they interconnect. • Use it to plan how you will move through the scope.

PARENT LETTER • A description of the content of the Parent Letter • Procedure and facilitation points that provide a time frame for distributing the Parent Letter and suggestions for encouraging parent participation in the at-home activity

Use the Parent Letter to explain math concepts to parents. • Has a brief overview of the concepts being taught • Includes vocabulary terms and deﬁnitions students need to know • Provides resources and activities students and parents can do together to practice the concepts Ideas for using this element: • Use it to keep parents informed about what their children are studying in math. • Send home a copy of the Parent Letter the week before to notify parents of upcoming concepts and ways to help at home. • Be prepared to explain activities as questions arise from parents.

USING STEMSCOPES

Using STEMscopes Introducing Content with the Engage Section The Engage section is all about laying the foundation for learning. You begin this section by pre-assessing students using the APK (Accessing Prior Knowledge) and ﬁlling knowledge gaps using the Foundation Builder. The Hook then lays out a storyline narrative to establish a purpose for learning and captures students’ attention with real-world connections. ACCESSING PRIOR KNOWLEDGE • A general description of the activity and how it relates to what is being taught in the scope • Materials and preparation needed to complete the activity • Procedure and facilitation points that take you step by step through the activity • Sample student responses to embedded discussion prompts • A handout students use to complete the activity, if needed

Use the APK activity to help determine what students already know about the content as well as any misconceptions they have before beginning the scope. • Activates students’ thinking about the concept and how it’s been presented to them previously • Gives students opportunities to display what they know • Identifies the need to use the Foundation Builder to fill any knowledge gaps • Reveals possible misconceptions Ideas for using this element: • Due to the nature of this element, it is suggested that you complete this activity before the Hook activity. • Students typically complete and discuss the activities in small groups. • Student misconceptions identified here can be addressed and corrected as students progress through the scope.

FOUNDATION BUILDER • A general description of the activity • Materials and preparation needed to complete the activity • Procedure and facilitation points that take you step by step through the activity • Sample student responses to embedded discussion prompts • Handouts, Slideshows, and any other printed materials students will use to complete the activity

Use the Foundation Builder to help ﬁll learning gaps and review and reinforce previously taught content before beginning the scope. • Reteaches content previously taught • Uses concrete materials students can manipulate to explore mathematical concepts and develop proﬁciency • Addresses vocabulary with multiple meanings to eliminate confusion Ideas for using this element: • This activity is intended to be a short teacher-guided intervention for use in small groups. • Student preconceptions are addressed and corrected during this activity. Notes

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

HOOK • A general description of the activity • Materials and preparation needed to complete the activity • Procedure and facilitation points that take you step by step through the activity • Pre- and Post-Explore sections • A video showing a real-world situation • Sample student responses to embedded discussion prompts • Handouts, videos, and any other printed materials students will use to complete the activity

Use the Hook to engage students using real-world contexts where speciﬁc math skills are needed. Here, students have their ﬁrst experience with the new content.

USING STEMSCOPES

Home

• Introduces a real-world problem that requires use of the skills that will be taught in the scope • Uses media to show the real-world situation in action • Give students the opportunity to see how math is used in a real-world situation • Is revisited and the problem is solved after students complete the Explore activities from the next section Ideas for using this element: • Explain the real-world situation while showing the video. • Facilitate a discussion about how the scope’s math concepts are used in the situation. • Return to the Post-Explore section to solve the problem after completing the Explore activities. • Students typically complete and discuss the Post-Explore activities in pairs or small groups.

USING STEMSCOPES

Using STEMscopes Student Learning Using the Explore Section This is where students dig into the meat of the content. The Explore section provides scaffolded hands-on activities that build toward mastery of the standards. Each Explore supplies prompts for rich discussion and student reasoning, a Student Journal, and an Exit Ticket. The Explore section also gives students access to Virtual Manipulatives and teachers access to Skill Basics lessons designed to reinforce basic concepts before introducing the Explores. EXPLORES • A suggestion of which Skill Basics to use before completing the Explore, if applicable • A general description of the activity • The Mathematical Thinking and Reasoning Standards addressed in the Explores • A brief setup video showing the materials and preparation needed and explaining the activity • Materials and preparation needed to complete the Explores • Procedure and facilitation points that take you step by step through the activity • A scenario involving a realworld situation students need to solve • Sample student responses to embedded discussion prompts • Math Chat questions at the end of each Explore

Use the Explores to focus on developing students’ conceptual understanding of speciﬁc math skills using relevant situations and manipulatives. As students work through the activities, they will develop more abstract thinking and better number sense. • Provides real-world problems to motivate students to ﬁnd solutions using the math skills covered in the scope • Involves hands-on learning, rich discussions, and collaboration that encourage students to use thinking and reasoning skills • Reduces dependence on manipulatives as students progress through the activities • Helps students acquire new mathematical vocabulary through academic language embedded in the activities Ideas for using this element: • Read and discuss the real-world situations. • Provide an opportunity for students to work through the activities with partners or in small groups. • As students collaborate, monitor and assess their understanding by asking guiding questions. • Guide and correct students through any misconceptions noted during discussions or on their Student Journals. • Provide a Math Chat time at the end of the activity for students to share their observations and learning. • Have students complete the Exit Ticket to formatively assess their understanding of the concepts. • Use students’ responses from the discussions, Student Journals, and Exit Tickets to guide future instruction. Notes

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

VIRTUAL MANIPULATIVES • The Virtual Manipulatives include components such as these: • Place Value Disks • Number Lines • Fraction Circles • Fraction Tiles • Color Tiles • Geoboard • XY Coordinate Board

Use the Virtual Manipulatives to provide each student with a limitless supply of manipulatives.

USING STEMSCOPES

Home

• Helps students explore mathematical concepts • Makes learning engaging and meaningful • Leads to more complex understanding of math concepts • Allows students to make visual connections between math concepts and the virtual manipulatives • Helps students develop mental models and abstract thinking Ideas for using this element: • Use the Virtual Manipulatives in the classroom or remotely in place of concrete objects. • Encourage students to use the Virtual Manipulatives to develop proﬁciency in math concepts. • Differentiate instruction by using the Virtual Manipulatives for Englishlanguage learners and for students who are struggling with the concepts. Students can also beneﬁt from visual models when learning new concepts. • Use the Virtual Manipulatives to help address and clarify student misconceptions.

USING STEMSCOPES

Using STEMscopes Guiding Students Using the Explain Section The Explain section offers a variety of resources that help connect the experiences of the Explore activities to the academic content students need to know. These resources include Anchor Charts, Picture Vocabulary, My Math Thoughts, Show What You Know, and, in some scopes, an Interactive Notebook that can be used to support the Explore activities and solidify student learning. ANCHOR CHARTS • A general description of each activity • An Anchor Chart for each Explore • Sample student responses to embedded discussion prompts • A printable sample Anchor Chart

Use the Anchor Charts during or after the Explore activities as a tool to anchor student learning of the concepts addressed in the scopes. • Provides large, poster-sized visuals of the most important content strategies • Helps students achieve mastery of skills and reinforce concepts throughout the year • Gives students access to the charts to use as resources when needed Ideas for using this element: • Create Anchor Charts during instruction or after the Explore activities. • Ask students guiding questions while interacting with the Anchor Chart to help reinforce students’ understanding of concepts. • Display Anchor Charts during instruction or throughout the year to review learning.

PICTURE VOCABULARY • A slideshow of each relevant vocabulary word • Starting in Grade 2, a ﬂash card option with either the picture and word or the picture and deﬁnition for each word • A printable copy

The Picture Vocabulary presents new vocabulary with pictures and studentfriendly definitions. • Includes a slideshow with a picture and written or visual definition for each vocabulary word • Clarifies the meaning of words used throughout the scopes • Gives students access to the vocabulary words to use as a resource when needed Ideas for using this element: • Directly teach math vocabulary using the Picture Vocabulary. • Refer to the Picture Vocabulary throughout the scope to reinforce students’ understanding of vocabulary terms. • If available, encourage students to use the flash card feature to learn relevant math vocabulary. Notes

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

SHOW WHAT YOU KNOW Use the Show What You Know to allow students to independently demonstrate their understanding and practice new skills after exploring concepts.

• A different Show What You Know activity to correspond with each Explore

• Allows students to apply the knowledge and skills they learned in the Explore activities to new situations

• A general description of the activity

• Correlates each activity piece with the same-number Explore. For example, Show What You Know – Part 1 allows students to practice the skills they developed in Explore 1.

• Materials and preparation needed to complete the activity

Ideas for using this element: • Assign the activity for students to complete independently after ﬁnishing the corresponding Explore. • Provide reading assistance if needed.

USING STEMSCOPES

Home

• Procedure and facilitation points that identify how to use the activity • A printable Student Handout and Answer Key

• Provide manipulatives, especially those used in the Explore, as needed. • Identify whether instruction needs to be adjusted based on student misconceptions before proceeding to the next Explore.

INTERACTIVE NOTEBOOK • A general description of the Interactive Notebook

Use the Interactive Notebook to allow students to take notes, express ideas,

• Materials and preparation needed to complete the activity

and/or process the information presented in class.

• Procedure and facilitation points that identify how to use the activity

Ideas for using this element:

• A printable Student Handout

• Provides students with the opportunity to solidify their learning

• Prepare an Interactive Notebook using a spiral or composition notebook for each student. • Precut or allow students to cut the pieces for each Student Handout according to the instructions. • Allow time for students to complete the activity and then glue the pieces in their Interactive Notebook. Notes

USING STEMSCOPES

Using STEMscopes Extending Learning with the Elaborate Section Workstations are a go! The Elaborate section makes differentiation a cinch with readymade activities—digital and paper-based games, Spiraled Review, Career Connections, literacy connections, and more—that are perfect for rotations! These activities allow students to continue learning while you make time for small-group interventions, reteaching, and independent projects to help both struggling and advanced learners. FLUENCY BUILDER • A description of the activity • Materials and preparation needed to complete the activity • Procedure and facilitation points that take you step by step through the activity • Printable Instruction Sheets and game materials

Use the Fluency Builder games to give students the opportunity to practice the skills they learned during the Explore activities. • Involves games designed to be motivating and entertaining • Increases focus and collaboration skills as students play with partners or in small groups • Allows students to continue to practice skills throughout the year using the games • Develops ﬂuency as students become more eﬃcient and accurate when using their math skills during game play Ideas for using this element: • Place students with partners or in small groups. • Read the game directions, and model the game if needed. • While students are playing the game, pull students for small-group intervention, for reteaching, or for supporting their work on independent projects.

SPIRALED REVIEW • A general description of a Spiraled Review • Preparation needed to complete the activity • Procedure and facilitation points that take you step by step through the activity • Multiple real-world questions that cover previously taught math concepts • Printable Student Handout and Answer Key

Use the Spiraled Review to allow students to continue to practice skills throughout the year. • Motivates students to use the math skills to find solutions for real-world problems • Allows students to review previous or current grade-level content based on the focal points set for each grade • Gives students the flexibility to use different processes and strategies to reach solutions • Develops fluency as the students become more efficient and accurate in solving problems Ideas for using this element: • Read the story to engage student interest before moving on to the questions. • Use the Spiraled Review as a warm-up in class or send it home for homework, but be sure to discuss answers and strategies with the class as a whole group. • Refer to the standard in the lower right-hand corner of each question box to assess the students’ content knowledge or need for further intervention.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

DATA SCIENCE Use data science to guide your students through analyzing data sets and finding correlations to data and the scope content. Also, lay the foundation for exploring trends and analytics. • Allows students an opportunity to see statistics presented at their grade level and in context of their learning. Ideas for using this element: • Use as a summary of learning at the end of a scope. • Have students work in groups to find data that supports a specific topic or opinion.

• A data set related to the topic being taught • Materials and preparation needed to complete the activity

USING STEMSCOPES

Home

• Procedure and facilitation points that take you step by step through the activity • Printable handouts

• Use as an extension activity.

INTERACTIVE PRACTICE Use the Interactive Practice to engage students in practice using technology. • Increases student participation and focus through graphics, sound, point accumulation, and engaging content • Develops ﬂuency as the students become more eﬃcient and accurate in solving problems

• An interactive online game • A “Show Answer” button • A feature that reads the questions • Sound and music that can be muted

Ideas for using this element: • Use the Interactive Practice as a workstation activity, or assign it as homework. • While students are working on the activity, pull students for small-group intervention, for reteaching, or for supporting their work on independent projects. Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________

USING STEMSCOPES

Using STEMscopes Assessing Using the Evaluate Section Get the data you need from the assessment tools provided in the Evaluate section. From performance-based assessments to Skills Quizzes and Observation Checklists, there are multiple evaluations to ensure students have mastered the standards. MATHEMATICAL MODELING TASK • A real-world prompt • Printable Student Handout and Answer Key

Use the Mathematical Modeling Task assessment to evaluate students’ ability to use mathematical evidence and reasoning in a realstic context. • Allows students to write out an argument in response to a relatable real-world prompt and provide support for their response • Focuses on real-world applications in new situations where complex reasoning and planning are necessary • Enhances critical thinking involved in problem solving and heightens students’ ability to make connections among mathematical ideas Ideas for using this element: • Review students’ responses to determine student mastery of math concepts. • Use students’ responses to determine whether students would beneﬁt from Intervention or Acceleration activities. Notes

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

SKILLS QUIZ • Multiple skills-based questions • Printable Student Handout and Answer Key

Use the Skills Quiz to identify which skills addressed throughout the scope students have mastered.

USING STEMSCOPES

Home

• Focuses on facts, details, deﬁnitions, and procedures with one correct answer Ideas for using this element: • Review students’ responses to determine student mastery of math skills. • Use students’ responses to determine whether students would beneﬁt from Intervention or Acceleration activities.

STANDARDS-BASED ASSESSMENT Use the Standard-Based Assessment to identify which concepts and skills presented throughout the scope students have mastered. • Focuses on applying skills and concepts in addition to answering how or why with one correct answer

• Multiple skills- and reasoning-based questions • Printable Student Handout and Answer Key

Ideas for using this element: • Review students’ responses to determine student mastery of math skills and concepts. • Use students’ responses to determine whether students would beneﬁt from Intervention or Acceleration activities.

USING STEMSCOPES

Using STEMscopes Using the Intervention and Acceleration Sections Useful during Elaborate or as an after-school support, Intervention contains a small handson activity designed to target students’ conceptual misunderstandings while building their math skills. The Intervention activities can also be used as a reteach or test-prep tool. In the Acceleration section, students connect the mathematical concepts to either science or engineering or relate what they’re learning to current events around the world. SKILL REVIEW AND PRACTICE • A general description of the activity • Materials and preparation needed to complete the activity • Procedure and facilitation points that take you step by step through the activity • Teacher Checklist to monitor students’ mastery • Depending on the scope, a Checkup and Answer Key, Student Handout, and other printed materials students will use to complete the activity

Use the Skill Review and Practice to revisit concepts to build student understanding. • It can be used flexibly as a review of previously learned concepts or as a tool for targeted intervention. • The process begins with a Quick Check. The Quick Check includes a brief set of questions that assess the individual skills covered by the scope. • Once these gaps are identified, they can be addressed using the corresponding activities on the Review. Each section of the review includes instructional guidance on the skill and an opportunity to practice it. • When the Review is complete, it is important to reassess students to determine whether the activity was effective. The Checkup is a ten-question quiz that can be used to evaluate what students know. Students could be asked to complete the entire assessment or just the specific questions that address the skills they needed to work on. Ideas for using this element: • Select small groups of students who need more support to develop mastery of math skills and concepts. • Provide ample opportunities for students to use manipulatives to explore mathematical concepts. • Ask guiding questions throughout the activity to assess students’ understanding and address misconceptions.

INTERACTIVE SKILL REVIEW • A general description of the activity • Materials and preparation needed to complete the activity

The Interactive Skill Review is an engaging digital game that allows students to practice vertically aligned skills from previous grade levels. • You can digitally assign the game to your whole class, a group of students, or an individual student as needed. • Provide students with an interactive way to review concepts that support their current learning. Ideas for using this element: • The games in the Interactive Skill Review element can be used for independent practice, for homework, or as a workstation in the classroom.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

CHOICE BOARD • Menu of options for students • Printable choice board • Handouts for each activity • Choice board selfassessment

Use Choice Boards to offer students an opportunity to decide and navigate their own learning extensions.

USING STEMSCOPES

Home

• Choice Boards offer a menu of activity options for students. Students are empowered to decide which items they would like to complete based on their unique interests. • There is a wide range of tasks to choose from in each scope, including connections to careers, culinary science, art, personal ﬁnance, noteworthy mathematicians, and more! These options provide opportunities for students to pursue a deeper understanding of the math concepts they are learning by connecting them to the real world. Ideas for using this element: • One option is to have each student choose an activity from the Choice Board to complete as a capstone project for the scope. This allows students to take everything they have learned and apply it in a way that interests them. • Another option is to have the Choice Board and the Activity Handouts ready in case students ﬁnish an assignment early. Or students could choose one of the activities to complete for homework that week. • Many of the options from the Choice Board have a corresponding Activity Handout to guide students through the task. The Choice Boards and the Activity Handouts can be downloaded from the Print Files section.

WOULD YOU RATHER Use Would You Rather activities to encourage critical thinking by asking students to choose between two options and justify their choice. • The Would You Rather activities are designed to encourage critical thinking by asking students to choose between two options and justify their choice.

• Description of the activity • Printable student handout • Answer Key

• Students will need to apply their math skills to decide on an answer to the prompt. Then, they will explain their thinking using precise mathematical language and reasoning. Ideas for using this element: • Promote new ways of looking at mathematical situations and applications • Illustrate that there’s more than one way to arrive at a correct conclusion • Facilitate classroom discussion and debate around math topics Notes __________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________

SCOPE 1

Add and Subtract Fractions Scope Introduction SCOPE SUMMARY Students will continue to build on the base they have from the introduction to adding and subtracting fractions from elementary school. In fourth grade students worked with added and subtracting fractions and mixed numbers with like denominators. Then, in ﬁfth grade, students learned to add and subtract fractions with unlike denominators, including mixed numbers. This will require students to ﬁnd and use common denominators. Students will select strategies to solve problems with like and unlike denominators. Student Expectations

6.NR.1.1 Fluently add and subtract any combination of fractions to solve problems.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In ﬁfth grade, students added and subtracted fractions with unlike denominators. Students know that a fraction is an equal part of a whole. They focused on adding and subtracting fractions by representing problems with models and solving. Students added and subtracted fractions and mixed numbers.

Later in sixth grade, students will learn to convert between any of the following: fractions, decimals, and percents. Students will use their understanding of fractions as division to apply to their proportional understanding. In seventh grade, students will need to solve multistep, contextual rational number problems, including converting between rational number forms when necessary.

Before beginning the lesson, students’ prior knowledge is assessed. The teacher assesses students’ ability to: •

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES

model and solve problems involving addition and subtraction of fractions.

At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope. If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK.

represent and solve addition and subtraction of fractions with unequal denominators.

Students move on to the Explore activities and come back at the end of the explorations to repeat the activity and discuss what they found.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Add and Subtract Fractions In this exploration, students will use student-selected strategies to add and subtract fractions. Students will: •

select a strategy of their choice to determine how much of the land is already planted.

Explore 2

Explore 1

EXPLORE ACTIVITIES Add and Subtract Mixed Numbers and Improper Fractions In this exploration, students will add and subtract mixed numbers and improper fractions using student-selected strategies. Students will: •

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

use fraction tiles, fraction towers, or fraction circles to solve problems.

ADD AND SUBTRACT FRACTIONS

Home

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve.

ADD AND SUBTRACT FRACTIONS

Add and Subtract Fractions Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will match numbered cards with lettered cards posted around the room to demonstrate their knowledge of the prior standard. This element is designed to uncover student misconceptions; it should not be taken for a grade. 5.NR.3.3 Model and solve problems involving addition and subtraction of fractions and mixed numbers with unlike denominators.

Materials

Preparation

Printed •

• •

1 Set of Match around the Room Cards (per class)

Print one set of Match around the Room Cards. Hang them in a random order around the room.

ADD AND SUBTRACT FRACTIONS

Home

Procedure and Facilitation Points 1. 2.

Have students write the numbers 1, 2, 3, and 4 on a sheet of paper. Instruct students to walk around the room with their papers. As they walk around the room, they need to see the numbered cards and match them with the lettered cards. Allow students to share their thinking with their neighbors. a.

Card 1 matches with Card C.

Card 2 matches with Card B.

Card 3 matches with Card D.

Card 4 matches with Card A.

Depending on your students and classroom space, consider distributing Cards A, B, C, and D to students at their desks. Have them make observations about the visual representations. Allow silent think time, then shoulder partner time to discuss. FACILITATION TIP

If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

Identifying Misconceptions •

FACILITATION TIP

Students may struggle ﬁnding a common denominator. Have students write out multiples of each denominator and circle the lowest common denominator.

After providing students time to think and share about Cards A, B, C, and D, project Cards 1, 2, 3, and 4 one at a time for students to look for possible matches. Guide a class discussion and welcome all answers and ideas. FACILITATION TIP If this scope is taught in the ﬁrst weeks of school, the Foundation Builder might be particularly useful. It provides additional opportunities to assess prior knowledge.

ADD AND SUBTRACT FRACTIONS

Add and Subtract Fractions Hook – Pizzas ACTIVITY PREPARATION Students will represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole, using objects and pictorial models and properties of operations.

Materials

Preparation

Printed •

• • •

1 Pizzas (per class)

Reusable •

1 Phenomena Video (per class)

Consumable •

Plan to show the video. Prepare to project Pizzas for the whole class to view. Prepare to introduce the scenario and to encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

Colored pencils (optional)

PROCEDURE AND FACILITATION POINTS Part I: Pre-Explore 1. FACILITATION TIP Project the text of this scenario for students. Guide them through a careful reading. Allow quiet think time for them to individually read, then shoulder partner time, and ﬁnally read it aloud together as a class. Model how to locate the essential math phrases and values.

FACILITATION TIP Encourage students to discuss the models and make observations about the fractions even if they can already intuit the solution. Being ﬂuent with visual models and noticing the details about fractions will help them succeed when more complex scenarios are presented.

4. 5.

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: Charlie’s family has 1 decided to have pizza for dinner. Charlie can eat __2 of a pizza. His sister Sarah can 3 3 1 eat __4 of a pizza. His dad usually eats __4 or a pizza, and his mom usually eats __8 of a pizza. The pizzas are divided into 8 pieces. We need to ﬁgure out how much pizza Charlie’s family will need to order. Ask students the following questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Student answers will vary. I notice that Dad and Sarah’s fractions have the same denominator. I wonder what type of pizza each person likes to eat. Project Pizzas for students to view. Explain to students that the fraction of pizza that each person can eat is given to us. We need to determine how much pizza the family will need to order. Discuss the following questions: a.

DOK-1 What do you notice about the fractions? Not all of the fractions have the same denominator.

DOK-2 What types of models can we use to represent fractions? Student answers will vary. We can use fraction circles, fraction strips, or number lines.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and restate the problem. Refer to Pizzas, and discuss the following questions: a.

DOK-1 How did you solve the problem? Student responses will vary based on the strategy selected.

DOK-2 What equivalent fractions did you use? The fraction __2 is the same

as __8, and __4 is the same as __8. 22

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

DOK-2 How many pizzas should they order? They need 15 slices of 15

.The pizzas are divided into 8 slices, so they need to order 2 pizza, or ___ 8

pizzas. There will be 1 slice (__8) left. 1

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ADD AND SUBTRACT FRACTIONS

Add and Subtract Fractions Explore 1 – Add and Subtract Fractions ACTIVITY PREPARATION Students will use student-selected strategies to add and subtract fractions.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.6 Attend to precision. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • •

•

1 Student Journal (per student) 1 Set of Scenario Cards (per group) 1 Exit Ticket (per student)

• •

Reusable • • • •

1 Resealable bag (per group) 2 Colored pencils, different colors (per student, optional) 1 Dry-erase marker, thin tip (per student, optional) 1 Ruler (per student, optional)

• •

Plan to divide the class into 2 groups to complete the activity in Part I. Place students in groups of three or four. Print Scenario Cards for each group on card stock (if possible), or laminate them so they can be reused. Place each set in a resealable bag. Print a Student Journal and an Exit Ticket for each student. Optionally, provide students with fraction towers, fraction circles, or fraction tiles to aid in learning.

PROCEDURE AND FACILITATION POINTS STEMscopes Tip The Standards list is located along the menu bar. Here, a keyword can be entered to locate each standard. The search will result in a list of standards and direct links to the scopes where those standards appear. The standards are organized by grade level as well. Clicking on a standard within a grade level will also provide direct links to the scopes. FACILITATION TIP Depending on your students, consider providing time with fraction manipulatives before presenting any Scenario Cards. Allow students to organize and sort pieces. Monitor and assess student knowledge by asking a few key questions from 6a–6f. before beginning the Scenario Cards.

2. 3. 4.

5. 6.

Read the following scenario to the class: Your aunt and uncle have a big farm where they grow crops. You enjoy being on their farm, so you have offered to help your uncle and aunt with the chores during the weekends. Uncle Tomas and Aunt Paulina have divided their farmland into several different sections. They have decided to plant some corn and wheat in one of the sections. They know they will have some land left in that section where they can plant beans, but they’re not sure how much land is left. They have asked for your help. Give a Student Journal to each student. Optionally, distribute two colored pencils and a ruler to each student. Give one bag of Scenario Cards to each group. Explain to students that they can select a strategy of their choice to determine how much of the land is already planted on and how much is left over from each Scenario Card. Have fraction towers, fraction circles, and fraction tiles available for students to use as necessary. As students are working, monitor and check for understanding. Ask the following questions: a.

DOK-2 Does the value of the fraction change when you create an equivalent fraction? No, it doesn’t, because they represent the same part of the whole.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 What does the denominator in a fraction show? It shows the total number of pieces in a whole or set.

DOK-2 Why do the denominators have to be the same to add or subtract fractions? The pieces of the whole or set have to be the same size; otherwise, you cannot combine them into one amount or take any away.

e. DOK-3 How can we ﬁnd a common denominator? Explain your thoughts. You can multiply the number of pieces in one fraction by the number of pieces in the other fraction. It gives us same-sized pieces for both fractions, and then we can add or subtract. f.

DOK-2 Why does multiplying the numerator and denominator by the same number create an equivalent fraction? A fraction that has the same numerator and denominator is equal to one. Anytime you multiply something by one, you get the same value.

DOK-2 What does the numerator represent? It represents the number of sections that are colored.

h. DOK-3 How did you determine how many pieces to color in Scenario 2? I knew 2 of the 5 sections were colored in the ﬁrst fraction. Since each section had been broken up into a group of 6, there were a total of 12 pieces I needed to color. I knew 1 of the 6 sections was colored for the second fraction. Since each of those sections had been broken up into a group of 5, there were a total of 5 more pieces I needed to color. 7. 8.

Allow students enough time to complete each Scenario Card and answer the reﬂection questions on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 What do you notice about the equivalent fraction denominators when compared to the two original fraction denominators? They are the products of the original denominators. • DOK-3 What relationship do you see between the original fraction numerators and the equivalent fraction numerators? You can multiply the original fraction’s numerator by the same number you multiplied the denominator by to create an equivalent fraction. • DOK-2 How can you use multiplication to find equivalent fractions? I can multiply the numerator and denominator by the same number of groups of pieces to get an equivalent fraction. • DOK-3 Describe some of the strategies you chose to use to find common denominators and solve each problem. I chose to draw a model and divide it into sections. I chose to multiply the denominators and use their product as the denominator for the equivalent fraction and then simplify. I chose to find the least common denominator using the least common multiple of the two denominators. •

Intervention

Acceleration

FACILITATION TIP Reinforce key vocabulary and check for understanding before moving on: numerator, denominator, equivalent, common. Use the images in Picture Vocabulary. STEMscopes Tip The Visual Glossary, located under the Scopes tab, provides a variety of instructional resources. Browse topics alphabetically in English or Spanish. Each topic includes a visual and/or video featuring key vocabulary and concepts. The visuals include text and a speech button with narration. The videos, featuring real-world examples, are 3–15 seconds in length.

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FACILITATION TIP Be prepared for a wide variety of strategies. Consider providing some criteria for success for showing work in the workspace. Some students may prefer just to write the solution and others may prefer to draw very detailed models. FACILITATION TIP Take time before this scope or during this Math Chat to provide some additional relevant real-world examples of when students will need to be able to add and subtract fractions. Engage students with situations from popular hobbies, local events, or common careers.

Post-Explore 1. 2. 3.

FACILITATION TIP Have students complete the Exit Ticket to formatively assess their understanding When previewing this Exit Ticket with of the concept. students, allow time for questions and read Complete the Anchor Chart as a class. aloud support. If this Exit Ticket is one Have each student complete their Interactive Notebook. of the ﬁrst of the school year, clarify your expectations for showing work, writing names, including dates and any other important criteria.

ADD AND SUBTRACT FRACTIONS

Add and Subtract Fractions Explore 2 – Add and Subtract Mixed Numbers and Improper Fractions ACTIVITY PREPARATION Students will add and subtract mixed numbers and improper fractions using student-selected strategies.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.6 Attend to precision. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • •

• •

1 Student Journal (per student) 1 Set of Scenario Cards (per group) 1 Exit Ticket (per student)

• •

Reusable • •

2 Fraction tile sets, fraction tower sets, or fraction circle sets (per group) 1 Resealable bag (per group)

Place students in groups of 3 or 4. Print and cut out a set of Scenario Cards for each group. Use card stock, or laminate the cards if you plan to reuse them. Place them in a resealable bag. Print a Student Journal and an Exit Ticket for each student. Prepare fraction towers, fraction circles, or fraction tiles for each group.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before projecting this scenario for students, engage them by asking about any driving trips they've been on with family or friends. Consider familiar local destinations. FACILITATION TIP When previewing this Student Journal, take time to address the improper fractions included in the scenarios. FACILITATION TIP Depending on your students, consider starting with a fraction addition scenario (Scenario 1) and guide them through it step by step. Next, complete the Grand Canyon scenario (subtraction) all together as a class. Finally, allow them to complete Scenarios 2–7 in small groups.

2. 3. 4. 5. 6.

Read the following scenario to the class: Several friends have planned some road trips and agreed to take turns driving. They are trying to keep the amount of time they each drive as even as possible. They are also trying to keep a record of how far they drive each day. Help the friends determine their driving times. Give each student a Student Journal. Have fraction tiles, fraction towers, and fraction circles available for students to use as necessary. Ask students to read through each scenario and write the number sentence on their Student Journals. Have students rewrite the number sentence using common denominators. 2 Ask students to model 2__6 with the fraction sets. Ask the following question: a.

DOK-2 What problem do you see with this number sentence? We can 3

take one whole away, but we can’t take __6 away from __6. 7.

Allow students to explore what they could do to ﬁnd the solution to 2__6 – 1__6. 6

Guide students into seeing that they can take one whole and regroup it as __6.

Once the whole is regrouped, they can complete the subtraction problem. Ask the following question: a.

8. 26

DOK-2 What did you do to be able to subtract __6 from __6? We used one of

6 8 3 the wholes and regrouped it as __6. Then, we had 1__6. We took away 1__6 and 5 had __6 left.

Have students write the new equation and a solution statement that answers the scenario question on their Student Journals.

10. 11.

Engage

Explore

Explain

Elaborate

Evaluate

Give each group a bag with the Scenario Cards. Tell students they will be working together to find the solutions to the rest of the road trip problems. Each problem will require them to find the common denominators of the two fractions. They might also need to regroup to solve the problems. Encourage students to use the fraction sets for support when finding common denominators and regrouping. As students are working, monitor and check for understanding. Ask the following questions: a.

DOK-1 What is the ﬁrst step when you’re adding or subtracting fractions or mixed numbers? Find the common denominators so the pieces of the whole are all the same size. 7

b. DOK-2 How did you ﬁnd the common denominator of __3 and __4 in the ﬁrst scenario? Student responses will vary based on the strategy used. I used an area model. First, I divided up each square into 3 horizontal sections. Then, I divided each section into groups of 4 equal pieces. I ended up with 12 pieces, which is the common denominator.

Intervention

Acceleration

STEMscopes Tip STEMcoach in Action, located under the Scopes tab, provides teachers with professional development for the STEM-centered classroom. Explore a variety of topics that are broken into 3–6 subtopics with overviews describing teacher, classroom, and student expectations; FAQs and resources; and/or video libraries.

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c. DOK-3 Scenario 3 is an addition problem with mixed numbers. How did you solve this? Student responses will vary based on strategy used. First, I drew 4 wholes. I shaded in 3 of the wholes and divided up the fourth whole into 8 horizontal sections and 3 vertical sections. That meant we had 24 equal pieces, so 24 was our denominator. Then, we shaded in 5 of the 8 sections, which was 15 pieces, so 15 was our numerator. Then, we drew 2 more wholes and shaded in 1 of them. We divided the other whole into 24 equal pieces, and we saw that 2 of the 3 sections had 16 more pieces, so we shaded those in. Then, we counted up how many wholes we had and how many pieces we had. d. DOK-2 How did you know you needed to regroup one of the wholes 6 5 in Scenario 4? We couldn’t take ___ from ___ . We had to take one of the 10 10 10

and then add it to ___ . That gave us ___ . wholes and regroup it into ___ 10 10 10 6

12. 13.

from ___ . Then, we could subtract ___ 10 10

When students have found solutions to the seven scenario problems, have them answer the reﬂection questions on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-3 Why is it necessary to find common denominators when adding or subtracting fractions? Whenever you add or subtract fractions, the pieces need to be the same size. You can’t add or subtract different-sized pieces. • DOK-2 Why don’t you always regroup a whole into a fraction when subtracting? Sometimes you don’t need to. If the first fraction is greater than the fraction that is being subtracted, you don’t have to regroup; you already have enough pieces to be able to take some away. You only need to regroup when the fraction that is being subtracted is greater than the first fraction. •

Post-Explore 1. 2. 3. 4.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook. Return to the Hook and instruct students to use their newly acquired skills to successfully complete the activity.

FACILITATION TIP Use this Math Chat to provide some additional relevant real-world examples. Engage students with situations from construction jobs, creative design projects, and student interests.

FACILITATION TIP When previewing this Exit Ticket with students, allow time for questions and read aloud support. Clarify where their solutions are to be written for which question. FACILITATION TIP If this Exit Ticket is one of the ﬁrst of the school year, clarify your expectations for showing work, writing names, including dates and any other important criteria. 27

ADD AND SUBTRACT FRACTIONS

Add and Subtract Fractions Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Add and Subtract Fractions Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Add and Subtract Mixed Numbers and Improper Fractions

Interactive Vocabulary

Interactive Notebook

Students form deﬁnitions of mathematical vocabulary words used throughout the scope.

A cut-and-glue activity to process learning that can be added to a notebook for future reference

Independent practice assignment that gives students an opportunity to demonstrate their learning

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Add and Subtract Fractions Independent and partner games and other activities that provide students with an engaging way to practice the new concept

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Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________

Intervention and Assessment SMALL-GROUP PLANNER 1

Use the Skill Review and Practice under the Intervention section to assess student mastery after this scope’s content has been taught. Distribute a copy of Quick Check to each student. Each student should complete the Quick Check independently.

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

ADD AND SUBTRACT FRACTIONS

Add and Subtract Fractions

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Students

Notes & Comments

 Distribute a copy of the Review to these students.  Meet with students individually or in a small group to assist them in working through the Review’s concepts.  Talk individually with each student about their thoughts in order to highlight strengths and roadblocks.  Look out for possible misconceptions.  Distribute a copy of the Review to these students.  Meet with students in a small-group to answer questions and identify areas where students are struggling.  Look out for moments of possible reteaching.  Release students to work independently once you see they’re nearing mastery of the concept.

 Distribute a copy of the Review to these students.  Conﬁrm that students are on the right track.  Direct students to work on materials in the Acceleration section, such as the Choice Board, while you work with the other students.

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER Use this template to decide how to assess your students for concept mastery. Depending on the format of the assessment, you can identify prompts and intended responses that would measure student mastery of the expectation. See the beginning of this scope to identify standards and grade-level expectations. Key Concepts I can solve problems involving adding any combination of fractions by using multiple strategies.

What prompts will be used?

What does mastery look like?

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I can solve problems involving subtracting any combination of fractions by using multiple strategies.

I can apply common denominator strategies to solve addition problems involving fractions with different denominators.

I can apply common denominator strategies to solve subtraction problems involving fractions with different denominators.

I can apply reasoning strategies to solve problems involving addition and subtraction of any combination of fractions.

I can use numerical reasoning to interpret problems involving addition and subtraction of any combination of fractions.

SCOPE 1

Multiplication and Division Problem Solving Using Fractions Scope Introduction SCOPE SUMMARY

Student Expectations

In this scope, students will be expected to model multiplication of fractions and whole numbers using visual fraction models and number lines. Students will use the models to reason about the products of fractions and whole numbers. They will be expected to explain why the product of a fraction that is less than one and a whole number is smaller than the whole number and the product of a fraction that is greater than one and a whole number is greater than the whole number. Students will also be expected to model and solve multiplication of whole numbers with mixed numbers and reason about the size of the products.

6.NR.1.2 Multiply and divide any combination of whole numbers, fractions, and mixed numbers using a studentselected strategy. Interpret products and quotients of fractions and solve word problems.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In ﬁfth-grade, students interpreted a fraction as division of the numerator by the denominator ( __a b = a ÷ b). ). Students also began to model and solve multiplication problems with whole numbers and fractions. After this work with solving, students explained why multiplying a whole number by a fraction that is greater than one results in a product that is larger than the whole number and why a whole number multiplied by a fraction of less than one results in a product that is smaller than the whole number.

Effective strategies for computation with positive rational numbers relate to a series of other sixthgrade scopes. Sixth graders will apply computation strategies when exploring measurement conversions, equations and inequalities, area, surface area, volume, and ratios. In Grade 7, students will extend operations to include negative numbers. Proﬁciency of number operations will effectively allow for seventh-grade students to study repeating decimals, proportional relationships and scale drawings, equations and inequalities, circle measurement, and probability and sampling.

Before beginning the lesson, students’ prior knowledge is assessed. The teacher assesses students’ ability to: •

model and solve problems involving multiplication of a fraction and a whole number.

•

model and solve problems involving division of a unit fraction by a whole number.

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope.

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

determine the solution to a division of fractions problem to solve a realworld dilemma.

Students move on to the Explore activities and come back at the end of the explorations to repeat the activity and discuss what they found.

If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. 32

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Multiply Fractions by Whole Numbers In this exploration, students will model multiplication of a fraction by a whole number and a whole number by a fraction. Students will: •

Explore 2

Explore 1

EXPLORE ACTIVITIES

use fraction tiles to model multiplication with fractions.

•

model division of fractions.

•

determine the quotients of fractions3

Explore 4

Explore 3

In this exploration, students will solve a real-world scenario involving helping determine how many individual packages of each dessert item can be made with information provided on Fundraiser Dessert Cards. Students will:

In this exploration, students will model multiplication of fractions. Students will: •

create area models.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment. Modeling Fraction Division

Multiply Fractions and Mixed Numbers

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

Division of Fractions In this exploration, groups of students will solve a scenario to determine how many tablecloths can be made using each color of fabric for fundraiser tables. Students will: •

use visual and algebraic representations to model division of a fraction.

•

model division of fractions using number lines.

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve. Notes

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

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MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

Multiplication and Division Problem Solving Using Fractions Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will read different student responses to a posed question on the prior standard, decide whether they agree or disagree with the student, and explain their reasoning. This element is designed to uncover student misconceptions; it should not be taken for a grade. 5.NR.3.4 Model and solve problems involving multiplication of a fraction and a whole number. 5.NR.3.6 Model and solve problems involving division of a unit fraction by a whole number and a whole number by a unit fraction.

Materials

Preparation

Printed •

•

Print both pages of Agree or Disagree for each student.

1 Agree or Disagree (per student)

Procedure and Facilitation Points 1. 2. 3. 4. 5. 6. 7.

Instruct students to complete the Agree or Disagree independently. Once students have completed the activity on their own, have them stand up. Instruct all students to walk around the classroom with their hand raised in a high-five position. On your instruction, students will stop and high-five the closest person. This will be their partner. Give students a couple of minutes to discuss their answers and justifications together. Repeat steps 3–5 as many times as you want with different partners. Discuss the responses as a class. Allow students to explain their reasoning for each problem. a.

Set 1 i. Disagree with Salia ii. Agree with Jin iii. Disagree with Bill

Set 2 i. Disagree with Arya ii. Disagree with Sam

FACILITATION TIP Consider breaking this Agree or Disagree activity into two big parts: Set 1 and Set 2. Have students complete Set 1 at their table groups and then have them do a walk around for Set 2. STEMscopes Tip The Planner, accessed along the menu bar, provides a calendar planning tool for teachers. Download, print, save, or share your plans. Use the Elements tab on the left to access grade-level scopes and virtual-learning options with embedded links to all scope elements. Drag the elements you want to implement into the calendar, and click on each element to enter element details and personal planning notes.

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

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iii. Agree with Zoey 8.

If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

Identifying Misconceptions • •

FACILITATION TIP This Foundation Builder provides a good review of several visual models. Depending on your students, it may be a good starting point for this scope for the whole class.

It may help students to work with Cuisenaire RodsTM, number lines, or other visual models to determine if the answer should be agreed with or not. Students may need help remembering when to use the multiplicative inverse or reciprocal in dividing fractions and not in multiplying. Again, using visual models can assist them in understanding the proper application. Notes

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

Multiplication and Division Problem Solving Using Fractions Hook – That Takes the Cake! ACTIVITY PREPARATION Students will determine the solution to a division of fractions problem to solve a real-world dilemma.

Materials

Preparation

Printed •

1 That Takes the Cake! (per class)

Reusable •

Plan to show the video. Prepare to project That Takes the Cake! for the whole class to view. Prepare to introduce the scenario and encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

• • •

1 Phenomena Video (per class)

PROCEDURE AND FACILITATION POINTS Part I: Pre-Explore 1. FACILITATION TIP Project this scenario in print and provide guided read aloud time. Support students as they locate the essential math phrases, terms, and values.

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: Stephan baked his famous “Almost Better Than Music” cakes to share with fellow pianists at the postrecital party. He made two full cakes, but when he came downstairs dressed and ready to leave for his piano recital with the cakes, he noticed that exactly half of one cake was missing. His mom informed him that his brothers had not known the cakes were for a party and had eaten half of one cake with their friends. Stephan had been planning to cut each cake into ten equal-sized pieces. He wondered now whether he would have enough cake to serve to the 12 performers and their piano teacher Mrs. Li. Ask students the following questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Student answers will vary. I notice that Stephan is partitioning the cake into equal-sized pieces, which are fractions. I wonder how many equal-sized pieces he will have with one and a half cakes. What will Stephan do if he does not have enough pieces of cake? I can use math to determine whether he will have enough pieces of cake by using division. Project That Takes the Cake! Notes

Explore

Explain

Elaborate

Evaluate

DOK-1 If Stephan has too many pieces, will he need to cut bigger or smaller pieces? He can either have leftovers or he can cut bigger pieces.

DOK-1 If Stephan has too few pieces, will he need to cut bigger or smaller pieces? He will need to cut smaller pieces.

DOK-2 If Stephan cuts bigger pieces of cake, what does that do to the denominator of the fraction that represents one piece of cake? As the piece of cake gets bigger, the denominator gets smaller. DOK-2 If Stephan cuts smaller pieces of cake, what does that do to the denominator of the fraction that represents one piece of cake? As the piece of cake gets smaller, the denominator gets larger.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Intervention

Acceleration

Explain to students that Stephan has to ﬁgure out whether he can still cut the cakes into tenths or whether he needs to partition the cake into a different number of pieces. Discuss the following questions:

Engage

Show the Phenomena Video again, and restate the problem. Refer to That Takes the Cake! and discuss the following questions: a.

DOK-1 How can you determine the number of pieces of cake that Stephan will have if he cuts each cake into tenths? We can divide the total number of cakes by the portion of each piece of cake.

DOK-1 What model can help you solve this problem? Answers will vary. A number line

DOK-1 What is the equation to ﬁnd how many pieces of cake Stephan will have if he divides the cakes into tenths? 1 1 1__2 ÷ ___ =? 10

DOK-1 If Stephan divides 1__2 cakes into pieces ___ the size of a cake, will 10 he have enough pieces to serve 13 people?

1 __

1 ___

3 __

1 ___

3 __

10 ___

30 ___

12 ÷ 10 = 2 ÷ 10 = 2 × 1 = 2 = 15. He will have 15 pieces. He will have enough cake to serve 13 people and have 2 pieces left over.

STEMscopes Tip Bookmarks and Notes, located on the Scopes home page, allow you to bookmark scopes or individual elements for quick and easy access and provide a place to digitally record personal planning notes. You may choose to set up folders by class, term, or semester to help with longterm planning and can alphabetize bookmarks for quick access.

FACILITATION TIP Continue to reinforce that this math sentence can be read as, "How many onetenths are in 1 and a half?".

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

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MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

Multiplication and Division Problem Solving Using Fractions Explore 1 – Multiply Fractions by Whole Numbers ACTIVITY PREPARATION Students will model multiplication of a fraction by a whole number and a whole number by a fraction. They will also reason about the size of the product in relation to both factors.

Standards for Mathematical Practice • • •

MP.4 Model with mathematics. MP.6 Attend to precision. MP.3 Construct viable arguments and critique the reasoning of others.

Materials Printed • • • •

Reusable

1 Student Journal (per student) 1 Set of Beverage Cards Part I (per group) 2 Sets of Beverage Cards Part II (per class) 1 Exit Ticket (per student)

• •

Consumable

2 Sets of fraction tiles (per group) 16 Rulers (per class)

• • •

4 Strips of manila paper measuring 3 × 18 inches (per student) 4 Strips of paper measuring at least 1 × 8 inches (per student) 2 Rolls of clear tape (per class)

Preparation • •

Plan to have students work in 8 groups to complete the activity. Print a Student Journal and an Exit Ticket for each student

Part I • •

Gather 2 sets of fraction tiles for each group. Print a set of Beverage Cards Part I for each group. Cut out and place each set of cards in a resealable bag labeled “Part I.” If desired, print on card stock and laminate for future use.

Part II • •

Print and cut out two sets of Beverage Cards Part II for the class. If desired, print on card stock and laminate for future use. Create 8 stations around the room. You will need to make 2 of each of the 4 stations below. • • • •

•

Grapeade pouches station: 2 strips of paper for each student and the Grapeade Pouches card. Kooky Kiwi Punch station: 2 strips of paper for each student and the Kooky Kiwi Punch card. Strawberry-banana smoothies station: 2 strips of manila paper for each student, 4 rulers, a roll of tape, and the StrawberryBanana Smoothies card. Wacky Watermelon Smoothies station: 2 strips of manila paper for each student, 4 rulers, a roll of tape, and a Wacky Watermelon Smoothies card.

Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Fraction Tiles)

PROCEDURE AND FACILITATION POINTS Part I 1.

Read the following scenario to the class: Sally’s school is going to have a fundraiser to raise money for a trip the sixth grade is taking this year. The fundraiser will be a school-wide carnival with various games, a silent auction, snacks, and more! Sally’s class is working on getting beverage stations ready. Help Sally’s class determine the amount of ingredients needed for different recipes. © Accelerate Learning Inc. - All Rights Reserved

2. 3. 4.

Engage

Explore

Explain

Elaborate

Evaluate

Give a Student Journal to each student. Distribute a set of Beverage Cards Part I and two sets of fraction tiles to each group. Explain to students that they will use fraction tiles to model multiplication with fractions to determine how much of the different ingredients are needed to make the Sassy Strawberry-Lemon Punch. As students are working, monitor and discuss what is happening to the product. a.

DOK-1 Which fraction tiles should we use to model lemon juice? We can use the ﬁfths fraction tiles to model lemon juice.

DOK-2 How can we model the lemon juice needed? We can make 5 2 2 models that each show __5 because each gallon will need __5 of a gallon of lemon juice.

DOK-2 What would you estimate the product to be for lemon juice? I would estimate that the amount of lemon juice needed is greater than 1.

d. DOK-1 Which fraction tiles can we use to model the number of pounds of strawberries? We can use the sixths fraction tiles to model strawberries. 5

e. DOK-2 How many __6 models are needed to represent the number of pounds of strawberries that are needed? 3 gallons need strawberries, so 5 I will need three models of __6. f.

DOK-3 What would you estimate as the number of pounds of strawberries needed? I estimate that the amount of strawberries needed is more than 2 pounds but less than 3 pounds because the amount of strawberries needed for each order is close to one pound.

Intervention

Acceleration

FACILITATION TIP Consider modeling the ﬁrst Beverage recipe from Part 1. After giving students some time to use the tiles, show them how to sketch an accurate model.

STEMscopes Tip The Assessment Builder, accessed under Assessments along the menu bar, allows you to build a customizable assessment. Choose to create a printable and/or digital assessment item bank. Search for English and Spanish items by standard, lesson, key words, topic, grade level, and question type. Assessments are saved in your private account for you to access or edit at any time.

After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat • • •

DOK-2 How did you write the multiplication expression for the lemon juice? 2 I wrote 5 × __5 for the multiplication expression for the lemon juice. 5 __

DOK-2 Why would you model the amount of strawberries with 3 sets of 6? 5 3 more gallons need to be made, so we need to make 3 groups of __6.

DOK-3 Explain how to ﬁnd the total amount of strawberries needed to make 3 5 more gallons of punch. I modeled __6 with the fraction tiles three times. Then, I rearranged the pieces of the fraction tiles to make groups of one. I had 2 groups 6 3 3 1 1 of __6, or 1, and __6 remaining. __6 is the same as __2, so they will need a total of 2__2 pounds of strawberries.

FACILITATION TIP

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

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During this Math Chat, take time to explain that multiplication is often expressed with the word "of". When "of" is between two values in a statement it often means "times" or "multiplied by". For example, 5 5 "3 groups of __6" can be read as "3 times __6."

Part II 1.

3. 4.

Read the following scenario to the class: Sally and her class have decided to make a few more types of drinks for the fundraiser. Help the class determine how much of each item is needed for the fundraiser. Explain to students that they will need to create either a strip diagram model or a number line model to determine the amount of each item needed at the stations. Students may use the fraction tiles to help visualize the scenarios. Determine where each group will start. Discuss with students how to rotate between each station. As students are working, monitor and discuss what is happening to the product. a.

DOK-2 What should you estimate the product to be? My estimate should be more than the fraction but less than the whole number.

DOK-1 How do you know it is greater than the fraction? The whole number is greater than 1, so it has to be greater than the fraction. We are taking more than one group of that fraction.

FACILITATION TIP Continue to encourage students to use the visual models even if they can intuit the solutions. Being ﬂuent with the models will support later success on more complex scenarios in math and real life.

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

Multiplication and Division Problem Solving Using Fractions Explore 1 – Multiply Fractions by Whole Numbers

STEMscopes Tip Each grade level includes a Daily Numeracy program. In it, teachers will ﬁnd an overview of Daily Numeracy and how it can be used in the classroom, a variety of short activities focused on developing students' mental math strategies and number sense, and resources that supplement the activities to build students' thinking and reasoning skills.

DOK-1 How do you know it is less than the whole number? A fraction is less than 1, so the product will be less than the whole number. We are only taking part of the whole number, not the entire whole number itself.

DOK-2 How can you use the fraction tiles to help you make a number line? You can use one of the whole fraction tiles.

e. DOK-2 How can you use the fraction tiles to mark the number line? You can put the tile under the number line, placing the left end at 0. The right end is the location of the whole interval. You can mark it on the number line. f.

DOK-3 How can you show the number of containers if it is less than a whole container? You can use the other fraction tile models, such as thirds, and put marks along the line. Or you could lay them along the top of the number line.

After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat

FACILITATION TIP Continue to remind students that multiplying a fraction by a whole number can be read using the word “of ”. It may help them visualize why the products are smaller than 2

the whole number. For example, “6 times __3” 2 __

2 __

is the same as “6 groups of 3” and “ 3 times 2

6” is the same as “ __3 of 6."

FACILITATION TIP When you preview this Exit Ticket with students, clarify your expectations for showing their thinking. Challenge students to solve the scenario in more than one way (repeated addition as well as multiplication).

DOK-2 Why is the product smaller than the whole number fraction? When multiplying a whole number by a fraction, the answer will be less than the whole number factor because we are only taking a part of it. We aren’t repeating it one time. We are multiplying a whole number by a fraction, which is less than 1. • DOK-2 Why is the product larger than the fraction factor? When we multiply the fraction by a number greater than 1, we are repeating the fraction more than one time, so the product will be greater than the factor we started with. • DOK-3 Explain why the denominator did not change. You are combining equal groups of fractional parts. The denominator just tells you how many it takes to make one whole. That does not change just because you are combining multiple parts. •

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

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MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

Multiplication and Division Problem Solving Using Fractions Explore 2 – Multiply Fractions and Mixed Numbers ACTIVITY PREPARATION Students will model multiplication of fractions. They will also reason about the size of the product in relation to both factors.

Standards for Mathematical Practice • • •

MP.4 Model with mathematics. MP.6 Attend to precision. MP.3 Construct viable arguments and critique the reasoning of others.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 5 Sets of Fun Run Trails (per class) 1 Exit Ticket (per student)

•

Reusable • • •

8 Rulers (per class) 6 Dry-erase markers (per class) 6 Fraction tiles or fraction circles sets (per class)

Divide the class into 10 groups. Print a Student Journal and an Exit Ticket for each student. Print and cut apart ﬁve sets of Fun Run Trails. If desired, print them on card stock and laminate them for future use. Create 10 stations around the room. You will have 2 of each station. Prepare the following items for each station: • •

Consumable • •

•

12 Strips of manila paper measuring 3 × 18 inches (per class) 4 Rolls of clear tape (per class)

Stations 1, 3, and 5 – A dry-erase marker and a set of Fun Run Trails Stations 2 and 4 – Three manila paper strips (one strip per group), one set of fraction tiles or fraction circles, two rulers, a roll of clear tape, and a set of Fun Run Trails Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Fraction Tiles or Fraction Circles)

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Project this scenario for students to read along with you as you introduce the Explore activity.

FACILITATION TIP

Read the following scenario to the class: Jake and his friends in the sixth-grade class decided they wanted to include running on fun run trails as part of their fundraising. They developed multiple fun run trails. Help determine the portion of each fun run trail that Jake and his friends ran. Explain that the ﬁrst ﬁve trails have been placed in stations around the room and students must determine the distance Jake and his friends ran on each trail. Students will use the materials at their stations to create a model. a.

In scenarios 1, 3, and 5, invite students to create area models with their dry-erase markers on their desks. If the dry-erase markers are not visible on the desks, wax paper or small whiteboards can be used. The models will help them determine the common denominator and the distance they travelled in each problem before they sketch them on their Student Journals.

In scenarios 2 and 4, invite students to use the strips of manila paper and draw number lines to create models for the problems. Follow these steps:

Post the expectations regarding scenarios 1, 3, and 5 vs scenarios 2 and 4.

FACILITATION TIP

i. Instruct groups to use one of the strips of manila paper.

Creating a labeled number line with accurate tick marks takes practice. Consider providing numbered strips or modeling how to eﬃciently create an effective number line.

ii. Next, have students use the ruler to draw an unlabeled number line in the middle of the strip along the full length of the strip. iii. Then, have them place a mark near the left side of the line for 0. © Accelerate Learning Inc. - All Rights Reserved

iv.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Have them use the whole fraction tile to mark __4 intervals in scenario 1 2 and __5 intervals in scenario 4 on the line by laying the 1 whole

tile under the number line, starting at 0 and using the other end to

mark __4 (or __5). They can move the left end of the tile to 1 and mark __4 1

(or 5 ), etc. 2 __

Have students continue until they have reached 1 whole. Use the fraction pieces to tick-mark the fractional parts in between each fraction interval.

c. In addition to each model, the groups will develop multiplication equations for the fun run trails and the distance the friends ran for each trail. d.

Challenge the groups to observe and compare the total distance of miles the friends ran (product) to the length of the total trail and the fraction of the trail they ran. Together, come to a conclusion on why the product is what it is.

If a group is struggling over area models, instruct the group to use a dry-erase marker and surface to make a rectangle and shade in the fractional part to represent the length of the trail. In the opposite direction (horizontally or vertically), shade in the fractional part of what they ran.

Depending on your students, consider modeling some or all of the scenarios together as a class. There are 5 scenarios and 6 trail distances.

a. How can you determine the answer based on the shaded sections? The part, or the rectangle, where the 2 shaded areas overlap 4.

If a group is struggling with the number line in scenario 2, instruct the group to model the length of each trail. Then, have them talk about separating the length of the trail into equal intervals based on what they are multiplying by. Use the following as an example: 5

a. Scenario 2 starts with __4 of a mile. How much did they run? __6 of the trail

Into what fraction do you need to separate each of the __4 intervals? Sixths

Have students separate each __4 interval into sixths.

d. How many fractional sections are there now? 24. Go to the __4, and ﬁnd __6 1 of the __4.

If a group is struggling with the scenario 4 number line, instruct the group to model the length of the trail. Then, have them talk about separating the length of the trail into equal intervals based on what they are multiplying by. Use the following as an example: 3

FACILITATION TIP

STEMscopes Tip Fact Fluency activities, located in each grade level under the Scopes tab, help develop students' addition and subtraction fact fluency in all grades and multiplication and division fact fluency in grades 3–5. Activities include mini-lessons, stations, games, and assessments to help address common fact-fluency groupings and strategies.

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a. Scenario 4 starts with __5 of a mile. How much did they run? __4 of the trail 1

b. Into what fraction do you need to separate each of the __5 intervals? Fourths c.

Have students separate each __5 interval into fourths.

d. How many fractional sections are there now? 20 e. 6.

. Go to the __5, How many 20ths are there in __5? 12 How big is __4 of __5? ___ 20 3 3 __ __ and ﬁnd 4 of the 5.

As students are working, monitor and discuss what is happening to the product. a. What do you notice about the product of a fraction and a fraction? It’s less than the length of the trail. b. Why would that be? When we multiply any number by 1, the product is always that number because we are repeating that number one time. When you multiply any number by a fraction that is less than 1, the product has to be less than the other factor because we are repeating it less than one time. We are only taking a part of it.

FACILITATION TIP Post and project this essential question on the board and refer to it often as students are working. Remind them that a fraction multiplied by a fraction means "a fraction OF 1 1 a fraction." For example, "__2 times __2" means 1 1 __ __ "2 OF 2." 43

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

Multiplication and Division Problem Solving Using Fractions Explore 2 – Multiply Fractions and Mixed Numbers

How can you prove it? When we built a model, we ﬁrst drew the fraction that represents the length of the track, but when we shaded in the fractional part we ran, it only covered part of the section that was already shaded.

Before you even start modeling or multiplying, what can you estimate the product to be when multiplying a fraction by a fraction? The product will be less than the fraction that represents the length of the fun run trail.

After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat FACILITATION TIP Ensure that students make some connections to some additional real-world applications for multiplying fractions and mixed numbers. Engage students with game scores, creative design projects, high interest collections, or hobbies.

DOK-2 What connections did you make during this activity? The ways I can model fractions are similar to ways of modeling other multiplication problems. It doesn’t matter if it is a fraction, it can be modeled the same as a whole number. • DOK-2 Why is the product smaller than the fraction length of the fun run trail? When multiplying a fraction by a fraction, the answer will be less than the factor because we are only taking a part of it. We aren’t repeating it one time. We are multiplying a fraction by a fraction, which is less than 1. • DOK-2 Explain why the denominator changed. You are separating the original fraction into more equal groups of fractional parts. The denominator tells you how many it takes to make one whole; that does change the denominator because you are changing the fractional parts into smaller pieces. •

Post-Explore FACILITATION TIP When previewing this Exit Ticket with students, clarify your criteria for success. Students are directed to draw a model, write a multiplication problem, and explain their reasoning. Be prepared for some students to use improper fractions in the multiplication problem and others to be resistant to drawing a model.

1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

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MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

Multiplication and Division Problem Solving Using Fractions Explore 3 – Modeling Fraction Division ACTIVITY PREPARATION Students will help determine quotients of fractions by modeling fraction division.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.7 Look for and make use of structure

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 2 Sets of Dessert Cards (per class) 1 Exit Ticket (per student)

•

Reusable • •

Plan to divide the class into 6 groups. Print a Student Journal and an Exit Ticket for each student. Print two sets of Dessert Cards for the class. If desired, laminate them for future use. You will be running 2 sets of 3 stations. Set up 2 of each station as follows: Station 1 – page 1 of the Dessert Cards (Chocolate Cake and Vanilla Cake cards) with light blue and light green Cuisenaire RodsTM Station 2 – page 2 of the Dessert Cards (German Chocolate Cake and Banana Bread cards) with light green, brown, and white Cuisenaire RodsTM Station 3 – page 3 of the Dessert Cards (Strawberry Cake and Red Velvet Cake cards) with dark green, light green, red, and white Cuisenaire RodsTM

•

1 Set of colored pencils (per student, optional) 1 Set of Cuisenaire RodsTM (per group)

• • • •

Optionally, make sure students have colored pencils to show each color of rod used in their models. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Cuisenaire Rods)

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before reading the scenario, ask the class 1) What is your favorite type of dessert?; 2) Why do you like this particular dessert?; 3) Can you buy your favorite dessert sold alone or does it come with other types of dessert? STEMscopes Tip Each scope includes a Home section accessed along the scope's menu bar. Here you will see student expectations as well as key concepts and fundamental questions. Each Home tab includes drop-down options to access the Scope Overview, Content Support, Content Unwrapped, Materials List, and Parent Letter pages.

2. 3. 4. 5.

6. 7.

Read the following scenario to the class: Each of the grade levels at Sally’s school will be in charge of some part of the fundraiser. Sally’s grade level is in charge of partitioning baked goods into individual packages to be sold at the dessert booth. The sixth-grade teachers prepared Dessert Cards with instructions on how each dessert should be partitioned. Determine how many individual packages of each dessert item can be made with the information provided on the Dessert Cards. Assign each group a station at which to begin working. Give a Student Journal to each student. Students will read the information on each Dessert Card at their stations. Have students discuss in their groups what they could do to ﬁnd how many individual packages can be made with the total amount of dessert. Students should determine that they can divide the total amount of dessert into fractional pieces based on the information provided to them.

Encourage students to use the provided Cuisenaire RodsTM to build a concrete model to solve. As students work, move from group to group, and scaffold as needed. Remind students to draw a pictorial model of the concrete model, write a solution © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

equation, and write a solution statement on their Student Journals. Optionally, have students use their colored pencils to show each different-colored rod that was used. a. Students may struggle to determine which pieces to use for each part. Guide students to using the following rods: i. Station 1 – Use the blue rod to equal one whole. ii. Station 2 – Use the light green rods to model eighths. iii. Station 3 – Use the dark green rod to equal one whole. 8.

9. 10. 11.

• • • •

•

Have students determine the fractional pieces their cake is being divided into and which Cuisenaire Rods™ represent that fraction based on the rods they have been instructed to use.

Monitor and assess students’ understanding as they collaborate by asking the following guiding questions: a.

DOK-1 How can you partition the chocolate cake into thirds? I will cut each of the three cakes into three pieces. 3 cakes divided into three pieces each will give me nine total pieces of chocolate cake to package.

DOK-1 What do you do with extra Cuisenaire RodsTM if you do not have one whole? If I start with a fraction of the whole, I will not need all of the pieces to make a whole, so I can cross off those pieces in my model.

DOK-1 What does a fraction in your answer represent? It represents a leftover portion that cannot be used to make another package because it is too small.

Have students rotate to the remaining stations, continue to build concrete models to solve, and record their thinking on their Student Journals. After students complete each station, have them answer the reﬂection questions on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat •

FACILITATION TIP

DOK-1 What do you notice about the quotient when you divide a fraction by a fraction? When you divide a fraction by a fraction, you will get a larger number than you started with. The quotient will be a mixed number or a whole number. DOK-3 Why would dividing by a fraction give you a larger number? You will get a larger number because you are dividing the original portion into smaller groups. DOK-3 What does a fraction in the quotient mean? A fraction in the quotient means you can make part of another dessert. 1 DOK-2 What patterns do you notice when you divide a whole number by __3? The whole number times the denominator (3) is the answer. 1 DOK-2 How does this pattern change when it is a fraction divided by __3? The numerator of the first fraction is multiplied by the denominator of the second fraction (3). This is the new numerator. The denominator of the first fraction is the new denominator. 2 DOK-2 What changes in the pattern when you divide a fraction by __3? The numerator of the first fraction is multiplied by the denominator of the second fraction (3). This is the new numerator. The denominator of the first fraction is multiplied by the numerator of the second fraction (2). This is the new denominator.

FACILITATION TIP Prompt students to use the Cuisenaire Rods™ to model dividing the chocolate cakes into thirds.

FACILITATION TIP If the quotient is a mixed number, have students determine what the whole number represents (the number of complete packages) and what the fraction represents. FACILITATION TIP Use a visible timer to time the stations. It may be a good idea to rotate the stations rather than the students. It makes rotating stations quicker. FACILITATION TIP

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

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It might help students to think about dividing a fraction by a fraction as asking, "How 4 1 many ? are in ?". For example, __6 divided by __3 1 4 __ 4 __ __ is really asking, "How many 3s are in 6?" 5 1

divided by __8 is really asking, "How many __8s 4 are in __5?"

FACILITATION TIP

Compare this to the quotient when whole numbers are divided by whole numbers. The quotient is smaller than the dividend because it is being divided into whole groups. FACILITATION TIP If students struggle to recognize the pattern, display the solution equations of the related problems from the Student Journal and discuss them.

Post-Explore 1. 2. 3.

FACILITATION TIP Have students complete the Exit Ticket to formatively assess their understanding of the concept. Determine your criteria for success on this Exit Ticket. What kind of model do students Complete the Anchor Chart as a class. need to draw? Is color coding required? Have each student complete their Interactive Notebook.

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

Multiplication and Division Problem Solving Using Fractions Explore 4 – Division of Fractions ACTIVITY PREPARATION Students will use visual and algebraic representations to show their division of fractions.

Standards for Mathematical Practice • • • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.6 Attend to precision. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Tablecloth Fabric Cards (per group) 1 Exit Ticket (per student)

Reusable • •

•

1 Resealable bag (per group) 1 Set of colored pencils (per student, optional)

PROCEDURE AND FACILITATION POINTS FACILITATION TIP

Before reading the scenario, ask the class 1) Who has a tablecloth or placement on your table at home?; 2) What does it look like?; 3) What is it made out of?

FACILITATION TIP The digital versions of Tablecloth Fabric Cards are in color; consider this detail when printing.

Read the following scenario to the class: Demarcus needs to determine if there will be enough fabric to create tablecloths for the tables at the fundraiser. Model division of fractions using number lines to determine how many tablecloths can be made using each color of fabric. Give one set of Tablecloth Fabric Cards to each group and one Student Journal to each student. Ask the class the following questions: a.

DOK-1 How can you model division of fractions using the pattern we discovered in the previous Explore activity? We can multiply the ﬁrst numerator by the second denominator and then multiply the ﬁrst denominator by the second numerator.

Explain new vocabulary to the class: Mathematicians call this the reciprocal or multiplicative inverse. The reciprocal is when you ﬂip the second fraction so that its numerator is now its denominator, and its denominator becomes its numerator.

FACILITATION TIP Provide a reference to help students recall. Display one of the problems from the previous Explore activity to aid with the discussion. FACILITATION TIP Provide an example of a reciprocal. Check for understanding by asking students to tell 3 1 you the reciprocal for __4 and __5. FACILITATION TIP

To support struggling students, consider preprinting number lines on the Student Journal to guide students modeling the division without having to create the number line. 48

Plan to divide the class into 6 groups. Print a Student Journal and an Exit Ticket for each student. Print a set of Tablecloth Fabric Cards for each group. Cut out the Tablecloth Fabric Cards, and place them in a resealable bag for each group. If desired, laminate them for future use. Optionally, students could use different-colored pencils to differentiate the dividend, division, and grouping to ﬁnd the quotient.

Have students work with their groups to determine the amount of tablecloths that can be made with each color of fabric. Students should start with the blue fabric card before moving on to the remainder of the Tablecloth Fabric Cards. Monitor and assess students’ understanding as they collaborate by asking the guiding questions listed below. (The questions align to the blue fabric scenario. Answers will vary for other scenarios.) a.

DOK-2 How can we model the blue fabric with a number line? We can draw a number line and number it from 0 to 5.

b. c. d.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How can we model __4 in the same diagram? We should divide each whole into four sections. 3

DOK-1 How many sections are needed for __4? We need 3 sections for __4. 3

e. DOK-2 How many groups of __4 are in 5?How many sections are remaining? There are 6 whole groups. There are 2 out of 3 sections remaining. f. 6.

Acceleration

FACILITATION TIP

DOK-1 What size fabric is needed to create one tablecloth? 3 One tablecloth needs __4 of a yard of fabric.

Intervention

DOK-2 How many blue tablecloths can be made? 6 blue tablecloths 2 and __3 of another tablecloth can be made.

After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Post and project questions 5a–5f so they can be used for each scenario as students work through the fabrics. Use these guiding questions when monitoring student understanding.

FACILITATION TIP Instruct students to show the 3 sections needed to make a tablecloth by circling them to make one group. Then, have them circle as many groups of 3 sections as possible.

Math Chat DOK-1 What is the reciprocal (multiplicative inverse)? The reciprocal is found when you ﬂip the second fraction so its numerator is now its denominator and its denominator becomes its numerator. • DOK-2 Why is it helpful to use the reciprocal (multiplicative inverse) when dividing? It would be helpful to use the reciprocal when you have larger mixed numbers or larger denominators in the fractions. If the numbers are too big to draw, reciprocals would be helpful. •

Post-Explore 1. 2. 3. 4.

FACILITATION TIP Determine your criteria for success on this Exit Ticket. Explain whether you will allow students to use other kinds of models, how students are to show their work, and if solutions must be simpliﬁed.

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

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MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

Multiplication and Division Problem Solving Using Fractions Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Multiply Fractions by Whole Numbers Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Multiply Fractions and Mixed Numbers Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 3

Students form deﬁnitions of mathematical vocabulary words used throughout the scope.

Modeling Fraction Division

Interactive Notebook

Show What You Know, Part 4

A cut-and-glue activity to process learning that can be added to a notebook for future reference

Division of Fractions

Independent practice assignment that gives students an opportunity to demonstrate their learning

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Multiply and Divide Fractions Independent and partner games and other activities that provide students with an engaging way to practice the new concept

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

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Intervention and Assessment SMALL-GROUP PLANNER 1

Students who are approaching mastery and need review

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

Multiplication and Division Problem Solving Using Fractions

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Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER Use this template to decide how to assess your students for concept mastery. Depending on the format of the assessment, you can identify prompts and intended responses that would measure student mastery of the expectation. See the beginning of this scope to identify standards and grade-level expectations. Key Concepts I can solve multiplication and division problems that include fractions with denominators including 2, 3, 4, 5, 6, 8, 10, and 12.

I can use numerical reasoning to interpret, represent, and solve mathematical situations that involve fractions, mixed numbers, and whole numbers.

I can use concrete models, visual fraction models, and a standard algorithm to represent and solve problems involving whole numbers, fractions, and mixed numbers.

What prompts will be used?

What does mastery look like?

MULTIPLICATION AND DIVISION PROBLEM SOLVING USING FRACTIONS

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I can apply reasoning strategies to use written methods to solve multiplication and division problems involving whole numbers, fractions, and mixed numbers.

I can ﬂexibly select a mathematical strategy to solve multiplication and division problems involving whole numbers, fractions, and mixed numbers using a variety of strategies.

SCOPE 1

Add and Subtract Decimals Scope Introduction SCOPE SUMMARY Students will use place value understanding to make sense of the standard algorithm to add and subtract decimal numbers. They may use models or other student-selected strategies to add and subtract decimal numbers in real-world problems.

Student Expectations

VERTICAL ALIGNMENT

6.NR.1.3 Perform operations with multi-digit decimal numbers ﬂuently using models and student-selected strategies.

Background Knowledge

Future Expectations

In fourth grade, students added and subtracted multi-digit whole numbers ﬂuently using place value understanding, properties of operations, and relationships between operations. Fifth grade was the ﬁrst time that students added and subtracted decimal numbers. Students added and subtracted numbers to the hundredths using a variety of strategies.

Students in seventh grade will reﬁne their ability to add and subtract rational numbers ﬂuently. Seventhgrade students will convert between rational number forms (fractions, decimals, and percentages) to solve problems as appropriate. They will also apply and extend previous understanding to solve problems using addition and subtraction of rational numbers.

Before beginning the lesson, the students’ prior knowledge is assessed to address any student needs and skills needed for content mastery. Students are assessed to determine if they can: •

add, subtract, multiply, and divide decimals to hundredths.

