STEMscopes Georgia Math Teacher Guide Grade 7 by acceleratelearning

7 Georgia Math Teacher Guide

Grade 7 Teacher Guide

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

ISBN: 979-8-88826-666-3

7 GEORGIA

MATH G7

GEORGIA

Teacher Guide: Grade 7 ISBN: 979-8-88826-666-3 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

Addition and Subtraction with Rational Numbers .................................................. 18

SCOPE 2

Multiplication and Division with Rational Numbers ............................................... 48

SCOPE 3

Rational Number Operations ............................................................................... 76

SCOPE 4

Proportional Relationships .................................................................................. 94

SCOPE 5

Understand Slope ............................................................................................. 116

SCOPE 6

Ratios, Rates, and Percents............................................................................... 134

SCOPE 7

Percent Application .......................................................................................... 152

SCOPE 8

Expressions...................................................................................................... 176

SCOPE 9

Solve Equations and Inequalities ....................................................................... 196

TABLE OF CONTENTS

Table of Contents

SCOPE 10 Scaling............................................................................................................. 214 SCOPE 11 Angles ............................................................................................................. 228 SCOPE 12 Angle Relationships .......................................................................................... 242 SCOPE 13 Circles ............................................................................................................. 258 SCOPE 14 Surface Area .................................................................................................... 282 SCOPE 15 Volume ............................................................................................................ 296 SCOPE 16 Probability ....................................................................................................... 314 SCOPE 17 Informal Inferences .......................................................................................... 334

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

Addition and Subtraction with Rational Numbers Scope Introduction SCOPE SUMMARY In this scope, students will be able to describe situations in which the inverse identity is present. They will understand absolute value and interpret sums of rational numbers through real-world examples. Students will gain a greater understanding of the additive inverse and its role in subtracting rational numbers. They will be able to perform the properties of operations with both positive and negative numbers. Student Expectations

7.NR.1.1 Show that a number and its opposite have a sum of 0 (are additive inverses). Describe situations in which opposite quantities combine to make 0.

7.NR.1.4 Show and explain subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference and apply this principle in contextual situations. 7.NR.1.5 Apply properties of operations, including part-whole reasoning, as strategies to add and subtract rational numbers.

Future Expectations

In previous grades, students linked line segments together on a number line to add and subtract rational numbers. In Grade 6, students learned about absolute value and the distance that two numbers are from one another regardless of direction. Students reasoned with properties of operations as they built the basic understanding of multiplication and division of rational numbers. This will be the foundation of this scope as students place terms and names to these properties and apply them to all rational numbers, not just integers.

By understanding how to properly add and subtract rational numbers, students can then explore the existence of irrational numbers in 8th grade. The idea of rational versus irrational numbers will be seen more as students begin to study the different types of functions.

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: •

•

examine a number, an absolute value, a number line; and, a realworld description to determine which option does not belong with the group. understand absolute value and the various concepts that relate to it.

Hook

7.NR.1.3 Represent addition and subtraction with rational numbers on a horizontal or a vertical number line diagram to solve authentic problems.

Background Knowledge

Accessing Prior Knowledge

7.NR.1.2 Show and explain p + q as the number located a distance |q| from p, in the positive or negative direction, depending on whether q is positive or negative. Interpret sums of rational numbers by describing applicable situations.

VERTICAL ALIGNMENT

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

determine the answer to a word problem involving the subtraction of integers by using counters and number lines.

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

Addition of Integers with Counters In this exploration, groups of students will solve a realworld scenario involving Mr Sanchez, a teacher, who is working to ﬁnd the atoms that have a zero charge. Students will: •

Explore 2

Explore 1

EXPLORE ACTIVITIES

use various mathematical methods, such as counters, number lines, and equations.

•

ﬁnd the differences of both positive and negative numbers through using various strategies.

•

solve problems using mats and counters, number lines, and equations.

Rational Number Subtraction with Number Lines In this exploration, groups of students will solve realworld problems involving subtraction a chef encounters at a restaurant. Students will: •

subtract positive and negative numbers.

•

understand the relationship between subtraction and corresponding additive inverses.

•

use number lines to determine answers to realworld subtraction problems and to ﬁnd equivalent and inverse expressions.

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

use number lines to solve various addition problems, including word problems, involving positive and negative rational numbers.

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

Explore 4

Explore 3

In this exploration, students will collaborate to solve issues involving Mr. Sanchez, a game developer who is coming to speak, and the need to help provide feedback to the game developer through solving various problems to determine how close scores are to encourage engagement and competition. Students will:

In this exploration, students will betasked with helping determine what could affect ﬁsh in Alaska by analyzing and solving situations to see which equate to zero and keep the balance of ﬁsh. 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.

Integer Subtraction with Counters and Number Lines

Rational Number Addition with Number Lines

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Home

Using the Properties to Solve In this exploration, students will solve a scenario about helping an event planner do a test run of one of her signature invitations, give her feedback, and help her solve some of her problems. Students will: •

use commutative, associative, additive, inverse, and distributive properties to solve problems.

•

solve problems that involve positive and negative integers and rational numbers.

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Addition and Subtraction with Rational Numbers 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 number, an absolute value, a number line, and a real-world description 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. 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.

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.

FACILITATION TIP

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. A (−3) does not belong, because all others are equivalent to 3.

4. 5.

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.

Identifying Misconceptions • •

Students may misunderstand that the real-world scenario describes the absolute values of both 3 and −3. Students may confuse absolute values with opposites. They do not understand that an absolute value represents the magnitude of a number or the distance from 0.

Direct students to focus on the mathematical content of the four options. Some students may notice the images or words to determine which item doesn’t belong. FACILITATION TIP Encourage students to use the sentence frame when they contribute to the discussion. “Letter __ does not belong because ______ .”

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

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FACILITATION TIP Find a place in the room to post a number line on the wall with with both positive and negative numbers, and keep asking “How far away from zero?” when discussing absolute value.

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Addition and Subtraction with Rational Numbers Hook – Let it Snow! ACTIVITY PREPARATION Students will determine the answer to a word problem involving the subtraction of integers by using counters and number lines.

Materials

Preparation

Printed •

• • •

1 Let It Snow! (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Let It Snow! 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) What is the weather like in the winter?; 2) Is it possible for the weather to change in a short period of time?; 3) Would you rather be outside in the winter when the temperature was above freezing or below freezing and snowing? FACILITATION TIP Review Celsius and Fahrenheit using a model of a thermometer. Explain that scientists and most countries use Celsius regularly.

FACILITATION TIP In order to detect misconceptions, take notes as students respond to each question one at a time. Create a quick chart with “Notice,” “Wonder,” and “Math” as headings.

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: It was Matteo and Sonia’s job to walk their dogs Bruno and Barbie every day before dinner. This was a cold job in winter. However, one winter evening, they put on their jackets and boots and realized the Sun was shining and it was not as cold as usual! Sonia checked the thermometer outside their back door while Matteo clipped the leashes onto the dogs’ collars. Sonia saw that it was above freezing at 4°C. Enjoying the nice weather, they walked farther than usual into the woods. Suddenly, they felt the wind pick up, and the temperature seemed to be dropping. They turned around and when they got back to the road, they realized the Sun was no longer shining and, in fact, it was snowing! They wondered how much the temperature had dropped and what the current temperature was. 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 the temperature is in degrees Celsius. I wonder how many degrees the temperature dropped. What is the current temperature? I can use math to determine the current temperature if I know how many degrees the temperature dropped. Project Let It Snow!

Engage

Explore

Explain

Elaborate

Evaluate

Explain to students that when Matteo and Sonia got home, Sonia checked the thermometer and the temperature had dropped 8�. When she told Matteo, he wondered what the current temperature was. Discuss the following questions: a.

DOK-2 Why might it be diﬃcult to solve this problem? This would be diﬃcult because the temperature dropped 8�.

DOK-2 How could having two color counters be useful, and what would each represent? One color could represent positive numbers, and one color could represent negative numbers. This could be helpful to determine how to solve problems with both negative and positive numbers.

DOK-1 What is another model that could be used to check your answer with the counters? Answers will vary. A number line is a model that can be used to check your answers.

Acceleration

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.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Intervention

FACILITATION TIP

Show the Phenomena Video again, and restate the problem. Refer to Let It Snow! and discuss the following questions: a.

DOK-1 Explain how to solve the problem by using two color counters. The yellow counters are positive, and the red counters are negative. Start with the 4 yellow counters because the temperature was 4� when they left the house. Then, we need to take away 8, so we will add 4 yellow and 4 red counters to create zero pairs and have enough counters to take away.

DOK-1 What is the equation? 4 – 8 = –4

DOK-1 What was the temperature when Matteo and Sonia got home? The temperature was –4�.

DOK-2 How can you use a number line to help you check your answer? I started at 4� and made 8 jumps backward because I was subtracting. I ended on –4�. Then, I knew the answer I got with the counters was correct.

Use this opportunity of restating the problem to have students identify what they are being asked to ﬁnd. If needed, direct them to the phrase, “They wondered how much the temperature had dropped and what the current temperature was.”

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

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ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Addition and Subtraction with Rational Numbers Explore 1 – Addition of Integers with Counters ACTIVITY PREPARATION Students will use counters, number lines, and equations to search for atoms with a zero charge, using models of atoms showing the number of protons and electrons.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. 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 Atom Cards (per group) 1 Horizontal Number Line (per group) 1 Vertical Number Line (per group) 1 Exit Ticket (per student)

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Reusable • • • •

1 Set of two-color counters (per group) 1 Resealable bag (per group) 2 Sheet protectors (per group) 1 Dry-erase marker (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 a set of Atom Cards for each group. Cut out the cards, and place each set of cards in a resealable bag for each group. If desired, print on card stock and laminate for future use. Add a set of two-color counters to each resealable bag with the Atom Cards. Print a Horizontal Number Line and a Vertical Number Line for each group. Consider printing them on card stock for durability. Then, place each number line inside a 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 FACILITATION TIP

After reading the scenario, ask the class 1) Who can explain what a pharmaceutical company does?; 2) What is the meaning of “zero charge?”; 3) What role does investigation play in a pharmacist’s job? FACILITATION TIP Most atom models that students have seen up until now will likely show equal protons and electrons. Students may be unaware that some atoms (ions) have unequal protons and electrons.

Read the following scenario to the class: Mr. Sanchez organized a Career Day for his class. The first speaker was a chemist doing research for a pharmaceutical company. His job is to try to find atoms that have a zero charge, which means an atom that has the same number of electrons (particles with negative charge) and protons (particles with a positive charge). He has to investigate hundreds of atoms. Today, your job is to help the chemist by investigating 8 atoms to see if they have a zero charge. Give a Student Journal to each student. Give a bag with Atom Cards and twocolor counters, a Horizontal Number Line inside a sheet protector, a Vertical Number Line inside a sheet protector, and a dry-erase marker to each group. Explain to students that they will be working in their groups to determine the charge of eight atoms. Students will choose an Atom Card, find the corresponding data table on the Student Journal, and investigate the charge of the atom. Explain to students that with two-color counters, the yellow side represents a proton with a +1 charge, and the red side represents an electron with a −1 charge. Tell students they will prove their data through pairing counters, utilizing a number line, and writing an equation. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Instruct students that they may use the Horizontal Number Line or Vertical Number Line and dry-erase marker to start with protons and make jumps to the left for the electrons they are adding. They may also do the opposite and start with the electrons and make jumps to the right for the protons they are adding. They can use the number lines to check problems or to prepare for drawing the jumps on the number lines on the Student Journal. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 What does the yellow side of a two-color counter represent? The yellow side represents a proton, which has a charge of positive one.

DOK-1 What does the red side of a two-color counter represent? The red side represents an electron, which has a charge of negative one.

DOK-2 How can you tell with the counters if an atom has a zero charge? If it has a zero charge, all of the red and yellow counters will pair up with none left over.

DOK-2 How do you use a number line to help you determine the charge of an atom? We start by marking the number of protons, which is a positive number. Then, we jump to the left the number of electrons to ﬁnd the ﬁnal charge.

Allow students enough time to write the number of electrons and protons, draw the two-color counters, record the electrons and protons on the number lines, and write the equations on their Student Journals for each Atom Card. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat •

•

DOK-2 How did using counters help you determine the charge of an atom? We looked at the cards and counted the number of protons and electrons. Then, we took out that many yellow and red counters. We paired them up. If they all paired up, the charge of the atom was zero. If they didn’t pair up, the charge of the atom was whatever was left over. One yellow counter means a charge of +1. One red counter means a charge of −1. DOK-2 How did using a number line help you determine the charge of an atom? We marked the number of protons on the number line on the positive side. Then, we added the number of electrons to it by jumping the number of electrons to the left on the number line to find the sum, which is the charge of the atom. DOK-2 What are the similarities and differences between horizontal and vertical number lines? Both types of number lines can be used to find the distance between numbers (for both addition and subtraction) by using jumps. Both types of number lines show positive and negative numbers. One difference is that a horizontal number line shows positive numbers to the right and negative numbers to the left, while vertical number lines show positive numbers up and negative numbers down. Adding is to the right and subtraction is to the left on a horizontal number line. Adding is up and subtraction is down on a vertical number line. DOK-1 How did you write the equation? We started with the number of protons and added the number of electrons. The leftover counters, or place on the number line, were the solution. DOK-2 Did it matter if you wrote the electrons or the protons first in the equation or when using the number line? No, it did not matter. The solution was the same. On the number line, if we started with the electrons, it was easy to jump to the right to add the protons. If we started with the protons, we jumped to the left to add the electrons since they were negative. DOK-2 What did you notice about adding electrons (negative numbers) to protons (positive numbers)? It was just like subtracting. This was especially easy to see when we were using the number line because we started with a positive number (protons) and then added the negative number (electrons). When we added a negative number, it was just like subtracting a positive number.

Intervention

Acceleration

FACILITATION TIP Take time to ensure that students are drawing the jumps correctly. Use a student work sample to show clearly drawn jumps with accurate starting and stopping points.

FACILITATION TIP Encourage students to use both tools; the counters and the number lines. Give them the opportunity to express which tool they prefer based on their learning. Remind them that the Exit Ticket will require them to use both tools.

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

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FACILITATION TIP Students might make the connection that when the counters are “all paired up,” it can be like a tie in a sports match or a video game.

FACILITATION TIP Have models of both vertical and horizontal number lines posted in the room. You can also hold a meterstick or ruler and demonstrate how it can be held horizontally or vertically to measure.

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Addition and Subtraction with Rational Numbers Explore 1 – Addition of Integers with Counters 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 __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

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ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Addition and Subtraction with Rational Numbers Explore 2 – Rational Number Addition with Number Lines ACTIVITY PREPARATION Students will use number lines to solve addition problems, including word problems, involving positive and negative rational numbers.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. 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 Alaska Cards (per pair) 1 Horizontal Number Line (per group) 1 Vertical Number Line (per group) 1 Exit Ticket (per student)

•

Reusable • • • •

• •

1 Resealable bag (per pair) 1 Box of crayons or colored pencils (per student) 1 Dry-erase marker (per pair) 2 Clear sheet protectors (per pair)

Plan to divide the class into groups of 2 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Alaska Cards for each pair. Cut out and place each set of cards in a resealable bag. If desired, print them on card stock, and laminate them for future use. Print the Horizontal Number Line and the Vertical Number Line for each pair. Consider printing them on card stock for durability. Then, place each number line inside a sheet protector for each pair. Gather enough dry-erase markers for each pair to receive one. Gather enough boxes of crayons or colored pencils for each student to have a set.

PROCEDURE AND FACILITATION FACILITATION TIP After reading the scenario, ask the class 1) What might happen if the ﬁsh population does not remain balanced in Alaska?; 2) What is the meaning of “equivalent absolute value”?; 3) What are some situations that could affect the ﬁsh population in Alaska?

Part I 1.

FACILITATION TIP Engage students with a quick review of Alaska’s location and size.

FACILITATION TIP Preview vocabulary that may be new for some students: conservationist, constant, released, waterways. FACILITATION TIP Preview each scenario before giving to students. Take time to ensure that students understand key words like deposit, debt, rectify, bonus, and checking account. 28

Read the following scenario to the class: Mr. Sanchez chose Nukilik, a fisherman and conservationist from Alaska, for his second Career Day speaker. One of Nukilik’s jobs is to ensure the fish population in Alaska remains constant. This requires him to keep track of data about pollution; temperatures; water quality; and how many fish were caught, hatched, and released into Alaskan waterways monthly. Fish caught versus fish hatched and released should be equal numbers with opposite signs. They should have an equivalent absolute value. Today, your job is to help Nukilik by investigating situations that could affect fish in Alaska by seeing which situations make zero and keep the balance. Give a Student Journal to each student. Give a bag of Alaska Cards, a Horizontal Number Line, a Vertical Number Line, and a dry-erase marker to each set of partners. Explain to students that they will be working with their partners to determine whether each word problem makes zero through an addition equation. Tell students they will read each scenario, write an equation, and circle yes or no to answer whether a zero balance was described. Encourage students to use the Horizontal Number Line or Vertical Number Line and a dry-erase marker to act out the word problems and determine solutions.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 What does the term absolute value mean? It means the distance from zero—either to the right (positive direction) or the left (negative direction)—with just a magnitude and no sign.

DOK-1 Was the ﬁrst addend always a positive or a negative number in the equations corresponding to the word problems? No, the ﬁrst addend could be either positive or negative.

DOK-1 In order to make zero, what did the signs of the addends have to be? They had to be the same number with opposite signs. One addend had to be positive, and one addend had to be negative.

Allow students enough time to use the Horizontal Number Lines, Vertical Number Lines, and dry-erase markers to write the corresponding equations and circle whether a situation results in a solution of zero on their Student Journals. After Part I, invite the class to a Math Chat to share their observations and learning.

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.