•

use concrete models based on place value, properties of operations, and/or the relationship between addition and subtraction.

•

explain the reasoning for a speciﬁc strategy.

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

add and subtract with multi-digit decimal numbers.

Students move on to the Explore activities and come back at the end of the explorations to complete the activity and discuss what they found.

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope. If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. 54

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Explore 1

EXPLORE ACTIVITIES Add and Subtract Multi-Digit Decimal Numbers In this exploration, students will solve a scenario to determine the prices and cost of each bake sale item at a fundraiser. Students will: •

determine how much money has been spent by each team.

•

add and subtract multi-digit decimal numbers using the standard algorithm.

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve.

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ADD AND SUBTRACT DECIMALS

Add and Subtract Decimals Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will examine a series of decimal models, all depicting addition and subtraction of decimals, and determine which option does not belong with the group. This element is designed to uncover student misconceptions; it should not be taken for a grade. 5.NR.4.4 Solve problems involving addition and subtraction of decimal numbers to the hundredths place using a variety of strategies.

Materials

Preparation

Printed •

• •

1 Does Not Belong (per student or per group)

Print both pages of Does Not Belong for each student. You may choose to place students in groups of two or three.

ADD AND SUBTRACT DECIMALS

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Procedure and Facilitation Points 1. 2. 3.

Distribute both pages of Does Not Belong to each student or group. Explain that each table on the handout contains four options. Three of the options go together, while one does not belong. Instruct students to determine which letter does not belong in each group and to explain their thinking. a.

4. 5.

Answer D does not belong because it is incorrect. The equation shows 0.98 + 0.28 = 0.118, but the equation labeling the model should show 0.98 + 0.28 = 1.18. Answer C does not belong because it is incorrect. The equation shows 0.8 – 0.59 = 0.51, but the equation labeling the model should show 0.8 – 0.59 = 0.21.

Conclude by leading a discussion. If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

FACILITATION TIP Students may not understand what they need to look for to determine which model does not belong. Have a brief discussion about things they should consider, such as the model, the equation, and the solution. FACILITATION TIP If having students complete this activity individually, implement this activity as Four Corners. This adds movement and a way to formatively assess student thinking.

Identifying Misconceptions •

•

Students may confuse whole numbers, tenths, and hundredths while using models and adding and subtracting decimals. They may revert to ones, tens, and hundreds learned in earlier grades. Students may choose to ignore or misalign the decimal points when completing the addition or subtraction. Notes

ADD AND SUBTRACT DECIMALS

Add and Subtract Decimals Hook – Our School FUNdraiser ACTIVITY PREPARATION Students will add and subtract with multi-digit decimal numbers.

Materials

Preparation

Printed •

• • •

1 Our School FUNdraiser (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Our School FUNdraiser for the whole class to view. Prepare to introduce the scenario and to encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION POINTS Part I: Pre-Explore 1. FACILITATION TIP

Project this scenario in print to the class. Conduct a guided read aloud with students. Coach them to ﬁnd the essential math words and values as you read and reread it together. FACILITATION TIP Continue to emphasize the math vocabulary used in scenarios. Take time to list words that are associated with common operations (total, addition, on average, more than, lower). FACILITATION TIP Students may need reading support for the ﬁve most popular orders. Consider projecting them one at a time and making notes on values and order amounts.

4. 5.

6. 58

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: Mrs. Jacoby is the chairperson of the King High School carnival. She keeps track of a lot of information so she can continue to improve the carnival each year. Last year, the carnival committee found that students do not want to spend, on average, more than $7.99 on food, drinks, and snacks. Mrs. Jacoby wants to know which popular food stand combinations students can afford. She also wants to determine how much to lower the combinations that may be too expensive. Ask students the following questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Student answers will vary. I wonder how to best write these mathematical equations to determine the value for each order. To ﬁnd the total, I know I will need to use addition, and to know how much to lower the expensive combinations, I will need to use subtraction. Project Our School FUNdraiser. Explain to students that Mrs. Jacoby has created the list of popular orders. Discuss the following questions: a.

DOK-1 Are all the values the same place value? No, some are whole numbers, while others are decimals that end in the tenths or hundredths place.

DOK-1 Is it possible to add or subtract whole numbers and decimals? If it is possible, how do you do it? Yes, it is possible. The decimal points and place values need to be aligned.

DOK-1 Using estimation, which orders do you think students will be able to purchase for $7.99 or less? They can buy the ﬁrst order for about $6, the second order for about $8, and the third order for about $6; orders four and ﬁve will be over $8.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and restate the problem. Refer to Our School FUNdraiser, and discuss the following questions: a.

DOK-1 How can you ﬁnd the total for each order? Align the place values, and add the cost of each item to calculate the total.

DOK-1 What is the cost for each of the top ﬁve orders? Order 1 is $5.93, order 2 is $7.85, order 3 is $6, order 4 is $8.24, and order 5 is $13.78.

DOK-1 How much more expensive are the orders greater than $7.99? Order 4 is $0.25 more, and order 5 is $5.79 more.

DOK-2 If a student brings $7.99 and orders everything from order 3, what other items could they still order? How do you know? $7.99 – 6 = $1.99, so the student can still order anything that costs less than $1.99. The student could order 1 hot dog, 2 popcorns, or 1 candy. They would not be able to combine items from diﬀerent stands and keep the cost less than $1.99.

e. DOK-2 A student wants to get the most items they can from the most stands they can. What stands can they purchase from in order to keep costs under $7.99? Would they be able to purchase multiples from the same stands and stay under $7.99? They would be able to purchase from the hot dog, lemonade, popcorn, and candy stands to get at least 4 different items that would cost them $5.54. They would be able to purchase multiples from the stands because they would have $2.45 left.

FACILITATION TIP Remind students that estimation is a powerful thinking tool. Encourage them to approximate before ﬁnding any exact answers. Some students may not feel comfortable with rounded answers.

ADD AND SUBTRACT DECIMALS

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FACILITATION TIP Question 2e. provides a good challenge for students to complete with partners. Encourage them to list their economical orders. Provide some constraints: must order from every stand, order from at least three stands, or have to buy mom and dad a pretzel...

ADD AND SUBTRACT DECIMALS

Add and Subtract Decimals Explore 1 – Add and Subtract Multi-Digit Decimal Numbers ACTIVITY PREPARATION Students will explore adding and subtracting multi-digit decimal numbers using the standard algorithm.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.6 Attend to precision. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • • •

• • • •

1 Student Journal (per student) 8 Sets of Grocery Ads (per class) 2 Sets of Bake Sale Cards (per class) 1 Exit Ticket (per student)

•

Consumable •

•

1 Sheet of graph paper (per student, optional)

Plan to divide the class into 8 groups. Print a Student Journal and an Exit Ticket for each student. Print 8 sets of Grocery Ads for the class. If desired, laminate them for future use. Print and cut out a set of the Bake Sale Cards for the class. If desired, print on cardstock, and laminate them for future use. Create 2 sets of 4 stations around the room. At each station, place a Grocery Ad and a Bake Sale Card. Optionally, provide students with graph paper to help keep place values lined up when adding and subtracting decimal numbers.

PROCEDURE AND FACILITATION POINTS Part I 1.

FACILITATION TIP

Open the Explore activity with the following class discussion, where students will discover that when adding decimal numbers, you ﬁrst need to line up the place values. a. DOK-1 What is the sum of 2,412 + 176? Accept all answers.

If multiple answers are provided, guide students through using estimation to determine the reasonableness of their answers.

Students should recall lining up place values before they begin adding. Allow students to revise their answers if needed. Come to a class consensus that 2,412 + 176 = 2,588.

DOK-1 What is the sum of 241.2 + 176? Accept all answers.

Students should recall lining up place values before they begin adding. Allow students to revise their answers if needed. Come to a class consensus that 241.2 + 176 = 417.2.

DOK-1 What is the sum of 241.2 + 17.6? Accept all answers.

Students should recall lining up place values before they begin adding. Allow students to revise their answers if needed. Come to a class consensus that 241.2 + 17.6 = 258.8.

DOK-1 What do you notice about the digits of the addends? You must line up the digits by their place values to add. For example, given 241.2 + 176, the digits in the ones place need to be lined up. When we do this, .2 in 241.2 does not have any digits lined up under it. Therefore, we must add a decimal and a placeholder zero. Once we line up the digits and add any placeholders that are needed, then we can add.

FACILITATION TIP If students are struggling with place value or lining up the decimals to correctly add, have them make a place value chart on their graph paper. They can use it to line up the numbers correctly. It is also helpful to have a place value chart as an anchor chart. FACILITATION TIP

To simplify this activity and to help students focus on the directions, complete Step 3 and 4 before having groups move to stations.

Have each group go to its starting station. Remind the class of station expectations.

4. 5.

Engage

Explore

Explain

Elaborate

Evaluate

Read the following scenario to the class: Sally’s homeroom is divided into four teams. Each team will provide a different item for the bake sale portion of the fundraiser. They are responsible for making one batch of their assigned item. Using a list of ingredients and their prices, determine the cost of each bake sale item. Give a Student Journal to each student. Have students collaborate with their groups to determine the amount each team will need to spend in order to purchase items for the bake sale. Students will use the Bake Sale Cards for the ingredients lists and the Grocery Ad for pricing. As students are working, actively monitor students, and provide support as needed.

Intervention

Acceleration

FACILITATION TIP Before reading the scenario, ask the class 1) Does anyone like to cook?; 2) If so, what can you cook?; 3) What ingredients go into the food you cook? FACILITATION TIP Establish if and when calculators are appropriate to use during this Explore activity and on the Exit Ticket.

ADD AND SUBTRACT DECIMALS

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a. DOK-1 What steps should we take to add decimal numbers together? We must ﬁrst line up the decimal points and each place value. We may need to add placeholder zeros when adding two decimal numbers that have different place values. b. DOK-1 What should we do when we line up decimal numbers with different place values? We will add placeholder zeros when adding two decimal numbers that have different place values. c.

DOK-1 What strategy could we use to check our answers? We could round the decimal numbers to estimate before we ﬁnd our answer.

FACILITATION TIP Encourage students to round each number to its highest place value.

d. DOK-1 How can you use graph paper to help you when adding decimal numbers? We can write each digit in a separate box to keep track of place value when adding. 7. 8.

Allow students time to answer the reﬂection questions at the end of Part I. After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 What do you have to do to add decimal numbers together? You have to make sure to line up the decimal point. • DOK-2 What is the total cost of all of the supplies? Include buying baggies for each team that will need them. $106.67 •

Part II 1. 2. 3.

4. 5.

6. 7.

Open Part II with the following class discussion, where students will discover that you must line up the place values when subtracting decimal numbers. DOK-1 What is 2,412 – 176? Accept all answers. Students should recall lining up place values before they begin subtracting. Allow students to revise their answers if needed. Come to a class consensus that 2,412 – 176 = 2,236. DOK-1 What is 241.2 – 176? Accept all answers. Students should recall lining up place values before they begin subtracting. Allow students to revise their answers if needed. Come to a class consensus that 241.2 – 176 = 65.2. DOK-1 What is 241.2 – 17.6? Accept all answers. Students should recall lining up place values before they begin subtracting. Allow students to revise their answers if needed. Come to a class consensus that 241.2 – 17.6 = 223.6. DOK-1 What do you notice about the digits when subtracting? You must line up the digits by their place values to subtract. For example, given 241.2 – 176, the digits in the ones place need to be lined up. When we do this, .2 in 241.2 does not have any digits lined up underneath. Therefore, we must add a decimal and a placeholder zero to the end of 176. Once we line up the digits and add any placeholders that are needed, then we can subtract.

FACILITATION TIP Have students describe different methods they could use to determine the total cost of all the supplies. Talk about the most eﬃcient method to use since they have already found the totals for the ingredients for each item for the bake sale. FACILITATION TIP Monitor students as they are subtracting. They sometimes forget to regroup when necessary. FACILITATION TIP If students provide multiple incorrect answers, remind them to use estimation to determine the reasonableness of their solutions.

ADD AND SUBTRACT DECIMALS

Add and Subtract Decimals Explore 1 – Add and Subtract Multi-Digit Decimal Numbers 9.

10. FACILITATION TIP Project the scenario in written text and coach students to ﬁnd the relevant details while you read it together.

11.

a. DOK-1 What operation will you use to determine how much money each team has remaining? Subtraction

FACILITATION TIP Because students will be subtracting from $50.00, it may be necessary to model how to regroup across multiple zeroes. An anchor chart could serve as a reminder for students of how to do this.

12.

FACILITATION TIP Before distributing this Exit Ticket, determine your criteria for success. How and where do you want students to show their work? Will students be allowed to use calculators to check their answers?

DOK-1 How much money was each team given for purchasing supplies? $50

DOK-1 What strategy will you use to solve? We will subtract the cost of supplies for one batch from $50.

Monitor and assess students’ understanding as they collaborate by asking the following questions: a. DOK-1 What strategy will you use to ﬁnd how much money is remaining? To ﬁnd the amount of money remaining, I will subtract the total spent from $50.

STEMscopes Tip The Scope Overview, located in the Home section of each scope, provides a colorful ﬂowchart that maps out the overall ﬂow of the scope. Activities contained in each of the 5E lessons are included, as well as the path for students who need additional support and acceleration activities for those who mastered the content.

Read the following scenario to the class: Each team is given $50 to purchase all necessary supplies. In order to purchase supplies for more batches, they will need to know how much money is remaining after supplies for one batch have been purchased. Can you help determine how much money each team has spent so far? Have students work collaboratively to determine the remaining amount each team has after buying supplies. Discuss the following questions with the class:

b. 13. 14.

DOK-1 What should you do when you need to subtract from a zero? You should regroup to make zero become a group of ten.

Allow students time to answer the reﬂection questions at the end of Part II. After Part II, invite the class to a Math Chat to share their observations and learning.

DOK-1 What do you need to do to subtract a number from 0? Students should know to regroup as needed to subtract from zero. • DOK-3 How could you check your answer to see if you made any mistakes? You can add your answer and the amount spent together. If the sum is 50 dollars, then you subtracted correctly. •

Post-Explore 1. 2. 3. 4.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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ADD AND SUBTRACT DECIMALS

Add and Subtract Decimals Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Add and Subtract Multi-Digit Decimal Numbers Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Interactive Notebook

A guide to facilitating the creation of a chart with students for each scope

A cut-and-glue activity to process learning that can be added to a notebook for future reference

Interactive Vocabulary Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Add and Subtract Decimals Independent and partner games and other activities that provide students with an engaging way to practice the new concept

ADD AND SUBTRACT DECIMALS

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Intervention and Assessment SMALL-GROUP PLANNER 1

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

ADD AND SUBTRACT DECIMALS

Add and Subtract Decimals

3 66

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER Use this template to decide how to assess your students for concept mastery. Depending on the format of the assessment, you can identify prompts and intended responses that would measure student mastery of the expectation. See the beginning of this scope to identify standards and grade-level expectations. Key Concepts I can use models to solve multidigit decimal number problems.

What prompts will be used?

What does mastery look like?

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I can use place value understanding for addition and subtraction of multi-digit decimal numbers.

I can ﬂexibly select mathematical strategies to solve addition and subtraction problems with multi-digit decimal numbers.

SCOPE 1

Multiply and Divide Decimals Scope Introduction SCOPE SUMMARY Students apply their knowledge of decimals to hundredths as they multiply or divide. Concrete models, strategies based on place value, and properties of operations are part of the learning. Students will be able to justify their strategies and solutions using their reasoning skills.

Student Expectations

6.NR.1.3 Perform operations with multidigit decimals numbers ﬂuently using models and student-selected strategies.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

Students in Kindergarten through third grade work on developing whole numbers in the baseten system. By third grade, all four operations are applied in problem-solving situations. Fourth grade introduced decimals to hundredths and compared decimals using visual models to develop decimal number sense.

Students in seventh-grade will reﬁne their ability to multiply and divide rational numbers ﬂuently. Seventh-grade students will convert between rational number forms, fractions, decimals, and percentages to solve problems as appropriate. They will also apply and extend previous understanding to solve problems using multiplication and division of rational numbers.

Before beginning the lesson, the students’ prior knowledge is assessed to address any student needs and skills needed for content mastery. Students are assessed to determine if they can: •

multiply multi-digit (up to 3-digit by 2-digit) whole numbers to solve authentic problems.

•

divide multi-digit whole numbers (up to 4-digit dividends and 2-digit divisors no greater than 25).

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope.

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

perform operations with multidigit whole numbers and decimals to hundredths to make models of the quantity.

•

discuss the different operations used and how to determine amounts.

Students move on to the Explore activities and come back at the end of the explorations to complete the activity and discuss what they found.

If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. 68

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Multiply Decimals – Place Value In this exploration, students will determine the amount of cake for different orders. Students will: •

make models using base ten blocks.

•

write the expression and solve.

•

determine the connection between whole number multiplication and decimal multiplication can affect the place value of the product.

Explore 2

Explore 1

EXPLORE ACTIVITIES

•

use area models and partial products to the standard algorithm when multiplying multi-digit decimal numbers.

Explore 4

Explore 3

In this exploration, students will use length and width to determine the area for bake sale table signs. Students will:

determine the placement of a decimal point in a product.

•

use arrays and area models.

•

solve area problems involving decimals.

•

create models.

Explore 6

Explore 5

In this exploration, students will be introduced to a problem scenario where students must determine how wide each booth at the school carnival will be and create models of the booths. Students will:

•

use area models and arrays to solve area problems involving decimals.

•

calculate how many square yards are needed.

•

calculate how many square yards are needed.

Divide Decimals – Place Value In this exploration, students will determine the amount of dog food that is placed in different bags. Students will: •

create a models.

•

write expressions and solve problems.

•

make the connection between whole-number division and decimal division.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment. Divide Decimals – Arrays and Area Models

In this exploration, groups of students will solve a scenario for a sign company. Students will:

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment. Multiply Multi-Digit Decimal Numbers

Multiply Decimals – Arrays and Area Models

MULTIPLY AND DIVIDE DECIMALS

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After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

Divide Multi-Digit Decimal Numbers In this exploration, groups of students will determine how many of each bake sale item was sold using the given total amount of money made for each item and each single item’s price. Students will: •

divide multi-digit decimal numbers using the standard algorithm.

•

determine the location of the decimal point in a quotient.

After solving the scenario, students discuss their learning, complete and Exit Ticket for assessment; then, revisit the Hook to apply their learning and solve. Notes

MULTIPLY AND DIVIDE DECIMALS

Multiply and Divide Decimals Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will read different student responses to a posed question on the prior standard, decide whether they agree or disagree with the student, and explain their reasoning. This element is designed to uncover student misconceptions; it should not be taken for a grade. 5.NR.2.1 Fluently multiply multi-digit (up to 3-digit by 2-digit) whole numbers to solve authentic problems. 5.NR.2.2 Fluently divide multi-digit whole numbers (up to 4-digit dividends and 2-digit divisors no greater than 25) to solve practical problems.

Materials

Preparation

Printed •

MULTIPLY AND DIVIDE DECIMALS

Home

•

Print both pages of Agree or Disagree for each student.

1 Agree or Disagree (per student)

Procedure and Facilitation Points 1. 2. 3. 4. 5. 6. 7.

Set 1 i. Disagree with Shalyn

STEMscopes Tip Content Support, found in the Home section of each scope, provides teachers who might need additional background knowledge with a complete explanation of student expectations, mathematical vocabulary, an explanation of the progression of the related standards learned, strategies for instruction, possible misconceptions and obstacles, and more.

ii. Disagree with Kiy iii. Agree with Agustin b.

Set 2 i. Agree with Shaun ii. Disagree with Levi iii. Agree with Bradley

If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

Identifying Misconceptions • •

It may help students to use graph paper or lined paper turned horizontally to place each digit in a separate box to keep work organized. Students may believe they have completed the problem after completing the multiplication or division for the ﬁrst place value. Remind students that the operation has to be completed for the entire problem and not just part of its value.

MULTIPLY AND DIVIDE DECIMALS

Multiply and Divide Decimals Hook – Slumber Party Shopping ACTIVITY PREPARATION Students perform operations with multi-digit whole numbers and decimals to the hundredths.

Materials

Preparation

Printed •

• • •

1 Slumber Party Shopping (per pair)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Slumber Party Shopping for the whole class to view. Prepare to introduce the scenario and to encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed. Be prepared to put students in partnerships. They will work together to solve the problems.

PROCEDURE AND FACILITATION POINTS Part I: Pre-Explore 1.

FACILITATION TIP

Project this scenario and read it aloud with students. Ask, "What do we know? What do we need to ﬁnd out?" 3.

4. 5.

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: A mom and her son are going shopping for a slumber party that he is going to have for his birthday. They are having fun while getting some items for the party. We need to determine how much the cost will be for certain items for the party or how many items of the same kind they can buy for a certain amount of money. You also want to know how much money they will have left when they are done with their shopping. Ask students the following questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Student answers will vary. I notice that there will be a slumber party. I wonder how many people will be invited. Project Slumber Party Shopping. Explain to students that the prices for different items have been listed. We need to determine the cost of purchasing different amounts of these items. Discuss the following questions:

FACILITATION TIP Listen to students' ideas to determine when each operation could be used. Adjust lessons if needed using Foundation Builder or Skill Basics activities.

6. 72

DOK-1 What do we know? We know that the mom and son are going to be buying some things for a slumber party. We don’t know what they will be buying or how many yet.

DOK-1 What operation could we use to ﬁnd the total amount of money spent at the store? I think we would use addition to ﬁnd the total amount of money spent. We could also use multiplication when more than one of the same item is bought. We could multiply the cost of the item by how many of that item was bought.

DOK-1 What operation do you think we would use when trying to see how many of the same kind of item we could buy for a certain amount of money? I think we should use division to equally split the amount we will spend on a group of the same kind of item by the cost of the item.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and restate the problem. Refer to Slumber Party Shopping, and discuss the following questions: a.

DOK-2 The mom and son want to buy some cake mix and frosting to make cupcakes for the party. They want to make a lot. They buy 3 boxes of cake mix and 4 cans of frosting. How much did they spend? Models selected to solve will vary. To ﬁnd the total money spent on cake mix, I multiplied $0.25 × 3 = $0.75. The cans of frosting were $0.50 × 4 = $2.00. They spent $2.75 altogether for the cupcakes.

DOK-2 They spent $5 on pizza for the slumber party. How many pizzas did they buy? Models selected to solve will vary. I divided $5.00 by $1.25 and got 4 as my solution. This means they bought 4 pizzas.

DOK-2 They also bought 7 liters of soda for the party. How much did they spend on soda? Models selected to solve will vary. I know that each liter of soda costs $0.50 and they bought 7 liters, so I multiplied $0.50 × 7 and got $3.50. They spent $3.50.

DOK-2 The last item needed for the party is ice cream. They bought 2 gallons of ice cream. How much did they spend on the ice cream? Models selected to solve will vary. I know that $4 × 2 = $8 and $0.75 × 2 = $1.50. To ﬁnd the total, I added those two values together: $8.00 + $1.50 = $9.50.

FACILITATION TIP Post the number of each item on the board for students to determine the total for each item and the total.

MULTIPLY AND DIVIDE DECIMALS

Home

FACILITATION TIP Have students share when they used each operation to solve the problem.

MULTIPLY AND DIVIDE DECIMALS

Multiply and Divide Decimals Explore 1 – Multiply Decimals – Place Value ACTIVITY PREPARATION Students will make a connection between whole-number multiplication and decimal multiplication as they model how the place value of the factors affects the place value of the product.

Standards for Mathematical Practice • • • • •

MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • •

• • • •

1 Student Journal (per student) 1 Exit Ticket (per student)

Reusable •

1 Set of base ten blocks (per group) •

Plan to have students work in groups of 4 for this activity. Print a Student Journal and an Exit Ticket for each student. Place the base ten blocks in a container for each student group. For students who need more support in recalling information, please see our Open Number Line, Base Tens, and Decimal Frame Supplemental Aids elements in the Intervention section. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students (Base ten blocks).

PROCEDURE AND FACILITATION POINTS 1.

Discuss the different ways we can think about base ten blocks. a. DOK-1 If a ﬂat is equal to one whole, what is the value of a rod? A unit? A rod is one-tenth of the whole, and a unit is one-hundredth of the whole.

FACILITATION TIP

Project this scenario and conduct a guided read aloud. Read through the essential parts more than once and coach students to locate the essential math terms and values. FACILITATION TIP Have students circle the orders on their Student Journals. Students will need to determine which order of operation is needed based on the statements.

3. 4. 5.

Read the following scenario to the class: Welcome to the Crazy Cake Factory! We specialize in creating the tastiest 10-by-10 in. sheet cakes. Our customers love them! However, our customers don’t always want the whole sheet cake. Sometimes they order just a part of a cake or parts of many different flavors of cake! To charge them the right price, we need to figure out how much cake is being ordered each time! Each flat represents one 10-by-10 in. cake or one whole cake. The students will work in groups and begin with base ten blocks to model the orders, using the flat to represent one whole cake. Students will record each expression using symbols, draw their models, and record the solutions for each order. As students are working, circulate around the room and discuss the following questions: a.

DOK-2 What patterns do you notice? The place value of the factors and products changes each time, even though the digits we multiply stay the same.

Explain

Elaborate

Evaluate

DOK-2 How do you ﬁnd one-tenth of one-tenth? I know it takes 10 units to make one rod. If one rod is one-tenth, then one unit is one-tenth of that rod. One unit is equal to one-hundredth of the whole, so one-tenth of one-tenth is 0.01.

DOK-1 As you move through the orders, what changed about the digit 5? The digit 5 moved one place value to the right each time. It was 10 times smaller every time. DOK-2 How did changing the place value of the digit 5 affect the place value of the product? I saw the product of 30 every time; it was just 30 different-sized pieces every time. Making the one factor 10 times smaller makes the product 10 times smaller.

Intervention

Acceleration

FACILITATION TIP Direct attention on the placement of the decimal point on the product of two decimals.

FACILITATION TIP Direct attention to the pattern of the orders and the decimal placement in the products.

Allow students to move on to the ﬁnal set of orders. Students should continue the same process of building each order with base ten blocks, drawing their models, and ﬁnding the products. a.

Explore

Have students continue on to the next set of orders. Students should ﬁnd the total amount of cake for the ﬁrst two orders without using the base ten blocks since they involve only whole numbers. The rest of the orders should be modeled using the base ten blocks. As students continue working through the orders, circulate around the room and discuss the following questions: a.

Engage

MULTIPLY AND DIVIDE DECIMALS

Home

If students get stuck on the ﬁnal order, tell them they can think of the multiplication symbol as the word of, meaning they need to ﬁnd fourtenths of three-tenths.

After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-3 How do you find the total amount of cake for the final order? We build a model of three-tenths since that is how much of each cake flavor we are starting with. The customer only wants four-tenths of that, so we break the three-tenths up into 10 equal groups and keep four of those groups. We are left with 12 hundredths. • DOK-1 When you multiply numbers with the following place values, what size will the final pieces in your model be? Whole number × whole number = whole pieces Whole number × tenths = tenths Whole number × hundredths = hundredths Tenths × tenths = hundredths • DOK-2 What do you need to do if you have more than 10 tenths or hundredths in your model product? You need to regroup 10 tenths for one whole or 10 hundredths for one-tenth. • DOK-3 What did you notice about your product when multiplying by decimals less than 1? Explain. When multiplying by 0.1, the product is ten times less than the factor being multiplied. When multiplying by 0.01, the product is a hundred times less than the factor being multiplied. I also noticed 0.10 times 0.10 is a hundredth, so I will have an answer in the hundredths place. If one of the factors is in the tenths place and the other factor is a whole number, the product will be in the tenths place because the decimal is part of a whole and is dividing by the whole number •

FACILITATION TIP In the Math Chat, have students determine strategies when multiplying decimals and the decimal placement of the products.

STEMscopes Tip Use the Content Unwrapped element in the Home section to see the instructional expectations clariﬁed. Here you will see what students should be doing, what students should know, and implications for instruction. Included in this element is a complete vertical alignment related to this topic that shows how student expectations span across applicable grade levels.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

MULTIPLY AND DIVIDE DECIMALS

Multiply and Divide Decimals Explore 2 – Multiply Decimals — Arrays and Area Models ACTIVITY PREPARATION Students will use both area models and arrays to solve area problems involving decimals.

Standards for Mathematical Practice • • • • •

Materials

Preparation

Printed • •

• • • •

1 Student Journal (per student) 1 Exit Ticket (per student)

Reusable • •

1 Set of base ten blocks (per group) 1 Dry-erase marker (per group)

•

Consumable • •

•

1 Roll of wax paper (per class) 1 Roll of tape (per class)

Plan to have students work in groups of 3 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Place base ten blocks in containers for each student group. Cut two large pieces of wax paper, and tape them together on the long end for each group. For students who need more support in recalling information, please see our Open Number Line, Base Tens, and Decimal Frame Supplemental Aids elements in the Intervention section. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students (Base Ten Blocks).

PROCEDURE AND FACILITATION POINTS Part I: Paper Measurement Tools FACILITATION TIP

Project this scenario and conduct a group read aloud. Coach students to locate essential math words and phrases as you read through it more than once. FACILITATION TIP Share with students that 1 yard is equal to three feet or 36 inches.

2. 3. 4.

Read the following scenario to the class: You are now working for Make-aStatement Sign Company! This company makes all kinds of signs and billboards, large and small. Its printers are able to print on sheets of paper that measure 1 yard long and 1 yard wide. Each flat will represent one sheet of paper. It is our job to figure out how many square yards of paper are needed for each sign or billboard order! Distribute materials to each student group. Students should begin by tracing a flat, a rod, and a unit on their Student Journal pages. Have students label the length, width, and area of each base ten block they traced. Explain that these are the measurement tools students will use to build and measure how much paper is needed for each sign. After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 What is the length, width, and area of the ﬂat? It is one whole sheet of paper, or 1 yard by 1 yard. The area is 1 square yard. • DOK-2 What is the length, width, and area of the rod? It is one-tenth of a sheet of paper, or 1 yard by 1 tenth of a yard (0.1 yard). The area is 0.1 of a square yard. •

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How do you know the width is one-tenth of a yard? It takes 10 of the rods to make one ﬂat, so the width of one rod is one-tenth. • DOK-2 What is the length, width, and area of one unit? It is one-hundredth of a sheet of paper, or 0.1 yard by 0.1 yard. The area is 0.01 of a square yard because it would take 100 of them to ﬁll 1 square yard. • DOK-1 How do you know the length is 0.1? It takes 10 units to make the length of one rod, and the length of the rod is 1 yard, so the length of the unit is one-tenth

Intervention

Acceleration

•

of a yard. Part II: Sign Station 1.

Distribute two large pieces of wax paper and a dry-erase marker to each group. Have students use the base ten blocks to build a model of each sign on top of the wax paper. Each sign should be a rectangle or square. The two strips of wax paper should be taped together on the long edge to ﬁt the base ten models. a.

2. 3. 4.

5. 6.

STEMscopes Tip A Parent Letter, located in the Home section, provides parents with a breakdown of the concepts being learned in school, as well as a choice board of related activities that students can complete at home. Sending home the Parent Letter at the start of each scope strengthens the family-school connection by keeping parents informed and included in the learning process.

MULTIPLY AND DIVIDE DECIMALS

Home

If needed, encourage students to think of the length and width as so many yards plus so many tenths or hundredths (expanded form). They should reason about the spaces they need to ﬁll with the paper measurement tools and use the dimensions they found in Part I to ﬁnd the right size of pieces. This helps students place the base ten blocks properly into the array.

Once a sign is built, students should draw their model on their Student Journals. Students will then use the dry-erase marker to trace around their models on the wax paper and label the dimensions. Students should remove one section from their model at a time, trace where the section was with their dry-erase marker, and write an equation that represents the blocks that were in that section. Students should continue this process until they are left with an area model of the sign drawn on the wax paper. Students should record the area model and the total area on their Student Journals. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 How did you know the size of pieces to put in each spot? I used the dimensions we found in Part I to figure out the size of pieces that could fit in each spot. • DOK-2 What did you notice about the arrays? Each array formed a rectangle. Each array was built using the length and width as a tool. Most of the arrays had a combination of different-sized pieces. • DOK-2 How did you use your model to find the final product? We had to add up the totals of all the pieces. If we had more than 10 of one piece, we needed to regroup them. • DOK-2 How is this process similar to or different from multiplying whole numbers? It is similar because the models and processes look similar. We can use arrays and area models to represent both kinds of multiplication. They are different because now we are multiplying parts of a whole instead of whole pieces. •

FACILITATION TIP The art teacher may be able to provide examples of how grids can be used in drawings.

FACILITATION TIP Allow the students time to brainstorm and attempt this process before you demonstrate it. Move around the room, observing their interactions and discussions. Address concerns during your demonstration. FACILITATION TIP Ask each group to share their justiﬁcation for the pieces they chose to use in their models. FACILITATION TIP Give each group an opportunity to provide peer evaluation and feedback to another group before discussing their ﬁndings with the whole class.

MULTIPLY AND DIVIDE DECIMALS

Multiply and Divide Decimals Explore 2 – Multiply Decimals — Arrays and Area Models Part III: Big Bucks Billboards 1. FACILITATION TIP

Discuss as a whole class how to set up the ﬁrst model. Allow groups to create their models as you monitor their discussions and address any concerns. After the groups have made their models, demonstrate the thought process that should be involved in creating the model.

FACILITATION TIP Use whole numbers to demonstrate distributive and commutative properties for any students who are struggling to understand these concepts. Mastery of these concepts is important for future math topics.

Students should look at each billboard order and draw an area model using the wax paper and dry-erase marker. Encourage students to imagine what the array would look like as they break apart the length and width into expanded form and ﬁnd the area of each piece. a.

If needed, allow students to revisit the base ten blocks to build an array.

After Part III, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 How are area models similar to arrays? They are both rectangles with a length and width. We can count the number of pieces in each section of the array, but we need to use multiplication to ﬁnd the total area of each section in the area model. • DOK-1 How is an area model helpful? You can multiply larger numbers with an area model. If you were to use large numbers with an array, you would need a lot of blocks. • DOK-2 How do you know the place value of the digits in your products? I know multiplying a whole number by a few tenths will give me a group of tenths. Multiplying a group of tenths by a group of tenths will give me a group of hundredths. I use what I know about place value to ﬁgure out the size of the pieces in each section. I remember the sizes from the arrays I built. •

Post-Explore FACILITATION TIP

When previewing this Exit Ticket with students, clarify criteria for success on the written answers for Questions 2 and 3.

2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

MULTIPLY AND DIVIDE DECIMALS

Home

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MULTIPLY AND DIVIDE DECIMALS

Multiply and Divide Decimals Explore 3 – Multiply Multi-Digit Decimal Numbers ACTIVITY PREPARATION Students will make connections between the area model and partial products to the standard algorithm when multiplying multi-digit decimal numbers.

Standards for Mathematical Practice • • • • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Area Model Template (per group) 1 Standard Algorithm Work Mat (per group) 1 Exit Ticket (per student)

•

Reusable • •

•

Plan to divide the class into groups of 2–4. Print a Student Journal and an Exit Ticket for each student. Print an Area Model Template for each group. Place the Area Model Template in a clear sheet protector so students can write on it with a dry-erase marker. Print a Standard Algorithm Work Mat for each group. Place the Standard Algorithm Work Mat in a clear sheet protector so students can write on it with a dry-erase marker. Gather enough dry-erase markers for each group to have one.

1 Dry-erase marker (per group) 2 Clear sheet protectors (per group)

PROCEDURE AND FACILITATION POINTS Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever made a poster or a sign before?; 2) If so, what was the poster or sign for?; 3) How big was it?

FACILITATION TIP Project this scenario in print for students to read along with you. FACILITATION TIP

3. 4.

Read the following scenario to the class: You are working for Make-a-Statement Sign Company again! Today, a sign needs to be made to represent each table at the upcoming bake sale. Use the length and width to determine the area for each table’s sign. Give a Student Journal to each student, and give each group an Area Model Template and a dry-erase marker. Have students work with their groups to determine the area of each table’s sign using the area model and partial products. As students are working, actively monitor them. Ask guiding questions of groups that may be struggling:

Determine which students are best supported by the Area Model Template. Consider allowing students to use an individual whiteboard.

DOK-1 What numbers do you write above each box of the area model? Each of the numbers given for the length and width must be decomposed by place value. Then, you write the numbers above the boxes from left to right starting with the largest place value.

FACILITATION TIP

Encourage students to use the area model template to help them place the decomposed numbers in the correct spot. Assist with decomposing the numbers if necessary.

DOK-1 How do you determine what factors to multiply together in each box of the area model? The factors you multiply together in each box come from the left and top of that box.

DOK-1 How do you ﬁnd the total area of each sign when you use the area model? Each of the products written in each box of the area model will need to be added together to give the total area of the sign. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

DOK-1 Do you need to line up the decimal points to multiply using partial FACILITATION TIP products? No, you do not line up the decimal points when using partial Ensure that students understand that they products to multiply. do not need to line up the decimal points to Part II multiply using partial products, but they do need to line up the decimals when adding the 1. Read the following scenario to the class: Summer realizes that the class forgot partial products to get the ﬁnal solution. to include sales tax in the total cost of bake sale supplies. She determines that

3. 4.

each team needs to include $0.083 for each dollar they spend. Use the standard algorithm for multiplying decimals to determine how much tax each team needs to add to the total for supplies. Discuss with the class how they can quickly include $0.083 for each dollar each team needs to spend. Students should recall that repeated addition is the same as multiplication. Therefore, they need to multiply the cost for each team’s supplies by 0.083. Give a Standard Algorithm Work Mat to each group. Students will work with their groups using the Standard Algorithm Work Mat and a dry-erase marker to determine the amount of tax that should be added to each team’s cost of supplies. As students are working, actively monitor them. Ask guiding questions of groups that may be struggling: a.

DOK-1 Do you need to line up the decimal points to multiply? No

DOK-1 Is it necessary to include this row of zeros (as pointed out in the picture below)? No, it will not change the product.

DOK-1 How do you know how many decimal places are needed in the total? You have to count how many decimal places in total are in the numbers being multiplied.

DOK-1 How many decimal places are included in money? 2 decimal places are included in money.

e. DOK-1 Can you just cut off all of the other digits here? No, you must round the hundredths place for money.

6. 7.

FACILITATION TIP Before reading the scenario, ask the class 1) Almost everything we buy has sales tax added on. What is sales tax?; 2) Who gets the money from the sales tax?

MULTIPLY AND DIVIDE DECIMALS

Home

FACILITATION TIP Some students may prefer to use an individual whiteboard rather than the Standard Algorithm Mat.

STEMscopes Tip Key Concepts, located under the Home tab, are "I can..." statements that describe what students will know and be able to do when they have mastered the standard(s) of the scope. During each Explore lesson, it is helpful to post these statements for students to reference at the start and end of the activity.

FACILITATION TIP Explain that two decimal places are included in money because money includes dollars and cents. Cents are hundredths of a dollar.

Allow students time to answer the reﬂection questions at the end of Part II. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

MULTIPLY AND DIVIDE DECIMALS

Multiply and Divide Decimals Explore 3 – Multiply Multi-Digit Decimal Numbers Math Chat DOK-1 How do you determine where the decimal point should go when multiplying two decimal numbers together? To ﬁnd out where the decimal will go in the product, you will need to count how many decimal places are in the factors combined. Then, count that many places from the right in the product (answer), and insert the decimal point in front of that number. • DOK-1 How many decimal places are used for money? Money only has two decimal places. • DOK-1 How do you determine what the digit in the hundredths place for money will be? To determine the digits in the decimals for money, you will need to round the hundredths place. •

FACILITATION TIP Demonstrate this by rounding the example above to the nearest hundredth or cent.

Post-Explore FACILITATION TIP

On this Exit Ticket, decide whether you will allow students to show their calculations on another paper rather than use the ﬁll in the blanks format provided.

2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook. Notes

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

MULTIPLY AND DIVIDE DECIMALS

Home

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MULTIPLY AND DIVIDE DECIMALS

Multiply and Divide Decimals Explore 4 – Divide Decimals – Place Value ACTIVITY PREPARATION Students will make a connection between whole-number division and decimal division as they model how the place value of the dividends affects the place value of the quotients.

Standards for Mathematical Practice • • • • •

Materials

Preparation

Printed • •

• • •

1 Student Journal (per student) 1 Exit Ticket (per student)

Reusable • • • •

•

1 Set of base ten blocks (per group) 1 Container (per group) 1 Blue marker (per student) 1 Yellow marker (per student)

•

Place students in groups of 3 or 4. Print a Student Journal and an Exit Ticket for each student. Place the base ten blocks and blue and yellow markers in a container for each student group. For students who need more support in recalling information, please see our Open Number Line, Base Tens, and Decimal Frame Supplemental Aids elements in the Intervention section. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Base Ten Blocks)

PROCEDURE AND FACILITATION POINTS Part I: Pet Food Patterns 1.

Discuss the different ways we can think about base ten blocks. a.

FACILITATION TIP

Project this scenario text for students to refer to as they work and collaborate.

3. FACILITATION TIP To help students ﬁnd the pattern when dividing decimals, have students circle each order and the expression.

FACILITATION TIP

While monitoring groups, ask students "What is the pattern in the products of the complete set of orders?"

DOK-1 If a ﬂat is equal to one whole, what is the value of a rod? What is the value of a unit? A rod is one-tenth of the whole, and a unit is onehundredth of the whole.

Read the following scenario to the class: The local pet store needs your help! They have decided to offer a special service to their loyal customers. Customers can order dog food divided up into custom-sized bags for their dogs. The orders are pouring in! The pet store needs you to help them ﬁgure out how much pet food goes into each size of bag they already have in stock. Each ﬂat represents 1 pound of dog food. The students will work in groups and begin with base ten blocks to model the amount of food in each size bag. Students will record each expression using symbols. They will build their models using the blue marker to draw base ten blocks that represent a pound of dog food and the yellow marker to divide the food into the bags. Students will then record the amount of food in each bag. Discuss with the students the following questions as you walk around the room to clarify any misconceptions: a.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How do you divide a 1 lb. bag of food into one-tenth lb. bags? One ﬂat is a pound. One-tenth of 1 pound is one rod. There are 10 rods in a ﬂat, so each bag holds one-tenth of a pound of food.

DOK-2 How do you divide a 1 lb. bag of food into one hundredth lb. bags? One ﬂat is a pound. One-hundredth of 1 pound is one unit cube. There are 100 units in a ﬂat, so each bag holds one-hundredth of a pound of food.

Part II: Pet Food Orders 1.

Read the following scenario to the class: The local pet store soon discovers that customers’ orders require bags that are different sizes than what they already have in stock. Once again, the pet store turns to you for help. They have asked you to ﬁgure out the size of the bag that is needed for each custom order. The students will work in groups and begin with base ten blocks to model the amount of food per order. Students should then divide that amount into the number of bags the customer has asked for to ﬁnd the size of bags that are needed. Ask students the following questions as you rotate around the room to verify and clarify any misconceptions: a.

DOK-2 How are the manipulatives related to the equation when dividing by whole numbers? We had 15 pounds of dog food to be divided up into 5 bags. We used 15 ﬂats, our dividend, and divided them up into 5 groups, the divisor. The equation is 15 divided by 5 is 3.

DOK-2 How are the manipulatives related to the equation when dividing by a decimal, such as 1.5 divided by 5? We used a ﬂat and 5 tens, our dividend. We noticed the ﬂat could not be divided up into 5 groups, our divisor, so we traded in for 10 tenths, giving a total of 15 tenths, our new dividend. We then divided up the 15 tenths into 5 groups. The equation is 1.5 divided by 5 is 0.3.

Students will record each expression using symbols, build their models using the blue marker to draw base ten blocks that represent the number of pounds of dog food and the yellow marker to divide the food into the number of bags the customer has asked for, and record the size of bags needed. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat •

•

• •

DOK-1 As you move through the orders, what changes about the digits 1 and 5 in the dividend? The digits move one place value to the right each time. They were 10 times smaller every time. DOK-2 How did changing the place value of the dividend affect the place value of the quotient? The quotient was 10 times smaller each time. Making the dividend 10 times smaller makes the quotient 10 times smaller. DOK-3 How did you ﬁnd the size of the bags needed for these orders? We built models to show how many pounds of food we were starting with. Then, we divided up that amount into the number of bags the customer asked for and found the size of bags we would need for these orders. DOK-2 What happened to the size of the bags when you had smaller amounts of food? As the amount of food decreased, the size of the bags needed to decrease. DOK-2 What do you think the size of the bags would be if you had 0.004 pounds of food separated into two bags? We would have 0.002 lb. bags.

Intervention

Acceleration

STEMscopes Tip The Accessing Prior Knowledge activity, located in the Engage section, helps teachers determine students' prior knowledge about a concept before engaging in the inquiry process. If students struggle with the task, the Foundation Builder, also found in the Engage section, helps to ﬁll the gaps in prior knowledge.

MULTIPLY AND DIVIDE DECIMALS

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FACILITATION TIP Use Virtual Base Ten Blocks to demonstrate how the manipulatives can be exchanged to complete the order.

FACILITATION TIP Direct attention on the placement of the decimal point on the dividend when dividing with decimals.

STEMscopes Tip The Engage section, located along the scope menu, is designed to activate student interest in the learning topic. Within the Engage section, activities to access students' prior knowledge about the topic, to build a strong foundation to bridge any gaps in understanding before diving into the new content, and to set the purpose for learning a new skill are included.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

MULTIPLY AND DIVIDE DECIMALS

Multiply and Divide Decimals Explore 5 – Divide Decimals — Arrays and Area Models ACTIVITY PREPARATION Students will use both arrays and area models to solve area problems involving decimals.

Standards for Mathematical Practice • • • • •

Materials

Preparation

Printed • •

• • • •

1 Student Journal (per student) 1 Exit Ticket (per student)

Reusable • • •

1 Set of base ten blocks (per group) 1 Dry-erase marker (per group) 1 Container (per group)

•

Place students into pairs. Print a Student Journal and an Exit Ticket for each student. Place base ten blocks in containers for each student group. Cut two large pieces of wax paper, and tape them together on the long end for each group. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Base Ten Blocks)

Consumable • •

1 Roll of wax paper (per class) 1 Roll of tape (per class)

PROCEDURE AND FACILITATION POINTS Part I: Booth Measurement Tools 1.

FACILITATION TIP The students should get your approval before moving forward to Part II.

2. 3. 4.

Read the following scenario to the class: Your school is having a carnival! The sixth-grade class has been asked to figure out the best location for each booth. You know how much total area is allowed for each booth, and the length of the booth is based on the length of the game that goes in the booth. It is up to you to figure out how wide each booth will be. Distribute materials to each student group. Students should begin by tracing a flat and a rod on their Student Journals. Have students label the length, width, and area of each base ten block they traced. Explain that these are the measurement tools they will use to build and measure the booth sizes. After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 What is the length, width, and area of the ﬂat? It is one whole section that is 1 yard by 1 yard. The area is 1 square yard. • DOK-2 What is the length, width, and area of the rod? It is one-tenth of a section, or 1 yard by one-tenth of a yard (0.1 yd). The area is 0.1 of a square yard. • DOK-1 How do you know the width is one-tenth of a yard? It takes 10 of the rods •

to make one ﬂat, so the width of one rod is one-tenth. 86

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II: Small Booths 1.

2. 3. 4.

5. 6.

Distribute two large pieces of wax paper and a dry-erase marker to each group. Have students use the base ten blocks to build a model of each booth with the speciﬁed area using the length measurement on top of the wax paper. Each booth should be a rectangle. The two pieces of wax paper should be taped together on the long edge to ﬁt the base ten models. Once a booth is built, students should draw their array on their Student Journals. Students will then use the dry-erase markers to trace around their models on the wax paper and label the dimensions. Students should remove one section from their models at a time, trace where the section was with their dry-erase markers, and write an equation that represents the total value of the blocks in that section divided by the length, or number of rows, in the array. This will give them the top dimension of that section. Students should continue this process until they are left with an area model of the booth drawn on the wax paper. Students should record the area model and solve the problem on their Student Journals. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat •

•

DOK-1 How did you know the size of pieces to put in each spot? I used the dimensions we found in Part I and what we’ve learned about place value to figure out the size of pieces that could fit in each spot. DOK-2 What did you notice about the arrays? Each array formed a rectangle. Each array was built using the total area and length of one side as a tool. The arrays had a combination of different-sized pieces. DOK-2 What did you notice when the divisor was less than one whole and you had to divide whole numbers? To make arrays when we were dividing wholenumber dividends by decimal divisors, we had to regroup the whole numbers into tenths and hundredths to make rectangular arrays and divide into equal groups. DOK-2 How did you use your model to find the final quotient? We had to divide up the pieces into a certain number of equal groups. If we needed more tenths, we needed to regroup a whole into tenths. DOK-2 How is this process similar to or different from dividing whole numbers? It is similar because the models and processes look similar. We can use arrays and area models to represent both kinds of division. They are different because now

FACILITATION TIP Consider assigning each group to represent a scenario on a large sheet of paper that will be posted in the room. FACILITATION TIP Consider setting up the grid for the ﬁrst scenario as a whole group. FACILITATION TIP

MULTIPLY AND DIVIDE DECIMALS

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Use a yardstick, meterstick, and ruler to help the students visualize the connections between the various units used to measure length. FACILITATION TIP Use manipulatives to help the students understand this pattern. It will aid them in their future number sense and ability to recognize reasonableness.

STEMscopes Tip The Foundation Builder, located in the Engage section, is used to bridge students' learning to the current concept by addressing foundational knowledge from previous grade levels. Foundation Builder activities use manipulatives to review prerequisite student knowledge. Possible student preconceptions about a topic, with suggested solutions on how to resolve the preconceptions, are also included.

we are dividing parts of a whole instead of whole pieces. Part III: Large Booths 1.

Students should look at each large-booth measurement and draw an area model FACILITATION TIP using the wax paper and dry-erase markers. Encourage students to imagine what Allow time for the students to brainstorm the the array would look like as they break the total area apart using the length to ﬁnd strategies that they will use to solve these the width of each booth. problems.

After Part III, invite the class to a Math Chat to share their observations and learning.

If needed, allow students to revisit the base ten blocks to build an array.

MULTIPLY AND DIVIDE DECIMALS

Multiply and Divide Decimals Explore 5 – Divide Decimals — Arrays and Area Models Math Chat DOK-2 How are area models similar to arrays? They are both rectangles with a length and width. We can count the number of pieces for the width of the array, but we have to use division to ﬁnd the width in the area model. • DOK-1 How is an area model helpful? You can divide larger numbers with an area model. If you were to use large numbers with an array, you would need a lot of blocks. • DOK-2 How did you know the place value of the digits in your products? I knew dividing a whole number by a whole number would give me a whole number, and dividing tenths by a whole number would give me tenths. I remembered the place values from the arrays I built. • STEMscopes Tip Transition students into the current concept by meeting them at their level with the Hook activity, found in the Engage section. These real-world scenario-based activities frame the overall learning throughout the scope and serve as both an introduction and concluding aspect of each concept. The Hook fosters personal growth.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

MULTIPLY AND DIVIDE DECIMALS

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MULTIPLY AND DIVIDE DECIMALS

Multiply and Divide Decimals Explore 6 – Divide Multi-Digit Decimal Numbers ACTIVITY PREPARATION Students will divide multi-digit decimal numbers using the standard algorithm.

Standards for Mathematical Practice • •

MP.6 Attend to precision. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Set of Bake Sale Cards (per group) 1 Bake Sale Menu (per group) 1 Exit Ticket (per student)

•

Plan to divide the class into groups of 2–4. Print a Student Journal and an Exit Ticket for each student. Print a set of Bake Sale Cards for each group. Cut out the cards, and place each set into a resealable bag. If desired, laminate them for future use. Print a Bake Sale Menu for each group. If desired, laminate them for future use.

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION POINTS FACILITATION TIP

Before reading the scenario, ask the class 1) What is a popular item that children want to own?; 2) Do you have one?; 3) How many of these items do you think a store sells each day? FACILITATION TIP Preview the Bake Sale Cards and Bake Sale Menu with students by projecting them and reading through them together before collaboration.

FACILITATION TIP Students sometimes struggle keeping the numbers in alignment as they complete long division problems. Provide graph paper to help them write numbers in the correct place. 90

2. 3.

4. 5. 6.

Read the following scenario to the class: Josefina asked her teacher how many of each bake sale item was sold. Her teacher told her the total amount of money made from each bake sale item and challenged her to find the answer to her original question. Josefina remembered she could use the Bake Sale Menu to determine how many of each item was sold. Help Josefina determine how many of each bake sale item was sold using the given total amount of money made for each item and each single item’s price. Give a bag of Bake Sale Cards and a Bake Sale Menu to each group. As a class, discuss taking out the decimal to solve using the Brownie Bites scenario. Allow student groups to have time to solve. a.

DOK-1 When we divide with no decimals, what do we get as our answer? I get 73.

DOK-1 We want to make the divisor be a whole number. How can I do this? We need to multiply the divisor by 100 to make it a whole number.

DOK-2 Do I need to do anything to the dividend? Yes, we also have to multiply the dividend by 100, or move the decimals two places to the right.

DOK-2 What do you notice about both quotients? The quotients are the same.

Students will then work with their groups to solve the Chocolate Chip Cookies scenario. Challenge students to complete the last two scenarios without completing the step of solving without decimals. After the Explore activity, invite the class to a Math Chat to share their observations and learning. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Math Chat DOK-2 How are the division problem with the decimals and the division problem without the decimals the same? The two division problems give the same quotient. • DOK-2 Why do you think we should move the decimal to make a whole-number divisor? It makes solving the problem easier to use a whole-number divisor. • DOK-1 Can you change the placement of only one decimal? No, you need to change both the dividend and divisor placement of the decimal so you are not changing the value of the quotient. • DOK-2 What if the dividend does not have a decimal in its number? If no decimal FACILITATION TIP is shown in the dividend, that means it is at the end of the number that is shown. Provide the following example. Suppose the Place the decimal there, and move it two places to the right. Add two zeros where total sales for brownie bites was $66. How you moved the decimal. many brownie bites would have been sold? Work with students to ﬁnd the solution. Post-Explore •

1. 2. 3. 4.

MULTIPLY AND DIVIDE DECIMALS

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Have students complete the Exit Ticket to formatively assess their understanding FACILITATION TIP of the concept. Before having students complete this Exit Complete the Anchor Chart as a class. Ticket, determine if they must show their work using a speciﬁc algorithm or method. Have each student complete their Interactive Notebook. Return to the Hook and instruct students to use their newly acquired skills to successfully complete the activity.

MULTIPLY AND DIVIDE DECIMALS

Multiply and Divide Decimals Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Picture Vocabulary

Show What You Know, Part 2

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Multiply Decimals – Arrays and Area Models Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 3

A guide to facilitating the creation of a chart with students for each scope

Multiply Multi-Digit Decimal Numbers Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 4

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Divide Decimals – Place Value Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Notebook

Show What You Know, Part 5

A cut-and-glue activity to process learning that can be added to a notebook for future reference

Divide Decimals – Arrays and Area Models

Show What You Know, Part 1

Show What You Know, Part 6

Multiply Decimals – Place Value

Divide Multi-Digit Decimal Numbers

Independent practice assignment that gives students an opportunity to demonstrate their learning

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Multiply and Divide Decimals Independent and partner games and other activities that provide students with an engaging way to practice the new concept

MULTIPLY AND DIVIDE DECIMALS

Home

Intervention and Assessment SMALL-GROUP PLANNER 1

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

MULTIPLY AND DIVIDE DECIMALS

Multiply and Divide Decimals

3 94

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER Use this template to decide how to assess your students for concept mastery. Depending on the format of the assessment, you can identify prompts and intended responses that would measure student mastery of the expectation. See the beginning of this scope to identify standards and grade-level expectations. Key Concepts I can use a variety of part-whole strategies, including area model, partial product, and partial quotient to solve multiplication and division problems with multi-digit decimal numbers.

What prompts will be used?

What does mastery look like?

MULTIPLY AND DIVIDE DECIMALS

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I can use place value understanding to solve multiplication and division problems with multi-digit decimal numbers.

I can ﬂexibly select mathematical strategies to solve multiplication and division problems with multi-digit decimal numbers.

SCOPE 1

Integers Scope Introduction SCOPE SUMMARY

Student Expectations

6.NR.3.1 Identify and compare integers and explain the meaning of zero based on multiple authentic situations. 6.NR.3.2 Order and plot integers on a number line and use distance from zero to discover the connection between integers and their opposites. 6.NR.3.3 Recognize and explain that opposite signs of integers indicate locations on opposite sides of zero on the number line; recognize and explain that the opposite of the opposite of a number is the number itself. 6.NR.3.5 Explain the absolute value of a rational number as its distance from zero on the number line; interpret absolute value as distance for a positive or negative quantity in a relevant situation.

Students will extend their knowledge of the number line to represent positive and negative numbers. They reason about the relationship between a number and its opposite as points that are equidistant from zero. Both horizontal and vertical numbers will be used, as experience with both types facilitate students’ movement from number lines to coordinate grids. Students use integers to represent real-world contexts and to understand the meaning of zero in each situation. They will also compare integers from real-world scenarios and write comparison statements. Students learn the absolute value symbol and that absolute value measures the distance from an integer to zero, and can be used to determine the distance between two numbers. Contextual problem solving, such as temperature, elevation, and banking help students relate their understanding of positive and negative values, opposites, and absolute value.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grade levels, students have worked exclusively with positive rational numbers to compare and order values in real-world contexts. Students have worked with numbers lines to locate positive rational numbers. They have also used comparison symbols (>, <, and =) to write comparison statements with positive rational numbers.

After this scope, students in sixth-grade apply their understanding of negative numbers to plot and locate points in all four quadrants of the coordinate plane. They also use their knowledge of absolute value to determine the distance of a horizontal or vertical line segment that crosses quadrants on the coordinate plane. In grade seven, a critical area of instruction is developing an understanding of operations with rational numbers. Seventh-grade students extend their understanding of positive and negative numbers to understand subtraction of rational numbers as adding the additive inverse, p − q = p + (−q). They will relate the distance between two rational numbers as the absolute value of their difference and apply this principle in real-world contexts. By the time students reach eighth grade, they are able to solve real-world and mathematical problems involving all four operations with rational numbers.

Before beginning the lesson, the students’ prior knowledge is assessed to address any student needs and skills needed for content mastery. Students are assessed to determine if they can: •

represent whole numbers as lengths from 0 to 100 on a number line.

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope.

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

determine the temperature using a thermometer.

•

describe the absolute value of the numbers represented.

Students move on to the Explore activities and come back at the end of the explorations to complete the activity and discuss what they found.

If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. 96

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

INTEGERS

Home

A Number and Its Opposite In this exploration, groups of students will solve a scenario and must assemble parts of a rock wall in the correct positions as customers request. Students will: •

use horizontal and vertical number lines.

•

determine a positive number is greater than zero and a negative number is less than zero.

Explore 2

Explore 1

EXPLORE ACTIVITIES

In this exploration, students will compare and order integers on a number line and use inequality symbols. Students will: •

recognize and explain that the opposite of the opposite of a number is the number itself.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

Explore 3

Compare and Order Integers

Absolute Value In this exploration, students will collaborate to solve a scenario about designing a football ﬁeld and ensuring the painting is accurately done with ﬁeld lines being equal. Students will: •

recognize that the absolute value of a positive or negative integer is the distance that number is away from zero on a number line.

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve. Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________

INTEGERS

Integers Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

INTEGERS

Home

ACCESSING PRIOR KNOWLEDGE Students will identify two truths and a lie by reading statements about the prior standard. This element is designed to uncover student misconceptions; it should not be taken for a grade. 2.MDR.5.5: Represent whole-number sums and differences within a standard unit of measurement on a number line diagram.

Materials

Preparation

Printed •

1 Two Truths and a Lie (per student or group)

• •

Print one Two Truths and a Lie for each student or each group. You may choose to put students in groups of two or three.

Procedure and Facilitation Points 1. 2. 3. 4. 5.

Read the prompt aloud to the class. Allow 2 minutes of thinking time for the students to read the three statements and determine which two statements are truths and which one is the lie. Ask students to share with shoulder partners how they marked their sheets and why. Allow 2–5 minutes of discussion. Ask students to justify their choices for the lie. a.

The second statement is the lie. The jumps show 30 being added to 20, not subtracted from 50. It starts with a jump to 20 and then adds another jump of 30, which is equal to 50.

If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

FACILITATION TIP Walk around and listen to student discussions. If students are having diﬃculties with the task, provide prompting questions about what they see on the activity and what they think is happening. FACILITATION TIP Display the prompt to the class. For each statement, poll the class to determine who believes the statement is true and who believes it is a lie. Allow students to explain their reasoning.

INTEGERS

Integers Hook – Five-Day Forecast ACTIVITY PREPARATION Students will look at thermometers and describe the absolute value of the numbers represented.

Materials

Preparation

Printed •

• • •

1 Five-Day Forecast (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Five-Day Forecast for the whole class to view. Prepare to introduce the scenario and encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) What type of weather do you like best: warm or cold?; 2) What do you like to do when it is cold outside?; 3) What temperature do you consider "cold"?

Part I: Pre-Explore 1.

FACILITATION TIP

Ask students which words are unfamiliar to them. Write those words on the board and explain that they will learn about those words during the scope. Revisit those words during the Post-Explore.

100

Engage

Explore

Explain

Elaborate

Evaluate

Explain to students that the meteorologist needs to ﬁnd the absolute value of each temperature as well as which temperatures are opposites. Discuss the following questions: a.

DOK-1 What is the difference between the numbers above zero and the numbers below zero? The numbers below zero have a subtraction symbol in front of them. The numbers above zero do not.

DOK-1 What do you notice about the numbers as they go away from zero? The numbers seem to get larger as they go away from zero.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Intervention

Acceleration

INTEGERS

Home

STEMscopes Tip Located along the scope menu is the Explore section. One to ﬁve inquirybased Explore activities are designed to be hands-on, teacher-facilitated lessons in which students collaborate to build conceptual understanding and reason mathematically. This section also contains Skill Basics lessons in Kindergarten through Grade 2 and Virtual Manipulatives.

Show the Phenomena Video again and restate the problem. Refer to Five-Day Forecast, and discuss the following questions: a.

DOK-1 What is the absolute value of Monday’s temperature? 10

DOK-1 What is the absolute value of Wednesday’s temperature? 10

DOK-1 If both numbers have the same absolute values, what does that tell you about them? They are opposites of each other.

DOK-1 Which days are opposites of each other? Monday and Wednesday are opposites. Tuesday and Friday are opposites.

FACILITATION TIP Ask students to explain the meaning of absolute value. Ask if absolute value can be either positive or negative and why. FACILITATION TIP If students struggle to answer the question, refer to the previous two questions. Write the temperatures for Monday and Wednesday on a number line and discuss their absolute values and what it means.

101

INTEGERS

Integers Explore 1 – A Number and Its Opposite ACTIVITY PREPARATION Students will recognize that a positive number is greater than zero and a negative number is less than zero. Students will use horizontal and vertical number lines to determine the location of a positive or negative number and its opposite.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics.

Materials

Preparation

Printed • • • • •

• • •

1 Student Journal (per student) 1 Set of Rock Wall Scenario Cards (per group) 1 Horizontal Number Line (per group) 1 Vertical Number Line (per group) 1 Exit Ticket (per student)

• •

Reusable • • •

• •

1 Dry-erase marker (per group) 1 Clear sheet protector (per group) 1 Resealable bag (per group)

Plan to divide the class into groups of 2–4 students. Print a Student Journal and an Exit Ticket for each student. Print a set of Rock Wall Scenario Cards for each group. Cut out the cards, and place them inside a resealable bag for each group. Print a Horizontal Number Line and a Vertical Number Line on card stock for durability. Place the Horizontal Number Line and Vertical Number Line double-sided inside a clear sheet protector for each group. Gather enough dry-erase markers for each group to receive one. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Number Lines)

PROCEDURE AND FACILITATION POINTS FACILITATION TIP

Print and project the scenario to read aloud with students.

FACILITATION TIP

Project and then distribute the two number lines ﬁrst. Take time to allow students to make observations about what they notice. Use guiding questions 3a–3g from Step 3 to help them analyze.

FACILITATION TIP Project and model Rock Wall Scenario 1 before having students work independently or collaborate.

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Read the following scenario to the class: You and your group work for the TopNotch Rock Wall Company. They are the top rock wall designers and creators in the country. They have hired your group to assemble different parts of multiple rock wall designs based on customer instructions. It is important to get the placement of each rock-wall stone in the correct spot so these designs accurately match the customers’ requests. Give a Student Journal to each student. Give a Horizontal Number Line, a Vertical Number Line, a set of Rock Wall Scenario Cards, and a dry-erase marker to each group. Have students quickly analyze the number lines, and ask the following questions: a. DOK-1 What is the starting point or middle point on each number line? Zero b.

DOK-1 What is the greatest number for each number line? 10

DOK-1 What is the smallest number for each number line? −10

d. DOK-1 On a vertical number line, do the numbers increase or decrease as you move farther above zero? Increase e.

DOK-1 On a vertical number line, do the numbers increase or decrease as you move farther below zero? Decrease

DOK-1 On a horizontal number line, do the numbers increase or decrease as you move to the right of the zero? Increase

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

Explain to students that they will be working with their groups to read each Rock Wall Scenario Card to determine the positive and negative numbers where the rock wall stones will be placed. Mathematicians call positive and negative whole numbers integers. Instruct students to use the number lines to represent their integers. Monitor and assess students as they collaborate by asking the following guiding questions: a. DOK-2 How can you determine the location of __________ (6 feet above 0, 3 feet to the left of 0, etc.) on the number line? Answers will vary depending on the scenario. I can start at 0 and ________ (move 6 lines above 0, move 3 lines to the left, etc.) and mark that location with a point. b.

DOK-2 How can you use that location to determine the opposite location? I can go back to 0 and move the same number of lines in the opposite direction on the number line to determine the opposite location. If I moved 6 lines above 0 for the ﬁrst number, then I will move 6 lines below 0 to ﬁnd the opposite number.

DOK-1 What location do you always start at in order to determine the correct numbers on the number line? We always start at 0.

Intervention

Acceleration

INTEGERS

Home

FACILITATION TIP Provide time for students to practice saying the word integers as a whole group, in partners, and independently. Use Picture Vocabulary to clarify the deﬁnition. FACILITATION TIP Students may not understand that numbers can be less than zero. Provide real-world examples of negative numbers such as a negative account balance. Ask students if they can provide other real-world examples.

d. DOK-1 What do you notice about the two points on the number line? They are the same distance from 0, just in opposite directions. e. DOK-3 How would you describe a positive integer? A positive integer does not have a negative symbol in front of it. f. 7. 8.

DOK-3 How would you describe a negative integer? A negative integer has a negative symbol in front of it.

Allow students enough time to complete all of the work for their scenario cards. Explain the following concepts to the class: The numbers you were working with today are known as integers. An integer is any whole number that is positive or negative, including zero. A negative integer is any whole number to the left of zero or less than zero and includes a negative sign in front of the number to represent that it is negative. A positive integer is any whole number to the right of zero or greater than zero. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat •

•

FACILITATION TIP Use questions 6e. and 6f. to create a twocolumn chart of words used to describe or infer positive and negative integers (debt, owe, up, below zero, above sea level, down, proﬁt, increase...).

STEMscopes Tip Each Explore activity includes a Student Journal that students complete collaboratively while participating in group work. Students use the journal to develop metacognitive skills by reﬂecting on how and what they are learning. Communicating mathematical thinking leads to a deeper conceptual understanding of the skills at hand.

DOK-3 How did you use a number line to ﬁnd the integers in each scenario? For each integer in the scenario, we needed to start with zero and then move the given number of spaces above, below, to the left of, or to the right of zero. To ﬁnd the opposite number, we needed to start back at zero and move the given number of spaces in the opposite direction. DOK-3 How can you determine if the positive and negative integers are opposites? I can know the positive and negative integers are opposites because they will be the exact same distance away from zero. DOK-2 Where are positive integers located on the number line? If the number line is horizontal, the positive integers are located to the right of zero. If the number line is vertical, the positive integers are located above zero. FACILITATION TIP DOK-2 Where are negative integers located on the number line? If the number line is horizontal, the negative integers are located to the left of zero. If the number It may help students to say, "Left is Less." line is vertical, the negative integers are located below zero. Take time to review left and right with physical responses. DOK-3 What is the opposite of zero? Explain. The opposite of zero is zero because zero moves zero places from zero, so the opposite number also doesn’t move from zero.

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INTEGERS

Integers Explore 1 – A Number and Its Opposite Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook. Notes

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

INTEGERS

Home

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INTEGERS

Integers Explore 2 – Compare and Order Integers ACTIVITY PREPARATION Students will compare and order integers on a number line and use inequality symbols. Students will recognize and explain that the opposite of the opposite of a number is the number itself.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Elevation Cards (per group) 1 Exit Ticket (per student)

Reusable •

•

1 Resealable bag (per group)

Plan to divide the class into groups of 2–4 students. Print a Student Journal and an Exit Ticket for each student. Print a set of Elevation Cards for each group on card stock. Laminate the cards for future use. Cut out the cards, and place them inside a resealable bag for each group. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Number Lines)

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Take time to assess student knowledge regarding elevation. Concepts about above and below sea level and how high mountain elevation affects matter may be new. FACILITATION TIP Depending on your students, consider modeling Card 1 with the whole class. After modeling, allow students to collaborate or work independently on Cards 2–6. FACILITATION TIP Preview with students how to pronounce each city and consider locating them on a map for students.

FACILITATION TIP Project this question so students can read it aloud and you can clarify, "the opposite of the opposite." 106

2. 3.

Read the following scenario to the class: The Beneﬁts of Elevation Society is putting together a presentation to show how living in higher altitudes affects your health. They’ve gathered all of the data, but now it is all mixed up. Each of their slides compares two different cities. Help them organize the data so that they are prepared for their big presentation. Give a Student Journal to each student. Give each group a set of Elevation Cards. Explain to students that they will be working with their group to compare the elevation levels of US cities using inequalities. Students will record the elevation of the two cities onto a number line. They will then use that information to order the elevations into an inequality. Monitor and assess students as they collaborate by asking the following guiding questions: a. DOK-2 How did you determine which city had the lowest elevation? Answers will vary depending on the scenario. The city with the smallest elevation will be the lowest. New Orleans has an elevation of −2, and Valdosta has an elevation of 66. Therefore, New Orleans has the lowest elevation. b.

DOK-2 How can you use that location to determine the opposite of the integers that represent the elevation for the cities? I know that the opposite of a positive number is a negative number, and the opposite of a negative number is a positive number. Since the integers are −2 and 66, the opposites will be 2 and −66.

DOK-2 What is the opposite of zero? The opposite of zero is still zero.

d. DOK-3 How can you determine the opposite of the opposite of a city’s elevation? First, you can ﬁnd the opposite of the city’s elevation, and then you can ﬁnd the opposite of that number. The opposite of the opposite is always the starting number. © Accelerate Learning Inc. - All Rights Reserved

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

Allow students enough time to complete all of the work for their Elevation Cards. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-3 How did you use a number line to compare the cities’ elevation levels? Once I plotted each city’s elevation on the number line, I was able to see which ones are higher and lower. From there, I am able to compare the lower levels to the higher levels. • DOK-2 How can you determine if a city’s elevation level belongs on the left or right side of the 0? If a city has a negative sea level, then it is below sea level and needs to be on the left side of the 0. • DOK-3 How are positive and negative integers used to represent a real-world context? In this scenario, the positive integers represent an elevation above sea level and negative numbers represent an elevation below sea level. • DOK-2 What would be the difference between the opposite of a below-sea-level city and its original? The opposite would mean that it would be above sea level since the opposite of a negative number is a positive, and the right side of the 0 is positive integers and the left side of the 0 is negative integers. •

Post-Explore 1. 2. 3.

Intervention

Acceleration

INTEGERS

Home

STEMscopes Tip The Math Chat, embedded in each Explore lesson outline as well as in printable form, provides a forum where students collaboratively discuss their ideas and strategies and develop their number sense, mathematical vocabulary, and math thinking skills. Discussing the concepts taught helps students formulate stronger reasoning and critical thinking skills.

FACILITATION TIP

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

When you preview this Exit Ticket, clarify what the slash mark means between temperatures. Also, model for students how to lightly cross out data values as they plot them on the number line. Lightly crossing out data can help students track where they are and go back to ﬁx graphs and plots as needed.

107

INTEGERS

Integers Explore 3 – Absolute Value ACTIVITY PREPARATION Students will use their knowledge of number lines to recognize that the absolute value of a positive integer or negative integer is the distance that number is away from zero.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics.

Materials Printed • • • • • •

1 Student Journal (per student) 1 Creating an Integer Football Field Instructions (per pair) 1 Integer Football Number Line (per group) 1 Set of Integer Football Scenario Cards (per group) 1 Absolute Value Symbol (per class, optional) 1 Exit Ticket (per student)

Reusable • • • • •

Consumable

1 Dry-erase marker (per group) 1 Clear sheet protector (per group) 1 Quart-sized resealable bag (per group) 1 Ruler (per pair) 1 Projector (per class, optional)

•

1 Large sheet of white construction paper (per pair)

Preparation •

Print a Student Journal and an Exit Ticket for each student.

Part I: Creating an Integer Football Field Number Line • • • •

Plan to divide the class into pairs. Print a Creating an Integer Football Field Instructions for each pair. Gather enough large construction paper and rulers so each pair can have one piece of construction paper and one ruler. Make sure to set up a projector if using the Absolute Value Symbol to show students the absolute value symbol.

Part II: Integer Football Penalties • • • • • •

Plan to divide the class into groups of 3 or 4 students. Print a set of Integer Football Scenario Cards on card stock for durability, and cut them apart. Place the Integer Football Scenario Cards into a quart-sized resealable bag, and label the bag “Part II.” Print an Integer Football Number Line for each group, on card stock for durability, and place the number line inside a sheet protector to create an erasable surface. Gather enough dry-erase markers so each group can have one. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Number Lines) Notes

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

INTEGERS

Home

PROCEDURE AND FACILITATION Part I: Creating an Integer Football Field Number Line 1. 2.

5. 6.

7. 8.

Remind students that in the previous Explore, they learned that every number and its opposite number are the same distance from zero on the number line. Explain the concept of absolute value to the class: Today, you will be exploring a new vocabulary word: absolute value. The absolute value of a number is the distance between that number and zero on the number line. We represent absolute value with a special symbol. (Draw the absolute value symbol, or project Absolute Value Symbol for students.) Read the following scenario to the class: The Fighting Dragons high school team just installed new turf on its football field. The team has hired you and your partner to paint the new yard lines on the football field. It is important that each line is perfectly spaced so one line isn’t closer or farther from zero than another line. The Fighting Dragons have asked that you submit a design plan before actually painting the field to make sure the painting job will be accurate and the lines will be equally spaced. Explain to students that the 50-yard line has been replaced by the number 0. The right side of the field is the positive side of the number line, so the yards are considered positive integers. The left side of the field is on the negative side of the number line, so its yards are considered negative integers. Give a set of Creating an Integer Football Field Instructions, a ruler, and a large piece of white construction paper to each pair. Instruct the students to work with their partners to read the instructions to create and design an integer football field. Encourage students to make sure the lines are labeled with positive and negative integers and are equally spaced from 0. Monitor students for accuracy as they create their integer football fields. Model and answer any questions related to the instructions as necessary. As students collaborate on their integer football field designs, ask the following guiding questions:

FACILITATION TIP When displaying the absolute value symbol, say it as well. State, "This means the absolute value of 4." Have students repeat it. Provide another example and have students say it. FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever watched a football game?; 2) If so, what teams played?; 3) What did the football ﬁeld look like? FACILITATION TIP There may be students who are not familiar with the sport of football and yard lines. For context, it would be helpful to display a football ﬁeld and explain yard lines.

FACILITATION TIP If multiple student pairs are struggling to create their number lines, model for the whole class how to get started creating the number line and using the ruler to evenly space out the yard lines

a. DOK-1 What number does the ruler start on for each number that is labeled? Zero b. DOK-3 Why does the ruler need to start on zero for each number that is labeled on the number line? Answers may vary. If we start on a different number each time, then the distance between each line on the number line would not be accurate or equal. c.

DOK-1 Are the numbers 1 and −1 the same distance from zero? Explain. Yes, both 1 and −1 are 1 inch away from zero.

d. DOK-1 What is the absolute value of 1 and −1? The absolute value is 1. e. DOK-3 How can the absolute value of 1 and −1 be the same? The absolute values for 1 and −1 are the same because they are the same distance from 0, just in opposite directions. 9. 10. 11. 12.

After students have completed their integer football ﬁelds, analyze their designs for accuracy and correct any misunderstandings. Give a Student Journal to each student. Students use their designs and newly acquired understanding to answer the Part I questions related to absolute value and their number lines. Once all students have ﬁnished answering the questions to Part I, review their answers and check for understanding.

Part II: Integer Football Penalties 1.

Read the following scenario to the class: For every football game that is played, there are referees who monitor the game and throw a penalty ﬂag against the offense or defense when they break the rules. This penalty will affect the placement of the football on the ﬁeld. The referees always place the ball on what is known as the line of scrimmage.

FACILITATION TIP Students may struggle to understand that the absolute value of –1 and 1 are the same even though they are on opposite sides of the zero. Explain to students that absolute value represents distance which is always a positive measurement. Directionality does not matter.

FACILITATION TIP Before reading the scenario, ask the class 1) In a football game, who watches what the players do to make sure they are following the rules?; 2) What does the referee do if a player breaks a rule? 109

INTEGERS

Integers Explore 3 – Absolute Value The line of scrimmage is like 0 on a number line. It is an imaginary line that separates the offense and defense, just like 0 is a point that separates positive and negative integers. If a team breaks the rules, the referees will throw a penalty ﬂag that will affect whether the ball will be moved forward or backward from the line of scrimmage. The referees need your help determining the absolute value, or distance from 0 and the line of scrimmage, that the ball needs to be moved for each penalty they call. Give a set of Integer Football Scenario Cards, an Integer Football Number Line, and a dryerase marker to each group. Instruct students to work with their groups to read each Integer Football Scenario Card and use the Integer Football Number Line to represent what is happening for each penalty the referee calls in the football game. Once the students agree on the representation of the penalties on the Integer Football Number Line, have them draw models and answer the absolute value questions related to the model on their Student Journals. Monitor students as they collaborate on their tasks, and ask the following guiding questions as necessary:

2. FACILITATION TIP Before allowing students to work independently, provide similar examples as what they will see in the football scenario cards. Allow them to practice placing integers on the football number line or by taking steps forward and backward.

a. DOK-2 What number on the number line represents the line of scrimmage? Zero

STEMscopes Tip The Exit Ticket is used as a quick formative assessment to determine whether students mastered the skills presented in the Explore or whether additional instruction is needed. It can also be used to reinforce the skills and concepts presented. Exit Tickets and Answer Keys are found in the print ﬁles on the right of the screen and can be downloaded and modiﬁed as needed.

b. DOK-1 What do you notice about the absolute value of a number and its opposite? A number and its opposite have the same absolute value. c.

d. DOK-3 How can a number line help you determine the absolute value of a number? Answers may vary. A number line can help ﬁnd the absolute value of a number because I can visually see the representation of a number and its distance from zero. 6.

Allow students enough time to complete all of the work required for the scenario cards and collaborate and answer the reﬂection questions on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

FACILITATION TIP

DOK-3 How can a positive and negative integer have the same absolute value? If they are the same distance away from zero, then a positive and negative integer will have the same absolute value.

Math Chat •

This is a good time to review vocabulary. Allow students to turn and talk with their partner to deﬁne the words integer, opposite, and absolute value. Then, discuss those vocabulary words as a whole group.

•

DOK-3 Whether an integer is positive or negative, what do you notice about the absolute value of a number and its opposite? An integer that is positive will have the same absolute value as its opposite number that is negative. DOK-3 How can a positive and negative integer have the same absolute value? If a positive and negative integer are the exact same distance from zero, their absolute value will be the same. DOK-4 Can the absolute value of a number ever be negative? Explain. No, the absolute value is the distance from zero, and when you count the number of units from that number to zero, it will always be positive. Your number could be on the right or left side of zero, but the number of units you count from zero to that number will always be positive. DOK-2 What is the absolute value of zero? The absolute value of 0 is zero.

Post-Explore FACILITATION TIP When you preview this Exit Ticket with students, consider reading the scenario aloud.

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1. 2. 3. 4.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

INTEGERS

Home

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INTEGERS

Integers Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

A Number and Its Opposite Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Compare and Order Integers Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 3

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Absolute Value Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Notebook A cut-and-glue activity to process learning that can be added to a notebook for future reference

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

INTEGERS

Home

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Integers

Can be done independently

Independent and partner games and other activities that provide students with an engaging way to practice the new concept

113

INTEGERS

Integers Intervention and Assessment SMALL-GROUP PLANNER 1

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

Students who are still acquiring the concept and need remediation

How to Use the Review

3 114

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

INTEGERS

Home

What prompts will be used?

What does mastery look like?

I can use numerical reasoning to explain positive and negative numbers used to describe quantities having opposite directions or values.

I can use positive and negative numbers to represent quantities and explain the meaning of zero based on each situation.

I can interpret relevant, mathematical problems related to positive and negative numbers.

I can order and plot integers on both horizontal and vertical number lines.

I can make connections between integers and their opposites using their distances on a number line.

I can explain that opposite signs of integers indicate locations on the opposite sides of zero on a number line.

I can explain what the opposite of the opposite of a number is.

I can explain the absolute value of an integer as its distance from zero.

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SCOPE 1

Rational Numbers Scope Introduction SCOPE SUMMARY Students reason about the order and absolute value of rational numbers on a number line. They use comparing and ordering of rational numbers to solve problems involving contexts such as thermometers, elevation, and banking.

VERTICAL ALIGNMENT

Student Expectations

6.NR.3.4 Write, interpret, and explain statements of order for rational numbers in authentic, mathematical situations. Compare rational numbers, including integers, using equality and inequality symbols. 6.NR.3.5 Explain the absolute value of a rational number as its distance from zero on the number line; interpret absolute value as distance for a positive or negative quantity in a relevant situation. 6.NR.3.6 Distinguish comparisons of absolute value from statements about order.

Background Knowledge

Future Expectations

Students in previous grades represented positive rational numbers as points on a number line and have located and plotted points in the ﬁrst quadrant of the coordinate plane. Students interpreted, ordered, and compared positive rational numbers in the context of real-world problems. Prior to this scope, sixth graders used horizontal and vertical number lines to reason about the values of positive and negative numbers. Students used integers to represent real-world contexts and to understand the meaning of zero in each situation. Sixth graders also learned that absolute value measures the distance from an integer to zero, and that absolute value can be used to determine the distance between two numbers. Students have related their understanding of positive and negative values, opposites, and absolute value to contextual problem solving, such as temperature, elevation, and banking.

In grade seven, a critical area of instruction is developing an understanding of operations with rational numbers. Seventh-grade students extend their understanding of positive and negative numbers to understand subtraction of rational numbers as adding the additive inverse, p − q = p + (−q). ). They will relate the distance between two rational numbers as the absolute value of their difference and apply this principle in real-world contexts. In addition, seventh-grade students learn how to multiply and divide negative numbers. Seventh-grade students continue to examine proportional relationships by graphing them on the coordinate plane. By the time students reach eighth grade, they are able to solve real-world and mathematical problems involving all four operations with rational numbers, and they are ready to apply their knowledge of rational numbers and graphing to study linear functions.

Before beginning the lesson, the students’ prior knowledge is assessed to address any student needs and skills needed for content mastery. Students are assessed to determine if they can: •

read, write, and compare decimals to the thousandths.

•

compare two decimals to the thousandths.

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope.

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

compare and order rational numbers.

•

determine the order of horses from fastest to slowest.

Students move on to the Explore activities and come back at the end of the explorations to complete the activity and discuss what they found

If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. 116

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Absolute Value of Rational Numbers In this exploration, students will recognize that the absolute value of a positive or negative rational number is the distance that number is away from zero. Students will:

Explore 2

Explore 1

EXPLORE ACTIVITIES

In this exploration, groups of students will help Jack compare information he recorded from an animal show. Students will:

•

determine how to partition the number lines.

•

determine where to place each value on the number lines.

compare rational numbers using comparison symbols.

•

compare rational numbers using number lines.

•

compare rational numbers using written explanations.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

Explore 3

Compare Rational Numbers

RATIONAL NUMBERS

Home

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

Model and Order Rational Numbers In this exploration, groups of students will solve a scenario to help Jack document information from his hiking trip that includes his earnings, spendings, and miles climbed each day. Students will: •

express a situation as a rational number.

•

locate and label rational numbers on horizontal and vertical number lines.

•

order rational numbers.

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve. Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________

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RATIONAL NUMBERS

Rational Numbers Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

118

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will listen to prompts about the prior standard, decide whether each prompt is fact or ﬁction, and communicate their decisions by walking to the designated sides of the classroom. This element is designed to uncover student misconceptions; it should not be taken for a grade.

RATIONAL NUMBERS

Home

5.NR.4.2:: Represent, compare, and order decimal numbers to the thousandths place based on the meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Materials

Preparation

Printed •

1 Set of Fact or Fiction Prompts

• •

Print one copy of Fact or Fiction Prompts to read aloud to students. Another option is to project the prompts by using a digital projector.

Procedure and Facilitation Points 1.

2. 3. 4. 5.

Designate one side of your room as the Fact side of the room and the other side as Fiction. Explain to students that they will decide whether they think each prompt is fact or ﬁction and then move to the corresponding side of the room. Read the prompt, and allow students to move to different sides of the room. FACILITATION TIP Have students discuss their reasoning among their peers. Review the greater than (>) and less than (<) Before reading the next prompt, allow students to move back to their starting points. symbols with students before starting the activity. A review of the symbols could be Repeat with another prompt. used as a practice round of Fact or Fiction. a.

Prompt 1 is false.

Prompt 2 is true.

Prompt 3 is true.

FACILITATION TIP

If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

After students discuss their reasoning with their peers, allow students from each group to explain their reasoning to the class. Use a sentence stem, "This is a fact/ﬁction because..." to help students express their thoughts.

Identifying Misconceptions • •

Students may believe that the number with more digits is greater. Students may not remember what the less-than (<) and greater-than (>) symbols mean. Notes

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RATIONAL NUMBERS

Rational Numbers Hook ACTIVITY PREPARATION Students will compare and order rational numbers.

Materials

Preparation

Printed •

• • •

1 And the Winner Is … (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project the And the Winner Is … for the whole class to view. Prepare to introduce the scenario and encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION POINTS Part I: Pre-Explore 1.

FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) Has anyone ever watched or participated in a race?; 2) What happens at the end of the race?; 3) How are the runners ranked in a race?

FACILITATION TIP Project the phrase fastest to slowest. Students commonly rush through comparison questions and put data in reverse order in their solutions. Encourage students to slow down as they read every scenario and note, "What are we trying to ﬁnd out?"

120

Engage

Explore

Explain

Elaborate

Evaluate

Acceleration

Explain to students that the race results have been posted. The numbers are not in any order but they should be. Discuss the following questions: a.

DOK-1 What do you notice about the times? The times are written as decimals and fractions.

DOK-1 If we’re going to order the numbers from fastest to slowest, what order should we put them in? Fastest to slowest is the same as least to greatest. The horse with the lowest time is the fastest time.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Intervention

Show the Phenomena Video again and restate the problem. Refer to And the Winner Is …, and discuss the following questions: a.

DOK-2 How can you compare and order numbers written in different formats? Answers will vary. You can plot them on a number line. You can convert them all to decimals.

DOK-2 Who are the top three racers? How do you know? The top three racers are Royal Roller, Real Player, and Astro Flight. They are the only three racers with times with 177 as the whole number.

FACILITATION TIP

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Students may struggle with the idea of the fastest time being the least because in games, the winner is usually the person with the greatest amount. Demonstrate this idea by having two student volunteers walk from the back of the classroom to the front.

FACILITATION TIP

If students respond with Magic Soul, Mystery Chance, and Crystal Secret, then they are still c. DOK-1 Order the racers from fastest to slowest. Real Player, Royal Roller, struggling with the idea of the fastest time Astro Flight, Target Royale, Solar Count, Prince Winner, Burning Empire, being the least value. Discuss which horses Crystal Secret, Mystery Chance, Magic Soul ran the race with the shortest time and how that equates to the fastest time. d. DOK-1 Which two horses had the same time? Mystery Chance and Magic Soul FACILITATION TIP Have students explain how they know the two horses have the same time when one of the times is written in decimal form and the other time is written as a fraction.

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Rational Numbers Explore 1 – Absolute Value of Rational Numbers ACTIVITY PREPARATION Students will use their knowledge of number lines to recognize that the absolute value of a positive or negative rational number is the distance that number is away from zero.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Blank Vertical Number Line (per group) 1 Set of Park Scenario Cards (per group) 1 Exit Ticket (per student)

• •

Reusable • • • •

• •

1 Thin dry-erase marker (per group) 1 Clear sheet protector (per group) 1 Quart-sized resealable bag (per group) 1 Projector (per class)

•

Print a Student Journal and an Exit Ticket for each student. Print a Blank Vertical Number Line for each group, on card stock for durability, and place the number line inside a sheet protector to create an erasable surface. Gather enough thin dry-erase markers for each group to have one. Print a set of Park Scenario Cards for each group on card stock for durability, and cut them apart. Place the Park Scenario Cards into a quart-sized resealable bag.

PROCEDURE AND FACILITATION POINTS 1.

2. FACILITATION TIP Take time to discuss sea level. Clarify that in a few locations the ground is below sea level. If the park was in the mountains, it's elevation would be higher. Consider showing students some local hiking trails or parks and their elevations.

Read the following scenario to the class: Jack was excited to explore the park today! He decided to climb up a tree to see how high different animals went. Then, Jack decided to go to the pond and see what animals he could ﬁnd there. After a long day of fun, Jack decided he should record what he saw. Help Jack record his ﬁndings from his day at the park! Explain to students that the park is at sea level. Ask students the following questions: a. DOK-3 What does it mean for the park to be at sea level? Sea level is zero, so this would be 0 on a vertical number line. b. DOK-2 Would the values of the animals in the tree be positive or negative? Everything in the tree would be positive because they are higher than the ground or sea level. This means the heights of the animals are positive.

3. 4.

Give one set of Park Scenario Cards, a Blank Vertical Number Line, and a dryerase marker to each group. Give a Student Journal to each student. Notes

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Engage

Explore

Explain

Elaborate

Evaluate

Explain to students that they will partition their number lines to represent the values for each scenario. Students will collaborate with their groups to determine how to partition the number lines and where to place each value on the number lines. Then, students will answer the questions corresponding to each scenario. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 Does the value increase, decrease, or stay the same as one goes down the vertical number line? It decreases.

DOK-1 Does the absolute value increase, decrease, or stay the same as one goes down the vertical number line? It increases.

10 ___

Intervention

Acceleration

FACILITATION TIP Consider modeling how to draw and partition the number lines on the Student Journal. Clarify criteria for success regarding the labels and ranges on number lines.

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FACILITATION TIP

10 ___

Absolute value is unique vocabulary. Determine ahead of time how much review or instruction students need on the deﬁnition, 14 14 12 ___ ___ notation, examples, and calculation. Use DOK-2 Where would ___ go on the number line? would be above 4, or 3 3 3 Picture Vocabulary as a reference. . DOK-1 Are the numbers 3 and − 3 the same distance from zero? 10

Explain. Yes. Both ___ and − ___ are ___ units away from zero. 3 3 3 d.

e. DOK-1 What would you write to compare the absolute value of −2 __8 and 7. 8.

1 1 1 2 ___ ? We would write −2 __8 > 2 ___ . 10 10

Allow time for students to complete the reﬂection questions at the end of their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

FACILITATION TIP The phrase equal magnitude is used on this scope's Exit Ticket. Be sure to include the words in your explanation of absolute value.

Math Chat •

DOK-1 What is the absolute value of __3 and − __4? The absolute value for both values

FACILITATION TIP

1 Project the Math Chat questions so students is __3. can read them as you guide the chat. • DOK-3 Whether a fraction is positive or negative, what do you notice about the absolute value of a fraction and its opposite? A fraction that is positive will have the same absolute value as its opposite fraction that is negative. STEMscopes Tip • DOK-3 How can a positive and negative rational number have the same absolute value? If a positive and negative rational number are the exact same distance Virtual Manipulatives are located from zero, their absolute value will be the same. under the Explore tab. Unlike concrete manipulatives, these digital • DOK-4 Can the absolute value of a fraction and a decimal number ever be the manipulatives require no setup and same? Explain. Yes. If the value of the fraction and the value of the decimal are easily accessed online at any time. number are equal in magnitude, the two values will have the same absolute value. Students can interact with a variety of virtual manipulatives to explore Post-Explore mathematical concepts anytime, 1. Have students complete the Exit Ticket to formatively assess their understanding anywhere. of the concept. 2. Complete the Anchor Chart as a class. 3. Have each student complete their Interactive Notebook.

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Rational Numbers Explore 2 – Compare Rational Numbers ACTIVITY PREPARATION Students will compare rational numbers using comparison symbols, number lines, and written explanations. In addition, they will ﬁnd absolute value to determine magnitude.

Standards for Mathematical Practice • • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Set of Animal Show Information Cards (per group) 1 Set of Blank Number Lines (per student, optional) 1 Exit Ticket (per student)

Reusable •

2 Resealable bags (per group)

Consumable • •

• •

• • • • •

Plan to divide the class into 6 groups. Print a set of Animal Show Information Cards for each group. Cut out all of the cards, and place them in a resealable bag for each group. If desired, laminate the cards for future use. Gather 60 index cards to put in a resealable bag for each group. Create one blank number line for each group on the ﬂoor using duct tape. Print a Student Journal and an Exit Ticket for each student. Make a copy of Blank Number Lines for struggling students to use. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Number Lines)

1 Roll of duct tape (per class) 60 Index cards (per group)

PROCEDURE AND FACILITATION POINTS FACILITATION TIP

Before reading the scenario, ask the class 1) What is your favorite wild animal?; 2) What do you know about your favorite animal?; 3) Have you ever seen your animal in its natural habitat? If so, what did you notice?

2. 3.

FACILITATION TIP After students have stated the symbols used to compare the heights, write the symbols on the board and have students identify the symbols. FACILITATION TIP Have students practice saying, writing and deﬁning the word inequality. 124

Read the following scenario to the class: Jack loved seeing all of the animals on his vacation! He decided to watch another animal show and record as much information as he could about it. Help Jack understand the information he recorded by creating number lines and using comparison symbols and absolute value to understand the information. Give a set of Animal Show Information Cards and one bag of index cards to each group. Have students look at the Jumping Contest card. Discuss the following questions with the class: a.

DOK-1 What comparison symbols can we use to compare the height each animal jumped? We can use less than, greater than, or equal to, to compare the heights each animal jumped.

DOK-1 What would you write to compare Sally’s jumping height to

Sindy’s jumping height? We can write 7.25 > 4 __2.

Explain the following to the class about inequalities: Mathematicians call this an inequality. We will use inequalities and words to compare Jack’s recordings at the animal show. © Accelerate Learning Inc. - All Rights Reserved

5. 6.

Explore

Explain

Elaborate

Evaluate

Explain to students that we have previously found absolute value. In this Explore activity we will need to ﬁnd the magnitude. Magnitude is found by taking the absolute value of a number. a.

DOK-1 Jack has a debt of $5. What is this value as a rational number? A debt of $5 would be written as −5 as a rational number.

DOK-1 Magnitude is found by taking the absolute value of a number. What is the magnitude of −5? The absolute value of −5 is 5. Therefore, the magnitude is also 5.

Give a Student Journal to each student. Students will work collaboratively with their groups and their giant number lines on the ﬂoor to create number lines using the information from each Animal Show Information Card. Students will need to determine the intervals for each number line and write the rational numbers on the index cards. Some cards will need to be reused for different sets; for example, write 0 on one index card, and use it for every number line created. Students will then use the giant number lines on the ﬂoor to plot each point from the Animal Show Information Cards. Then, students will complete their Student Journals, write inequalities, and answer the questions for each Animal Show Information Card. As students are working, actively monitor each group. For students who are struggling, you can provide the Blank Number Lines to help them. Ask the following guiding questions: a.

Engage

DOK-1 Where are negative numbers on a horizontal number line? On a vertical number line? Negative numbers are on the left side of a horizontal number line. On a vertical number line, negative numbers are below zero.

DOK-1 Where are positive numbers on a horizontal number line? On a vertical number line? Positive numbers are on the right side of a horizontal number line. On a vertical number line, positive numbers are above zero.

DOK-1 How can you determine the interval to use for each scenario? Student responses will vary. You can determine the interval to use on the number line by looking at the fraction or decimal part of the rational numbers in the scenario.

DOK-1 What should you do to plot both fractions and decimal numbers on the number line? Student responses will vary. You can change all of the fractions to decimals or all of the decimals to fractions.

Intervention

Acceleration

FACILITATION TIP Prompt students to ﬁnd absolute value by asking, "How far away from zero is the number?"

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FACILITATION TIP Specify what is meant by rational number. State that rational numbers could include fractions, decimals, or integers. FACILITATION TIP Provide a few more examples to ensure students' understanding of magnitude. It might be helpful to display the examples on the board and read them aloud so that students can connect absolute value to magnitude. FACILITATION TIP Consider having a giant horizontal number line with integers from −20 to 20 on your wall. Keep it up to use anytime you need to discuss rational numbers, fractions, and decimals in between or absolute value. FACILITATION TIP Students often confuse vertical and horizontal. Provide an example of a horizontal number line and an example of a vertical number line. FACILITATION TIP When you have students review horizontal and vertical, have them use arms/ hands to show the difference.

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RATIONAL NUMBERS

Rational Numbers Explore 2 – Compare Rational Numbers Math Chat

FACILITATION TIP Students may confuse the number with the greatest magnitude and the number with the greatest value. If so, remind students that magnitude is based on absolute value, or distance from zero.

•

DOK-1 How does using a number line help determine which value is smaller when both values are negative numbers? All negative numbers are on the left side of zero. The farther the number is to the left of zero, the smaller the number is. The number that is farther to the left of zero is the smaller number.

•

DOK-2 In the Animal Show Information Card titled “Above and Below,” which animal had the greatest magnitude? Bobby the blue crab had the greatest magnitude because he has a distance of 38.3 feet away from zero. Grant the 1

green sea turtle is 24 __2 feet from zero, and Alec the Atlantic puﬃn is 12 __5 feet

away from zero. DOK-2 Look at the animal show’s proﬁts and losses in your Student Journal. Which lines show debts of more than $10.00? Lines 1, 2, and 5 show debts of more than $10.00. • DOK-2 Compare spending $2.50 to earning $5.25 using words. Spending $2.50 would be written as −2.50 as a rational number, and earning $5.25 is written as 5.25. −2.50 is less than 5.25. •

FACILITATION TIP If students struggle to provide the desired response, prompt them to state the symbols or words that are used to compare numbers. Then, discuss the rational numbers associated with spending $2.50 and earning $5.25. FACILITATION TIP To encourage detailed explanations for this Exit Ticket, provide sentence frames for students who may need them.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Rational Numbers Explore 3 – Model and Order Rational Numbers ACTIVITY PREPARATION Students will express a situation as a rational number and locate and label rational numbers on horizontal and vertical number lines. Students will order rational numbers.

Standards for Mathematical Practice • • •

MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision.

Materials Printed • • • •

1 Student Journal (per student) 1 Blank Vertical Number Line (per group) 1 Blank Horizontal Number Line (per group) 1 Exit Ticket (per student)

Preparation • • • • • •

Reusable • •

Plan to divide the class into groups of 3 or 4 students. Print a Student Journal and an Exit Ticket for each student. Print one Blank Horizontal Number Line and one Blank Vertical Number Line on card stock for each group of students. Place each number line into a sheet protector to create erasable surfaces. Gather enough dry-erase markers for each group to have one. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Number Lines)

2 Sheet protectors (per group) 1 Dry-erase marker (per group)

PROCEDURE AND FACILITATION POINTS Part I: Hiking Trip FACILITATION TIP

Before reading the scenario, ask the class 1) Does anyone like to hike?; 2) If so, where do you go to hike?; 3) What's the longest hike you have ever been on?

Read the following scenario to the class: Jack was getting ready to go on a weekend hiking trip! While on his trip, he documented his earnings and spendings and the miles he climbed up and down the mountain each day. Help Jack create number lines of his recordings and order these numbers. Remind students that they have worked with positive fractions and decimal numbers in elementary school. Inform them that they will use positive and negative numbers that include fractions and decimals. Mathematicians call these numbers rational numbers. a. b.

FACILITATION TIP Students may have trouble articulating how to locate numbers on a number line. Display an open number line and ask students how to locate two-ﬁfths on the number line.

3. 4. 5.

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DOK-1 What is an example of a rational number? Answers will vary. 1

, −__ 2.5, 3 ___ 10 4

DOK-1 How can we locate rational numbers on a number line? Answers will vary. First, we can ﬁnd the two whole numbers that the rational number is in between. Then, we partition the space between the numbers depending on the fraction or decimal part of the rational number.

Give a Student Journal to each student. Divide the class into groups. Distribute a Blank Vertical Number Line, Blank Horizontal Number Line, and dry-erase marker to each group. Students will collaborate with their groups to read each situation and represent the situation on the given number line. © Accelerate Learning Inc. - All Rights Reserved

7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

Instruct students to use the group’s Blank Horizontal Number Line and Blank Vertical Number Line to partition, locate, and label each rational number on the correct number line. Once the group agrees on the location, they will record their work on the number line on their Student Journals. Monitor students as they collaborate by asking the following guiding questions: a.

DOK-1 What does it mean to earn money? Answers may vary. It means we are receiving money for a job.

DOK-2 Will the rational number be positive or negative when you earn money? Explain. The rational number will be positive because you will have a number greater than zero.

DOK-1 What does it mean to spend money? Answers may vary. It means you are paying some of your money for a product or service.

DOK-2 Will the rational number be positive or negative when you climb up the mountain? Climb down the mountain? Explain. The rational number will be positive when you climb up the mountain because you will have a number greater than zero or greater than the starting point of the hike. The rational number will be negative when you climb down the mountain because you will have a number less than zero or a number less than the starting point of the zero.

Intervention

Acceleration

STEMscopes Tip The Explain section, located along the scope menu, has a variety of elements designed to solidify students' understanding of the content presented in the Explore section. Each scope's Explain section includes a Picture Vocabulary, independent practice assignments, anchor charts, journal prompts, and interactive notebook activities.

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e. DOK-2 Will the rational number be positive or negative when you spend money? Explain. The rational number will be negative because you will have a number less than zero. f. DOK-3 How can you locate rational numbers on a horizontal number line? Answers may vary. Positive rational numbers will be to the right of zero. Negative rational numbers will be to the left of zero. I can ﬁnd the two whole numbers the rational number is in between. Then, I can determine how to partition the space between the whole numbers according to the fraction or decimal part of the rational number. g. DOK-3 How can you locate rational numbers on a vertical number line? Answers may vary. Positive rational numbers will be above zero. Negative rational numbers will be below zero. I can ﬁnd the two whole numbers that the rational number is in between. Then, I can determine how to partition the space between the whole numbers according to the fraction or decimal part of the rational number. 9.

2. 3.

Have students identify the horizontal number line that was provided to their group. Choose one of the rational numbers from the activity and have the students explain how to locate it on the number line.

Allow students enough time to discuss and record their responses for each table on their Student Journals.

Part II: Fishing Trip 1.

FACILITATION TIP

Read the following scenario to the class: Jack had so much fun on his hiking trip that he decided to take another weekend trip to go ﬁshing! Help determine the location of each rational number Jack recorded on his trip. Then, order the rational numbers. Students will collaborate with their groups to read each situation and represent the situation on the given number line. Instruct students to use the group’s Blank Horizontal Number Line and Blank Vertical Number Line to partition, locate, and label each rational number on the correct number line. Once the group agrees on the location, they will draw their models of the number line on their Student Journals.

FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever been ﬁshing?; 2) If so, who did you go with?; 3) How many ﬁsh did you catch? FACILITATION TIP Use the directions "ordering the rational numbers" as an opportunity to review different directions for ordering. Prepare students to watch for: highest to lowest, coldest to warmest, least to greatest, etc. Students need to be reminded to carefully read the scenarios before they begin the work of ordering.

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Rational Numbers Explore 3 – Model and Order Rational Numbers 5.

Monitor students as they collaborate by asking the following guiding questions: a.

DOK-3 How can you represent rational numbers with fractions and rational numbers with decimals on the number line together? Student responses will vary. You can change all of the rational numbers to either be decimals or fractions. Once all rational numbers are the same, then you can plot the numbers on the number line.

DOK-3 What does zero represent on the number line? Zero represents where the boat is still on the water, or sea level.

DOK-3 How can you locate rational numbers on a horizontal number line? Answers may vary. Positive rational numbers will be to the right of zero. Negative rational numbers will be to the left of zero. I can ﬁnd the two whole numbers the rational number is in between. Then, I can determine how to partition the space between the whole numbers according to the fraction or decimal part of the rational number.

DOK-3 How can you locate rational numbers on a vertical number line? Answers may vary. Positive rational numbers will be above zero. Negative rational numbers will be below zero. I can ﬁnd the two whole numbers the rational number is in between. Then, I can determine how to partition the space between the whole numbers according to the fraction or decimal part of the rational number.

FACILITATION TIP Some students may struggle to convert rational numbers to decimals or fractions. Remind them that both fractions and decimals are parts of a whole and they can use benchmarks to help them place numbers on the number line. FACILITATION TIP Take time to quickly assess which students need support with clarifying left and right. Use total physical response to help them review (point left, raise your right pinky, who sits to your right etc.). Some students might be helped by the phrases "left is less" (horizontal) and "less is lower" (vertical).

6. 7.

Allow students enough time to collaborate and record all of their work and answer the reﬂection questions on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat FACILITATION TIP If students are having diﬃculty distinguishing between rational numbers and whole numbers, provide an example of each for students to describe how plotting them on a number line is similar. FACILITATION TIP Have students create their own number line to locate 2.8. Allow students to compare their number lines with their group members. As a class, discuss how the number lines are similar and how they are different. FACILITATION TIP Prompt students to describe how they can use a number line to order rational numbers from greatest to least. Ask if there was a problem in the activity that required the numbers to be ordered from greatest to least and how they knew. FACILITATION TIP Before assigning the Exit Ticket, allow students to read the scenario and ask clarifying questions. Alternatively, emphasize the phrase warmest to coldest vs coldest to warmest.

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DOK-3 How is plotting rational numbers on a number line similar to plotting whole numbers on a number line? Plotting decimals, fractions, and whole numbers on a number line are similar to each other because you will always start at zero and move up or to the right for a positive number and down or to the left for a negative number. When plotting rational numbers, you will not mark on the whole number, but instead you will mark at the location between two whole numbers. • DOK-2 How do you create your own number lines for rational numbers? Students should describe how to partition number lines so they are able to determine the location of rational numbers with fractions and decimals. • DOK-3 How can you represent rational numbers with fractions and rational numbers with decimals on the number line together? You can change all of the rational numbers to be either decimals or fractions. Once all rational numbers are the same, then you can plot the numbers on the number line. • DOK-3 How can a number line help to order rational numbers? Number lines can help order rational numbers because you can start at the left (or bottom) and write the numbers down in order from left (bottom) to right (top). •

Post-Explore 1. 2. 3. 4.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Rational Numbers Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Absolute Value of Rational Numbers Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Compare Rational Numbers Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 3

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Model and Order Rational Numbers Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Notebook A cut-and-glue activity to process learning that can be added to a notebook for future reference

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Compare Rational Numbers

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Independent and partner games and other activities that provide students with an engaging way to practice the new concept

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Rational Numbers Intervention and Assessment SMALL-GROUP PLANNER 1

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

Students who are still acquiring the concept and need remediation

How to Use the Review

3 134

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

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What does mastery look like?

I can interpret and explain the meaning of numerical statements of inequality and their relative position of two integers positioned on a number line. I can recognize that rational numbers are numbers that can be written as a fraction with integers for both the numerator and the denominator. I can create a comparison statement of rational numbers. I can compare rational numbers. I can order rational numbers. I can use inequality symbols appropriately in solutions. I can compare numbers according to their absolute values. I can place rational numbers on a number line according to their values. I can identify rational numbers and their absolute values. I can interpret statements as comparison statements or order statements.

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SCOPE 1

Equivalent Numerical Expressions Scope Introduction SCOPE SUMMARY A critical area of instruction in 6th grade is writing, interpreting, and evaluating expressions and equations that represent real-world problems. Sixth-grade students apply the order of operations to evaluate expressions with parentheses and exponents. Students use their knowledge of factors and multiples to compute ﬂuently, to identify common multiples and common factors, and to explore prime factorization. They determine the least common multiple (of numbers up to 12) and the greatest common factor (of numbers up to 100). Student Expectations

6.PAR.6.1 Write and evaluate numerical expressions involving rational bases and whole-number exponents. 6.PAR.6.2 Determine greatest common factors and least common multiples using a variety of strategies to make sense of applicable problems. 6.PAR.6.4 Evaluate expressions when given values for the variables, including expressions that arise in everyday situations.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grade levels, students gained experience writing expressions to model contextual situations. Beginning in grade three, students applied the properties of operations to add, subtract, multiply, and divide whole numbers and positive rational numbers. In fourth grade, students listed factor pairs of numbers up to 100 and used visual array models and corresponding factor pairs to determine if a number is prime or composite. In ﬁfth grade, students wrote expressions, without exponents, to represent multistep problems, and they used the order of operations to evaluate them. Students have also encountered square units when solving problems involving area, and cubic units when solving problems involving volume.

In the next sixth-grade scope, students will model and evaluate algebraic expressions and develop the ability to use the distributive property ﬂexibly. Students will simplify algebraic expressions using exponents and by collecting like terms, and they will use properties of operations to determine whether or not two expressions are equivalent. Sixth grade marks a foundational year for building the bridge between concrete concepts of arithmetic and the abstract thinking of algebra. Visual representations and concrete models can help students develop understanding as they move toward using abstract symbolic representations in seventh grade. Students build on their understanding of multiples, factors, and mathematical properties to generate and use the arithmetic of rational numbers. Seventh- and eighth-grade students continue to interpret, write, simplify, and solve expressions and equations.

Before beginning the lesson, the students’ prior knowledge is assessed to address any student needs and skills needed for content mastery. Students are assessed to determine if they can: •

explain patterns.

•

justify their reasoning.

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope.

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

ﬁnd equivalent expressions.

•

create a numerical expression equivalent to the expression given by using order of operations.

Students move on to the Explore activities and come back at the end of the explorations to repeat the activity and discuss what they found.

If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. 136

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Greatest Common Factors In this exploration, students will work with their groups to solve scenarios involving helping a party planning company determine the largest number of balloon arrangements that can be made. Students will: •

Explore 2

Explore 1

EXPLORE ACTIVITIES

ﬁnd factors, common factors, and greatest common factors.

•

use manipulatives such as linking cubes and balloon cards.

•

determine how many of each color of balloons will be in the arrangement.

•

determine the relationship between prime factors and the last common multiple.

•

use Cuisenaire® Rods and prime factors to ﬁnd the least common multiple of different sets.

•

help place the order for the items for upcoming parties.

Explore 4

Explore 3

In this exploration, students will solve a scenario where they must help a party planner determine how many packages of each item are needed to have equal amounts of items for upcoming parties. Students will:

•

use strategies to ﬁnd prime factors.

•

write prime factorization expressions.

•

use the distributive property in association with greatest common factors.

Exponents In this exploration, students will learn through expanding from the previous party planning scenario. This scenario is about cupcake orders. Students will: •

rewrite and simplify expressions using exponents.

•

write the prime factorization expression in expanded form.

•

rewrite a cupcake order as an exponential expression.

After completion of the exploration, students will discuss their learning and then complete an Exit Ticket for an assessment.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

Explore 5

In this exploration, groups of students will solve a scenario where they help ﬁnd the prime factorization for each set of cookies to determine the greatest number of cookie snack bags that can be made. Students will:

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment. Least Common Multiples

Prime Factorization

EQUIVALENT NUMERICAL EXPRESSIONS

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Order of Operations In this exploration, students will solve a scenario involving helping employees at a catering company determine which expression will provide them with the correct amounts of orders. Students will: •

match each order expression with its correct number order.

•

use the order of operations to match expressions with their solutions.

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve.

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Equivalent Numerical Expressions Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will read different student responses to a posed question on the prior standard, decide whether they agree or disagree with the student, and explain their reasoning. 5.NR.1.2: Explain patterns in the placement of digits when multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10, up to 10³.

Materials

Preparation

Printed •

•

Print one Agree or Disagree for each student.

1 Agree or Disagree (per student)

Procedure and Facilitation Points 1. 2. 3. 4. 5. 6. 7.

Instruct students to complete the Agree or Disagree independently. Once students have completed the activity on their own, have them stand up. Instruct all students to walk around the classroom with their hand raised in a high-five position. On your instruction, students will stop and high-five the closest person. This will be their partner. Give students a couple of minutes to discuss their answers and justifications together. You may then continue as many times as you want with different partners. Discuss the responses as a class. Allow students to explain their reasonings for each problem. a.

Disagree with Arya

Disagree with Sarah

Agree with Zoey

If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

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FACILITATION TIP Play music while students are walking around. When the music stops, students high ﬁve the closest person. Time the student discussions. Resume music to signal to the students to ﬁnd a new partner. FACILITATION TIP Take a poll to see who agrees or disagrees with each statement. This can be done with thumbs up or thumbs down. Hear reasoning from students who agree and students who disagree.

Identifying Misconceptions •

•

It may help students to think that multiplying by 10 and powers of 10 and so on means that the product has a greater value. This means that the decimal point is shifted to the right. Similarly, it may help students to think that dividing by 10 and powers of 10 means that the product has a smaller value. This means that the decimal point is shifted to the left. Notes

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Equivalent Numerical Expressions Hook – Who Got It? ACTIVITY PREPARATION Students will create a numerical expression equivalent to the expression given by using order of operations.

Materials

Preparation

Printed •

• • •

1 Who Got It? (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Who Got It? for the whole class to view. Prepare to introduce the scenario and to encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) When you have something you need to do, do you ask your friends for help?; 2) If so, which friends do you choose to help you?; 3) Why do you choose those particular friends? FACILITATION TIP Project the scenario and highlight the phrase equivalent expression. FACILITATION TIP When you project Who Got it? to the class, initially cover up the suggested solutions. Give students some time to independently consider how to solve, and then work with a shoulder partner. Depending on your students, consider challenging them to suggest different answers students might come up with in error. FACILITATION TIP

Part I: Pre-Explore 1.

4. 5.

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: Cho is attempting to ﬁnd an equivalent expression. She has asked her friends for help. Which of her friends would be the best person to ask for help? Ask students the following questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Student answers will vary. I notice they are working together. One person is explaining something to the other two. Project Who Got It? to the class. Explain to students that Cho has decided to ask three friends for their help. The problem is that they all have different answers. Discuss the following questions: a.

DOK-1 Why would all three people have different answers? They each used a different order to ﬁnd an equivalent expression.

DOK-1 How do we know which person to ask for help? Have them each show their way to see whether it follows the order of operations.

Complete the Explore activities.

Before asking the questions, allow students to brainstorm what the word equivalent means. Have them turn and talk with their neighbor to discuss their thoughts. Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________

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Explore

Explain

Elaborate

Evaluate

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and restate the problem. Refer to Who Got It? Discuss the following questions: a.

c. d.

DOK-1 Which step should be taken first when finding an equivalent expression? They should complete the operation in the parentheses 5 1 1 first, which says __6 – __3. This is equal to __2.

DOK-1 What is the second step that should be taken when ﬁnding an 1 equivalent expression? The exponent should be applied to __2, which 1 gives us __4. DOK-1 What is the third step in ﬁnding this equivalent expression? 1 You would divide 48 and __4, or multiply 48 and 4 to get 192.

DOK-1 What is the fourth step in ﬁnding this equivalent expression? You would multiply 4 by 2 to get 8.

e. DOK-1 What is the ﬁnal step to ﬁnd the equivalent expression? You simply add 192 + 8 to get 200.

Intervention

Acceleration

FACILITATION TIP After the Explore activity, students should be comfortable stating the entire order of operations quickly and ﬂuently. Be sure to take time to use acronyms, music, and chants to solidify the order. FACILITATION TIP Before working through the given problem, have students explain the order of operations. FACILITATION TIP Stretch students' thinking. Have the student groups discuss what the friends in the scenario did to get an incorrect answer.

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Equivalent Numerical Expressions Explore 1 – Greatest Common Factors ACTIVITY PREPARATION Students will ﬁnd factors, common factors, and greatest common factors.

Standards for Mathematical Practice • • •

MP.4 Model with mathematics. MP.6 Attend to precision. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Balloon Arrangements Cards (per group) 1 Exit Ticket (per student)

Reusable • • • •

•

12 Red linking cubes (per group) 18 Blue linking cubes (per group) 1 Resealable bag (per group) 1 Projector/document camera (per class)

•

Plan to divide the class into groups of 4. Print a Student Journal and an Exit Ticket for each student. Print page 2 of the Balloon Arrangements Cards for each group. If desired, print the cards on card stock and laminate them for future use. Cut out the cards, and place them in a resealable bag labeled “Part II” for each group. Have a projector ready to project page 1 of the Balloon Arrangements Cards for the class. Gather 12 red linking cubes and 18 blue linking cubes for each group.

PROCEDURE AND FACILITATION POINTS

FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever had or been to a party that had balloons?; 2) How many balloons were there?; 3) What color were they? FACILITATION TIP Project this scenario in print form and help students read through it a few times. These types of factor and multiple problems can be confusing. Encourage students to recognize and look for cue words and try to visualize the stories.

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Part I: Greatest Common Factors 1.

2. 3. 4.

Read the following scenario to the class: BEST Party Planning is getting ready for upcoming parties they have booked. Danny is in charge of getting balloon arrangements together for each party. There must be at least one balloon of each color in each arrangement. Help Danny determine the largest number of balloon arrangements that can be made with none left over for each party. Then, determine how many of each color will be in the arrangement. Give a Student Journal to each student. Project page 1 of the Balloon Arrangements Cards for the class. Have students work through page one of their Student Journals as the class completes the class discussion below. Discuss the following questions with the class: a.

DOK-1 Use your linking cubes to determine the different arrangements we could make of red balloons where there are the same number of red balloons in each group. (As students answer, list the groups on the board.) 1 group of 12, 2 groups of 6, 3 groups of 4, 4 groups of 3, 6 groups of 2, and 12 groups of 1

DOK-1 What do you notice about the number of groups? These groups are the factors of 12.

DOK-1 What is a factor? Factors are the numbers that are multiplied together to get another number.

Engage

Explore

Explain

Elaborate

Evaluate

e. DOK-2 What are the different arrangements we could make of blue balloons where there are the same number of blue balloons in each group? (As students answer, list the groups on the board.) 1 group of 18, 2 groups of 9, 3 groups of 6, 6 groups of 3, 9 groups of 2, and 18 groups of 1 f.

DOK-1 What do you notice about the number of groups? These groups are the factors of 18.

DOK-1 What is a factor? Factors are the numbers that are multiplied together to get another number.

h. DOK-1 What are the factors of 18? 1, 2, 3, 6, 9, and 18 i. DOK-2 The same number of red balloons needs to be in each arrangement, and the same number of blue balloons needs to be in each arrangement. If all arrangements need to be identical, what factors do the red balloon arrangements and the blue balloon arrangements have in common? 1, 2, 3, and 6 are the factors that red and blue balloons have in common. j. Explain the following to the class: Mathematicians call these common factors.

Intervention

Acceleration

FACILITATION TIP Instruct students to use their linking cubes to determine the different arrangements that could be made with 18 blue balloons. Allow them to record their answers ON the Student Journal before discussing as a whole group. FACILITATION TIP Circle the common factors in each list as students identify them. Encourage the students to circle the common factors on their Student Journals.

EQUIVALENT NUMERICAL EXPRESSIONS

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FACILITATION TIP

Suggested strategy to help with vocabulary development: after saying the phrase k. DOK-2 How can we use factors to group the balloons? Answers will vary. common factors, have students repeat Students should determine that finding the factors will help us find how it. Then, clarify the definition of common many balloon arrangements can be made. factors and point to the common factors on the board. l. DOK-2 What arrangements could be made of red balloons and blue balloons? Model these arrangements with your linking cubes. We could have one arrangement with 12 red balloons and 18 blue balloons. We could have two arrangements that would each have 6 red balloons and 9 blue balloons. We could have three arrangements that would each have 4 red balloons and 6 blue balloons. Or we could have 6 arrangements that would each have 2 red balloons and 3 blue balloons. m.

DOK-2 What is the greatest number of identical arrangements that can be created? The greatest number of arrangements we could create is 6 arrangements. Each arrangement will have 2 red balloons and 3 blue balloons in it.

n. Explain the following to the class: Mathematicians call this the greatest common factor. o.

DOK-2 Why would we want the greatest number of groups of balloons that can be created with the same number of red balloons and the same number of blue balloons? We want the greatest number of groups because we need to make the most balloon arrangements possible.

Part II: Balloon Arrangements 1. 2. 3.

Give the Part II cards to each group. Have students collaborate with their groups to determine the common factors and greatest common factors of each set of balloons. As students are working, actively monitor each group. Ask the following questions: a.

DOK-1 What are the common factors of 30 and 48? (Numbers depend on which set of balloons they are working on.) The common factors of 30 and 48 are 1, 2, 3, and 6.

DOK-2 What strategy are you using to ﬁnd common factors? Answers will vary. I wrote out the factors for each number of balloons. Then, I circled the factors that were the same between the pair.

After the Explore activity, invite the class to a Math Chat to share their observations and learning.

FACILITATION TIP Before moving on to Part II, take time to provide some extra practice if needed for ﬁnding and listing factors and determining common ones. Decide whether you want to review prime numbers. FACILITATION TIP As an accommodation, provide a multiplication chart for students who need additional support. This will help students identify factors. FACILITATION TIP Ensure that students have listed all of the factors for the given numbers. If students are struggling to identify all of the factors for a number, encourage them to use the divisibility rules. Have an anchor chart of those rules available as a reference. 143

EQUIVALENT NUMERICAL EXPRESSIONS

Equivalent Numerical Expressions Explore 1 – Greatest Common Factors Math Chat DOK-1 What is a factor? A factor is a number that divides into another number without a remainder. • DOK-1 How can you determine what the factors are for each number given? ou can use the divisibility rules to help. • DOK-2 Explain how finding all of the factors of each number helps you determine the greatest common factor. When you find all of the factors for each number, you can see what factors they have in common. Then, you can find the greatest common factor. • DOK-3 Explain why you would want to use the greatest common factor for a set of numbers and not any of the other common factors. We want to have the biggest number of balloon arrangements possible. Therefore, we must use the greatest common factor. • FACILITATION TIP Be prepared to allow a variety of methods: listing factors for each number, listing factor pairs for each number, using a times table to hunt for factors, the ladder method, etc. FACILITATION TIP Consistently review the definitions and examples for factors and products. Be sure they are posted on the word wall/anchor chart.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook. Notes

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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Equivalent Numerical Expressions Explore 2 – Prime Factorization ACTIVITY PREPARATION Students will use strategies to ﬁnd prime factors and write prime factorization expressions. Students will use the distributive property in association with greatest common factors.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics.

Materials

Preparation

Printed • • • • • •

1 Student Journal (per student) 1 Set of Cookie Scenario Cards (per group) 1 Set of Table Scenario Cards (per class) 1 Factor Tree Diagram (per group) 1 Factor Tree Discussion Card (per class) 1 Exit Ticket (per student)

Reusable • • • •

1 Clear sheet protector (per group) 1 Dry-erase marker (per group) 1 Resealable bag (per group) 1 Projector or document camera (per class)

• • •

•

Plan to divide the class into groups of 4. Print a Student Journal and an Exit Ticket for each student. Print one Factor Tree Diagram per group. Place each Factor Tree Diagram inside a clear sheet protector for students to write on. Optionally, print the diagram on card stock and laminate it for future use. Print one set of Cookie Scenario Cards per group. If desired, print the cards on card stock and laminate them for future use. Cut out the Cookie Scenario Cards, and place them in a resealable bag labeled “Part II” for each group. Have a projector or document camera ready to project the Factor Tree Discussion Card for Part I and the ﬁrst Table Scenario Card for a class discussion in Part III.

PROCEDURE AND FACILITATION POINTS Part I: Factor Trees 1. 2. 3. FACILITATION TIP If students struggle to determine two factors of 120, work with them to use the divisibility rules to list the factors. FACILITATION TIP Encourage students to provide examples of prime numbers. Write them on the board, along with their factors. Do the same for composite numbers. FACILITATION TIP Have students copy the deﬁnition for prime and composite numbers and list examples on their Student Journal. Be sure to include this vocabulary on your word wall or anchor chart. 146

Display the Factor Tree Discussion Card for students. Give a Student Journal to each student. Discuss the following concepts and questions with the class: a.

Look at the diagram shown on the board.

DOK-1 What are two factors that will result in 120? Answers will vary. 12 and 10

Explain to students that these two factors, twelve and ten, will be the ﬁrst two branches of the factor tree. Show students where to write the factors twelve and ten on the Factor Tree Discussion Card.

DOK-1 What is a prime number? A prime number is a whole number greater than one with exactly two factors: one and itself.

e. DOK-1 What is a composite number? Composite numbers are whole numbers greater than one with more than two factors. f.

DOK-1 Are twelve and ten prime numbers or composite numbers? Twelve and ten are both composite numbers.

DOK-1 Can twelve and ten be decomposed into more factors? Yes, because twelve and ten are both composite numbers, we can ﬁnd two more factors of each number.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

i. DOK-1 Are either of the factors six or two prime numbers? Two is a prime number, and six is a composite number. j. DOK-1 Should we continue to decompose prime numbers like two? A prime number can only be decomposed into one and itself. Therefore, once we reach a prime number, we do not need to decompose that factor again. k.

Explain to students that once we get a prime number, we will circle it so we know that we do not need to continue to ﬁnd factors for that number.

l. DOK-1 What are two factors of 10? Answers will vary. Two and ﬁve are factors of ten. m.

DOK-1 Are either of the factors two or ﬁve prime numbers? Both two and ﬁve are prime numbers, so we will circle both factors.

STEMscopes Tip The Picture Vocabulary, located in the Explain section, can be made into a word wall that students reference throughout the scope. Add vocabulary to the wall during the Math Chat or an Explore lesson as a means of solidifying conceptual understanding and of modeling precision in language and mathematical communication.

n. DOK-1 What are two factors of 6? Two factors of six are two and three.

4. 5. 6.

DOK-1 Are either of these factors prime? Yes, both two and three are prime numbers, so we will need to circle both numbers.

DOK-1 What do you notice about the factors that are circled? All of the factors that are circled are prime numbers.

DOK-1 How can you write these prime numbers as an expression? Responses may vary. We can write the expression 2 × 2 × 2 × 3 × 5.

Explain the following to the class: Mathematicians call this expression the prime factorization. Mathematicians write prime factorization expressions from least to greatest.

Allow time for students to write their factor trees on their Student Journals in the ﬁrst box. Give one Factor Tree Diagram and one dry-erase marker to each group. Discuss the following items with the class: a. Think about what the prime factorization expression would be if we started the factor tree with different factors than twelve and ten. b.

Have students continue another factor tree for 120 using two factors different from twelve and ten. Remind students to circle the prime numbers as they complete their factor trees. Discuss the following question with the class: a.

DOK-1 What are two other factors of 120? Answers will vary. 60 and 2

DOK-1 Is the prime factorization expression for this factor tree the same as or different than the ﬁrst one? Explain. Explanations may vary. Both prime factorization expressions are the same because they are made up of only the prime numbers that result in 120 when multiplied.

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FACILITATION TIP Further explain that prime factorization is the process of writing a number as a product of its prime factors. Show this by evaluating the expression 2 × 2 × 2 × 3 × 5 to get a product of 120.

FACILITATION TIP Allow students to predict whether the prime factorization will be the same if the factor tree was started with different factors of 120. Have students turn and talk to a partner. Discuss their predictions as a class. FACILITATION TIP Suggestion: assign a different factor pair of 120 for each group to start their factor trees. Once all groups have completed their factor trees, allow them to compare their prime factorization.

After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-3 What is a prime number? A prime number is a whole number greater than one with exactly two factors: one and itself. • DOK-3 How do you write the prime factorization expression? You write a multiplication expression using all of the prime numbers from the factor tree from least to greatest. • DOK-3 Explain why the prime factorization of a number is the same regardless of what two factors you decide to use in the ﬁrst step of the factor tree The prime factorization of a number will be the same regardless of what two factors you start the factor tree with because each number will always have the same set of prime numbers it is composed of. • DOK-2 Why do we not include one in the factor tree? One is not a prime number and does not help us get to the prime numbers in a factor tree. •

FACILITATION TIP Find time to review and a method that works to help students clearly memorize/ differentiate between the deﬁnitions of prime and composite numbers. FACILITATION TIP Discuss why 1 is not a prime number. Encourage students to list the factors of one. Explain that 1 can only be divided by itself. Since it only has one factor, it is not a prime number.

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Equivalent Numerical Expressions Explore 2 – Prime Factorization Part II: Cookies 1.

FACILITATION TIP If students struggle to determine the greatest common factor of two numbers using prime factorization, have them underline the prime factors that the two numbers have in common. Then, multiply those prime factors.

2. 3.

Read the following scenario to the the class: Lizzy is creating cookie snack bags for some parties. She thinks she can find the prime factors of each cookie type in the set to determine the greatest number of cookie bags she can make containing each type of cookie. Help find the prime factorization for each set of cookies to determine the greatest number of cookie snack bags that can be made. Give a set of Cookie Scenario Cards to each group. Explain to students that they will collaborate with their groups to create a factor tree to help them write the prime factorization for each type of cookie. Then, students will multiply the common prime factors together to find the greatest number of cookie bags that can be made. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat •

•

DOK-2 Explain how ﬁnding the prime factors helps to determine the greatest common factor. When you ﬁnd the prime factors, you can use all of the prime factors the two numbers have in common and multiply them together to get the greatest common factor. DOK-2 Explain how to write the prime factorization of a number. You write all of the prime numbers multiplied by each other from least to greatest. For example, 30 would be written as (2)(3)(5).

Part III: Tables and Chairs FACILITATION TIP Before reading the scenario, ask the class 1) When you have a family gathering with your relatives, what do you do together?; 2) Is food served?; 3) Do you all sit together or at separate tables to eat?

2. 3.

FACILITATION TIP

Discuss different ways to determine the greatest common factor for 30 and 24. Allow students time to use their preferred method.

DOK-1 What is the greatest common factor of 30 and 24? The greatest common factor of 30 and 24 is 6.

DOK-2 What does the greatest common factor of six represent in this scenario? Explain. The greatest common factor, 6, represents the number of people who will sit at each table. Six is the number of people because this is what needs to be the same for the adult tables and children tables.

FACILITATION TIP

For students who are struggling to complete the area model, provide a worked example of an area model using different numbers. Explain how the numbers in the area model are related to each other.

5. 6.

7. 148

Read the following scenario to the class: Sai is one of the party planners. He is in charge of setting up tables for each event. There are tables for adults and smaller tables for children. Each table should have the same number of people sitting at it. Help Sai determine how many people will sit at the tables and how many tables are needed for adults and children. Project the Table Scenario Card Party A for the class. Discuss the following questions with students:

Allow students time to collaborate with their groups to create an area model using the greatest common factor. Discuss the following questions with the class: a.

Sai knows there will need to be 30 adult chairs and 24 children chairs. He also knows that 6 guests will be at each table. He wrote an expression for the number of chairs to be 30 + 24.

DOK-1 Using your area model and the distributive property, determine how Sai can write an equivalent expression to show the number of adult tables needed and the number of children tables needed. Responses may vary. Sai can write an equivalent expressions of 6(5 + 4).

Allow time for students to write the number of tables for adults and children and the number of guests at each table on their Student Journals. Explain to students that they will work with their groups to determine the equivalent expressions and number of tables and chairs for the remaining two parties. After Part III, invite the class to a Math Chat to share their observations and learning. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Math Chat •

•

DOK-2 Explain how many people are at each table and how many tables are needed for adults and for children for party A. The greatest common factor is the number of people at each table. The number you multiply the greatest common factor by to get the original number is how many tables are needed. For example, when there are 30 adults and 24 children, there will be 6 people at each table. We will need 5 tables for adults and 4 tables for children. DOK-2 Explain how you could use the distributive property to add the numbers 96 and 36. I know that 96 and 36 have a greatest common factor of 12. So, 96 and 36 can be rewritten as 12 (8 + 3).

FACILITATION TIP

Allow time for students to discuss their thoughts with their group. Encourage them to use paper or a dry erase surface to Post-Explore show their work to determine the greatest 1. Have students complete the Exit Ticket to formatively assess their understanding common factor and write their expression. of the concept. FACILITATION TIP 2. Complete the Anchor Chart as a class. Before students complete this Exit Ticket, 3. Have each student complete their Interactive Notebook. explain whether they are required to use

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the factor tree model or if they can use any method as long as they show their work.

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Equivalent Numerical Expressions Explore 3 – Least Common Multiples ACTIVITY PREPARATION Students will use Cuisenaire Rods™ and prime factors to ﬁnd the least common multiple of different sets.

Standards for Mathematical Practice • • •

MP.4 Model with mathematics. MP.6 Attend to precision. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Set of Food Supply Cards (per group) 1 Hot Dogs Scenario (per class) 1 Exit Ticket (per student)

• •

Reusable • • •

1 Set of Cuisenaire Rods™ (per group) 1 Resealable bag (per group) 1 Projector or document camera (per class)

•

Plan to divide the class into groups of 3 or 4 students. Print a Student Journal and an Exit Ticket for each student. Print a set of Food Supply Cards for each group. If desired, print the cards on card stock and laminate them for future use. Cut out the cards, and place them in a resealable bag for each group. Gather a set of Cuisenaire Rods™ for each group. Have a projector or document camera ready to project the Hot Dogs Scenario for the class in Part I. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Cuisenaire Rods™)

PROCEDURE AND FACILITATION POINTS Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone had a birthday party recently?; 2) How many friends did you invite?; 3) What types of food did you have? STEMscopes Tip

The Anchor Charts element, located in the Explain section, guides teachers and students in creating a summary to showcase strategies, skills, and concepts learned during each Explore. An included printable sample anchor chart can be referenced for ideas on how to highlight key learning. 3. 4.

Read the following scenario to the class: Party planner Chao is in charge of buying food supplies for upcoming parties. He wrote down all the supplies he needs to buy and how many of each supply comes in one package. He began to notice a problem, though. The number of items in each package is not the same. He decided he is going to need to buy multiple packages of items so there are an equal number of items. Help Chao determine how many packages of each item is needed to have equal amounts of both items. Project Hot Dogs Scenario. Discuss with the class why we would want to purchase multiple packages of each item. a.

DOK-1 Would you want to purchase one package of each item? No, because not everyone would get a hot dog and a hot dog bun.

DOK-1 Would two packages of each item be enough? Two packages is not enough. That would be 20 hot dogs and only 16 hot dog buns.

DOK-2 How could I ﬁnd out how many packages of each item we should purchase so there are equal amounts of both items? You would need to ﬁnd the smallest multiple of 10 and 8.

Give a set of Cuisenaire Rods™ to each group. Explain to students that we can use Cuisenaire Rods™ to model the packages we would purchase. Allow students time to discover how much each rod would represent. Model with students how to compare a set of numbers using the rods. a.

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Explain

Elaborate

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DOK-1 What color rod should we use to model the pack of hot dogs? We should use the orange rod.

DOK-1 What color rod should we use to model the pack of hot dog buns? We should use the brown rod.

DOK-2 How will you know when to stop adding rods onto your model? You can stop adding rods once the two models are equal.

Give a Student Journal to each student. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a.

7. 8.

Engage

Intervention

Acceleration

FACILITATION TIP Discuss why the orange rod should be used to represent the pack of hot dogs. Work with students to compare the length of 10 white rods and an orange rod. Do the same for the hot dog buns. FACILITATION TIP

Work with students to find a solution to the problem. As students are modeling the hot dogs and hot dog buns with the Cuisenaire DOK-1 What are some of the multiples of 7 and 9? (Answers will depend Rods™, record the total numbers of hot dogs on which scenario you ask about.) Multiples of 7 are 7, 14, 21, 28, 35, 42, and buns represented by the models to help students connect the models to the math. 49, 56, 63. … Multiples of 9 are 9, 18, 27, 36, 45, 54, 63. ... DOK-1 How do you know when to stop finding multiples of two numbers? You can stop finding multiples once you reach a multiple that both numbers have in common.

Have students work with their groups to complete each Food Supply Card. After Part I, invite the class to a Math Chat to share their observations and learning.

FACILITATION TIP

Provide a multiplication chart for students with documented needs. Explain to them that the multiplication chart shows the multiples of each number.

EQUIVALENT NUMERICAL EXPRESSIONS

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Math Chat DOK-1 How did you make 12 with your rods? Answers vary depending on how students make 12. Two 6s (greens) is one example. • DOK-1 How did you know when to stop adding rods to your models? You can stop adding rods once the two models are equal. • DOK-3 Why would we need to ﬁnd the least common multiple when buying food for a party? You need the LCM to make sure you have the same amount of items that go together (hot dogs and buns, for example). •

Part II 1.

Read the following scenario to the class: Camila wants to help Chao place the order from the vendor. Camila says you can use prime factors of both numbers to ﬁnd the least common multiple. Examine Camila’s work to help Chao determine the relationship between prime factors and the least common multiple. Students work collaboratively with their groups to examine Camila’s work with prime factors and determine the relationship between prime factors and least common multiples. Then, students will use Camila’s strategy to ﬁnd the least common multiple for the cupcakes and cupcake toppers scenario. Monitor and talk with students as needed to check for understanding by using the following guiding question: a.

DOK-1 How did Camila use prime factors to ﬁnd the least common multiple? Camila determined which prime factors the two numbers have in common and then multiplied the common prime factor(s) by the remaining prime factors.

After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 How can finding the prime factors of each number help us find the least common multiple? When you find the prime factors, you can multiply the common prime factors by each of the individual prime factors that they do not have in common. • DOK-3 Why would we need to find the least common multiple when buying food for a party? You need the LCM to make sure you have the same amount of items that go together (hot dogs and buns, for example). •

FACILITATION TIP Allow students to work with their groups to explore different ways to model 12 with the rods. Have students share out their ideas. Model their ideas as they share out.

FACILITATION TIP Before reading the scenario, ask the class 1) What is a job or chore you are responsible for?; 2) Do you do the job alone or does someone help you with it?

STEMscopes Tip Students take notes, express ideas, and/or process the information presented in class using the Interactive Notebook element, located in the Explain section of each scope. These cut-and-glue activities provide an interactive way for students to showcase the concepts and skills learned in the Explore activities and can be added to a notebook for future reference.

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Equivalent Numerical Expressions Explore 3 – Least Common Multiples • FACILITATION TIP Allow students time to work and compare their solutions with their group's solution. Alternatively, have students write about the process of using prime factors to determine the least common multiple. FACILITATION TIP This Exit Ticket requires modeling with Cuisenaire Rods™. Determine whether students need to draw in color or if they can sketch in black and white.

DOK-3 Explain how to use Camila’s strategy to determine the least common multiple for 20 and 8. First, ﬁnd the prime factors of 8 and 20. Prime factors for 8 are 2 · 2 · 2, and the prime factors for 20 are 2 · 2 · 5. Next, ﬁnd the common prime factors, which are 2 and 2. Multiply the common prime factors by all remaining factors to get the least common multiple: 2 · 2 · 2 · 5 = 40.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

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Equivalent Numerical Expressions Explore 4 – Exponents ACTIVITY PREPARATION Students will rewrite and simplify expressions using exponents.

Standards for Mathematical Practice • • •

MP.4 Model with mathematics. MP.6 Attend to precision. MP.7 Look for and make use of structure.

Materials Printed • • • •

1 Student Journal (per student) 1 Cupcake Order Form (per class) 2 Sets of Order Cards (per class) 1 Exit Ticket (per student)

Preparation •

Part I: Cupcake Orders • • •

Reusable •

1 Projector or document camera (per teacher)

Print a Student Journal and an Exit Ticket for each student.

Plan to divide the class into pairs to complete Part I of this activity. Print out a Cupcake Order Form for the class. Make sure to have a projector or document camera to project the Cupcake Order Form to the class for a class discussion.

Part II: Orders • •

Plan to divide the class into 6 groups to complete Part II of this activity. Print and cut out two sets of Order Cards. You will be running 2 sets of 3 stations at the same time. Create 6 stations around the room with 1 Order Card at each station.

PROCEDURE AND FACILITATION POINTS Part I: Cupcake Orders FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone eaten a cupcake recently?; 2) What ﬂavor of cupcake did you eat?; 3) Were you having cupcakes to celebrate something? If so, what were you celebrating?

2. 3.

FACILITATION TIP Students will be ﬁnding the prime factorization of various numbers. They should have scratch paper or a dry erase surface to create factor trees. Consider providing each group with the factor tree diagram from Explore 2. FACILITATION TIP Ask students to explain the method they should use to determine the prime factorization of a number. Discuss if they need to complete a factor tree to determine the prime factorization of 2. Discuss that since 2 is already a prime number, they do not need a factor tree. 154

Read the following scenario to the class: BEST Party Planners are ready to turn in their order forms for cupcakes to their boss. Their boss has asked them to write each order as an exponential expression. Use the information on the Cupcake Order Form to write the prime factorization expression as the order’s expanded form, and then rewrite it as an exponential expression to turn in to the boss. Give a Student Journal to each student. Project the Cupcake Order Form for the class. Model simplifying expressions using exponents with students. Discuss the following items with the class: a.

BEST Party Planners needs to order two red velvet cupcakes.

DOK-2 What is the prime factorization of 2? The prime factorization of 2 is 2. Show students how to write this out on the red velvet cupcakes row in the second column of the Cupcake Order Form.

DOK-1 How many twos are in this prime factorization expression? There is one 2. Write this in the third column of the red velvet cupcake row.

BEST Party Planners needs to order 4 fudge cupcakes.

e. DOK-2 What is the prime factorization of 4? The prime factorization of 4 is 2 × 2. Show students how to write this out on the fudge cupcakes row in the second column of the Cupcake Order Form. f.

DOK-1 How many twos are in this prime factorization expression? There are two 2s. Write this in the third column of the fudge cupcake row.

Engage

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h. DOK-2 What is the prime factorization of 8? The prime factorization of 8 is 2 × 2 × 2. Show students how to write this out on the confetti cupcakes row in the second column of the Cupcake Order Form. i. DOK-1 How many twos are in this prime factorization expression? There are three 2s. j. DOK-2 What do you notice about the number of twos that are in the prime factorization expression each time? Student responses will vary. I notice that each time there is one more two than in the row before it. Each row has at least one 2 in the prime factorization expression. 4. 5.

Allow students time to complete the second and third columns for the last three cupcake types. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a.

DOK-1 What is the prime factorization of 32? The prime factorization of 32 is 2 × 2 × 2 × 2 × 2. (Answers will vary based on the number of cupcakes.)

DOK-1 How many twos are in this prime factorization? Answers will vary based on the prime factorization asked about. There are ﬁve 2s in the prime factorization for 32.

DOK-2 What number is used in every prime factorization? Two is used for each prime factorization.

Each number of cupcakes was decomposed into an expression involving some number of 2s. DOK-1 For the confetti cupcakes, how many twos need to be multiplied together to get 8? Three 2s are multiplied together to get 8.

Explain the following to the class: Mathematicians call the number that we are repeatedly multiplying together the base.

We write the base 2 in our cupcakes example in the last column. Model for students writing a 2 in the last column for the confetti cupcakes.

Write a smaller 3 at the top right side of the base 2. Model for students how to write the exponent.

Explain the following to the class: Mathematicians call this smaller number on the top right the exponent or power.

h. DOK-1 How many twos are multiplied together in the prime factorization expression for 2? There is one 2 in this prime factorization expression. i. DOK-2 What will be the base for red velvet cupcakes? The base for red velvet cupcakes is 2. j. DOK-2 What will be the exponent for red velvet cupcakes? The exponent for red velvet cupcakes is 1 because there is only one two in the prime factorization expression.

Some students may follow the pattern of just adding an additional 2 to the prime factorization of each number. If so, have them verify the prime factorization by using a factor tree.

Discuss the following questions with the class:

e. Look at the confetti cupcakes row again. DOK-1 How many twos are in the prime factorization expression? There are three 2s in the prime factorization expression for confetti cupcakes.

FACILITATION TIP

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Allow students time to work with their partners to write the exponential expression for each of the remaining cupcake orders on their Student Journals. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a.

DOK-1 How many twos are in the prime factorization expression for strawberry cupcakes? (Answers will vary based on which cupcake type you are asking about.) There are ﬁve 2s in the expression for strawberry cupcakes.

FACILITATION TIP Further explain that 2 is the base in this situation because it is repeatedly being multiplied. Point to the row that has 2 × 2 × 2 and read it.

FACILITATION TIP Explain that the exponent, or power, represents how many times the base was multiplied. Point to the three 2s in the column that has 2 × 2 × 2. Then, point to the exponent 3.

FACILITATION TIP Before starting Step 7, take time to have students record the deﬁnition or visual examples of base, exponent, expanded form, and exponential form. Page 5 of the Student Journal has space for students to note these details on Questions 1 and 2.