Math Chat •

•

DOK-2 How can you tell, on a number line, if the situation is balanced and makes zero? We begin by marking the first addend. Then, we jump the magnitude and direction of the second addend. +8 moves eight spots to the right, while +(−8) moves eight spots to the left. Where we land is the solution. DOK-1 How can you tell, on a number line, if the situation is balanced and makes zero? No matter where the starting point is, the jumps will end at zero as the solution. DOK-2 If the first addend is a negative number, does the second addend need to be negative or positive to make zero? Why? The second addend needs to be positive because if you add a negative number to a negative number, the answer will be a negative number that is even farther away from zero on the number line. If you add a positive number, it moves back to the right. DOK-2 What are some action pairs in real-life situations that lead to opposites combining to make zero? Spending and saving, increasing and decreasing in elevation, having particles with opposite charges, temperatures rising and falling, etc. DOK-2 Why might a conservationist like to keep a population growth at zero like Nukilik wanted to do with the fish? If the population growth is negative, the fish may become endangered or extinct. If the population growth is positive, the environment may not be able to sustain that many fish.

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

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FACILITATION TIP Find a way to connect with real-life situations that students are familiar with; a popular game or activity in their daily lives or one played in the classroom.

Part II 1.

Read the following scenario to the class: Nukilik appreciated your investigative skills when trying to determine whether the balance was being kept in Alaska with fish populations and the environment. The last couple of months have had some unseasonable weather and a lot of tourists. Some preliminary data has shown some imbalances, and Nukilik is concerned. Nukilik wants to see if you and your partner can detect imbalances as well as balances. He wants to test your skills before putting you to work researching Alaskan data. Today, your job is to help Nukilik by solving word problems that add both positive and negative numbers that result in positive and negative sums to prove you are ready to help him with an important project dealing with imbalances on-site in Alaska. Explain to students that they will be working with their partners to match a word problem with an equation and a number line model. Tell students once they find the three matching parts, they will color the three boxes the same color. Each matching triplet will be colored a different color. Encourage students to use the Horizontal Number Line or Vertical Number Line and a dry-erase marker to act out the word problems and find matching number line models and equations.

FACILITATION TIP After reading the scenario, ask the class 1) Why might increased tourism affect the balance of the fish population?; 2) How might unseasonable weather affect the balance of the fish population?; 3) What might be done to restore the balance of the fish population in Alaska? FACILITATION TIP Ensure that students understand that the equation and number lines next to the word problems do not match.

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Addition and Subtraction with Rational Numbers Explore 2 – Rational Number Addition with Number Lines 4.

FACILITATION TIP

Offer support to students who need it by directing them to ﬁnd the ﬁrst addend in the story and then look for that starting point on the various number lines. If it’s a match, have them check to the rest of the story with that number line to see if it is a representation of that story.

Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 What direction is positive on a horizontal number line, and what direction is positive on a vertical number line? To the right is positive on a horizontal number line, and up is positive on a vertical number line.

DOK-1 What direction is negative on a horizontal number line, and what direction is negative on a vertical number line? To the left is negative on a horizontal number line, and down is negative on a vertical number line.

Allow students enough time to use the Horizontal Number Lines, Vertical Number Lines, and dry-erase markers to ﬁgure out corresponding equations and color matching components of word problems, equations, and number line models on their Student Journals. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat •

DOK-1 Which way do you move on the number line in each of the following situations? Negative number + negative number Negative number + negative number moves left • Negative number + positive number Negative number + positive number moves right • Positive number + negative number Positive number + negative number moves left • Positive number + positive number Positive number + positive number moves right •

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.

•

FACILITATION TIP

•

Encourage students to further explore this question on their own outside of class; have them report back ideas the next day.

DOK-2 Can you tell if the answer will be positive or negative before you actually ﬁnd the sum of the numbers? Explain. Yes, we can tell which way we will move on the number line. If both addends are positive, the answer will be positive. If both addends are negative, the answer will be negative. If one addend is positive and one addend is negative, the addend farthest from zero on the number line (the number with the greatest absolute value) will give its sign to the sum. DOK-2 Temperature, altitude, and bank account balances are all examples of real-world situations that can have negative values. What are some examples of real-world situations that cannot have negative values? Explain. Weight or mass cannot be negative because something cannot have a negative weight or mass or it would not be matter. Distance cannot be negative because it is length, and a measurement of length cannot be negative. Age cannot be negative because age begins at birth, so you cannot have a negative number of years. DOK-2 Look at the sums when you added a negative number. What kind of operation is this equivalent to? Explain. When I added a negative number, it was the same as subtracting a positive number. This is because both situations result in moving a certain number left on the number line to ﬁnd the sum or difference. The solution is the same.

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

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

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ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Addition and Subtraction with Rational Numbers Explore 3 – Integer Subtraction with Counters and Number Lines ACTIVITY PREPARATION Students will ﬁnd the differences of both positive and negative numbers. They will employ various strategies as they use mats and counters, number lines, and equations to solve problems.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.5 Use appropriate tools strategically. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • • • •

• • • •

1 Student Journal (per student) 1 Subtraction Counters Mat (per pair) 1 Horizontal Number Line (per pair) 1 Vertical Number Line (per pair) 1 Exit Ticket (per student)

• •

Reusable • • • • •

1 Pair of standard dice (per pair) 1 Set of two-color counters, 24 per set (per pair) 1 Resealable bag (per pair) 1 Dry-erase marker (per pair) 2 Clear sheet protectors (per pair)

•

Plan to have students work in groups of 2 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Gather enough dice for each pair of students to receive a pair of dice. Gather enough red and yellow two-color counters for each set of partners to have a set of 24 counters, 12 for each partner. Put the dice and counters in a resealable bag for each set of partners. Print the Subtraction Counters Mat for each pair. Print the Horizontal Number Line and the Vertical Number Line for each pair. Consider printing them on card stock for durability. Then, place each number line in a sheet protector for each group. Gather enough dry-erase markers for each pair to receive one.

PROCEDURE AND FACILITATION FACILITATION TIP

Before reading the scenario, ask the class 1) What is the main goal of a game developer?; 2) What steps are involved in developing a game?; 3) What do game players want in a game? FACILITATION TIP Encourage students to restate the problem. Provide time to put game developer’s wants into their own words. Have some students share with the class. FACILITATION TIP Be sure to clarify these directions in the journal “Even numbers rolled are positive; odd numbers rolled are negative. Subtract the second roll from the ﬁrst roll.” Review odd and even numbers as needed.

2. 3. 4.

Read the following scenario to the class: The third speaker Mr. Sanchez arranged for Career Day is a game developer. Her job is to create and test games to see which are the most popular and should be produced and sold. Today, your job is to help the game developer by playing her newest game creation, called Optimist versus Pessimist. She wants feedback on the game, which involves subtracting positive and negative numbers from positive and negative numbers using mats and counters, number lines, and equations. She wants to see how close scores are to encourage engagement and competition. Give a Student Journal to each student. Give a Subtraction Counters Mat, a Horizontal Number Line, a Vertical Number Line, a dry-erase marker, and a bag of dice and counters to each partnership. Explain to students that they will be playing a dice game called Optimist versus Pessimist with their partners. Model how to play the dice game using these steps: a.

Students will choose a team identity, either optimists (positive numbers) or pessimists (negative numbers).

Students will take turns rolling dice and subtracting the second roll from the ﬁrst roll. They will use counters and mats, number lines, and equations to ﬁnd solutions in different sections of the game. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

6. 7.

Introduce students to the Subtraction Counters Mat. Students will each roll the dice and count out the corresponding number of positive or negative counters. Then, they will perform the subtraction problem. The leftover counters go in the Leftover Yellow Counters (+) or Leftover Red Counters (−) columns. The leftover counters are the solution. b. Teach students that they will add equal numbers of counters to the mat (positive and negative zero pairs) if necessary because that is the same as adding zero and does not alter the problem. c. Remind students that they will keep score for each section and will add all the scores of all the sections together to determine the winner at the end. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 What are the parts of a subtraction equation? The minuend is the number being subtracted from. The subtrahend is the number being subtracted. The difference is the answer/solution.

DOK-2 In what situations did adding the same amount of red and yellow counters help you solve the problem? When there were not enough counters to take away, I added the necessary counters and made sure to add the same number of each color. Then, I could take away the counters that were supposed to be subtracted and ﬁnd the solution.

DOK-1 Which section made it easiest for you to ﬁnd the solution? Why? Answers will vary. The counters section was easiest because the solution was the number of leftover counters, and the sign was negative if the counters were red and positive if the counters were yellow.

DOK-1 Which section was the fastest to ﬁnd the solution? Why? Answers will vary. The section with just equations was the fastest. Once I had discovered the rules and patterns in the ﬁrst two sections, I could quickly write the equations and solutions. If I needed to use counters or a number line to check my answers, I could do that without recording my work.

Intervention

Acceleration

FACILITATION TIP Take time to model a few practice rounds by having a student take turns with you to show how the game works.

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.

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

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Allow students enough time to finish playing all three sections of the game. Also, allow them time to find the total scores and answer the reflection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 How did the counters and the number lines help you make sure your equation was correct? It was easy to check our equation if we were using counters because we started with the first roll and subtracted the second roll. Sometimes we had to add the same number of red and yellow counters to be able to subtract the subtrahend. The solution was the number and color (sign) of the leftover counters. It was also helpful to use the number line to check our equations. We used the first roll as our starting point. Then, we jumped left or right (depending upon whether the second roll was negative or positive) the number rolled on the second roll. The endpoint was the difference and our solution. • DOK-2 What did you do when you did not have enough counters to subtract the subtrahend you rolled? When there were not enough counters to take away, I added the necessary counters (either red or yellow) and made sure to add the same number of each color. Then, I could take away the counters that were supposed to be subtracted and find the solution. • DOK-2 How did the counters help you understand that subtracting a negative number is the same as adding a positive number? It was easy to see when we added the same number of each color of counter (with a value of zero) to the counters on the mat. Then, we took away the number of negative counters indicated in the subtrahend. All of the positive (yellow) counters were left. Then, it was just like adding two positive numbers together. •

FACILITATION TIP Provide students with an opportunity to share which tool they currently prefer, but remind them that they should be comfortable modeling situations with counters, number lines, and equations.

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Addition and Subtraction with Rational Numbers Explore 3 – Integer Subtraction with Counters and Number Lines •

DOK-2 Did you notice any rules or patterns that helped you successfully subtract numbers with different signs and values? Give examples. •

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.

• • • •

When we subtract a lesser positive number from a greater positive number, the solution is a positive number. 9 – 3 = 6 When we subtract a greater positive number from a lesser positive number, the solution is a negative number. 2 – 4 = −2 When we subtract a negative number from a positive number, the solution is a positive number. 5 – (−3) = 8 When we subtract a negative number of lesser value from a negative number of greater value, the solution is a positive number. −2 – (−7) = 5 When we subtract a negative number of greater value from a negative number of lesser value, the solution is a negative number.−6 – (−4) = −2 When we subtract a positive number from a negative number, the answer is a negative number. −3 – 5 = −8

Post-Explore FACILITATION TIP

As an extension, ask students to give feedback to the game developer on the back of their Exit Ticket.

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

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

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ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Addition and Subtraction with Rational Numbers Explore 4 – Rational Number Subtraction with Number Lines ACTIVITY PREPARATION Students will use number lines to determine answers to subtraction problems of both negative and positive numbers with both negative and positive solutions, including word problems, and to ﬁnd equivalent subtraction and inverse addition expressions.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. 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 Addition and Subtraction Cards (per pair) 1 Exit Ticket (per student)

Reusable • • •

2 Resealable bags (per pair) 1 Blue crayon or colored pencil (per student) 1 Green crayon or colored pencil (per student)

• • •

•

Plan to divide the class into groups of 2 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of the Addition and Subtraction Cards for each set of partners. Cut out and place the Part I cards in a resealable bag labeled “Part I.” Cut out and place the Part II cards in a resealable bag labeled “Part II.” If desired, print them on card stock, and laminate them for future use. Gather enough blue and green crayons or colored pencils for each student to have one of each.

PROCEDURE AND FACILITATION Part I: Matching FACILITATION TIP

Before reading the scenario, ask the class 1) Besides cooking, what are some other responsibilities an executive chef might have?; 2) When might an executive chef encounter positive and negative numbers? FACILITATION TIP If needed, use the Picture Vocabulary in Explain to review additive inverse. FACILITATION TIP

2. 3. 4.

After explaining the rules, but before passing out the supplies, model the turn taking with a student. Clarify if making a match leads to an extra turn or not.

Read the following scenario to the class: The fourth speaker Mr. Sanchez arranged for Career Day is an executive chef at one of the top restaurants in town. Her job involves not only cooking, but also purchasing, keeping track of inventory, keeping track of food temperatures for safety and successful recipes, and keeping track of profits and losses of different menu items. Today, your job is to show the executive chef that you can subtract positive and negative numbers and that you understand the relationship between subtraction and corresponding additive inverses. Give a Student Journal to each student. Give a bag of Part I Addition and Subtraction Cards to each pair. Instruct students to take out the Part I Addition and Subtraction Cards, shuffle them, and put them facedown in a 6 × 4 array. Students should take turns flipping over two cards. Both students should independently determine whether the two cards show equivalent expressions, one with subtraction and the other with the corresponding addition of the inverse. If they do, both students should record the match on their Student Journals, following directions to record expressions, complete number lines, and report solutions. The student who made the match keeps the cards and takes another turn. If there is no match, the student’s partner takes a turn. Monitor and assess students as they are working by asking the following guiding questions: a. DOK-1 What do you think the term additive inverse means? Additive inverse means adding the opposite (or negative) of a number, like 5 + (−7).

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 If you ﬂipped over a subtraction expression and an addition expression, how could you tell if they were equivalent and a match? First, we solved each expression to see if they had the same solution. Then, we checked to see if the numbers were the same so the subtraction and the inverse addition have the same starting point and change the same amount.

DOK-2 What were clues that you did not have a match? If we ﬂipped two cards with subtraction or two cards with inverse addition or if the cards had different numbers on them, the cards were not a match.

DOK-1 Since the number line was only marked in whole numbers, what would you do if you had to locate a fraction or decimal on the number line? We would estimate where on the number line the fraction or decimal would go.

Intervention

Acceleration

FACILITATION TIP Look for fractions and decimals on the cards that students may need support estimating.

Allow students time to complete their Student Journals, including the reﬂection questions. After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat •

• •

•

DOK-2 Once you made a match, how did using the number lines help you check your answers? The number lines helped us see that the expressions on the cards were a match. The expressions should look identical on the number line. They should have the same starting and ending points and jump the same distance from the starting point going the same direction. DOK-1 What do you need to do to locate a fraction or decimal on the number line? We would estimate where on the number line the fraction or decimal would go. DOK-2 What rule did you learn about subtraction expressions and the corresponding expressions using additive inverses? The solutions are the same because subtracting a positive number has the same result as adding a negative number. DOK-2 Give an example of a pair of corresponding subtraction and inverse addition expressions. 9 – 2 and 9 + (−2) DOK-2 What did you learn about subtracting negative numbers? Explain what you learned. Subtracting a negative number has the same result as adding a positive number. This is because the two negatives (the minus sign and the negative sign) combine to make a positive. DOK-2 Based on your experience with the matching game, explain which way to move on the number line in the following situations: • • • •

Adding a positive number: right Adding a negative number: left Subtracting a positive number: left Subtracting a negative number: right

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

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FACILITATION TIP Use this opportunity to remind students about decimals and fractions in between on a number line. Rulers are good concrete tools to show how number lines can display fractions and decimals.

FACILITATION TIP Have students point left and right when answering these prompts to detect any misconceptions.

Part II: Subtraction Word Problems 1.

2. 3.

Read the following scenario to the class: The executive chef was very impressed with your abilities to subtract positive and negative numbers from positive and negative numbers. Now she is ready to have you help her and her chef friends with real-world situations they have encountered with inventory, temperatures, proﬁts, and losses. Your job today is to solve real-world word problems involving subtraction that the chef encounters at her restaurant. Give a bag of Part II cards to each pair. Instruct students to take out the Part II Addition and Subtraction Cards, shuﬄe the cards, and then take turns drawing cards to solve. After reading a card, students should match the card with the corresponding number line. Then, they

FACILITATION TIP After reading the scenario, ask the class 1) What might be a real-world situation an event planner would encounter when dealing with menu items?; 2) How can problems with menu items affect an event planner’s business?

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Addition and Subtraction with Rational Numbers Explore 4 – Rational Number Subtraction with Number Lines

FACILITATION TIP Encourage student pairs to read the Word Problem cards out loud to each other. Having both students read the problem aloud may provide better understanding of the context.

5. FACILITATION TIP Select some student-generated answers to the reﬂection questions to display and discuss.

FACILITATION TIP Display this quote for students to read silently, out loud, and all together before discussing. 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.

Clarify or review how students are to use the blue and green colors on the Exit Ticket.

DOK-1 What is inverse addition? Inverse addition means adding the opposite (or negative) of a number, like 5 + (−7).

DOK-2 How can you ﬁgure out inverse addition when given a subtraction expression? We leave the ﬁrst number in the expression the same. We change the sign of the operation from subtraction to addition. We leave the second number of the expression the same, but we change the sign.

Allow students time to complete their Student Journals, including the reﬂection questions. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 What do you think “the difference of two numbers can be positive or negative, but the distance between two numbers is always positive” means? The difference depends on which number is being subtracted from which number, so it can be positive or negative. 1 – 3 is −2, while 3 – 1 is +2. Distance can never be negative because it is a measure of length, and length cannot be negative since it is a magnitude. • DOK-2 How did you use the number line to ﬁgure out the subtraction and addition expressions? We could see the starting point and the ending point. If the arrow went to the left, we knew it subtracted a positive number or added a negative number. If the arrow went to the right, we knew it subtracted a negative number or added a positive number. We just had to use the number line to determine the value of the number added or subtracted. • DOK-2 For which real-life situations did the chef have to use subtraction of negative and positive numbers with solutions less than zero? What other similar real-world situations can you think of? The chef had to subtract negative and positive numbers when she was missing supplies, with temperatures of food, costs of supplies, and sales of products. Another situation we can think of is gaining and losing elevation below sea level. •

Post-Explore 1.

FACILITATION TIP

should circle the subtraction expression that describes the word problem and ﬁll in the missing addend that makes it the additive inverse of the subtraction expression. Then, students should record the solution and explain what the number means. Monitor and assess students as they are working by asking the following guiding questions:

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

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

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ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Addition and Subtraction with Rational Numbers Explore 5 – Using the Properties to Solve ACTIVITY PREPARATION Students will identify and use commutative, associative, additive inverse, and distributive properties to solve problems that involve positive and negative integers and rational numbers.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. 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 Who Owns That Property? Cards (per pair) 1 Exit Ticket (per 2 students)

Reusable •

• • •

•

Plan to divide the class into groups of 2 to complete this activity. Print a Student Journal for each student. Print a set of the Who Owns That Property? Cards for each pair. Cut out and place each set of cards in a resealable bag. If desired, print them on card stock, and laminate them for future use. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student has one.

1 Resealable bag (per pair)

PROCEDURE AND FACILITATION Part I: Who Owns That Property? FACILITATION TIP

Address the alternative deﬁnition of property being used in this Explore activity. Point out the difference between owning property and using math properties as rules. Review the math properties being used in this activity. FACILITATION TIP Before reading the scenario, ask the class 1) When might someone use an event planner?; 2) What do you think are the speciﬁc responsibilities of an event planner? After reading the scenario, ask the class 3) What would happen if Ms. Shelley’s budgeting calculations were incorrect? FACILITATION TIP Encourage both partners to examine each card. If students are struggling, have them read the cards more than once to each other.

2. 3.

Read the following scenario to the class: The fifth speaker Mr. Sanchez arranged for Career Day is an event planner. Her job is to find out a client’s vision for a social event and make it happen—from invitations to food to entertainment to locations. Events can include birthday parties, graduations, business luncheons, town fiestas, etc. Ms. Shelley, the event planner, takes a client’s vision and makes it come to life through budgeting, hiring, decorating, organizing, setup and takedown, and purchasing goods and services. Today, your job is to help the event planner by doing a test run on one of her newest signature invitations and giving her feedback about whether clients will like this new trend. Give a Student Journal to each student. Give a bag of Who Owns That Property? Cards to each pair. Instruct students to take turns reading the cards aloud, examining the equation on each card, and sorting it into the correct property category. Then, have students record each equation under its category in the table. Write the solution. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 What is the commutative property? A property stating that the order in which we add numbers does not affect the sum.

DOK-1 What is the associative property? A property stating that when three or more numbers are added, the way in which they are grouped does not affect the sum.

DOK-1 What is the additive inverse property? A property stating that every number has another number that can be added to it, resulting in a sum of zero. © Accelerate Learning Inc. - All Rights Reserved

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

Allow students time to complete their Student Journals and reﬂection questions. After Part I, invite the class to a Math Chat to share their observations and learning.

DOK-2 Explain which of the properties work with subtraction: Commutative Associative Additive inverse Distributive The commutative property does not work because order matters. 2–7≠7–2 The associative property does not work because order matters. 10 – (5 + 2) = 3 is not equivalent to (10 – 5) + 2 = 7. The additive inverse property can include subtraction when used in the context 10 – 7 = 10 + (−7) to understand that subtraction and inverse addition are equivalent. The distributive property works because multiplying factors and subtracting are equivalent to subtracting and then multiplying, so order doesn’t change.. 2(5 – 2) = (2 × 5) – (2 × 2) = 2(3) = 10 – 4 = 6 • DOK-1 Does the sign of a number affect whether a property will work? No, any rational number will work with a property. Rational numbers include both positive and negative numbers. • DOK-1 Which property do you ﬁnd the easiest to identify and use? Commutative is the easiest because it simply involves changing the order of the addends. • • • • •

Part II: Property Management

2. 3.

Read the following scenario to the class: Ms. Shelley was impressed with the decoding ability of you and your partner. Now she wants to show you some of the situations she encounters in her job that require her to solve problems using the commutative, associative, additive inverse, and distributive properties. Today, your job is to help the event planner by solving some of her problems. Students should use their math skills to ﬁnd and circle appropriate equations, identify properties, and record solutions. Monitor and assess students as they are working by asking the following guiding questions: a.

4. 5.

Acceleration

DOK-1 What is the distributive property? A property stating that multiplying the sum or difference of two numbers gives the same solution as multiplying each factor separately and then adding or subtracting the products.

Math Chat

Intervention

DOK-1 What is a rational number? A rational number is any number that results from one integer being divided by another integer that is not zero. This includes fractions and decimals.

DOK-1 Is an integer a rational number? Yes, all integers are rational numbers.

DOK-1 Is a decimal an integer or a rational number? A decimal is a rational number but not an integer.

DOK-1 Is a fraction an integer or a rational number? A fraction is a rational number but not an integer.

FACILITATION TIP Have students copy notes and examples for each property on their Student Journals. Create a chart displaying the deﬁnition, examples, and how to identify each property. You may need to prompt students to focus on the operational symbols, parentheses, and order of numbers rather that the actual digits and numbers.

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

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FACILITATION TIP Consider letting students vote for their favorite math property and have a few volunteers explain their reasons. FACILITATION TIP After reading the scenario, ask the class 1) What types of problems might Ms. Shelley encounter in her event planning business?; 2) What situations might Ms. Shelley encounter where she needs to use a property to solve problems? FACILITATION TIP Preview the directions on the Student Journal for Part II. You may need to clarify “correspond” and “being showcased.” FACILITATION TIP It is important to take time to ensure that students understand the vocabulary in these guiding questions. Consider creating a chart with examples of integers and rational numbers.

Allow students time to complete the Student Journal and reﬂection questions. After Part II, invite the class to a Math Chat to share their observations and learning.

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Addition and Subtraction with Rational Numbers Explore 5 – Using the Properties to Solve Math Chat DOK-2 What is the difference between a rational number and an integer? An integer is a rational number, but rational numbers also include decimals and fractions. An integer is any whole number—positive, negative, and zero. A rational number is any number that results from one integer being divided by another. • DOK-2 Did the commutative, associative, additive inverse, and distributive properties work for all rational numbers? Yes, the properties worked for all rational numbers, not just integers. • DOK-1 Besides being an event planner, can you think of any other careers that use properties such as commutative, associative, additive inverse, distributive, or others as part of the job? Stockbrokers, physicists, mathematicians, pharmacists, teachers, chemists, professors, business executives, accountants, etc. • 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. FACILITATION TIP Consider that some students may need the support of their Math Chat notes to demonstrate their understanding of the concept.

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.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

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ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Addition and Subtraction with 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

Addition of Integers with Counters 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

Rational Number Addition with Number Lines 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

Integer Subtraction with Counters and Number Lines

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

Rational Number Subtraction with Number Lines

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

Show What You Know, Part 5 Using the Properties to Solve 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 Integers Independent and partner games and other activities that provide students with an engaging way to practice the new concept

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

Notes __________________________________________________________________________________________________________________________________________________

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

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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

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

Addition and Subtraction with Rational Numbers

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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 add and subtract integers and rational numbers by using strategic thinking and a variety of tools.

I can describe situations that represent additive inverses.

I can interpret sums of rational numbers.

What prompts will be used?

What does mastery look like?

ADDITION AND SUBTRACTION WITH RATIONAL NUMBERS

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I can represent addition and subtraction of rational numbers on a horizontal or vertical number line diagram.

I can use absolute value to represent the distance between two rational numbers on a number line.

I can discover integer rules by exploring the signs of integers and their meanings.

I can apply properties of operations to add and subtract rational numbers.

SCOPE 1

Multiplication and Division with Rational Numbers Scope Introduction SCOPE SUMMARY In this scope, students will apply their understanding of operations to multiplication and division as they relate to negative numbers. Students will explore the distributive property as it applies to negative numbers. Students will expand their knowledge to negative fractions, recognizing them as division problems in real-world contexts.

Student Expectations

7.NR.1.6 Make sense of multiplication of rational numbers using realistic applications. 7.NR.1.7 Show and explain that integers can be divided, assuming the divisor is not zero, and every quotient of integers is a rational number. 7.NR.1.8 Represent the multiplication and division of integers using a variety of strategies and interpret products and quotients of rational numbers by describing them based on the relevant situation.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In Grade 6, students understood that decimals and fractions are part of the same number system as whole numbers. Students learned the basics of dividing fractions as well as the relationship between multiplication and division. They have a basic understanding of positive and negative numbers and their locations on the number line as well as number opposites. This will be the foundation of this scope as students apply these concepts to expand their knowledge of multiplication and division.

As students enter 8th grade, they will expand their knowledge of multiplication and division to the idea of irrational numbers. As they enter high school, these functions will carry on as either rational or irrational graphs as exponential, logarithmic, and power functions. Understanding the basic rules of multiplication and division operations will be the building blocks for further exploration.

7.NR.1.9 Apply properties of operations as strategies to solve multiplication and division problems involving rational numbers represented in an applicable scenario.

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, compute, and solve realworld problems that involve the division of fractions.

•

match numbered cards with lettered cards posted.

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: •

identify which property of operations was used to solve a problem and determine the correct answer.

•

identify the property that shows two scores are equivalent and determine the scores.

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. 48

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

In this exploration, groups of students will solve a realworld scenario involving a middle school’s field day. Students are first tasked with determining how many cups of water each racer has added or lost during the bucket race; then, figure out which team won the race. Students will: •

multiply integers, with a focus on using the distributive property.

•

use concrete models and connect actions to the standard algorithm.

Explore 2

Integer Multiplication with Counters

In this exploration, students will be tasked with solving another scenario involving events for the ﬁeld day mentioned in the previous exploration. Here, students are tasked with calculating the points earned or lost for the completion or incompletion of an obstacle course, and ﬁgure out how many more games a team has to play. Students will: •

multiply integers.

•

use numbers lines.

•

make connections to the standardized algorithm.

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

Integer Division with Counters

Rational Number Division with Number Lines

In this exploration, students will be tasked with tallying the number of water balloons caught by each team and helping scores figure out how many players were in each team to figure out how many awards to give out for the water-balloon toss challenge at the middle school field day. Students will: •

solve real-world division problems with integers.

•

generalize division rules.

•

discover formal rules for operations when applying positive and negative numbers.

In this exploration, students will collaborate with peers to solve a cup-stacking scenario where they must calculate the score earned by each player in the cup-stacking challenge and ﬁnd a score card that does not match so the player’s score can be recorded. Students will: •

solve division problems with integers on a number line.

•

generalize rules.

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

Rational Number Multiplication with 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.

Explore 4

Explore 3

Explore 1

EXPLORE ACTIVITIES

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

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Using Properties to Solve In the ﬁnal exploration, students complete and solve the last part of the ﬁeld day scenario. Students will: •

apply commutative, associative, and distributive properties to multiply and divide integers and rational numbers.

•

calculate, match, and group cards and numbers based on properties.

•

verify calculations through recalculating.

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

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Multiplication and Division with Rational Numbers 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. 6.NR.1.2 Multiply and divide any combination of whole numbers, fractions, and mixed numbers using a student-selected strategy. Interpret products and quotients of fractions and solve word problems.

Materials

Preparation

Printed •

• •

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 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 a neighbor. a. Card 1 matches with Card A.

Card 3 matches with Card B.

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.

Alternatively for Step 2, project the lettered cards for students to view from their desks and consult with partners as they think.

FACILITATION TIP

Identifying Misconceptions •

Before walking around, students could fold their paper into thirds, and copy the expressions from the numbered cards (1, 2, 3) into separate sections to provide space and time for them to show their thinking. FACILITATION TIP

b. Card 2 matches with Card C. c.

FACILITATION TIP

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Home

Students may forget that dividing is the same as multiplying by the reciprocal. Practice the algorithm, but also demonstrate with whole numbers how dividing is the same as multiplying by the reciprocal.

Reciprocal is a key concept. Have students say the word, write the deﬁnition, and note some examples on paper. Refer to the Visual Glossary for support.

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Multiplication and Division with Rational Numbers Hook – Settle a Score ACTIVITY PREPARATION Students will identify the property that shows two scores are equivalent and then students will determine the score.

Materials

Preparation

Printed •

• • •

1 Settle a Score (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Settle a Score 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) What happens during a diving meet where athletes are judged on their dives?; 2) What types of things are the divers judged on?; 3) Why do you think there are multiple judges that score divers in a diving meet?

FACILITATION TIP Be sure you are projecting Settle a Score before you ask the questions in Step 3. Direct students to discuss what they notice about the judges’ math rather than the video content.

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. Adams is the head meet scorer at a diving meet. One of her jobs is to ensure judges’ scores are correct. Another responsibility is to train the meet scorers. Her new trainee, Mr. Worth, looked at the two judges’ scores for a dive and wondered why they were different. Mrs. Adams explained they were equivalent scores because of a property of operations. Can Mr. Worth ﬁgure out the property and also the correct score for the diver? 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 Mrs. Adams is using properties of operations. I wonder whether the judges’ scores are equivalent. What property is being used? I can use math to determine the property of operation and to ﬁgure out the score of the diver. Project Settle a Score.

Explore

Explain

Elaborate

DOK-1 What is the associative property? The rule says when three or more numbers/terms are added or multiplied, the sum or the product is the same no matter how they are grouped.

DOK-1 What is the commutative property? The rule says that changing the order of numbers/terms does not change the value of the solution when performing addition or multiplication.

DOK-1 What is the distributive property? The rule says that multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number/term and then adding the products together.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Evaluate

Explain to students that Mrs. Adams has double checked her math and insists the scores from the two judges are equivalent. She informs Mr. Worth the property of operation being used is the associative, commutative, or distributive property. Discuss the following questions: a.

Engage

Show the phenomena video again, and restate the problem. Refer to Settle a Score, and discuss the following questions: a.

DOK-1 Are the scores from the two judges equivalent? Yes DOK-2 What property of operation is being used? How can you tell? The commutative property is being used because the expressions 1 __ 2

(–30 ÷ –5) and 0.2(–66 ÷ –6) are in different orders on the judges’

Intervention

Acceleration

FACILITATION TIP To engage students, have them vote on which property they think is being used. Keep a tally and save it to revisit in Part II: Post-Explore. FACILITATION TIP Refer to the Picture Vocabulary under the Explain tab to provide clear visual deﬁnitions and examples of these properties for students if needed.

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.

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Home

scorecards. b.

DOK-1 What is the score the diver received? 5.2 Notes

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Multiplication and Division with Rational Numbers Explore 1 – Integer Multiplication with Counters ACTIVITY PREPARATION Students will be multiplying integers and focusing on the distributive property using concrete models and will connect those actions to the standardized algorithm.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. 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 Racer Cards (per group) 1 Set of Team Cards (per group) 1 Integer Operations Work Mat (per group) 1 Exit Ticket (per student)

Reusable • • •

1 Set of two-color counters (per group) 2 Resealable bags (per group) 1 Projector or document camera (per class)

• • •

•

• • •

Plan to divide the class into groups of three or four to complete this activity Print a Student Journal and an Exit Ticket for each student. Print the Integer Operations Work Mat for each group. If desired, print it on card stock, and laminate it for future use. Print a set of Racer Cards for each group. Cut out and place each set in a resealable bag labeled “Part I.” If desired, print them on card stock, and laminate them for future use. Print a set of Team Cards for each group. Cut out and place each set in a resealable bag labeled “Part II.” If desired, print them on card stock, and laminate them for future use. Gather a set of two-color counters for each group. Be prepared to project the Integer Operations Work Mat for the class to see. 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. (Two-Color Counters)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What is a school field day?; 2) What is the purpose of a field day?; 3) What events take place during a field day?

Part I 1.

FACILITATION TIP To create extra engagement, you can use the name of your school and have students imagine themselves or classmates in this race. FACILITATION TIP To support struggling students throughout these scopes, you can keep the colors that represent positive and negative numbers consistent. For example, red = negative and yellow = positive.

2. 3.

Read the following scenario to the class: Jackson Middle School is having its annual field day! The first event is the bucket race! The first part is for single racers. Each racer tries to collect cups of water while going uphill toward the finish line, but they have to be careful not to spill the water. There are markers on the hill to help racers know how far they have gone in the race. Your group will help figure out how many cups of water each racer has added or lost during the race. Give a Student Journal to each student. Give a set of two-color counters and an Integer Operations Work Mat to each group. Project the Integer Operations Work Mat for students to see. Have the following discussion with the class: a.

Which color will we use to represent positive and negative numbers? Have the whole class agree on which colors represent positive and negative numbers. For this Explore activity, we will use:

Engage

Explore

Explain

Elaborate

Evaluate

Explain the following to the class: The first factor in an expression is important in modeling integer multiplication with counters. If the first factor is positive, it means to add that many sets of the second factor. If the first factor is negative, it means to remove that many sets of the second factor.

DOK-1 What clues in the expression 5 · −2 tell how to set up the counters? The ﬁrst factor is positive 5, which means we add 5 sets of −2 counters.

DOK-1 How would we represent the expression 5 · −2? Have the students discuss their ideas, and project this on the screen.

Intervention

Acceleration

FACILITATION TIP To ensure students understand the model, demonstrate the process with both positive and negative factors being ﬁrst. Students may need to see several examples before using the tools in small groups.

e. Explain the following to the class: When the ﬁrst factor is negative, we need to remove counters, but there are no counters here to start with. To have counters to remove, we need to add sets of zero pairs. f.

DOK-1 How would we represent the expression −5 · −2? We know that 5 times 2 is ten, so the first step is to add ten pairs that equal zero. Now we can remove the five sets of –2 to find the answer. Have the students discuss their ideas, and project this on the screen. STEMscopes Tip

4. 5.

7. 8.

Give a set of Racer Cards to each group. Instruct students to use the two-color counters to model the numbers being multiplied on the Integer Operations Work Mat to solve the problems on the Racer Cards. Actively monitor students as they are working with their groups to model the problem using two-color counters. Ask questions such as the following: a.

DOK-2 How can you determine if the integers described in the scenario are positive or negative? The word add is for positive integers, and the words lost and spilled are for negative integers.