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Equivalent Numerical Expressions Explore 4 – Exponents b. 9.

After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 What is an exponent? An exponent is the number of times you multiply the base by itself. • DOK-2 What is the expanded expression for 28? The expanded expression is 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. • DOK-3 What is the exponential expression for 12? The exponential expression for 12 is 22 × 3. •

FACILITATION TIP Discuss with students how the prime factorization of 12 written as an exponential expression is different from the examples provided with the cupcakes. The number 12 has an additional prime factor, 3, that should still be included in the prime factorization when written as an exponential expression.

Part II: Orders 1.

FACILITATION TIP

Before reading the scenario, ask the class 1) If you had a party, would you like to plan it yourself or have someone else plan it?; 2) What types of things would you have at your party?

FACILITATION TIP

To help students know which order cards to complete, differentiate the sets by color. Either copy the sets on a different color paper or place a speciﬁc color sticker on each set. Assign student groups to work a speciﬁed color set of order cards.

Read the following scenario to the class: Jazmine is in charge of orders for cookies, chair rentals, and flowers for the party planning company. Her boss has asked her to rewrite the orders using exponential expressions. Can you help Jazmine rewrite the cookie orders, chair rental orders, and flower orders as exponential expressions to turn in to her boss? Discuss with the class how to rotate through the Order Cards. Each card is seen around the room twice, but students will only need to complete each card one time. Discuss with the class that they will be given both expanded form expressions and exponential expressions. Students will need to find either the expanded form or the exponential expression and find the total number of items for each order. Students will work collaboratively with their groups to determine the missing information for each order. As students are working, ask guiding questions to groups that may be struggling:

FACILITATION TIP While explaining the directions, point to one of the order cards and point to an expression written in expanded form and one written as an exponential expression. FACILITATION TIP Some students might struggle to understand that there could be more than one base when a number is written as an exponential expression. Have students identify the different prime factors in a number written in expanded form. Explain that each different prime factor is a base.

6. 7.

DOK-1 What is the base? The base in an exponential expression is the number that is being multiplied repeatedly.

DOK-1 What is the power (or exponent)? The power (or exponent) is the number of times the base is multiplied by itself.

DOK-2 How can you use the repeated multiplication expression to ﬁnd the base and the exponent? You can see what number is repeated in the multiplication, and that will be the base. Then, count how many times it is repeated to ﬁnd the exponent.

DOK-1 What effect does a decimal have in the repeated multiplication? The product will be less than the factors because the decimal point moves to the left with every repeated factor.

Once students have rotated to each of the Order Cards, they will answer the reﬂection questions in Part II of their Student Journals. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat • • •

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DOK-1 What is the exponent for vanilla cupcakes? The exponent for vanilla cupcakes is 4.

DOK-1 What is the base in an exponential expression? The base in an exponential expression is the number that is being multiplied repeatedly. DOK-1 What is the power in an exponential expression? The power (or exponent) is the number of times the base is multiplied by itself. DOK-1 How do you write an expression of repeated multiplication when given an exponential expression? The base of the exponential expression tells you the number that is being repeatedly multiplied together. The exponent (or power) tells you how many times to write the base down to be multiplied together. DOK-2 What does having an exponent of 1 mean? The exponent is the number of times the base is repeated. An exponent of 1 would mean the base number is only there one time. © Accelerate Learning Inc. - All Rights Reserved

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Acceleration

DOK-2 Are rational exponential expressions always going to have a product greater or less than the factors? Explain. The product will always be less. 1 1 1 For example, __2 squared is __2 of a __2, so it is being made smaller.

Post-Explore 1. 2. 3.

FACILITATION TIP Have ave students complete the Exit Ticket to formatively assess their understanding of the concept. On this Exit Ticket, items C and E include the number 1 as an exponent. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

Notes __________________________________________________________________________________________________________________________________________________

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Equivalent Numerical Expressions Explore 5 – Order of Operations ACTIVITY PREPARATION Students will use the order of operations to match expressions with their solutions.

Standards for Mathematical Practice • • •

MP.4 Model with mathematics. MP.6 Attend to precision. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • • •

•

1 Student Journal (per student) 1 Which Order Is Correct? (per class) 1 Set of Catering Order Cards (per group) 1 Exit Ticket (per student)

• •

Reusable • •

•

1 Resealable bag (per group) 1 Projector or document camera (per class)

Plan to divide the class into groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print one set of Catering Order Cards per group. If desired, print the cards on card stock and laminate them for future use. Cut out the Catering Order Cards, and place them in a resealable bag. Prepare to have a projector or document camera to project Which Order Is Correct? for the class.

PROCEDURE AND FACILITATION POINTS

FACILITATION TIP Before reading the scenario, ask the class 1) What is your favorite restaurant to go to?; 2) What food do you eat at the restaurant?

Part I 1.

2. 3. FACILITATION TIP Discuss which operations are seen in the expressions being displayed. Remind students of the different ways that multiplication could be represented, including with parentheses or the multiplication dot.

Read the following scenario to the class: Sidney’s Catering Company is getting ready for a big dinner event they will be catering next month. Three employees have each created an order card, but none of the expressions match! Help the employees determine which expression will provide them with the correct amount of orders. Give a Student Journal to each student. Project Which Order Is Correct? for students to see. Have the following discussion with the class: a. Remind students that when we have expressions with different operations, we need to solve the order of operations. b.

DOK-1 What is the order of operations? Parentheses, exponents, multiply or divide from left to right, add or subtract from left to right

Look at the order cards. What do you notice about these expressions that we have not used before in order of operations? Accept all answers without giving feedback. Come to a class consensus that some of these expressions have exponents and some have brackets in them.

DOK-1 What do brackets look similar to? Brackets look similar to parentheses. Tell students that these are both types of grouping symbols. Brackets and parentheses can be interchanged.

FACILITATION TIP As students say the steps in the order of operations, write them on the board. This will serve as a reference for students who struggle to remember them. FACILITATION TIP Consider using hand signals, choral response, or song to help students memorize the Order of Operations. 158

e. We also have exponents to include in our order of operations. f.

Where do you think exponents belong in the order of operations? (Accept all answers without giving feedback.) After one minute, tell students that exponents are after parentheses in the order of operations. © Accelerate Learning Inc. - All Rights Reserved

6. 7.

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DOK-1 Which operation is ﬁrst in order card A? Answers will vary based on order card letter. First, solve inside the parentheses, 2 + 4.

DOK-2 Explain what (6)2 means. This means to multiply six by six, which gives us 36.

DOK-2 What number is added to 5 for the last step in order card B? 324 is added to 5 in the last step of order card B. This gives us 329 for the solution.

DOK- 3 Explain how you know order card C is the expression to get 225 plates. I know order card C is correct because I solved it following the order of operations. You must solve everything inside the brackets before solving the exponent. • DOK- 2 How are the expressions on card A and card B different? In card A, you have 3(2 + 4)2. First, solve 2 + 4 inside the parentheses, and then square its answer. In card B, you have [3(2 + 4)]2. Here, you ﬁrst solve the parentheses of (2 + 4) to get 6 and then multiply 6 by 3 before you square the answer. DOK-2 How do you know when to solve the exponent? The exponent is second in order of operations after parentheses. This means you must solve all of the operations within the parentheses before you can solve the exponent. • DOK-3 What strategy did you use to help you keep track of solving order of operations? I solved each step under the step before it to keep track of where I was in the order of operations. •

Part II

5. 6.

FACILITATION TIP If students are struggling to follow the order of operations when solving problems, have them write down the order of operations before solving each problem. Have them cross out each step as they complete it or if it is not needed.

Allow time for students to complete the reﬂection questions at the end of Part I. After Part I, invite the class to a Math Chat to share their observations and learning.

•

2. 3.

Acceleration

Allow students time to collaborate with their groups about each of the three order cards. Students should solve each expression together on their Student Journals, following the order of operations to determine which order card they believe is written correctly to represent 225 plates. As students collaborate, monitor their work and use the following guiding questions to assess student understanding:

Math Chat

Intervention

Read the following scenario to the class: Now everyone at Sidney’s Catering Company knows how to write and solve the orders! Help the team match each order expression with its correct order number. Give the Catering Order Cards to each group. Have students work together to solve and match each expression to its correct order number. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 Which operation should you solve ﬁrst? Answers will vary based on the order form expression. In the expression 275 – (32 – 7)(2) – 12, the ﬁrst step is to solve 32 – 7.

DOK-1 What does the fraction bar mean in the expression 15+(8−3)/2? In this expression, the fraction bar means to divide the solution for the top by 2.

DOK-1 What is the solution to __4 (36)? __4 (36) is equal to 9.

EQUIVALENT NUMERICAL EXPRESSIONS

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FACILITATION TIP Write the expression from card C on the board. Have students explain the steps used to solve it. As they are explaining, solve the problem following their steps.

FACILITATION TIP To deepen students' understanding of using the order of operations to simplify expressions, rewrite one of the expressions and remove the parentheses. Ask students what operation they would start with since there are no parentheses. FACILITATION TIP After reading the scenario, ask the class 1) What is an expression? STEMscopes Tip Fluency Builders, located in the Elaborate section, are partner or smallgroup student-led games that engage students in practicing the skills and concepts addressed in the scope. These games come with studentfriendly instruction sheets. All the materials used in the games are found in the print ﬁles on the right side of the screen.

Allow time for students to complete the reﬂection questions at the end of Part II. After Part II, invite the class to a Math Chat to share their observations and learning.

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Equivalent Numerical Expressions Explore 5 – Order of Operations Math Chat DOK-2 Why does order matter when solving expressions? Student responses will vary. Everyone needs to get to the same end result, so we must all follow the same steps to get the same result. • DOK-2 In the expression 23 · 15 – (8 + 14), what would be the first step? What would be the last step? The first step would be to solve inside the parentheses. The last step would be subtraction. • DOK-2 Explain how you determined whether a rational number influenced the order you needed to complete steps or not. In order card 10, the last step was dividing by two because the numerator had multiple steps. In order card 9, the fractions were within the parenthesis and then had an exponent applied; this means they had to be done first. • DOK-2 Explain how to solve the expression that matched order card number 32. First, find 22, which is 4. Next, multiply 7.5 by 4, which is 30. Last add 2 + 30, which is 32. •

FACILITATION TIP Provide time for students to simplify this expression individually. Then, have them compare their solutions with a shoulder partner and explain their process.

FACILITATION TIP On this Exit Ticket, consider providing the number of steps needed for each item. For example, for students to successfully show complete understanding on the ﬁrst item, they must show six steps, but on the third item there are only three steps.

Post-Explore 1. 2. 3. 4.

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Equivalent Numerical Expressions Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Greatest Common Factors Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Prime Factorization Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 3

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Least Common Multiples

Interactive Notebook

Show What You Know, Part 4

A cut-and-glue activity to process learning that can be added to a notebook for future reference

Exponents

Independent practice assignment that gives students an opportunity to demonstrate their learning

Show What You Know, Part 5 Order of Operations Independent practice assignment that gives students an opportunity to demonstrate their learning

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Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Order of Operations

Can be done independently

Independent and partner games and other activities that provide students with an engaging way to practice the new concept

Notes

EQUIVALENT NUMERICAL EXPRESSIONS

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Intervention and Assessment SMALL-GROUP PLANNER 1

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

EQUIVALENT NUMERICAL EXPRESSIONS

Equivalent Numerical Expressions

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Students

Notes & Comments

Engage

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Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER Use this template to decide how to assess your students for concept mastery. Depending on the format of the assessment, you can identify prompts and intended responses that would measure student mastery of the expectation. See the beginning of this scope to identify standards and grade-level expectations. Key Concepts I can determine the greatest common factor of two whole numbers less than or equal to 100.

What prompts will be used?

What does mastery look like?

EQUIVALENT NUMERICAL EXPRESSIONS

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I can use the order of operations to translate written expressions into numerical expressions written correctly. I can determine the least common multiple of two whole numbers less than or equal to 12. I can use the distributive property to present the sum of two whole numbers as the product of a common factor multiplied by the sum of two whole numbers with no common factor. I can write an expression of repeated multiplication as an exponent. I can write an exponent in expanded form as an expression of repeated multiplication. I can deﬁne an exponent as having a base value and a power. I can evaluate an exponent by performing the process of repeated multiplication of a rational number. I can solve expressions by using the order of operations.

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SCOPE 1

Algebraic Expressions Scope Introduction SCOPE SUMMARY

Student Expectations

A critical area of instruction in sixth grade is writing, interpreting, and evaluating expressions and equations that represent real-world problems. In this scope, students model and write expressions with variables. Visual models and manipulatives can be used for conceptual understanding of mathematical properties and algebraic notation. Students investigate the product of sums or differences using the distributive property in relation to area models. Once the concept of the distributive property has been established, students generate equivalent numerical expressions using the distributive property. Next, students evaluate expressions using substitution and the order of operations. Students learn how to simplify expressions using exponents by collecting like terms and applying the distributive property. Finally, students determine whether two expressions are equivalent by simplifying and applying properties of operations.

6.PAR.6.3 Write and read expressions that represent operations with numbers and variables in realistic situations. 6.PAR.6.4 Evaluate expressions when given values for the variables, including expressions that arise in everyday situations. 6.PAR.6.5 Apply the properties of operations to identify and generate equivalent expressions.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grade levels, students gain experience writing expressions to model contextual situations. Beginning in third grade, students apply the properties of operations to add, subtract, multiply, and divide whole numbers and positive rational numbers. In fifth grade, students write expressions to represent multistep problems, and they use the order of operations to evaluate them. In fifth grade, students also utilize the distributive property when they multiply multi-digit numbers by connecting arrays and area models with partial products and the standard algorithm. Prior to this scope, sixth-grade students apply the order of operations to evaluate expressions with parentheses and exponents. Sixth-grade students also use their knowledge of factors and multiples to compute fluently, identify common multiples and common factors, and explore prime factorization. They determine the least common multiple (of numbers up to 12) and the greatest common factor (of numbers up to 100).

Sixth grade marks a foundational year for building the bridge between concrete concepts of arithmetic and the abstract thinking of algebra. Visual representations and concrete models can help students develop understanding as they move toward using abstract symbolic representations in seventh grade. Examples where students evaluate the same expression by substituting several different variable values lead to reasoning about the concept of a function. Students build on their understanding of multiples, factors, and mathematical properties to generate and use the arithmetic of rational numbers. Seventh- and eighth-grade students will continue to interpret, write, simplify, solve, and graph expressions and equations.

Before beginning the lesson, the students’ prior knowledge is assessed to address any student needs and skills needed for content mastery. Students are assessed to determine if they can: •

write simple expressions that record calculations with numbers.

•

interpret numerical expressions without evaluating them.

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope.

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

determine the number of minutes represented in equations.

Students move on to the Explore activities and come back at the end of the explorations to repeat the activity and discuss what they found.

If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. 166

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Write Expressions In this exploration, groups of students will solve a scenario about Melbourne, Australia’s Kangaroo Jump Fundraiser. Students will: •

Explore 2

Explore 1

EXPLORE ACTIVITIES

read verbal sentences and translate them into algebraic expression.

•

use the commutative properties of addition and multiplication.

•

write an algebraic expression for each kangaroos jump.

Explore 4

Explore 3

In this exploration, students will explore through solving a scenario about Australia’s parrot sanctuaries to help determine how much food to put on each tray using equivalent expressions. Students will:

In this exploration, students will explore solving different scenarios about the National Zoo of Australia and the number of birds and animals seen. Students will: •

simplify expressions.

•

determine the perimeter of each enclosure for animals.

•

use expressions given or written to build a model to represent the expression

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment. Simplify Using Properties

Simplify Expressions

Evaluate Expressions In this exploration, students will solve a scenario involving the same Australia Zoo to determine if groups are spending more or less than $40 on lunch. Students will: •

evaluate expressions and verbal expressions of groups’ lunch orders.

evaluate area formulas for given values.

•

use substitution to simplify algebraic expressions.

use area models to determine if expressions are equivalent.

•

apply the order of operations to simplify algebraic expressions.

•

use the distributive property to simplify expressions.

• •

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After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve. Notes

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Algebraic Expressions Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will match numbered cards with lettered cards posted around the room to demonstrate their knowledge of the prior standard. This element is designed to uncover student misconceptions; it should not be taken for a grade. 5.NR.5.1 Write, interpret, and evaluate simple numerical expressions involving whole numbers with or without grouping symbols to represent actual situations.

Materials

Preparation

Printed •

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1 Set of Match around the Room Cards (per class)

• •

Print one set of the Match around the Room Cards. Hang them in a random order around the room.

Procedure and Facilitation Points 1. 2.

Have each student write the numbers 1, 2, and 3 on a sheet of paper. Instruct students to walk around the room with their papers. As students walk around the room, they need to see the numbered cards and match them with the lettered cards. Allow students to share their thinking with their neighbors. a.

Card 1 matches with Card C.

Card 2 matches with Card B.

Card 3 matches with Card A.

Before students think with their neighbors, give them some think time by themselves to quietly read the prompts and note key vocabulary. You could project the letter cards for students prior to the walk around. FACILITATION TIP

If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

Identifying Misconceptions • •

FACILITATION TIP

Students may struggle to understand what the words product and sum mean. Students may struggle to identify when parentheses are needed.

To help students visually differentiate between the lettered and numbered cards, print them on different colors. FACILITATION TIP In addition to the Foundation Builder, frequently review and post operational and computational vocabulary in your classroom (triple, double, three times as large, less than, etc.) to help students expand their ﬂuency. FACILITATION TIP Quickly assessing and reviewing key vocabulary words (product, sum, difference, and quotient) before completing this APK may help you evaluate other knowledge gaps more eﬃciently.

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Algebraic Expressions Hook – The First Wealth Is Health ACTIVITY PREPARATION Students will evaluate an algebraic expression to solve a real-world problem.

Materials

Preparation

Printed •

• • •

1 The First Wealth Is Health (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project The First Wealth Is Health for the whole class to view. Prepare to introduce the scenario and encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION POINTS Part I: Pre-Explore 1.

FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) What are some healthy habits people have?; 2) Choose one healthy habit you have. How long does it take you to do it?; 3) Do you do it regularly or just once in awhile? FACILITATION TIP

Be sure to project The First Wealth Is Health table before asking these engaging questions. FACILITATION TIP If students need prompting, ask "What do we call unknown values in math? What do we call letters in mathematical expressions or equations?"

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: Lucy is working on developing healthy habits. In her health class she learned about several healthy habits she intends to perform on a daily basis and about how many minutes it takes to perform each habit. Lucy has even created a table with this information to inspire her to keep working to achieve her goal of being healthy. Each healthy habit in the table is assigned a letter. Every day Lucy writes an equation by using those letters to see how much time she spent on healthy habits that day. Ask students the following questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Student answers will vary. I notice that Lucy is working with time in minutes. I also notice that Lucy is using variables, letters to represent the number of minutes. I wonder what healthy habits Lucy is performing. What letters will Lucy use to represent each healthy habit? I can use math to determine the number of minutes represented in the equation so I can tell how many minutes Lucy devoted to healthy habits in a day. Project The First Wealth Is Health.

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Engage

Explore

Explain

Elaborate

Evaluate

Acceleration

Explain to students that Lucy has decided on the healthy habits she will focus on and she has put them in her table. She has also shared the variable she is using to represent each healthy habit. In addition, she has a reasonable time allotted for each habit. Discuss the following questions: a.

DOK-1 What are the healthy habits Lucy is focusing on right now? Brush teeth, shower, eat a healthy meal, exercise, and relaxation/deep breathing

DOK-1 What are the letters she is using to represent each habit? t is brush teeth; s is shower; m is eat a healthy meal; e is exercise session; r is relaxation/deep breathing.

Complete the Explore activities. a.

DOK-1 What does the equation mean? It means that twice during the day, Lucy brushed her teeth and twice during the day she did relaxation and deep breathing. It means that three times during the day she ate a healthy meal. Once a day she both exercised and took a shower.

Part II: Post-Explore 1. 2.

Intervention

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Show the Phenomena Video again, and restate the problem. Refer to The First Wealth Is Health, and discuss the following: a.

DOK-2 Evaluate the equation. How many minutes did Lucy devote to healthy habits during the day? 2(t + r) + 3(m) + e + s 2(2 + 5) + 3(20) + 30 + 10 2(7) + 3(20) + 30 + 10 14 + 60 + 30 + 10 114 minutes devoted to healthy habits

DOK-1 What are some other healthy habits that Lucy could have used? Answers will vary. Washing hands, drinking water, etc.

FACILITATION TIP This question provides a great opportunity to remind students that translating words into math is an essential skill. There are many real-world examples where students can see the beneﬁts (gym memberships, phone contracts, rental agreements).

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Algebraic Expressions Explore 1 – Write Expressions ACTIVITY PREPARATION Students will read verbal sentences and translate them into algebraic expressions. Students will begin to explore equivalent expressions by using the commutative properties of addition and multiplication.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Statement Cards (per group) 1 Exit Ticket (per student)

Plan to divide the class into groups of 2 or 3 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print one set of Statement Cards for each group. Cut out the cards, and place each set into a resealable bag. If desired, laminate them for future use.

Reusable •

1 Resealable bag (per group)

Consumable • •

1 Blank sheet of paper (per group) 1 Sheet of chart paper (per class)

PROCEDURE AND FACILITATION POINTS Part I FACILITATION TIP Have students complete this quadrant activity on a paper they can keep to refer to in the future. Perhaps, consider using the back side of a Student Journal page that they will keep.

1. 2. 3.

4. FACILITATION TIP Before reading the scenario, ask the class 1) What do you know about kangaroos?; 2) Have you ever seen a kangaroo in the zoo or in the wild?; 3) If so, what was it doing?

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Divide the class into groups. Give each group a blank sheet of paper. Instruct students to divide the paper into four quadrants and label each quadrant with one operation symbol: +, –, ×, ÷. Give each group 3 minutes to brainstorm words and phrases to match each operation. Share the following example:

When students are ﬁnished brainstorming, invite group members to share their ideas and create a class anchor chart of math operation terms. This will be a tool for students to refer back to throughout the Explore activity. Read the following scenario to the class: Every summer, in Melbourne, Australia, the nation’s largest kangaroo rescue center holds a Kangaroo Jump Fundraiser to raise funds for the center. Kangaroos are known for consistency in the lengths of their jumps. The announcer at Melbourne’s Kangaroo Jump Fundraiser announces the distance each kangaroo jumps. The distances are announced in relation to the average jump length for a kangaroo. Because this length varies each year, the announcer uses the variable x to represent the average jump length. Read the announcer’s statements to write an algebraic expression for each kangaroo’s jump. © Accelerate Learning Inc. - All Rights Reserved

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

Give a Student Journal to each student. Have students work with their groups to read the announcer’s statements, analyze the models, and use their math operation terms to write algebraic expressions to represent the length of each kangaroo’s jump in Part I of their Student Journals. Allow students to complete Part I of their Student Journals with their groups. As they work, ask the following questions: a.

DOK-1 Do you know the value of x? No, the value of x is not given. We are creating expressions in relation to x, so x will be a part of the expressions.

DOK-1 How do you determine which operation to use for each expression? I decide whether the word represents addition, subtraction, multiplication, or division. I can also refer to our math operation terms for help if needed.

Intervention

Acceleration

FACILITATION TIP Demonstrate how these bar models relate to the equations. Be speciﬁc about when the "x" is placed underneath the bar model or inside of the bar model. Alternatively, students may create different original models that support their own thinking and translating.

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FACILITATION TIP This is a good time to review expression vs equation with students. Refer to Picture Vocabulary in this scope.

After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat •

DOK-1 Group 1 wrote 3 · x as their expression, while group 2 wrote 3( 3(xx). Which group is correct? Both groups are correct. Both expressions represent 3 times x. Explain the following to the class: Mathematicians also represent 3 times x using the expression 3x. 3 . If there is no mathematical symbol, it is understood to multiply.

•

DOK-1 Why should we not use an x to represent multiplication? Since we are using variables such as x, it would be very confusing to use x to represent multiplication.

FACILITATION TIP If needed, clarify how mathematicians represent multiplication without a symbol. Provide some more examples that do not use "x" as the variable.

Explain the following to the class: I noticed different expressions to represent the average length, x, divided by 3. Some groups wrote x ÷ 3. Mathematicians also x

represent this expression as __3. Think about the model for the fraction __4. The model would be one whole divided into 4 equal pieces, or 1 ÷ 4. •

DOK-2 How would the expression change if it were the quotient of 3 and x? In

FACILITATION TIP

division, order matters. This would be __x . The dividend is the numerator, and the

Take time to review dividend, divisor, numerator, and denominator. Include these terms and examples on a word wall in your classroom if you have one.

divisor is the denominator.

Explain the following to the class: I also noticed some groups represented 1

“the average length, x,, divided by 3” with the expression __ . 3x •

DOK-1 Are the expressions __ and __3 equivalent? Explain. I can tell by the model 3x 1 __

that “x divided by 3” is showing 3 of x, so they must be equivalent. Also, 1 whole 1 __

divided into 3 equal pieces is 3.

Part II 1. 2.

Give one set of Statement Cards to each group. Instruct students to read each statement on the Statement Cards. Students will write an algebraic expression to represent the length of each kangaroo’s jump in their Student Journals. Have students work with their groups to complete Part II of their Student Journals.

FACILITATION TIP Some students need help understanding that "of" can be translated into "times" or "multiply". Showing some simple examples with common fractions and whole numbers will help illustrate the concept. (For 1 example: __2 * 10 can be verbally stated, "one half of ten"). FACILITATION TIP Clarify whether students are to read Scenario Cards or Statement Cards.

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Algebraic Expressions Explore 1 – Write Expressions 4.

Encourage students to continue referring and adding to the math operation terms anchor chart as needed. Students should create models similar to the models in Part I to help visualize the jump lengths.

As students work, ask the following questions:

FACILITATION TIP As students work, post these additional questions, "When does order matter in an expression? Why does order matter in some expressions and not in others? What operations require you to pay attention to the order?" Encourage them to discuss these focus questions as they collaborate. FACILITATION TIP Support students as they create word phrases for the subtraction expression on the reﬂection questions. Many students ﬁnd it easier to translate and solve addition equations rather than subtraction. Slow down and emphasize which verbal phrases communicate subtraction.

FACILITATION TIP Be prepared with some more examples to allow students to practice creating verbal expressions from algebraic expressions. Have volunteers share their answers. As an extension, have some students create (with constraints) algebraic expressions and translations to challenge the class.

6. 7.

DOK-2 Does it matter which term is written ﬁrst in an addition/ multiplication expression? No, addition and multiplication are commutative, so the order of the terms does not matter.

DOK-2 Does it matter which term is written ﬁrst in a subtraction/division expression? Yes, order matters with subtraction and division. You have to pay attention to the order of the algebraic sentence in order to know the order to write the terms.

Upon completion of Part II, have students answer the reﬂection questions on their Student Journals. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 Does the order of the terms matter when writing expressions? Order matters when you are writing expressions involving subtraction and division. Addition and multiplication are commutative. 5 • DOK-1 Can “the quotient of x and 5” be written as __x ? Explain why or why not. No, x the quotient of x and 5 is represented by the expression __5. Order matters with division. The dividend is the numerator, and the divisor is the denominator. • DOK-3 Give an example of a verbal expression that represents the algebraic expression 2xx – 5. Two times some number minus 5; take 5 from a number times two; double a number minus 5. •

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Algebraic Expressions Explore 2 – Simplify Expressions ACTIVITY PREPARATION Students will discover the parts of an expression and use the appropriate term for each part. Students will combine like terms to write expressions in simplest form.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Animal Count Cards (per group) 1 Exit Ticket (per student)

•

Reusable • •

1 Set of linking cubes – 10 red, 10 blue, and 10 yellow (per group) 1 Resealable bag (per group)

Plan to put students into groups of 2 or 3 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Animal Count Cards for each group. Cut out the cards, and place each set into a resealable bag. If desired, laminate them for future use. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Linking Cubes)

PROCEDURE AND FACILITATION POINTS Part I FACILITATION TIP Projecting the expression using different colors may help students see the coefficients, variables, and operations. FACILITATION TIP After guiding students through these questions, solidify their understanding by projecting the Picture Vocabulary in the scope and have them record examples and definitions for essential vocabulary: like terms, term, variables, algebraic expression, coefficient, and combining like terms.

Show students the expression 6r 6r – 4r + 3b + r + 4b.. Lead students through a discussion to discover the parts of the expression by asking the following questions: a.

How many terms are in this expression? (Accept all answers without giving feedback.) After one minute, tell students that terms are things we add or subtract, and then ask if any students want to revise their answers. Come to a class consensus that there are 5 terms in this expression.

What is the ﬁrst term in this expression? (Accept all answers without giving feedback.) Come to a class consensus that the ﬁrst term is 6r. 6

What is the second term in this expression? (Accept all answers without giving feedback.) Students should give 4rr and −4r as options. Tell students that the plus or minus sign stays with the term that follows, and then ask if any students want to revise their answers. That sign tells you whether the term is positive or negative. Come to a class consensus that the second term is −4r. −4

What is the third term in this expression? Come to a class consensus that the third term is 3b.

e. What is the coefficient of the first term? (Accept all answers without giving feedback.) After one minute, tell students that a coefficient is the numerical part of a term, and then ask if any students want to revise their answers. Come to a class consensus that 6 is the coefficient of the first term. 176

Engage

Explore

Explain

Elaborate

Evaluate

What is the coeﬃcient of the second term? Students may give 4, −4 or −4r −4r as options. Tell students that the plus or minus sign tells you whether the term is positive or negative, and then ask if any students want to revise their answers. Come to a class consensus that −4 is the coeﬃcient of the second term.

What is the coeﬃcient of the third term? Come to a class consensus that 3 is the coeﬃcient of the third term.

Intervention

Acceleration

h. What is the variable of the ﬁrst term? (Accept all answers without giving feedback.) If a consensus of r is not reached, tell students that a variable is a letter that stands for an unknown number. Ask if any students want to revise their answers, and come to a class consensus that r is the variable of the ﬁrst term.

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i. What is the variable of the second term? Come to a class consensus that r is the variable of the second term. 2.

3. 4.

5. 6.

Read the following scenario to the class: A group of students from South Sydney Middle School went to the National Zoo of Australia on a ﬁeld trip. While there, students were asked to record the number of red birds and blue birds that came and went from the aviary during their snack break. The students recorded their ﬁndings using the expression 6r – 4r + 3b + r + 4b, using the variable r to represent red birds and the variable b to represent blue birds. Build a model to represent the students’ expression. Then, simplify the expression by combining like terms. Distribute the sets of linking cubes to student groups. Instruct students to use their linking cubes to determine the total number of red birds and the total number of blue birds at the aviary at the end of the snack break. Allow students time to explore with the linking cubes. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How did you model 6r 6r – 4r? I started with 6 red cubes and took away 4 red cubes, leaving 2 red cubes.

DOK-1 What does 6r 6 – 4r mean in our scenario? It means that 6 red birds came to the aviary, and then 4 red birds left.

DOK-1 How did you model the term r? There wasn’t a coeﬃcient, so I just put 1 red cube into the group.

Have students share their totals with neighboring groups and then share totals with the class. 3 red birds and 7 blue birds a.

FACILITATION TIP Before reading the scenario, ask the class 1) Have you seen any birds lately?; 2) If so, what did the birds look like?; 3) What is your favorite type of bird? FACILITATION TIP If time and supplies are limited, demonstrate with the linking cubes and then have students come up and model the scenario. STEMscopes Tip Spiraled Review, located in the Elaborate section, provides students with a contextual scenario used to solve four different problems. This activity helps students maintain essential knowledge, see how mathematical skills connect from one topic to the next, and experience real-world applications of previously learned skills.

DOK-2 What is the simpliﬁed algebraic expression for 3 red birds and 7 blue birds? The simpliﬁed algebraic expression is 3r + 7b.

Share with students that when they combine like terms, they are simplifying an expression. Mathematicians always want to simplify expressions to be eﬃcient.

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Algebraic Expressions Explore 2 – Simplify Expressions Part II 1.

FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever been to the zoo?; 2) Did the animals live alone in a habitat or with other animals?; 3) What did the animals' habitats look like?

2. 3.

FACILITATION TIP

Read the following scenario to the class: The zookeeper at the National Zoo of Australia was tasked with reporting the total number of each animal in each habitat that was visible to guests at peak hours. He monitored each habitat for 30 minutes and documented the number of animals that came and went from the viewing deck. The animals that came into view were represented by positive integers, and the animals that left the viewing area were represented by negative integers. Use the information recorded by the zookeeper to write expressions, build models, and combine like terms to simplify the expressions representing the animals he saw come and go. Give a Student Journal to each student and a set of Animal Count Cards to each group. Instruct students to use the Animal Count Cards to create expressions, model with linking cubes, and combine like terms to simplify the expressions. Students will record the expressions and ﬁnal animal counts on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding:

Preview the Animal Count Cards with students to explore any unfamiliar animal vocabulary (goanna, platypus, ﬂamingo, cockatoo...) before they begin collaboration.

DOK-1 How do you know whether to add or subtract two terms? I look at the sign in front of the term. If it’s a plus sign, I add. If it’s a minus sign, I subtract.

DOK-2 Why do you think this is called “simplifying expressions”? We are taking a long expression and making it shorter or simpler.

DOK-1 Why does your answer for habitat C not have any goannas? What does this mean in terms of the context? When I simpliﬁed the expression, I got 0g. This means there were 0 goannas in view at the end of the 30 minutes.

DOK-1 What is the difference between the terms k and 1k? These terms represent the same amount. The coeﬃcient of k is 1.

FACILITATION TIP Address the coeﬃcients of 0 and 1 here and clarify whether it is acceptable to include them in simpliﬁed expressions.

Allow students time to complete the zookeeper’s strategy for habitat E by underlining the terms for cockatoos and circling the terms for magpies. Then, students will combine like terms and write out each animal count.

Part III FACILITATION TIP

Before reading the scenario, ask the class 1) Why might animals at a zoo be placed in their own enclosures? After reading the scenario, ask the class 2) What is meant by the perimeter of a enclosure? FACILITATION TIP Students may need consistent review of the terms area and perimeter and how they are calculated. Take time to quickly assess.

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3. 4.

Read the following scenario to the class: The National Zoo of Australia decided to split up some of the species into their own enclosures. Kookaburras, platypuses, and crocodiles were now going to have their own fenced-in spaces. The design team has drawn a sketch of each ﬁnalized enclosure and labeled each side’s length. Use this information to write an expression to represent the perimeter of each enclosure, and then simplify each expression to pass along to the building team at the zoo. Instruct students to look at each animal enclosure’s sketch. They will use each side’s measurement to write an expression to represent the perimeter of each enclosure and simplify each expression by combining like terms on their Student Journals. Encourage students to continue modeling with the linking cubes as an option. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What are the terms in this expression? g + h + g + 2h + g + h + 9g +h

DOK-1 What is the coefficient of g in the expression g + 2h? The coefficient is 1. When the coefficient is 1, mathematicians just write the variable because the 1 is assumed.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

DOK-1 How do you combine like terms? I just add the coeﬃcients of the terms with the same variable.

e. DOK-1 Does it matter which terms you combine ﬁrst? No, as long as I make sure to combine like terms, the order does not matter. f. 5. 6.

DOK-1 How do you ensure you don’t leave out any terms when simplifying? I cross out the terms as I combine them.

Upon completion of Part III, have students answer the reﬂection questions on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

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FACILITATION TIP

Math Chat DOK-2 Explain how to simplify expressions. Use the words coefficient, variable, variable and term in your explanation. You simplify expressions by combining like terms. Like terms have the exact same variable. You combine them by adding the coefficients of the like terms. • DOK-1 Does it matter which terms you combine first when simplifying? Why or why not? Order does not matter when combining like terms, as long as you ensure you are combining terms that are alike and you do not leave out any terms. • DOK-1 What is a coefficient? A coefficient is the numerical part of a term. • DOK-2 What is the perimeter of the platypus enclosure in its simplest form in Part III? The perimeter of the platypus enclosure is 10x + 8y + 2z. •

Before concluding this Math Chat and continuing to the Exit Ticket, ensure that students understand that their goal is to simplify expressions. Some students may still focus on trying to solve for the variables, get one numerical answer, or simplify down to one term.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding FACILITATION TIP of the concept. Before the Exit Ticket, remind students Complete the Anchor Chart as a class. that the simplest form may not be a single Have each student complete their Interactive Notebook. number or term. You may need to provide struggling students with a scaffolded Exit Ticket. For example, you might tell them how many terms are in the simpliﬁed expression or what operations are used. Notes

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Algebraic Expressions Explore 3 – Simplify Using Properties ACTIVITY PREPARATION Students will use the distributive property to simplify expressions. Students will identify equivalent expressions. Students will evaluate area formulas for given values.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Expression Cards (per group) 1 Exit Ticket (per student)

•

Reusable • •

Plan to have students work in groups of 2 or 3 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Expression Cards for each group. Cut out the cards, and place each set in a resealable bag. If desired, laminate them for future use. Gather enough sets of algebra tiles for each group to have one to use as needed.

1 Resealable bag (per group) 1 Set of algebra tiles (per group)

PROCEDURE AND FACILITATION POINTS Part I 1.

Ask students to solve the multiplication problem 8(62) using an area model. After about 20 seconds, draw or project the ﬁrst steps to give struggling students a jumping-off point.

After one minute, have students turn and talk to their neighbors about how to complete the area model. Invite a student to the board to complete it.

STEMscopes Tip The Evaluate section, found along the scope menu, contains assessment tools designed to help teachers gather the data they need to determine whether intervention or acceleration is warranted. From standards-based assessments to an open-ended reasoning prompt, there is an evaluation for every student’s learning style.

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Read the following scenario to the class: Australia is home to one of the largest parrot sanctuaries in the world. Employees are constructing rectangular feeding trays for the parrots. The employees need to know the area of each feeding tray to help them determine how much food to put on each tray. Help the employees write two equivalent expressions to ﬁnd the area of each feeding tray. Give a Student Journal to each student. Direct students’ attention to the Adolescent Parrot Feeding Tray section on their Student Journals. Guide the students through the setup for the Adolescent Parrot Feeding Tray section. a.

DOK-1 How will you ﬁnd the area of this rectangular tray? I will multiply the length and the width.

Have students write an expression on their Student Journals to represent the area.

Point out that mathematicians write the 5 in front of the parentheses. 5( + 4) 5(x

Think about how we used the area model to multiply 8(62). Place 5 and x + 4 on the area model.

e. In order to multiply, we must split up the terms in x + 4 and create two smaller area models.

7. 8.

Find the area of each box.

DOK-1 What is the sum of the areas? 5x + 20

Divide the students into groups. Students will work with their groups to complete area models for the Adult Parrot Feeding Tray section. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 Which part of the expression will go to the left of the area model? The length or the number outside the parentheses will go to the left of the area model.

DOK-1 How do you decompose x + 7? When you decompose x + 7, one section of the area model will be x and the other section will be 7.

Intervention

Acceleration

FACILITATION TIP Before reading the scenario, ask the class 1) What do you know about parrots?; 2) What do parrots need to have to survive? After reading the scenario, ask the class 3) What is meant by the area of a feeding tray? FACILITATION TIP When ﬁrst printing the Student Journal, don't copy page 5. Project these reﬂection questions to guide student collaboration and then complete page 5 together when students are ready.

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FACILITATION TIP Consistent review of area vs perimeter will support student vocabulary. Take time to clarify again. FACILITATION TIP As you illustrate the area model for this expression, be sure to pause and ask, "What questions do you have?" Students may struggle to see the abstraction of the labels especially if they have limited experience with the area model with simple whole numbers. FACILITATION TIP Project these guiding questions to support student collaboration. Emphasize to students to use the vocabulary in their discussions (area, parentheses, decompose, equivalent expression, distributive). As you monitor, positively reinforce their mathematical language.

Upon completion of Part I, have the following discussion with the class: a.

Post the equivalent expressions from Part I on the board. 5( + 4) = 5x 5(x 5 + 20 and 12( 12(xx + 7) = 12x 12 + 84

DOK-1 What do you notice about these equivalent expressions? Accept all answers.

Come to a class consensus that a number in front of parentheses means to multiply that number by each term inside the parentheses.

Explain the following to the class: Mathematicians call this the distributive property.

e. Encourage students to think about the property name, as this will be asked about in the reﬂection.

FACILITATION TIP Slow down here to repeatedly demonstrate multiplying the number in front of the parentheses by all terms inside the parentheses. Draw arrows to show how you are distributing. A common error is to only distribute to the ﬁrst term.

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Algebraic Expressions Explore 3 – Simplify Using Properties Part II 1. FACILITATION TIP Before reading the scenario, ask the class 1) How do parrots get from place to place?; 2) How far do you think parrots can ﬂy? FACILITATION TIP Note that the expression cards all include the same variable, x.

2. 3. 4. 5.

Read the following scenario to the class: Two employees at the parrot sanctuary recorded algebraic expressions to represent the flight distances of recently rescued parrots. You think the two employees recorded the distances using different but equivalent expressions. Determine which expressions are equivalent by using area models to prove your thinking. Give a set of Expression Cards and a set of algebra tiles to each group. Explain to students that they will be sorting through expressions to find equivalent matches of a lettered card and a numbered card. Students will work with their groups to complete area models for the algebraic expressions. Direct students’ attention to the first row of the table on Part II of their Student Journals, which is completed as an example. Ask students the following questions:

STEMscopes Tip The Standards-Based Assessment is found within the Evaluate section. Students demonstrate mastery of the concepts covered in the scope using multiple-choice and gridded response questions aligned to the scope standard(s). This assessment can be assigned and scored digitally, printed, or edited to meet students’ individual needs.

DOK-2 What are the similarities and differences between this area model and the area model we created earlier for 8 · 62? This example has variables and numbers. They are both in a 1 × 2 array. The top of both area models represents the total of one of the factors.

DOK-1 How can we calculate the product of 3 and x? We can think about it as 3 groups of x. We can relate it to using algebra tiles. Three x tiles is the same as 3x.

Allow students to collaborate with their groups to complete Part II. Remind students that they are welcome to use algebra tiles to help complete the area models. As they work, ask the following questions: a.

DOK-1 How do you know when to use the distributive property? When there is a number just to the left of the parentheses, I need to multiply it by each term inside the parentheses.

DOK-1 How do you know when an expression is in its simplest form? An expression is in its simplest form when no other terms can be combined.

Part III FACILITATION TIP

Before reading the scenario, ask the class 1) Parrots are very intelligent birds. What do you think parrots can be trained to do? 2. 3.

FACILITATION TIP 1 Another way to visualize multiplying by __2 is 1 1 to think, "__2 of a number.” (For example, __2 of 1 1 1 6 is 3, __2 of __2 is __4). Remind students that the

Read the following scenario to the class: The parrot sanctuary does not keep the parrots in enclosures; rather, they have trained the parrots to stay in certain zones designated by different-shaped canopies. The material for each canopy needs to be replaced. Help determine the area of each canopy so the parrot sanctuary can order enough material to replace the old canopies. Have students work with their groups to evaluate the areas of the canopies needed for the different zones. Instruct students to collaborate with their groups to complete Part III of their Student Journals. As they work, ask the following questions: a.

DOK-1 What does the exponent 2 mean? That means to multiply the base by itself twice.

DOK-1 What does it mean in mathematics when you have two letters side by side? If there is no symbol in between, that means we need to multiply.

DOK-1 What is an equivalent way to multiply something by __2? Multiplying

1 by __2 is the same as dividing by 2.

word "of" between two numbers often means multiply. 182

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Explain

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Evaluate

Math Chat DOK-2 What does it mean to decompose or split an expression? The term decompose means “to break into smaller parts.” For example, x + 9 would be split into x and 9 when decomposed. • DOK-1 Does order matter when simplifying expressions? It does not matter which terms I combine ﬁrst as long as I ensure I am combining like terms. Typically, terms are written in alphabetical order in simplest form. • DOK-1 What does the term distribute mean in relation to math? The term distribute means “to multiply the term just to the left of the parentheses by everything inside the parentheses.” For example, 2(4x + 1) means (2 · 4x) + (2 · 1). • DOK-1 What do mathematicians mean when they say to simplify? The term simplify means “to write the mathematical expression in the simplest possible way.” Mathematicians want expressions to be easy to use. •

•

Intervention

Acceleration

FACILITATION TIP Reﬂection question number one addresses a commonly asked question about ordering simpliﬁed expressions using alphabetization. Clarify with students if it is required that they use alphabetization consistently. Perhaps, draw their attention to this standard form when you see it in examples in class.

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FACILITATION TIP Sometimes, it may help to connect the math meaning of distribute with the every day meaning. Ask, "Would it be fair if I only distributed a cookie to the ﬁrst person in line?"

DOK-2 Use the distributive property to simplify 3(2x 3(2x + 3). 6x + 9

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding FACILITATION TIP of the concept. As an extension on the Exit Ticket, allow Complete the Anchor Chart as a class. students to generate other expressions Have each student complete their Interactive Notebook. that are also equivalent. If needed, provide struggling students with more than one opportunity (provide more expressions to simplify) to demonstrate understanding. Notes

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Algebraic Expressions Explore 4 – Evaluate Expressions ACTIVITY PREPARATION Students will use substitution and apply the order of operations to simplify algebraic expressions. Students will also write algebraic expressions as a means of representing solutions to real-world scenarios.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.6 Attend to precision.

Materials Printed • • •

1 Student Journal (per student) 1 Menu (per group) 1 Exit Ticket (per student)

Preparation • • •

Plan to divide the class into groups of 2 or 3 to complete this activity. Print a Student Journal and an Exit Ticket (2 per page) for each student. Print a Menu for each group. If desired, laminate it for future use.

PROCEDURE AND FACILITATION POINTS Part I 1.

Post the expressions 2 · 3 and 2(3) on the board. Ask the following question: a.

Post the expressions 2(3)and 2(x)on the board. Ask the following question: a.

In addition to these multiplication problems, you might start with some simple addition and subtraction problems to model substitution. FACILITATION TIP Before reading the scenario, ask the class 1) Have you ever been on a school ﬁeld trip?; 2) Where did you go?; 3) What did you do there? FACILITATION TIP Print and project the details of this scenario so that volunteers can read it aloud and students can write down the essential information. 184

DOK-1 Are these expressions equivalent? No, they both represent 2 times a number, but in the second expression, that number is unknown.

Post 2(3) and “2( “2(x) when x = 3” on the board. Ask the following question: a.

FACILITATION TIP

DOK-1 Are these expressions equivalent? Yes, they both represent 6—2 times 3 and 2 groups of 3.

DOK-1 Are these expressions equivalent? Yes. 2(x) when x = 3 means you can replace the x with a 3.

Explain to students that this is called substitution: Substitution is when you replace a variable in an algebraic expression with a known value. Check for understanding by asking the following questions: a.

DOK-1 What is the value of 3( 3(x)when x = 4? 3(4) = 12

DOK-1 What is the value of 5xx when x = 2? 5x means 5(x). 5(2) = 10

Read the following scenario to the class: A group of students from South Sydney Middle School went to the National Zoo of Australia on a ﬁeld trip. The students were split into groups, each chaperoned by an adult. At lunchtime, groups examined the menu and compiled their orders into an algebraic expression, which included a tip of $3. Each group had a $40 limit to spend on lunch. Prior to placing the orders, the chaperones wanted to ensure that each group’s order was under budget. Evaluate the expressions representing each group’s order to determine their total and whether they were under or over their $40 limit.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

FACILITATION TIP Give a Student Journal to each student and a Menu to each group. Explain to students that they will use the prices on the Outback Snack Shack Read over the Outback Snack Shack Menu Menu to evaluate the expressions for each group to determine the total cost of with students. Clarify the variables (grilled lunch. Remind students of the $40 limit. cheese sandwich is on two lines). 8. Allow students to collaborate with their groups to complete Part I of their Student FACILITATION TIP Journals. As they work, ask the following questions: Be aware that some students may notice that these expressions are not written using a. DOK-1 How do you know which mathematical operation to perform alphabetization of the variables. Assure ﬁrst? After I substitute the values for the variables, I use the order of students that they are still valid and that operations to simplify. what order the terms are in is not critical. b. DOK-1 What does the variable s represent? The variable s represents However, remind them to precisely use order salad. of operations to simplify. c. DOK-1 What is the total cost for Group 5’s order? $43.50 FACILITATION TIP d. DOK-1 How do you determine what to remove from an order if a group is Use Group 1 (or create your own example) over the $40 budget? I calculate the amount over $40 the group totaled to model to students how you want them and ﬁnd an item on the menu that equals that amount. to show their work with substitutions and Part II evaluations. Determine how and where to include dollar signs (only in solution?). 1. Read the following scenario to the class: Groups 7 and 8 lost their lunch orders, 6. 7.

3. 4.

but they know how much they spent in relation to group 6. They read their verbal descriptions to their chaperones to determine how much each group spent on their lunch orders. Explain to students that they will use the verbal descriptions in Part II on their Student Journals to write algebraic expressions to represent the total spent by each group and then ﬁnd the total of all three groups. Upon completion of Part II, have students answer the reﬂection questions on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat •

•

DOK-1 In Part I, why was there no variable on the last term, 3, in each expression? There was no variable because the group was only paying this $3 tip one time. It did not need to be multiplied by anything. DOK-1 The order for group 1, 3s 3 + g + 2b + r + 3, represented 3 salads, 1 grilled cheese, 2 bottles of water, 1 root beer, and a $3 tip. Create an order that would match the expression 4(h 4( + f + d). 4 students each ordered a hamburger, fries, and a sports drink. DOK-2 In what types of expressions were you making mistakes? • Errors multiplying with decimals • Errors in using the order of operations DOK-1 What is substitution? Substitution is when you replace a variable in an algebraic expression with a known value.

Post-Explore 1. 2. 3. 4.

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FACILITATION TIP

Project a few of these guiding questions to focus student collaboration. FACILITATION TIP Before reading the scenario, ask the class 1) If you have ever gone on a school ﬁeld trip, did the place you visited have food you could buy?; 2) Did you buy food there or bring your own food?; 3) Were you able to buy any snacks?

FACILITATION TIP Encourage students to attend to precision after substituting in values for variables. Some students may substitute successfully, but rush through the calculations. FACILITATION TIP Consider using connections to the everyday uses of the word substitution (recipes, teachers, sports) and discuss the similarities and differences to mathematical substitution. FACILITATION TIP On the Exit Ticket, reinforce to students that they need to show their steps to successfully demonstrate their understanding.

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Algebraic Expressions Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Write Expressions Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Simplify Expressions Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 3

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Simplify Using Properties

Interactive Notebook

Show What You Know, Part 4

A cut-and-glue activity to process learning that can be added to a notebook for future reference

Evaluate Expressions

Independent practice assignment that gives students an opportunity to demonstrate their learning

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Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Match Equivalent Algebraic Expressions

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Independent and partner games and other activities that provide students with an engaging way to practice the new concept

Interactive Practice

Lock & Key

The Cryptex

A game to practice the skills established by the standards in the scope

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Intervention and Assessment SMALL-GROUP PLANNER 1

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

Students who are still acquiring the concept and need remediation

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Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

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What does mastery look like?

I can define parts of expressions using mathematical language and terms such as sum, difference, term, product, factor, quotient, coefficient, variable, and constant. I can write expressions in which a letter stands for a number. I can read expressions in which a letter stands for a number. I can evaluate algebraic expressions for a given value of a variable by using order of operations. I can understand that variables are letters that represent unknown numbers. I can understand that math operations also apply to variables. I can evaluate expressions at specific values for the given variable(s). I can perform arithmetic operations including those involving whole-number exponents in the conventional order when parentheses are not provided. I can apply the properties of operations to generate equivalent expressions, including the distributive property. I can combine like terms to generate equivalent expressions. I can explain that when two expressions are equivalent, they name the same number regardless of the value that is substituted into them.

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SCOPE 1

Equations and Inequalities Scope Introduction SCOPE SUMMARY In this grade level, students write, model, and solve equations and inequalities. Students will learn that a variable can represent an unknown number or any number in a speciﬁed set. They will use substitution to determine whether an equation or inequality is true. Students will also write and graph inequality statements that represent real-world contexts in which there is a constraint with inﬁnite solutions. Student Expectations

6.PAR.7.2 Write one-step equations and inequalities to represent and solve problems; explain that a variable can represent an unknown number or any number in a speciﬁc set. 6.PAR.7.3 Solve problems by writing and solving equations of the form x ± p = q, px = q and x/p = q for cases in which p, q and x are all nonnegative rational numbers.

VERTICAL ALIGNMENT Background Knowledge In previous grade levels, students have written numerical expressions, equations, and inequalities to represent and solve real-world problems, and have used variables to represent unknown values. Students have also learned to write and solve numerical expressions using the order of operations. In ﬁfth grade, students have explained that the meaning of fractions is division of the numerator by the denominator.

6.PAR.7.4 Recognize and generate inequalities of the form x > c, x ≥ c, x < c, or x ≤ c to explain situations that have inﬁnitely many solutions; represent solutions of such inequalities on a number line.

Future Expectations Students in seventh-grade construct and solve multistep algebraic equations and inequalities with rational number coeﬃcients, including negative numbers. When constructing equations and inequalities, students will use them to solve practical problems leading to the equation px + q = r or p(x (x + q) = r.. Students will also interpret their (x solutions. Eighth-grade students create, analyze, and solve linear equations and inequalities within relevant applications.

Accessing Prior Knowledge

ENGAGE ACTIVITIES Before beginning the lesson, the students’ prior knowledge is assessed to address any student needs and skills needed for content mastery. Students are assessed to determine if they can: •

•

use parentheses, brackets, or braces within numerical expressions. evaluate expressions that contain these symbols.

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope.

Hook

6.PAR.7.1 Solve one-step equations and inequalities involving variables when values for the variables are given. Determine whether an equation and inequality involving a variable is true or false for a given value of the variable.

At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

compare the cost of pizza using two different equations.

•

determine the best deal for pizza.

Students move on to the Explore activities and come back at the end of the explorations to repeat the activity and discuss what they found.

If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. 190

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Add and Subtract Equations In this exploration, students will deﬁne variables to write, model, and solve addition and subtraction equations. Students will: • •

Explore 2

Explore 1

EXPLORE ACTIVITIES

use the balance scale and algebra tiles as you solve the problem together. model equations using tape diagrams.

Explore 4

Explore 3

In this exploration, students will be presented with a scenario about a local amusement park to determine which day the park has better deals for parking and pizza. Students will:

In this exploration, students will deﬁne variables to write, model, and solve multiplication and division equations. Students will: •

use algebra tiles to help model and solve equations.

•

model equations using tape diagrams.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment. Write and Solve Equations

Multiply and Divide Equations

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Write and Model Inequalities In this exploration, students will collaborate to solve a scenario involving determining how many tickets will be used on different rides at an amusement park to make sure enough tickets are purchased. Students will:

•

write and solve equations.

•

write and model inequalities.

•

use properties to ﬁnd the value of each variable.

•

determine if a given value falls in the range of solutions for each inequality.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve.

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Equations and Inequalities Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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ACCESSING PRIOR KNOWLEDGE Students will read different student responses to a posed question on the prior standard, decide whether they agree or disagree with the student, and explain their reasoning. 5.NR.5.1: Write, interpret, and evaluate simple numerical expressions involving whole numbers with or without grouping symbols to represent actual situations.

Materials

Preparation

Printed •

•

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Print one Agree or Disagree for each student.

1 Agree or Disagree (per student)

Procedure and Facilitation Points 1. 2. 3. 4. 5. 6. 7.

Instruct students to complete the Agree or Disagree independently. Once students have completed the activity on their own, have them stand up. Instruct all students to walk around the classroom with their hand raised in a high-five position. On your instruction, students will stop and high-five the closest person. This will be their partner. Give students a couple of minutes to discuss their answers and justifications together. You may then continue as many times as you want with different partners. Discuss the responses as a class. Allow students to explain their reasoning for each problem. a.

Disagree with Sanjay

Agree with Kai

Disagree with Miko

FACILITATION TIP After giving time for students to work independently (showing their reasoning and steps in the Agree or Disagree), take a minute to review the Order of Operations. After reviewing (using verbal and physical responses if needed) with students, give them another chance to work independently. FACILITATION TIP Using a minute timer for steps 1–6 may help facilitate the process. You can set it for 1–3 minutes depending on your students.

If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

Identifying Misconceptions • •

Students may do math from left to right like they read. It may help to remind students of the order of operations when they are doing problems. Students may not recognize that a number directly in front of a parenthesis is performing the operation of multiplication. Notes

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Equations and Inequalities Hook – Pick Your Pizza Promo! ACTIVITY PREPARATION Students will determine the cost of a pizza by using equations. Students will compare the costs of pizzas in two different equations to determine the best deal for pizza.

Materials

Preparation

Printed •

• • •

1 Pick Your Pizza Promo! (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Pick Your Pizza Promo! for the whole class to view. Prepare to introduce the scenario and to encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION POINTS Part I: Pre-Explore 1. FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) What is your favorite type of pizza?; 2) How often do you eat pizza?; 3) Do you or your family make pizza, or do you order it from a pizza place?

FACILITATION TIP Consider projecting Pick Your Pizza Promo! before asking the guiding questions. Give students some independent think time and some partner think time before asking students to share.

4. 5.

FACILITATION TIP Be prepared for some students to quickly compare unit rates (36/3 vs 65/5) for $ per pizza rather than consider an equation with cost of soda and a variable, p. 194

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: Tony loves the handmade pizza from Giovanni’s. He wants to order it for his Romano family dinner on Sunday, but Tony knows it is more expensive than regular pizza, so he looks for coupons. He wonders which promo is the best deal for the price of a pizza. Once Tony determines that, he will present the coupon to his parents and try to get them to order his favorite pepperoni pizza from Giovanni’s. Ask students the following questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Student answers will vary. I notice that Tony is ﬁnding the costs of pizzas. I wonder what the promotional coupons are. What will the cost of each pizza be? How will Tony ﬁnd out the costs of the pizzas? I can use math to determine the costs of the pizzas from the two coupons. I can also use math to compare the prices of the pizzas and see which coupon is the best deal, offering the least expensive pizza. Project Pick Your Pizza Promo! Explain to students that Tony has narrowed it down to the two promo coupons he thinks best suit the needs of his large family. Now he needs to decide which coupon gives the better deal for the price of a pizza. Discuss the following questions: a.

DOK-1 What is important about the pizzas listed on all of the coupons? The types and sizes of pizzas are the same on all of the coupons, so it is fair to compare their prices.

DOK-1 Does having soda listed make it impossible to determine the cost of the pizzas in the ﬁrst coupon? No, that doesn’t make it impossible to determine because the cost of the soda is listed separately.

DOK-1 How would we represent the cost of the pizza in the equations for each coupon? The cost of the pizza will be a variable such as p.

Complete the Explore activities.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and restate the problem. Refer to Pick Your Pizza Promo! Discuss the following questions: a.

DOK-1 How can you determine the cost of the pizza with a coupon? Write an equation with the cost of the pizza represented by a variable, and then solve the equation.

DOK-2 What is the equation for the ﬁrst coupon? 3p + 2 = 38

DOK-2 What is the equation for the second coupon? 5p = 65

DOK-2 What is the equation for the third coupon? p ÷ 10 = 1.25

e. DOK-2 What is the equation for the fourth coupon? p – 5 = 18 f.

DOK-1 Solve all equations. What is the cost of a pizza with each coupon? Coupon 1: 3p + 2 = 38 → 3p + 2 – 2 = 38 – 2 → 3p = 36 → 3p ÷ 3 = 36 ÷ 3 → p = $12 Coupon 2: 5p = 65 → 5p ÷ 5 = 65 ÷ 5 → p = $13 Coupon 3: p ÷ 10 = 1.25 → (10/p)(10/p)= 1.25(10) → p = $12.50 Coupon 4: p – 5 = 18 → p + (–5 + 5) = 18 + 5 → p = $23

DOK-1 Which coupon supplies the best deal for large pizzas? Why? The ﬁrst coupon gives the best deal because large pepperoni pizzas are $12 each, whereas they are $13 each with the second coupon, $12.50 each with the third coupon, and $18 each with the fourth coupon. $12 is less than all of the other values.

FACILITATION TIP If time allows, challenge students to bring in local pizza coupons or gather some on your own to use as an engagement tool. STEMscopes Tip The Skills Quiz, located in the Evaluate section, is a short standardsbased assessment where students demonstrate their computational ﬂuency. These assessments include a variety of question types and can be used to formatively evaluate students' knowledge about topics covered in the scope or to review the content.

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Equations and Inequalities Explore 1 – Add and Subtract Equations ACTIVITY PREPARATION Students will deﬁne variables to write, model, and solve addition and subtraction equations.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Set of Amusement Park Cards (per group) 1 Balance Scale (per group) 1 Exit Ticket (per student)

•

Reusable • • •

• •

1 Resealable bag (per group) 1 Set of algebra tiles (per group) 1 Projector or document camera (per class)

•

Plan to have students work in groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a Balance Scale for each group. If desired, print it on card stock and laminate it for future use. Print one set of Amusement Park Cards for each group. Cut out and place the cards in a resealable bag for each group. If desired, print the cards on card stock and laminate them for future use. Gather a set of algebra tiles for each group. Make sure to have a projector or document camera available to project a scenario for the class. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Algebra Tiles)

PROCEDURE AND FACILITATION POINTS FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever been to an amusement park?; 2) Where did you go?; 3) Did you have to pay or buy tickets to enter the park?

FACILITATION TIP It's possible that most students have not had any experience with real-world balance scales, and they are not familiar with how they work. Many schools have balance scales that you can use to demonstrate. Take some time to show them what a balanced and unbalanced scale looks like. Have students come up and attempt to balance it or unbalance it.

FACILITATION TIP To help students focus, clarify with the class how to use algebra tiles and the Balance Scale handout before distributing algebra tiles.

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Read the following scenario to the class: Six families want to visit their nearby amusement park this summer. The families want to get the most out of their money! The amusement park offers different admission specials throughout the week. Help determine the cost of popcorn or the total cost each family spent on snacks. Divide the class into groups. Give one Balance Scale and a set of algebra tiles to each group. Instruct students to take out a few algebra tiles to view. Discuss with the class how to use algebra tiles and the Balance Scale. a.

These are called algebra tiles. We can use algebra tiles to help model and solve equations. The ones that look like small squares each have a value of one. Today, we will use the yellow side to model positive numbers.

Find an algebra tile that looks like a rectangle. These tiles model the variable. Today, we will use the green side to model positive variables.

Project the Johnson family card for the class. Read the Johnson family card together. Then, model how to use the balance scale and algebra tiles as you solve the problem together. Discuss the following questions with the class: a.

DOK-1 What should our variable be for the popcorn scenario? Answers will vary. We can use t to represent the total amount of money each family began with.

DOK-2 What will the constant be on the left side of the equal sign with the variable? 8 will be the constant because we are taking 8 away from the total. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

DOK-3 What will the constant be on the right side of the equal sign? 5 will be after the equal because the card states that after they paid 8 dollars, there were 5 dollars left over. DOK-3 What will the equation be for the Johnson family popcorn scenario? The equation for the Johnson family popcorn scenario is t – 8 = 5.

e. DOK-1 What is the coeﬃcient of t? There was only one family total, so the coeﬃcient is 1. f.

DOK-2 How many xs do we need to use to model the family’s total with our algebra tiles? We will only need one x to model.

DOK-1 How many ones do we need to show how much they spent for that day? They spent 8 dollars, so we need 8 ones.

h. DOK-2 Would these ones be on the same side as the variable or the other side of the balance scale? They would go on the same side as the variable because they are being subtracted from the total they started with. i. DOK-1 What was the change they received after they paid? They received 5 dollars. j. DOK-1 How many ones should we use? We should use 5 ones because this is how much was left over. k.

DOK-1 Will this go on the same side as the x and 8 ones or on the other side of the balance scale? The 5 ones will go on the other side of the balance scale because it is showing how much money was left over after spending the 8 dollars on snacks.

Intervention

Acceleration

FACILITATION TIP As you discuss the algebra tiles and how to use them, monitor students' prior experience. Be prepared with some example equations to demonstrate and provide some practice for students. Conﬁrm that students know how to use them eﬃciently before they begin collaborating in groups. FACILITATION TIP

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To become fluent, students may need to consistently be reminded about the standard use of coefficients and dots rather than an "x" to show multiplication. A coefficient of 1 is rarely shown. STEMscopes Tip Supplemental Aids, located in the Intervention section, provide materials that will meet the needs of diverse learners. These materials include graphic organizers, handouts, and manipulatives that can further support students.

l. DOK-3 How could we determine the value of the variable? (Note: Model with the students adding 8 yellow tiles on each side.) Answers may vary. We could cancel out the 8 by making zero pairs. Since we had to add 8 to one side, we need to add 8 to the other side as well. m.

DOK-2 What is the value of one x? The value of one x is 13.

n. DOK-3 What does this mean in our scenario? This means the Johnson family started with 13 dollars. 5. 6.

7. 8.

Have students draw the algebra tiles model on their Student Journals for Monday’s total amount of money they brought for snacks. Discuss with the class how to model equations using tape diagrams. a.

DOK-1 How can I use models to represent and solve the expression t – 8? Draw two rectangles, one on top of the other. The one on top represents the whole, which is unknown, so we put a t. The rectangle on the bottom is separated in two sections, one to show the 8 dollars they spent and the other to represent 5, the change the family received back.

DOK-1 How can we determine the value of t? Answers may vary. We can determine the value of t by adding the 8 and 5 together.

Have students draw the tape diagram model on their Student Journals for Monday’s popcorn cost. Next, read the following scenario to the class: The Guiterrez family bought a slushie for $4 and a family bucket of popcorn. Their total for snacks was $8. How much is the bucket of popcorn? Project the Guiterrez family card for the class. Discuss the following questions with the class: a.

DOK-1 What should our variable be for the popcorn scenario? Answers will vary. We can use b to represent the price per popcorn bucket.

DOK-2 What will the constant be on the left side of the equal sign with the variable? 4 will be the constant. It was the cost of the slushie.

FACILITATION TIP Remind students that working with models can improve their problem-solving abilities. Students may want to just solve the problems in their head, but the learning task in this activity is to create models. Encourage them to evaluate the effectiveness of all models; have them be prepared to select one that they prefer (in tha they are required to use one in the future). FACILITATION TIP It may simplify the procedure to choose to show one model (tape, tiles, or scale) at a time and allow for practice rather than having students practice several at a time in this one activity. Consider breaking this activity into two or three sessions or steps.

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Equations and Inequalities Explore 1 – Add and Subtract Equations c.

DOK-3 What will the constant be on the right side of the equal sign? 8 will be the constant on the right side of the equal sign because the card says the total cost was 8 dollars.

DOK-3 What will the equation be for the Gutierrez family popcorn scenario? The equation for the Gutierrez family popcorn scenario will be 4 + b = 8.

e. DOK-1 What is the coeﬃcient of b? There was only one bucket of popcorn, so the coeﬃcient is 1. f.

DOK 2 How many xs do we need to use to model the popcorn bucket with our algebra tiles? We will only need one x to model.

g. DOK-1 How many ones do we need to show how much they spent on the slushie? They spent 4 dollars on the slushie, so we will need 4 ones.

STEMscopes Tip The STEMscopes Teacher Toolbox, located under the Scopes tab on the menu bar, features a variety of resources and tools to help teachers get the get most out of their STEMscopes experience, including essentials like lesson-planning documents, intervention strategies, monitoring tools, mathematical discourse strategies, and data resources.

9. 10.

FACILITATION TIP Even if students can intuit the solutions, continue to encourage them to draw models.

11.

12.

13. 14.

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DOK-2 Would these ones be on the same side as the variable or the other side of the balance scale? They would go on the same side as the variable because they are being added to the popcorn cost.

DOK-1 What was the total cost? The total cost was 8 dollars.

DOK-1 How many ones should we use? We should use 8 ones because that was the total.

DOK-1 Will this go on the same side as the x and 8 ones or on the other side of the balance scale? The 8 ones will go by themselves on the right side of the balance scale because they were the sum of the popcorn and slushie.

DOK-3 How could we determine the value of the variable? (Note: Model with the students taking away 4 yellow tiles on each side.) Student responses may vary. Take the 4 yellow tiles away so the variable is alone. Then, take 4 away from the other side.

DOK-2 What is the value of one x? The value of one x is 4.

DOK-3 What does this mean in our scenario? This means the Gutierrez family spent 4 dollars on a family bucket of popcorn.

Have students draw the algebra tiles model on their Student Journals for Tuesday’s price per popcorn bucket. Discuss with the class how to model equations using tape diagrams. a.

DOK-1 How can I use tape diagrams to represent and solve the equation 4 + b = 8? Draw a rectangle. Divide it into 2 pieces. Write 4 on one side to represent the 4 dollars spent on the slushie. On the other piece, write a b to represent the unknown popcorn price. Draw a line to show the 4 and b together are 8.

DOK-1 How can we determine the value of b? Answers may vary. We can determine the value of b by subtracting 4 from 8.

Students will continue to work collaboratively with their groups to determine the total amount of money each family took for snacks or the price per popcorn bucket. As students are working together, monitor their learning, and ask guiding questions for any students who are struggling with algebra tiles and questions in step 4 for students struggling with tape diagrams. (Answers will vary based on the scenario the student is working on.) Allow students enough time to complete their work and record their observations and reﬂections on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Engage

Explore

Explain

Elaborate

Evaluate

Math Chat •

• •

•

DOK-1 How can you find the value of the variable when using models? In subtraction equations, the value of the variable can be found by making zero pairs to cancel out the ones on the left side to get the variable alone. Then, add the same amount on the right side to find the total. In addition equations, you take away from both sides. DOK-1 What operation are you doing to find the value of the cost per popcorn bucket? You are subtracting the constant from the total cost of snacks. DOK-1 What operation are you doing to find the value of the total amount of money each family took for snacks? You are adding the constant to the difference to find the value of the variable. DOK-1 Examine all equations. What is similar about all of these equations? To solve the equation, you have to use the opposite operation. DOK-2 For any addition equation, x + p = q, how can you find the value of x? For any subtraction equation, x – p = q, how can you find the value of x? For the addition equation, you will subtract p from q to determine x. For example, if the equation was x + 4 = 24, we would subtract 4 from 24 and get that x = 20. For the subtraction equation, you will add p to q to determine x. For example, if the equation was x – 4 = 24, we would add 4 to 24 and get x = 28. DOK-3 Josie says that the equation x + 9 = 12 is equal to the equation 9 + x = 12. Is this true? Explain your reasoning. Both equations are equal. I can subtract 9 from 12 and get 3 in either equation. The commutative property was used to move where the x and the 9 are located in the equation.

Explain the following to the class: Mathematicians write the variable ﬁrst in equations. Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

Intervention

Acceleration

FACILITATION TIP Ask, "What questions do you still have about using models to ﬁnd variables?" Reinforce that the following Exit Ticket requires drawing a model to show solutions. FACILITATION TIP If students need to review the term constant take time to deﬁne and explain.

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FACILITATION TIP Take time to guide students to this conclusion. Model using inverse operations for students on a few simple equations. FACILITATION TIP When you explain this to the class, determine whether you will expect students to write the variable ﬁrst in equations. Also be prepared for some students to monitor all equations for this standard as you progress through these scopes. FACILITATION TIP Prior to having students complete this Exit Ticket, think about whether you want students to draw a speciﬁc model, use both or allow them to use their own methods to solve the equations. Some students may already want to use inverse operations.

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Equations and Inequalities Explore 2 – Multiply and Divide Equations ACTIVITY PREPARATION Students will deﬁne variables to write, model, and solve multiplication and division equations.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Set of Amusement Park Cards (per group) 1 Balance Scale (per group) 1 Exit Ticket (per student)

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Reusable • • •

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1 Resealable bag (per group) 1 Set of algebra tiles (per group) 1 Projector or document camera (per class)

•

PROCEDURE AND FACILITATION POINTS 1.

FACILITATION TIP Select the essential guiding questions and project them and record answers as you discuss.

Read the following scenario to the class: Six families want to visit their nearby amusement park this summer. The families want to get the most out of their money! The amusement park offers different admission specials throughout the week. Help determine the cost of one ticket for the day of the week each family attended. Project the Johnson family’s Amusement Park Card for the class. Discuss the following questions with the class: a.

DOK-1 What is a variable? A variable is a letter that represents an unknown number.

DOK-1 What letter should we use to represent the price per ticket? Student responses will vary. We can use p to represent the price per ticket.

DOK-1 What is a coeﬃcient? A coeﬃcient is the number directly in front of the variable.

DOK-2 How can I find what the coefficient for our variable is for the Johnson family tickets scenario? Student responses may vary. In this scenario, the coefficient is the number of family members that are buying tickets.

STEMscopes Tip Depth of Knowledge (DoK) Levels are found on the Lesson Planning Resources page in the Essentials section of the Teacher Toolbox. A printable document lists the DoK levels for all elements of the scope. This resource gives teachers the ability to choose the appropriate DoK-leveled assignments to help students expand and deepen their mathematical thinking and reasoning. 200

e. DOK-1 Where is the total in an equation? Student responses may vary. The total in an equation is on one side of the equal sign by itself. f.

Engage

Explore

Explain

Elaborate

Evaluate

Give one Balance Scale and a set of algebra tiles to each group. Instruct students to take out a few algebra tiles to view. Discuss with the class how to use algebra tiles and the Balance Scale for the ticket scenario for the Johnson family. a. These are called algebra tiles. We can use algebra tiles to help model and solve equations. The ones that look like small squares each have a value of one. Today, we will use the yellow side to model positive numbers. b.

Find an algebra tile that looks like a rectangle. These tiles model the variable. Today, we will use the green side to model positive variables.

Our variable is p. DOK-1 What is the coeﬃcient of p? The coeﬃcient is 4.

Since our coeﬃcient is 4, we will need to place four green xs (rectangles) on one side of our Balance Scale.

Intervention

Acceleration

FACILITATION TIP If students were successful with algebra tiles in Explore 1, they should only need a quick review of expectations for this Explore activity.

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e. DOK-1 What was the total cost of the tickets? The total cost of the tickets was $24. f.

DOK-1 How many ones should we use? We should use 24 ones because this is our total.

DOK-1 Will this go on the same side as the xs or on the other side of the Balance Scale? These 24 ones will go on the other side of the Balance Scale because they are the total the Johnson family spent on tickets.

h. Now, we need to determine what the value of our variable is. i. DOK-3 How could we determine the value of the variable? Answers may vary. We could distribute the ones out evenly to each x to ﬁnd out what each x is worth. Model with the students distributing the ones to the variables. j. DOK-2 What is the value of one x? The value of one x is 6. k. 4.

DOK-3 What does this mean in our scenario? This means that the value of one ticket on Monday when the Johnson family attended is $6.

Discuss with the class how to model equations using tape diagrams.

STEMscopes Tip Blackline Masters, located in the Essentials section of the Teacher Toolbox, provide teachers with frequently needed instructional print materials. There are a wide variety of printables, including an analog clock, coordinate plane, fraction strips, hundreds charts, assorted number lines, sharing mats, and ten frames.

a. We have used tape diagrams to model expressions. DOK-1 What would I draw to model the expression 4p? Draw a rectangle, and divide it into four sections. Label each section “p.”

DOK-1 How can I show that all of these sections together equal 24? Answers will vary. Draw a line/box to show that from the beginning of the rectangle to the end of the rectangle together, there will be 24.

DOK-1 How can we determine the value of p? Answers may vary. Divide the total (24) by four since there are four sections labeled “p.” 24 ÷ 4 = 6

Now, project the Nguyen family’s Amusement Park Card for the class. a. First, discuss the difference in information provided on this scenario card. b.

DOK-1 What are we trying to determine in this scenario? Total cost for the tickets

DOK-1 What letter should we use to represent the total cost? Answers will vary. We can use c to represent the total cost.

DOK-1 Where will we find our coefficient in this scenario? Answers may vary. In this scenario, the coefficient is the number of family members that are buying tickets.

e. DOK-2 What number will we be using for the coeﬃcient? 6 because there are 6 family members. f.

DOK-3 For F this problem, how will our coeﬃcient be written with the variable since we are looking for a total? The coeﬃcient will be written c 1 as __6. In this case, __6.

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EQUATIONS AND INEQUALITIES

Equations and Inequalities Explore 2 – Multiply and Divide Equations g.

Student Goal Setting, located in the Essentials section of the Teacher Toolbox can be used by students to self-evaluate. Included in this section is a student goal-setting sheet on which students identify a math goal, write or draw "I can" statements, describe what they will do to reach the goal, and evaluate whether they have met their goal.

h. DOK-2 What will the equation be for the Nguyen family’s total cost? __6 = 6

Students will still use the Balance Scale and a set of algebra tiles. You will now discuss how to use algebra tiles for the Nguyen family’s total cost. a.

We will use the yellow side to model positive numbers.

Find an algebra tile that looks like a rectangle. These tiles model the variable. Today, we will use the green side to model positive variables.

Our variable is c. DOK-1 What is the coefficient of c? The coefficient is __6 because we will use it to find a total.

Since our coeﬃcient is __6, we will need to place one green x (rectangles) on one side of our Balance Scale.

DOK-1 Because we are determining how much it is for 6 people, we will need to break the x into pieces. How many pieces do we need to break the x into? 6 pieces, to represent the 6 family members

DOK-1 On the right side of the Balance Scale, how many ones should we use? We should use 6 ones to represent the 6 dollars per ticket.

Now, we need to determine what the value of our variable is.

j. DOK-3 What does this mean in our scenario? This means that the value for x, the total cost, is $36. 7.

Discuss with the class how to model equations using tape diagrams. a. We have used tape diagrams to model expressions. DOK-1 What would c I draw to model the expression __6? Draw two rectangles on top of each other, one to represent the total cost, which is unknown, so we label it c. Then, the bottom rectangle is divided into 6 parts to represent the 6 family members. In each part, we will write 6 to represent 6 dollars for each person.

Reinforce that the following Exit Ticket requires drawing a model to show solutions.

i. DOK-2 Now, looking at your x as a whole and when you count all the ones on the right side of the Balance Scale, how many ones are there? There are 36 ones.

FACILITATION TIP

h. DOK-3 How could we determine the value of the variable? Answers may vary. We could distribute the 6 ones out evenly to each part of the x to ﬁnd out what the whole x is worth. Model with the students distributing the ones for each piece of the x.

STEMscopes Tip

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DOK-1 How much is each ticket? Each ticket is $6.

10.

11.

DOK-1 How can we determine the value of c? Answers may vary. Multiply the 6 family members by the price per ticket, $6, to give you a total of $36.

Give a Student Journal to each student and a set of Amusement Park Cards to each group. Instruct students to ﬁrst work with their groups on the ticket scenarios. Allow time for students to show their work for the algebra tiles model and the tape diagram model on their Student Journals for Monday’s and Tuesday’s tickets. Have students continue to work collaboratively with their groups to determine the cost per ticket for each remaining day. Allow students to choose which model to draw on their Student Journals. As students are working together, monitor their learning and ask the following questions to check for understanding. Answers will vary based on the scenario the student is working on; answers provided are for Monday tickets with the Johnson family. a.

DOK-2 What is the equation? The equation is 4p = 24.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How many sections should we partition in the tape diagram? We should use 4 sections.

DOK-1 How many ones should we use (if using algebra tiles)? We should use 24 ones because this is our total.

e. DOK-1 What number should we label the total? We should write “24” because this is our total. f.

DOK-1 Will this go on the same side as the xs or on the other side of the Balance Scale (if using algebra tiles)? These 24 ones will go on the other side of the Balance Scale because they are the total the Johnson family spent on tickets.

DOK-3 How could we determine the value of the variable? Student responses may vary. When using algebra tiles, we can distribute the ones out evenly to each x to ﬁnd out what each x is worth.

Intervention

Acceleration

STEMscopes Tip The Interventions section is found in the Teacher Toolbox. It provides teachers with intervention strategies for students who need support with a variety of roadblock behaviors. Included are detailed methods to help students with their communication, physical, cognitive, social and emotional, and adaptive development.

EQUATIONS AND INEQUALITIES

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h. DOK-2 What is the value of our variable? The value of the variable is 6. i. DOK-3 What does this mean in our scenario? This means the value of one ticket on Monday when the Johnson family attended is $6. 12. 13.

Allow students enough time to complete their work and record their observations and reﬂections on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning. FACILITATION TIP

Math Chat • • •

• • • •

•

• •

DOK-1 What is a coefficient? The coefficient is the number that is multiplied by the variable. DOK-1 What is a variable? The variable is a letter that represents an unknown number. DOK-1 How can you find the value of the variable when using models? The value of the variable can be found by distributing all of the ones (squares) to the x rectangles equally until all of the ones have been used. DOK-1 What operation are you doing to find the value of the variable in the equation for tickets? You are dividing to find the value of the variable. DOK-1 What operation are you doing to find the value of the variable in the equation for total cost? You are multiplying to find the value of the variable. DOK-1 Examine all equations. What is similar about all of these equations? To solve the equation, you have to use the opposite operation. DOK-2 For any multiplication equation, px = q, how can you find the value of x? You will divide q by p to determine x. For example, if the equation was 4x = 24, we would divide 24 by 4 and get that x = 6. x DOK-2 DOK2 For any division equation, __p = q, how can you find the value of x? You will multiply q by the reciprocal of p to determine x. For example, if the equation x 1 was __4 = 6, since the 4 is under the fraction bar, it is technically __4, so we would 1 __ multiply 4 by its reciprocal 4, which cancels out and leaves the variable alone. Then, multiply the 6 by 4, and conclude x = 24. DOK-2 Are the expressions 4 4xx and x · 4 equal? Explain your reasoning. Yes, I can put a 2 in for x in each expression. 4(2) equals 8, and 2 · 4 equals 8. 4

DOK-2 Are the expressions __x and __4 equal? Explain your reasoning. No, because if I 4

put 8 for x in each expression, _8 is a fraction for _4 , while __4 is a whole number of 2.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

Select some essential Math Chat questions to project while students are working and collaborating. As you monitor, direct students to the key vocabulary and encourage them to use it during their work time and the Math Chat.

FACILITATION TIP To become ﬂuent, students may need to consistently be reminded about the standard use of coeﬃcients and dots rather than an "x" to show multiplication. FACILITATION TIP When you preview this Exit Ticket with students, clarify expectations for showing models. Consider that some students may only want to write solutions or use inverse operations. 203

EQUATIONS AND INEQUALITIES

Equations and Inequalities Explore 3 – Write and Solve Equations ACTIVITY PREPARATION Students will write and solve equations using properties to ﬁnd the value of the variable.

Standards for Mathematical Practice • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively.

Materials

Preparation

Printed

•

• 1 Student Journal (per student) • 1 Set of Daily Deals Cards (per group) • 1 Exit Ticket (per student)

• •

Reusable • •

1 Projector or document camera (per class) 1 Resealable bag (per group)

•

Plan to divide the class into groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print one set of Daily Deals Cards per group. If desired, print the cards on card stock and laminate them for future use. Cut out the Daily Deals Cards, and place them in a resealable bag for each group. Have a projector or document camera ready to project Tuesday’s Daily Deals.

PROCEDURE AND FACILITATION POINTS Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever found a good deal on something you wanted to buy?; 2) What was it?; 3) Why was it such a good deal?

FACILITATION TIP After Explore 1 and 2, students may still need explicit instruction on using inverse operations to solve equations. Be prepared with some sample equations to demonstrate and provide student practice.

FACILITATION TIP Have students record the deﬁnition and examples of inverse operations for the four basic operations. Show some real-world examples with simple whole numbers to demonstrate why it works. (For example, "If I have 84 candies, how many would each of you get?") Write an equation and use inverse operations to solve. "If we have 20 students signed up for a ﬁeld trip and a bunch more (x) show up and get on the bus before the teacher counts students. When the teacher counts students, she gets 32." Write an equation and use inverse operations to solve. 204

3. 4. 5.

Read the following scenario to the class: The local amusement park offers parking and pizza deals every Tuesday through Friday. Montrell’s family wants to see which day has the best deals for parking and pizza. Help Montrell’s family determine which day has the better deal by writing and solving equations. If needed, revisit the following Math Chat discussion questions from previous Explore activities to review using inverse operations to solve equations. a.

DOK-2 For any addition equation, x + p = q, how can you ﬁnd the value of x? You would subtract p from q to determine x. For example, if the equation was x + 4 = 24, we would subtract 4 from 24 and get that x = 20.

DOK-2 For any subtraction equation, x − p = q, how can you ﬁnd the value of x? You would add p to q to determine x. For example, if the equation was x − 4 = 24, we would add 4 to 24 and get that x = 28.

DOK-2 For any multiplication equation, px = q, how can you ﬁnd the value of x? You would divide q by p to determine x. For example, if the equation was 4x = 24, we would divide 24 by 4 and get that x = 6.

DOK-2 For any division equation, __p = q, how can you ﬁnd the value of x? You would multiply q and p to determine x. For example, if the equation x was __2 = 24, we would multiply 24 by 2 and get that x = 48.

Give a Student Journal to each student. Project Tuesday’s parking price scenario for the class to see. Have a class discussion about using fractions and decimals in equations. Model writing and solving equations with students as you ask the class the following questions: a.

DOK-1 How can I write an equation to represent the price of parking for 1 Tuesday? The equation would be __4h = 5.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How can I get the variable h by itself? Student responses will 1 vary. To get the variable h by itself, we will need to divide by __4 on both sides of the equation.

DOK-1 Does it matter which order I write the division equation on the other side of the equal sign? Yes, the order matters in how you write 1 1 the equation. You should write 5 ÷ __4. If you write __4 ÷ 5, you will get a different answer.

DOK-1 How do you solve 5 ÷ __4? To solve, we will need to multiply 5 by

Intervention

Acceleration

FACILITATION TIP Be prepared to model solving several equations. Solve equations that use all inverse operations. Be clear about how you show "getting the variable by itself." Encourage students to show all of their steps so you can monitor for errors and confusion when you look at their Student Journals.

the reciprocal of __4, which is __1, or 4.

e. DOK-1 What is the price for one hour of parking? The price for one hour of parking would be $20. f.

Project Tuesday’s pizza price scenario for the class to see.

DOK-1 How can I write an equation to represent the price for pizza on Tuesday? 2.15 + p = 15.42

h. DOK-1 What step should we take to get the variable p by itself? To get p by itself, we will need to subtract 2.15 from both sides. i. DOK-1 Does it matter which order I write the subtraction equation on the other side of the equal sign? Yes, the order matters in how you write the equation. You should write 15.42 − 2.15. If you write 2.15 − 15.42, you will get a different answer.

EQUATIONS AND INEQUALITIES

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FACILITATION TIP Some students may need a review of how to line up the decimals and to rename when subtracting. Determine whether you will allow calculators.

j. DOK-1 What is 15.42 − 2.15? 13.27 k. 6. 7. 8.

9. 10.

DOK-1 What is the price for the pizza on Tuesday? $13.27

Give a set of Daily Deals Cards to each group. Students will work collaboratively to write and solve an equation for each of the remaining daily deals. As students are working together, monitor their learning, and ask the following questions to check for understanding: a.

DOK-1 What operations do you use to solve the parking price deals? You will need to divide by the coeﬃcient to solve the parking price deals or multiply by the groups.

DOK-1 What operations do you use to solve the pizza price deals? You will need to add or subtract to solve the pizza price deals.

DOK-2 What process would you use when your coeﬃcient is a fraction to get the variable by itself? To take the fraction to the other side of the equation, we need to divide. Since we are dividing by a fraction, we will use the multiplicative inverse.

FACILITATION TIP The Daily Deal Cards include color images that are not essential to the equations. FACILITATION TIP Project questions 8a–8c for students to see while they collaborate. Follow up by asking them to be prepared for the Math Chat by discussing these prompts with their groups.

Allow students enough time to work with their groups to complete their work and reﬂections on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 Which day had the best parking deal? Wednesday had the best parking deal. • DOK-2 Which day had the best pizza deal? Tuesday had the best pizza deal. • DOK-2 How do you divide a fraction by a fraction? To divide one fraction by another fraction, you will need to multiply by the inverse. • DOK-2 If the equation is using addition, why do you need to use subtraction in order to solve it? In order to solve equations, you must use the inverse operation in order to move numbers from one side of the equal sign to the other. Therefore, if there is an addition sign, in order to undo it, you must use the inverse of addition, which is subtraction. •

FACILITATION TIP These ﬁrst four questions are identical to the Student Journal Reﬂection questions. Consider projecting these questions before and during student collaboration and the Math Chat rather than printing that page for each student.

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Equations and Inequalities Explore 3 – Write and Solve Equations DOK-1 If the equation is using multiplication, why do you need to divide to solve it? In order to solve equations, you must use the inverse operation to move numbers from one side of the equal sign to the other. If the operation is multiplication, we will need to use division to undo it. • DOK-1 Summarize how to solve equations with variables. To solve equations with variables, you need to use the inverse operation of what is being done to the variable. For example, if you have x – 4 = 12, you will add 4 to both sides to solve for x. • DOK-3 Sam says the equation for the pizza deal on Wednesday is −3.25 + p = 10.50. Maria says the equation is p – 3.25 = 10.50. Who is correct? Explain your reasoning. Both Sam and Maria are correct. The two equations are equivalent. Maria’s equation of p – 3.25 = 10.50 is the standard form for writing equations. Sam’s equation −3.25 + p = 10.50 is using the commutative property of addition to write the same equation. •

FACILITATION TIP Be prepared with some common everyday examples of using equations to calculate some relevant real-world experiences using equations. (How many points needed to win video game/sports game? Average score needed to improve grade? How many hours of yard work or babysitting to reach a goal?) FACILITATION TIP

Explain the following to the class: Mathematicians write the variable ﬁrst in equations. •

DOK-3 Where outside of the classroom would you use an equation or inequality? You can ﬁnd how much you can spend on one item if you know the total money you had and how much you have already spent.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

Student experience with inequalities may be limited at this point. Consider keeping your examples related to equations using positive integers only for now. FACILITATION TIP Clarify how student understanding will be assessed on this Exit Ticket. Will students need to show inverse operations or can they show the solutions? Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________

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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Equations and Inequalities Explore 4 – Write and Model Inequalities ACTIVITY PREPARATION Students will write and model inequalities. Students will determine if a given value falls in the range of solutions for each inequality.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics.

Materials

Preparation

Printed • • • •

•

1 Student Journal (per student) 1 Set of Ticket Scenario Cards (per group) 1 Number Line (per student) 1 Exit Ticket (per student)

• •

Reusable • • • •

•

1 Dry-erase marker (per student) 1 Sheet protector (per student) 1 Projector or document camera (per class) 1 Resealable bag (per group)

•

Plan to divide the class into groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print one set of Ticket Scenario Cards for each group. Cut out the cards, and place them in a resealable bag for each group. Optionally, print them on card stock, and laminate them for multiple uses. Print one Number Line per student. Laminate it or put it inside a sheet protector for students to use dry-erase markers on it. Have a projector ready to model an example with students.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before reading the scenario, ask the class 1) If you have ever been to an amusement park, what rides did you go on?; 2) Did you have to buy tickets to ride the rides?; 3) What was your favorite ride? Why? FACILITATION TIP Use this number line to assess students' prior knowledge about inequalities and teach as needed. Use the Picture vocabulary to show the use of shaded and unshaded points on a number line. Students may also need practice determining which direction the ray points as related to the inequality symbol. FACILITATION TIP If needed, before asking the class questions 3a–3g, provide student practice reading several simple inequalities. Once students can read the inequalities out loud, move on to having them graph them on the number line. Students will be more successful if they are able to both graph an inequality and write an inequality before applying it to the real-world Ticket Scenario Cards. 208

2. 3.

Read the following scenario to the class: Emily and Sam just got tickets for the amusement park! They want to determine how many tickets they will use on different rides to make sure they purchased enough tickets. Help Emily and Sam write inequalities to represent how many tickets they want to use in each scenario. Give a Number Line and a dry-erase marker to each student. Project Emily’s circular rides scenario (Ticket Scenario Card 1) for the class to see. Ask the class the following questions: a.

DOK-1 Does Emily want to use more or less than 15 tickets? She wants to use less than 15 tickets.

DOK-1 Would she use exactly 15 tickets? She does not want to use exactly 15 tickets.

DOK-1 How can we write an inequality to show she will use less than 15 tickets? We can write t, for tickets, is less than 15. c < 15

Model 15 on your Number Line. We want to show that Emily would use less than 15 tickets but not exactly 15 tickets.

e. DOK-1 How would I represent exactly 15 tickets being used? Student responses will vary. There would be a dot at 15 if I wanted to represent exactly 15. f.

DOK-2 If a dot represents 15, how could we represent up to 15 but not exactly 15 tickets? Student responses will vary. We can use an unshaded circle at 15.

4. 5.

7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

Give a set of Ticket Scenario Cards to each group. Have students work collaboratively to read each Ticket Scenario Card to determine whether the number of tickets in the scenario is less than, greater than, less than or equal to, or greater than or equal to a given number. Then, have students write an inequality, graph the inequality on a Number Line to describe the scenario, and answer a question related to the model. As students are working together, monitor their learning, and ask the following questions to check for understanding: a.

DOK-1 How do you represent greater than or equal to on a number line? To represent greater than or equal to on a number line, we would put a closed circle at the number and then an arrow pointing to the right of the number.

DOK-1 How do you represent less than or equal to on a number line? To represent less than or equal on the number line, we would put a closed circle at the number and then an arrow to the left of the number.

Allow students enough time to work with their groups to complete each scenario and reﬂection question on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Intervention

Acceleration

FACILITATION TIP To support struggling students, read the scenarios out loud with the class and highlight key words to emphasize the inequalities. (No more than, at least, or more...). FACILITATION TIP Some students may be ready to write and graph their own scenarios while struggling students may need to practice one skill at a time (reading, writing then graphing) using inequalities.

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FACILITATION TIP As students practice, ﬁnd ways to help them remember how to represent with open/closed circles and left or right. Some students may need to hear, "Less is left" or some other cues to support success.

Math Chat DOK-1 When do you use an open circle to model an inequality, and when do you use a closed circle to model an inequality? Open circles are when the variable will not equal that amount, only more or less than that amount. Closed circles are used for when the variable will equal that amount or more or less. • DOK-1 How are inequalities different from equations? Equations equal one number, but inequalities can equal more than one number. • DOK-2 What is a possible number of tickets Emily could have used in ticket scenario 4? Accept all answers greater than or equal to 2 tickets. • DOK-2 What is a number of tickets that Sam could not have used in ticket scenario 9? Explain your reasoning. Sam could not use 40 tickets on food because he must use less than 25 tickets, and 40 is more than 25. •

Post-Explore 1. 2. 3. 4.

FACILITATION TIP When you prepare students to complete the Exit Ticket, consider offering some students copies with numbers already placed on the number lines.

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EQUATIONS AND INEQUALITIES

Equations and Inequalities Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Add and Subtract Equations Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Multiply and Divide Equations Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 3

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Write and Solve Equations

Interactive Notebook

Show What You Know, Part 4

A cut-and-glue activity to process learning that can be added to a notebook for future reference

Write and Model Inequalities

Independent practice assignment that gives students an opportunity to demonstrate their learning

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Equations and Inequalities Independent and partner games and other activities that provide students with an engaging way to practice the new concept

EQUATIONS AND INEQUALITIES

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211

Intervention and Assessment SMALL-GROUP PLANNER 1

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

EQUATIONS AND INEQUALITIES

Equations and Inequalities

3 212

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER Use this template to decide how to assess your students for concept mastery. Depending on the format of the assessment, you can identify prompts and intended responses that would measure student mastery of the expectation. See the beginning of this scope to identify standards and grade-level expectations. Key Concepts I can use substitution to determine whether a given number in a speciﬁed set makes an equation or inequality true.

What prompts will be used?

What does mastery look like?

EQUATIONS AND INEQUALITIES

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I can explain that a variable can represent an unknown number or a number in a speciﬁed set. I can write equations of the form x ± p = q, px = q and x/p = q including the use of positive rational numbers to represent and solve mathematical problems. I can write inequalities of the form x > c, x ≥ c, x < c, or x ≤ c to represent and solve mathematical problems. I can use concrete models or drawings and strategies based on place value when solving onestep equations. I can use strategies based on the properties of operations and the relationships between addition and subtraction and between multiplication and division when solving one-step equations. I can solve equations involving positive rational numbers using number sense, the four operations and maintaining equality of both sides of the equation. I can interpret a solution in the original context and assess the reasonableness of results. I can represent mathematical situations using inequalities involving variables. I can create mathematical situations from speciﬁc inequalities. © Accelerate Learning Inc. - All Rights Reserved

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SCOPE 1

Ratios, Rates, and Unit Rates Scope Introduction SCOPE SUMMARY

Student Expectations

In this scope, students understand the concept of a ratio as an association between two quantities. Students use ratio language to describe ratio relationships. Tape diagrams, double number lines, tables, and graphs are featured strategies and representations for generating and analyzing equivalent ratios. Students learn that a rate is a ratio that uses two different units, and that a unit rate is a rate per one. Contextual situations such as paint mixtures, recipes, constant speed, and pricing will help students reason about ratios, equivalent ratios, rates, and unit rates.

6.NR.4.1 Explain the concept of a ratio, represent ratios, and use ratio language to describe a relationship between two quantities. 6.NR.4.2 Make tables of equivalent ratios relating quantities with whole-number measurements, ﬁnd missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. 6.NR.4.3 Solve problems involving proportions using a variety of student-selected strategies. 6.NR.4.4 Describe the concept of rates and unit rate in the context of a ratio relationship.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

Students were introduced to the concept of rate and multiplicative comparisons beginning in fifth grade. In fifth grade, students calculated unit conversions between measurements involving length, weight, mass, liquid volume, and time. Students built on multiplicative comparisons to add and subtract fractions with unlike denominators by finding equivalent fractions using the least common multiple. Fifth-grade students plotted points in the first quadrant on the coordinate plane, and they generated and graphed numerical patterns. Experiences with unit conversions, tables, and graphs provided the foundation for understanding ratio relationships.

Sixth graders will apply ratio and rate reasoning in upcoming scopes that explore percents and measurement conversions. In Grade 7, students will extend their understanding of ratios, rates, and percents to explore proportional relationships, scale drawings, using similar triangles to compare slope, and calculating unit rates.

6.NR.4.5 Solve unit rate problems including those involving unit pricing and constant speed.

Before beginning the lesson, students’ prior knowledge is assessed. The teacher assesses students’ ability to: •

demonstrate and explain the relationship between equivalent fractions.

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope.

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

determine a ratio of a part to a whole and express the ratio by using different ratio language.

Students move on to the Explore activities and come back at the end of the explorations to repeat the activity and discuss what they found.

If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. 214

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Ratios In this exploration, students will describe ratio relationships between two quantities. Students will: •

determine the ratio between each type of fruit brought to the market each week.

Explore 2

Explore 1

EXPLORE ACTIVITIES

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

Ratio Tables and Graphs In this exploration, students will reason, analyze, and create tables and graphs of ratios. Students will: •

determine how many of each type of fruit bush comes in one pack.

•

determine the ratio for raspberry bushes to blueberry bushes and the ratio for strawberry bushes to blackberry bushes.

RATIOS, RATES, AND UNIT RATES

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Rates and Unit Rates In this exploration, students will discuss how they can determine the unit rate of a ratio. Students will: •

determine the rate and unit rate for each type of berries using ratio tables, double number lines, and tape diagrams.

Explore 4

Explore 3

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

In this exploration, students will will use multiple representations of ratios and rates to compare quantities and make predictions. Students will: •

compare ratios of berries in each berry bag.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

Explore 5

Comparing Ratios and Using Rates to Make Predictions

Solving Proportions In this exploration, students will use equivalent ratios and rates to solve proportions. Students will: •

solve proportions using ratios.

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve.

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RATIOS, RATES, AND UNIT RATES

Ratios, Rates, and Unit Rates Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

216

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will look at prompts about the prior standard and communicate whether they feel the prompts are fact or ﬁction by walking to the designated sides of the classroom. This element is designed to uncover student misconceptions; it should not be taken for a grade. 4.NR.4.1: Using concrete materials, drawings, and number lines, demonstrate and explain the relationship between equivalent fractions, including fractions greater than one, and explain the identity property of multiplication as it relates to equivalent fractions. Generate equivalent fractions using these relationships.

Materials

Preparation

Printed •

•

1 Set of Fact or Fiction Prompts (per class)

•

RATIOS, RATES, AND UNIT RATES

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Print one set of Fact or Fiction Prompts to read aloud to your students. Another option is to project the prompts by using a digital projector.

Procedure and Facilitation Points 1.

2. 3. 4. 5.

Designate one side of your room as the Fact side of the room and the other side as Fiction. Instruct students to move to one side of the room based on whether they think the prompt is fact or ﬁction. Read the prompt, and allow students to move to different sides of the room. Have students discuss their reasoning among their peers. Before reading the next prompt, allow students to move back to their starting points. Repeat with another prompt. a.

Prompt 1 is false.

Prompt 2 is true.

Prompt 3 is true.

If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

Identifying Misconceptions • •

FACILITATION TIP If space is limited, consider using anonymous voting for true of false (hands close to chest or heads down or small ballots). FACILITATION TIP Provide sentence frames for student discussion. For example, "I think that ________ is/is not equivalent to ________ because I see/know that _____________." FACILITATION TIP This Foundation Builder has fractions and visual models on the Math Match cards that may provide a good review activity for the whole class if needed.

Students may struggle to identify equivalent fractions. Students may struggle to identify what fractions the model represents. Notes

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RATIOS, RATES, AND UNIT RATES

Ratios, Rates, and Unit Rates Hook – He's a Baller! ACTIVITY PREPARATION Students will determine a ratio of a part to a whole and express the ratio by using different ratio language.

Materials

Preparation

Printed •

• • •

1 He’s a Baller! (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project He’s a Baller! for the whole class to view. Prepare to introduce the scenario and to encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION POINTS Part I: Pre-Explore 1. FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) Has anyone ever kicked or thrown a ball on a roof?; 2) If so, what type of ball was it?; 3) How did you get it down?

STEMscopes Tip Use the Communicate Math – Discourse page, found under the Communicate Math tab in the Teacher Toolbox, to learn strategies that can be used to model expectations and appropriate interactions students need to follow during productive math discussions with partners, in small groups, or with the whole class.

4. 5.

FACILITATION TIP Post these questions for students to see and read as you discuss. Be prepared for students to need support pronouncing ratio and explaining the deﬁnition.

6. 218

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: Mr. Smith is everyone’s favorite custodian at Lincoln Middle School. Every few weeks, he gets out his ladder and climbs on top of the school roof to collect the balls that have gotten stuck up there during PE, recess, and post-lunch games and sports. Mr. Smith has decided to evaluate the data he collected to determine which balls get stuck most often. Once he knows, he will put the data in the form of a ratio, demonstrating which type of ball makes up the biggest share of total balls stuck on the roof. Then, he will recommend that students play sports and games that use those balls farther from the school building at locations such as the ﬁeld or the track. Ask students the following questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Student answers will vary. I notice that Mr. Smith is using ratios. I wonder which balls will be found on the roof. What type of ball will Mr. Smith ﬁnd to be most common on the roof? How many total balls will be found on the roof? I can use math to compare quantities of balls found on the roof in the form of ratios. Project He’s a Baller! Explain to students that Mr. Smith found 12 balls on the roof. The types of balls include soccer balls, footballs, and tennis balls. Each type of ball was found in a different quantity. a.

DOK-1 What is a ratio? A ratio is a relationship that compares two numbers.

DOK-2 What ratios could be created from the balls visible on the slide? The ratios could compare the number of one type of ball to the number of another type of ball. The ratios could compare the number of a type of ball to the total number of balls found.

DOK-1 How many balls were found in all? There were 12 balls found on the roof.

DOK-1 How many of each type of ball were found? There were two footballs, four soccer balls, and six tennis balls.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and restate the problem. Refer to He’s a Baller! Discuss the following questions: a.

DOK-1 What are some different ways you can write a ratio? You can use a colon, a fraction bar, and words.

DOK-2 What two quantities are you comparing? Why? We are comparing the number of tennis balls (6) and the total number of balls (12). We are comparing these two quantities because tennis balls are the most common type of ball found on the roof, which is what Mr. Smith wants to know.

DOK-1 What is the ratio of the number of the most common type of ball found to the total number of balls found? Express this in two different ways. The ratio of the number of most common ball to the number of 6 total balls is 6 : 12 or ___ or 6-to-12. There are 6 tennis balls found for 12 every 12 total sports balls.

DOK-1 Explain what this ratio means. It means that out of every twelve balls stuck on the roof, six of them were tennis balls.

e. DOK-2 Can you create two other equivalent ratios for this situation? Answers will vary. 1 : 2, 3 : 6, and 12 : 24 f.

DOK-2 What would be the simplest way to write the ratio 6 : 12? How do you know that is the simplest? If you divide 6 and 12 by the greatest common factor of 6, you get 1 : 2. This would be the simplest form. I know this is the simplest form because there are no other common factors.

DOK-1 What is a ratio where one of the parts is a 1 called? A unit rate

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FACILITATION TIP After completing the scopes, students should feel comfortable stating the meaning of ratios with this sentence structure. However, struggling students may still need a sentence frame to prompt them: "For every_________ ball on the roof, there are _________ tennis balls." Or maybe just, "For every ___, there are_____."

h. DOK-1 Can you predict how many tennis balls would be on the roof if there were 48 total balls found? Yes, based on the rate of 6 : 12, if we multiplied the entire ratio by 4, there would be 24 tennis balls on the roof.

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RATIOS, RATES, AND UNIT RATES

Ratios, Rates, and Unit Rates Explore 1 – Ratios ACTIVITY PREPARATION Students will describe ratio relationships between two quantities.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Fruit Stand Cards (per group) 1 Exit Ticket (per student)

Reusable • • •

• •

1 Resealable bag (per group) 1 Set of 10 pink, 10 red, 10 blue, and 10 purple linking cubes (per group, optional) 1 Projector or document camera (per teacher)

Plan to divide the class into groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print and cut one set of Fruit Stand Cards per group. If desired, print the cards on card stock and laminate them for future use. Place them in a resealable bag. Plan to project the Fruit Stand Card for week 1. Optionally, have sets of linking cubes ready for struggling students. Have students model the different fruit with linking cubes to provide support in creating the tape diagram for each ratio. Each group would need one set of ten pink linking cubes for strawberry, one set of red linking cubes for raspberry, one set of blue linking cubes for blueberry, and one set of purple linking cubes for blackberry.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP

Before reading the scenario, ask the class 1) What types of fruit do you like?; 2) Have you ever picked fresh fruits off trees or plants?; 3) If so, what types of fruit did you pick?

FACILITATION TIP Emphasize the importance of order for ratios. Consider using Picture Vocabulary to show students. Have them record examples and the deﬁnition on their Student Journal. Note that the Picture Vocabulary Ratios includes three ways to write ratios (fraction bar included). 220

Read the following scenario to the class: Trixie is picking fruit from her fruit farm to sell at the local farmers’ market fruit stand. Each week, she picks different fruits depending on that week’s demand. She wants to keep a log of each week’s demand by representing the numbers of each type of fruit in the form of a ratio. Help her determine the ratio between each type of fruit she brings to the market each week. Display the Fruit Stand Card for week 1. Discuss the following questions with the students: a.

How many blueberries are there for week 1? There are 6 blueberries.

How many strawberries are there for week 1? There are 8 strawberries.

We want to show the relationship of strawberries to blueberries in week 1. (Write out, “There are ______ strawberries for every ______ blueberries.”) To model the relationship between strawberries and blueberries, what would we need to ﬁll in the blanks with? In the ﬁrst blank, we would need to put 8 because there are 8 strawberries. In the second blank, we would need to write 6 because there are 6 blueberries.

Explain the following to the class: Mathematicians call this relationship a ratio. Mathematicians write ratios in the order the ratio asks. In this scenario, we want to represent strawberries to blueberries; therefore, we write the number for strawberries in the ﬁrst spot and the number for blueberries in the second spot. We can also write the ratio for the information shown on week 1’s Fruit Stand Card in the following ways: “For every 8 strawberries, there are 6 blueberries”; “8 : 6”; “8-to-6”; or “There are 8 strawberries for every 6 blueberries.” © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

e. How can we draw a tape diagram to represent the relationship between strawberries and blueberries? Draw one tape diagram with 8 spaces to represent the strawberries. Below, draw a tape diagram that is only 6 spaces to represent the 6 blueberries. Model for students how to draw the tape diagram model for 8 strawberries to 6 blueberries. f. 3. 4.

DOK-1 What does each number in your ratio represent? Answers will vary based on the week the student is working on. For week 2, the 5 represents blackberries, and the 7 represents raspberries.

DOK-1 How would the ratio of fruit for week 3 change if I added one more strawberry to the picture? The number for the strawberries would be 6 instead of 5.

DOK-1 Does the larger number always have to go second in a ratio? No, the order depends on the order the fruit is listed in the ratio in the question. The number of items goes with the correct fruit listed.

DOK-1 How do you know which number to put first in the ____ : ____ ratio? Whichever fruit is listed first in the ratio goes in the first blank, and whichever fruit is listed second goes in the second blank.

e. DOK-2 Is your ratio comparing part to part or part to whole? Explain how you know. Answers will vary based on which scenario students are working on. In week 5, the ratio of blackberries to strawberries is a part-to-part ratio. We are comparing two different parts of the whole set of fruit.

7. 8.

Acceleration

FACILITATION TIP When you model the tape diagram, you can add letters in the spaces (s for strawberry and b for blueberry) for students who need a more concrete model.

Explain the following to the class: As we work through each week’s scenario, we will learn other ways to represent ratio relationships.

Give a Student Journal to each student and one set of Fruit Stand Cards to each group. Students will work collaboratively to represent the information on each Fruit Stand Card. They will record the number of each type of fruit, draw a tape diagram to show the number of each type of fruit, and then write the ratio describing their models in two different ways. Provide linking cubes to groups that are struggling. Have students model the ratio with linking cubes by linking the correct amount together for each fruit in the scenario. Then, students can draw a model of their linking cubes as a tape diagram on their Student Journals. As students are working together, monitor their learning, and ask the following questions to check for understanding:

Intervention

DOK-2 How would this ratio change if you were comparing part to whole instead of part to part? Answers will vary. When comparing part-towhole ratios, you will have the same ﬁrst number in the ratio but will need to add all of the fruits together to ﬁnd the total for the second number in the ratio.

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FACILITATION TIP Before having students collaborate on their Student Journals, model a few scenarios for them with the linking cubes.

FACILITATION TIP Before asking this yes or no question, ask "How does order of the numbers affect a ratio?"

FACILITATION TIP When explaining comparing part to whole, slow down to ensure that students understand how to calculate "the whole" and determine the correct "part" in these ratios.

Allow students enough time to collaborate with their groups to complete the remaining ratios and the reﬂection questions on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning. Notes

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Ratios, Rates, and Unit Rates Explore 1 – Ratios Math Chat • •

•

FACILITATION TIP Be prepared with some relevant everyday uses for ratios to engage students. (Dollars per hour, pizza slices per student, points scored per quarter in a game, teachers to students on a ﬁeld trip)

•

DOK-1 Describe in your own words what a ratio is. A ratio is a comparison relationship between two different things. DOK-1 List two different ways to represent a ratio. Students should recognize that they can represent a ratio using a verbal description, such as “For every…, there are…” or “There are ___ out of ____.” They can also use “_____ to ____” as well as “____:___.” DOK-2 If you were given a total of two fruits and the exact number of one fruit, how could you create a ratio between the two different fruits? First, subtract the number of the given fruit from the total in order to ﬁnd out the number of the second fruit. Then, use the number you just found and the number of the ﬁrst fruit in order to make a ratio between the two different fruits. DOK-2 Explain the difference between a part-to-part ratio and a part-to-whole ratio In a part-to-part ratio, we are looking for one part of the whole and then another part of the whole. In a part-to-whole ratio, we are comparing one part of the whole to the whole set. DOK-3 Where outside of the classroom would you use ratios? Ratios are used when mixing ingredients together for baking.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

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Ratios, Rates, and Unit Rates Explore 2 – Ratio Tables and Graphs ACTIVITY PREPARATION Students will reason, analyze, and create tables and graphs of ratios.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • •

1 Student Journal (per student) 1 Set of Purchasing Fruit Bushes Cards (per group) 1 Exit Ticket (per student)

• • •

Plan to divide the class into groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print one set of Purchasing Fruit Bushes Cards for each group. If desired, print the cards on card stock and laminate them for future use.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever visited a farm?; 2) If so, what was grown on the farm?; 3) What did you do on the farm?

Part I 1.

2. 3. STEMscopes Tip Communicate Math – Making Connections is located under the Communicate Math tab of the Teacher Toolbox. Students learn mathematical concepts by linking them to their prior knowledge and experiences. Teachers can emphasize the connections from this page to help students bridge their knowledge from concept to concept. Examples of possible connection types are provided.

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Read the following scenario to the class: Trixie wants to expand her fruit farm. She is preparing to plant more fruit bushes in order to create a bigger harvest. Trixie visits the local gardening store and sees that packages of fruit bushes are sold in different quantities. She must buy a full pack when purchasing. Help Trixie determine the amount of packages of bushes she needs to buy. Give a Student Journal to each student and the Purchasing Fruit Bushes Cards – Part I to each group. Students will work collaboratively to use the information on the Purchasing Fruit Bushes Cards – Part I to determine how many of each type of fruit bush comes in one pack. Then, they will answer questions and record their solutions in the tables on Part I of their Student Journals. As students are working together, monitor their learning, and ask the following questions to check for understanding: a.

DOK-1 What do the numbers in the ﬁrst row of your table represent? The numbers in the ﬁrst row of the table represent the number of bushes that come in one pack.

DOK-1 What is the ratio of raspberry bushes to blackberry bushes in one pack? The ratio of raspberry bushes to blackberry bushes in one pack is 6 to 4.

DOK-1 Why do both sides of the ratio need to change each time? Trixie wants to keep the ratios equivalent to the original ratio. Therefore, both sides must be increased by the same multiple.

DOK-2 By looking at the table, how can you determine if two numbers are in the same equivalent ratio? The two numbers must be in the same row to be in the same equivalent ratio.

Engage

Explore

Explain

Elaborate

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Part II 1.

2. 3.

Read the following scenario to the class: Trixie wants to use the ratios of fruit bushes in her current farm to determine the number of bushes she should plant in the expansion. Using the Purchasing Fruit Bushes Cards – Part II, determine the ratios for each comparison to help Trixie ﬁnd equivalent ratios of fruit bushes that she could plant at the farm. Give the Purchasing Fruit Bushes Card – Part II to each group. Students will work collaboratively to use the information on the Purchasing Fruit Bushes Cards – Part II to determine the ratio for raspberry bushes to blueberry bushes and the ratio for strawberry bushes to blackberry bushes Trixie has in her garden. Then, they will use these ratios to complete the ratio tables for each comparison and answer the questions that follow on their Student Journals. As students are working together, monitor their learning, and ask the following questions to check for understanding: a.

DOK-1 What is the ratio of raspberry bushes to blueberry bushes? The ratio of raspberry bushes to blueberry bushes is 9 to 10.

DOK-1 What is the ratio of strawberry bushes to blackberry bushes? The ratio of strawberry bushes to blackberry bushes is 8 to 6.

DOK-2 How can you determine the missing information in each table? To find the missing information, I can find the multiple used to increase one fruit bush in the row and use that number to multiply the other fruit bush to find the new number of bushes in that row.

DOK-1 How do you write ordered pairs from a table? To write the ordered pairs, you write the number in the first row of the first column and then write the number in the second row of the first column. This will give you the first ordered pair.

e. DOK-1 What do you notice about the graph of the ratio? The graph is a straight line that goes through the origin, (0, 0).

Intervention

Acceleration

FACILITATION TIP Before reading the scenario, ask the class 1) What types of fruit grows on bushes?; 2) Have you ever planted fruit bushes? 3) If so, what fruit bushes did you plant? FACILITATION TIP A quick review of vocabulary will support student success on Part II: ordered pair, origin, x-axis, y-axis, and linear. FACILITATION TIP

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Purchasing Fruit Bushes Cards are in color in the digital slide. Before printing copies, consider the image clarity.

FACILITATION TIP

Some students may need help with labeling the graph and writing the numbers f. DOK-1 How can you determine if a ratio is part of the equivalent ratios correctly. Determine beforehand how you for the comparison on the graph? The ratio would be on the line if it is an can support precise representations on the equivalent ratio. graphs. 5. Allow students enough time to complete their work and record their solutions on Part II of their Student Journals. 6. Allow students time to complete the reﬂection questions at the end of Part II. STEMscopes Tip 7. After the Explore activity, invite the class to a Math Chat to share their The Communicate Math – Questioning observations and learning. page, found under the Communicate Math Chat Math tab of the Teacher Toolbox, includes questioning strategies • DOK-1 If you were given a ratio of 3 strawberries for every 5 blueberries, how teachers can use to help challenge and could you determine the number of blueberries if the number of strawberries stimulate students' ability to clarify and increased to 12? I would need to determine the factor that 3 was multiplied by extend their mathematical thinking. to get 12, which is 4. Then, I would multiply 5 blueberries by 4 to get 20. This will Examples of possible questioning give an equivalent ratio of 12:20. types are provided. • DOK-2 How can you determine from a table if two numbers you are looking for are an equivalent ratio to what you have? In Part I, if the two numbers are not in the same row in a table, then they are not an equivalent ratio. In Part II, if the two numbers are not in the same column in a table, then they are not an equivalent ratio. FACILITATION TIP • DOK-2 How can you determine from a graph if a pair of numbers you are looking Challenge students to think of some for is part of an equivalent ratio? If the pair of numbers you are looking for is on the line, then it will be an equivalent ratio. If they are not on the line or only one is examples of the use of ratios in addition to recipes. Be prepared with some of your own on the line, then they are not an equivalent ratio. to help them notice ratios in the real world. • DOK-3 Where in the real world would you want to ﬁnd equivalent ratios? Encourage them to look for ratios over the Equivalent ratios can be helpful when looking at making larger quantities of a next few days and be prepared to share with recipe. the class. © Accelerate Learning Inc. - All Rights Reserved

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Ratios, Rates, and Unit Rates Explore 2 – Ratio Tables and Graphs Post-Explore 1. FACILITATION TIP Clarify your criteria for success on this Exit Ticket. Consider labels on the graph, graphing all ordered pairs listed, and/or including parentheses on the listed ordered pairs.

2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

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Ratios, Rates, and Unit Rates Explore 3 – Rates and Unit Rates ACTIVITY PREPARATION Students will discuss how they can determine the unit rate of a ratio.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • •

•

1 Student Journal (per student) 1 Exit Ticket (per student)

• •

Reusable •

Plan to divide the class into groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Have a projector ready to project Part I for a class discussion.

1 Projector (per class)

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before reading the scenario, ask the class 1) What types of berries grow on plants?; 2) Has anyone ever picked berries?; 3) If so, was it easy to pick the berries? Why or why not?

Part I 1.

2. 3.

STEMscopes Tip The Standards list is located along the menu bar. Here, a keyword can be entered to locate each standard. The search will result in a list of standards and direct links to the scopes where those standards appear. The standards are organized by grade level as well. Clicking on a standard within a grade level will also provide direct links to the scopes.

FACILITATION TIP Double number lines can be a very effective visual tool for modeling ratios. Consider taking time to review a separate example before starting with this scenario. 228

Read the following scenario to the class: Trixie spends a lot of time picking berries to sell at the market. Some berries are harder to pick than others. She is trying to decide if she needs to hire someone to help her with this task. You will be helping Trixie determine how long it takes her to pick each type of berry and the rate at which she picks them compared to the others. Give a Student Journal to each student. Project the table at the top of page 1 of the Student Journal for the class to discuss. Look at the row for strawberries. a.

DOK-1 How many baskets of strawberries did Trixie pick? Trixie picked nine baskets of strawberries.

DOK-1 How many minutes does it say it took Trixie to pick the baskets of strawberries? It took Trixie 45 minutes to pick the baskets of strawberries.

Explain the following to the class: Rate is a comparison with two different units of measure. We are looking at baskets per number of minutes, so we will ﬁnd Trixie’s rate.

DOK-1 If it took Trixie 45 minutes to pick all of the baskets of strawberries, what would be her rate of picking strawberries? Trixie’s rate is 9 baskets per 45 minutes.

e. Let’s ﬁnd how many minutes it took Trixie to pick one basket of strawberries by creating a ratio table. 4.

Discuss with students how to create a double number line for baskets of strawberries and minutes. They will use this information to determine how many minutes it would take Trixie to pick one basket. Model with students while discussing. a.

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Explain

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Acceleration

FACILITATION TIP One line is labeled “Baskets of strawberries,” and the other line is labeled “Minutes.” Label the double number line with “Baskets of strawberries” While some students may be able to solve and “Minutes.” the ratio questions without the double number line in these scenarios, remind them b. DOK-1 How many baskets of strawberries did Trixie pick? Trixie picked that it is a useful math tool that they need 9 baskets of strawberries. Model for students writing 9 on the double for future standards that may have more number line for “Baskets of strawberries.” complex data. Other students may have c. DOK-1 How many minutes did it take Trixie to pick 9 baskets of trouble spacing the numbers appropriately strawberries? It took Trixie 45 minutes to pick 9 baskets of strawberries. on the lines (show them how to ﬁnd half, Model for students writing 45 for the minutes on the double number line. third, fourth, etc.). d. We want to determine how many minutes it took Trixie to pick one basket of strawberries. Where would we write one basket on your double number line? On the double number line, one would be written before the 9 on the number line for baskets of strawberries. Model writing 1 basket of strawberries on the double number line.

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e. DOK-1 What do we need to do to ﬁnd how many minutes it took Trixie to pick one basket of strawberries? Since Trixie picked 9 baskets in 45 minutes, we would need to divide both numbers by 9 so we can get 1 basket. f.

DOK-1 How many minutes did it take Trixie to pick one basket of strawberries? Trixie picked one basket of strawberries in ﬁve minutes. Model for students where to write ﬁve minutes on the double number line model. Students will write the rate per one basket of strawberries under the double number line.

h. Explain the following to the class: This is a special type of rate. Mathematicians call this rate the unit rate. i. DOK-2 Why do you think this is called the unit rate? Accept all answers for one minute. Then, come to a class consensus that unit rate is a special rate where one of the units has a value of one. 5. 6.

Students will collaborate with their groups to determine the rate and unit rate for each type of berries using ratio tables, double number lines, and tape diagrams. As students are working together, monitor their learning, and ask the following question to check for understanding: a.

STEMscopes Tip The Visual Glossary, located under the Scopes tab, provides a variety of instructional resources. Browse topics alphabetically in English or Spanish. Each topic includes a visual and/or video featuring key vocabulary and concepts. The visuals include text and a speech button with narration. The videos, featuring real-world examples, are 3–15 seconds in length.

DOK-1 When ﬁnding the rate per one basket, why do you divide each part by the number of fruits rather than by the number of minutes? You are trying to ﬁnd how long it takes for one basket of fruit. Therefore, fruit needs to be divided so there is only a 1 representing it.

Students will complete the models showing the unit rate for each type of berry and then answer the reﬂection questions at the end of Part I on their Student Journals. After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 What is your deﬁnition of a unit rate? A unit rate must have 2 different units, and one of the values must be a 1. • DOK-3 Which strategy do you prefer using to solve for unit rate? Explain. Responses and explanations will vary. • DOK-2 What is the unit rate for picking a basket of blackberries? The unit rate for picking a basket of blackberries is 5 minutes per basket. • DOK-3 How could you determine how many minutes it took Trixie to pick 7 baskets of blackberries? Multiply the number of baskets times the unit rate for picking blackberries, which is 5 minutes. 5 × 7 = 35, so it took Trixie 35 minutes to pick 7 baskets of blackberries. •

FACILITATION TIP Emphasize to students that they need to be able to create a model on the Exit Ticket for this scope (table, tape diagram, or double number line). Encourage students to evaluate which model works best in which situations. Every tool is not for everybody, and every student may not master every tool.

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Ratios, Rates, and Unit Rates Explore 3 – Rates and Unit Rates Part II 1.

FACILITATION TIP

Read the following scenario to the class: Trixie wants to know how much she earns for each basket of berries she picks and the total amount she earns for each type of fruit. Help Trixie determine the rate and unit rate for each basket of berries using ratio tables, tape diagrams, and double number lines. Students will collaborate with their groups to determine rates and unit rates for each berry type using ratio tables, tape diagrams, and double number lines. As students are working together, monitor their learning, and ask the following questions to check for understanding:

This ﬁrst question is an important one for students who may get overwhelmed by the information. It gives them a good starting spot. Ask, "What do you know?" FACILITATION TIP Be prepared for a wide variety of successful strategies that students use to solve for unit rates.

FACILITATION TIP As a follow-up question to this one about 15 baskets, ask "How can ﬁnding a unit rate help you make smart money decisions in real life?" 4.

FACILITATION TIP As a follow-up, help students determine if the unit of 1 must be in speciﬁc order in a unit rate (ﬁrst or second).

FACILITATION TIP Before having students complete this Exit Ticket, clarify your criteria for success. Some students might be able to show the solution with more than one model. Some students may calculate the unit rate without a model. Consider whether students can choose which model to use and how it is labeled.

DOK-1 What information are you given? I know the number of baskets Trixie sold and the total amount she made.

DOK-2 What is the unit rate for selling one basket of raspberries? The unit rate for one basket of raspberries is $2.50.

DOK-3 Explain the strategy you are using to solve for the amount of money earned by selling one basket of blueberries. Answers will vary based on the strategy they use. I used a ratio table to solve for the unit rate of one basket of blueberries. I know that 18 baskets cost $67.50, and that 18 ÷ 18 = 1 basket. Therefore, I need to divide $67.50 by 18 to ﬁnd the unit rate. 67.50 ÷ 18 = $3.75, so one basket of blueberries costs $3.75.

DOK-3 If I know the unit rate for the amount of money earned for one basket of strawberries, how can I ﬁnd the total amount of money earned for 15 baskets of strawberries? If I know the amount of money earned by selling one basket, I can multiply that amount by 15 to ﬁnd the total she would earn when selling 15 baskets of strawberries.

Students will complete the models showing the unit rate for selling baskets of each type of berry and then answer the reﬂection questions at the end of Part II on their Student Journals. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 What is the difference between a rate and a unit rate? A rate is a comparison of two different units of measure. A unit rate is a special rate where one of the units has a value of one. • DOK-1 How can you use a rate to determine the unit rate? You can use the rate to see what you need to divide by to get the unit with a value of one to be one. Then, you divide the other part of the rate by the same number. • DOK-3 Explain where in the real world you might ﬁnd unit rates necessary. Unit rates are helpful when ﬁnding the better price on items at the store. •

1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

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Ratios, Rates, and Unit Rates Explore 4 – Comparing Ratios and Using Rates to Make Predictions ACTIVITY PREPARATION Students will use multiple representations of ratios and rates to compare quantities and make predictions.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Berry Farms Cards (per group) 1 Exit Ticket (per student)

Reusable •

Divide the class into groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Berry Farms Cards for each group of students. Cut out and place each set in a resealable bag for easy distribution. If desired, print the cards on card stock and laminate them for future use.

1 Resealable bag (per group)

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Consider asking students if they have purchased picked berries from the farmers market. Ask, "How are they usually packaged?"

Part I: Comparing Ratios 1.

2. 3. FACILITATION TIP Determine ahead of time how much support or coaching your students may need to determine how to ﬁnd the missing numbers on the tables.

4. 5.

FACILITATION TIP

Consider posting this question about the order of values in a ratio. Discuss the reasons before students begin collaborating.

DOK-1 Is the order of the numbers in the ratio important? Yes. The order of the numbers shows how the quantities in the ratio are being compared.

DOK-2 How can you use the table to determine the ratio of blueberries to strawberries in one berry bag? You can make a ratio using the number of blueberries and strawberries from any row in the table. By simplifying the ratio, you can determine how many blueberries and strawberries are in one of the berry bags.

DOK-2 How can you use ratios to determine which berry bag has more blueberries per strawberry? Writing the ratio of blueberries to strawberries for each berry bag and then ﬁnding the equivalent ratios will allow you to compare the ratios.

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Read the following scenario to the class: Trixie wants to sell her fruit at the local farmers’ market. She packages the berries into bags that contain a mixture of fruit. You will be helping her determine how many berries she sells so she will know how many bags to prepare for the next time she goes to the farmers’ market to sell her fruit. Give a Student Journal to each student. Explain to students that they will be working with their groups to compare ratios of berries in each berry bag. Have students ﬁll in the missing numbers in the tables and answer the questions in the tables. Then, students will use the information to compare the berry bags. Monitor and assess student understanding as each group collaborates by asking the following guiding questions:

Engage

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Intervention

Acceleration

Part II: Making Predictions 1.

2. 3.

6. 7.

Read the following scenario to the class: Trixie and Aanya are working with Blooming Berry Ranch and Snazzy Fruit Farm, and they are each preparing for the annual berry festival at the end of the summer. They have to prepare berry bags and deliver the bags to each farm. Trixie needs your help predicting how many miles she can drive and Aanya needs your help predicting how many miles she can bike in a given time. Give a set of Berry Farms Cards to each group of students. Explain to students that they will be working with their groups to read the scenarios on the Berry Farms Cards and will use the scenarios to make predictions. Have students read the scenarios on the Berry Farms Cards and collaborate with their groups to represent the scenarios using double number lines. Students will also use the information to make predictions. Actively monitor and assess students’ understanding as they are collaborating with their groups by asking the following questions: a.

DOK-2 How is Trixie’s rate for two hours represented on the double number line in Berry Farms Card 1? How can you use this to determine the unit rate? Trixie’s rate for two hours is represented as 60 miles, and the unit rate would be 30 miles per 1 hour.

DOK-2 What operation is being used to ﬁll in the remaining ratios on the double number line? Explain. Multiplication is being used to create equivalent ratios that are placed on the double number line.

Allow students enough time to record all of their work and answer the reﬂection questions on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 What is the scale factor? What do scale factors represent in this situation? The scale factor is the number that is multiplied by both values of a ratio to create an equivalent ratio. • DOK-2 How did you use the double number line to make predictions about the rates? I used the rate that was given in the scenario and created the double number line using the scale factor to get the rest of the numbers that needed to be included to create equivalent ratios. • DOK-2 Give an example of why someone would want to make predictions about rates. If someone needed to complete a task or drive a certain amount of miles in a speciﬁed time period, they could use equivalent ratios to determine if they would be able to complete their task on time. •

FACILITATION TIP Depending on time, consider doing Part II more collectively as a class. There are only two Berry Farm Cards; they could be projected for students to work on independently or in their table groups.

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FACILITATION TIP Reflection Question 3 could provide a useful quick check to monitor how students are progressing toward mastering the concept. FACILITATION TIP Scale factor is likely a new term for many students. Take time to define, model how to find it, and use it. Refer to Picture Vocabulary for a visual representation.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook. Notes

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Ratios, Rates, and Unit Rates Explore 5 – Solving Proportions ACTIVITY PREPARATION Students will use equivalent ratios and rates to solve proportions.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Set of Farmers’ Market Cards (per group) 1 Set of Berry Festival Cards (per group) 1 Exit Ticket (per student)

•

Reusable •

2 Resealable Bags (per group)

Divide the class into groups of 3 or 4 students. Print a Student Journal and an Exit Ticket for each student. Print a set of Farmers’ Market Cards for Part I for each group of students. Cut the cards apart, and place them inside a resealable bag labeled “Part I.” If desired, print the cards on card stock and laminate them for future use. Print a set of Berry Festival Cards for Part II for each group of students. Cut the cards apart, and place them inside a resealable bag labeled “Part II.” If desired, print the cards on card stock and laminate them for future use.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Print and project this scenario for students. Conduct a read aloud and coach students to read through it more than once. Guide them to locate the essential phrases, terms, and values. FACILITATION TIP Consider color coding the parts of the Student Journal for Part I and Part II. Distribute the pages as needed to students (Part I: page 1 and 2, Part II: pages 3–5)

Part I: Solving Proportions Using Ratios 1.

2. 3. 4.

FACILITATION TIP Depending on your students, provide some simple examples and model how to solve proportions before students collaborate.

Read the following scenario to the class: Trixie is working with two local farms to set up booths at this weekend’s farmers’ market. Each farm is selling the same types of berries to their customers. However, each farm has a different number of berries in each pack, and they have different prices for each type of berry. While shopping at the farmers’ market, you want to ﬁnd the best deal. You will need to compare each pack of berries and their cost to determine which farm is the best one to purchase berries from. Give a Student Journal to each student. Give a set of Farmers’ Market Cards to each group. Explain to the class that they will be working with their groups to solve proportions using ratios. Inform students that in the previous Explore activities, they worked with equivalent ratios and rates and that a proportion is a type of equation that shows that two ratios or rates are equivalent. Have students work in groups and read the Farmers’ Market Cards. They will use the information on the Farmers’ Market Cards to solve proportions about berries and the farmers’ market. Students will take what they have learned in the previous Explore activities and will use scale factor to solve the proportion. (Note that students may need to use models to help them solve proportions). Actively monitor and assess students’ understanding as they are collaborating with their groups by asking the following questions: a.

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DOK-2 How can we use the information in the Farmers’ Market Cards to set up a proportion? A ratio is similar to a fraction in that it shows a relationship between two quantities. I can identify the quantity that will represent the numerator and the quantity that will represent the denominator. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

2. 3.

6. 7.

Acceleration

DOK-2 How can we use a proportion to solve for Shaniya’s and Perry’s profits? I know that we are solving for Perry’s profits if Shaniya’s profits are $36. We can set up a proportion using the information we have; the ratio given in the scenario could be the first ratio, and we can use the variable p to represent what we are looking for and include it in the second ratio. After we include the information in the equivalent ratios, we will use scale factor to find Perry’s profits.

Allow students enough time to record their work for each table on their Student Journals and to record their solutions.

Part II: Solving Proportions Using Unit Rates 1.

Intervention

Read the following scenario to the class: Since the farmers’ market was a success, Trixie is now working with Blooming Berry Ranch and Snazzy Fruit Farm, and they are each preparing for the annual berry festival at the end of the summer. They will be packaging the fruit that each farm grows and will sell the packages of fruit. Trixie needs your help to determine how many packs of strawberries, raspberries, blueberries, and blackberries they will have produced and which farm will offer the best price on fruit for the berry festival. Give a set of Berry Festival Cards to each group of students. Explain to the students that they will be working with their groups to solve proportions using unit rate to determine the number of strawberries, raspberries, blueberries, and blackberries each farm produces. Have students solve proportions using unit rate and determine the cost of strawberries, raspberries, blueberries, and blackberries per pack. After students determine the cost per pack, they should determine which farm offers the best deal on each fruit. Actively monitor and assess students’ understanding as they are collaborating with their groups by asking the following questions: a.

DOK-2 How can you use a proportion to solve for the unit rate? Find the unit rate of the rate you know, and then write equivalent rates.

DOK-2 How can you determine which farm is offering the best deal on each fruit? The farm that is offering the lowest price per berry is the farm that is offering the best price on the fruit.

FACILITATION TIP

RATIOS, RATES, AND UNIT RATES

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Project this scenario for students to read along with you before distributing the Student Journal pages for Part II.

FACILITATION TIP Select one of the proportions from the Berry Festival Cards to model how to solve for students.

FACILITATION TIP Engage your students with some more relevant real-world applications that they can relate to. Unit rates can illuminate hidden costs. Unit rates can also clarify rates of pay and speeds. Students may be passionate about fairness, and unit rates can help them avoid being "tricked" or "ripped off."

Math Chat DOK-2 How are proportions helpful in finding an equivalent ratio or rate? Proportions are helpful in finding an equivalent ratio or rate because I can turn the rate or ratio into a fraction and use multiplication or division to find an equivalent value. This will make it easier to compare or predict rates and ratios because then the numerator or denominator can have the same value as another proportion. • DOK-2 What strategies were used to solve the proportions? First, I used the given ratios to set up as fractions and used scale factors. Then, when I was solving for the unit rate, I used division to find the unit rate, or the missing value. • DOK-2 Give an example of how proportions are used in the real world. An example of using proportions in real life is when we put gas in our cars. There is a relationship between the number of gallons of fuel we put in the tank and the amount of money we will have to pay. In other words, the more gas we put in, the more money we’ll pay. •

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Ratios, Rates, and Unit Rates Explore 5 – Solving Proportions Post-Explore 1. 2. 3. 4.

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Ratios, Rates, and Unit Rates Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Ratios Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Ratio Tables and Graphs Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 3

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Rates and Unit Rates

Interactive Notebook

Show What You Know, Part 4

A cut-and-glue activity to process learning that can be added to a notebook for future reference

Comparing Ratios and Using Rates to Make Predictions

Independent practice assignment that gives students an opportunity to demonstrate their learning

Show What You Know, Part 5 Solving Proportions Independent practice assignment that gives students an opportunity to demonstrate their learning

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review A quick story to engage student interest along with four problems over previously learned skills

RATIOS, RATES, AND UNIT RATES

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Intervention and Assessment SMALL-GROUP PLANNER 1

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

RATIOS, RATES, AND UNIT RATES

Ratios, Rates, and Unit Rates

3 240

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER Use this template to decide how to assess your students for concept mastery. Depending on the format of the assessment, you can identify prompts and intended responses that would measure student mastery of the expectation. See the beginning of this scope to identify standards and grade-level expectations. Key Concepts I can interpret and explain the meaning of numerical statements of inequality and their relative position of two integers positioned on a number line.

What prompts will be used?

What does mastery look like?

RATIOS, RATES, AND UNIT RATES

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I can recognize that rational numbers are numbers that can be written as a fraction with integers for both the numerator and the denominator. I can create a comparison statement of rational numbers. I can compare rational numbers. I can order rational numbers. I can use inequality symbols appropriately in solutions. I can compare numbers according to their absolute values. I can place rational numbers on a number line according to their values. I can identify rational numbers and their absolute values. I can interpret statements as comparison statements or order statements.

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SCOPE 1

Percents Scope Introduction SCOPE SUMMARY

Student Expectations

6.NR.4.6 Calculate a percent of a quantity as a rate per 100 and solve everyday problems given a percent.

Students will reason about the meaning of percentages used in contextual situations. They will gain proficiency with benchmark fractions and percent equivalents, (such as __1 = 25% 4 and __1 = 50%). Students will solve percent problems that involve finding the percent of a 2 whole amount, finding the whole when given a part and the percent, and finding the percent when given the whole and a part. They will use tape diagrams, tables, and double number lines as strategies which relate ratios to percents and that reinforce that a percent is a rate per 100.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In ﬁfth grade, students built on multiplicative comparisons to add and subtract fractions with unlike denominators by ﬁnding equivalent fractions using the least common multiple. Fifth-grade students plotted points on the coordinate plane, and they generated and graphed numerical patterns. Prior to this scope, sixth-grade students studied the concept of a ratio as an association between two quantities, and they used ratio language to describe ratio relationships. Tape diagrams, double number lines, tables, and graphs are featured strategies applied when solving mathematical and real-world problems. Experiences with ratio and unit conversion tables provide the foundation for understanding percents, which begins in sixth grade.

Sixth graders will apply ratio and rate reasoning in an upcoming scope that explores measurement conversions. In seventh grade, students will extend their understanding of ratios, rates, and percents to explore proportional relationships, scale drawings, using similar triangles to compare slope, and calculating unit rates.

Before beginning the lesson, the students’ prior knowledge is assessed to address any student needs and skills needed for content mastery. Students are assessed to determine if they can: •

interpret multiplication as scaling.

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope.

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

determine the sale price when given a percent.

Students move on to the Explore activities and come back at the end of the explorations to repeat the activity and discuss what they found.

If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. 242

Engage

Explore

Explain

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Intervention

Acceleration

PERCENTS

Home

Represent Percents Using a Hundreds Grid In this exploration, students will solve a scenario about determining how many lawns David is moving in each neighborhood. Students will: •

deﬁne and understand percentages as expressions of rate per 100.

•

use a hundreds grid as a model to represent percents and will determine equivalent fractions and decimals.

Explore 2

Explore 1

EXPLORE ACTIVITIES

Explore 3

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

Solving Percent Problems Using Benchmark Fractions and Percents In this exploration, groups of students will solve a scenario about helping a lawn care company understand their customer’s request so that they can make each person’s yard look more attractive. Students will: •

solve percent problems using benchmark fractions and percentages.

•

represent the problems with tape diagrams and double number lines.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

Finding the Price and Discount In this exploration, students will solve a scenario helping David, from the previous exploration, to determine the original price, the sale price, and the discount price of items that are for sale to determine if he is being offered a good deal. Students will: •

use various strategies to ﬁnd the whole, percent, and part when solving percent problems about price and discount.

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve. Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________

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PERCENTS

Percents Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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ACCESSING PRIOR KNOWLEDGE Students will dialogue with classmates about their understanding of the prior standard through a Four Corners discussion. This element is designed to uncover student misconceptions; it should not be taken for a grade. 5.NR.3.5: Explain why multiplying a whole number by a fraction greater than one results in a product greater than the whole number, and why multiplying a whole number by a fraction less than one results in a product less than the whole number and multiplying a whole number by a fraction equal to one results in a product equal to the whole number.

Materials Printed •

1 Set of Four Corners Slides (per class)

Preparation • •

Print one set of the Four Corners Slides. Hang the slides in four separate areas of the classroom, easily visible to all.

Procedure and Facilitation Points 1. 2. 3. 4.

Ask the students to look at the four corners and think about which corner image best explains multiplication as scaling. Allow 2 minutes of thinking time. Ask students to move to the corner image they chose. Ask each group to discuss why they chose the image with each other. Allow 2–5 minutes of discussion at the corner images. After students have discussed why they chose their answer, talk about the answer with the class, and allow students to explain their representation of the problem. If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

• • •

In addition to hanging the slides in the corners, project them for students to view from their seats during the 2-minute think time. Engage students by having them record some thoughts on paper before they move around the room. FACILITATION TIP Depending on your students, you might choose to review scaling using the Visual Glossary before or after this APK. FACILITATION TIP

Identifying Misconceptions •

FACILITATION TIP

Slide 1: Students who choose slide 1 think that a product is always greater than the multiplicands. They don’t understand that multiplying a whole number with a fraction less than 1 will result in a product less than the whole number. Slide 2: This is the correct slide. 6 Slide 3: The students who choose this slide don’t understand that __6 is actually equal to 1, and the product of a whole number and 1 is the same whole number. Slide 4: The students who choose this slide are multiplying the whole number with both the numerator and the denominator, which is an incorrect way of multiplying whole numbers and fractions.

Project some guiding questions or sentence frames for students to use as needed for corner discussions. "I chose this image because _____ scaling means__________." FACILITATION TIP In addition to the Foundation Builder, model for students how multiplication between to numbers can often be visualized as "of." 4 In these examples,you could say "7 'of' __6 is 6 __ less than 7," "5 'of' 6 is less than 5." Meaning 6 4 7 groups of __6 or 5 groups of __6. 1

A very simple example is "__2 of 8 is 4 or 1 __ *8 = 4". 2

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Percents Hook ACTIVITY PREPARATION Students will determine the sale price when given a percent.

Materials

Preparation

Printed •

• • •

1 It’s Your Choice (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project It’s Your Choice for the whole class to view. Prepare to introduce the scenario and encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION POINTS Part I: Pre-Explore 1. FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) Do you own a game system?; 2) If so, what brand is it?; 3) Why did you want that particular game system?

FACILITATION TIP Project this essential question for students before showing It's Your Choice. When you project It's Your Choice, provide students a chance to vote (silently with hand signals, tell them to record their vote on their own paper as well) for the best deal.

246

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Explore

Explain

Elaborate

Evaluate

Explain to students that Oakley found two stores that have a sale on the game system. She cut out their ads. Help her decide which is the best deal. Discuss the following questions: a.

DOK-1 What difference do you see with how the two sales are advertised? One store has a dollar amount off, whereas another has a percentage off.

DOK-1 What information will we need to know in order to ﬁnd the best price? We will need to know the original price, the discount, and the discounted price.

Acceleration

FACILITATION TIP To further engage students, consider these questions: "Why do you think some stores show percent off rather than dollar amount off? Which sale price is easier for most consumers to calculate mentally? How often do you or your family use a calculator when out shopping? How can being able to quickly compute percentages help you in the world outside of math class?" FACILITATION TIP

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Intervention

PERCENTS

Home

Show the Phenomena Video again, and restate the problem. Refer to It’s Your Choice, and discuss the following questions: a.

DOK-1 How can you determine the discounted price at E-Buy? You would subtract the discount amount from the total. $300 – $75 = $225

DOK-1 How can you determine the discounted price at S Mart? Answers will vary. First, you need to ﬁnd the amount of the discount. You can use a double number line to ﬁnd that 30% of 300 is 90. Then, we would subtract that from the total. $300 – $90 = $210

DOK-1 Which store has the better price? With a sale price of $210, S Mart has the better price.

Some students may note that in addition to this information, we will need to know shipping costs and times.

FACILITATION TIP Ask students to explain how to ﬁnd the solutions more than one way. Challenge them to ﬁnd as many ways possible to prove their answers.