DOK-2 How can you use the two-color counters to show the opposite of an integer? Use the two-color counters to model the number. You can ﬂip the two-color counters to show its opposite.

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.

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Home

Allow students enough time to complete their work and record their observations and reﬂections on their Student Journals. After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 How did you know how many groups you needed for each situation? The ﬁrst factor tells you how many groups there need to be. • DOK-2 When multiplying integers, what type of integers will result in a negative product? If one factor is positive and the other factor is negative, the product is negative. • DOK-2 When multiplying integers, what type of integers will result in a positive product? If both of the factors have the same sign, the product is positive. • DOK-3 Can you come up with a rule to determine whether the product is going to be positive or negative? If the numbers being multiplied have the same sign, the product will be positive. If the numbers being multiplied have different signs, the product will be negative. •

FACILITATION TIP Have students copy this rule on their Student Journals to refer to throughout this scope; include some examples.

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Multiplication and Division with Rational Numbers Explore 1 – Integer Multiplication with Counters Part II 1. FACILITATION TIP After reading the scenario, ask the class 1) How would a bucket race in teams be played?; 2) What diﬃculties might a team bucket race encounter that a single-racer bucket race would not?; 3) Would you prefer to be a part of a team or a single racer in a bucket race?

2. 3. 4.

Read the following scenario to the class: The second part of the bucket race is in teams! The results of the team races are shown in expressions. 3 teams had the exact same score, and one team came in the lead. Figure out which team won the race. Give a set of Team Cards to each group. Remind students to continue using two-color counters to represent the integers in the expressions. Actively monitor students as they are working with their groups to model the problem using two-color counters. Ask the following types of questions:

FACILITATION TIP Be sure to use the same colored counters for both Part I and Part II to create consistency. If you use red and white in Part II, you should also use red and white in Part I.

5. 6.

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 Before having students complete the Exit Ticket, show a practice example so they know what your expectation is for the workspace.

DOK-2 How do you use the color counters when you have numbers inside parentheses? I use the number outside the parentheses to determine how many times I have to use the color counters to represent the numbers inside the parentheses.

DOK-2 How do you ﬁnd the solution to the expression? I pair up a red and white counter, and I take them out or cross them off. I count the number of color counters I have that do not have pairs.

Allow students enough time to complete their work and record their observations and reﬂections on their Student Journals. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 Why do you cancel out or cross off a pair of red and white counters? The red = −1, and the white = 1. −1 + 1 = 0, so they cancel each other out. • DOK-2 Why do team 1 and team 2 have the same solution? Team 2 is the expanded form of team 1’s expression. 3 was distributed to multiply by 7 and by –4. • DOK-2 What happens to a multiplication or division expression when you add a negative sign? Give an example. When I add a negative sign to a multiplication or division expression, the answer is the opposite of the original answer. Example: 4 × (−3) = −12 and (−4) × (−3) = 12. • DOK-3 How does the distributive property of multiplication work? The distributive property takes the number outside the parentheses, multiplies it by each term inside the parentheses, and simpliﬁes as needed. •

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 WITH RATIONAL NUMBERS

Home

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Multiplication and Division with Rational Numbers Explore 2 – Rational Number Multiplication with Number Lines ACTIVITY PREPARATION Students will be multiplying integers using number lines and will connect those actions to the standardized algorithm.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. 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 Team Cards (per group) 1 Horizontal Number Line (per student) 1 Exit Ticket (per student)

• •

•

Reusable • • • • •

1 Clear sheet protector (per student) 1 Dry-erase marker (per student) 1 Dry-erase eraser(per student) 1 Projector or document camera (per class) 1 Resealable bag (per group)

• • •

Plan to divide the class into groups of three or four to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Team Cards for each group. Cut out and place each set inside a resealable bag. If desired, print them on card stock, and laminate them for future use. Print a Horizontal Number Line for each student. Place each number line inside a clear sheet protector. If desired, print it on card stock, and laminate it for future use. Gather enough dry-erase markers and erasers for each student to receive one. Be prepared to project the Horizontal Number Line for the class to see. 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 FACILITATION TIP After reading the scenario, ask the class 1) Have you ever participated in an obstacle course? If so, what are some of the obstacles that you found were easy? Diﬃcult?; 2) What strategies could you use to calculate points and ﬁgure out how many more games a team has to play? FACILITATION TIP Alternatively, make copies with several horizontal number lines for each student. They can use pencil, pen, or colored markers to show their work. FACILITATION TIP Before asking questions 4a–4e, display the problem 5(–2) with room to add labels for the multiplicand and multiplier. Have students copy and label the parts of the expression on their Student Journal. 58

2. 3. 4.

Read the following scenario to the class: The second event for ﬁeld day is the obstacle course! Four teams are participating this year. Each team’s goal is to complete the course and earn points. Your task is to calculate the points earned for completion, calculate the points lost for incompletion, and ﬁgure out how many more games a team has to play. Give a Student Journal, Horizontal Number Line, dry-erase marker, and dry-erase eraser to each student. Project the Horizontal Number Line for students to see. Have students represent the problem 5(−2) on their Horizontal Number Lines, ask the following questions, and draw the arrows as they answer the questions: a.

DOK-1 How do we determine the length of each arrow? The multiplicand (−2) tells us the length of each arrow.

DOK-1 How do you determine the number of arrows? The multiplier 5 tells us the number of arrows.

DOK-2 How do we determine the direction of the arrows? If the multiplier is positive, the arrows will move in the same direction as the arrow of the multiplicand. If the multiplier is negative, the arrows would be the same number of arrows in the opposite direction of the multiplicand. © Accelerate Learning Inc. - All Rights Reserved

7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How do you determine the product? The product is wherever you end on the number line after drawing the correct number of arrows of the multiplicand in the correct direction.

Have students collaborate with their groups to answer the questions from the Team Cards. Students should use their Horizontal Number Lines to discuss as a group how to model the problem prior to representing the problem on their Student Journals. As students are working, monitor their progress and ask questions such as the following: a.

DOK-2 How do you determine the sign of the product? I use the direction of the arrows. If the arrows are going toward the right, then the sign of the product is positive.

DOK-2 What terms help you determine the signs of the numbers in the scenarios? The term completion indicates a positive number, while the terms incompletion and missing indicate negative numbers.

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 What pattern of the products did you notice after using number lines? I noticed that when the signs are the same, the product is positive. When the signs are opposite, the product is negative. • DOK-2 Give an example of a real-world situation of multiplying two negative integers that results in a positive integer. An example of multiplying two negative integers is paying off $10 in debt each month for the last 2 months. −10 (−2) = $20. I do not owe $20 anymore after 2 months. • DOK-3 What is the sign of the product of two or more integers with an even number of negative signs? Why? The sign of the product will always be positive since an even number of negative signs will always have pairs that have positive products. • DOK-3 What is the sign of the product of two or more integers with an odd number of negative signs? Why? The sign of the product will always be negative since an odd number of negative signs will always have an extra negative factor that will not be in a pair to make it positive. •

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

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.

FACILITATION TIP Have students copy the pattern in their reﬂection journal to support understanding. It is important for students to know how signs affect the product. FACILITATION TIP

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Home

Write some examples of expressions with more than two integers for students to attempt and then model how the signs affect the product. Demonstrate in steps how the sign changes as you solve. FACILITATION TIP Write some examples of expressions with more than two integers for students to attempt and then model how the signs affect the product. Demonstrate in steps how the sign changes as you solve. FACILITATION TIP Before giving students the Exit Ticket, show an example problem so they know how and where to draw the number line and locate the score.

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Multiplication and Division with Rational Numbers Explore 3 – Integer Division with Counters ACTIVITY PREPARATION Students will solve real-world division problems with integers using two-color counters. Students will also generalize the rules for division while discovering the formal rules for all operations as they apply to positive and negative numbers.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. 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 Score Cards (per group) 1 Integer Operations Work Mat (per group) 1 Exit Ticket (per student)

Reusable • • •

1 Set of two-color counters (per group) 2 Resealable bags (per group) 1 Projector or document camera (per class)

• • • •

• • •

Plan to divide the class into groups of two to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print the Integer Operations Work Mat for each group. If desired, print it on card stock, and laminate it for future use. Print a set of Score Cards for each group. Cut out and place each set in a resealable bag labeled “Part I.” If desired, print them on card stock, and laminate them for future use. Gather a set of two-color counters for each group. Be prepared to project the Integer Operations Work Mat for the class to see. 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. (Two-Color Counters)

PROCEDURE AND FACILITATION Part I 1. FACILITATION TIP Project this scenario and read it aloud with students. Depending on students’ sports and golf experience, take time to clarify these rules.

FACILITATION TIP If available, you can use individual whiteboards as Integer Operations Mats.

2. 3. 4.

Read the following scenario to the class: Foxﬁre Golf Club is having a tournament for the youth of the area. The whole community is invited. In golf, each hole has an expected score known as “par.” If you get above that score, it is represented by a positive number. If you get below that score, it is represented by a negative number. For example, if a hole is a par 4 and you make the ball in the hole in 3 hits, you get a score of −1. If a hole is a par 3 and you make the ball in the hole in 4 hits, you get a score of +1. Help the students ﬁnd their average scores. Give a Student Journal to each student. Give a set of two-color counters and an Integer Operations Work Mat to each group. Project the Integer Operations Work Mat for students to see. Have the following discussion with the class: a.

Review the following information: DOK-1 Which colors will we use to represent positive and negative integers? Have the whole class agree on which colors represent positive and negative integers. For this Explore activity, we will use the following color designations:

5. 6.

8. 9.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How would we represent the expression −10 ÷ 2? Have the students discuss their ideas, and project this on the screen.

DOK-1 How many groups were you able to make? 5 negative groups

DOK-2 How is this different from how the expression 10 ÷ (−2) is represented? With this expression, you’re starting with a positive number or yellow counters. When you divide by a negative, you’re doing the opposite, so you’ll have to ﬂip the counters over to make negative groups.

Intervention

Acceleration

FACILITATION TIP Project several examples on the screen and have students draw or use counters as you represent the expressions.

FACILITATION TIP Challenge students to display their thinking in more than one way. Could this expression also be modeled with two groups of –5?

Give a set of Score Cards to each group. Instruct students to use the two-color counters to model the numbers being divided on their Integer Operations Work Mats to solve the problems on the Score Cards. Actively monitor students as they are working with their groups to model the problem using two-color counters. Discuss the following questions: a.

DOK-2 How do you determine which counter you have to use? I use the sign of the dividend.

DOK-2 How do you use the divisor? The divisor is either the number of groups or the number in each group. If the divisor is negative, I ﬂip the counters to change it to the opposite of the sign of the dividend.

Math Chat

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.

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Home

DOK-2 How did the signs of the dividend and divisor affect the sign of the quotient? The integers that had one positive and one negative sign gave me a negative quotient. The integers that had 2 positive signs gave me a positive quotient. • DOK-2 Explain why the expressions −(12 ÷ 4), (−12) ÷ 4, and 12 ÷ (−4) all have the same quotient. Adding a negative means the opposite of the value. All 3 expressions represent the opposite of 12 ÷ 4. • DOK-2 How are the rules in multiplying and dividing integers similar? They both have the same rules for the ﬁnal product or quotient. If the signs are the same, the product or quotient is positive. If the signs are opposite, the product or quotient is negative. •

Part II 1.

2. 3.

FACILITATION TIP

Read the following scenario to the class: There will be entertainment and a food truck circle where guests can keep a tab to pay for their food. Each tab must be repaid before the end of the tournament. Help everyone settle up their tab before the event is over. Remind the students to continue to use two-color counters to ﬁnd the missing value from the equation. Actively monitor students as they are working with their groups to model the problem using two-color counters. Ask the following types of questions: a.

DOK-2 How do you determine the number of circles (groups) you have to make? I use the value of the quotient as the number of circles to use.

DOK-2 How do you determine the number of counters in each circle? I divide the dividend by the quotient.

The process of solving problems by looking for and using the structure is an important mathematical practice. Encourage students to persevere to see the solution. Some students may think of this as “working backward” to solve. FACILITATION TIP Find a place in the room to keep track of words in scenarios that direct students to use positive or negative numbers. A T-chart where you can add words as you work through this scope would support students’ learning. 61

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Multiplication and Division with Rational Numbers Explore 3 – Integer Division with Counters c.

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.

FACILITATION TIP Demonstrate how students are expected to draw the model on the Exit Ticket. They may need to indicate the color of the counters with the letters R or Y.

4. 5.

DOK-2 How do you determine the sign of the divisor? When the quotient is positive and the dividend is positive, the divisor is also positive. When the quotient is positive and the dividend is negative, then the divisor is negative. When the quotient is negative, the divisor’s sign is the opposite of the dividend.

Math Chat DOK-2 How does the quotient help in finding the sign of the divisor? If the quotient is positive, the dividend and divisor have the same sign. If the quotient is negative, the dividend and divisor have different signs. • DOK-2 How do you model dividing by negative numbers when using number counters? If the dividend is positive, you start with yellow counters. If the dividend is negative, you start with red counters. If the divisor is positive, you keep whatever you started with. If the divisor is negative, you flip the counters over to show the opposite value. Then, you either separate the counters into the equal groups of the divisor or into that many groups to find the quotient. •

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 WITH RATIONAL NUMBERS

Home

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Multiplication and Division with Rational Numbers Explore 4 – Rational Number Division with Number Lines ACTIVITY PREPARATION Students will solve real-world division problems with integers using number lines. Students will also generalize the rules for division while discovering the formal rules for all operations as they apply to positive and negative integers.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Use appropriate tools strategically. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Horizontal Number Line (per student) 1 Set of Solution Cards (per group) 1 Exit Ticket (per student)

•

Reusable • • • • • • •

1 Clear sheet protector (per student) 1 Dry-erase marker (per student) 1 Dry-erase eraser (per student) 1 Pair of scissors (per student) 1 Projector or document camera (per class) Consumable 1 Glue stick (per student)

•

• •

Plan to divide the class into groups of three or four. Print a Student Journal and an Exit Ticket for each student. Print a set of Solution Cards for each group. Cut out and place each set in a resealable bag labeled “Part II.” If desired, print them on card stock, and laminate them for future use. Print a Horizontal Number Line for each student. Place the Horizontal Number Line inside a clear sheet protector. If desired, print it on card stock, and laminate it for future use. Gather enough dry-erase markers, dry-erase erasers, glue sticks, and pairs of scissors for each student to receive one. Have these materials ready on the tables before the activity. Be prepared to project the Horizontal Number Line for the class to see. 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 Part I 1.

FACILITATION TIP Print and project this scenario for students and read it aloud with them. Note that some students may connect negative yardage as a “loss of yards” in football.

Read the following scenario to the class: The fourth graders are doing a football challenge. The football challenge consists of different challenges each student must complete. There are running challenges and throwing challenges. If the player runs or throws in the wrong direction, they are given negative yardage. Use each scenario to create an expression and help ﬁnd the solution to each problem. Project the Horizontal Number Line for students to see. Have a marker and eraser ready. Have the following discussion with the class: a.

DOK-1 How do we determine where to start on the number line? The dividend tells us the starting point.

DOK-2 How do we determine the length and direction of each arrow? The divisor tells us the length and direction of each arrow. If the divisor is −2, then the arrow will be 2 units long pointing to the left. If the divisor is 2, then the arrow will be 2 units long pointing to the right.

FACILITATION TIP Use this opportunity to write some expressions and label the dividend, divisor, and quotient to review for students. Use Stemscopes Visual Glossary if needed. This scope includes Picture Vocabulary for quotient in Explain. 64

4. 5. 6.

Explain

Elaborate

Evaluate

DOK-2 How do you determine the quotient? I count the total number of arrows and look at the direction of the arrows as well as which side of the number line they’re on. If the arrows are pointing to the right, the quotient will have the same sign as the side of the number line they’re on. If the arrows are pointing to the left, the quotient will have the opposite sign as the side of the number line they’re on. Therefore, 2 arrows pointing right on the positive side of the number line are the quotient 2. Two arrows pointing left on the positive side of the number line are the quotient −2.

DOK-2 How would you represent player 1’s scenario, −10 ÷ −2? Have the students try this out on their own, and project the player 1 box from the Student Journal as a reference.

Give a Student Journal, Horizontal Number Line, dry-erase marker, dry-erase eraser, pair of scissors, and glue stick to each student. Give a set of Solution Cards to each group. Allow students time to cut each of the Solution Cards. Have students collaborate with their groups to match the expressions with the Solution Cards. As students are working, monitor their progress and ask the following types of questions:

Explore

Engage

DOK-2 How do you determine the sign of the quotient? I check the direction of the arrows. If the arrows are pointed to the left, then the quotient will have the opposite sign as the side of the number line they’re on. If the arrows are pointed to the right, then the quotient will have the same sign as the side of the number line they’re on. DOK-2 What could be an easy mistake to make when using number lines for division? Student answers may vary. Forgetting to use the dividend as the starting point or putting the starting point on the wrong side of the number line would be an easy mistake to make.

Intervention

Acceleration

FACILITATION TIP Discussing division as repeated subtraction is a good opportunity to connect to students’ prior learning. Review and remind students that they already have this skill and can understand the concept; no , they just apply it to rational numbers.

STEMscopes Tip

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

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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.

Math Chat •

•

DOK-2 How did the signs of the dividend and divisor affect the sign of the quotient? I noticed that when the signs are the same, the quotient is positive. When the signs are opposite, the quotient is negative. DOK-2 What is the sign of the quotient of two or more integers with an even number of negative signs? Why? The sign of the quotient will always be positive since an even number of negative signs will always have pairs that also have positive quotients. DOK-2 What is the sign of the quotient of two or more integers with an odd number of negative signs? Why? The sign of the quotient will always be negative since an odd number of negative signs will always have an extra negative divisor that will not be in a pair to make it positive.Part II

Part II 1.

Read the following scenario to the class: For the second part of the football challenge, students were to return kickoffs. They were timed and judged based on how many yards they ran. Again, if they ran the wrong direction, they were given negative yards. Help the organizers determine who returned the ball the most yards per second.