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Percents Explore 1 – Represent Percents Using a Hundreds Grid ACTIVITY PREPARATION Students will deﬁne and understand percents as expressions of rate per 100. They will use a hundreds grid as a model to represent percents and will determine equivalent fractions and decimals.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.6 Attend to precision.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Percent Work Mat (per group) 1 Set of Mowing Cards (per group) 1 Exit Ticket (per student)

•

Reusable • • • •

1 Dry-erase marker (per group) 1 Clear sheet protector (per group) 2 Quart-sized resealable bags (per group) 1 Set of colored pencils (per group)

• • •

Plan to divide the class into groups of 3 or 4 students. Print a Student Journal and an Exit Ticket for each student. Print a Percent Work Mat for each group on card stock for durability. Place it inside a clear sheet protector to create an erasable surface. Print a set of Mowing Cards for each group of students. If desired, laminate the cards for future use. Cut the cards apart. Place the Part I cards inside a quartsized resealable bag labeled “Part I” and the Part II cards inside a quart-sized resealable bag labeled “Part II.” Gather enough dry-erase markers for each group to have one. Prepare a set of colored pencils for each group. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore

PROCEDURE AND FACILITATION POINTS drop-down menu and can be digitally assigned to students. (Hundreds Board) Part I: Understanding Percents Using a Hundreds Grid FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever thought about starting a business?; 2) If so, what type of business would you start?; 3) Do you think you would enjoy having a business of your own or would it be too much work?

2. 3.

FACILITATION TIP

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Depending on your students, you might break this down into separate steps. Some students will need the simple reminder about decimal points in percents. Guide students through visually modeling values as decimals several times and then as a percent several times.

4. 5.

Read the following scenario to the class: David just started a lawn-mowing business. He schedules 100 customers each week in various neighborhoods, with some neighborhoods having a higher percentage of lawns to mow than others. David needs your help determining how many lawns he is mowing in each neighborhood. Give the Percent Work Mat, a dry-erase marker, and a set of colored pencils to each group. Explain to students that a percent is a special ratio that means “out of 100.” It is measured by the number of units as compared with 100. A percent is just another way to show a part-to-whole ratio. A percent can be represented as a fraction, decimal, or ratio since all of these represent a part-to-whole. We will be using the hundreds grid on the Percent Work Mat to represent percents. Guide students in representing a fraction and decimal as a percent. Have 1 students represent __4 and 0.03 on the Percent Work Mat. Actively monitor and assess students’ understanding as they collaborate with their groups by asking the following questions: a.

Engage

Explore

Explain

Elaborate

Evaluate

is worth 1 hundredth (____ , and the word percent means “per hundred.” 100 )

Intervention

Acceleration

PERCENTS

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DOK-2 How did you represent __4 on the Percent Work Mat? I shaded 1 25 out of every 4 squares. There were 25 out of 100 (____ squares that are 100 ) shaded on the hundreds grid.

DOK-2 What percentage of the hundreds grid was shaded? Since 1 percent is equal to the value of one unit in the grid and 25 units were shaded, 25% of the hundreds grid was shaded.

DOK-2 How did you represent 0.03 on the Percent Work Mat? I shaded 3 3 squares, and there were 3 out of 100 (____ squares that were shaded on 100 ) the hundreds grid.

FACILITATION TIP If needed have students shade a few simple 1 examples before having them shade __4 and 0.03. Start with 0.1, 0.01, 1.0, 0.5, 0.05, etc to assess prior knowledge.

e. DOK-2 What percentage of the hundreds grid was shaded? Since 1 percent is equal to the value of one unit in the grid and 3 units were shaded, 3% of the hundreds grid was shaded. 6. 7. 8.

10. 11.

Give a Student Journal to each student. Give a set of Part I Mowing Cards to each group. Explain to the class that they will be working with their groups to read each Mowing Card and will use the information on the card to create a model using a hundreds grid. Encourage students to use the Percent Work Mat to assist them with determining the percent that is represented for each scenario. They will then collaborate with their groups to determine how the percent is written as a part-to-whole ratio in fraction and decimal form. Once they shade their models on the Percent Work Mat, they will show their work by shading the models found on the Student Journal. They will then use their models to complete the corresponding table of information based on the Mowing Card. Instruct students to simplify each fraction into an equivalent fraction, if possible. Actively monitor and assess student understanding as they collaborate with their groups by asking the following questions: a.

DOK-1 What percent is being represented in this scenario? Answers will vary based on the Mowing Card. Green Lawn Care mowed 76% of lawns.

DOK-2 How can we represent the lawn care company’s percentage of lawns mowed using the hundreds grid? Answers will vary based on the Mowing Card. Students should represent the percent of lawns mowed by the lawn care company on the hundreds grid. They should shade in 1 square, or 1%, for each lawn mowed.

DOK-2 How can we represent the lawn care companies’ number of lawns mowed as a part-to-whole ratio in fraction form? Answers will vary based on the Mowing Card. The model shows me that 76 hundredths out of 100 hundredths is being represented. The numerator of the fraction is 76 since that is the part, and the denominator will be 100 since that is the whole of a percent. That would make the 76 fraction ____ . 100

DOK-2 Can you simplify the fraction into an equivalent fraction? Explain. 76 Answers will vary based on the Mowing Card. Yes, the fraction ____ can 100 19 be simpliﬁed to ___ if we divide the numerator and denominator by 4. 25

e. DOK-2 How can we represent the percent of lawns mowed as a part-towhole ratio in decimal form? Answers will vary based on the Mowing Card. Since a decimal represents hundredths, 76 hundredths would be represented as 0.76 in decimal form. 12.

Allow students enough time to discuss the information on each Mowing Card

FACILITATION TIP Part I Mowing Cards have colorful digital images. Before printing several copies, consider editing or projecting them on your own screen. FACILITATION TIP Have a student volunteer come up to demonstrate how to precisely shade the models on paper. FACILITATION TIP Before instructing students to simplify, be prepared to do a quick assessment of ﬂuency with this skill and/or conduct a review. STEMscopes Tip The Planner, accessed along the menu bar, provides a calendar planning tool for teachers. Download, print, save, or share your plans. Use the Elements tab on the left to access grade-level scopes and virtual-learning options with embedded links to all scope elements. Drag the elements you want to implement into the calendar, and click on each element to enter element details and personal planning notes.

FACILITATION TIP Student experience with ratios in fraction form may be limited. They may need to be prompted to the connection between decimals/fractions and ratios.

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Percents Explore 1 – Represent Percents Using a Hundreds Grid and to record their work and answer the reﬂections questions at the end of Part I on their Student Journals. Part II: Using a Hundreds Grid to Represent Percents FACILITATION TIP Take time to demonstrate a shading strategy or have a student come up to the front to show accurate shading skills. Colored markers and pencils may cause mistakes to be diﬃcult to erase and symbols or letters will take some time for students to draw.

1. 2.

Give a set of Part II Mowing Cards to each group. Explain to students that they will work with their groups to read each Mowing Card and will use the hundreds grid to interpret the percentage into the number of lawns David is mowing in each neighborhood each week before recording their individual work on their Student Journals. As students are collaborating on their work, monitor their understanding by asking the following guiding questions: a.

DOK-1 What does the hundreds grid represent in relation to this scenario? The hundreds grid represents the 100 lawns David mows each week.

FACILITATION TIP

Be sure to eliminate confusion between the different uses of the word “block” in this scenario. Some students may be thinking a “block” is like a “neighborhood.”

DOK-1 What does each block in the hundreds grid represent? Each block represents a lawn David mows.

DOK-3 How can we represent a percentage of a whole using the hundreds grid? Since there are 100 blocks on the grid and 100 lawns that need to be mowed, I will shade in the percentage for each neighborhood on the hundreds grid. For example, if David mowed 24% of the lawns for that week in a given neighborhood, then I will shade in 24 boxes. If David mowed 62% of the lawns in a given neighborhood, then I will shade 62 boxes.

DOK-2 How can you represent each neighborhood on the hundreds grid? I can shade each neighborhood with a different design or color. For example, one neighborhood can be solid. Another neighborhood can have dots, and the last neighborhood can have stripes. Another option is that we can write the ﬁrst letter of the name of each neighborhood in the number of boxes it represents.

e. DOK-3 How can we determine the missing percentage of the week for the remaining neighborhood? After we shade or design the boxes for the ﬁrst two neighborhoods, the percentage of the lawns in the last neighborhood will be the remaining number of boxes. f.

FACILITATION TIP If you have whiteboards with 10X10 grids, consider letting students shade each scenario on those and then record only the units, fraction, and decimal form on paper.

DOK-2 How can we represent the percentage as a fraction? The fraction will be the part of the lawns mowed that we shaded out of the 100 total lawns. For example, if the percent is represented with 24 boxes shaded 24 out of 100, then the fraction will be ____ . 100

DOK-1 How can we ﬁnd the number of lawns mowed in each neighborhood each week using the hundreds grid? The number of lawns mowed in each neighborhood will be represented by the number of boxes shaded in the hundreds grid for that neighborhood. If 24 boxes, or 24%, of the hundreds grid was shaded for a given neighborhood, that means David mowed 24 lawns in that neighborhood.

Allow students enough time to solve each of the Mowing Cards and to represent all of their individual work and reﬂections at the end of Part II on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 How do you represent a fractional percent between 0% and 1%? You shade part of one square since a fractional percent means a percent value (a number out of 100) in mixed number or fraction form. • DOK-2 How do models help in understanding percents? The hundreds grid allows •

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you to see what part of the whole is represented by the percent and how it relates to a fraction or decimal ratio when the whole is made up of 100 hundredths and the part is represented by the shaded model. DOK-2 How are percents, fractions, and decimals related to one another? Percents, fractions, and decimals are all related to each other in that they each show part-to-whole relationships or ratios. DOK-3 How did the value of the whole change when using a hundreds grid to represent percents, decimals, and fractions? The value of the whole for a hundreds grid changed from 100 to 1 when representing percents, fractions, and decimals because each unit in the grid becomes 1 hundredth to show that 100 100 hundredths equals 1 whole, 100 percent, or ____ . 100

Post-Explore 1. 2. 3.

FACILITATION TIP Have students complete the Exit Ticket to formatively assess their understanding of the concept. Be sure students read the percent question at the bottom of the Exit Ticket before they Complete the Anchor Chart as a class. begin shading and calculating. Have each student complete their Interactive Notebook. Notes

251

PERCENTS

Percents Explore 2 – Solving Percent Problems Using Benchmark Fractions and Percents ACTIVITY PREPARATION Students will use benchmark fractions and percents to solve percent problems and will represent the problems with tape diagrams and double number lines.

Standards for Mathematical Practice • • • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.6 Attend to precision.

Materials

Preparation

Printed • • • • •

• • •

1 Student Journal (per student) 1 Set of Landscaping Mansions Cards (per group) 1 Tape Diagram Work Mat (per group) 1 Double Number Line Work Mat (per group) 1 Exit Ticket (per student)

•

Reusable • • •

•

1 Dry-erase marker (per group) 2 Clear sheet protectors (per group) 1 Resealable bag (per group)

Plan to divide the class into groups of 3 or 4 students. Print a Student Journal and an Exit Ticket for each student. Print a set of Landscaping Mansions Cards for each group. Cut out the cards, and place them in a resealable bag for each group. If desired, print the cards on card stock and laminate them for future use. Print a Tape Diagram Work Mat for each group. Place the Tape Diagram Work Mat in a clear sheet protector. If desired, print it on card stock and laminate it for durability. This will be used for Part I and Part II of the Explore activity. Print a Double Number Line Work Mat for each group. Place the Double Number Line Work Mat in a clear sheet protector. If desired, print it on card stock and laminate it for durability. This will be used for Part II of the Explore activity. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Number Lines)

PROCEDURE AND FACILITATION POINTS Part I: Benchmark Fractions and Percents Using Models FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever done any yard work?; 2) If so, what did you do?; 3) Did you enjoy working in the yard? Why or why not?

2. 3. FACILITATION TIP Use Picture Vocabulary to help students record examples and diagrams of benchmark fractions and percents. Continuously review that 100% = the whole of a sample. 252

Read the following scenario to the class: Mowing Mansions Lawn-Care Company offers mowing services and landscaping services. Landscaping is a process of making a yard more attractive by altering the yard’s design. This can be done by removing or planting new shrubs, trees, or bushes. Some of Mowing Mansions Lawn-Care Company’s customers have hired the company to do some landscaping and have written down their requests. Help Mowing Mansions Lawn-Care Company understand each customer’s request so they can make each person’s yard look more attractive. Give a Student Journal to each student and a Tape Diagram Work Mat to each group. Explain to students that they will be collaborating with their groups to ﬁnd the benchmark fractions and percents. Have students label the tape diagram on the Tape Diagram Work Mat with the multiples of the benchmark percent 10% and the multiple of its corresponding 1 fraction ___ , and ask the following questions: 10 a.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 What is a benchmark percent? A benchmark percent is a common percent like 1%, 5%, 10%, 25%, 50%, or 75%.

DOK-2 How did you divide the tape diagram when you labeled each part by 10%? I divided the tape diagram into 10 parts.

DOK-1 What percent does the entire bar represent? 100%

Intervention

Acceleration

PERCENTS

Home

e. DOK-2 If three parts of the tape diagram are shaded, what fraction and 3 percent of the tape diagram is shaded? 30%, ___ . 10

5. 6. 7.

DOK-2 How can you use benchmark percents to solve percent problems? You can ﬁnd a common percent and then use scaling to ﬁnd the answer.

DOK-2 How can you ﬁnd 10% of 400 using a tape diagram? 10% of 400 1 is ___ of 400, so I can take 400 and divide it into 10 parts to represent 40 10 for each part.

Explain to the class that they will collaborate with their groups to determine the fraction and percent that are represented in the tape diagram. Encourage students to simplify each fraction into an equivalent fraction if possible. Actively monitor and assess students’ understanding as they collaborate with their groups by asking the following questions: a.

DOK-1 What percent is being represented in this scenario? Answers will vary based on the mansion. Fairﬁeld Castle had 20% of its lawn mowed.

DOK-2 How can we use our knowledge of benchmark percents to calculate the cost of lawn-care service for each scenario using the tape diagram? Answers will vary based on the mansion. Students should calculate the percent of lawns that are being cared for by the Mowing Mansions Lawn-Care Company by using the tape diagram.

Allow students enough time to discuss the information and record their work in the Student Journal.

Part II: Solving Percent Problems Using Models 1. 2.

Give a set of Landscaping Mansions Cards and a Double Number Line Work Mat to each group. Explain to students that they will work with their groups to read each Landscaping Mansions Card and will represent the information on the card on the group’s tape diagram or double number line. Use the Tape Diagram Work Mat and the Double Number Line Work Mat to point out to the class that the tape diagram is similar to a double number line. While you show students the Double Number Line Work Mat, inform them that the top number line will represent the total number of plants. They will need to separate the number line into equal parts to represent each part that makes the whole. The bottom number line represents the percents. 0% and 100% are already represented. They will need to determine the missing percents that are represented on the number line based on the total number of parts the number line is split into. Students will then work together using their Tape Diagram Work Mats and Double Number Line Work Mats to determine the answer to their Landscaping Mansions Card and will record their work on their Student Journals. As students are collaborating on their work, monitor their understanding by asking the following guiding questions:

FACILITATION TIP Remember that a helpful hint for solving problems with the word "of" in between two numbers, is to replace the "of" with "times" or "multiply." For example, "3 groups of 8 means 3 * 8" "10% of 400 is .1 *400." FACILITATION TIP Encourage students to shade accurately even if they already know the fraction and decimal. Remind them that even if these models seem simple, tape diagrams are a useful visualization tool for more complex problems (the Exit Ticket includes a tape diagram). FACILITATION TIP Note that these tape diagrams have the sections already marked for students to shade. Creating well-spaced parts to shade accurately may be diﬃcult for some students later. FACILITATION TIP When reading through the Landscaping Mansion Cards, consider preteaching pertinent vocabulary: acre, manor, estate, residence. FACILITATION TIP If needed, provide some additional practice with separating tape diagrams and double number lines into equal parts. Give some simple relevant examples related to class favorites (candy colors, classes, lunchroom choices). Demonstrate, use student samples, and review the use of these tools for students. FACILITATION TIP

Clarify the relationship between the number a. DOK-1 How will the Mowing Mansions Lawn-Care Company provide of lines drawn and the number of speciﬁc landscaping services to this mansion? Answers will vary depending parts needed. For example to cut a model on the Landscaping Mansions Card. The Mowing Mansions Lawninto 10 parts, you draw 9 lines; to cut a Care Company will mow ____ acres. The Mowing Mansions Lawn-Care Company will remove ____ (bushes, trees). The Mowing Mansions Lawn- model into 3 parts, you draw 2 lines. Care Company will replace ____ shrubs. © Accelerate Learning Inc. - All Rights Reserved

253

PERCENTS

Percents Explore 2 – Solving Percent Problems Using Benchmark Fractions and Percents

DOK-2 What is the total number of equal parts that we need to divide the tape diagram or number line into? Explain. Answers will vary based on the card. We need to split the tape diagram into _____ equal parts because there are a total of _____ (trees, shrubs, bushes, acres).

DOK-3 How can you determine what percents need to be written on the tape diagram or number line? Answers may vary. Since there are _____ total parts, we need to divide 100% by the total parts to determine what percents need to be written. For example, if we have 4 total parts, then we will divide 100% by 4, which equals 25%. This means each part represents a multiple of 25%. We will label the parts with 25%, 50%, and 75%.

DOK-2 How can you ﬁnd 10% of 400 using a tape diagram? You can divide a tape diagram into 10 equal parts and label the value for all of the percents. Each part of the tape diagram represents 40, so 10% of 400 is 40.

e. DOK-2 How can you determine what 30% of a number represents? You can divide the tape diagram or number line into 10 parts and ﬁnd the value of three parts. f. 6.

DOK-2 What does a percent that is greater than 100% represent? A percent greater than 100% represents a number greater than 1.

Allow students enough time to discuss the information on each Landscaping Mansions Card, to record their work, and to answer the reﬂection questions at the end of Part II of their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat •

•

FACILITATION TIP On the Exit Ticket, consider clarifying your expectations on how students are to "use the model." In order to be successful, do they need to label the top and bottom to visually represent the problem, or can they just solve the problem? 254

DOK-2 How do benchmark percents help with calculating percents? Benchmark percents can be used to perform mental estimation and calculation of percents. Values of benchmark percents can be added and subtracted to calculate the value of other percents. DOK-3 How can a tape diagram model a part-to-whole relationship between percents and a total quantity? A tape diagram is a visual representation of how a total can be split into equal parts and how those equal parts relate to the size of various percents. When we shade a percent of the whole, it shows us what part is equal to that percent. DOK-1 Why is 10% a useful benchmark percent to use when solving percent problems? Since 10% is a common percent, it is easier to scale up to solve the problem. DOK-1 When would it be useful to use a benchmark percent that is less than 10%? It would be useful when you are solving a percent problem and the percent is not a multiple of 10, such as 1% or 5%. DOK-1 How do you ﬁnd the amount for percents that are greater than 100%? Percents that are greater than 100% are fractions (or decimals) greater than 1. When you are ﬁnding the amount for a percent that is greater than 100%, the tape diagram and double number line should be extended past 100% to represent the amount.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

PERCENTS

Home

255

PERCENTS

Percents Explore 3 – Finding The Price and Discount ACTIVITY PREPARATION Students will use various strategies to ﬁnd the whole, percent, and part when solving percent problems about price and discount.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.6 Attend to precision.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Percent Model Work Mat (per station) 1 Set of Sale Cards (per class) 1 Exit Ticket (per student)

•

Reusable • •

•

1 Dry-erase marker (per station) 1 Clear sheet protector (per station)

Plan to divide the class into 6 groups. Print a Student Journal and an Exit Ticket for each student. Print a set of Sale Cards for the class. Place each Sale Card around the room in a different location to create 6 stations. Print a Percent Model Work Mat for each station. Place each Percent Model Work Mat inside a clear sheet protector to create an erasable surface. Gather enough dry-erase markers for each station to have one marker.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before reading the scenario, ask the class 1) What was the last product you bought?; 2) Did you feel like you got a good deal on your purchase? Why or why not? FACILITATION TIP If time allows, engage students with some local real-world advertisements/sale prices. Project and discuss them before passing out the Student Journal.

2. 3.

FACILITATION TIP

Encourage students to continue to practice using these visual tools even when they can determine these prices and discounts in their head. These tools are effective for more complex problems they will encounter later.

FACILITATION TIP Have a timer set with a speciﬁc amount of time for rotations. You can always assess and ask, "How much more time do you need?" and add a minute or two.

256

6. 7.

Read the following scenario to the class: David decides to go to a local shopping center that is having a sale. He decides he needs to ﬁnd the original price, sale price, and discount percent for items he is planning to purchase so he can make comparisons to determine if he is being offered a good deal. Help David determine the original price, sale price, and discount percent of each item that is for sale. Give a Student Journal to each student. Explain to the class that they will be working with their groups as they rotate through each station to read the information about each item on sale. They will use the discount and sale price to determine the original price, use the original price and sale price to determine the discount, or use the original price and discount to determine the sale price. Assign each group a Sale Card at which to begin. Instruct students to use the Percent Model Work Mat at each station to solve using either a double number line or tape diagram to determine the original price, sale price, or discount. Allow students enough time at each station to collaborate and record their work as a group and individually on their Student Journals. Instruct the class on when to rotate from one station to the next. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 How can we represent the discounted price as a fraction? Since the discounted price is a part of the whole price, the discounted price will be the numerator. The original price is the whole, so that will be the denominator. We don’t know the original price yet, so we can represent it with a letter to show that it is unknown. © Accelerate Learning Inc. - All Rights Reserved

Explore

Explain

Elaborate

Evaluate

DOK-3 How can we use the given part of the price and simplest form of the percent fraction to determine the original price? I can make another fraction where the part of the original price is the numerator. I can then determine the factor to multiply the numerators by to turn the percent numerator into the part of the whole price numerator. The factor that we multiply the numerators by is the factor we need to multiply the denominators by.

DOK-2 How can you ﬁnd the sale price when given the discount and the original price? You can divide the tape diagram or double number line into 10 parts. The original price represents 100%. Subtract the discount from 100%. For example, if the discount is 30%, 100% – 30% = 70%. The amount that represents 70% of the original price is the sale price. You can multiply 70% times the original price to get the sale price.

Engage

DOK-3 How can we use the information from the Sale Card to divide the percent bar in order to ﬁnd the original price? Answers will vary depending on the scenario. Since the percent is _____, I can divide the diagram by _____ (5s, 10s, 20s, 25s, etc). I will place the discounted price on the _____ percent line since the price represents that percent. I can then divide the discounted price by the number of parts the percent bar is divided into that equal the discounted price. That will determine what each part of the percent bar is worth. I can then add the worth of each part together to ﬁnd the original price. For example, $375 is 3/4 of the original price. $375 ÷ 3 = $125. $125 + $125 + $125 + $125 = $500 or $125 × 4 = $500. Therefore, the original price was $500.

DOK-3 How are models useful when solving percent scenarios? A model can provide a visual representation to display a percent and how it relates to a part and a whole. We can then use that model to relate that percent to a given part or a given whole in order to ﬁnd the missing part or whole. • DOK-2 How can you represent the percent as a fraction in order to solve the problem? I can represent the percent as a fraction by using the given percent as the numerator and 100 as the denominator since the whole of a percent is 100. We can then convert that fraction into the simplest form. • DOK-3 Why is it important to know multiple strategies to solve various percent problems? Knowing various strategies to solve percent problems is important because you can use a second strategy to check to see if your answer is correct from the ﬁrst strategy. Also, one strategy may require a lot more steps than another strategy, while another strategy may require fewer steps. In this case, the strategy with fewer steps could solve a problem faster than another strategy. •

Post-Explore

2. 3. 4.

Acceleration

FACILITATION TIP Consistently remind students that when they see the word "of" between two values in a problem, they can often replace it with "times" or multiply. Question 7c. provides a great example of "70% of the original price" or ".7 * p."

Have students complete each station and record their work and responses to the reﬂection questions on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat

Intervention

PERCENTS

Home

FACILITATION TIP Emphasize to students that these models can help them solve problems in their head while on the go in the real world. A visualization tool is a thinking tool that may help them think on their feet when trying to find the best deal or not get ripped off. FACILITATION TIP As a follow up, help students understand that different strategies may be more efficient in different scenarios. Also, different people may be (and become) more efficient with different strategies depending on thinking styles and experience/practice. FACILITATION TIP For this Exit Ticket, clarify your success criteria for "show your work." Some students may need some scaffolding to show a visual model (blank double number line drawn in for them), and some students may not want to draw any representation.

257

PERCENTS

Percents Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Represent Percents Using a Hundreds Grid Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Solving Percent Problems Using Benchmark Fractions and Percents Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 3

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Finding the Price and Discount Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Notebook A cut-and-glue activity to process learning that can be added to a notebook for future reference

258

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

PERCENTS

Home

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Equivalent Fractions, Decimals, and Percents Independent and partner games and other activities that provide students with an engaging way to practice the new concept

Fluency Builder Percent Applications Independent and partner games and other activities that provide students with an engaging way to practice the new concept

259

PERCENTS

Percents Intervention and Assessment SMALL-GROUP PLANNER 1

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

Students who are still acquiring the concept and need remediation

How to Use the Review

3 260

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

PERCENTS

Home

What prompts will be used?

What does mastery look like?

I can calculate the percentage of a number by using proportional reasoning.

I can calculate the percent of a quantity as a rate per 100.

I can solve problems involving ﬁnding the whole when given a part.

I can solve problems involving ﬁnding the part when given the whole.

I can determine the percentage that one number is of another number.

261

SCOPE 1

Measurement Conversions Scope Introduction SCOPE SUMMARY Students are introduced to ratios and work on explaining the concept of a ratio, representing a ratio, and using ratio language to describe a relationship between two quantities. Sixth graders are using this knowledge of ratios and their prior knowledge of unit conversions to solve problems that exist in everyday life to convert ratios within the customary and metric measurement systems. Students will be working to solve one-step conversions, two-step conversions, and multistep conversions between systems of measurement. Student Expectations

6.NR.4.7 Use ratios to convert within measurement systems (customary and metric) to solve authentic problems that exist in everyday life.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

Fifth grade is the ﬁrst time that students are expected to solve problems in which they must convert between different units in the same measurement system. By the time students reach sixth grade, they understand multiplication by a whole number or by a fraction as scaling up or down, they have solved multistep word problems involving unit conversions, and they have experience using tables, equations, and tape diagrams to represent part-part-whole relationships. Prior to this scope, sixth graders explored ratios, rates, unit rates, and percentages. The ability to generate equivalent ratios and to determine unit rates will now extend as students explore unit conversions.

In seventh grade, students compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. Seventh-grade students also identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and with verbal descriptions, and they represent proportional relationships using equations.

Before beginning the lesson, the students’ prior knowledge is assessed to address any student needs and skills needed for content mastery. Students are assessed to determine if they can: •

convert and solve problems involving different-sized standard measurement units within a given measurement system.

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope.

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

convert measurement units of kilograms to pounds.

Students move on to the Explore activities and come back at the end of the explorations to repeat the activity and discuss what they found.

If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. 262

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

One-Step Measurement Conversions

Explore 2

Explore 1

EXPLORE ACTIVITIES

In this exploration, students will solve a scenario involving the Happy Trails Petting Zoo to convert measurements into different units in order to give them to a design team. Students will: •

solve one-step word problems using ratio reasoning.

•

convert measurement units within the standard measurement system.

•

convert measurement units within the metric measurement system.

In this exploration, groups of students will help Old Mac’s Petting Zoo order supplies from a new company through converting measurements into different quantities and units to give to its ordering department. Students will: •

solve one-step word problems using ratio reasoning to convert measurement units within the standard measurement system.

•

solve one-step word problems using ratio reasoning to convert measurement units within the metric measurement system.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

Explore 3

Two-Step Measurement Conversions

MEASUREMENT CONVERSIONS

Home

Multistep Conversions between Systems of Measurement In this exploration, students will solve a scenario involving a local petting zoo to determine new equivalent units of measure for supplies gathered that have wrong units. Students will: •

solve multistep word problems using ratio reasoning to convert between the standard measurement system and the metric measurement system.

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve. Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________

263

MEASUREMENT CONVERSIONS

Measurement Conversions Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

264

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will read different student responses to a posed question on the prior standard, decide whether they agree or disagree with the student, and explain their reasoning. 5.MDR.7.3: Convert among units within the metric system and then apply these conversions to solve multistep, practical problems. 5.MDR.7.3

Materials

Preparation

Printed •

•

Print one Agree or Disagree for each student.

1 Agree or Disagree (per student)

Procedure and Facilitation Points 1. 2. 3. 4. 5. 6. 7.

FACILITATION TIP

Instruct students to complete the Agree or Disagree independently. Once students have completed the activity on their own, have them stand up. Instruct all students to walk around the classroom with their hand raised in a high-five position. On your instruction, students will stop and high-five the closest person. This will be their partner. Give students a couple of minutes to discuss their answers and justifications together. You may then continue as many times as you want with different partners. Discuss the responses as a class. Allow students to explain their reasonings for each problem. a.

Agree with Sonia

Disagree with Amber

Disagree with Simone

• •

Set a one- or two-minute timer for independent think time, then again for walk around time, and ﬁnally for partner time. FACILITATION TIP As students discuss, monitor them closely. If needed, display physical measurement tools to support students’ thinking. If time allows, gather a meterstick, a yardstick, a few graduated cylinders and beakers, gram stackers, US cooking tools, and labeled drink containers (maybe your "to go" coffee cup from the morning). FACILITATION TIP

If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

Identifying Misconceptions •

MEASUREMENT CONVERSIONS

Home

Students may not recall how to convert within the metric and customary systems. Students may need to review which unit abbreviation is used for each unit. Students may not recall the basic measurement conversion.

Project sentence frames for students to use as they collaborate with partners and as they respond to the whole class. For example, “ _________ is correct because _________.” Or, “We agree with _________ because _________.” Or, “We are wondering about _________ because _________.” FACILITATION TIP In addition to the Foundation Builder, you might assess student knowledge of these systems with separate activities to determine if students need more time speciﬁcally on the metric or the customary US system before moving on to the scope. FACILITATION TIP Be prepared to conduct a brief refresher of customary and metric terminology before or immediately after this activity depending on your students.

265

MEASUREMENT CONVERSIONS

Measurement Conversions Hook ACTIVITY PREPARATION Students will convert kilograms to pounds to help solve a problem.

Materials

Preparation

Printed •

• • •

1 The Great British Baking Flour (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project The Great British Baking Flour for the whole class to view. Prepare to introduce the scenario and encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP

Part I: Pre-Explore

Before showing the video and reading the scenario, ask the class 1) Does anyone like to cook?; 2) If so, what do you like to cook?; 3) What are the ingredients you use to cook with?

FACILITATION TIP Take a moment to point out where the United Kingdom is located, and explain the common practice of using metrics around the world.

FACILITATION TIP Project the speciﬁc details of this scenario for students to read with you. Have students write down the measurements; 20 lb (review lb as notation for pound) and 5 kg (review kg as notation for kilogram).

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: Ada is making scones for a huge international studies project at school where she is representing Great Britain. She needs 20 pounds of special British flour to make the scones authentic. Her aunt lives in London and sends her two bags of special flour. Each bag is 5 kg. Ada does not have a scale. Will Ada have enough flour for her scones? Ask students the following questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Student answers will vary. I notice that Ada is converting measurement units. I notice that Ada is converting between the metric and customary measurement systems. I wonder how many pounds are in a kilogram. Once I know how many pounds are in a kilogram, I can use math by using multiplication to figure out whether Ada has enough flour. Project The Great British Baking Flour.

266

Explore

Explain

Elaborate

Evaluate

DOK-2 How might Ada compare the mass of the bags of ﬂour to the amount she needs? She could look for a scale that has both systems of measurement on it. She could ﬁnd out how many pounds are in one kilogram. DOK-2 How might Ada organize her information ? Ada might make a chart with equivalent measurements on each side–kilograms on one side and pounds on the other side. Ada might draw out a model.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Intervention

Acceleration

Explain to students that Ada must determine whether she has enough ﬂour for her recipe. Discuss the following questions: a.

Engage

Show the Phenomena Video again, and restate the problem. Refer to The Great British Baking Flour, and discuss the following questions: a.

DOK-2 How can you start to determine whether Ada will have enough ﬂour? You must ﬁnd out how many pounds are in 1 kilogram. There are 2.2 pounds per kilogram.

DOK-1 How many pounds are in each bag of ﬂour? 2.2 lb./kg × 5 kg = 11 pounds. There are 11 pounds of ﬂour in each bag.

DOK-1 How many pounds of British ﬂour does Ada have altogether? 11 lb./bag × 2 bags = 22 pounds. Ada has 22 pounds of ﬂour altogether.

DOK-2 Does Ada have enough flour for the large quantity of scones she must make? 22 lb. – 20 lb. = 2 lb. Yes, Ada needs 20 pounds and she has 22 pounds, so she has enough flour for scones plus 2 leftover pounds of flour.

FACILITATION TIP Establish that students are clear regarding the differences between measurements for mass and volume (US ounces can be confusing). Most cooking recipes rely on measurements using volume. FACILITATION TIP Continue to remind students that these visual tools (charts and models) can improve thinking. If they can become ﬂuent in drawing and sketching these concepts, they should be able to mentally solve problems more quickly.

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FACILITATION TIP Project questions 2b and 2c and have students show how they solve. Select some student work samples to share under the camera.

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Measurement Conversions Explore 1 – One Step Measurement Conversions ACTIVITY PREPARATION Students will solve one-step word problems using ratio reasoning to convert measurement units within the standard measurement system or within the metric measurement system.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Petting Zoo Cards (per class) 1 Exit Ticket (per student)

Plan to divide the class into 8 groups to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print one set of Petting Zoo Cards for the class. If desired, print the cards on card stock and laminate them for future use. Cut out each of the cards.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever been to a petting zoo?; 2) If so, what types of animals did you get to pet?; 3) What animals would you have liked to pet that weren't at the petting zoo?

• 1.

FACILITATION TIP Be prepared with additional real-world, relevant examples for when conversions are necessary. (For example: Purchasing supplies for construction projects, designing and conducting a science experiment, deciding on the best deal at the store, deciding whether to sign up to run a 5k) FACILITATION TIP Use a timer for the rotations. You can also have students stay in groups and have the Petting Zoo Cards be rotated by passing them from group to group. FACILITATION TIP Clearly post the options for which models students can create. Clarify your expectations for how those models are labeled or shown. Consider showing a student work sample or creating your own example to demonstrate.

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Create 8 stations around the room with a card at each station. Read the following scenario to the class: Happy Trails Petting Zoo wants to make some updates to different areas around their property. They will be taking measurements and providing these measurements in two different units to give to a design team who will be making the improvements. They have already measured everything in one set of units. Help the petting zoo convert each measurement into a different unit to give to the design team. Discuss the following question with the class: a.

3. 4. 5.

DOK-1 What does the word convert mean? Convert means “to change.” Students will need to convert a given measurement in one unit into an equivalent measurement of a different unit.

Give a Student Journal to each student. Assign each group of students a station at which to begin. Discuss with the class how to rotate through each Petting Zoo Card station. Students will work collaboratively to read each scenario on the Petting Zoo Cards at their stations. They will use this information to ﬁll in the measurement table on their Student Journals. Then, they will create a model (equivalent ratio table, double number line, proportion, tape diagram, or equation) to solve for the equivalent measurement. As students are working together, monitor their learning, and ask the following questions to check for understanding: a.

DOK-1 Which strategies can you use to solve measurement conversion problems? I can use equivalent ratios, tape diagrams, double number lines, proportions, and equations to solve measurement conversion problems.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 What units will you be converting to? Answers depend on the scenario. I will convert from meters to kilometers.

DOK-1 What is the equivalent measurement for the given measurement in the scenario? Answers depend on the scenario. The equivalent measurement is 8 kilometers.

e. DOK-1 How can you tell if you need to multiply or divide? Answers will vary. If you need to go from a bigger number to a smaller number, you will divide. If you are going from a smaller number to a bigger number, you will multiply. 7. 8. 9.

Have students rotate through the stations to complete their Student Journals. Allow students enough time to answer the reﬂection questions on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat •

•

• 1. 2. 3.

DOK-1 How are conversions between measurements the same as equivalent ratios? Measurement conversions are equivalent ratios because we are taking a given ratio and finding a new ratio of the same proportion. DOK-2 How is finding equivalent units in the metric system different from finding equivalent measurements in the standard system? The metric system is different because you are always changing by a multiple of ten. DOK-2 How can you tell if you need to multiply or divide? If you need to go from a bigger number to a smaller number, you will divide. If you are going from a smaller number to a bigger number, you will multiply. DOK-2 If you are given the measurement of 4,000 milligrams, how can you determine how many grams this will be? There are 1,000 milligrams in one gram. I should divide 4,000 by 1,000 to determine that there are 4 grams in 4,000 milligrams. DOK-4 Where might you need to use conversions in the real world? I may need to convert units of measurement when building a bird feeder.

Intervention

Acceleration

FACILITATION TIP Question 6e is an essential skill for conversions. Slow down and do a few examples to demonstrate thinking skills students can use to determine when to multiply or divide. Encourage students to always analyze their solutions and ask, "Does that make sense?"

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STEMscopes Tip Each scope includes a Home section accessed along the scope's menu bar. Here you will see student expectations as well as key concepts and fundamental questions. Each Home tab includes drop-down options to access the Scope Overview, Content Support, Content Unwrapped, Materials List, and Parent Letter pages.

FACILITATION TIP

Before having students complete this Exit Have students complete the Exit Ticket to formatively assess their understanding Ticket, determine your criteria for how you want students to demonstrate that they of the concept. "Solve by creating a model using ratio Complete the Anchor Chart as a class. reasoning to convert." Have each student complete their Interactive Notebook. Notes

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MEASUREMENT CONVERSIONS

Measurement Conversions Explore 2 – Two-Step Measurement Conversions ACTIVITY PREPARATION Students will solve two-step word problems using ratio reasoning to convert measurement units within the standard measurement system or within the metric measurement system.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Old Mac’s Petting Zoo Cards (per class) 1 Exit Ticket (per student)

•

Plan to divide the class into 8 groups to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print one set of Old Mac’s Petting Zoo Cards for the class. If desired, print the cards on card stock, and laminate them for future use. Cut out each of the cards. Create 8 stations around the room with a card at each station.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP

Before reading the scenario, ask the class 1) What types of supplies would a petting zoo need?; 2) What types of supplies would be needed for different animals, like a goat versus a chicken?

FACILITATION TIP Before placing the Old Mac's Petting Zoo Cards around the room, read parts of the cards together with students to clarify any confusion. To prevent students from trying to solve as your read them, cover up the values with sticky notes (For example, "The ducks at Old Mac’s Petting Zoo can eat _______grams of food..."). Covering up the values may help students to focus on methods rather than quick solutions. FACILITATION TIP Depending on your students, you could have students use the speciﬁc model already included on all scenarios or ask them to create more than one model for each. FACILITATION TIP Questions 7b and 7c are excellent prompts to help struggling students get started. Encourage students with "What do we know?" and "What do we need to ﬁnd out?" 270

3. 4. 5. 6.

Read the following scenario to the class: Old Mac’s Petting Zoo is in the next town over. His petting zoo has decided to use a new company to order its supplies from. They gathered the information for each supply needed, but the new company has different supply quantities! Help Old Mac’s Petting Zoo convert each measurement into a different quantity and unit to give to its ordering department. Remind students that the word convert means “to change.” They will need to write equivalent measures using a different unit provided. Give a Student Journal to each student. Assign each group of students a station at which to begin. Discuss with the class how to rotate through each Old Mac’s Petting Zoo Card station. Have students work collaboratively to read each scenario. They will use this information to ﬁll in the measurement table on their Student Journals. Then, they will create a model (equivalent ratio table, double number line, tape diagram, proportion, or equation) to convert units of measurement from one measurement system to another by using the given ratios to ﬁnd new quantities of each of the supplies. As students are working together, monitor their learning, and ask the following questions to check for understanding: a.

DOK-1 What information are you given in the scenario? Answers depend on the scenario. The information given is 5 inches.

DOK-1 What units will you be converting to? Answers depend on the scenario. I will convert from inches to feet.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

e. DOK-1 What can you do to help convert from 5 inches to 12 inches? I can convert 12 inches to 1 inch and then convert from 1 inch to 5 inches. f.

8. 9.

DOK-1 What is the equivalent measurement for the given measurement in the scenario? Answers depend on the scenario. The equivalent 5 measurement is ___ of a foot. 12

Allow students enough time to complete their work and record their observations and reﬂections on their Student Journals. After the Explore, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 Which model do you prefer to use to solve measurement conversions? Explain. I prefer using a ratio table because I can see all the measurements in one row/column. • DOK-2 Explain how to change the supply amount from 5 inches to an equivalent amount in feet. We know that 12 inches equals 1 foot, but I can’t get from 12 to 5. •

of a So I ﬁrst divide 12 inches by 12 to get 1 inch and divide 1 foot by 12 to get ___ 12 foot. Now I can ﬁnd how many feet are in 5 inches because I can multiply by 5 to 1

of a foot to ___ of a foot. change 1 inch to 5 inches and ___ 12 12 •

DOK-4 Explain where in the real world you might need to convert measurements. Converting measurements could be used when baking a cake.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

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FACILITATION TIP Continue to support students as they explore their preferred methods. Some students may like to say, "I just do it in my head." Encourage these students to try to identify exactly how they are thinking the problem out. Communicate to students that many assessments (and future jobs) require explanations or models in detail. FACILITATION TIP Be prepared with additional real-world, relevant examples for when conversions are necessary.

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Measurement Conversions Explore 3 – Multistep Conversions Between Systems of Measurement ACTIVITY PREPARATION Students will solve multistep word problems using ratio reasoning to convert between the standard measurement system and the metric measurement system.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

•

1 Student Journal (per student) 1 Set of Petting Zoo Cleanup Day Cards (per group) 1 Exit Ticket (per student)

Reusable •

•

1 Resealable bag (per group)

Plan to divide the class into groups of 4 to complete this activity. Print a set of Petting Zoo Cleanup Day Cards for each group. If desired, print the cards on card stock and laminate them for future use. Cut out the cards, and place them in a resealable bag for each group. Print a Student Journal and an Exit Ticket for each student.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before reading the scenario, ask the class 1) Where are petting zoos often located?; 2) What do you think is involved in getting a petting zoo ready to open?; 3) Would you enjoy working at a petting zoo? Why or why not? FACILITATION TIP Note that the Petting Zoo Cleanup Day! Cards have some estimating ("1 liter is about 2 pints") and rounding. Review rounding and ensure students that it is okay that answers are not exact in these scenarios.

272

2. 3.

Read the following scenario to the class: The local petting zoo is ready to get cleaned up for opening day! Supplies have been measured to be gathered. Tommi is leading the way but ﬁnds some of the supplies are measured in the wrong units! Help Tommi determine the new equivalent units of measure for each item so he can get started. Give a Student Journal to each student and one set of Petting Zoo Cleanup Day Cards to each group. Have students work collaboratively to read each scenario on the Petting Zoo Cleanup Day Cards. Have them use this information to ﬁll in the measurement table on their Student Journals. Then, have them each create a model of their choice (equivalent ratio table, double number line, tape diagram, proportion, or equation) to solve for the equivalent measurement. As students are working together, monitor their learning, and ask the following questions to check for understanding:

FACILITATION TIP

Project question 4a for students to read and consider as they collaborate. Encourage them to be prepared to explain it in their own words and use an example.

DOK-1 How can you tell if you need to multiply or divide? Answers will vary. If you need to go from a bigger number to a smaller number, then you will divide. If you are going from a smaller number to a bigger number, then you will multiply.

DOK-2 What strategy can you use to convert 75 liters to the number of pints needed in the ticket booth-painting scenario? Answers will vary. If I multiply 1 liter by 75, I can get 75 liters. I know 1 liter is equal to 2 pints. So to get pints, I can multiply 2 by 75 to ﬁnd my answer of 150 pints. © Accelerate Learning Inc. - All Rights Reserved

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

Math Chat DOK-2 Explain what is different when converting between the standard measurement system and the metric system. Converting between the standard measurement system and the metric system does not give friendly numbers to work with. All of the conversion ratios include fractions or decimal numbers. • DOK-2 Describe how you transformed the units of measure for the paint for the ticket booth. Since 1 liter is 2 pints, I multiplied 2 times 75 to get 150 pints. • DOK-4 Where in real life would you need to convert between the metric and standard measurement systems? Some tools are in metric units. When using these tools, we may need to convert from metric to standard measurement system to determine the correct size tool to use. •

1. 2. 3. 4.

Intervention

Acceleration

FACILITATION TIP The ﬁrst reﬂection question about when to multiply and divide creates a good opportunity to help students solidify when to use which method. Some students will intuitively select which operation to use when, but others may need some memorization tools. Have students use illustrative notes (BIG to small=divide; small to BIG=multiply). Be aware that some students may be thinking of the size of the measurement rather than the value of the number.

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FACILITATION TIP This Exit Ticket leaves room for students to choose the model. Consider providing some students with predrawn number lines or blank charts. Challenge some students to solve each problem using more than one method.

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Measurement Conversions Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

One-Step Measurement Conversions Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Two-Step Measurement Conversions Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 3

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Multistep Conversions between Systems of Measurement Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Notebook A cut-and-glue activity to process learning that can be added to a notebook for future reference

274

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Convert between Measurement Systems Independent and partner games and other activities that provide students with an engaging way to practice the new concept

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Fluency Builder Convert within a Measurement System Independent and partner games and other activities that provide students with an engaging way to practice the new concept

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Intervention and Assessment SMALL-GROUP PLANNER 1

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

MEASUREMENT CONVERSIONS

Measurement Conversions

3 276

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER Use this template to decide how to assess your students for concept mastery. Depending on the format of the assessment, you can identify prompts and intended responses that would measure student mastery of the expectation. See the beginning of this scope to identify standards and grade-level expectations. Key Concepts I can use strategic thinking to manipulate and transform units appropriately when multiplying or dividing quantities to solve math problems.

What prompts will be used?

What does mastery look like?

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I can convert measurement units when given a conversion factor within one system of measurement.

I can convert measurement units when given a conversion factor between two systems of measurement.

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SCOPE 1

Coordinate Planes Scope Introduction SCOPE SUMMARY In this grade level, students are expected to interpret, locate, and graph points in all four quadrants of the coordinate plane. Students use the signs of integers within an ordered pair to determine which quadrant a point is located in, and they recognize that when two ordered pairs differ only by signs, that the location of the points are related by a reﬂection across an axis. Student Expectations

6.PAR.8.1 Locate and position rational numbers on a horizontal or vertical number line; ﬁnd and position pairs of integers and other rational numbers on a coordinate plane. 6.NR.8.2 Show and explain that signs of numbers in ordered pairs indicate locations in quadrants of the coordinate plane and determine how two ordered pairs may differ based only on the signs.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grades, students understood positive rational numbers as points on a number line. Fifth grade is the ﬁrst time that students graph ordered pairs on a coordinate plane. Fifth-grade students have represented real-world and mathematical problems by graphing and interpreting points in the ﬁrst quadrant of the coordinate plane.

Sixth-grade students will solve problems by graphing points in all four quadrants of a coordinate plane. They will use absolute value to find the distance between points that share an x-coordinate or yy-coordinate. -coordinate. Sixth grade students will also draw polygons on a coordinate plane given points for the vertice and find the length of the polygon’s side with shared x- or yy-coordinate -coordinate values. In seventh grade, students will apply their understanding of coordinate planes to proportional relationships on graphs and finding the constant of proportionality. Seventh-grade students will also graph and interpret two-step inequalities.

Before beginning the lesson, the students’ prior knowledge is assessed to address any student needs and skills needed for content mastery. Students are assessed to determine if they can: •

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES

ﬁnd x and yy-coordinate on a coordinate system.

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope.

At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

ﬁnd coordinates on a map.

•

compare maps to coordinate planes.

Students move on to the Explore activities and come back at the end of the explorations to repeat the activity and discuss what they found.

If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. Notes _____________________________________________________________________________________________________________ _____________________________________________________________________________________________________________

278

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Number Lines and Coordinate Planes In this exploration, groups of students will solve a scenario about helping plan a new city. Students will: •

make connections between number lines and coordinate planes.

•

use the location cards and number lines to develop the layout of main buildings in the city.

Explore 2

Explore 1

EXPLORE ACTIVITIES

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

Reﬂections on a Coordinate Plane In this exploration, students will collaborate with their peers where they must add new buildings to accommodate city growth. Students will: •

ﬁnd locations of buildings to determine the relationship between the signs of numbers and locations in quadrants on a map.

•

determine ordered pairs that show reﬂections across one of the axes.

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After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve.

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COORDINATE PLANES

Coordinate Planes Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

280

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

COORDINATE PLANES

Home

5.PAR.6.2: Represent problems by plotting ordered pairs and explain coordinate values of points in the ﬁrst quadrant of the 5.PAR.6.2 coordinate plane.

Materials

Preparation

Printed •

• •

1 Four Corners (per class)

Print one set of the Four Corners. Hang the Four Corners pages in four separate areas of the classroom, easily visible to all.

Procedure and Facilitation Points 1. 2. 3. 4.

FACILITATION TIP

Ask the students to look at the Four Corners and think about which corner image best explains the locations of the bank, library, and post oﬃce. Allow 2 minutes of thinking time. Ask students to move to the corner image they chose. Ask each group to discuss with one another why they chose their image. Allow 2–5 minutes of discussion at the corner images. After students have discussed why they chose their answer, talk about the answer with the class, and allow students to explain their reasoning. If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

Identifying Misconceptions • • • •

Slide 1: Students who chose this slide have reversed the x- and y-coordinates. y Slide 2: Students who chose this slide found the correct coordinates, but they don’t correspond with the correct locations. Slide 3: This is the correct slide. Slide 4: Students who chose this slide do not have the correct x-coordinate; they used the location of the yy-coordinate -coordinate for both coordinates.

Students may need a closer look at the slides. Consider projecting them one at a time or distributing one copy of each to each table set or group for them to examine before being directed to move. FACILITATION TIP Provide a specific time frame, such as 30 seconds, for movement. As the groups are discussing why they chose their corner, move from group to group to monitor their discussions. Ask clarifying questions to understand their reasoning for choosing their answer. FACILITATION TIP During the class discussion, listen for key ideas such as starting at the origin, moving horizontally first, and then moving vertically. If these ideas are not mentioned, ask guiding questions to lead students toward these key ideas. FACILITATION TIP Depending on your students, continue to review x and y, first and second, horizontal and vertical, side to side and up and down, north/south, east/west, and left and right. Use physical responses with students. Their physical responses (hand signals, arm motions) will help them review, but will also help you assess misconceptions and gaps.

281

COORDINATE PLANES

Coordinate Planes Hook ACTIVITY PREPARATION Students will determine where ordered pairs lie on a coordinate plane.

Materials

Preparation

Reusable •

• •

1 Phenomena video (per class)

Plan to show the video. Prepare to introduce the scenario and encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION POINTS Part I: Pre-Explore 1. FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) Have you ever needed to use a map to get to a location?; 2) Where were you going?; 3) What features of the map helped you get there?

FACILITATION TIP Provide an example of a map of the city or state for student to look at. This is a good time to implement a think, pair, share.

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video: Maribel is going to visit her parents out of state. She decided that she needed to use her road map to help navigate her way to her parents’ house. Maribel finds her current location on the map and then finds the town where her parents live. She can use the lines on the map to find the best route to visit her parents. Ask students the following questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Student answers will vary. I notice that there are many lines on the map. I wonder what math there is in maps. I notice that there are numbers on the dashboard of the car. Explain to students that maps can be helpful to find directions to places we want to go, but do not know how to get there. Discuss the following questions: a.

Find a map from students' social studies or science class to help students connect these concepts to a map they are already familiar with. Project this map and review longitude and latitude if needed. You can also demonstrate with a digital mapping tool.

DOK-1 Where do you see math in maps? Allow students to share all ideas. Student answers will vary. There are numbers to show locations on a map.

DOK-1 Thinking back to what you have learned in math prior to this, how could we use math with maps? Allow students to share all ideas. Student answers will vary. We could use maps to ﬁnd distance.

DOK-1 Where might you ﬁnd coordinates on a map? We can ﬁnd coordinates on a map when looking for different locations, such as the town that Maribel will be driving to.

FACILITATION TIP

DOK-1 Where is longitude and latitude shown on a map? Latitude is shown on the lines that are horizontal, and longitude is shown on the lines that are vertical.

FACILITATION TIP

Make note of the fact that latitude and longitude will not always be seen on a map. Discuss how latitude and longitude are related to coordinates on a map or coordinate grid.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Show the Phenomena Video again and restate the problem. Discuss the following questions: a.

282

DOK-1 What do you notice about maps and coordinate planes? Allow students to share all ideas. Student answers will vary. A map is laid out on a coordinate plane. We can use coordinates to ﬁnd locations on a map. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How are maps and coordinate planes different? The lines on coordinate planes are vertical and horizontal, but the lines on maps do not have to be vertical or horizontal because they show the path that the roads take. DOK-1 What strategies can you use to determine where points are located on a coordinate plane? The coordinates of an ordered pair ﬁrst show the x value, and then the y value. If we are looking for the point (2, 5) we would ﬁrst move right to 2 and then move up to 5. If we had a negative x value, like (–2, 5), then we would move left to 2, and then up to 5. If we had a negative y value, like (2, –5), we would move right to 2, and then down to negative 5.

Intervention

Acceleration

COORDINATE PLANES

Home

283

COORDINATE PLANES

Coordinate Planes Explore 1 – Number Lines and Coordinate Planes ACTIVITY PREPARATION Students will make connections between number lines and coordinate planes.

Standards for Mathematical Practice • •

MP.5 Use appropriate tools strategically. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • • • • •

•

1 Student Journal (per student) 1 Set of Quadrant Cards (per group) 1 Horizontal and Vertical Number Lines (per group) 1 Set of Main Buildings Cards (per group) 1 Set of Buildings Cards II (per group) 1 Exit Ticket (per student)

•

Reusable • • • •

• •

2 Gallon-sized resealable bags (per group) 1 Permanent marker (per class) 1 Set of colored pencils (per group) 1 Dry-erase marker (per group, optional)

•

Plan to have students work in groups of 4 or 5 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Main Buildings Cards and Horizontal and Vertical Number Lines for each group. If desired, print them on card stock and laminate them for durability. Cut out and place these cards in a resealable bag labeled “Part I” with a permanent marker. Print a set of Buildings Cards II and the Quadrant Cards for each group. If desired, print them on card stock and laminate them for durability. Cut out and place the cards in a resealable bag labeled “Part II.” Gather a set of colored pencils for each group. If the Horizontal and Vertical Number Lines and Quadrant Cards are laminated, you will need to gather dry-erase markers for each group instead of a set of colored pencils.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP

Part I: Number Lines to Quadrants

Before reading the scenario, ask the class 1) What do the buildings in our city look like?; 3) What types of businesses are in the buildings?; 3) Have you noticed any new buildings being built in our city?

2. FACILITATION TIP For students who are struggling, provide them with an image of a compass rose to help them use the cardinal directions to place the number lines. They can also draw on their knowledge of horizontal and vertical number lines from the previous scope.

5. 6. 7. 284

Read the following scenario to the class: You recently decided to become a city developer and are now being tasked with planning a new city! You want to center your work around the courthouse and start with the main intersection to determine where to place important buildings in the city. You will need to work with your team to use the location cards and number lines to develop the layout of main buildings in the city. Give a set of colored pencils and the Part I bag containing a set of the Main Buildings Cards and Horizontal and Vertical Number Lines to each group. Have students work in their groups to determine how the number lines fit together with the courthouse being at the center of the number lines. Students can use the cardinal directions to help determine how the number lines fit together. Then, have students mark the other 6 building locations using their colored pencils on the number lines and label each building with its name. If the Horizontal and Vertical Number Lines are laminated, have students label using their dry-erase markers. Give a Student Journal to each student. Allow time for students to complete Part I of their Student Journals and its reflection questions. After Part I, invite the class to a Math Chat to share their observations and learning. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Math Chat DOK-1 How are number lines and coordinate planes similar? Coordinate planes are one horizontal number line and one vertical number line placed together to intersect at 0 on both number lines. • DOK-1 What is the ordered pair for the courthouse? The courthouse is located at zero horizontally and zero vertically; therefore, the ordered pair is (0, 0). • DOK-2 Do you think that one number in the ordered pair must always be zero? Students should pull from prior knowledge of ﬁrst-quadrant graphing to understand that zero does not have to be in the ordered pair. •

Part II: Making the Map 1.

2. 3. 4.

5. 6. 7.

Read the following scenario to the class: Your development team will need to put a map together to ﬁnd the best location for other buildings around the city. You will start with four pieces of a coordinate plane. Determine how to put the coordinate plane together to form the city map. Then, locate and label the other buildings around the city! Distribute the Part II bag containing the Buildings Cards II and Quadrant Cards to each group. Have students work in their groups to put the coordinate plane city map together. Have students locate the next four buildings to be added in the city. Instruct students to use their colored pencils to mark and label the buildings on the coordinate plane city map. If the Quadrant Cards are laminated, have students use their dry-erase markers. After each group locates all of the buildings on their maps, have students mark and label the buildings on the coordinate grid on their Student Journals. Allow time for students to complete the reﬂection questions at the end of Part II. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat •

• •

DOK-1 When finding ordered pairs on the map, you must first move _____ or _______, and then you will move _____ or _______. When finding ordered pairs on the map, you must first move left/west or right/east; and then you will move up/ north or down/south. How do you know which way to move? You will know which way to move by looking at the signs in front of the x- and y-coordinates. DOK-2 What do you notice about all of the ordered pairs that were located in Quadrant I? In Quadrant I, both the x- and y-coordinates were positive. DOK-2 What do you notice about all of the ordered pairs that were located in Quadrant II? In Quadrant II, the x-coordinates were negative and the y-coordinates were positive. DOK-2 What do you notice about all of the ordered pairs that were located in Quadrant III? In Quadrant III, both the x- and y-coordinates were negative. DOK-2 What do you notice about all of the ordered pairs that were located in Quadrant IV? In Quadrant IV, the x-coordinates were positive and the y-coordinates were negative. DOK-1 What is the name of the location (0, 0) on a coordinate plane? The location (0, 0) is called the “origin.” DOK-2 A library is being built at (−6, 0). What quadrant would the library be located in? Coordinates on the x-axis and y-axis do not lie in any quadrant.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

FACILITATION TIP Ask students to state the coordinates of some of the other buildings. As they state them, write them down. Ask students what they notice about the coordinates or what they have in common.

COORDINATE PLANES

Home

FACILITATION TIP Explain that when zero is in the ordered pair, the point will be located on an axis. If the ﬁrst number in the ordered pair is zero, the point is located on the y-axis. If the second number in the ordered pair is zero, the point is located on the x-axis. FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever seen a map of our city?; 2) If so, what does it look like?; 3) Who might use a map of our city? FACILITATION TIP This is a good time to explain that the coordinate grids that students are working with have four quadrants created by the x- and y-axes. Explain that the quadrants are numbered 1–4 in a counterclockwise direction.

FACILITATION TIP Prompt students to specify which direction a negative or positive x-coordinate tells them to move and which direction a negative or positive y-coordinate tells them to move.

FACILITATION TIP If students have trouble identifying that the library is located on the x-axis and not a quadrant, have them locate it on their coordinate plane city map. FACILITATION TIP Before having students complete the Exit Ticket, show them how to use a straight edge (sometimes a clear ruler is very helpful) to follow the coordinate grid lines as they slide left/right or up/down. 285

COORDINATE PLANES

Coordinate Planes Explore 2 – Reflections on a Coordinate Plane ACTIVITY PREPARATION Students will work with a city map and find locations of buildings to determine the relationship between the signs of numbers and locations in quadrants. Students will determine ordered pairs that show reflections across one of the axes.

Standards for Mathematical Practice • •

MP.5 Use appropriate tools strategically. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • •

• •

1 Student Journal (per student) 1 Set of Building Location Cards (per group) 1 Exit Ticket (per student)

• •

Reusable • •

2 Quart-sized resealable bags (per group) 1 Permanent marker (per class)

•

Consumable • •

1 Set of colored pencils (per group) 1 Large piece of white butcher paper (per group)

•

Plan to divide the class into groups of 4. Prepare a large coordinate plane for each group on a large piece of white butcher paper with a permanent marker. Label the coordinate plane from −12 to 12 on both axes. Print a Student Journal and an Exit Ticket for each student. Print a set of the Building Location Cards for each group. If desired, print the cards on card stock and laminate them for durability. Cut out the Building Location Cards for Part I, and place them in a resealable bag labeled “Part I” for each group. Cut out the Building Location Cards for Part II, and place them in a resealable bag labeled “Part II” for each group. Gather a set of colored pencils for each group.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before reading the scenario, ask the class 1) What new buildings do you see being built in our city?; 2) What do you think the new buildings will be used for?; 3) How would you locate the new buildings on a map? FACILITATION TIP To avoid drawing large coordinate planes by hand in permanent marker, take some time to look around for alternatives. Some schools or districts have large collaborative whiteboards with grids already drawn on them. Many schools have individual whiteboards with grids printed permanently on them as well. You could also have some special poster-size coordinate planes printed.

Part I: Graphing in All Four Quadrants 1.

2. 3.

b. 5. 6.

286

Read the following scenario to the class: Growth in the city is at an all-time high! Your city planning committee wants to add new buildings to accommodate the growth! Plot and label each of the buildings from the Building Location Cards for Part I onto the city map. Give a set of colored pencils, the coordinate plane on butcher paper, and a Part I bag of the Building Location Cards to each group. Students will work in their groups to determine and mark the locations of the new buildings on their group coordinate plane. Label each building with the color on the card and its name. While students are working, actively monitor students. Ask the following questions: DOK-2 How did you determine that Little Tot Park is located in that spot? 1 Student answers will vary. The ﬁrst number in the ordered pair is 5__2, so 1 __ I knew I needed to move right to 52 on the x-axis. The second number is 1 −8, so I knew I had to move down from 5__2 eight spaces.

DOK-1 What quadrant is the water park located in? The water park is in quadrant three.

Give a Student Journal to each student. Have each student record their group’s location of each building on their Student Journals. Allow time for students to complete Part I of their Student Journals, including the reﬂection questions. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 Which quadrant has positive x values and positive y values? Quadrant I DOK-1 Which quadrant has negative x values and negative y values? Quadrant III DOK-3 How can the signs of ordered pairs help you determine whether you correctly plotted the point on the graph? The signs of ordered pairs help direct which way you should move on the graph. Negative x values will be to the left of the y-axis, and positive x values will be to the right of the y-axis. Negative y values will be below the x-axis, and positive y values will be above the x-axis.

Part II: Reﬂections 1.

3. 4. 5. 6. 7.

Read the following scenario to the class: Your development team wants to add a few more buildings to the city that are reﬂections of locations of current buildings in the city. Help determine the locations of these additional buildings. Have a class discussion about reﬂections. Ask the following questions: a.

DOK-2 The Fill Up Gas Station is currently at what location? (8, −2)

DOK-2 You want to put a coffee shop in the same location but north of the main street (x-axis). What would the ordered pair of the coffee shop be? (8, 2)

DOK-2 What is a reflection? Reflections are mirror images of an object. Reflections on a graph are points that are on opposite sides of one or both axes.

Give a Part II bag with Building Location Cards to each group. Have students work in their groups to find the locations of four new buildings using the reflection of locations of buildings currently on the map. After each group locates all of the new buildings on the map, have students mark and label the buildings on their Student Journals. Allow time for students to complete the reflection questions at the end of Part II. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-3 How do signs in ordered pairs differ in reflections across the x-axis? The sign for the x value stays the same, but the sign for the y value changes to the opposite. If the sign for the y value was positive, it becomes negative. • DOK-3 How do signs of ordered pairs differ in reflections across the y-axis? The sign for the x value changes to the opposite, but the sign for the y value stays the same. If the sign for the x value was positive, it becomes negative. • DOK-2 What is the ordered pair for the reflection of (−3, 7) across the x-axis? (−3, −7) • DOK-2 What is the ordered pair for the reflection of (8, 12) across the y-axis? (−8, 12) •

Post-Explore 1. 2. 3. 4.

Acceleration

After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat • • •

Intervention

FACILITATION TIP Be sure to have examples of the four quadrants with labels recorded in the classroom for students to use as reference. If you haven't reviewed the Roman numerals commonly used, take time to reinforce them.

COORDINATE PLANES

Home

FACILITATION TIP Prompt students to reflect on the previous two Math Chat Questions and how the signs of the ordered pairs can help them determine if the points are in the correct quadrant. FACILITATION TIP Before reading the scenario, ask the class 1) When you look in a mirror, what do you see?; 2) Where else would you see a reflection besides a mirror? FACILITATION TIP It might help to model reflections by folding a quadrant along the x-axis and/or y-axis to demonstrate how the reflections match up. You could even poke a hole through the folded plane. Show students where it goes through and matches up depending on which axis it is reflected over. FACILITATION TIP Display a coordinate grid for students to see, and plot the locations of the gas station and the coffee shop. This will allow them to visualize the two locations to better understand reflections. FACILITATION TIP Take time to explain that the location of the coffee shop from the previous question is a reflection of the Fill Up Gas Station's location across the x-axis. Point out that the gas station is located 8 units east and 2 units south of the courthouse, but the coffee shop is 8 units east and 2 units north of the courthouse. FACILITATION TIP Have students identify locations on the map that represent reflections across the x-axis. Do the same for the y-axis. FACILITATION TIP If students struggle to name the ordered pair of the reflection, allow them to plot the points on a coordinate grid. 287

COORDINATE PLANES

Coordinate Planes Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Number Lines and Coordinate Planes Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Reﬂections on a Coordinate Plane

Interactive Vocabulary

Interactive Notebook

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

A cut-and-glue activity to process learning that can be added to a notebook for future reference

Independent practice assignment that gives students an opportunity to demonstrate their learning

288

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Coordinate Planes

Can be done independently

COORDINATE PLANES

Home

Independent and partner games and other activities that provide students with an engaging way to practice the new concept

289

COORDINATE PLANES

Coordinate Planes Intervention and Assessment SMALL-GROUP PLANNER 1

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

Students who are still acquiring the concept and need remediation

How to Use the Review

3 290

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

COORDINATE PLANES

Home

What does mastery look like?

I can explain that a rational number is a point on the number line.

I can ﬁnd and position integers and other rational numbers on a coordinate plane.

I can extend coordinate axes to represent points in the plane with negative number coordinates.

I can use numerical and graphical reasoning to plot points in all four quadrants on the coordinate plane.

I can interpret points in all four quadrants on the coordinate plane based on the signs.

I can show and explain the relationship between ordered pairs and their locations on the coordinate plane.

291

SCOPE 1

Coordinate Plane Problem Solving Scope Introduction SCOPE SUMMARY Students will be able to solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. They will use the coordinates and absolute value to ﬁnd distances between points with the same ﬁrst coordinate or the same second coordinate. Students will connect graphed ordered pairs to form polygons and will determine the lengths of their sides. They will also graph rectangles, and using their knowledge of the length of each side, they can determine area and perimeter. Student Expectations

6.PAR.8.3 Solve problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to ﬁnd distances between points with the same x-coordinate or the same y-coordinate. 6.NR.8.4 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to ﬁnd the length of a side joining points with the same x-coordinate or the same y-coordinate.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

Students in previous grades understand positive rational numbers as points on a number line. Fifth grade is the ﬁrst time that students graph ordered pairs on a coordinate plane. In a previous sixth grade scope, they have represented real-world and mathematical problems by graphing and interpreting points in all four quadrants of the coordinate plane.

In seventh grade, students will apply their understanding of coordinate planes to proportional relationships on graphs and ﬁnding the constant of proportionality. They will graph and interpret the unit rate as the slope of a graph and compare proportional relationships. Seventh-grade students will also graph and interpret two-step inequalities. By the eighth grade, students will use their understanding of graphing proportional relationships and inequalities on a coordinate plane to graph and interpret linear functions and linear inequalities.

Before beginning the lesson, the students’ prior knowledge is assessed to address any student needs and skills needed for content mastery. Students are assessed to determine if they can: •

represent, solve, and interpret math problems through graphing points on a coordinate plane.

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope.

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

determine the distance between two points using a coordinate plane.

Students move on to the Explore activities and come back at the end of the explorations to repeat the activity and discuss what they found.

If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. 292

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Distances between Points In this exploration, students will use the provided information to plot Spring Egg-Toss team results on a coordinate plane and determine the winner of the Spring Egg-Toss competition. Students will: •

plot coordinates that share the same ﬁrst coordinate or same second coordinate on a graph.

•

calculate the distance between the points.

•

ﬁnd missing coordinates of a rectangle when given three vertices.

Explore 2

Explore 1

EXPLORE ACTIVITIES Polygons on a Coordinate Plane In this exploration, students will solve a scenario about upgrading Pecan Park. Students will: •

draw polygons on coordinate planes with given coordinates for the polygon vertices.

•

solve problems about area and perimeter of a graphed polygon.

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

COORDINATE PLANE PROBLEM SOLVING

Home

293

COORDINATE PLANE PROBLEM SOLVING

Coordinate Plane Problem Solving Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

294

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will match numbered cards with lettered cards posted around the room to demonstrate their knowledge of the prior standard. This element is designed to uncover student misconceptions; it should not be taken for a grade. 6.PAR.8.1 Locate and position rational numbers on a horizontal or vertical number line; ﬁnd and position pairs of integers and other rational numbers on a coordinate plane.

Materials

Preparation

Printed •

• •

1 Set of Match around the Room Cards (per class)

Print one set of Match around the Room Cards. Hang them in a random order around the room.

Procedure and Facilitation Points 1. 2.

Have students write the numbers 1, 2, and 3 on a sheet of paper. Instruct students to walk around the room with their papers. As students walk around the room, they need to see the numbered cards and match them with the lettered cards. Allow students to share their thinking with their neighbors. a.

Card 1 matches with Card B.

Card 2 matches with Card A.

Card 3 matches with Card C.

•

FACILITATION TIP Before students begin moving, have them record the ordered pairs from Cards A, B, and C on their sheet of paper. FACILITATION TIP Before students begin walking around, project the numbered cards so students can see them clearly as needed. FACILITATION TIP

If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

Identifying Misconceptions •

COORDINATE PLANE PROBLEM SOLVING

Home

Students may struggle to identify that the coordinate pairs are written in the format ((x, y). y Students may struggle to recall that you move along the x-axis ﬁrst and then the yy-axis -axis to plot a point.

To get a sense of the answers students chose, perform a quick formative assessment. Hold up a letter card and have students hold up one, two, or three ﬁngers to indicate which number card they think is its match. Discuss their thinking as a whole group.

295

COORDINATE PLANE PROBLEM SOLVING

Coordinate Plane Problem Solving Hook– Road Trip ACTIVITY PREPARATION Students will ﬁnd the distance between two points on a map and calculate the total perimeter of a trip.

Materials

Preparation

Printed •

• • •

1 Road Trip (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Road Trip for the whole class to view. Prepare to introduce the scenario and encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION POINTS Part I: Pre-Explore 1. FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) Has anyone ever taken a road trip?; 2) If so, where did you go?; 3) What did you see? STEMscopes Tip Content Support, found in the Home section of each scope, provides teachers who might need additional background knowledge with a complete explanation of student expectations, mathematical vocabulary, an explanation of the progression of the related standards learned, strategies for instruction, possible misconceptions and obstacles, and more.

4. 5.

DOK-1 How do we ﬁnd the location of a point on the coordinate plane? The number of places along the x-axis is the ﬁrst coordinate in an ordered pair, and the number of places along the y-axis is the second coordinate in an ordered pair.

DOK-2 What does the key on the bottom of the map tell us? The key tells us that each box represents 10 miles.

DOK-2 Why would he want to drive in a rectangular shape? It would be a shorter distance and save on gas.

Complete the Explore activities. Notes

296

Engage

Explore

Explain

Elaborate

Evaluate

Acceleration

FACILITATION TIP

Part II: Post-Explore 1. 2.

Intervention

Show the Phenomena Video again, and restate the problem. Refer to Road Trip, and discuss the following questions: a.

DOK-1 How can you determine the distance between two points? Answers will vary. You can count the number of spaces between them. You can subtract the coordinates that are different.

DOK-1 How far did Omar drive to get to the hiking trail? Explain. The distance from his house to the hiking trail is 8 units. The key tells us each unit is 10 miles. He drove 80 miles.

DOK-1 What was the last stop Omar made before he went back home? He drove in a rectangular shape, so the last stop was the museum.

DOK-2 How far did Omar travel over his trip? How did you ﬁnd this number? He traveled a total distance of 340 miles. I found the perimeter of the rectangle he traveled by adding up all 4 sides of the rectangle.

As students share ideas, they may decide that Omar should see all of the sights rather than just drive in a rectangular shape. Before your class discussion, consider "Can he go to the Dairy Farm or the Amusement Park?" FACILITATION TIP If students struggle to explain how to find the location of a point, ask them to explain how to find the location of one of the points on the map. FACILITATION TIP Discuss why it is important to pay attention to the key and how to use it when finding the distance between two points.