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Multiplication and Division with Rational Numbers Explore 4 – Rational Number Division with Number Lines 2. 3. 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.

4. 5.

Instruct students to use the number lines to model the numbers being divided on the number line and ﬁnd the score for each scenario. Actively monitor students as they are working with their groups to model the problem using number lines. Ask the following types of questions: a.

DOK-1 How can you show 2.5 divided on a number line? Since our number lines are in whole numbers, you need to use two wholes and half of another to show 2.5.

DOK-1 What does the divisor represent in all of these expressions? The divisor represents the time it took to run the ball.

DOK-2 How are score cards 2 and 4 similar? Each expression represents running 4 yards per second.

Math Chat FACILITATION TIP Have students record the patterns or rules for dividing and multiplying integers in their notebooks or on their Student Journals. FACILITATION TIP Encourage students to thoughtfully consider this question about what method or mathematical tool they prefer. Provide the option to create their own problem and use their favorite model to solve it on the back of their Exit Ticket.

DOK-2 What is the sign of the quotient when both the dividend and divisor are negative? The sign of the quotient is positive because when you divide numbers with the same sign, the quotient will always be positive. • DOK-2 Which of the two methods of dividing integers helped you understand the rules better? I prefer the two-color counters because I can group them more easily. •

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 WITH RATIONAL NUMBERS

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MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Multiplication and Division with Rational Numbers Explore 5 – Using Properties to Solve ACTIVITY PREPARATION Students will apply commutative, associative, and distributive properties to multiply and divide integers and rational numbers.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. 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 Matching Scorecards (per student) 1 Set of Team Scorecards (per group) 1 Exit Ticket (per student)

Reusable • • • • •

1 Pair of scissors (per student) 1 Resealable bag (per group) 1 Projector or document camera (per class) Consumable 1 Glue stick (per student)

• • •

• •

Plan to divide the class into groups of three or four to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Matching Scorecards for each student. Print a set of Team Scorecards for each group. Cut out and place each set in a resealable bag labeled “Part II.” If desired, print them on card stock, and laminate them for future use. Gather enough glue sticks and scissors for each student to receive one of each. Have these materials ready on the tables before the activity. Be prepared to project the Matching Scorecards for the class to see.

PROCEDURE AND FACILITATION Part I 1.

2. 3.

Read the following scenario to the class: It is time for the awards ceremony. The first award is for the player of the day. The player with the most points at the end of the day will win the award. There are two scorecards for each player. One is the initial score, and the second scorecard is the verified score. 6 of the players are qualified for the award, while 2 of them are not. Help calculate the scores, match scorecards 1 and 2, and group the cards based on their properties. Give a Student Journal and a set of Matching Scorecards to each student. Project the two cards shown below for students to see.

Have the following discussion with the class:

FACILITATION TIP Before reading the scenario, ask the class 1) What typically occurs at the end of a school field day?; 2) What types of awards were presented at the field days you have attended? FACILITATION TIP You can edit the Student Journal Google form to create space for students to copy definitions for each property before or during the discussion.

DOK-1 Why do these two cards match? They match because they have the same value when multiplied.

DOK-1 What is the product of the two cards? 24

Instruct the students to write down the product or quotient as the score.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

DOK-1 What property do these two cards show? They show the commutative property.

e. Remind students that the commutative property of multiplication states that the order of the numbers being multiplied does not affect the product.

5. 6.

7. 8.

DOK-1 What is the associative property? It is the property stating that when multiplying three numbers, the placement of the grouping symbols does not affect the product.

DOK-1 What is the distributive property? It is the property stating that multiplying the sum of a group of terms by a number or variable is the same as multiplying each term by a number or variable and then adding the products.

Students will cut each of the Matching Scorecards and collaborate with their groups to match each scorecard 1 with the corresponding scorecard 2. As students are working, monitor their progress, and ask the following questions: a.

DOK-1 What operation do the cards that do not match have? They have division.

DOK-2 When distributing a number with a negative sign, what is one thing you have to check? I have to check that the negative sign is distributed correctly, especially when the second term is also a negative number.

FACILITATION TIP As an alternative, have the cards already cut for the groups to save time for students to collaborate. FACILITATION TIP Repeat questions 6a. and 6b. a few times, allowing different students to answer in their own words. Write sample expressions on the board, highlighting the negative sign and using arrows to demonstrate correct distribution.

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Math Chat DOK-1 How many points did the player with the most points get? 42 DOK-2 Does division have a commutative property? Why or why not? No, it does not. When I change the order of a division problem, the quotient changes as well. • DOK-2 Does division have an associative property? Why or why not? No, it does not. When I change the grouping of a division problem, the quotient changes as well. • DOK-2 When multiplying terms using the commutative or associative property, does the sign of the ﬁnal product change based on the grouping or the order? No, it does not change the sign of the ﬁnal product. The sign depends on the number of integers multiplied and their signs. • •

Part II 1.

2. 3. 4.

Read the following scenario to the class: The second award to be given out is the team of the day! Scorer 1 did the ﬁrst calculations for each team’s scores. Your job is to verify whether scorer 1 did the calculation correctly and recalculate the scores if the scorer calculated it incorrectly. Give a set of Team Scorecards to each group. Instruct students to go over each calculation and verify which calculations are correct and which ones need to be recalculated. Actively monitor students as they are working with their groups. Ask and review the following questions: a.

DOK-1 What is the order of operations? Parentheses, exponents, multiplication/division (left to right), and addition/subtraction (left to right)

DOK-1 What property of multiplication is shown in team 1’s expression? It is showing the distributive property.

FACILITATION TIP Before moving on to Part II, spot check students for understanding of each property. Call on students to state the properties or have them explain them orally to partners. FACILITATION TIP After reading the scenario, ask the class 1) What might qualify a team to receive the “Team of the Day” award? 2) Why is it important to check and recheck calculations? FACILITATION TIP Before passing out the Team Score Cards, review the Order of Operations with students. Post these steps in the room and point them out to students who need reminders.

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Multiplication and Division with Rational Numbers Explore 5 – Using Properties to Solve

5. 6.

DOK-1 What is the ﬁrst step in simplifying the expression for Team 3? I need to divide the numbers inside the parentheses ﬁrst.

DOK-2 Why is it important to verify the calculations? We want to make sure the calculations are correct so each team gets the points they earned.

Math Chat FACILITATION TIP Take time to ensure that students can state and apply these rules with ﬂuency. Ask for volunteers to summarize the rule in their own words and ﬁnd a place to post a student-generated version of the rule in the room. FACILITATION TIP The Order of Operations is a skill that will apply over the course of a student’s math career; take time to monitor and assess. Engage students by pointing out the similarities between successful coding and the correct use of Order of Operations. FACILITATION TIP If students are struggling, allow them to use their notes on their Student Journal.

DOK-1 For which team(s) did scorer 1 get the calculations correct? Team 3 DOK-2 What rules do rational numbers have when multiplying or dividing negative numbers? Since integers are rational numbers, all rational numbers follow the same rule. When the numbers have like signs, the product/quotient is positive. When the numbers have opposite signs, the product/quotient is negative. • DOK-2 When simplifying multistep expressions, what is a common mistake students make? When dealing with multistep problems, we do all the calculations but sometimes forget to put the sign correctly. • DOK-2 What is an important reminder when solving multistep problems involving properties of rational numbers? Always follow the order of operations. • •

Post-Explore 1. 2. 3. 4.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

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MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Multiplication and Division with 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

Integer Multiplication with Counters 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

Rational Number Multiplication with Number Lines 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

Integer Division with Counters

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

Rational Number Division with Number Lines

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

Show What You Know, Part 5 Using Properties to Solve 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 Rational Numbers Independent and partner games and other activities that provide students with an engaging way to practice the new concept

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

Notes __________________________________________________________________________________________________________________________________________________

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

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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

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

Multiplication and Division with Rational Numbers

3 74

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 multiply and divide rational numbers by using number lines and counters.

I can represent every quotient of integers as a rational number.

I can use various strategies to multiply rational numbers and interpret their products.

What prompts will be used?

What does mastery look like?

MULTIPLICATION AND DIVISION WITH RATIONAL NUMBERS

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I can use various strategies to divide rational numbers and interpret their quotients.

I can use properties of operations to discover rules for multiplying signed numbers.

I can determine the direction on a number line when representing multiplication and division of two rational numbers.

I can multiply and divide rational numbers through applying the properties of operations.

SCOPE 1

Rational Number Operations Scope Introduction SCOPE SUMMARY In this scope, students will be able to use long division to convert rational numbers to decimals. They will understand that a rational fraction written in decimal form will either terminate or repeat. Students will discover the possibility of an inﬁnite decimal through repeating decimals. They will use these skills to solve problems pertaining to real-world scenarios with all four operations. Student Expectations

Background Knowledge

Future Expectations

In Grade 6, students used standard algorithms in order to add, subtract, multiply, and divide multi-digit decimals ﬂuently. Students learned to represent fractions as a decimal with a denominator of 10 or 100. They began to relate fractions and decimals to whole numbers as integers on a number line.

Students will use their knowledge of rational numbers as they pertain to fractions that terminate and repeat in upcoming years. In 8th grade, they will build on these concepts to introduce irrational numbers. Students will approximate irrational numbers as they pertain to the number line and eventually functional graphs.

ENGAGE ACTIVITIES Accessing Prior Knowledge

7.NR.1.11 Solve multistep, contextual problems involving rational numbers, converting between forms as appropriate, and assessing the reasonableness of answers using mental computation and estimation strategies.

VERTICAL ALIGNMENT

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

Hook

7.NR.1.10 Convert rational numbers between forms to include fractions, decimal numbers and percentages, using understanding of the part divided by the whole. Know that the decimal form of a rational number terminates in 0s or eventually repeats.

•

add multi-digit numbers; and, add, subtract, multiply, and divide multi-digit decimals using standard algorithms.

•

determine if statements or claims are always true, sometimes true, or never true.

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

solve real-world problems that involve complex fractions.

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. Notes _____________________________________________________________________________________________________________ _____________________________________________________________________________________________________________

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Convert between Forms In this exploration, groups of students will learn concepts through helping solve a real-world scenario involving a pie shop and the need to prove students can convert fractions to decimals in order to help with the county fair. Students will: •

use long division to convert fractions to decimals.

•

differentiate between terminating and repeating decimals.

Explore 2

Explore 1

EXPLORE ACTIVITIES

In this exploration, students will collaborate with peers to solve a real-world scenario involving the pie shop from the previous exploration. Here, students are tasked with helping Priscilla determine and pull our the correct packaging, both the container and number, and package the pies. Students will: •

solve expressions and multistep word problems where fractions are divided by fractions.

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

Solving With Complex Fractions

RATIONAL NUMBER OPERATIONS

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Multistep Rational Number Operations In the ﬁnal exploration of this scope, students will complete another scenario involving the pies and Priscilla. Students are tasked with helping Priscilla evaluate her expenses and sales at last year’s events so she is better prepared to maximize her proﬁts this year. Students will: •

solve problems involving all four operations.

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

RATIONAL NUMBER OPERATIONS

Rational Number Operations 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 play Always, Sometimes, Never to determine if statements or claims are always true, sometimes true, or never true. This element is designed to uncover student misconceptions; it should not be taken for a grade. 6.NR.1.3 Perform operations with multi-digit decimal numbers ﬂuently using models and student selected strategies.

Materials

Preparation

Printed • •

•

1 Set of Always, Sometimes, Never Cards (per student or per group) 1 Always, Sometimes, Never Statements (per teacher)

• •

Print one set of Always, Sometimes, Never Cards for each student. Note: The print document includes both a color version and a black-and-white version. Select the version that works best for your classroom. Cut the cards apart, and consider laminating the cards for repeated use. Plan to read or project the Always, Sometimes, Never Statements.

RATIONAL NUMBER OPERATIONS

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Procedure and Facilitation Points 1. 2.

3. 4.

Distribute the Always, Sometimes, Never Cards to students. Tell students that you are going to read/project a series of statements to them, and they will determine whether each statement is always true, sometimes true, or never true. After thinking time, students hold up cards with their responses and must be able to justify their responses. Ask students to justify their responses to their elbow partners. Choose volunteers to explain their reasoning to the whole group. (For the “sometimes” statement, can you explain when they are true? Alternatively, ask students how they would rewrite “sometimes” statements so that they are always true or never true.) Scenario statements and answers are provided below. • •

• • • • 6.

When dividing with decimals, you can multiply the dividend by 100 and leave the divisor alone. Never When multiplying factors with decimals, the product will have the same number of digits to the right of the decimal point as both factors combined. Always Answers should be rounded to the hundredths place. Sometimes When dividing with decimals, we use the standard algorithm method. Sometimes When adding or subtracting with decimals, we line up the decimal points and then add or subtract. Always In division, when you move the decimal two places to the right in the divisor, you move the decimal two places to the left in the dividend. Never

FACILITATION TIP In order to engage students, create a few high-interest statements unrelated to math. Use these engaging statements to have them practice voting using the hand symbols or cards. For example, “Mr. Smith wears glasses, Mr. Smith starts class with a warm-up...” 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.

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. Notes

RATIONAL NUMBER OPERATIONS

Rational Number Operations Accessing Prior Knowledge Identifying Misconceptions • FACILITATION TIP These prior skills and ﬂuency with decimals are critical to student success in this scope. Be sure to preview the Skills Quiz in order to be prepared to support students appropriately as you move through the scope.

• •

Remind students to line up decimal points when adding or subtracting decimals but not when multiplying or dividing decimals. Make sure students understand that the number of decimal places in the product is equivalent to the sum of the number of decimal places in the factors. Students must be aware that whatever power of ten they multiply a divisor by, they must also multiply a dividend by in order to keep the problem equivalent and the quotient the same.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

RATIONAL NUMBER OPERATIONS

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RATIONAL NUMBER OPERATIONS

Rational Number Operations Hook – Be a Smart Cookie ACTIVITY PREPARATION Students will solve a real-world word problem that involves complex fractions.

Materials

Preparation

Printed •

• •

1 Be a Smart Cookie (per class)

Reusable •

•

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Be a Smart Cookie 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) Have you ever seen cookies the size of pizzas?; 2) What type of containers do you think a bakery puts giant cookies in when they sell them to customers?; 3) How could you serve a cookie this big to your friends so that each friend gets the same amount of cookie?

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: Monica worked in her family bakery called Smart Cookies. It sold only large cookie cakes the size of large pizzas in many different flavors. The cookies were cut into eighths and could be sold by the piece or as whole cookie cakes. One week, the bakery ran out of whole boxes and individual boxes! They only had boxes left that would fit half of a cookie cake inside. For every purchase, Monica had to determine how many boxes the order needed before she would box up the 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 Monica is partitioning the cookie cakes into fractions (eighths). I notice the boxes hold a different fraction (halves). I wonder how many pieces of cookie cake are ordered in a large order. How will Monica divide fractions to figure out how many boxes to use — a model or an algorithm? I can use math to determine the total number of cookie cakes ordered and how many boxes are needed for an order. Project Be a Smart Cookie. Notes

Engage

Explore

Explain

Elaborate

Evaluate

Explain to students that Monica has received an online order that she needs to 3 __

7 __

get boxed up. It includes 8 of a sugar cookie cake, 4 of a chocolate chip cookie 3

cake, and __8 of a peanut butter cookie cake. She must determine how many 1

boxes she needs because each box ﬁts __2 of a cookie cake. Discuss the following questions:

DOK-1 What is the ﬁrst step to take to solve this problem? First, I need to ﬁnd the total number of cookie cakes being purchased in the order.

DOK-1 How can you find the total number of cookie cakes being purchased in the order? First, I need to find a common denominator, and then I need to find the sum of the three different cookie cakes.

Acceleration

FACILITATION TIP Note that some students may decide to find the total number of halves in the cookie cake order as a first step. Allow for different strategies and have students share their thinking. FACILITATION TIP Refer students to the visual glossary for the complex fraction definition.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Intervention

RATIONAL NUMBER OPERATIONS

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FACILITATION TIP

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

This is a good opportunity to help students visualize ways to think about dividing by a fraction. In this example, “How many halves 16 are in ___ ?” or “How many halves are in 2?” 8

DOK-1 What is a complex fraction? It is a fraction that also has a fraction in either its numerator, its denominator, or both.

b. DOK-1 What was the total number of cookie cakes in the order? 16 7 3 3 7 6 3 16 ___ because __8 + __4 + __8 = __8 + __8 + __8 = ___ 8 8 c.

FACILITATION TIP

DOK-2 After determining the total number of cookies ordered, what is the next step? The total number of cookies must be divided by how 16 ___ 8

many cookies each box can hold. This creates a complex fraction. __ 1

Add or draw the image of the solution in question 2e. to your projected Be a Smart Cookie to illustrate the boxes and slices in the four boxes.

DOK-2 How is a complex fraction solved? Explain the steps. It is 16 __ 1 changed into a division expression: ___ ÷ 2. Then, we can ﬁnd the 8 reciprocal of the second fraction and multiply the two fractions 16 __ 1 together: ___ × 2. Then, it is solved and reduced. 8 16

e. DOK-1 How many boxes are needed for the order? 4 boxes ___ × __1 = ___ =4 8 8 Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________

RATIONAL NUMBER OPERATIONS

Rational Number Operations Explore 1 – Convert between Forms ACTIVITY PREPARATION Students will convert fractions and percents using long division. Students will differentiate between terminating and repeating decimals.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.5 Use appropriate tools strategically.

Materials

Preparation

Printed • • •

1 Student Journal (per student) 1 Set of Fraction, Percent, and Sign 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 a set of the Fraction, Percent, and Sign Cards for each group. Cut out and place each set of cards in a resealable bag for each group. If desired, print them on card stock, and laminate them for future use. Gather enough calculators for each student to have one.

1 Resealable bag (per group) 1 Calculator (per student)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What activities take place at county fairs?; 2) What types of desserts are sold at a county fair? After reading the scenario, ask the class 3) What would be the beneﬁt to Priscilla’s Pies and their customers to base pie prices on decimals, rather than fractions? FACILITATION TIP

2. 3.

This real-world application of decimals will help students see a connection between costs, decimals and fractions. FACILITATION TIP Using three different colors of paper will clarify the piles of cards. You could also simplify this activity by saving the Sign Cards for later and allowing students to focus on calculations until the Math Chat.

Read the following scenario to the class: Your group wants to help Priscilla’s Pies during the county fair. Priscilla makes the best pies in the county, and she makes many different flavors and types. Based on the sizes of her pies, she cuts and sells different fractions of the pies. Before she will let you help her, she needs to make sure you understand percents and fractions and how to convert them into other forms of fractions and decimals because she bases her pie prices on the various equivalencies. Today, you will work with your group to prove you can convert between forms. Give a Student Journal to each student. Distribute the Fraction, Percent, and Sign Cards. Have students separate the cards into two piles (one pile of Fraction/Percent Cards and one pile of Sign Cards) and place them facedown. Instruct students to take turns flipping over a card from each pile. Students should flip over one Sign Card for each Fraction/ Percent Card. Have all students then independently convert each fraction, decimal, and percent to the equivalent forms. Instruct students to record the card, the sign, the equivalent forms, their computations, the solution, and whether the decimal terminates or repeats. Finally, have students check their solutions with calculators. Monitor and assess students as they are working by asking the following guiding questions:

FACILITATION TIP

Be sure to include dividend, divisor, and quotient as vocabulary on your class anchor chart.

DOK-1 What is a dividend? The number you divide into; the quantity that is to be divided by another quantity

DOK-1 What is a divisor? The number you divide by; the quantity by which another quantity is to be divided

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How does using a calculator help you? It helps me check my long division and make sure I did not make a mistake with computation or alignment.

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

Math Chat •

• • •

•

• •

DOK-1 What clue do you have that a decimal is a repeating decimal? There is a remainder that keeps repeating. It does not have to be the same numeral or numerals that keep repeating in the quotient. DOK-1 What clue do you have that a decimal is a terminating decimal? The remainder is zero. DOK-2 When you convert a fraction to a decimal using long division, what are some of the errors that can occur? Computation errors and alignment errors DOK-1 How do you determine whether a decimal converted from a fraction is positive or negative? Ignore the sign until the long division is complete. Then, put the correct sign in front of the decimal. DOK-1 What was your strategy for converting the percent into a fraction and decimal? I ﬁrst rewrote it as a fraction with a denominator of 100 and then simpliﬁed it. I then completed the long division to determine the decimal. DOK-2 Was long division necessary to convert the percent into a decimal, or was that a way to help you? It is not necessary because when I rewrote it as a fraction with a denominator of 100, I knew the decimal place needed to be shifted to the left two places to write the fraction as a decimal. The long division helped me check my work, like the calculator. DOK-1 How does using a calculator help you? I use it to check my long division, numerator divided by denominator. DOK-2 In what other instances might you need to convert fractions to decimals in the real world? If I want to know what percent a fraction is, I need to convert the fraction to a decimal. For example, if a school’s goal is to have 50% of 7th graders attend Math Night, and 65 of 125 7th graders do attend Math Night, the school met its goal because 52% of seventh graders attended.

2. 3.

Acceleration

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.

FACILITATION TIP Some students may question what the positive and negative signs mean in the described pie scenario. Remind them that it is still good practice to be sure to check your signs for solutions when you complete your calculations.

FACILITATION TIP Be prepared with some real-world examples of your own that your students can connect with. Find a relevant sport, video game, or a recent graded test score that you can include to jump start students to think of their own instances. FACILITATION TIP

Post-Explore 1.

Intervention

RATIONAL NUMBER OPERATIONS

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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.

To check for understanding, before giving students this Exit Ticket, use individual whiteboards to allow them to complete long division decimal problems and do some partner checks, then calculator checks.

RATIONAL NUMBER OPERATIONS

Rational Number Operations Explore 2 – Solving with Complex Fractions ACTIVITY PREPARATION Students will solve expressions and multistep word problems where fractions are divided by fractions.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. 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 Word Problem Cards (per pair) 1 Exit Ticket (per student)

Plan to divide the class into pairs. Print a Student Journal and an Exit Ticket for each student. Print a set of Word Problem Cards for each pair. Cut out and place each set in a resealable bag. If desired, print the cards on card stock, and laminate them for future use.

Reusable •

1 Resealable bag (per pair)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What jobs would employees at Priscilla’s Pies need to do to get the pies from the showcase to the customer?; 2) What skills would be important for employees to have to work at Priscilla’s Pies?; 3) How does having eﬃcient and knowledgeable employees beneﬁt Priscilla’s Pies? FACILITATION TIP Complete a practice example of a Word Problem card to model how students should show their thinking on the Student Journal. You could also have students complete some problems with you and some in partners.

2. 3. 4.

FACILITATION TIP Note that on the Word Problem cards, the question “How many boxes will her pie ﬁll?” may prompt students to round or only answer in whole numbers. Before you present the problems, explain that the answers may include fractions, and that some containers may be partially full.

Read the following scenario to the class: Your group has been hired by Priscilla for your excellent math skills. As your last bit of training, you are shadowing Priscilla herself as she helps customers. She sells the pies and then packages them in different containers that hold different fractions of pies. While you train today, she wants you to listen to the orders and packaging requests. As she checks out the customers on the cash register, she expects you to pull out the correct packaging (both container and number of containers) and package the pies for the customers. With your help, she can keep the line moving, serve more customers in less time, and improve her customer satisfaction. Give a resealable bag with the Word Problem Cards to each pair. Give a Student Journal to each student. Discuss the following with the class: The problems that we will be working with are called complex fractions. Complex fractions are fractions that have a fraction in the numerator, the denominator, or both the numerator and denominator. Explain to students that they will read each scenario. Then, students should work together to write the complex fraction, division expression, corresponding multiplication expression, and solution in the table on their Student Journals. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 How can you write Word Problem Card 1 as a complex fraction 1 1 expression? Word Problem Card 1 can be written as __2 ÷ __4 as a division 1 __ 2

problem, so the complex fraction would be _1 . __

7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How can you be sure two fractions are inverse fractions? If we multiply them together, the product will be 1.

DOK-2 How are complex fractions different from regular fractions? Complex fractions have a fraction as the numerator and/or denominator.

Intervention

Acceleration

FACILITATION TIP As you monitor and assess, provide prompts to the class if needed about inverse fractions. Be sure to include inverse on the class anchor chart.

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

Math Chat DOK-2 How are complex fractions different from regular fractions? Complex fractions have a fraction in the numerator, denominator, or both. • DOK-1 How can you check to see if two fractions are truly inverse fractions? The numerator and denominator of the original fraction will be reversed in the inverse fraction. Also, when multiplied together, the two fractions will have a product of 1. • DOK-2 How did you solve problems that had mixed numbers for the total amount of pie purchased? We changed the mixed number to an improper fraction. This made multiplying it by the inverse of the divisor easier since it was only one step. • DOK-2 What do you notice about the relationship between the value of the divisor and the value of the quotient? We noticed that when the dividend stayed the same, then the lesser the value of the divisor, the greater the value of the 4 1 4 2 quotient. For example, __5 ÷ __5 = 4, but __5 ÷ __5 = 2. •

•

DOK-3 When have you or someone in your family needed to divide a fraction by 3

a fraction? We had __4 of my cousin’s birthday cake left after his party. It was his

favorite cake, so he wanted to take it all home. My mom had containers that could

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FACILITATION TIP Remind students that division is repeated subtraction. It is a way to ﬁnd how many divisors are in the dividend. In this example, 4 1 the expression __5 divided by __5 is asking, 1 4 __ __ “How many 5 are in 5 ?”

ﬁt exactly __4 of the cake in each container. My mom asked me to get containers out

of the pantry, so I had to ﬁgure out how many containers she needed. 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 Be aware that this Exit Ticket includes a multistep problem. Students will need to add fractions with unlike denominators before dividing the fractions.

RATIONAL NUMBER OPERATIONS

Rational Number Operations Explore 3 – Multistep Rational Number Operations ACTIVITY PREPARATION Students will solve problems involving all four operations.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. 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 Scenario Cards (per group) 1 Exit Ticket (per student)

Reusable •

•

1 Resealable plastic bag (per group)

Plan to divide the class into student groups of 4 to complete this activity. Print a Student Journal for each student. Print a set of Scenario Cards for each group. Cut them out, and place the cards in a resealable bag. If desired, print the cards on card stock, and laminate them for future use. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student has one.

PROCEDURE AND FACILITATION FACILITATION TIP After reading the scenario, ask the class 1) What types of expenses would Priscialla’s Pies have?; 2) Besides providing ﬁnancial data, what other types of information might the sales records from the previous year provide?; 3) Why would you need to follow the order of operations to determine the correct information? FACILITATION TIP These scenario cards provide a great opportunity to point out to students that fractions in word form and number form can be included in real-world problems. A common error for struggling students is to scan for numerical digits in story problems and miss the values in word form. Highlight some of the math operation words included in these scenarios (triple, in half, quadruple).

2. 3.

Read the following scenario to the class: Your group was hired by Priscilla to help her business at the county fair and the farmers’ market. Before she sells pies at these events this year, she wants your group to help her evaluate her expenses and sales at last year’s events so she is better prepared to maximize her proﬁts this year. Your job today is to look at her ﬁnancial records from last year and answer questions. She reminds you that to determine the correct information, you need to remember to follow the order of operations. Distribute the Student Journals and Scenario Cards to students. Tell students to work together in their groups to create and solve expressions to answer the questions from the Scenario Cards. Remind them to follow the order of operations. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What is the correct order of operations? How do you remember it? Student responses will vary. Anything in parentheses ﬁrst and then any exponents, followed by multiplication/division and then addition/ subtraction

DOK-2 When writing your expressions, when should you use positive integers? When money is earned because it is added to what you have

DOK-2 When writing your expressions, when should you use negative integers? When money is spent because it is taken away

FACILITATION TIP If the Order of Operations is not posted somewhere in the classroom, be sure to write it on the board (or at least an acronym like GEMDAS).

6. 88

When groups have ﬁnished answering the questions on their Student Journals, for each question, allow one group of students the opportunity to share their thought processes while creating and solving expressions with the class. Give student groups time to discuss and evaluate their own ideas and processes after each question has been presented by a group. Allow students time to complete the Student Journal reﬂection questions. © 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 When did you use addition and when did you use subtraction setting up and solving the problems? We used addition when money was earned and subtraction when money was spent. • DOK-2 When did you multiply setting up and solving the problems? We multiplied when there was a repeat expense or a repeat earning because it is the same number repeated a certain number of times. • DOK-2 When did you divide setting up and solving the problems? We divided any time we needed to split something into equal groups. • DOK-2 Was there only one way to write each expression? Can you give an example of an expression written a different way? No, for Scenario Card 8, you can group the products that had the same amount. 2(3.80 + 3.65) + 3(4.60 + 2.25 – 2.50) + 4(4.35) •

Post-Explore 1. 2. 3. 4.

Intervention

Acceleration

FACILITATION TIP Use a T-chart or add to one already in the classroom to list words used for positive or negative integers (or maybe both). These fair scenario cards include expenses, wages, proﬁt, and bonuses. FACILITATION TIP Allow enough time for student to complete these scenarios. The details and information provided about the County Fair and the Farmer’s Market must be applied to different cards. Struggling students may need to be prompted to sort the cards by event, perhaps folding the chart in half as they search for relevant information.

RATIONAL NUMBER OPERATIONS

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FACILITATION TIP Encourage students to show each step of the Order of Operations on the Exit Ticket. If needed, direct them to solve within the grouping symbols to start, and then apply the rest of the Order of Operations. Showing their steps will help you and the students see where they are successful.

PROPORTIONAL RELATIONSHIPS

Proportional Relationships 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

Proportional vs. Non-Proportional 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

Unit Rates 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

Proportional Relationships with 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

Describe Points on a Graph

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

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

PROPORTIONAL RELATIONSHIPS

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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

PROPORTIONAL RELATIONSHIPS

Proportional Relationships

3 92

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 convert between fractions, decimals, and percentages.

What prompts will be used?

What does mastery look like?

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I can represent every rational number as a ratio.

I can describe and represent the decimal form of a rational number as a terminating decimal or a repeating decimal.

I can solve multistep problems involving rational numbers.

I can determine if an answer is reasonable by using mental computation and estimation strategies.

SCOPE 1

Proportional Relationships Scope Introduction SCOPE SUMMARY In this scope, students will dissect proportional relationships by using tables, graphs, equations, and diagrams to identify the unit rate. They will use this concept to determine if two quantities are proportionate to one another. Students will understand how to read and write proportional equations, relating them to lines on a graph given different situations.

Student Expectations

7.PAR.4.2 Determine the unit rate (constant of proportionality) in tables, graphs (1, r), equations, diagrams, and verbal descriptions of proportional relationships to solve realistic problems. 7.PAR.4.3 Determine whether two quantities presented in authentic problems are in a proportional relationship.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In Grade 6, students gained an understanding of the concepts and language of ratios to form relationships between quantities. They learned what a unit rate was and were able to use unit rates and ratios to solve real-world and mathematical problems.

Computing unit rates will be important as the students continue through high school math. As the basis for the study of functions, unit rates allow the students to discover the average rate of change represented as a slope on a graph. In 8th grade, students will use these concepts to understand the connections between proportional relationships, lines, and linear equations.

7.PAR.4.4 Identify, represent, and use proportional relationships. 7.PAR.4.5 Use context to explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

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

play Always, Sometimes, Never to determine whether statements or claims are always true, sometimes true, or never true.

•

represent quantities in problems that change in relation to one another.

•

write equations to express quantities.

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 whether the data represented on a line graph demonstrates a proportional or a non-proportional relationship.

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.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

In this exploration, students will be tasked with helping Jake determine how much money per house he could receive at each job he got job offers from; and help Jania see if any of the jobs have signing bonuses. Students will:

Explore 3

•

Explore 2

Proportionality vs. Non-Proportional

Unit Rates In this exploration, students will be tasked with helping Jake, from the last exploration, ﬁnd out which runner can run the most miles in 1 hour. Students will: •

learn how to determine whether two quantities in tables and graphs are in proportional relationships.

identify the unit rate in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

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

Proportional Relationships with Equations

Describe Points on a Graph

In this exploration, students will be tasked with helping Jake once again. Here students help ﬁnd the total registration fees for each camp given the number of students who registered. Then, determine the constant of proportionality and the equations needed to calculate the total camp fees he will be collecting. Students will: •

Explore 4

Explore 1

EXPLORE ACTIVITIES

In this exploration, students will learn through solving and completing a ﬁnal scenario helping Jake plan his ﬁrst track meet through looking at various scenarios and graphs. Students will: •

ﬁnd the equations of proportional relationships from tables, graphs, and verbal descriptions.

PROPORTIONAL RELATIONSHIPS

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explain the different points on graphs showing proportional relationships.

Students conclude the exploration with an Exit Ticket and revisit the Hook to solve using their learning from the explorations.

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

PROPORTIONAL RELATIONSHIPS

Proportional Relationships 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 play Always, Sometimes, Never to determine whether statements or claims are always true, sometimes true, or never true. This element is designed to uncover student misconceptions; it should not be taken for a grade. 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.

Materials Printed • •

Preparation •

1 Set of Always, Sometimes, Never Cards (per student or per group) 1 Always, Sometimes, Never Statements (per teacher)

• •

Procedure and Facilitation Points 1. 2.

3. 4.

Distribute the Always, Sometimes, Never Cards to students. Tell students you are going to read/project a series of statements to them, and they will determine whether each statement is always true, sometimes true, or never true. After thinking time, students hold up cards with their responses and must be able to justify their responses. Ask students to justify their responses to an elbow partner. Choose volunteers to explain their reasoning to the whole group. (For the “sometimes” statement, can you explain when they are true? Alternatively, ask students how they would rewrite the “sometimes” statements so that they are always true or never true.) Scenario statements and answers are provided below. • • • • •

• 6.

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The independent variable can be found on the x-axis and the dependent variable can be found on the yy-axis -axis of a graph. Always The independent and dependent variables are equal. Sometimes The dependent variable is a given value, and the value of the independent variable is calculated based on the dependent variable’s value. Never Juan saved $10 per week for 10 weeks. The dependent variable is time in weeks. Never The price of movie tickets is $12 each. c is the total cost of tickets and n is the number of tickets purchased. The equation c = 12n solves for c, which is the dependent variable. Always The independent variable is found on the top row or ﬁrst column of a table. Always

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 To save time, have students use hand signals for Always, Sometimes, Never rather than copying and distributing cards (thumbs up, thumbs sideways, thumbs down). FACILITATION TIP Ask for volunteers or spot check for students being able to justify their responses. Although this APK is not for a grade, it is a good opportunity to determine student strengths and gaps for this scope.

FACILITATION TIP See Supplemental Aids in Intervention. Reviewing prior grade skills regarding the x axis, y axis and origin will support success with the standards in this scope.

Identifying Misconceptions • • •

Students may confuse independent and dependent variables. Students may not understand that an independent variable is found on the x-axis of a graph or the top row/left-hand column of a table. Students may not understand that an equation represents the relationship between the dependent and independent variables.

PROPORTIONAL RELATIONSHIPS

Proportional Relationships Hook – Break a Leg! ACTIVITY PREPARATION Students will determine whether the data represented on a line graph demonstrates a proportional or a non-proportional relationship.

Materials

Preparation

Printed •

• • •

1 Break a Leg! (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Break a Leg! 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 FACILITATION TIP In addition to the Foundations Builder, students would beneﬁt from reviewing graphing on a coordinate plane (See Explore 4). Engage students by providing an opportunity to graph points with rational numbers on four quadrants. Reviewing prior grade skills regarding the x-axis, y-axis, and origin will support success with the standards in this scope. Also see Supplemental Aids in Intervention.

FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) Why do schools put on plays?; 2) If your school put on a play, what could be done to encourage people to come to the performance?; 3) Do you think offering to give the ﬁrst few people in line free admission would boost ticket sales? Why or why not?

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. Fernandez, the drama teacher, coordinated with the business club and chose a small group of students to run marketing and ﬁnances for the school’s latest production. The students ran a promotion advertising that the ﬁrst ten people in line to buy tickets to each show would get free admission. The rest of the tickets would be sold for $20 apiece. In addition, tickets would only be sold for one show at a time. When that show sold 75 tickets and sold out, tickets for the next performance would go on sale. Mr. Fernandez was unsure whether this tactic would work, and he wondered whether the relationship between tickets sold and dollars earned would be proportional. 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 business club students are trying to earn the most money possible with a ticket price of $20 per ticket. I wonder how many performances will sell out. How much money is made at each performance? I can use math to determine whether the relationship between tickets purchased and money earned is proportional. Project Break a Leg! Notes

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Explain to students that the marketing strategy worked! Many people hurried to be first in line to buy tickets. If they did not receive a free ticket, they still decided to buy a ticket. Shows were selling out and more shows were being added. The business club students in charge of marketing and finances showed Mr. Fernandez a graph of ticket sales for the first show and told him many more shows would also generate the same graph. Discuss the following questions: a.

DOK-1 What is the title of the graph? Ticket Sales

DOK-1 What does the graph represent? The graph shows how many tickets were sold, how much money was made per ticket sold, and how much money was made if the show sold out by selling 75 tickets.

DOK-1 What do you notice about the graph? Answers will vary. The line is a straight line. The line has a positive slope. The tickets sold have a scaled interval of 5 and the money earned has a scaled interval of 20.

Complete the Explore activities.

FACILITATION TIP You may need to direct students to the labels on the x-axis and y-axis. These details will help them see what the graph represents. Prompt by asking, “Who is earning the money? Who is purchasing tickets?” if necessary.

PROPORTIONAL RELATIONSHIPS

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Part II: Post-Explore 1. 2.

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

DOK-1 How much money is made at a sold-out performance of the show? $1,300

DOK-1 If each ticket costs $20 and 75 tickets are sold, how much money should be earned? $1,500

DOK-1 Why are the answers to the ﬁrst two questions not equivalent dollar amounts? As part of the promotion, the ﬁrst ten people in line to purchase tickets to each show get in for free.

DOK-2 Look at the graph. Is the relationship between tickets sold and money earned proportional? Explain. No, the relationship is not proportional. The straight line seems to indicate a proportional relationship, but because the ﬁrst ten tickets were free, the ratios of points on the line are not equal. For example: $0 for 10 tickets, $200 for 20 tickets, and $1,300 for 75 0 200 1300 ___ 52 tickets are not equal. ___ = 0; ____ = 10; _____ = 3 = 17.3 0 ≠ 10 ≠ 17.3 75 10 20

FACILITATION TIP Consider asking for volunteers to restate the problem before restating it yourself.

FACILITATION TIP Questions d and e will provide good opportunities for discussion after the Explore activities are complete. Students may need time to see that this linear graph is not showing a proportional relationship.

e. DOK-2 What would have to change for the relationship to be proportional? Answers may vary. If the price of each ticket were kept at $20 and no tickets were given away for free, the relationship would be proportional because the ratio from every point on the line on the graph would equal 20.

PROPORTIONAL RELATIONSHIPS

Proportional Relationships Explore 1 – Proportional vs. Non-Proportional ACTIVITY PREPARATION Students will determine whether two quantities in tables and graphs are in a proportional relationship.

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 Job Offer Cards (per group) 1 Set of Job Listing Cards (per group) 1 Exit Ticket (per student)

•

Reusable • • •

2 Resealable bags (per group) 1 Ruler (per student) 1 Marker (per group)

• •

Plan to divide the class into groups of 3 or 4. Print a Student Journal and an Exit Ticket for each student. Print a set of Job Offer Cards for each group. Cut out and place each set in a resealable bag labeled “Part I.“ If desired, print them on card stock, and laminate them for future use. Print a set of Job Listing Cards for each group. Cut out and place each set in a resealable bag labeled “Part II.” If desired, print them on card stock, and laminate them for future use. Gather enough rulers for each student to have one. Gather enough markers for each group to have one.

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) If you wanted to get a job after school, what companies would you like to work for?; 2) What does working “part-time” mean?; 2) After receiving multiple job offers, how might you determine which job offer to accept?

2. 3.

Read the following scenario to the class: Jake applied for a part-time assistant coaching position at multiple organizations where he can work up to 10 hours per week. He received 4 job offers! Using the graphs Jake created, help him determine how much money per hour he would receive at each job. Give a set of Job Offer Cards to each group. Discuss the following question with the class: a.

FACILITATION TIP Engage students before handing out the job offer cards by telling them to read carefully for any job “signing bonus.” Does the signing bonus make it worthwhile for Jake? FACILITATION TIP This information about ratios is review from sixth grade, but is critical for student success. Have students copy these deﬁnitions and examples in their journals: ratio, equivalent ratio, and unit rate. Refer to the Visual Glossary.

4. 5.

Give a Student Journal to each student. Discuss the following questions with the class: a.

DOK-1 What is a ratio? A ratio is a comparison of two quantities that shows their sizes in relation to one another.

DOK-1 How do you write ratios? You can write ratios using the word to; a using a colon, like a:b; or written like a fraction: __b .

c. d.

100

DOK-1 How can I create a table using the graphs provided on the Job Offer Cards? I can ﬁll in the table with the values from the graph. The number of hours worked will be shown in the x values, and the salary will be shown in the y values.

DOK-1 How can you determine if you have two equivalent ratios? I can ﬁnd the unit rate, which is the rate per 1, or use an equivalent ratio table. Explain the following to the class: When all of the ratios in a table have the same unit rate, the relationship is proportional. If all of the ratios in the table do not have the same unit rate, these relationships are nonproportional. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Explain the following to the class: We are going to use tables and graphs with different ratios for the four job offers Jake received. We need to determine if the ratios of money earned to hours worked form equivalent ratios (proportional relationships) or if the ratios do not form equivalent ratios (non-proportional relationships). Use the graphs on the Job Offer Cards to complete each table in your Student Journal. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What operation do you use to ﬁnd the ratio of the two quantities? I use division.

DOK-1 What do you notice about the starting amounts in the tables? 2 tables have 0 for starting amounts, and 2 tables have different starting amounts.

DOK-2 How did you differentiate between proportional and nonproportional relationships in tables? Tables that have a proportional relationship have a value of 0 for a signing bonus or starting amount. The tables that have a non-proportional relationship have a starting bonus or starting amount.

Intervention

Acceleration

FACILITATION TIP Before students begin their work, determine if they need calculators to complete the division calculations.

PROPORTIONAL RELATIONSHIPS

Home

Allow students enough time to complete the tables in Part I.

Part II 1.

2. 3.

Read the following scenario to the class: Jake encouraged his friend Jania to apply to assistant coaching jobs, too! She started looking up job listings online. The job listings she saw put the pay rates for a given time frame but did not show whether they have signing bonuses. Help Jania graph the rates to see if any of the jobs have signing bonuses. Give a set of Job Listing Cards to each group. The students will collaborate to ﬁnd the signing bonuses by plotting the points from the Job Listing Cards on the graphs in their Student Journals. Remind students that they have to plot at least two points and extend the line until it connects to the y-axis. y As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What values are represented on the x-axis? The x-axis represents the number of hours.

DOK-1 What values are represented on the y-axis? y-axis? The y-axis represents the pay rate.

DOK-2 How did you ﬁnd the signing bonus? bonus I looked at the y-axis at 0 hours and found the pay rate.

DOK-1 Do proportional graphs show the signing bonus? No, the nonproportional graphs show the signing bonus.

e. DOK-2 How can you differentiate proportional graphs from nonproportional graphs? Proportional graphs start at (0, 0), while nonproportional graphs do not. 5. 6.

Math Chat •

DOK-2 What did you notice about the graphs of the offers with proportional relationships? Both offers do not have signing bonuses, and the graphs begin at (0, 0), or the origin.

FACILITATION TIP After reading the scenario, ask the class 1) What is a signing bonus?; 2) How could looking at pay rates for a given time frame help determine whether organizations were offering signing bonuses? FACILITATION TIP Complete a practice example for a Job Listing, demonstrating how to graph carefully. Students may need to plot more than two points to extend the line with accuracy. Encourage them to use the rulers (or any straight edge) with precision. FACILITATION TIP Call on several students to check for understanding on questions 4a and 4b as students work or after completing Part II.

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.

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PROPORTIONAL RELATIONSHIPS

Proportional Relationships Explore 1 – Proportional vs. Non-Proportional DOK-2 How did you determine which tables have proportional relationships? I found the ratio for each pair of values on the tables. The tables with equivalent ratios have proportional relationships. • DOK-2 What does the point on the y-axis represent? It represents the starting bonus amount. • DOK-2 What does it mean when the starting rate is at (0, 0)? It means that at 0 hours worked, you earn $0. There is no starting bonus. •

FACILITATION TIP On the Exit Ticket, provide extra graph paper for students if they want to graph the job offers with tables or create tables for the job offers with graphs.

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

PROPORTIONAL RELATIONSHIPS

Home

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PROPORTIONAL RELATIONSHIPS

Proportional Relationships Explore 2 – Unit Rates ACTIVITY PREPARATION Students will identify the unit rate in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

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 Exit Ticket (per student)

Plan to divide the class into groups of 3 or 4. Print a Student Journal and an Exit Ticket for each student.

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) Why is it important for coaches to analyze athletes’ performance records?; 2) What types of performance records would runners have?; 3) How might runners’ data be displayed? FACILITATION TIP To support student engagement and comprehension, consider projecting this scenario in print for students to see.

2. 3. 4.

Read the following scenario to the class: Jake accepted an assistant coaching position at Sports Unlimited. He is assigned to analyze the runner records at the track. His first task is to figure out which runner can run the most miles in an hour. He is given the runner records in graphs, diagrams, tables, equations, and descriptions. Help Jake find out which runner can run the most miles in 1 hour. Give a Student Journal to each student. Have students collaborate to find the unit rate for each runner on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

Consider assigning different runners to different groups or individual students. You could split the ten runners in half and assign to each half of the class.

DOK-2 When trying to ﬁnd the unit rate, what is your goal when doing calculations? My goal is to ﬁnd the number of miles for one hour.

FACILITATION TIP

DOK-1 When looking at the number of minutes, what are the easiest values to use to ﬁnd the number of miles per hour? I would use 15 minutes, 20 minutes, and 30 minutes, as they multiply well to go into 60 minutes.

DOK-1 What operation do you use when the number of hours is greater than one? I use division to ﬁnd the number of miles for one hour.

DOK-2 When given equations, how do you ﬁnd the unit rate? The rate is the number multiplied by the number of hours to ﬁnd the number of miles.

FACILITATION TIP

Questions 4a and 4b may need to be addressed before students work independently. Prompt students that they are looking for miles/hour as a unit rate. Review how many minutes are in an hour.

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

Math Chat •

• 104

DOK-2 How is a ratio different from a unit rate? A unit rate is a special ratio that has a denominator of 1 and shows how many units of the ﬁrst type relate to one unit of the second type. DOK-2 How do you determine the unit rate given a ratio? I reduce the fraction to make the denominator equal to 1. © Accelerate Learning Inc. - All Rights Reserved

•

Engage

Explore

Explain

Elaborate

Evaluate

2. 3.

Acceleration

DOK-2 How do you determine the unit rate in an equation? I substituted x with 1 for one hour and found the number of miles. I tried it with other hours, and it was a constant change. DOK-2 For which representation was ﬁnding the unit rate the easiest? I found it easier to ﬁnd the unit rates in tables and graphs because the points are already given to me and I just have to make proportions.

Post-Explore 1.

Intervention

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 Note that the Exit Ticket includes ratios with both minutes and hours. Be sure to clear up any misconceptions about this common ratio before distributing.

PROPORTIONAL RELATIONSHIPS

Home

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PROPORTIONAL RELATIONSHIPS

Proportional Relationships Explore 3 – Proportional Relationships with Equations ACTIVITY PREPARATION Students will ﬁnd the equations of proportional relationships from tables, graphs, and verbal descriptions.

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 Camp Flyers (per group) 1 Set of Camp Fee Cards (per group) 1 Exit Ticket (per student)

•

Reusable •

Plan to divide the class into groups of 3 or 4. Print a Student Journal and an Exit Ticket for each student. Print a set of Camp Flyers for each group. If desired, print them on card stock, and laminate them for future use. Print a set of Camp Fee Cards for each group. Cut them out, and place them in a resealable bag. If desired, print them on card stock, and laminate them for future use.

1 Resealable bag (per group)

PROCEDURE AND FACILITATION FACILITATION TIP

Part I

Before reading the scenario, ask the class 1) How would offering a track and field camp be beneficial to runners?; 2) What activities would take place at a track and field camp?; 3) What would registration fees at a track and field camp cover?

FACILITATION TIP

2. 3.

Note that the Exit Ticket includes ratios with both minutes and hours. Be sure to clear up any misconceptions about this common ratio before distributing the Exit Ticket. FACILITATION TIP Alternatively, project the Camp Flyers to the whole class. FACILITATION TIP Reviewing independent and dependent variable is essential vocabulary for this scope. Post the deﬁnitions in the room, have students copy them on their Student Journal, and ask for several volunteers to share in their own words how they remember the difference between the two. 106

Read the following scenario to the class: Jake’s second assignment as an assistant coach is to set up two-week track and field camps for middle school and high school students! His first task for this assignment is to collect the registration for both camps. Help Jake figure out what the total registration fees are for each camp given the number of students that registered. Give a set of Camp Flyers to each group. Have students look at the Camp Flyers. Discuss the following questions with the class: a.

DOK-1 What is an independent variable? What is the independent variable in this scenario? An independent variable is the variable that is already given in a situation and does not change. In this scenario, the number of students registered for camp is the independent variable.

DOK-1 What is a dependent variable? What is the dependent variable in this scenario? The value of the dependent variable changes depending on the independent variable. In this scenario, the total registration fees collected are the dependent variable.

DOK-1 How can I determine the dependent variable, the total registration fees collected, if I know the independent variable? The value of the dependent variable will be the product of the number of students registered for summer camp times the cost of registration fees.

Give a Student Journal to each student.

Engage

Explore

Explain

Elaborate

Evaluate

As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How would you solve for the total registration fees? I would multiply the number of students by the registration fee.

DOK-2 What does having a value of (0, 0) from the table mean? It means that when you have 0 students register, you also have 0 registration fees.

DOK-1 What value (independent variable) was given in the table? The number of students was already given.

DOK-1 What value (dependent variable) do you need to ﬁnd? We need to ﬁnd the total registration fees.

e. DOK-2 Compare the registration fees for middle school and high school. How does the difference affect the total? The middle school registration fee is $25 cheaper than the high school fee. The total registration fee for middle school will be lower by 25%. 6. 7.

Allow students enough time to complete Part I and answer the reﬂection questions that follow. After Part I, invite the class to a Math Chat to share their observations and learning.

Intervention

Acceleration

FACILITATION TIP Post questions 5a–5d for students to view while they work. Encourage them to be prepared to discuss them during the 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.

PROPORTIONAL RELATIONSHIPS

Home

Math Chat • • • •

•

DOK-1 What value was constant for each camp? The registration fee was equal to the constant of proportionality for each camp. Explain the following to the class: Mathematicians call this value the constant of proportionality. It can also be called the unit rate. DOK-2 What does it mean for the variable to be dependent? The variable is dependent if you need another value to ﬁnd its value. DOK-2 If you were only given the table and not the registration fee, how would you ﬁnd the registration fee per student (constant of proportionality)? I would divide the total registration fees by the number of students for each column and see if the quotients are constant. DOK-4 Give examples of real-life situations with independent and dependent variables. The independent variable is the number of hours worked, and the dependent variable is the amount earned. The independent variable is the number of guests, and the dependent variable is the amount of pizza ordered.

FACILITATION TIP Emphasize to students that constant of proportionality and unit rate may both be used to describe a ratio with a one in the denominator. They can note this on their Student Journal by highlighting “constant of proportionality” and writing “unit rate” above it in Part I.

FACILITATION TIP Before the Math Chat, create your own list of real-life situations that students can relate to.

Part II 1.

2. 3. 4.

Read the following scenario to the class: Now that the students have registered for track and ﬁeld camp, the students will be able to choose which event they want to participate in. Help Jake determine the constant of proportionality and the equations he will need to calculate the total camp fees he will be collecting depending on the number of students. Give a set of Camp Fee Cards to each group. The students will collaborate on ﬁnding the independent variable, dependent variable, and constant of proportionality for each event. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What is the independent variable in each event? The number of students enrolled

DOK-1 What is the dependent variable in each event? The total camp fees

DOK-2 How can you determine if a graph shows a proportional relationship? The graph passes through the origin, (0, 0).

FACILITATION TIP After reading the scenario, ask the class 1) What is the meaning of constant of proportionality?; 2) What do you predict the proportional relationship between the number of students registering for the track and ﬁeld camp and the camp fees collected will be? FACILITATION TIP Post these questions and preview them before students begin working to encourage them to be prepared for the Math Chat.

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PROPORTIONAL RELATIONSHIPS

Proportional Relationships Explore 3 – Proportional Relationships with Equations 5. FACILITATION TIP Take time here to be sure students are able to write the equations and understand the different terms. Slow down to spot check for understanding. FACILITATION TIP Remind students that constant of proportionality can also be called the unit rate. FACILITATION TIP Connect these real-world applications to students by explaining that knowing how to solve for unit rates can help them save money and prevent bad decisions when spending.

Math Chat DOK-1 When looking at a graph or table, how do you find the constant of proportionality? I find the ratios of the dependent variables to the independent variables. I also make sure they are all equal to determine whether they have a proportional relationship. • DOK-1 What is the product of the independent variable and the constant of proportionality equal to? It is equal to the dependent variable. • DOK-2 How is knowing the equation of a graph, table, or verbal description beneficial to real-world situations? Knowing the equation allows us to solve for greater quantities without having to calculate one small quantity at a time. •

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.