COORDINATE PLANE PROBLEM SOLVING

Home

297

COORDINATE PLANE PROBLEM SOLVING

Coordinate Plane Problem Solving Explore 1 – Distances Between Points ACTIVITY PREPARATION Students will plot coordinates that share the same ﬁrst coordinate or same second coordinate on a graph and calculate the distance between the points. Students will also ﬁnd the missing coordinates of a rectangle when given three vertices.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.5 Use appropriate tools strategically. MP.6 Attend to precision.

Materials

Preparation

Printed • •

•

1 Student Journal (per student) 1 Exit Ticket (per student)

• •

Reusable •

Plan to divide the class into 6 groups (4 or 5 students in each group). Print a Student Journal and an Exit Ticket for each student. Gather enough sets of colored pencils for each student to have one set.

1 Set of colored pencils (per student)

PROCEDURE AND FACILITATION POINTS FACILITATION TIP

Part I: Partner Competition

Before reading the scenario, ask the class 1) Have you ever participated in a competition?; 2) If so, what did you have to do?; 3) Did you compete with a partner, a group, or by yourself?

FACILITATION TIP For context, allow students to participate in a paper ball toss. Discuss how students could determine the distance between them and their partner. Have them consider conventional measuring tools and unconventional measuring tools.

2. 3.

FACILITATION TIP If students are struggling, provide an example of partner locations and model how to ﬁnd the distance between two points using a coordinate plane blackline master in the Teacher Toolbox. As students are working, have them label each point they plot with the partner number for each team.

298

Read the following scenario to the class: Pecan Park was holding its annual Spring Egg-Toss Competition. In the partner competition, teams consisted of a pair of participants who continued to move apart until they dropped their egg. The coordinates of each member of four different teams were recorded in a table. The coordinates reﬂected the positions of the partners during their last successful egg toss. The team with the greatest distance between partners was the winner of the competition. Use this information to plot each team on the coordinate plane provided and calculate the distance between the team members to determine the winner of the competition. Give a Student Journal to each student. Students will work in their groups to plot the location of each partner on the coordinate plane by using their colored pencils to connect each team’s coordinates with its corresponding team color. Then, students will calculate the distance between each set of partners on the map and record the distance in the table. Monitor student collaboration, and use the following guiding questions to assess understanding: a.

DOK-2 What do you notice about the coordinates of the orange team and the purple team? The pairs share an x-coordinate.

DOK-1 What do these coordinates look like when graphed? They are on a vertical line segment.

DOK-1 Why are none of the distances negative? Distance cannot be negative. No matter what direction you are tossing the egg, it is still traveling a positive distance.

DOK-1 What is a mathematical term that represents a number’s distance from 0? Absolute value

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 What do you notice about the coordinates for each pair on a team? What does this tell you about the location of the pair on the coordinate plane? The coordinates for each team share either an x-coordinate or a y-coordinate. This means that the pair of coordinates will either be vertically or horizontally aligned. • DOK-1 How are the coordinates related to the distance? The distance between coordinate points is the difference between the absolute values of the aligned coordinates. •

Part II: Group Showcase

3. 4.

Read the following scenario to the class: After the partner competition was over, groups formed rectangles to see how long they could toss the egg before dropping it. This showcase was just for fun. The coordinates of three members of each group were recorded in a table. Plot each member’s coordinates, and determine the location where the fourth member stood to complete the rectangle egg toss. Students will work in their groups to plot the location of each group member on the blank coordinate grid in Part II of their Student Journals. Students will use their colored pencils to connect each group’s coordinates with their corresponding color. Then, they will determine the coordinates for the fourth member that would complete the rectangle and record the coordinates in the table on page 3. Remind students to record the fourth member’s coordinates in the table. Monitor student collaboration, and use the following guiding questions to assess understanding: a.

DOK-1 What is the x-coordinate for member 4 of the blue group? 6

DOK-1 What is the y-coordinate for member 4 of the blue group? 2

DOK-2 What do you notice about the x-coordinates in the pink group? There are two pairs of matching x-coordinates in the group.

DOK-1 Is that true for each team? Yes

e. DOK-2 What does this tell you about the x-coordinates of a rectangle? Every rectangle will have two pairs of vertices with the same x-coordinate. 5.

Acceleration

Once students are ﬁnished plotting the coordinates and completing the table, they will answer the questions in their Student Journals. After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat

Intervention

Once students are ﬁnished plotting the coordinates and completing the table, they will answer the questions in their Student Journals. Allow time for students to complete the reﬂection questions at the end of Part II. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 If two ordered pairs share an x-coordinate, what does that tell you about the location of the points on the coordinate plane? If two ordered pairs share an x-coordinate, then they are vertically aligned (they are on the same vertical line). • DOK-1 If two ordered pairs share a y-coordinate, what does that tell you about the location of the points on the coordinate plane? If two ordered pairs share a y-coordinate, then they are horizontally aligned (they are on the same horizontal line). • DOK-1 How is absolute value related to finding the distance between two ordered pairs that share either an x-coordinate or a y-coordinate? To find the distance between points on a shared axis, find the difference in the absolute values of their aligned coordinates. |a – b| or |b – a| •

FACILITATION TIP Point out either a vertical line or horizontal line from the coordinate plane. Prompt students to explain which coordinate is the same for the points and how it relates to the type of line that is created.

FACILITATION TIP

COORDINATE PLANE PROBLEM SOLVING

Home

Before reading the scenario, ask the class 1) Would you like to participate in an egg toss?; 2) Why or why not?; 3) How far do you think you would be able to toss an egg to someone without the egg breaking?

FACILITATION TIP To encourage deeper thinking, ask what students notice about the y-coordinates. Hold a brief class discussion about why. FACILITATION TIP To encourage deeper thinking, ask students to observe if the same is true for the y-coordinates. Hold a brief discussion about why that is true for a rectangle. STEMscopes Tip Use the Content Unwrapped element in the Home section to see the instructional expectations clariﬁed. Here you will see what students should be doing, what students should know, and implications for instruction. Included in this element is a complete vertical alignment related to this topic that shows how student expectations span across applicable grade levels.

299

COORDINATE PLANE PROBLEM SOLVING

Coordinate Plane Problem Solving Explore 1 – Distances Between Points •

DOK-2 If you were asked to ﬁnd the fourth coordinate of a square, would the process change? Explain. The process would not change. A square is a rectangle, so it would still have two matching x-coordinates and two matching y-coordinates.

Post-Explore 1.

FACILITATION TIP

2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

During the Exit Ticket, provide students with a small coordinate grid to sketch on to help them solve the scenario. Also, note that one of the coordinate pairs (on the digital version) is split up between text rows.

300

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

COORDINATE PLANE PROBLEM SOLVING

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COORDINATE PLANE PROBLEM SOLVING

Coordinate Plane Problem Solving Explore 2 – Polygons on a Coordinate Plane ACTIVITY PREPARATION Students will draw polygons on the coordinate plane given coordinates of the vertices. Students will also solve problems involving area and perimeter of graphed polygons.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.5 Use appropriate tools strategically. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

•

1 Student Journal (per student) 1 Set of Pecan Park Cards (per group) 1 Exit Ticket (per student)

• •

Consumable •

2 Resealable bags (per group)

Plan to divide the class into 6 groups (4 or 5 students in each group). Print a Student Journal and an Exit Ticket for each student. Print a set of Pecan Park Cards for each group. Cut out the Part I: Pecan Park East cards, and place them in a resealable bag labeled “Part I” for each group. Cut out the Part II: Pecan Park West cards, and place them in a resealable bag labeled “Part II” for each group.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before reading the scenario, ask the class 1) Have you ever been to a public park?; 2) If so, what activities did the park offer?; 3) What was your favorite activity to do?

FACILITATION TIP Consider providing straight edges for students to use to connect coordinates. Demonstrate how to slide the straight edge side to side and up and down to accurately locate and mark the points.

Part I: Pecan Park East 1.

2. 3. 4.

FACILITATION TIP Some students may have diﬃculty connecting the points in order to create the polygons. Encourage them to connect the points as they go to ensure they create the correct ﬁgure.

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Read the following scenario to the class: Pecan Park received funding to upgrade some of their equipment and facilities. The work is being done in two phases. In Phase 1, Pecan Park East is redesigning the layout of the climber dome, slides, baseball ﬁelds, and swings. The park chairperson wants to ensure that none of the equipment or facility areas overlap. Each piece of equipment or facility area is represented by a polygon, and the coordinates of the vertices of each area are given. Plot the coordinates of the vertices, and connect them to mark each area on the coordinate plane to determine whether any areas overlap. Give a Student Journal to each student. Give a set of the Part I: Pecan Park East cards to each group. Students will work cooperatively to read each card and plot the vertices for each piece of equipment or facility area on the coordinate plane on page 1 of the Student Journal. Once the vertices from each piece of equipment or facility area have been plotted, students will draw a polygon by connecting the points. Students will then use the completed park map to determine whether any of the equipment or facility areas overlap. Monitor student collaboration, and use the following guiding questions to assess understanding: a.

DOK-1 What polygon is the climbing dome shaped like? The climbing dome is shaped like an octagon.

FACILITATION TIP

Take this opportunity to reinforce the characteristics of the four quadrants. Ask which quadrant has positive x-coordinates and positive y-coordinates. Ask which quadrant has negative x-coordinates and negative y-coordinates.

DOK-1 How are the coordinates for the slide different from the coordinates of the other equipment or areas? All of the vertices for the slide are positive.

DOK-1 How do you plot the coordinate (−2.5, −2.5)? You have to think of axes like number lines. −2.5 is halfway between −2 and −3. I move to −2.5 on the x-axis, and then move down −2.5 units on the y-axis. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

FACILITATION TIP DOK-1 Why is it important for these areas not to overlap? The equipment and facilities need to be spread out so there is enough space for each If students are struggling to determine item. If two pieces overlap, those areas would be built on top of each where –2.5 is on the axes, prompt them to other. label the halves on the axes, starting with the positive axes. Allow time for students to complete the questions at the end of Part I on their d.

Student Journals.

Part II: Pecan Park West 1.

2. 3.

Read the following scenario to the class: Pecan Park West will hold the basketball court, the sandpit, and the picnic area. Each of these facilities requires additional work to complete the upgrade. Each facility is represented by a polygon, and the coordinates of the vertices are given. Plot the coordinates, and connect them to mark each area on the coordinate plane. Then, help the chairperson determine the additional calculations required to complete the upgrades. Give a set of the Part II: Pecan Park West cards to each group. Have students work in their groups to plot the vertices for each facility area on the coordinate plane on page 2 of their Student Journals. Once the vertices from each facility area have been plotted, students will draw a polygon by connecting the points. Students will then use the completed park map to determine the additional calculations required to complete the upgrades on their Student Journals. Monitor student collaboration, and use the following guiding questions to assess understanding: a.

DOK-1 What is the distance between −5 and −2? 3

DOK-1 Why are there 6 grid boxes on the graph between the coordinates (−5, −2.5) and (−2, −2.5)? There are 6 boxes because the scale of the graph is counting by 0.5. So each box represents 0.5 instead of 1.

DOK-1 How do you ﬁnd the area of a rectangle? The area of a rectangle is calculated by multiplying the length times the width.

DOK-1 How do you ﬁnd the perimeter of a rectangle? The perimeter of a rectangle is found by adding all of the sides together or by using the formula P = 2l + 2w.

FACILITATION TIP Before reading the scenario, ask the class 1) If you could design a public park, what areas would you include? For example, areas might be a picnic area, skateboard area, etc.

FACILITATION TIP Discuss with students how this affects how they plot points on the coordinate grid. Instead of counting how many units horizontally and vertically they need to move, students need to focus more on the numbers on the axes.

e. DOK-1 How do you determine when to calculate the area and when to calculate the perimeter? The area is calculated when you are covering or ﬁlling in the inside of an object. The perimeter is calculated when you need to measure the distance around the outside of an object.

FACILITATION TIP

DOK-1 How do you determine the length of a rectangle graphed on a coordinate plane? I count the units between two vertices on a horizontal line segment. • DOK-1 How would you determine the side length of a rectangle given the vertices (−2, 4) and (4, 4)? You would look at the coordinate in each pair that is different, ﬁnd the absolute value of those numbers, and then add. –2 = 2 and 4 = 4, so 2 + 4 = 6 units.

FACILITATION TIP

5. 6.

COORDINATE PLANE PROBLEM SOLVING

Home

Create an anchor chart to help students distinguish between perimeter and area and how to calculate them. Reference the Allow time for students to complete the Part II questions and reﬂection questions anchor chart when necessary. at the end of their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat •

Post-Explore 1. 2. 3. 4.

It is important to pay attention to how the axes are scaled. If the axes are not counting by ones, students will need to subtract the coordinates of the vertices. Use the coordinate plane in Part II to make that point.

FACILITATION TIP Have students complete the Exit Ticket to formatively assess their understanding Depending on students' needs, provide a of the concept. coordinate plane for students to model the Complete the Anchor Chart as a class. problems on the Exit Ticket. Have each student complete their Interactive Notebook. Return to the Hook and instruct students to use their newly acquired skills to successfully complete the activity.

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COORDINATE PLANE PROBLEM SOLVING

Coordinate Plane Problem Solving Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Distances between Points Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Polygons on a Coordinate Plane Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Interactive Notebook

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

A cut-and-glue activity to process learning that can be added to a notebook for future reference

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Coordinate Plane Problem Solving Independent and partner games and other activities that provide students with an engaging way to practice the new concept

Notes

COORDINATE PLANE PROBLEM SOLVING

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Intervention and Assessment SMALL-GROUP PLANNER 1

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

COORDINATE PLANE PROBLEM SOLVING

Coordinate Plane Problem Solving

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Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER Use this template to decide how to assess your students for concept mastery. Depending on the format of the assessment, you can identify prompts and intended responses that would measure student mastery of the expectation. See the beginning of this scope to identify standards and grade-level expectations. Key Concepts I can use absolute value on a coordinate plane to ﬁnd distance.

What prompts will be used?

What does mastery look like?

COORDINATE PLANE PROBLEM SOLVING

Home

I can solve problems with polygons when given coordinate pairs with or without a coordinate grid.

I can create a polygon on a coordinate plane when I am given coordinates that create the vertices.

I can solve real-world problems with polygons on a coordinate plane.

I can use given coordinates to ﬁnd side lengths that join with the same x-coordinate or the same y-coordinate.

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SCOPE 1

Area and Volume Scope Introduction SCOPE SUMMARY

Student Expectations

Students in sixth grade continue to understand area as the number of squares needed to cover a plane figure. They find the area of triangles, quadrilaterals, and other polygons by decomposing these shapes, rearranging pieces, and relating the shapes to rectangles. As students compose and decompose shapes to determine area, they learn that area is conserved. For example, students decompose trapezoids into triangles and/or rectangles and use this reasoning to determine formulas for the area of a trapezoid. Students find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths and show that the volume is the same as would be found by multiplying the edge lengths of the prism.

6.GSR.5.1 Explore area as a measurable attribute of triangles, quadrilaterals, and other polygons conceptually by composing or decomposing into rectangles, triangles, and other shapes. Find the area of these geometric ﬁgures to solve problems. 6.GSR.5.3 Calculate the volume of right rectangular prisms with fractional edge lengths by applying the formula, V = (area of base) × (height).

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In sixth grade, students extend their previous understanding of length, area, and volume as they solve problems by applying formulas for the area of triangles and parallelograms and the volume of rectangular prisms. Students build on their work with area from previous grade levels by reasoning about relationships between shapes to determine area and volume. They continue to understand area as the number of squares needed to cover a plane ﬁgure. In prior grades, students calculate the volume of right rectangular prisms using whole-number edges and understand that doing so means ﬁnding the number of unit cubes within a solid shape. In sixth grade, students extend this work to unit cubes with fractional edge lengths.

After this scope, students in sixth grade will explore the surface area of ﬁgures using nets made up of rectangles and triangles. In seventh grade, students will extend their knowledge to solve problems involving the volume and surface area of two- and three-dimensional objects composed of triangles, circles, quadrilaterals, polygons, cubes, cylinders, and right prisms. This work will lead to the upcoming concepts of scale drawings, rates, ratios, and similar shapes.

Before beginning the lesson, the students’ prior knowledge is assessed to address any student needs and skills needed for content mastery. Students are assessed to determine if they can: •

ﬁnd the volume of a right rectangular prism with unit cubes.

•

show how to ﬁnd volume.

•

represent products as volumes.

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope.

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

determine the volume of the new planter box.

•

determine the amount of soil that will need to be purchased.

Students move on to the Explore activities and come back at the end of the explorations to repeat the activity and discuss what they found.

If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. 308

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Discovering Area Formulas In this exploration, groups of students will solve a scenario where they pretend they work for a landscaping company and must determine how to ﬁnd the area of spaces through decomposing and rearranging different shapes. Students will: •

decompose triangles, trapezoids, and parallelograms.

•

rearrange their parts to form rectangles.

•

make connections to the ﬁgures’ area formulas, the formula for area of a rectangle, and use those formulas to calculate area.

Explore 2

Explore 1

EXPLORE ACTIVITIES Finding the Area of Quadrilaterals

AREA AND VOLUME

Home

In this exploration, students will solve a real-world problem involving helping the X-traordinary Landscaping Company determine the area of their customers’ gardens for blueprints. Students will: •

decompose and rearrange ﬁgures to ﬁnd the area of the quadrilaterals.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

Finding the Area of Triangles In this exploration students will be tasked with helping the landscapers determine how to ﬁnd the area of customers’ gardens. Students will: •

Explore 4

Explore 3

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

decompose and rearrange ﬁgures to ﬁnd the area of triangles.

In this exploration, groups of students will help the landscaping company again to determine the area of gardens. Students will: •

determine the area of composite ﬁgures.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

Explore 5

Finding the Area of Composite Figures

Volume of Rectangular Prisms In this exploration, students will be faced with a scenario The Farmer’s Company to help the company measure their produce cartons. Students will: •

ﬁnd the volume of cartons.

•

ﬁnd the volume of different shipping boxes.

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve.

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AREA AND VOLUME

Area and Volume Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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Explore

Explain

Elaborate

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Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will examine a series of tables and determine which option does not belong with the group. This element is designed to uncover student misconceptions; it should not be used as a summative assessment.

AREA AND VOLUME

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4.GSR.8.3:: Solve problems involving area and perimeter of composite rectangles involving whole numbers with known side lengths. 4.GSR.8.3 5.GSR.8.4:: Discover and explain how the volume of a right rectangular prism can be found by multiplying the area of the base times the height to solve authentic, mathematical problems.

Materials

Preparation

Printed •

•

1 Does Not Belong (per student or per group)

•

If not assigning the APK digitally, print one Does Not Belong for each student. Plan to divide the class into groups of two or three.

Procedure and Facilitation Points 1. 2. 3.

4. 5.

Pass out the Does Not Belong to each student or group. Explain that each table on the handout contains four options. Three of the options go together, while one does not belong. Instruct students to determine which letter does not belong in each group and to explain their thinking. a.

Set 1: B does not belong because it is the only option involving volume. The other options involve area.

Set 2: C does not belong because it is the only option involving surface area. The other options involve volume.

Set 3: D does not belong because it describes an area of 18 square inches, and the other options describe an area of 16 square inches.

Set 4: A does not belong, because it describes a volume of 60 cubic inches, and the other options describe a volume of 64 cubic inches.

Identifying Misconceptions •

•

FACILITATION TIP Set a timer for student groups to collaborate. When the time is up, have each group vote using hand signals for A, B, C or D. After each vote, call on groups to explain their thinking. Lead a discussion about their reasoning about each set as you move along. FACILITATION TIP Be prepared for students to ﬁnd alternative logical reasons for why other letters don't belong; allow them to explain their thinking. FACILITATION TIP While you discuss each set, draw students' attention to key words that may help them decipher what measurement is involved in each set (ﬁll, wrap, outside, inside, etc).

Students may not realize that area measures the number of square units that cover the top of a two-dimensional figure, and volume measures the number of cubic units that fill the inside of a three-dimensional figure. Students may not know that the area of a rectangle is length times width, and the volume of a rectangular prism is length times width times height. Notes

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AREA AND VOLUME

Area and Volume Hook – Garden Plots ACTIVITY PREPARATION Students will determine the area of parallelograms and triangles and the volume of rectangular prisms with fractional dimensions.

Materials

Preparation

Printed •

• • •

1 Garden Plots (per class)

Reusable •

1 Phenomena Video (per class)

•

Plan to show the video. Prepare to project the Garden Plots slide for the whole class to view. Prepare to introduce the scenario and encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION POINTS Part I: Pre-Explore 1. FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) Does anyone have a garden at your house?; 2) If so, what is planted in your garden?; 3) What shape is your garden?

FACILITATION TIP To stimulate students' answers to the notice, wonder and math questions, project the Garden Plots slide after the video and before asking the questions.

4. 5.

FACILITATION TIP Students may comment that they notice that one garden will be much smaller (inches) than the other. FACILITATION TIP This is a good time to review pyramids vs. prisms.

FACILITATION TIP Project the original scenario and call on students to restate the problem. Ask, "What do we know? What do we need to ﬁnd out?"

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Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the following situation while showing the video: Sarah is working to create new garden styles. She wants each of her new gardens to be a new unique shape. She has even decided to create a high-rise planter box! Once she decides on a shape, she will need to determine the area of the new gardens. She also needs to determine the volume of the new planter box to ﬁnd how much soil she will need to purchase. Ask students the following questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Student answers will vary. I notice that Sarah is ﬁnding area and volume. I wonder what shapes Sarah will use to create her new unique garden. What will Sarah plant in the garden? I can use math to determine the area of the garden and volume of the planter box. Project Garden Plots. Explain to students that Sarah has decided on the shapes for her new unique garden and planter box! She shared her ideas of what the dimensions should be for each new plot. Discuss the following questions: a.

DOK-1 Why might the unique garden have two measurements for length? The bottom of the garden is longer than the top of the garden so there are two different-sized lengths.

DOK-1 What is the shape of the unique garden? The unique garden is a trapezoid.

DOK-1 What shape is the planter box? The planter box is in the shape of a rectangular prism.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and restate the problem. Refer to Garden Plots, and discuss the following questions: a.

DOK-1 How can you determine the area of the unique garden? The unique garden is a trapezoid. We can determine the area of the unique 1 garden by using the formula A = __2(b1 + b2)(h). © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 What is the area of the unique garden? The area of the unique garden is 21.3 square feet.

DOK-1 How can you determine the volume of the planter box? The planter box is shaped like a rectangular prism. I can ﬁnd the volume of the planter box by using the formula V = lwh.

DOK-1 What is the volume of the planter box? The volume of the planter box is 60 cubic inches.

Intervention

Acceleration

FACILITATION TIP After completing the Explore activities, students may know two volume formulas: V = Bh and V = lwh.

AREA AND VOLUME

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Area and Volume Explore 1 – Discovering Area Formulas ACTIVITY PREPARATION Students will decompose triangles, trapezoids, and parallelograms and rearrange their parts to form rectangles. Students will make connections to the ﬁgures’ area formulas and the formula for area of a rectangle and use those formulas to calculate area.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 2-D Figures Work Mat (per group) 1 Set of Shapes (per group) 1 Exit Ticket (per student)

• • •

Plan to divide the class into groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Gather the tangrams, and place the following shapes in a resealable bag for each group: • 4 small triangles • 1 medium triangle • 2 squares

•

Print a set of Shapes for each group. Cut out and place each set in the resealable bags with the tangrams. If desired, print the shapes on card stock and laminate them for future use. Print one 2-D Figures Work Mat per group. Place it inside a clear sheet protector for students to use dry-erase markers on it. If desired, print it on card stock and laminate it for future use. Gather enough dry-erase markers for each student to have one. Optionally, have a projector or document camera ready in case students need help decomposing shapes or identifying base and height.

Reusable • • • • • •

1 Projector (per class, optional) 1 Set of tangrams (per group) 1 Dry-erase marker (per student) 1 Dry-erase eraser (per student) 1 Clear sheet protector (per group) 1 Resealable bag (per group)

•

• •

PROCEDURE AND FACILITATION POINTS

FACILITATION TIP Before reading the scenario, ask the class 1) What do landscaping companies do?; 2) Have you ever seen landscapers working?; 3) If so, what types of jobs were they doing? FACILITATION TIP

Part I: Decomposing and Rearranging Figures Using Tangrams to Explore Area 1.

2. 3.

Consider asking students, “Who has professional landscapers in their family?”

314

DOK-1 What are perpendicular lines? Lines that meet at right angles

FACILITATION TIP

DOK-1 What is a right angle? An angle that measures 90°

Use two contrasting colors for the tangrams and the 2-D Figure Work Mat. The contrast will help you monitor students' creations.

DOK-1 What are parallel lines? Lines that are always the same distance apart from each other

Explain to students that they will use tangrams to decompose and rearrange figures to create other 2-D figures. Note that when decomposing figures and rearranging figures, students should understand that 1 square represents 1 square unit. Students should be encouraged to explore a variety of ways to create © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

2-D ﬁgures that help support their understanding of area. Encourage students to create triangles, squares, rectangles, parallelograms, and trapezoids. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 What 2-D ﬁgures can be used to decompose a square? Answers will vary. You can decompose a square using 2 small triangles.

Intervention

Acceleration

FACILITATION TIP As you lead students through this discussion, write down the deﬁnitions and draw simple examples for them to copy onto their Student Journal or another notebook. Refer to the Visual Glossary.

AREA AND VOLUME

Home

FACILITATION TIP While allowing students to explore, clarify some constraints for students if needed. For example, “Can the shapes overlap? Can they be folded or cut?”

DOK-1 What 2-D ﬁgures can be used to decompose a rectangle? Answers will vary. You can decompose a rectangle using 2 small triangles and a square.

FACILITATION TIP Print and project some of the essential guiding questions for students to see while they are working with the shapes. Some students will need the extra structure. FACILITATION TIP

DOK-2 How is your rectangle similar or different from another group’s rectangle? Answers will vary. I used 2 small triangles and a square to compose a rectangle, and another group used 4 small triangles to compose a rectangle. Although different 2-D ﬁgures were used to compose a rectangle, when we put our rectangles on top of one another, we noticed that the length and width of both rectangles are the same.

DOK-1 What is a parallelogram? Include attributes of a parallelogram. Answers will vary. A parallelogram is a type of quadrilateral that has two sets of parallel sides. A rectangle is a special parallelogram and has opposite sides that are equal and four right angles. A parallelogram with base b and height h can be divided into a trapezoid and a right triangle and rearranged into a rectangle.

e. DOK-2 What 2-D ﬁgures can be used to decompose a parallelogram? Answers will vary. I can decompose a parallelogram using 1 medium triangle and 2 small triangles.

Be sure to have names of multiple 2-D ﬁgures listed and labeled on a class word wall, anchor chart, or included in students' notes.

DOK-2 How can you rearrange the parallelogram to create a rectangle? Answers will vary. I can rearrange the parallelogram to create a rectangle by moving the medium triangle to the middle and putting a small triangle on the left of the medium triangle and 1 small triangle on the right of the medium triangle.

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Area and Volume Explore 1 – Discovering Area Formulas

DOK-2 How is a parallelogram similar to a rectangle? Answers will vary. If you decompose a parallelogram and rearrange the parts, you can compose a rectangle. Since the parallelogram and rectangle can be composed of the same parts, the area is the same. The parallelogram and the rectangle have the same base and same height. Since base × height gives the area of the rectangle, we can use the same measurements on the parallelogram to compute its area. If a square is 1 square unit, then the parallelogram and the rectangle both have an area of 2 square units.

h. DOK-2 What 2-D ﬁgures can be used to decompose a trapezoid? Answers will vary. I can use a square and 2 small triangles to decompose a trapezoid.

i. DOK-2 How many square units does the triangle represent? If a square 1 is 1 square unit, then a triangle is __2 of a square unit because I used 2 triangles to compose the square. FACILITATION TIP

j. DOK-2 What is the area of a trapezoid? I can use 1 square and 2 triangles to compose a trapezoid. Since a square has an area of 1 square unit, a trapezoid has an area of 2 square units.

In addition to asking questions 5a–5l, challenge students by projecting shapes (without the decomposed lines showing) for them to construct. How many different rectangles can they make? What is the largest triangle they can construct? Can they create a parallelogram without hints?

Have the students look at just the paper Shapes cutouts.

l. DOK-2 What other 2-D ﬁgures can be used to decompose a trapezoid? Answers will vary. I can use two triangles to make a trapezoid.

FACILITATION TIP

As you explain parallelograms to the class, draw one so they can copy and label the details you are identifying. Perhaps demonstrate folding one or cutting one to illustrate the principle.

Explain the following to the class: Mathematicians say that in a parallelogram, the length of the vertical cut segment is also the length of the vertical side of the rectangle. The height is perpendicular (at right angles) to the base, and the height can be drawn outside of a parallelogram as long as it is drawn at a 90° angle to the base. After Part I of the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat •

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DOK-2 How do the areas of the parallelogram and trapezoid compare to the area of a rectangle? Using the tangrams, I noticed that a parallelogram and a trapezoid are both composed of the same parts (2-D ﬁgures) that can be used to compose a rectangle. Since a parallelogram and trapezoid are composed of the same parts (2-D ﬁgures), they will also have the same area. © Accelerate Learning Inc. - All Rights Reserved

• • •

•

Engage

Explore

Explain

Elaborate

Evaluate

2. 3.

Acceleration

DOK-2 How is the decomposed figure similar to the newly created figures? Each decomposed figure and newly created figure has the same area. DOK-1 How can you decompose a medium triangle? I can decompose a medium triangle using 2 small triangles. DOK-1 How can you determine the base and height in a parallelogram? In a parallelogram, any side can serve as the base, but the height is always perpendicular to the side chosen as the base. DOK-1 How can you determine the base and height in a triangle? The base of a triangle is any one of the sides, and the height of the triangle is the length of the height from the opposite vertex to that base. A right triangle has three sides: the hypotenuse, height, and base of the triangle. The base and height of a right triangle are always the sides adjacent to the right angle, and the hypotenuse is the longest side. DOK-1 How can you determine the base and height in a trapezoid? The parallel sides represent the bases. The base and height of a trapezoid are perpendicular to each other. The perpendicular distance between the two parallel sides of a trapezoid is the height.

Part II: Decomposing and Rearranging Figures to Discover Area Formula 1.

Intervention

Read the following scenario to the class: X-traordinary Landscaping Company works with different-shaped areas to develop beautiful designs. The shapes of these areas are not always rectangular, so workers must determine other ways to find the total area they have to work with. Today, you will help the landscapers determine how to find the area of each space. Give a Student Journal to each student. Give a dry-erase marker to each group. Explain to students that they will decompose and rearrange garden figures on a grid to create rectangles. Note, it is important for students to use the grid to determine the base and height and then find the area. Students should not count the squares in the grid to find the area of 2-D figures. Have students use the 2-D Figures Work Mat, tangrams, and a dry-erase marker to create and label the base and height of each 2-D figure. Have students collaborate with their groups to discuss where the base-height pairs are located on a parallelogram. Note, if students have difficulties identifying the base and height, project the 2-D Figures Work Mat, and discuss with the class how to identify the base and height on parallelograms, triangles, and trapezoids. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 How is the garden shaped in garden 1? The garden is shaped like a parallelogram.

DOK-2 How can you decompose and rearrange garden 1 to create a rectangle? Answers will vary. Students should notice that if they cut one triangular piece off of one side and add it to the other side, then they will have a rectangle. (See image below.)

DOK-2 How can you rearrange garden 2 to create a rectangle? Answers will vary. Students should notice that if they double the triangle, they can make a rectangle.

AREA AND VOLUME

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FACILITATION TIP Before reading the scenario, ask the class 1) If you were designing a garden, how would knowing the total area of the garden help you when purchasing materials and supplies?; 2) What types of materials and supplies would you need to buy for a garden?; 3) How could you ﬁnd the area of the garden if it wasn't a rectangle? FACILITATION TIP To help with precision, students can use pencil and paper to create and label the 2-D ﬁgures; dry erase markers are not as accurate. Students might want to keep the pencil drawings as reference later.

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Area and Volume Explore 1 – Discovering Area Formulas d.

FACILITATION TIP When you explain these formulas to the class, project them with added images of ﬁgures with labels. Use the Visual Glossary if needed. Have students copy the formulas and take notes on the ﬁgures.

7. 8.

If students are struggling to relate these two formulas, project the them and ask, "What is the same about the trapezoid area formula and the triangle area formula? What is different?" Write down student responses.

Explain the following to the class: Mathematicians use the following area formulas to ﬁnd the area of 2-D ﬁgures: for a rectangle, A = lw or A = bh; for a parallelogram, 1 1 A = bh; for a triangle, A = __2bh; and for a trapezoid, A = __2(b1 + b2)h. Allow students enough time to record all their work for Part II of the Explore activity on their Student Journals. After Part II of the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat • •

•

FACILITATION TIP

DOK-2 Could I use more than one shape to make a rectangle or a parallelogram? What would I need to add to the formula if I use more than one? Yes, you can use 2 triangles to make a rectangle/ parallelogram, and you can use two trapezoids to make a parallelogram. 1 I will need to add __2 to the formula if I use more than one shape.

•

DOK-1 How do you calculate the area of a rectangle? Multiply the base of the rectangle by its height. DOK-2 Why is the area formula for a parallelogram the same as the area formula for a rectangle? A parallelogram can be decomposed to create a rectangle without changing the length of the base or the height. DOK-3 How did you determine the area formula for the triangles? Each triangle has only one base. We can take two of the same triangle and rearrange them to create one rectangle. Since we are using 2 triangles to create one rectangle, we will need to find half of the rectangle’s area to get the triangle’s area. This is the same for non-right triangles, except that they will form a parallelogram instead of a rectangle. We would still need to find half of the parallelogram’s area since it is made from 2 triangles, and we only want the area of one non-right triangle. DOK-3 How did you determine the area formula for the trapezoids? Trapezoids have two bases that are different sizes. When the trapezoids were doubled to create a parallelogram, we had to take half of the area because we made our parallelogram out of two trapezoids. DOK-2 How does the formula for the area of a trapezoid relate to the formula for a triangle? A trapezoid can be cut diagonally into two triangles. Then, you find the area of each triangle, which added together would give you the formula for a 1 trapezoid, __2 (b1 + b2)h. DOK-2 How does the area formula for a rectangle help you understand how the area of other figures are determined? Since we know that the area of a rectangle is A = l × w or A = bh and saw that the rectangle and parallelogram are composed of the same figures, then we can also use A = bh to find the area of a parallelogram.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

318

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Area and Volume Explore 2 – Finding the Area of Quadrilaterals ACTIVITY PREPARATION Students will use their understanding of decomposing and rearranging ﬁgures to ﬁnd the area of quadrilaterals.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • •

•

1 Student Journal (per student) 1 Exit Ticket (per student)

•

Plan to divide the class into groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student.

PROCEDURE AND FACILITATION POINTS Part I: Determining the Area of Quadrilaterals on Grids FACILITATION TIP

Before reading the scenario, ask the class 1) Does anyone have a garden at their house?; 2) If so, what is planted in your garden?; 3) What would you like to add to your garden (or plant in a garden if you don't currently have one)?

2. 3.

FACILITATION TIP Students may need clariﬁcation on some of the scenarios on the Student Journal. Depending on your classes, you can read through them together. The phrases "around," "throughout," and "only include" may cause some confusion.

Read the following scenario to the class: X-traordinary Landscaping Company is working with customers to design gardens. The company is showing each of its customers a blueprint of their garden with the flowers that will be planted. Today, you will help the landscapers determine how to find the area of each customer’s garden. Give a Student Journal to each student. Explain to students that they will find the area of quadrilaterals on grids. Note that students will find the area of a variety of 2-D figures (parallelograms, trapezoids, rectangles, and squares). Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 How did you determine the area of a parallelogram? When decomposing and rearranging, I moved the right triangle from one side of the parallelogram to the other side to form a rectangle. This showed me the height of the parallelogram. I know that I can use the height found here to multiply by the base to determine the area of a parallelogram.

DOK-2 What strategy did you use to find the area of each customer’s garden? Student responses may vary. First, find the area of the entire garden, and then find the area of the fountain. Next, subtract the area of the fountain from the area of the entire garden.

FACILITATION TIP Refer to your anchor chart, word wall, or student notebooks to review the definitions of quadrilateral and the other 2-D figures. Call on students to state the definitions and explain their attributes. 5.

Allow students enough time to record all their work for Part I of the Explore activity on their Student Journals.

Part II: Area of Quadrilaterals 1. FACILITATION TIP To simplify this activity, the Student Journal could be printed and distributed in sections: Part I (pages 1–6) and Part II (pages 7–10). Using two different colors may help students keep them organized. 320

2. 3.

Explain to students that they will apply their understanding of the area of quadrilaterals to determine the area of parallelograms and trapezoids using the formulas. Students should still have their Student Journals from Part I of the Explore activity. Students will calculate the area of parallelograms and trapezoids using the formula and without the grid to determine the area of the garden where ﬂowers will be planted. © Accelerate Learning Inc. - All Rights Reserved

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 How did you identify the height of the parallelogram? Answers may vary. The dotted line on the parallelogram helped me determine that the measurement that was included was the height.

DOK-2 What strategy did you use to find the area of each garden? Answers may vary. I used the parallelogram formula for area, which is base times height, to find the area of the garden where flowers will be planted. If the garden was a trapezoid, I used the formula base one plus base two times height divided by two. When there was more than one area where flowers will be planted, I determined the area for each garden.

Allow students enough time to record all of their work for Part II of the Explore activity on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-3 What happens to the area of a parallelogram if the height doubles but the base stays the same? If the height of a parallelogram doubles and the base stays the same, the area of the parallelogram will also double. • DOK-2 What is the relationship between the decomposed figure and its newly created figures? Each decomposed figure and newly created figure included the use of the formula A = bh, the same area formula that is used for rectangles and parallelograms. • DOK-2 How does the area of a rectangle formula help you understand how the area of a parallelogram and square are determined? Since a parallelogram and rectangle are composed of the same parts, area = base times height can be used to calculate area, which is similar to area = length times width. Since a square has sides that are all equal length, we can still use area = base times height to find the area of a square, which is similar to area = side times side. • DOK-2 Does it matter which of the trapezoid’s bases is substituted for b1 and which is substituted for b2? Explain. It doesn’t matter which trapezoid base is substituted for b1 and b2 because of the commutative property of addition. The sum of the bases will still be the same amount. •

Post-Explore 1. 2. 3.

Intervention

Acceleration

FACILITATION TIP If students are struggling to determine where –2.5 is on the axes, prompt them to label the halves on the axes, starting with the positive axes.

AREA AND VOLUME

Home

FACILITATION TIP This question could be supported with a visual example to remind students of why b1 and b2 are interchangeable and to clarify what b1 and b2 stand for in the formula.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

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Area and Volume Explore 3 – Finding the Areas of Triangles ACTIVITY PREPARATION Students will use their understanding of decomposing and rearranging figures to find the area of the figure.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • •

•

1 Student Journal (per student) 1 Exit Ticket (per student)

•

Plan to divide the class into groups of four to complete this activity. Print a Student Journal and an Exit Ticket for each student.

PROCEDURE AND FACILITATION POINTS

FACILITATION TIP Before reading the scenario, ask the class 1) What is your favorite type of vegetable to eat?; 2) What do you like about it?; 3) What types of vegetables have you or someone you know tried to grow? FACILITATION TIP On page 1 and 2 of the Student Journal, remind students that squares and rectangles can be called parallelograms. The Area workspace is titled parallelogram rather than square or rectangle.

Part I: Determining the Area of Triangles on Grids 1.

2. 3.

FACILITATION TIP Project questions 4a and 4b for students to view. Have students think independently about each answer, then pair with a shoulder partner, and finally be prepared to share with the whole class. As a follow-up question, ask students to state the formula for finding the area of a triangle. Encourage them to state the formula in more than one way (base times height divided by two, one half times base times height, .5 times base times height...). FACILITATION TIP Determine how you would like students to clearly answer, "How did you find the area..." on the Student Journal. Consider using a student sample as an example or provide a sentence frame, " I found the area by first... next... finally..." 322

Read the following scenario to the class: X-traordinary Landscaping Company is working with the Summerville Farm to create a map of their vegetable gardens. Each vegetable garden is shaped like a triangle. They are showing the owners of the farm where the vegetables will be planted. Today, you will help the landscapers determine how to find the area of each customer’s garden. Give a Student Journal to each student. Explain to students that they will decompose and rearrange 2-D figures to find the area of triangles. Note that students will find the area of a variety of 2-D figures (parallelograms, rectangles, and squares) and will use those figures to find the area of a triangle. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 How can you identify the base of the triangle? Answers will vary. Since any side can be the base, I looked at the measurements that represent the side and its length.

DOK-1 How can you identify the height of the triangle? Answers will vary. The corresponding height of a triangle is the length of a perpendicular segment from the base to the vertex opposite of it.

DOK-2 What strategy did you use to ﬁnd the area of each garden on the grid? Answers may vary. First, determine the number of units that represents the base and the number of units that represents the height. I used the formula for area of a parallelogram and then multiplied the 1 answer times __2 to get the answer for the area of a triangle.

Allow students enough enough time to record all of their work for Part I of the Explore activity on their Student Journals.

Part II: Using the Formula for the Area of a Triangle 1. 2.

Explain to students that they will apply their understanding of the area of triangles to determine the area of the gardens without the grid. Students will calculate the area of triangles using the formula and without the grid to determine the area of the garden where vegetables will be planted. © Accelerate Learning Inc. - All Rights Reserved

4. 5.

Engage

Explore

Explain

Elaborate

Evaluate

Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 What strategy did you use to find the area of each garden? Answers may vary. First, find the area of the entire garden, and then find the area of the garden that will include the vegetables that will be planted. Use the A = bh formula to find the area of the entire garden, and then take half of the area to get the area of the triangle.

DOK-1 What does the dotted line represent on a triangle? Answers will vary. The dotted line represents the height.

DOK-2 How did you compose a parallelogram to find the area of the triangle? Answers will vary. I added an additional right triangle that had the same base and height and composed both right triangles to create a parallelogram. To find the area of a triangle, I found the area of the parallelogram and then multiplied the area times one-half to find the area of a triangle.

Math Chat DOK-2 How is the area formula for parallelogram connected to the area formula for a triangle? Since a triangle takes up half of the area of a parallelogram with the same base and height, there is one-half in the area formula for a triangle. • DOK-3 How many possible bases does a triangle have? Every side of a triangle can be a base, so there are three possible bases. • DOK-2 In the problem about the pea garden, how can we use the area of one shaded triangle to calculate the area of the entire shaded part of the garden? The height of the garden represents 12 feet, and each base for the shaded triangles represents half of the base of the garden. To calculate the shaded part of the garden, use one-half of 27 feet, which is 13.5 feet, and multiply it times 12 to get 162 square feet. 1 1 Since the area of a triangle formula is A = __2 bh, you would multiply 162 times __2 to get 81 square feet. You can then multiply the area of one triangle times 2 to ﬁnd the area of the shaded parts (triangles). •

Intervention

Acceleration

FACILITATION TIP Reassure students that most real-world problems (as well as assessments) about area do not include a grid. In addition, images are often not drawn to scale. Encourage students to focus on the critical labels. Sometimes, assessment questions include irrelevant labels that can add to the confusion.

AREA AND VOLUME

Home

FACILITATION TIP Struggling students may need support calculating the area on the peas garden. Demonstrate with colored shading or labels how to decompose the parts to calculate the shaded areas and encourage students to take notes on their Student Journal.

Post-Explore 1. 2. 3.

FACILITATION TIP Have students complete the Exit Ticket to formatively assess their understanding of the concept. Before distributing the Exit Ticket, read the problem with students and allow time for Complete the Anchor Chart as a class. clarifying questions. Students may not be Have each student complete their Interactive Notebook. aware that Tamika has two gardens, or may not be able to determine that cucumber shapes are included in both images.

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Area and Volume Explore 4 – Finding the Area of Composite Figures ACTIVITY PREPARATION Students will determine the area of composite ﬁgures.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • •

•

1 Student Journal (per student) 1 Set of Garden Cards (per group) 1 Exit Ticket (per student)

•

Reusable •

Plan to divide the class into groups of four to complete this activity. Print a set of Garden Cards for each group. Cut out the cards, and place them in a resealable bag for each group. If desired, print the cards on card stock and laminate them for future use. Print a Student Journal and an Exit Ticket for each student.

1 Resealable bag (per group)

PROCEDURE AND FACILITATION POINTS FACILITATION TIP After reading the scenario, ask the class 1) Where in our classroom do you see a composite figure?; 2) Where have you seen other real-world examples of composite figures outside of our classroom? FACILITATION TIP Start with a quick review of the definitions of these figures as well as composite figures. In prior Explore activities, squares and rectangles were sometimes referred to as parallelograms. As students tally the shapes that create the composite figures, be sure that they know which category each shape belongs in.

2. 3. 4.

FACILITATION TIP For students to clearly see how to decompose these ﬁgures, it's probably best for each one to have their own copy. They can use light pencil markings to explore and show their decomposition of each ﬁgure. FACILITATION TIP Be sure students have access to standard area formulas either on an anchor chart, word wall, or in a journal. 324

Read the following scenario to the class: X-traordinary Landscaping Company just got some recent requests for gardens that are composite figures. This means the gardens are each in a shape that is a combination of rectangles, triangles, and/or parallelograms. They need you to use your knowledge of area formulas for each of these polygons to determine the areas of the gardens shaped like composite figures. Give a bag of Garden Cards to each group. Give a Student Journal to each student. Explain to students that they will be working with their groups to determine the area of each of the eight customers’ gardens. They will decompose and rearrange the gardens into a variety of figures to help them determine the areas. Have students use the Garden Cards to determine the area of each customer’s garden. If the Garden Cards are laminated, students can draw on the cards with dry-erase markers to show their decomposition and rearrangement of the composite figure. They will write how many of each 2-D figure the composite figure can be decomposed into. They will also use the area formulas to determine the area of each customer’s garden. As students are working together, monitor their learning, and ask the following questions to check for understanding: a.

DOK-2 How can you decompose this garden? Answers will vary. For garden A, I can decompose to make two rectangles and then add the two areas together to ﬁnd the total area of the garden.

DOK-2 Why do you need to divide by 2 to find the area for a triangle? For a trapezoid? You have to use two triangles to make a rectangle/ parallelogram, but you only need the area of one, so you will find half of the total area. You have to use two trapezoids to make a parallelogram, but you only need the area of one, so you will find half of the total area. © Accelerate Learning Inc. - All Rights Reserved

7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

Allow students enough time to record all of their work on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-3 What strategy can you use to find the area of a composite figure? I can decompose the shape into smaller areas that I can find and then add all of the areas together to find the total area. I can also compose the shape into a bigger figure and then subtract the portion that is not in the original figure to find the area of the given figure. • DOK-2 Describe the process that you used to find the area of garden E. I decomposed garden E into 2 triangles and 1 parallelogram, found the area of each figure, and determined the sum of the areas. • DOK-3 What is an example of why we would need to find the area of a composite figure? If we were painting a house and painting a wall with windows, we could view the wall as a composite figure and find the area of the composite figure. •

Intervention

Acceleration

FACILITATION TIP Provide an example or have a student work sample for students. Show how you want them to share their process for each solution. If students are using calculators, consider requiring them to still show their steps so you can assess.

AREA AND VOLUME

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FACILITATION TIP

Take some time to find a few more relevant real-world examples (images) of composite figures and area. Examples might include: Post-Explore turf replacement for a local ball park, carpet 1. Have students complete the Exit Ticket to formatively assess their understanding for the new library, new tiles in the cafeteria, or flooring in a gym. of the concept. 2. 3.

Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

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Area and Volume Explore 5 – Volume of Rectangular Prisms ACTIVITY PREPARATION Students will ﬁnd the volume of rectangular prisms.

Standards for Mathematical Practice • • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • •

•

1 Student Journal (per student) 1 Set of Produce Carton Cards (per group) 1 Exit Ticket (per student)

• •

Reusable • •

40 Linking cubes (per group) 1 Resealable bag (per group)

•

Plan to divide the class into groups of 3 or 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print and cut out a set of Produce Carton Cards for each group. Place each set of Produce Carton Cards in a resealable bag. If desired, print the cards on card stock and laminate them for durability. Gather 40 linking cubes for each group.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before reading the scenario, ask the class 1) What are products typically shipped in?; 2) How is the size the boxes need to be determined? FACILITATION TIP Before giving a set of Produce Carton Cards and cubes to each group, clarify and project the process for building. Does every student need to build every carton? Should students keep each carton complete until ﬁnished with all of the cartons (Are there enough cubes?)? Step 5 mentions building with all one color; be certain you have enough cubes of different colors if needed.

Part I 1.

2. 3.

Read the following scenario to the class: The Farmer’s Company is looking at its inventory of produce cartons to update their website. All of the cartons will need to show the total amount of produce they can hold. The Farmer’s Company uses smile units for measurements, so we would need to use a scale to convert the measurements to smile units. Can you use the Farmer’s Company measurements to build a model of each carton and use the model to find the volume of each carton? Give a set of Produce Carton Cards and linking cubes to each group. Explain to students that the linking cubes represent the produce that fits in the carton. They should understand that one smile unit is a length equal to two linking cubes. Instruct students to read the first card together. After the card has been read, give students time to discuss how they could build a model of the carton with their groups. a.

FACILITATION TIP Reassure students that a "smile unit" is a ﬁctitious unit.

326

DOK-1 How can you create the produce cartons with the linking cubes? You can connect the number of linking cubes needed to make the length, width, and height of each of the cartons.

Instruct students to build Produce Carton A, a cube using 8 linking cubes of one color. Allow students to build the ﬁrst layer, and then discuss what the ﬁrst layer represents. a.

DOK-1 How many cubes are in the ﬁrst layer? 4

DOK-2 What does the ﬁrst layer represent? The base of the produce carton that is shaped like a square

6. 7.

8. 9.

10.

Engage

Explore

Explain

Elaborate

Evaluate

Explain to students that ﬁnding the area of the base is a part of the process that we use to ﬁnd the volume of a rectangular prism. Have students continue to build Produce Carton A. Explain to students that the cube that was built represents a smile unit cube. Students should notice that the smile unit dimensions all represent 1. Give a Student Journal to each student. Have students work collaboratively to read each Produce Carton Card and use their linking cubes to build a model of the carton using the information from each card. Then, they will draw a model of their cartons and record the length, width, and height in cubes and smile units on their Student Journals. Actively monitor students as they are working with their groups to build the produce cartons with linking cubes. Ask the following questions: a.

DOK-1 What information are you given about this carton? I am given the length, width, and height.

DOK-1 What are the dimensions of this carton? Answers will vary. The dimensions of this carton are 2 cubes and 1 smile unit for length, width, and height.

DOK-2 What strategy are you using to build your model of the carton? Answers will vary. I am using the dimensions on the Produce Carton Card to build the cube.

DOK-2 Compare carton A and carton B. Which carton do you think will hold more produce? Answers will vary. I think carton B will hold more because the height is bigger.

e. DOK-2 How does the number of cubes you can put inside the carton relate to the length? How does it relate to the width? How does it relate to the height? How does it relate to the total cubes? The number of cubes that fit across one row at the bottom is the length. The number of cubes that fit in one row from front to back is the width. The number of cubes that fit in one column from bottom to top is the height. The total number of cubes needed to build the carton is the volume. 11.

12.

Allow time for students to build models and complete their Student Journals for all remaining Produce Carton Cards. Students will determine each carton’s volume in smile units. If necessary, briefly review the formula for finding volume. When students are finished, collect the linking cubes, and move on to Part II.

Part II 1.

2. 3. 4.

Read the following scenario to the class: Now that the Farmer’s Company has gotten your help in building a model of each carton and recording each one’s measurements, they need you to find the volume of different shipping boxes so they will know how much produce each box can hold. Direct students’ attention to Part II of their Student Journals. Explain to students that they will be working with their groups to determine the volume of each type of box using the information that is provided. Have students work collaboratively to use each box’s given dimensions to calculate its volume in two ways. First, they will find the volume using the formula of length multiplied by width multiplied by height. Then, they will calculate the area of each box’s base and use this information to find the volume using the formula of base multiplied by height. Students should compare both answers for volume for each box and see that they are the same. As students are working together, monitor their learning, and ask the following questions to check for understanding: a.

DOK-1 What do I need to do to multiply fractional measurements? I will change the mixed number to be an improper fraction or change a whole number to be a fraction.

Intervention

Acceleration

AREA AND VOLUME

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FACILITATION TIP Take some time to model how to draw a 3-D ﬁgure before having students attempt on their own. Consider that some students will struggle with accuracy and spacial awareness on these sketches, and some students will want to be very precise and overly exact. FACILITATION TIP Ensure that students understand that they need to label their hand drawings on the Student Journal in “smile units.” Students can underline or highlight that speciﬁc part of the directions on the Student Journal.

FACILITATION TIP After reading the scenario, ask the class 1) How can you find the volume of different box sizes? FACILITATION TIP These dimensions include fractions of feet and meters. Some of the fractions can easily be converted into inches rather than fractions of a foot. Be prepared to explain how you want students to calculate and label solutions. FACILITATION TIP Depending on students' experiences with the two different volume formulas, consider taking time to demonstrate how to use them. Perhaps select one of the more challenging produce box problems to show the steps for both formulas. Have students copy your process so they understand how to show their work. FACILITATION TIP Although many students should be fluent in multiplying fractional measurements, you may need to demonstrate changing mixed numbers into improper fractions and then multiplying numerators and denominators. Determine and clarify whether solutions need to be simplified. 327

AREA AND VOLUME

Area and Volume Explore 5 – Volume of Rectangular Prisms b.

DOK-1 What does the B mean in the formula V = Bh? (Note that they should compare both volume formulas to determine B.) When comparing both volume formulas, I noticed that length times width represents the area of the base, so B, “big b,” is the area of the base.

DOK-2 Will the formula l × w × h give you a different solution than the formula Bh? Why? No, both formulas will give you the same solution. B is ﬁnding the area, which is length times width, and you multiply the area of the base times the height.

STEMscopes Tip Transition students into the current concept by meeting them at their level with the Hook activity, found in the Engage section. These real-world scenario-based activities frame the overall learning throughout the scope and serve as both an introduction and concluding aspect of each concept. The Hook fosters personal growth.

6. 7. 8.

Students will collaborate with their groups to complete the remaining questions on their Student Journals. Allow students enough time to complete their work and record their observations and reﬂections on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 How does the number of cubes you can put inside a box relate to the length? How does it relate to the width? How does it relate to the height? How does it relate to the total cubes? The number of cubes that fit across the bottom is the length. The number of cubes that fit from front to back is the width. The number of cubes that fit bottom to top is the height. The total number of cubes is the volume of the shipping box. • DOK-2 Is there any difference between finding the volume using linking cubes and using the dimensions? When you find the volume using linking cubes and when you find the volume using dimensions, the volume will still be the same. The only difference is the method that is chosen to find the volume. • DOK-1 Explain how to multiply fractional measurements. In order to multiply fractional measurements, I need to convert the mixed number to an improper fraction or change the whole number to a fraction. •

FACILITATION TIP Before having students complete the Exit Ticket, determine your criteria for success. Demonstrate how students should "show their work" and what kind of labels and simplifying are expected, if any.

Post-Explore 1. 2. 3. 4.

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Area and Volume Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Discovering Area Formulas Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Finding the Area of Quadrilaterals Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 3

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Finding the Area of Triangles

Interactive Notebook

Show What You Know, Part 4

A cut-and-glue activity to process learning that can be added to a notebook for future reference

Finding the Area of Composite Figures

Independent practice assignment that gives students an opportunity to demonstrate their learning

Show What You Know, Part 5 Volume of Rectangular Prisms Independent practice assignment that gives students an opportunity to demonstrate their learning

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Area and Volume

Can be done independently

AREA AND VOLUME

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Independent and partner games and other activities that provide students with an engaging way to practice the new concept

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Area and Volume Intervention and Assessment SMALL-GROUP PLANNER 1

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

Students who are still acquiring the concept and need remediation

How to Use the Review

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Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

AREA AND VOLUME

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What does mastery look like?

I can use knowledge of the area of a rectangle to determine the area of a triangle.

I can use the formula for the area of a triangle.

I can determine the area of polygons by composing or decomposing into other shapes.

I can calculate the volume of a right rectangular prism with fractional edge lengths.

I can show that the volume is the same as multiplying the edge lengths of a right rectangular prism.

I can apply the formula for volume of a right rectangular prism.

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Surface Area Scope Introduction SCOPE SUMMARY Students explore properties of nets that form three-dimensional figures. They will represent and interpret three-dimensional figures using nets composed of rectangles and triangles. Students will use these nets to find and justify the surface area of such figures. They will apply these techniques to solve real-world problems involving surface area. Student Expectations

6.GSR.5.2 Given the net of three-dimensional ﬁgures with rectangular and triangular faces, determine the surface area of these ﬁgures.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grades, students have composed, decomposed, and identified two- and three-dimensional figures. They explored the area and perimeter of rectangles using arrays and by relating the process of counting square units to repeated addition and subtraction. In fourth grade, students used formulas for the area and perimeter of a rectangle to solve mathematical and real-world problems involving whole-number and fractional side lengths. In a previous sixth-grade scope, students composed, decomposed, and rearranged parts of a polygon into triangles and quadrilaterals to find the area of the polygon.

In seventh grade, students extend their knowledge to solve problems involving surface area of threedimensional objects composed of triangles, circles, and quadrilaterals. This work will lead to concepts of scale drawings and similar shapes.

Before beginning the lesson, the students’ prior knowledge is assessed to address any student needs and skills needed for content mastery. Students are assessed to determine if they can: •

recognize area as additive.

•

ﬁnd the area of rectilinear ﬁgures.

•

apply techniques learned to solve problems.

Students who are successful will be ready to proceed to the Hook activity. Any students who need to build on prior knowledge can utilize the Foundation Builder within this scope.

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

determine the surface area of a cube.

•

use the solution to determine the answer to a real-world problem.

Students move on to the Explore activities and come back at the end of the explorations to repeat the activity and discuss what they found.

If your students are struggling with previously taught concepts, use the Foundation Builder activity in this scope to reinforce ideas presented in the APK. 334

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Nets In this exploration, students will solve a scenario that involves involves matching patterns of tents to correct tent models to help a company determine how much it will cost if they make their tents out of new fabric. Students will: •

use nets made up of triangles and rectangles to represent three-dimensional ﬁgures.

•

ﬁnd the attributes of each tent including the number of edges, faces, bases, and vertices.

Explore 2

Explore 1

EXPLORE ACTIVITIES

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Finding the Surface Area of 3-D Figures In this exploration, students will work to help Tent-tastic, the tent company, determine each tent’s dimensions to ﬁgure out how much total material is needed for each sample tent. Students will: •

use nets to ﬁnd the surface area of threedimensional ﬁgures.

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment. Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________

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Surface Area Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

336

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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE

SURFACE AREA

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Students will identify two truths and a lie by reading statements about the prior standard. This element is designed to uncover student misconceptions; it should not be taken for a grade. 3.GSR.7.2: Determine the area of rectangles (or shapes composed of rectangles) presented in relevant problems by tiling and 3.GSR.7.2: counting.

Materials

Preparation

Printed •

•

1 Two Truths and a Lie (per student or group)

•

If not assigning the APK digitally, print one Two Truths and a Lie for each student or each group. You may choose to put students in groups of two or three.

Procedure and Facilitation Points 1. 2. 3. 4. 5.

Read the prompt aloud to the class. Allow 2 minutes of thinking time for the students to read the three statements and determine the two truths and one lie. Ask students to share with a shoulder partner how they marked their sheets and why. Allow 2–5 minutes of discussion. Ask students to justify their choice for the lie. a.

The second statement is incorrect because the area of a composite figure is found by partitioning the figure into two rectangles, finding the area of each rectangle, and then adding the two areas together to get the total area of the composite figure.

If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

FACILITATION TIP Some students may be curious about the unit squares surrounding the image of this composite ﬁgure. Consider sketching in the rest of the unit squares to ﬁll the shape. It might help clarify that students are to focus on area vs perimeter. FACILITATION TIP Uncover the statements one at a time to allow students time to consider each one individually. FACILITATION TIP Provide a way for students to vote silently or privately. They could mark it on their paper, hold a hand signal close to their shirt front, or do a heads down vote.

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Surface Area Hook – All Taped Up ACTIVITY PREPARATION Students will determine the surface area of a cube and use the solution to determine the answer to a real-world problem.

Materials

Preparation

Printed •

• • •

1 All Taped Up (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project All Taped Up for the whole class to view. Prepare to introduce the scenario and encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION POINTS Part I: Pre-Explore 1.

FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) When you give a gift to someone, how do you wrap it?; 2) Do you prefer to get a gift in a bag or box? Why?; 3) Have you ever tried to make a box to put a gift in?

FACILITATION TIP Project the details of this scenario for students to read together with you. Help students locate the essential information. Ask, "What do we know? What do we need to ﬁnd out?"

4. 5.

FACILITATION TIP If you have time, print out a cube pattern (this image may work) on card stock. Demonstrate folding it and ﬂattening it for students to see the net. 6. 338

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: Amira is decorating a gift box she will assemble that is in the shape of a cube with edges measuring 5 inches in length. She bought pretty duct tape to cover the outside of the box instead of wrapping paper. Amira bought one roll of tape that has 144 square inches. Will she have enough tape or does she need to buy a second roll? Ask students the following questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Student answers will vary. I notice that the box is a cube with 6 congruent faces. I notice that the units of measurement are square inches. Will Amira have enough tape to cover the outside of the gift box or will she need to buy more tape? I can use math to determine the surface area of the gift box and compare it to the area of the duct tape. Project All Taped Up. Explain to students that Amira must determine the surface area of the gift box to compare the surface areas of the tape and the box. Discuss the following questions: a.

DOK-1 How does Amira know the surface area of the roll of tape? It was given to her.

DOK-1 What is special about the measurements of the gift box? It is a cube, so each side is a congruent square.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and restate the problem. Refer to All Taped Up, and discuss the following questions: a.

DOK-1 How can you determine the surface area of the gift box? I can determine the area of one face by using the formula A = s2. Then, I can multiply the area of that face by six because there are six congruent faces on the cube.

DOK-1 What is the surface area of the gift box? The surface area of the gift box is 150 square inches. 5 × 5 = 25; 25 × 6 = 150

DOK-2 How can you determine whether Amira has enough tape? I can compare the two surface areas.

DOK-1 Does Amira have enough decorative tape to cover the gift box? No, she will need to buy another roll of tape. 144 < 150

SURFACE AREA

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FACILITATION TIP Some students may comment (based on their own experiences) that Amira would likely overlap some of the tape when wrapping the box.

e. DOK-1 How many square inches of tape does Amira have left over or how many square inches of tape is she short? 144 – 150 = –6 Amira is 6 square inches short of having enough tape. Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________

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Surface Area Explore 1 – Nets ACTIVITY PREPARATION Students will use nets made up of triangles and rectangles to represent three-dimensional ﬁgures.

Standards for Mathematical Practice • • •

MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision.

Materials

Preparation

Printed • • • • •

•

1 Student Journal (per student) 1 Set of Student Journal Cutouts (per student) 1 Set of Tent Model Cards (per group) 1 Set of Pattern Cards (per group) 1 Exit Ticket (per student)

• • • • • •

Reusable • •

1 Pair of scissors (per student) 1 Set of markers (per group)

Note: We recognize that the triangular pyramid net is a tetrahedron, so the rules that are applied using the Pythagorean theorem will not be applied to the nets that are included in this Explore activity. The focus of this Explore activity is on ﬁnding surface area with given dimensions. The Pythagorean theorem will not be introduced until 8th grade.

Consumable • •

Plan to divide the class into groups of 2–4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Student Journal Cutouts for each student. Print a set of Tent Model Cards for each group. If desired, print the cards on card stock and laminate them for future use. Print a set of Pattern Cards for each group. Gather a pair of scissors for each student. Gather a roll of tape, a set of markers, and glue for each group.

1 Roll of tape (per group) 1 Bottle of glue (per group)

PROCEDURE AND FACILITATION POINTS

FACILITATION TIP Before reading the scenario, ask the class 1) Have you ever stayed in a tent before?; 2) If so, what shape was the tent?; 3) What material was the tent made out of? FACILITATION TIP

Part I: Nets 1.

2. 3.

Before distributing supplies, model for students how to accurately cut, fold, and tape the nets. It may help to have an adult cut widely around the nets ahead of time. This prep can eliminate a lot of extra student time cutting to get to the actual lines of the nets. FACILITATION TIP Print the Pattern Cards on a heavier card stock and in different colors for each ﬁgure.

340

Read the following scenario to the class: Tent-tastic is looking into making all of its tents out of new polyester fabric. Before the company can ﬁnd out how much this will cost, Tent-tastic must create the patterns for each of the tents. Tent-tastic is challenging you to help match the pattern for each tent to its correct tent model. Distribute Tent Model Cards, Pattern Cards, scissors, and tape to each group. Have students work with their groups to cut out each Pattern Card and build their tents by folding and using tape. Then, students will match each Tent Model Card to its three-dimensional model that was created from each Pattern Card. Note that each student in the group should be instructed to cut and tape a different Pattern Card so that all of the tents are created in a timely manner. a.

Discuss with students that these patterns are called nets. Explain that nets are the two-dimensional shapes used to create three-dimensional ﬁgures.

While students are working, actively monitor their progress. Ask the following guiding questions: a.

DOK-2 What are the attributes of this ﬁgure? Answers will vary. This ﬁgure has 8 vertices, 12 edges, and 6 rectangular faces.

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 What is the relationship between the faces on each tent and the two-dimensional figures you see on its matching Pattern Card? The faces on each tent are the same 2-D figures that the two-dimensional figures on the matching Pattern Card are composed of.

Give a Student Journal and a set of Student Journal Cutouts to each student. Have students cut out the Student Journal Cutouts. Students should match the Student Journal Cutouts to their models of the tent and pattern (net). Once the cards are matched, have students glue the Student Journal Cutouts in their Student Journals. After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 What two-dimensional figures did you identify in the nets? There are triangles, squares, and rectangles in the nets for the tents. • DOK-2 How are the tent model and its matching net similar? Each of the twodimensional figures on the net makes one face of the three-dimensional figure (tent model). • DOK-4 How could recognizing nets be helpful not only in math but in everyday life? Recognizing nets can help you identify three-dimensional figures in math and in everyday life. In addition, they can be used to build objects such as toys. • DOK-3 What can each net, or pattern, tell us about the attributes of each tent? The net can tell us about the surface of each tent. It can tell us about how many faces, edges, and vertices it has. •

Part II: Attributes of Three-Dimensional Figures 1.

2. 3.

Read the following scenario to the class: Tent-tastic will need to know the number of edges, faces, bases, and vertices of each tent to form the metal poles for each tent. Tent-tastic is challenging you to determine the number of each attribute for each tent model. Give a set of markers to each group. Students will work collaboratively with their groups to label attributes on the 3-D models they made in Part I. Students should use the markers to label the 3-D models as follows: a.

Green – trace the edges

Red – dot each vertex

Yellow – outline the base

Blue – number the faces

Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 How many edges does this three-dimensional ﬁgure have? Answers will vary. The three-dimensional ﬁgure has 12 edges.

DOK- 1 How many faces does this three-dimensional ﬁgure have? Answers will vary. The three-dimensional ﬁgure has 6 faces.

DOK-1 How many vertices does this three-dimensional ﬁgure have? Answers will vary. This three-dimensional ﬁgure has 8 vertices.

DOK-1 How many bases does this three-dimensional ﬁgure have? Answers will vary. The three-dimensional ﬁgure has 1 base.

e. DOK-1 What two-dimensional ﬁgure is each face shaped like? Answers will vary. Each face on this three-dimensional ﬁgure is shaped like a _____ (triangle, rectangle, etc.). f.

Intervention

Acceleration

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FACILITATION TIP Take time to break the written directions on the Student Journal into very clear steps for students. The Student Journal is 7 pages; print the ﬁrst 6 for students, and then use page 7 to guide your Math Chat. FACILITATION TIP If time is an issue, have the Student Journal Cutouts precut, so that all students need to do is match and glue.

FACILITATION TIP Before reading the scenario, ask the class 1) What are the parts of a tent?; 2) What steps need to be taken to set up a tent? FACILITATION TIP Use a 3-D figure to demonstrate and define the attributes you want students to be ready to count (edges, faces, vertices, bases). Refer to the Visual Glossary if needed. FACILITATION TIP Many schools have sets of 3-D figures. These plastic prisms and pyramids may be more stable for students to explore, trace and manipulate. STEMscopes Tip Each Explore activity includes a Student Journal that students complete collaboratively while participating in group work. Students use the journal to develop metacognitive skills by reflecting on how and what they are learning. Communicating mathematical thinking leads to a deeper conceptual understanding of the skills at hand.

DOK-2 What is the name of this three-dimensional ﬁgure? Answers will vary. This three-dimensional ﬁgure is a ____ (square pyramid, triangular prism, etc.).

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Surface Area Explore 1 – Nets 5. 6.

As groups complete step 3, have students record their learning on Part II of their Student Journals. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat • STEMscopes Tip The Math Chat, embedded in each Explore lesson outline as well as in printable form, provides a forum where students collaboratively discuss their ideas and strategies and develop their number sense, mathematical vocabulary, and math thinking skills. Discussing the concepts taught helps students formulate stronger reasoning and critical thinking skills.

• • •

•

DOK-1 How many bases does a prism have? A prism has 2 bases that are the same shape. DOK-1 How many bases does a pyramid have? A pyramid has one base. DOK-1 Which three-dimensional ﬁgure has the most edges? A cube or rectangular prism has the most edges. Each of these prisms has 12 edges. DOK-2 How are pyramids and prisms similar? How are they different? Both prisms and pyramids are three-dimensional, which means they have length, width, and height. A prism has two identical bases that are parallel, while a pyramid has one base. DOK-2 What is the difference between a triangular prism and a triangular pyramid? A triangular prism has 2 bases that are triangles and 3 other faces that are connecting the bases together that are all rectangles. A triangular pyramid has 1 base that is a triangle and 3 other faces connected to the base that are all triangles. DOK-2 What attribute did all of the tent models have in common? None of the tent models had curved surfaces.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook. Notes

342

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Explain

Elaborate

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Intervention

Acceleration

Notes

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Surface Area Explore 2 – Finding the Surface Area of 3-D Figures ACTIVITY PREPARATION Students will ﬁnd the surface area of three-dimensional ﬁgures using their corresponding nets.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

• • • •

1 Student Journal (per student) 1 Set of Nets (per group) 1 Exit Ticket (per student)

Reusable •

1 Pair of scissors (per student, optional)

Consumable •

1 Roll of tape (per group, optional)

Plan to divide the class into groups of 2–4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print one set of Nets for each group. Optionally, for struggling students, gather a pair of scissors and a roll of tape so they can build the 3-D model.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before reading the scenario, ask the class 1) If you and your family were going to buy a tent, what features would it have?; 2) What material would it be made out of? FACILITATION TIP

2. 3.

Choose one of the Nets (maybe A) to show how to find the surface area step by step. Have students complete the steps with you, recording as you model the process. FACILITATION TIP Depending on your class, consider breaking the directions on the Student Journal into concrete steps. For example, 1) Look at each net. 2) Name the 3-D figure. 3) Identify the 2-D figures in the net. 4) Determine the missing measurements. 5) Calculate the area of the 2-D net. 6) Find the surface area of the 3-D figure. FACILITATION TIP For struggling students, provide a premade net and/or 3-D solid for them to explore rather than having them cut, fold, and glue. 344

4. 5. 6.

Read the following scenario to the class: Tent-tastic is making a sample of each tent to show to customers. They will need to cut enough material to make each tent. They need your help determining each tent’s dimensions so they can figure out how much total material is needed for each sample tent. Give a set of Nets to each group. Allow students time to discuss with their groups how they can use the given measurements to find how much total material is needed. Encourage each group to share their strategies with the class. Students should come to the conclusion that they can find the area of each two-dimensional figure in the three-dimensional figure. Then, they can add all of the areas together to find the total material needed. Explain the following to the class: Surface area is the total area of each of the faces and curved surfaces of a solid figure. Give a Student Journal to each student. Students should work with their groups to find the surface area of each tent. They will record the two-dimensional figures seen on each net, calculate each two-dimensional figure’s area, and find the sum of all the areas to determine the surface area of the tent on their Student Journals. For struggling students, encourage them to cut and fold each net to see how the dimensions relate to each part. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a. What two-dimensional figures do you see in this net? Answers will vary. I see rectangles, triangles, etc. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

b. What measurements are provided? Answers will vary. One of the 1 measurements on this three-dimensional ﬁgure is 4__2 feet, 3 meters, etc. c.

How can you use the provided measurements to determine the measurements of the other two-dimensional ﬁgures? Answers will vary. Since a square has 4 equal sides, if I know the length of one side of a 1 1 square is 4__2 feet, then all of sides of the square are also 4__2 feet.

d. What formulas for area will you use to determine the area of each twodimensional figure? Answers will vary. I will use the formula A = base × height to find the area of a rectangle. e. What strategy will you use to find the surface area of the entire tent? Answers will vary. The strategy I will use to find the surface area of the entire net is to find the sum of the areas of all faces. 8. 9.

Allow students time to ﬁnd the surface area for each of the tent models and answer the reﬂection questions on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat

SURFACE AREA

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FACILITATION TIP Direct students to where they can reference the standard area formulas (word wall, Visual Glossary, anchor chart, Student Journal, or other notes).

DOK-3 How can you use a net of a three-dimensional figure to determine the figure’s surface area? Each face of a three-dimensional figure is a twodimensional figure. You can find the area of each two-dimensional figure and add them together to get the total surface area for the three-dimensional figure. • DOK-2 What did you do differently to find the surface area of the square pyramid compared to the surface area of the cube? When finding the surface area of a square pyramid, since a square pyramid is composed of one square and four triangles, I had to find the sum of the areas using the formulas for area of a 1 triangle, A = __2base × height, and area of a square, A = s2. When finding the surface area of a cube, since a cube is composed of six squares, the sum of the areas would include the area of six squares. I had to find the sum of the areas using the formula for area of a square, A = s2. • DOK-2 How were your strategies for finding the surface area of the cube and rectangular prism similar? When finding the surface area of a cube, since each face is a square and all sides are equal on a square, finding the area of each face using A = base × height is the same as finding the area of each square using A = s2. When finding the surface area of a rectangular prism, I need to find the area of each face using A = base × height. FACILITATION TIP • DOK-3 Describe a scenario in everyday life where knowing how to calculate Take some time to find some additional surface area might be helpful. An example of how we use surface area in everyday life examples to prompt students' everyday life is wrapping a present and needing to know how much wrapping thinking (cereal box, shipping container paper to use. costs, shoe box design and decor, speaker shapes, etc). Post-Explore •

1. 2. 3. 4.

FACILITATION TIP

As students complete the Exit Ticket, challenge early ﬁnishers to consider how they could construct a net for a cylinder. Have them try to draw it on the back of the Exit Ticket. How many other nets can they draw from memory?

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Surface Area Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Nets Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Finding the Surface Area of 3-D Figures

Interactive Vocabulary

Interactive Notebook

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

A cut-and-glue activity to process learning that can be added to a notebook for future reference

Independent practice assignment that gives students an opportunity to demonstrate their learning

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

SURFACE AREA

Home

Can be done independently

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Surface Area Using Nets Independent and partner games and other activities that provide students with an engaging way to practice the new concept

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Surface Area Intervention and Assessment SMALL-GROUP PLANNER 1

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

Students who are still acquiring the concept and need remediation

How to Use the Review

3 348

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER

SURFACE AREA

Home

Use this template to decide how to assess your students for concept mastery. Depending on the format of the assessment, you can identify prompts and intended responses that would measure student mastery of the expectation. See the beginning of this scope to identify standards and grade-level expectations. Key Concepts

What prompts will be used?

What does mastery look like?

I can use pictures or physical models of nets made up of triangles and rectangular faces to determine the surface area of three-dimensional ﬁgures.

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SCOPE 1

Represent and Interpret Data Scope Introduction SCOPE SUMMARY

Student Expectations

Students represent and interpret data using dot plots, histograms, box plots, and bar graphs. This is the ﬁrst time that students are introduced to histograms and box plots. Students will learn about the characterization of data distributions by analyzing shape and spread. They will also describe variability by observing dot plot representations. Dot plots enable students to examine the distribution of a data set and to identify attributes such as the center, spread, and overall shape of the data set. A dot plot representing data collected from a statistical question can help to visually see the variability in the data set. Students learn that data sets contain many values that can be summarized with a single number. They will discover how to create histograms from frequency tables to look at peak, shape, and spread. Students will compare data sets and box plots to discover how the ﬁve-number summary is determined. They will continue to work with bar graphs to display distribution of categorical data.

6.NR.2.1 Describe and interpret the center of the distribution by the equal share value (mean).

6.NR.2.4 Design simple experiments and collect data. Use data gathered from realistic scenarios and simulations to determine quantitative measures of center (median and/or mean) and variability (interquartile range and range). Use these quantities to draw conclusions about the data, compare different numerical data sets, and make predictions. 6.NR.2.5 Relate the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. 6.NR.2.6 Describe the impact that inserting or deleting a data point has on the mean and the median of a data set. Create data displays using a dot plot or box plot to examine this impact.

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Background Knowledge

Future Expectations

In fourth grade, students created line plots and dot plots to represent measurement data and used line and dot plots to solve relevant problems. In fifth grade, students used line plots to solve problems in relation to data sets. Students also create and interpret bar graphs to represent categorical data in fifth grade. Fifth grade is the first time that students were exposed to measurements of data including mean, median, mode, and range.

Students in seventh grade use random sampling to draw inferences about populations, and they investigate chance processes, and make generalizations. They develop, use, and evaluate probability models. In eighth grade, students investigate patterns of association in bivariate data. This work extends into high school, where students continue to interpret categorical and quantitative data, and then explore conditional probability and the rules of probability.

make a line plot to display a data set in fractions.

•

solve problems involving addition and subtraction of information on a number line.

Hook

6.NR.2.3 Interpret numerical data to answer a statistical investigative question created. Describe the distribution of a quantitative (numerical) variable collected, including its center, variability, and overall shape.

VERTICAL ALIGNMENT

Accessing Prior Knowledge

At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

make a box plot.

Students move on to the Explore activities and come back at the end of the explorations to repeat the activity and discuss what they found.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Dot Plots In this exploration, students will work with groups to solve a real-world scenario helping Survey Tiger, a survey company, match statistical questions to dot plots, explain what each describes, and describe vocabulary terms so other people can understand the research. Students will: •

•

Explore 2

Explore 1

EXPLORE ACTIVITIES

analyze a statistical question in order to determine a dot plot and an explanation that matches the variable responses for that question.

•

analyze information given in a table.

•

analyze information from a box plot.

•

discover where the values of the ﬁve-number. summary are located on a box plot.

Explore 4

Explore 3

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

In this exploration, students solve a scenario involving a Straw Tower Challenge and the need to analyze the data correctly to report the results to the school’s newspaper. Students will:

In this exploration, groups of students will solve a scenario involving helping two friends interpret data from a histogram that was created based on points earned in a game. Students will: •

create a frequency table and histogram.

•

represent and interpret data using histograms.

•

draw inferences.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

explain the spread and shape of a given dot plot based on a given description for the displayed data.

Box Plots

Histograms

REPRESENT AND INTERPRET DATA

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Bar Graphs In this exploration, students will analyze information given in a table and a bar graph. Students will: •

display categorical data using a bar graph.

After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

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Represent and Interpret Data Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will match numbered cards with lettered cards posted around the room to demonstrate their knowledge of the prior standard. This element is designed to uncover student misconceptions; it should not be taken for a grade. 4.MDR.6.3:: Create dot plots to display a distribution of numerical (quantitative) measurement data. 4.MDR.6.3

Materials

Preparation

Printed •

• •

1 Set of Match around the Room Cards (per class)

Print one set of the Match around the Room Cards. Hang cards in a random order around the room.

REPRESENT AND INTERPRET DATA

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Procedure and Facilitation Points 1. 2.

Have students write the numbers 1, 2, and 3 on a sheet of paper. Instruct students to walk around the room with their papers. As they walk around the room, they need to see the numbered cards and match them with the lettered cards. Allow students to share their thinking with their neighbors.

FACILITATION TIP Students may need a closer look at the numbered cards; consider printing each student a copy. You can also project them beforehand so students can take notes on their numbered paper.

Card 1 matches with Card A.

Card 2 matches with Card C.

FACILITATION TIP

Card 3 matches with Card B.

To facilitate the Match Around the Room, print the lettered cards on a different color than the numbered cards.

If students are struggling to complete this task, move on to do the Foundation Builder in order to ﬁll this gap in prior knowledge before moving on to other parts of the scope.

Identifying Misconceptions •

Students may confuse the number on the number line with the frequency.

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Represent and Interpret Data Hook – Triple Dice Roll ACTIVITY PREPARATION Students will determine the values of a box plot, including the minimum, Q1, median, Q3, and the maximum values for a data set.

Materials

Preparation

Printed •

• • •

1 Triple Dice Roll (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Triple Dice Roll for the whole class to view. Prepare to introduce the scenario and encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION POINTS

FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) When rolling several dice, how do you determine the greatest value?; 2) How do you determine the least value?; 3) What do you do that helps you determine the minimum and maximum values? FACILITATION TIP

Part I: Pre-Explore 1.

Post the scenario question for students to read together with you. Have three dice ready to roll to demonstrate the game for students. Have a few students roll dice while you record the totals. FACILITATION TIP Decide how much you want to share about the meaning of maximum, minimum, Q1, Q3, and median at this point. You could have students guess what the "Q" might stand for in math and share ideas about the other terms.

4. 5.

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: Mr. Abramson wants to make a box plot with his first-period class. He wants to know the minimum, Q1, median, Q3, and maximum values. To collect data, his students will roll three dice at one time and record the value of the dice rolled by each student. Then, he wants his students to work together to create a box plot for him. Ask students the following questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Student answers will vary. I notice that the class is finding minimum and maximum values. I wonder what median, Q1, and Q3 values are and how to figure them out. What will the values be? I can use math to determine potential dice rolls for the number line. I can also use math to determine the necessary values for the box plot. Project Triple Dice Roll. Explain to students that Mr. Abramson has 16 students in his first-period class and they will each roll one of three dice. Discuss the following questions: a.

DOK-1 What do you think the minimum value is? The lowest value rolled by a student.

DOK-1 What do you think the maximum value is? The highest value rolled by a student.

Complete the Explore activities. Notes

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and restate the problem. Refer to Triple Dice Roll, and discuss the following questions: a.

DOK-2 What is the ﬁrst step after collecting data? Why? The ﬁrst step is to put the data in order from least to greatest. It is much easier to see and analyze the data when the numbers are ordered.

DOK-2 What is the median and how did you ﬁnd it? The median is the center number in the middle of a list of ordered numbers. We counted in from the least and the greatest. We met in the middle and the answer was 12.

DOK-1 What was the minimum value? The minimum value was 4. That was the lowest dice roll and the ﬁrst number in the list of ordered numbers.

DOK-1 What was the maximum value? The maximum value was 18. That was the highest dice roll and the last number in the list of ordered numbers.

FACILITATION TIP As a follow up or extension, engage students by taking time to record 16 dice rolls from your classes and complete the steps for ordering the data from least to greatest, ﬁnding the median, and creating the box and whisker plot.

REPRESENT AND INTERPRET DATA

Home

e. DOK-2 What was the Q1 and how did you ﬁnd it? The Q1 was 8. I found it because it is halfway between the minimum value and the median. Halfway between 4 and 12 is 8. 4 + 12 = 16; 16 ÷ 2 = 8 f.

DOK-2 What was the Q3 and how did you ﬁnd it? The Q3 was 15. I found it because it is halfway between the median and the maximum value. 12 + 18 = 30; 30 ÷ 2 = 15 Notes

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Represent and Interpret Data Explore 1 – Dot Plots ACTIVITY PREPARATION Students will analyze a statistical question in order to determine a dot plot and an explanation that matches the variable responses for that question. Students will explain the spread and shape of a given dot plot based on a given description for the displayed data.

Standards for Mathematical Practice • • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Set of Data and Question Cards (per group) 1 Set of Research Analysis Cards (per class) 1 Exit Ticket (per student)

Reusable •

•

Print a Student Journal and an Exit Ticket for each student.

Part I: Match Dot Plots and Explanations to Statistical Questions • •

Plan to divide the class into groups of 3 or 4 to complete this part of the activity. Print a set of Data and Question Cards for each group of students. If desired, print the cards on card stock and laminate them for durability. Cut the cards apart, place them inside a gallon-sized resealable bag for each group, and label the bag “Part I.”

Part II: Describe Distribution of Dot Plots (spread and shape)

1 Gallon-sized resealable bag (per group)

Plan to divide the class into 6 groups to complete this part of the activity. Print a set of Research Analysis Cards for the class. If desired, print the cards on card stock and laminate them for durability. Cut them apart. • Create 6 stations around the room with a Research Analysis Card at each station. • •

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever had something they were working on get mixed up?; 2) What happened?; 3) How were you able to put everything back in the correct order?

Part I: Match Dot Plots and Explanations to Statistical Questions 1.

FACILITATION TIP Post the essential parts of the scenario text for students to read together. Ask, “What are we trying to do for Survey Tiger?”

2. 3.

FACILITATION TIP If needed, direct students to the word wall, anchor chart, or Student Journal for an accurate deﬁnition/example of a statistical question. FACILITATION TIP

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Take time to reinforce the importance of creating an accurate data table before trying to create a dot plot.

Read the following scenario to the class: Survey Tiger received the data back from statistical question surveys randomly asked of the general public. A Survey Tiger manager wrote an explanation, described the center of the data, and had an employee take the data they received and place that information into a dot plot. Unfortunately, the dot plots got mixed up from the statistical questions and explanations from which they were originally created. Survey Tiger needs your help matching statistical questions to the dot plots and explanations they describe. Give a bag of Data and Question Cards to each group of students. Quickly review the following information by asking the following questions: a.

DOK-1 What is a statistical question? A statistical question is a question that can be answered with data, and the data or information collected will vary.

DOK-2 How can a statistical question be answered using a dot plot? A statistical question can be answered using a dot plot by displaying the data as a range on a number line and graphing the data points as dots on the dot plot.

Explain to the students that they will be collaborating with their groups to analyze and match the statistical questions, dot plots, and explanations to each other. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Monitor and assess each group’s understanding of the task by asking the following guiding questions: a.

DOK-2 What are some possible responses to this statistical question? Answers will vary depending on the question.

DOK-2 How could these responses be represented on a dot plot? Answers will vary depending on the question. I think students will mostly speak one language ﬂuently, but it is possible for students to speak more than one language ﬂuently. These responses might be represented on a dot plot with most of the dots above the number one, and some of the other dots above two or three.

DOK-3 How can a dot plot provide useful information? A dot plot can show a number line that displays a range of numbers, along with a dot above locations on the number line to represent a given response to the statistical question.

DOK-2 How would you describe the range of the data responses? The range of a data response represents the values of the lowest number given for data responses all the way to the highest number given for the data responses. Other responses can fall anywhere between this range.

e. DOK-3 How can you explain why you chose this dot plot and explanation to match with this statistical question? Answers may vary depending on the question. I chose this dot plot because I felt like it best represented the data responses that could be given for this statistical question. I chose this explanation because I felt like it most accurately describes the dot plot and the possible data responses that could be given for this statistical question. 6. 7. 8.

Give a Student Journal to each student. Allow students enough time to collaborate and record their responses on their Student Journals for each statistical question. Quickly discuss as a class the dot plot and explanation the groups chose, and encourage groups to explain their thinking in their decisions.

Part II: Describe Distribution of Dot Plots (spread and shape) 1.

2. 3.

Read the following scenario to the class: After some surveys for Survey Tiger were completed and the data was displayed on a dot plot, researchers then analyzed the data and described their findings. Often these researchers will publish their findings to the general public so their analysis of the data can help people who may have had the same question. Some researchers will use certain vocabulary terms to describe the data that not all readers of the data may understand. Survey Tiger needs your help defining and describing these vocabulary terms to the public so other people can better understand the given research. Explain to the class that they will be working with their groups to analyze dot plots and statements made about each dot plot. Encourage students to describe the vocabulary the researchers used to explain the dot plot based on what they noticed about the shape and spread of the data points. Quickly review the following questions with the class: a.

DOK-1 Based on prior knowledge, how would you deﬁne the words symmetrical and asymmetrical? Symmetrical means that the shape of an object is the same on both sides of a center point, and asymmetrical means that the shape of an object is not the same on both sides of the center point.

Intervention

Acceleration

FACILITATION TIP To simplify this part of the activity for students, break it into steps. Have students ﬁrst select the matching dot plots together. Stop and have a class discussion about which dot plots match which questions and why. Next, distribute the Explanation Cards and have groups match those.

FACILITATION TIP Range is an essential vocabulary word that students need to be familiar with and be able to use fluently. Take time to define and have students note the definition. Include it on your word wall or anchor chart (Range= high minus low).

REPRESENT AND INTERPRET DATA

Home

FACILITATION TIP Determine ahead of time if there are exact correct matches for each dot plot, explanation, and question. If so, students’ explanations should include speciﬁc data rather than “I felt like.” FACILITATION TIP Copy the Student Journal in two different colors and distribute separately as you are ready to complete each part. Before distributing Part II, be prepared to support students’ as they look for vocabulary terms that the general public may need help understanding. Students could scan and highlight the terms together or with your help before diving into the analysis.

FACILITATION TIP After giving students some time to ﬁnd, deﬁne, and describe these vocabulary terms, clarify each word. Depending on prior experience and knowledge, determine if your classes need a quick review or explicit instruction before rotating through the Research Analysis Card Stations. FACILITATION TIP To quickly review symmetry using a visual cue, fold some shapes, graphs, or dot plots on their center point.

DOK-2 How would you describe the spread of data? Explain. Answers may vary. When I think of the word spread, I think of spreading or covering an object with something, like spreading butter. When it comes to data, I think it talks about how far the data points are spread out.

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Represent and Interpret Data Explore 1 – Dot Plots 5. 6. FACILITATION TIP Students can use Picture Vocabulary to record accurate deﬁnitions and explanations.

7. 8.

Assign each group to a Research Analysis Card station around the room. Allow each group enough time to analyze the data and discuss what they think the vocabulary for shape and spread describes at each station. Students will record their deﬁnitions and explanations on their Student Journals. Monitor and assess each group as they collaborate by asking the following guiding questions:

FACILITATION TIP In addition to the 8a–8c questions ask, “How do we determine the ‘center’?” Be sure that students are aware that the dot plot may have a physical center in the middle, this is not the “center” of the data. In other words, “the middle of the line is not necessarily the ‘center’ of the data.”

DOK-3 What do you think is being analyzed when the researchers described the shape of the data? Answers may vary. I think the researchers were analyzing the number of peaks in the data points and the shape the data makes across the dot plot number line.

DOK-3 What do you notice about the shape of data that is skewed left? What do you notice about the shape of data that is skewed right? I notice that the shape of data that is skewed left has data points that peak on the right side but has outliers or data that tails off to the left. I notice that the shape of data that is skewed right has data points that peak on the left side but has outliers or data that tails off to the right.

DOK-3 What do you notice about the spread of data that has a small deviation? What do you notice about the spread of data that has a large deviation? I notice that data that has a small deviation doesn’t spread out very far from the center. I notice that data that has a large deviation spreads out very far from the center.

FACILITATION TIP Be aware that some students may need to physically review left and right. Use total physical response to reinforce left and right. You could make a quick game of it to allow you to assess who needs extra support (For example, “Raise your left hand, Point to the right wall, Tap your right foot, Tip your left ear to your left shoulder...”). FACILITATION TIP Students could include a quick small sketch for Questions 1 and 2 for the reﬂection questions.

9. 10. 11.

Allow students enough time to complete and record their work at each station before rotating to a new station. After students have recorded all their work for each station, they will answer the reﬂection questions at the end of their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-3 How would you describe the difference between the shape of data that is skewed left versus being skewed right? Data that is skewed left has a peak of data on the right side but tails off and has outliers on the left side of the number line, while data that is skewed right has a cluster of data that peaks on the left side and tails off to the right. • DOK-3 How would you describe the difference between a spread of data that has a small deviation and a spread of data that has a large deviation? Data that has a small deviation doesn’t spread out very far from the mean, while the spread of data with a large deviation is spread out far from the mean. • DOK-3 Why is it important to have a question that allows for variability in the response? Answers may vary. It is important for a question to allow for variability in its response because it allows the data to more accurately represent the responses of a group of individuals. •

FACILITATION TIP On this Exit Ticket, consider challenging some students to identify the center of the data when they answer the second question.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

REPRESENT AND INTERPRET DATA

Home

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Represent and Interpret Data Explore 2 – Histograms ACTIVITY PREPARATION Students will discover how to create and use histograms to represent sets of data and draw inferences.

Standards for Mathematical Practice • • • • •

Materials

Preparation

Printed • •

• •

1 Student Journal (per student) 1 Exit Ticket (per student)

Plan to divide the class into pairs to complete this activity. Print a Student Journal and an Exit Ticket for each student.

PROCEDURE AND FACILITATION POINTS

FACILITATION TIP Before reading the scenario, ask the class 1) What is your favorite board game?; 2) How do you play it?; 3) What is the goal of the game? FACILITATION TIP Some students will struggle to read the lists of data and sort with accuracy and precision. Encourage the use of colorcoded highlighters or gentle/light pencil crossing out of data as they sort it. Remind students to not heavily scribble out values, just in case they need to go back and recount the frequencies.

Part I: Tori’s Points 1.

2. 3.

a. This graph is called a histogram. DOK-2 How are histograms similar to bar graphs? Answers may vary. Histograms and bar graphs both have bars to represent data.

FACILITATION TIP Consider your classes’ prior experience with histograms, dot plots, bar graphs, frequency tables, intervals, and/or peaks before distributing the Student Journal. Decide whether to preteach some of the vocabulary terms or embed them into this activity. FACILITATION TIP If needed, direct students to note the height of the histogram bar prior to listing the speciﬁc data points on 5a–5e. This number will help them to know how many values need listing. 360

Read the following scenario to the class: Tori and Jakob are playing mystery games. In their ﬁrst game, they will use two spinners. They must add the values from both spinners to represent the points that round. The player with the most points is the winner. Tori created a list and histogram of the points she earned. Answer the questions related to her data. Give a Student Journal to each student. Direct students’ attention to the histogram on their Student Journals. Discuss the following questions with the class:

DOK-2 How are histograms different from bar graphs? Answers may vary. Histograms show data in ranges. Bar graphs have a bar for each number in the data set.

DOK-2 How are histograms different from dot plots? Answers may vary. Dot plots have a dot to represent each individual data point. Histograms do not show each individual data point, only the number of data points within a range of values.

Students will use the list and histogram to collaborate with their partners to discover how data is shown in a histogram. Then, students and their partners will answer the questions about Tori’s points. As students are working together, monitor their learning, and ask the following questions to check for understanding: a.

DOK-1 How are Tori’s points shown in the graph? Tori’s points are grouped in intervals. Each interval has a bar that shows how many times the values in that interval were scored by Tori.

c. 6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How can you determine which interval has the most values in it? I can look at the height of each bar and ﬁnd the tallest one.

Allow students time to complete Part I of their Student Journals and its reﬂection questions. After Part I, invite the class to a Math Chat to share their observations and learning.

Intervention

Acceleration

FACILITATION TIP An essential question that could be added to the reﬂection is: What is the center of this data?

Math Chat DOK-1 How can you determine the frequency of an interval on the histogram? I can look at the height of each bar for each interval and ﬁnd the frequency of that bar on the left side of the histogram. • DOK-1 What does the tallest bar in the histogram represent? The tallest bar in the histogram represents the interval with the most values in it. •

Explain the following to the class: Mathematicians call the tallest bar the peak. DOK-2 How is data represented on the histogram when the interval has a frequency of zero? If the interval did not have any values in it, then the height of the bar would be zero, or nothing would be ﬁlled in. • DOK-2 Describe the spread of the data. The data shows a small deviation from the center because the data does not spread far out from the center. • DOK-2 Describe the shape of the data. The shape of the data is symmetrical because about the same number of data points are on either side of the center. •

Part II: Jakob’s Points 1.

2. 3.

4. 5.

Read the following scenario to the class: Jakob wants to represent the points he earned in the mystery game like Tori did. Help Jakob create a frequency table and histogram using his recorded data. Students will collaborate with their partners to create a frequency table and a histogram to represent the points Jakob earned. As students are working together, monitor their learning, and ask the following questions to check for understanding: a.

DOK-2 How do you determine the frequency of each sum? Count the number of times that sum occurs in Jakob’s sums.

DOK-1 How do you create the bars in a histogram? Draw a bar extending from the lower value of each interval to the lower value of the next interval. The height of each bar should be equal to the frequency of its corresponding interval.

Allow students enough time to complete Part II and the reﬂection questions in their Student Journals. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat

STEMscopes Tip Virtual Manipulatives are located under the Explore tab. Unlike concrete manipulatives, these digital manipulatives require no setup and are easily accessed online at any time. Students can interact with a variety of virtual manipulatives to explore mathematical concepts anytime, anywhere.

REPRESENT AND INTERPRET DATA

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FACILITATION TIP After reading the scenario, ask the class 1) How are frequency tables and histograms alike?; 2) How are they different? FACILITATION TIP It might help students to have one partner (or the teacher) slowly read the values aloud while others keep tally on the table. Again, encourage students to use a highlighter or light pencil to keep track of values as they record them. It takes precision and patience to accurately tally a list of values. FACILITATION TIP If students are to eventually create their own histograms, be sure to establish some clear criteria for elements of a successfully created histogram. How many bars? How much if any space between bars? What size intervals? X-axis label? Y-axis label?

DOK-1 What do the intervals on the xx-axis of your histogram represent? The intervals on the x-axis represent the ranges of point values Jakob earned. • DOK-1 What do the numbers on the y-axis of your histogram represent? The numbers on the y-axis represent the frequency of each range of point values. • DOK-2 What are some of the advantages of using a histogram? A histogram lets me quickly compare the frequency of each interval by looking at the heights of each bar. The taller the bar, the more frequent the spinners’ sums were on the interval. The smaller the bar, the less frequent the spinners’ sums were on the interval. It also organizes the values from least to greatest according to the horizontal number line. • DOK-3 How is a histogram different from a bar graph? A histogram has data grouped into intervals. In a bar graph, each number has its own bar. •

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Represent and Interpret Data Explore 2 – Histograms DOK-3 How is a histogram different from a dot plot? A dot plot shows each individual data point, but a histogram just shows how many points are within a range. • DOK-4 Describe a scenario outside of school where a histogram might be the best way to represent a set of data. A histogram might be used to look at data over decades. •

FACILITATION TIP This Exit Ticket provides the intervals on both the x- and y-axis for the histogram. As an extension, give students an opportunity to create their own histogram where they can determine the intervals.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

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Acceleration

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Represent and Interpret Data Explore 3 – Box Plots ACTIVITY PREPARATION Students will analyze information given in a table and a box plot. Students will discover where the values of the ﬁve-number summary are located on a box plot.

Standards for Mathematical Practice • • • • •

Materials

Preparation

Printed • • •

•

1 Student Journal (per student) 1 Set of Class Data Set Cards (per student) 1 Exit Ticket (per student)

• •

Reusable • • • •

2 Resealable bags (per class) 1 Set of colored pencils (per group) 1 Glue bottle (per group) 1 Projector or document camera (per class)

• •

Plan to divide the class into groups of 4 students to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print one set of the Class Data Set Cards for each student. Cut out each of the cards. Place all of the Class A Box Plot Cards in one resealable bag and all of the Class A Data Cards in a different resealable bag. Gather a set of colored pencils and glue for each group. Be prepared to project the Class A Box Plot Card for the class.

PROCEDURE AND FACILITATION POINTS

FACILITATION TIP

Part I 1.

Before reading the scenario, ask the class 1) Has anyone ever participated in a challenge?; 2) If so, what did you have to do?; 3) Did you win the challenge? FACILITATION TIP

Some other types of graphs students may include are: linear graphs, bar graphs, inequalities on a number line, graphing on a coordinate plane, etc.

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Read the following scenario to the class: You are reporting on the Straw Tower Challenge for your school newspaper. One sixth-grade class that is participating in the Straw Tower Challenge has given you their data. Every pair of students in the class recorded the height of their straw tower on a class data chart. You will need to analyze the data given to you so you can correctly report the results for the school newspaper. Discuss the following concepts and questions with the class: a.

DOK-1 What are some of the types of graphs we have worked with so far? We have worked with histograms and dot plots.

DOK-2 How were these graphs similar and different? Dot plots show each individual data point on the graph. Histograms have intervals where the number of data points in each interval is recorded.

Display the Class A Box Plot Card for the class.

Explain the following to the class: This is another type of graph that we can use called a box plot. This graph also has a unique way of organizing data. You will be examining how a box plot is set up and how it specializes in communicating measures of data in ways the other two graphs do not.

6. 7.

8. 9. 10.

11.

Engage

Explore

Explain

Elaborate

Evaluate

Students will glue the box plot in the top box on their Student Journals. Students will collaborate with their teams to explore how box plots are set up and what measures of data we can ﬁnd in a box plot. Each group will collaborate to make predictions about the data from the box plot and record their thinking on their Student Journals. As students are working, actively monitor the students. Ask guiding questions such as the following: a.

DOK-1 What is the lowest data value? The lowest value is 21.

DOK-1 What is the highest highest data value? The highest value is 33.

DOK-1 What do you predict is the middle value of this data? I predict the middle value is 29.

Allow students enough time to complete the predictions about the data from the box plot. Read the following scenario to the class: Class A’s teacher found the data from the Straw Tower Challenge! Use these data points to determine the values of different measures of data for the Straw Tower Challenge for your report. Give the Class A Data Card to each student. Have students glue the Class A Data Card in the box on their Student Journals. Instruct students to collaborate with their teams to determine the measures of data that they previously made predictions about. Have students circle each measure of data on the data card. a.

DOK-1 What is the height of the shortest straw tower? The height of the shortest straw tower is 21 inches.

DOK-1 What is the height of the tallest straw tower? The height of the tallest straw tower is 33 inches.

DOK-1 What is the height of the straw tower that is in the middle of the data? The height of the straw tower in the middle of the data is 29 inches.

4. 5.

Acceleration

FACILITATION TIP To save time, have students refer to the Class A Box Plot Card as needed without having to glue it to the Student Journal.

FACILITATION TIP Project these guiding questions 5a–5c for students to see and read as they collaborate. Monitor student discussions for accurate responses and prompt as needed. Record and project student answers to these questions before moving on to Class A Data Cards in Step 8.

FACILITATION TIP To save time, as with the Class A Box Plot Card, consider allowing students to refer to the card as needed without gluing it to their Student Journal.

Allow students enough time to complete their work and record their observations in Part I of their Student Journals.

Part II 1.

Intervention

REPRESENT AND INTERPRET DATA

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Read the following scenario to the class: Another class has decided to join in giving their information from the Straw Tower Challenge! Use the box plot and ordered list of data to analyze class C’s data set. Students will collaborate with their teams to determine the location of the smallest value, largest value, middle value, lower middle value, and upper middle value. Students will label these locations in the colors listed on their Student Journals. As students are working, actively monitor the students. Ask guiding questions such as the following:

FACILITATION TIP After reading the scenario, ask the class 1) What types of information can be gleaned by analyzing a box plot? FACILITATION TIP

If colored pencils are limited (or to save time), use symbols or letters to code each value. For example, mark each data point with a special check mark, star, circle, arrow, a plus sign, or other simple symbol or letters a. DOK-1 How can I ﬁnd the value in the middle of the data set? Once the rather than colors. Note these changes on values are in order from least to greatest, you can cross off one value on reﬂection questions 2–7 on page 2 and each side until you reach the middle value. adjust on page 3 as well. b. DOK-2 How can I determine the middle value in the lower portion of the data? I can look at all the values below the number in the middle and cross off one value from each side until I reach the middle of this portion of data.

Allow enough time for students to complete Part II and the reﬂection questions on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

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Represent and Interpret Data Explore 3 – Box Plots Math Chat • FACILITATION TIP Students should be familiar with minimum and maximum from prior scopes. Review as needed and include these terms on your anchor chart, word wall, and Student Journal.

Explain the following to the class: Mathematicians call the smallest value in a data set the minimum. Above your label “smallest value,” write “minimum” in red. •

As you ask students about the location of the values on the box plot, have them move their pencil tips to each location. For example, you could say, “Touch the middle value,” or “Point to the middle value," or “Move your pencil tip to the middle value of the lower portion (Q1).”

DOK-1 Where is the largest value located on the box plot? The largest value on the box plot is located farthest right and labeled with a dot above it.

Explain the following to the class: Mathematicians call the largest value in a data set the maximum. Above your label “largest value,” write “maximum” in blue. •

FACILITATION TIP

DOK-1 Where is the smallest value located on the box plot? The smallest value on the box plot is the leftmost point labeled with a dot above it.

DOK-2 The middle value of the data set is labeled where on a box plot? The middle value of a data set is the middle line, which is located inside the box.

Explain the following to the class: Mathematicians call the middle value in a data set the median. Above your label “middle value,” write “median” in green. •

DOK-2 The middle value of the lower portion of data is shown in the box plot where? The middle value of the lower portion of data is the left side of the box on the box plot.

Explain the following to the class: Mathematicians call the lower middle value in a data set quartile 1, or Q1. Above your label “lower middle value,” write “Q1” in purple. •

DOK-2 The middle value of the upper portion of data is shown at what location on the box plot? The middle value of the upper portion is shown as the right side of the box in a box plot.

Explain the following to the class: Mathematicians call the upper middle value in a data set quartile 3, or Q3. Above your label “upper middle value,” write “Q3” in orange. •

FACILITATION TIP Reassure your classes that data scientists frequently use box and whisker plots to display data. Although students may not see these graphs a lot in their day to day life, they will need to be able to read and create them for math and science classes as they advance in academics.

DOK-3 Each of the four sections in the box plot are called quartiles. What do you think this term tells about the percentage of data in each of these four sections? Quartiles sound like quarters. There are four quarters in a dollar, just like there are four quartiles in the box plot. One quarter is worth 25 cents, or 25% of the whole dollar.

Explain the following to the class: In a box plot, each quartile represents 25% of all of the data values in the set. Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook. Notes

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Acceleration

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Represent and Interpret Data Explore 4 – Bar Graphs ACTIVITY PREPARATION Students will analyze information given in a table and a bar graph. Students will display categorical data using a bar graph.

Standards for Mathematical Practice • • • • •

Materials

Preparation

Printed • •

• •

1 Student Journal (per student) 1 Exit Ticket (per student)

Plan to divide the class into groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student.

PROCEDURE AND FACILITATION POINTS STEMscopes Tip The Explain section, located along the scope menu, has a variety of elements designed to solidify students' understanding of the content presented in the Explore section. Each scope's Explain section includes a Picture Vocabulary, independent practice assignments, anchor charts, journal prompts, and interactive notebook activities.

Part I 1.

3. 4.

FACILITATION TIP

To support student focus as they collaborate, project guiding questions 5a–5c. Follow up with additional points: What is the title of the graph? Are the axes' labeled? What do you notice about the intervals on the x-axis? The y-axis? 6. 368

Read the following scenario to the class: The yearbook staff is collecting data to ﬁnd the favorites for the 6th-grade class. They have surveyed the 6th-grade students during their lunch periods. Look at the data displayed on the bar graphs, and help them decide which are the favorites for the 6th-grade class. Discuss the following concepts and questions with the class: a.

DOK-1 What are some of the types of graphs we have worked with so far? Histograms, dot plots, and box plots

DOK-2 How were these graphs similar and different? Dot plots show each individual data point on the graph. Histograms have intervals where the number of data points in each interval is recorded. Box plots show ranges of data.

Give a Student Journal to each student. Students will collaborate with their teams to explore how bar graphs are set up and what measures of data we can ﬁnd in a bar graph. Each group will collaborate to make predictions about the data from the bar graph and record their thinking on their Student Journals. As they are working, actively monitor the students. Ask the following guiding questions: a.

DOK-1 What is the lowest data value? The lowest value is 20.

DOK-1 What is the highest data value? The highest value is 63.

DOK-2 What do you notice about the relationship in period 4’s thoughts on Fast Cars 9 to period 5’s thoughts on Fast Cars 9? Period 5 liked Fast Cars 9 the most, but in period 4, it was only the 3rd most liked movie.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II 1.

2. 3.

4. 5.

Read the following scenario to the class: For the next class favorite, favorite song, the yearbook staff took the survey but didn’t have time to display the data. Take the data they’ve collected, and create a bar graph to display the data. Answer the questions that follow. Students will collaborate with their teams to create a bar graph with the data provided. Students will analyze the data and answer questions. As they are working, actively monitor the students. Ask the following guiding questions: a.

DOK-1 What are the categories I should use to separate the data? Each song title should be its own bar on the bar graph.

DOK-2 How can you differentiate the two lunch periods on the bar graph? I can use two bars, one for each lunch period, for each category.

DOK-2 The song “Hover” has the highest number on the table. Does that make it the class favorite song? Explain. Not necessarily. We need to ﬁnd the song that has the highest total combined.

Math Chat DOK-1 When creating a bar graph, how do you decide the units needed for the y-axis? I ﬁnd the highest value and make my y-axis taller than that. Then, I use units to count between, depending on the highest number. • DOK-2 How does displaying categorical data help you draw conclusions about the data? Using a bar graph helps you see the high and low numbers easier as well as calculate differences in the data. •

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FACILITATION TIP After the Explore activity, consider engaging students by conducting a simple survey with deﬁned categories. Create a data table with your class responses. Allow students to graph the results. FACILITATION TIP

Creating an appropriate scale for a graph is a complex process for most students. Provide some practice for students. Choosing appropriate ranges and intervals Post-Explore support the creation of readable unbiased 1. Have students complete the Exit Ticket to formatively assess their understanding graphs. of the concept. FACILITATION TIP 2. Complete the Anchor Chart as a class. Take time to locate some relevant real-world 3. Have each student complete their Interactive Notebook. bar graphs to engage students. Consider 4. Return to the Hook and instruct students to use their newly acquired skills to using their science/social studies text successfully complete the activity. books, social media surveys, or popular hobby marketing data.

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Represent and Interpret Data Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Dot Plots Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Histograms Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 3

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Box Plots

Interactive Notebook

Show What You Know, Part 4

A cut-and-glue activity to process learning that can be added to a notebook for future reference

Bar Graphs

Independent practice assignment that gives students an opportunity to demonstrate their learning

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ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Represent and Interpret Data Independent and partner games and other activities that provide students with an engaging way to practice the new concept

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Intervention and Assessment SMALL-GROUP PLANNER 1

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

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Represent and Interpret Data

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Students

Notes & Comments

Engage

Explore

Explain

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Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER Use this template to decide how to assess your students for concept mastery. Depending on the format of the assessment, you can identify prompts and intended responses that would measure student mastery of the expectation. See the beginning of this scope to identify standards and grade-level expectations. Key Concepts I can create dot plots and box plots to analyze the results of an investigation.

What prompts will be used?

What does mastery look like?

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I can describe and interpret data displayed.

I can identify each quartile presented in a box plot and how much of the data is represented.

I can interpret data from a histogram.

I can determine the number of observations from the context or diagram.

I can describe the distribution of a quantitative (numerical) variable collected, including its center (median, mean).

I can analyze the shape of a data distribution.

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SCOPE 1

Summarize Numerical Data Scope Introduction SCOPE SUMMARY

6.NR.2.2 Summarize categorical and quantitative (numerical) data sets in relation to the context: display the distributions of quantitative (numerical) data in plots on a number line, including dot plots, histograms, and box plots and display the distribution of categorical data using bar graphs. 6.NR.2.3 Interpret numerical data to answer a statistical investigative question created. Describe the distribution of a quantitative (numerical) variable collected, including its center, variability, and overall shape. 6.NR.2.4 Design simple experiments and collect data. Use data gathered from realistic scenarios and simulations to determine quantitative measures of center (median and/or mean) and variability (interquartile range and range). Use these quantities to draw conclusions about the data, compare different numerical data sets, and make predictions. 6.NR.2.5 Relate the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. 6.NR.2.6 Describe the impact that inserting or deleting a data point has on the mean and the median of a data set. Create data displays using a dot plot or box plot to examine this impact.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In fourth grade, students created line plots and dot plots to represent measurement data and used line and dot plots to solve relevant problems. In fifth grade, students used line plots to solve problems in relation to data sets. Students also create and interpret bar graphs to represent categorical data in fifth grade. In previous sixth-grade scopes, students learned how to describe variability by observing the distribution of data on dot plots, histograms, box plots, and bar graphs. Students compared data sets and box plots to discover how the five-number summary is determined.

make a line plot to display measurement data in fractions.

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use operations to solve problems.

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examine a line plot, a data table, and statements.

Hook

6.NR.2.1 Describe and interpret the center of the distribution by the equal share value (mean).

Accessing Prior Knowledge

Student Expectations

Students summarize data by further analyzing measures of center and spread. In this scope, students learn about mean absolute deviation (MAD), used to interpret deviation from the mean. Students will analyze different graphs used for the same data set and will make decisions about which graph(s) best communicate measures of center and spread for that data set. They determine median (middle value) and mean (average value) as a measure of center, and they reason about the context of a data set in deciding which measure of center most accurately summarizes it. The center of a graph usually summarizes a typical value. If the data set contains an outlier, or if there is a wide range of values, the mean and median result in different values, and in those cases, students determine which measure more accurately summarizes the data. Students describe the impact that inserting or deleting data has on the mean and the median. They also interpret variability by determining range as a measure of spread. Students observe that data sets with wider ranges of values have greater variability, and the data is more spread out.

At the beginning of the scope, students are introduced to the Hook. Through completing the Hook, students will demonstrate their ability to: •

determine the mean absolute deviation from a given set of data.

Students move on to the Explore activities and come back at the end of the explorations to repeat the activity and discuss what they found.

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Acceleration

Mean and Median In this exploration, groups of students will solve a scenario to help José, a baseball player, ﬁnd the mean as a balance point to determine the number of home runs and the number of total hits he would have to have in each series. Students will: •

determine the center (mean and median).

•

determine the shape of data using dot plots and histograms.

Explore 2

Explore 1

EXPLORE ACTIVITIES

Explore 4

Explore 3

In this exploration, students will continue solving a scenario from the story in the previous exploration to determine the mean age of the baseball players and how far the ages deviate from the mean. Students will:

In this exploration, students will help a league of baseball teams record their baseball data through a box plot to help them analyze the data and compare it to other teams. Students will: •

use given data to create box plots.

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determine the spread (range and interquartile range) of data sets.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment. Mean Absolute Deviation

Range and Interquartile Range (IQR)

Comparing Different Representations of the Same Data In this exploration, groups of students will solve another part of the baseball scenario to see where players can improve before their next training season begins. Students will:

determine the mean absolute deviation when given data sets and/or dot plots.

•

determine which representation is better for a speciﬁc type of statistical information.

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment.

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analyze data.

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After completion of the activity, students share their learning, complete an Exit Ticket; then, revisit the Hook to solve.

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Summarize Numerical Data Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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ACCESSING PRIOR KNOWLEDGE Students will examine a line plot, a data table, and statements interpreting data, and will determine which option does not belong with the group. This element is designed to uncover student misconceptions; it should not be taken for a grade. 5.MDR.7.2: Ask questions and answer them based on gathered information, observations, and appropriate graphical displays to solve problems relevant to everyday life.

Materials

Preparation

Printed •

• •

1 Does Not Belong (per student or per group)

Print one Does Not Belong for each student. You may choose to place students in groups of two or three.

Procedure and Facilitation Points 1. 2. 3.

STEMscopes Tip

C is the incorrect answer. If all the sugar were combined, it would be 1 7 __2 cups. If that were distributed evenly among ﬁve recipes, each recipe 1

4. 5.

would receive 1 ½ cups because 7 __2 ÷ 5 = 1 __2.

Identifying Misconceptions • • • •

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FACILITATION TIP

Students may confuse the frequency (number of dots on line plot) with the value on the number line. When problem solving, students may add the number of dots rather than the total values. Students may confuse the number of the recipe from the table with the value/ amount from the table. Students may not know which operations to use when solving problems based on the data table and line plot.

Another misconception may occur when students read the number line and do not notice that the fractions in between the integers do not include the whole number. For 1

example, students may see the __2 between 1 1

and 2 and misread it as __2 rather than 1 __2.

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Summarize Numerical Data Hook – Go Team Go! ACTIVITY PREPARATION Students will determine the mean absolute deviation from a given set of data.

Materials

Preparation

Printed •

• • •

1 Go Team Go! (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Go Team Go! for the whole class to view. Prepare to introduce the scenario and encourage students to think about how to solve it. Be prepared to move to the Explore activities, returning to the Hook activity with newly gained knowledge after the Explore activities have been completed.

PROCEDURE AND FACILITATION Part I: Pre-Explore 1. FACILITATION TIP

Before showing the video and reading the scenario, ask the class 1) Is anyone on a sports team?; 2) If so, what sport is it?; 3) What ages of children are on your team? FACILITATION TIP You could explain that sometimes teachers and schools use the mean absolute deviation of certain test scores to help them make better decisions about test questions, instruction, and testing protocols. FACILITATION TIP Consider projecting Go Team Go! before asking the questions in Step 3. Students may also be curious about why Coach Betsy wants to ﬁnd the mean absolute deviation of the ages on her team.

4. 5.

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Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: Coach Betsy has just made the selections for her competitive gymnastics team. She only chooses ten dedicated and talented girls each year. Girls on the team must be at least six years old and cannot be older than 18. Coach Betsy is happy that she has a good mix of ages on her team. She decides to find out the mean absolute deviation of the ages on her team. Ask students the following questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Student answers will vary. I notice that the youngest team member can be six and the oldest can be 18, so team members can be as much as 12 years apart in age. I wonder what the average age (or mean) of the team members is. What is the mean absolute deviation? I can use math to determine the mean age of the gymnasts on the team by adding up all the ages and dividing the sum by ten because there are ten team members. Project Go Team Go! Explain to students that coach Betsy has released the names and ages of the new competitive team members. She has shared her idea that the first step to finding the mean absolute deviation is to find the mean of the ages of the girls on the team. Discuss the following questions: a.

DOK-1 What does the word deviation mean in math? It means the difference between one value (such as the mean of the girls’ ages) and another value.

DOK-1 What does the word absolute mean in math? It refers to the distance from a value (usually zero) either greater than or less than it.

Engage

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Explain

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Intervention

Acceleration

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and restate the problem. Refer to Go Team Go! and discuss the following questions: a.

DOK-1 What steps are taken to ﬁnd the mean? The ages of the girls are added together. Then, the sum is divided by the number of girls on the team, which is ten. The answer is the mean.

DOK-1 What is the mean absolute deviation? The mean absolute deviation is the average distance (also called spread) from the mean.

DOK-1 What will the mean absolute deviation tell us about the ages of the girls on the competitive gymnastic team? It will tell us the average number of years difference from the average age of the girls on the team.

DOK-1 What is the mean value of the girls’ ages on the competitive gymnastics team? 11.6 years old (16 + 12 + 6 + 8 + 8 + 7 + 17 + 14 + 10 + 18 = 116; 116 ÷ 10 = 11.6)

FACILITATION TIP Depending on students’ prior experience with mathematical deviation and absolute, be prepared with Visual Glossary examples and deﬁnitions before leading this discussion. Otherwise, reassure students that they will learn these terms as they progress through the scope.

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e. DOK-1 What is the next step in ﬁnding the mean absolute deviation? Find the distance from the mean for the age of each girl. For example, 16 − 11.6 = 4.4. The distances for each girl are as follows: 4.4, 0.4, 5.6, 3.6, 3.6, 4.6, 5.4, 2.4, 1.6, and 6.4. f.

DOK-1 What is the ﬁnal step that determines the mean absolute deviation of the girls’ ages on the competitive gymnastics team? Find the average of the differences from the mean for the girls. (4.4 + 0.4 + 5.6 + 3.6 + 3.6 + 4.6 + 5.4 + 2.4 + 1.6 + 6.4 = 38; 38 ÷ 10 = 3.8) DOK-1 What is the mean absolute deviation of the girls’ ages for the competitive gymnastics team? 3.8 years

h. DOK-1 What does this information mean in the real world? The value of MAD is 3.8, which tells us that the spread between the ages of the competitive gymnasts on the team is on average 3.8 years from the mean age of 11.6 years old.

FACILITATION TIP Be prepared with some other relevant real-world examples of where the value of MAD might be used. One example could be calculating the variance of “likes” on a social media post. Finding out how spread out the data is might help inﬂuencers determine the best time of day or the best day of the week to post.

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Summarize Numerical Data Explore 1 – Mean and Median ACTIVITY PREPARATION Students will determine the center (mean and median) and shape of data using dot plots and histograms.

Standards for Mathematical Practice • • •

MP.4 Model with mathematics. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

1 Student Journal (per student) 1 Set of Baseball Scenario Cards (per group) 1 Exit Ticket (per student)

Reusable • •

• • •

•

Plan to divide the class into groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print one set of Baseball Scenario Cards for each group. José’s cards will be used for Part I, and Sammy’s cards will be used for Part II. If desired, print the cards on card stock and laminate them for future use. Place 15 linking cubes into a resealable bag for each group.

1 Gallon-sized resealable bag (per group) 1 Set of 15 linking cubes (per group)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) Does anyone play on a sports team?; 2) If so, how do you earn points for the team?; 3) Is there anyone who keeps track of how many points you earn?

Part I: Mean as Balance Point 1.

2. 3. FACILITATION TIP Project the deﬁnition and some examples of balance point vs mean and median. Use the Visual Glossary if needed.

4. 5.

FACILITATION TIP For some students, it might simplify and clarify the data if you add a letter or word to name each series. The amount of numbers may overwhelm students. Consider adding an S to the label, so it says S1 rather than just a 1. You could also label series 1 as series “A”, or “April”, series 2 as “M”, or “May,” etc.

7. 380

Read the following scenario to the class: José is tracking his total number of home runs and total number of hits for several series of baseball games he played in. José wants to determine how many home runs and how many total hits he would have to have in each series to have balanced out his hits and home runs evenly. Help José ﬁnd the mean as a balance point to determine the number of home runs and the number of total hits he would have to have in each series to balance the home runs and total hits. Give a Student Journal to each student. Divide the class into groups. Give a set of José’s Baseball Scenario Cards and linking cubes to each group. Have students discuss what the term balance point means. Explain to students that the balance point is the mean of a data set. Have students use José’s Home Runs scenario card to record the number of home runs hit in each series on the table in their Student Journals. Students will also create a dot plot to represent the number of home runs hit in each series. They will use the linking cubes to represent the data points for José’s home runs. Note that one linking cube represents one data point (table). For example, in series 1, José has 3 home runs = 3 linking cubes. Have students stack the linking cubes to represent each series. Note that series 1 should have 3 cubes, series 2 should have 1 cube, series 3 should have 5 cubes, series 4 should have 5 cubes, and series 5 should have 1 cube. Once the correct number of linking cubes is stacked for each series, students should ﬁgure out how many cubes each series should have to have an equal number of home runs hit by redistributing the stacks so that each stack has an equal number of cubes. Students will only use cubes for José’s Home Runs scenario card. Actively monitor students as they are working with their groups using the cubes to understand mean as the balance point. Ask the following questions: © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How can you use the linking cubes to determine the balance point? I can move the linking cubes around so each stack of cubes has the same number of cubes in it. The number of cubes in one stack will tell me the balance point of the data. DOK-2 How do you think the balance point would be affected if we added more data points above the balance point? That current number would no longer be the balance point. The balance point would have to increase. DOK-2 How can you interpret mean as the balance point using a dot plot? When viewing the data on the dot plot, we can ﬁnd the distance between every point and the mean, add the distances on each side of the mean, and compare the two sums, and we can see this balancing.

Allow time for students to input the data from both of José’s Baseball Scenario Cards and create their dot plots. They will analyze each dot plot by answering questions and then answer the Part I reﬂection questions. After Part I, invite the class to a Math Chat to share their observations and learning.

DOK-1 How did your group equally distribute the number of cubes to make sure each stack of cubes is balanced? Using José’s Home Runs card, I created 5 stacks using the linking cubes since there are 5 data points in the data set. Each of the stacks had 3 cubes. • DOK-2 Is the balance point always toward the middle part of the graph? No, the balance point is not always toward the middle part because data that is much larger or smaller than the other numbers has an impact on the mean. • DOK-2 How does adding or taking away a data point affect the balance point of the data? Adding or taking away one data point would move the balance point for the data. •

Part II: Finding the Center and Shape of Data

2. 3. 4.

Acceleration

FACILITATION TIP As an alternative to linking cubes, you could use square counting tiles ﬂat on desks. Tiles may be easier to keep ﬂat on desks and the visibility of the stacks may be more clear. FACILITATION TIP Take time and slow down while explaining “How can you interpret mean as the balance point using a dot plot?” Step-bystep instructions may support student success. 1) Find the mean. 2) Find the distance between each point on the plot and the mean. 3) Add the distances of the points below the mean. 4) Add the distances of the points above the mean. 5) Compare the two sums. FACILITATION TIP

Math Chat

Intervention

SUMMARIZE NUMERICAL DATA

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Read the following scenario to the class: Sammy wants to track the total number of times he is up to bat and the total number of hits he has for the past seven series of baseball games he has played in. Sammy will use his data to determine the middle value and the average number of times he is up to bat as well as the middle value and average number of hits he has per series. Help Sammy ﬁnd the center and shape of data using the histogram. Give a set of Sammy’s Baseball Scenario Cards to each group. Encourage students to discuss their observations with their groups as they work through Sammy’s Baseball Scenario Cards. Explain to students that they will use Sammy’s Baseball Scenario Cards to record Sammy’s total times at bat and Sammy’s total hits in the frequency tables on their Student Journals. They will use the frequency tables to create histograms. Monitor and assess students as they are working by asking the following questions: a.

DOK-2 Why do you have to arrange the numbers from least to greatest? It is important to arrange the numbers from least to greatest because you are looking for the middle value. You eliminate the minimum value and the maximum value. You eliminate the 2nd minimum value and the 2nd maximum value and so on until you are left with the middle value.

DOK-1 How do you ﬁnd the middle value if there are two middle numbers? You add the two middle numbers and divide them by two. It is like getting their average.

DOK-1 How can you determine the mean in a data set? Find the sum of the data points and divide it by the number of data points.

Ensure that students clearly label their dot plots before they create them. FACILITATION TIP On reﬂection question 1 on page 2, determine if the balance point can also be called the mean for this data. Clarify for students. FACILITATION TIP As an extension, some students might connect with the term outlier as a way to visualize data that is much larger or smaller. FACILITATION TIP Before reading the scenario, ask the class 1) If you play in a sports team, is any data about your playing abilities kept?; 2) If so, what type of information is kept about you?

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Summarize Numerical Data Explore 1 – Mean and Median d.

FACILITATION TIP After accepting all answers on Step 6, slow down to clarify and review all of the most common measures of center for students. “The middle value” changes if you are using the mean, the median, or the mode. Be sure that each term (include range and IQR) is deﬁned and posted on a word wall, anchor chart, or on Student Journals. FACILITATION TIP On the reﬂection questions for Part II, take time to clarify the vocabulary on pages 4 and 5 questions 2 and 4. Be sure students are able to interchangeably use mean and median as needed for “evenly distributed” and “middle value.”

Ask students, “What other term might you use for the middle value?” Accept all reasonable answers. Inform students that mathematicians call the middle value the median. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat •

•

• FACILITATION TIP This Exit Ticket includes the terms median and mean. You might want to highlight or underline these words with students when you distribute the Exit Ticket. Also, some students may need to be reassured that this is just a ﬁctitious scenario; getting 30 hits or more per game is not realistic.

DOK-1 How can you identify the shape of the data distribution? If most of the data is clustered in the middle and there is a similar number of data points on each side of the balance point, then the shape of the data is symmetrical. When data is mostly clustered toward the left-hand or right-hand side of the graph, then the data is skewed. When there are no clear peaks in the graphs, the data is uniform.

DOK-1 In the Sammy’s Total Times at Bat scenario, what data point would you remove to make the mean and the median the same or closer in value? I would take away the number 20 so there are only 6 numbers in the data set, and the mean would be 37.5, which is closer to the median, 40. DOK-2 How does the shape of the graph affect the mean and the middle value? When the shape of the data is symmetrical like José’s cards, the mean and middle value are equal or very close together. When the shape of the data is skewed to the left, the tail of the graph is pulled toward the left like in Sammy’s Total Times at Bat card. When the tail of the data is skewed to the right like in Sammy’s Total Hits card, the tail of the graph is pulled toward the right, and the mean and the middle value are not equal. DOK-2 Is the mean or the median a better indicator of the center of data? Explain. Either the mean or median can be used to determine the center of the data when it is symmetrical. When the data is skewed, the middle value is a better indication of the center. DOK-2 What other term might you use for the middle value? Accept all reasonable student answers. After a few students have answered, tell students that mathematicians call the middle value the median.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook. Notes

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

SUMMARIZE NUMERICAL DATA

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Summarize Numerical Data Explore 2 – Range and Interquartile Range (IQR) ACTIVITY PREPARATION Students will use given data to create box plots. Students will determine the spread (range and interquartile range) of data sets.

Standards for Mathematical Practice • • •

MP.4 Model with mathematics. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Set of Baseball Team Cards (per group) 1 Box Plot Information Card (per class) 1 Exit Ticket (per student)

• • •

•

Plan to divide the class into groups of 2–4 students. Print a Student Journal and an Exit Ticket for each student. Print a set of Baseball Team Cards for each group. Cut them out, and place them inside a resealable bag. If desired, print the cards on card stock and laminate them for future use. Be prepared to project the Box Plot Information Card for the class.

Reusable • •

1 Projector or document camera (per class) 1 Resealable bag (per group)

PROCEDURE AND FACILITATION Part I: Understanding Box Plots Using Measures of Center and Range FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever played baseball?; 2) If so, what position did you play?; 3) Are you able to hit the baseball most of the time, some of the time, or none of the time?

FACILITATION TIP Be prepared for some students to be curious about the baseball data collected throughout the season. They may wonder if it is for the whole season, per game, or per series.

Read the following scenario to the class: A league of baseball teams is tracking the number of hits throughout the season from the players on their teams. They have recorded this data and need your help representing it in the form of a box plot. The box plots will help each team analyze its data and compare it to the data of other teams in the league. Project the Box Plot Information Card. Lead students through a discussion to remind students of box plot terms: a.

What is the ﬁrst step to creating a box plot? First, order the data values from least to greatest.

Remind students that mathematicians call the number in the middle of a set of data the median. What did we do to find the median of the data set? To find the median, cross off the first and last value. Then, cross off the second and second-to-last value. Continue this pattern until you reach the middle.

What do you do if there are two numbers in the middle? If there are two numbers in the middle, ﬁnd the average of those two numbers.

Mathematicians call the middle value of the numbers below the median Quartile 1, or Q1, and the middle value of the numbers above the median Quartile 3, or Q3. What steps do we take to find Q1 and Q3? Box all of the numbers to the left of the median. Cross off the first and last values until you reach the median of the lower half of the data. Then, box all of the numbers to the right of the median. Cross off first and last values until you reach the median of the upper half of the data. Then, repeat those steps for Q3 using the numbers to the right of the original median.

FACILITATION TIP Consider printing the colored Box Plot Information Card for each student to look at and use for note taking during this discussion. Print or project these discussion prompts (2a–2f) to help students take notes and record the steps and required vocabulary they need to create a box plot. Students could write the steps on the Student Journal.

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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

e. To create a box plot, we must ﬁnd these numbers. f. 3. 4.

Explain the following to the class: Mathematicians call the minimum, maximum, median, Q1, and Q3 the ﬁve-number summary.

Give a set of Baseball Team Cards to each group and a Student Journal to each student. Have students work collaboratively by reading the scenario and data on the Baseball Team A card. They will answer the ﬁrst set of questions on page 1 of their Student Journals and then represent the data by creating a box plot. Monitor and assess students as they are working by asking the following questions: a.

DOK-1 What is the minimum number of hits by a single player on team A? 1

DOK-1 What is the maximum number of hits by a single player on team A? 30

DOK-1 How could we determine the range of hits by the baseball team? To determine the range, we could subtract the least number of hits from the greatest number of hits or the maximum minus the minimum.

DOK-2 If there are a total of 20 values and 5 values are below Q1, how can you find the percentage of data below Q1? To find the percentage of data below Q1, you can write the values below Q1 as a fraction, which would be 5/20. Then, find the equivalent fraction out of 100 since percents are out of 100. This would be 25/100, so the percentage below Q1 is 25%.

e. DOK-2 If you know the percentage below Q1 and the percentage above Q3, what percentage of the data should be inside the box? 50% f.

6. 7.

DOK-1 How could we determine the range of hits by the middle 50% of batters? To determine the range of the middle 50% of batters, we could subtract Q1 from Q3.

Explain the following to the class: Mathematicians call the range of the middle 50% the interquartile range, or IQR. Allow students enough time to record all of their work for Part I on their Student Journals.

FACILITATION TIP If students are struggling to work collaboratively through page 1 of the Student Journal, consider guiding them through the steps together. They should then be prepared to work more independently/collaboratively on page 3.

SUMMARIZE NUMERICAL DATA

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FACILITATION TIP Take time to ensure that students record range deﬁnition and examples on their Student Journal and include it on the class word wall/anchor chart. FACILITATION TIP Slow down and take time to help students see that you are counting the “number of numbers” when you say “write the values below Q1 as a fraction.” Show them how to physically tally the values (lightly cross out with pencil perhaps). FACILITATION TIP Encourage students to refer to the Box Plot information card or project the digital again for them.

Part II: Understanding Box Plots Using Measures of Spread 1.

3. 4.

Read the following scenario to the class: Team B, team C, and team D think it is a great idea to track their hits this season! Help each team determine their range and IQR of hits so far this season. Monitor and assess students as they are working by asking the following questions: a.

DOK-1 What step should you take ﬁrst to ﬁnd range and IQR? First, I will need to order the number of hits from least to greatest.

DOK-1 What information do you need to determine to create a box plot? I will need to determine the minimum, maximum, median, Q1, and Q3 to create a box plot.

DOK-1 How do you calculate the range? To calculate range, subtract the minimum value from the maximum value.

DOK-2 How do you calculate the interquartile range? The difference between Q3 (upper quartile) and Q1 (lower quartile) is the interquartile range (IQR).

FACILITATION TIP After reading the scenario, ask the class 1) What is range?; 2) What is IQR?; 3) How would having this information beneﬁt the players and team? FACILITATION TIP Emphasize for students how to carefully create an accurate list from least to greatest using light pencil marks to cross off as they transfer the data to their list. Another method might be for one student (or teacher) to read the data out loud as students record it in order.

Allow students enough time to record all of their work for Part II on their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

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Summarize Numerical Data Explore 2 – Range and Interquartile Range (IQR) Math Chat DOK-1 What is the difference between the range and the interquartile range? The range calculates the difference between the highest and lowest values in a data set. The IQR only takes the maximum and minimum of the middle numbers of the data set. • DOK-1 When would you use the IQR instead of the range? The IQR is a more accurate idea of the spread around the middle values. It is not affected by very low or high numbers like the range, which takes all the data points into the calculations. • FACILITATION TIP Another reason why scientists and mathematicians use the IQR is because it helps them focus on outliers. Range is a helpful measure, but it doesn’t necessarily reveal strong outliers. FACILITATION TIP Struggling students may need their notes or the Box Plot Information Card to refer to as they complete the Exit Ticket. Determine your success criteria for showing work on this Exit Ticket.

Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook. Notes

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

SUMMARIZE NUMERICAL DATA

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Summarize Numerical Data Explore 3 – Mean Absolute Deviation ACTIVITY PREPARATION Students will determine the mean absolute deviation given data sets and dot plots.

Standards for Mathematical Practice • • •

MP.4 Model with mathematics. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • •

• •

1 Student Journal (per student) 1 Exit Ticket (per student)

Plan to divide the class into groups of 2–4 students. Print a Student Journal and an Exit Ticket for each student.

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) If you play on a sports team, are the rest of the players on the team around the same age as you?; 2) Do you think professional sports players are all about the same age? FACILITATION TIP

2. 3. 4.

Project questions 4a–4d to support student collaboration and guide your discussion for the Math Chat.

FACILITATION TIP Show an example or have a student bring up a work sample to share. Demonstrate how much work you want students to show as they are working with this data.

5. 6.

DOK-1 What does ﬁnding the mean of this set of data tell you? In this data set, ﬁnding the mean would tell you the average age of baseball players.

DOK-1 How do you ﬁnd the mean of a set of data? To ﬁnd the mean, we must add all of the ages together and then divide by the total number of ages in the data set.

DOK-1 How do you ﬁnd the distance each age is from the mean? To ﬁnd the distance each age is from the mean, we would count the spaces between the age you are looking at and the mean on the number line in the dot plot.

DOK-1 Describe how to ﬁnd the mean of these distances. When ﬁnding the mean of these distances, we would add all of the distances together and divide by the total number of distance values that we have.

Allow time for students to record their learning on their Student Journals and answer the reﬂection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat •

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Read the following scenario to the class: Baseball fan Billi is interested in ﬁnding the ages of baseball players. Billi thinks that all of the players will be around the same age, with not much spread in their ages. Use the dot plot Billi has created to determine the mean age of baseball players and how far the ages deviate from the mean. Give a Student Journal to each student. Have students work with their groups to complete their Student Journals. Monitor and assess students as they are working by asking the following questions:

DOK-1 Why would we ﬁnd the absolute value of the distance between each point and the mean? We would ﬁnd the absolute value because distance cannot be negative. © Accelerate Learning Inc. - All Rights Reserved

•

• •

•

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 What are the steps we took to determine the mean of the absolute value of distances? First, find the mean of the data set. Next, calculate the distance between each value and the mean of the data set. Last, find the mean of the distances between each value and the mean of the data set. Allow students time to reflect on the steps they took through this process. After a minute, tell the students that mathematicians call this value that we found the mean absolute deviation, or MAD. DOK-1 Why would the mean and the median not be equal? The mean and median would not be equal if there is a wide variability in the data set. DOK-1 Describe the strategy you used to find mean absolute deviation. First, find the mean of the data set. Next, calculate the distance between each value and the mean of the data set. Last, find the mean of the distances between each value and the mean of the data set. DOK-2 What would a greater value represent when you find the mean absolute deviation? The greater the MAD value, the farther apart the spread of the data will be. DOK-1 How does MAD communicate variability? Mean absolute deviation describes variation in a data set and is the average distance between each data value and the mean.

Intervention

Acceleration

FACILITATION TIP If students don’t already have the steps noted for ﬁnding the MAD, take time to record them on the Student Journal.

STEMscopes Tip Fluency Builders, located in the Elaborate section, are partner or smallgroup student-led games that engage students in practicing the skills and concepts addressed in the scope. These games come with studentfriendly instruction sheets. All the materials used in the games are found in the print ﬁles on the right side of the screen.

SUMMARIZE NUMERICAL DATA

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Post-Explore 1. 2. 3.

Have students complete the Exit Ticket to formatively assess their understanding of the concept. Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook. Notes

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Summarize Numerical Data Explore 4 – Comparing Representations of the Same Data ACTIVITY PREPARATION Students will determine which representation is better for a speciﬁc type of statistical information.

Standards for Mathematical Practice • • •

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

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Baseball Cards (per group) 1 Exit Ticket (per student)

Plan to divide the class into groups of 2–4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print one set of Baseball Cards for each group. If desired, print the cards on card stock and laminate them for future use.

PROCEDURE AND FACILITATION FACILITATION TIP

Before reading the scenario, ask the class 1) Does anyone collect or have seen baseball cards?; 2) If so, what statistics about players are on the cards?; 3) Why would people want to know a player’s statistics?

2. 3. 4.

FACILITATION TIP Depending on your students, take some time to project each Baseball card one at a time before passing them out. Support student comprehension of what each graph is displaying by pointing out the labels, the titles, the intervals, etc. You might want to conﬁrm that most students can state what the graphs are communicating before they begin collaborating.

5. 6.

FACILITATION TIP Project the essential information from this scenario so students can read it with you. Emphasize the three important data sets. Some students may need additional explanation regarding RBIs, hits, and at bats. Perhaps you can make a connection between shots on goal and goals scored.

Read the following scenario to the class: A local baseball team has collected and recorded data about its players throughout the past season. They want to use this data to see where they can improve before their next training season begins. Can you help analyze each data set? Give a Student Journal to each student. Divide the class into groups. Give one set of Baseball Cards to each group. Explain the following to the class: Baseball has many different ways of collecting and recording data for each player. Three of the important data sets that baseball players and fans want to know are the runs batted in, or RBIs, a player has; the total number of hits for each player; and the total number of times a player is at bat. These sets of data help determine which players have a higher overall batting average. Use the graphs provided to determine what measures can be found from each data set and which representations relate better to which scenarios. Have students collaborate with their groups using the Baseball Cards to answer the corresponding questions on their Student Journals. Monitor and assess students as they are working by asking the following questions: a.

DOK-1 How did you determine the spread of the data? To determine spread, we needed to ﬁnd the range or the interquartile range.

DOK-2 Which graphs help you easily determine median? Explain. Box plots show you the median, as it is the middle line in the box. You could also ﬁnd the median of a dot plot, but it would take more work, as it is not given straight to you.

DOK-2 Which graph tells you what values are included in the data and what values are not included in the data? Explain. Dot plots are the only graphs that give you each of the data points in their representation.

DOK-1 Which graphical representation shows if there are gaps or clusters in the data? You can easily see gaps and clusters in dot plots.

FACILITATION TIP Select some of these guiding questions to project as students are collaborating. Direct students to be prepared to discuss the reﬂection questions during the Math Chat. 390

Allow time for students to analyze each Baseball Card, answer the corresponding questions, and answer the reﬂection questions on their Student Journals. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat •

• • • •

DOK-2 Describe what each graph is better at representing. The histogram shows the distribution of the data. The dot plot gives you all of the data values, so there is more statistical data that you can gain from the dot plot. The box plot shows you the information for each quartile and can easily show you the distribution and shape of the data. DOK-1 Which graph best represents gaps and clusters in data sets? Dot plots best represent gaps and clusters in data sets. DOK-1 Which graph shows the median? The box plot shows the median. DOK-1 Which graph makes it easier to ﬁnd the interquartile range? The box plot is the easiest graph to use to ﬁnd the interquartile range. DOK-2 Can all graphs give you the same measures of data? Give one example. No, not all graphs can give you all of the measures of data. Histograms give you the data values on intervals of the data; therefore, you can not determine some measures of data from a histogram.

Post-Explore 1. 2. 3. 4.

Intervention

Acceleration

STEMscopes Tip Spiraled Review, located in the Elaborate section, provides students with a contextual scenario used to solve four different problems. This activity helps students maintain essential knowledge, see how mathematical skills connect from one topic to the next, and experience real-world applications of previously learned skills.

SUMMARIZE NUMERICAL DATA

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FACILITATION TIP As an extension, some students might want to answer a fourth question about the real-world use of data representations. For example, “How might a company use different graphs of the same data in different situations? Could a company take a batch of survey results and present it in different ways? List some reasons why it is important for us to be able to read and interpret these statistical representations.”.

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Summarize Numerical Data Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

A slide presentation of important vocabulary terms along with a picture and deﬁnition

Mean and Median Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

A guide to facilitating the creation of a chart with students for each scope

Range and Interquartile Range (IQR) Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 3

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Mean Absolute Deviation

Interactive Notebook

Show What You Know, Part 4

Independent practice assignment that gives students an opportunity to demonstrate their learning

A cut-and-glue activity to process learning that can be added to a notebook for future reference

Comparing Different Representations of the Same Data Independent practice assignment that gives students an opportunity to demonstrate their learning

392

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

A quick story to engage student interest along with four problems over previously learned skills

Mean Absolute Deviation Independent and partner games and other activities that provide students with an engaging way to practice the new concept

SUMMARIZE NUMERICAL DATA

Home

Fluency Builder Measures of Data Independent and partner games and other activities that provide students with an engaging way to practice the new concept

393

Intervention and Assessment SMALL-GROUP PLANNER 1

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

SUMMARIZE NUMERICAL DATA

Summarize Numerical Data

3 394

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

What does mastery look like?

SUMMARIZE NUMERICAL DATA

Home

I can understand the equal-share concept and how it relates to the formula for mean. I can determine the mean of a data set by using the equal-share method. I can summarize quantitative data in relation to context. I can describe the interquartile range of a data set. I can describe the mean absolute deviation, the range, and the overall shape of a data set. I can use prior knowledge of absolute value to help interpret data. I can apply understanding of the measures of center (mean and median). I can apply understanding of interquartile range and range. I can draw conclusions about data, compare different numerical data sets, and make predictions using data gathered from realistic scenarios and simulations. I can determine which measures of center and variability best describe the data based on the shape of the data. I can analyze the shape of a data distribution and determine the impact a single data point has on the data set represented visually. I can create dot plots, box plots, and histograms to analyze the results of an investigation. I can identify each quartile presented in a box plot and how much of the data is represented. I can describe the distribution of a quantitative (numerical) variable collected, including its center (median, mean). I can analyze the shape of a data distribution.

395

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6 Georgia Math Teacher Guide

Grade 6 Teacher Guide

STEMscopes.com ISBN: 979-8-88826-716-5

ISBN: 979-8-88826-665-6

6 GEORGIA

MATH G6