108

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

PROPORTIONAL RELATIONSHIPS

Home

109

PROPORTIONAL RELATIONSHIPS

Proportional Relationships Explore 4 – Describe Points on a Graph ACTIVITY PREPARATION Students will explain the different points on graphs showing proportional relationships.

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 Track Meet Scenario Cards (per group) 1 Exit Ticket (per student)

Plan to divide the class into groups of 3 or 4. Print a Student Journal and an Exit Ticket for each student. Print a set of the Track Meet Scenario Cards for each group. Cut out and place each set in a resealable bag for easy distribution. If desired, print them on card stock, and laminate them for durability.

Reusable • • • •

1 Resealable bag (per pair) 1 Box of crayons or colored pencils (per student) 1 Dry-erase marker (per pair) 2 Clear sheet protectors (per pair)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What would a coach need to do to organize a track meet? After reading the scenario, ask the class 2) What information might be graphed that would be important in organizing a track meet?; 3) How might looking at data and graphs help an event run smoothly?

2. 3. 4.

FACILITATION TIP Consider using this Explore activity earlier in the scope. Students may beneﬁt from the review of points on a graph. FACILITATION TIP Post these questions or similar versions for students to view as they work. Being able to answer questions 5a–5g is essential for success with this scope.

Read the following scenario to the class: Jake has been doing a great job as an assistant coach. The head coaches have noticed his efforts and assigned him the lead to organize the fall track meet! This is Jake’s first big event, and he was to get everything organized. He is looking at 4 different scenarios with their graphs to make sure his first major event runs smoothly. Help Jake plan his first track meet! Give a Student Journal to each student. Give a set of Track Meet Scenario Cards to each group. Have students collaborate to answer the questions about each graph in their Student Journals using the Track Meet Scenario Cards. Students should explain each answer using points from the graph. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 Where is the origin on the graph? It is at (0, 0).

DOK-1 How do you determine the x and y values on the graph? The x values are the ones on the horizontal line, and the y values are the ones on the vertical line.

DOK-2 Can you have a point with only one number (for example, point 3)? No, each point is always an ordered pair, (x, y).

DOK-2 How do you ﬁnd the rate from a point? I compare or make a y fraction from __x .

e. DOK-2 How do you determine the unit rate from the points on the graph? I ﬁnd the y value when the x value is 1. 110

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How do you ﬁnd the unit rate if the point where x = 1 is not plotted on the graph? I look for the other points and divide the y and x to get the value when the denominator is 1.

DOK-2 How do you use the graph to determine the value that is not graphed? I ﬁnd the unit rate from the points on the graph and use the unit rate to ﬁnd other values on the graph.

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

Math Chat DOK-2 What does the origin mean in the graphs from the Track Meet Scenario Cards? Having the point (0, 0) on the graphs means that if none of the schools participated, none of the student athletes would have attended or would have been given any awards. Guests would not have been present, either. Having 0 events also meant 0 minutes for each event. • DOK-2 What point on the graph shows the unit rate? The unit rate is always when the x value is at 1. • DOK-2 How is knowing the values of the points on the graph of proportional relationships beneficial in the real world? Knowing the values of points on the graph helps us when we need to determine the unit rate or the other rates that depend on time, number of people, and other factors. • DOK-2 How do you find the number of students, guests, awards, or minutes if it is not shown on the graph? I would find the unit rate, or constant of proportionality, and create an equation or table to help me find the other values. •

Intervention

Acceleration

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.

PROPORTIONAL RELATIONSHIPS

Home

FACILITATION TIP Continue to remind students that a unit rate can also be called constant of proportionality.

Post-Explore 1. 2. 3. 4.

FACILITATION TIP Using this Exit Ticket and Explore activity earlier in the scope may help you detect some student misconceptions.

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RATIONAL NUMBER OPERATIONS

Rational Number Operations 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

Convert between Forms 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 with Complex Fractions 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 Rational Number Operations 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

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

RATIONAL NUMBER OPERATIONS

Home

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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

RATIONAL NUMBER OPERATIONS

Rational Number Operations

3 114

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 explain that unit rate and constant of proportionality are the same.

What prompts will be used?

What does mastery look like?

RATIONAL NUMBER OPERATIONS

Home

I can analyze proportional reasonings using the graph on a coordinate plane.

I can analyze proportional reasonings by determining whether a graph represents a straight line that passes through the origin.

I can use verbal descriptions, tables, graphs, and equations to identify, represent, and use proportional relationships.

I can create a representation of a proportional relationship from another given representation.

I can use tables and graphs to model the constant rates in mathematical relationships.

I can represent proportional relationships by using equations.

I can explain the meaning of a point on the graph of a proportional relationship in terms of the situation.

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SCOPE 1

Understand Slope Scope Introduction SCOPE SUMMARY Students develop an understanding of slope using similar right triangles with hypotenuses on the same line graphed on a coordinate plane. They relate slope to unit rate by graphing proportional relationships. Students are able to determine the rate of change or slope from tables and graphs.

VERTICAL ALIGNMENT

Student Expectations

7.PAR.4.7 Use similar triangles to explain why the slope, m, is the same between any two distinct points on a nonvertical line in the coordinate plane. 7.PAR.4.8 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

Background Knowledge

Future Expectations

In previous grade levels, students build foundational knowledge by acquiring a conceptual understanding of fractions and decimals. In sixth grade, students develop a foundation for understanding proportions through the development of ratio and rate reasoning and part-whole computational strategies related to fractions, decimals, and percents.

Students will build on this foundation and extend their knowledge of rate of change and slope as they begin to write proportional and non-proportional linear equations from verbal descriptions, tables, and graphs. As students progress into high school algebra, they will determine the rate of change or slope given two points on a line or equations written in various forms.

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

•

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES

solve problems involving proportions using a variety of student-selected strategies. solve unit rate problems including those involving unit pricing and constant speed. identify two truths and a lie by reading statements about the prior standard.

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

determine the yy-intercept -intercept of several examples and ﬁnd the one that is not equivalent.

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.

116

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Similar Triangles In this exploration, students will use similar triangles to explain the slope of the lines. Students will: •

use similar triangles to understand why the slope is the same between any two points.

Explore 2

Explore 1

EXPLORE ACTIVITIES Determine the Rate of Change In this exploration, students will calculate the rate of change from a graph, table, linear function, or verbal description using the rate of change formula. 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 tables and graphs to ﬁnd the rate of change.

•

use the rate of change formula to compute the rate of change in a variety of situations.

UNDERSTAND SLOPE

Home

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. Slope In this exploration, students will derive the slope-intercept form of an equation by using ordered pairs to identify m and b in an equation. Students will: •

compare proportional relationships represented in different ways.

Students conclude the exploration with an Exit Ticket and revisit the Hook to solve using their learning from the explorations. Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________

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UNDERSTAND SLOPE

Understand Slope Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

118

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

UNDERSTAND SLOPE

Home

6.NR.4.3 Solve problems involving proportions using a variety of student-selected strategies. 6.NR.4.5 Solve unit rate problems including those involving unit pricing and constant speed.

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.

FACILITATION TIP

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 ﬁrst statement is the lie because the unit rate of the donut in the dozen is $0.75.

Consider projecting the Two Truths and a Lie to the whole class rather than distributing to groups. Reveal the prompt and the statements one at a time and conduct a read aloud. After revealing the whole page, allow for thinking and shoulder partner talk time. FACILITATION TIP Clarify that students know what a dozen is. Some students may also wonder if this prompt is referring to a “baker’s dozen.”

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 yy-intercept -intercept with x-intercept. -intercept. Students may not understand that rates consist of two different units; in this case the units are donuts and dollars (12 donuts per $9). Notes

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Understand Slope Hook – Treat Training ACTIVITY PREPARATION Students will determine the yy-intercept -intercept of several examples and ﬁnd the one that is not equivalent.

Materials

Preparation

Printed •

• • •

1 Treat Training (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Treat Training 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 Part I: Pre-Explore 1. FACILITATION TIP

Print and project the text of this scenario and read it aloud with students.

FACILITATION TIP To encourage engaging observations, project Treat Training before asking the notice, wonder, and math questions.

4. 5.

FACILITATION TIP Before this Explore activity, rate of change, slope, and slope-intercept equation will likely be new concepts for many students. Consider encouraging students to just make observations about the different representations.

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: Jerry was given a present of dog treats from the local dog bakery. He decided to use the treats to train his puppies. The results were very positive, so he decided to buy a set number of treats each week to use as he continues to train his puppies in more advanced tricks. Jerry decided he should keep track of his purchases. He tried using several different models to see which worked best for him. However, he made a mistake with one of the models. When his sister told him one of the models didn’t represent his data, he studied them all to see which model did not belong. Show the Phenomena Video. Ask the class 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. Answers will vary. I notice that Jerry is using different models to represent data. I wonder what types of models Jerry is using. How many treats is Jerry using every week? I can use math to determine the y-intercept for the data. Project Treat Training. Explain to students that Jerry was originally given a pound of dog treats. Because 1 they worked so well for dog training, he began buying __2 of a pound of dog treats each week. Discuss the following questions: a.

DOK-1 What are the ways this data is represented? It is represented in a table, a story, a slope-intercept equation, and a line graph.

DOK-1 What is the rate of change? Accept all reasonable answers. Rate of change is equivalent to the slope. It is the ratio of the vertical change divided by the rate of horizontal change.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and repeat the scenario. Refer to Treat Training, and discuss the following questions: a.

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DOK-1 In order for the different models to be equivalent, what must be the same in the different models? The slope (or rate of change) has to be the same. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 What is the origin? The origin is the point (0, 0) on the coordinate plane.

DOK-1 What is the correct rate of change (or slope)? The rate of change 1 is __2.

DOK-1 Which of the models is incorrect? Explain. Answer choice 3 is incorrect because it includes the wrong slope. The correct slope1 1 intercept equation is y = __2x, not y = –__2x.

Intervention

Acceleration

FACILITATION TIP Use this opportunity of restating the problem to have students identify what they are being asked to ﬁnd. If needed, direct them to the phrase, “They wondered how much the temperature had dropped and what the current temperature was.”

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Understand Slope Explore 1 – Similar Triangles ACTIVITY PREPARATION Students will use similar triangles to explain the slope of the lines.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. 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 Similar Triangles Cards (per group) 1 Exit Ticket (per student)

• • •

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 one set of Similar Triangles Cards for each group. Cut out and place each set of cards in a resealable bag. If desired, print them on card stock, and laminate them for future use.

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION FACILITATION TIP Project the scenario in word form and read it aloud together. Engage students with some images of public transport in New York and some familiar local cities. FACILITATION TIP Quickly review horizontal and vertical if needed before distributing the Student Journal FACILITATION TIP Depending on your students’ prior experience with calculating slope using graphs, consider modeling the process using one of the Similar Triangle Cards.

2. 3. 4.

FACILITATION TIP

Read the following scenario to the class: Isabella and her friends are recording the amount of money spent on public transportation in New York. They will record their information on graphs. Help Isabella and her friends ﬁnd out how much money is spent on public transportation. Give a Student Journal to each student. Give a set of Similar Triangles Cards to each group. Explain to students that they will collaborate with their groups on how to use similar triangles to understand why the slope, m,, is the same between any two points. Have students use the information on the Similar Triangles Cards to ﬁnd the ratio of the length of the vertical side to the length of the horizontal side. They will also determine if the slope is positive or negative. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-2 What do the ratios of the vertical side to the horizontal side represent? Student responses will vary. The ratios represent the slope, which is the ratio of the number of units that represent the vertical side of the triangle to the number of units that represent the horizontal side between any two points.

DOK-2 What method can you use to ﬁnd the slope? Student responses Vertical side will vary. I can use the formula m = ____________.

Some students may be familiar with the phrase “rise over run” to describe the ratio.

7. 8. 122

Horizontal side

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Math Chat DOK-2 What can you conclude about the slope between any two points on a line? The slope of a straight line between two points (x1, y1) and (x2, y2) can be easily determined by ﬁnding the difference between the coordinates of the points. • DOK-2 How can you determine if a slope is positive or negative? If a line has a positive slope (m > 0), then y always increases when x increases and y always decreases when x decreases. So, the graph of the line starts at the bottom left and goes toward the top right. If a line has a negative slope (m < 0), then y always decreases when x increases and y always increases when x decreases. •

Post-Explore 1. 2. 3.

FACILITATION TIP

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When discussing how to determine if a slope is positive or negative, include some visual references. Students will need to be prompted to see the connection between the increasing and decreasing values of x and y and the line created on the graph.

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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Understand Slope Explore 2 – Determine the Rate of Change ACTIVITY PREPARATION Students will calculate the rate of change from a graph, table, linear function, or verbal description using the rate of change formula.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. 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 Food Bank Volunteer Cards (per group) 1 Rate of Change Work Mat (per group) 1 Rate of Change Card (per teacher) 1 Exit Ticket (per student)

• • •

•

Reusable • • • •

1 Dry-erase marker (per group) 2 Resealable bags (per group) 1 Clear sheet protector (per group) 1 Projector or document camera (per class)

• • •

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 a set of Food Bank Volunteer Cards for each group. Cut out and place each set of Part I cards in a resealable bag labeled “Part I.” Cut out and place each set of Part II cards in a resealable bag labeled “Part II.” If desired, print them on card stock, and laminate them for future use. Print a Rate of Change Work Mat for each group. Place each work mat in a clear sheet protector to create an erasable surface. If desired, print it on card stock, and laminate it for future use. Gather a dry-erase marker for each group. Print a Rate of Change Card for each teacher. If desired, print it on card stock, and laminate it for future use. Be prepared to project the Rate of Change Card for the class to see.

PROCEDURE AND FACILITATION FACILITATION TIP Project the scenario and read it along with students. Consider reading through it more than once and coaching students to locate the speciﬁc information that will be included on the tables and graphs.

Part I: Determining Rate of Change from a Table and a Graph 1.

2. 3. 4.

FACILITATION TIP Consider modeling how to use the Student Journal with the Food Bank Volunteer Cards. Students may need more support with rate of change and input/output tables to successfully collaborate with each other. 124

Read the following scenario to the class: The Young Leaders Community Service Organization will now have an opportunity to volunteer at the local food bank. The food bank has decided to create tables and graphs to show the number of hours each volunteer works at the food bank and will also include information that shows the monetary donations and canned goods received while volunteers work. Help Javier use the tables and graphs to determine the rate of change. Give a Student Journal to each student. Give a set of Part I Food Bank Volunteer Cards, the Rate of Change Work Mat, and a dry-erase marker to each group. Explain to students that they will explore the rate of change for a linear function. They will use tables and graphs to ﬁnd the rate of change for the food bank volunteer hours. They will use the rate of change formula to compute the rate of change in a variety of situations. Project the Rate of Change Card. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 What does the term rate of change mean? (Note that students should understand the three formulas that can be used to ﬁnd the rate of change.) Student responses will vary. The ratio of the amount of change in the output to the amount of change in the input © Accelerate Learning Inc. - All Rights Reserved

8. 9.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How do you find the rate of change from a table? Student responses will vary. Choose two pairs of x and y values. Find the 2 difference in y values divided by the difference in x values. __1 = 2 DOK-2 How do you find the rate of change from a graph? graph? Student responses will vary. The rate of change formula can be used to find two points on the graph and find how far apart the two points are vertically 2 and then horizontally. __1 = 2

Intervention

Acceleration

FACILITATION TIP Post the answers to questions 5a and 5b. on a word wall or chart for students to refer to throughout this scope. Some students may be very skilled at using a table, but need support using a graph to calculate rate of change.

UNDERSTAND SLOPE

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Have students ﬁnd the rate ate of change using the tables and graphs that represent the food bank volunteer hours and the monetary donations and canned goods the food bank receives. Encourage students to use the Rate of Change Work Mat. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 What is the rate of change in this situation? Student responses will vary. The rate of change is 2. The number of donations increased by 2 for every hour spent volunteering at the food bank.

DOK-2 What are the independent and dependent variables? The number of volunteer hours is the independent variable, and the dependent variable is the number of donations.

Explain the following to the class: You worked with determining the rate of change today, and this is also known as “slope.” Slope is the rate of change between any two points on a line. It is the measure of the steepness of a line. Allow students enough time to complete their work and record their observations and reﬂections on their Student Journals. After Part I, invite the class to a Math Chat to share their observations and learning.

FACILITATION TIP Independent and dependent variables are likely new terms and may require explicit instruction or review.

Math Chat DOK-2 What happens when the rate of change is positive? The linear function is increasing, which shows that as the x value increases, the y value increases. • DOK-2 What happens when the rate of change is negative? The linear function is decreasing, which shows that as the x value increases, the y value decreases. • DOK-2 How do you choose the points to calculate the rate of change from graphs and tables? You can choose any two points. The rate of change is constant, so it does not matter which two points are chosen. •

FACILITATION TIP Use Picture Vocabulary to introduce linear function.

Part II: Determining Rate of Change Using Verbal Descriptions and Linear Functions 1.

2. 3.

Read the following scenario to the class: Javier is still keeping track of volunteer hours, canned goods, and monetary donations that are sent to the food bank. He will now use linear functions and verbal descriptions to represent data from the food bank. Help Javier use the data to determine the rate of change. Give a set of Part II Food Bank Volunteer Cards to each group. Explain to students that they will use verbal descriptions to create linear functions and will determine the rate of change. (Note that students should understand that the equation of a straight line is a linear function.) Have students use verbal descriptions to create linear functions and determine the rate of change as it relates to donations, supplies, and volunteer hours at the food bank. a.

DOK-1 What is the equation of a straight line? What is another name for this linear function? y = mx; Another name for this line is the slope-intercept form of a line.

DOK-1 What does m represent? m is the slope.

DOK-2 How can you determine the slope when writing the linear function? I can determine the slope by identifying the constant rate of change.

FACILITATION TIP Linear functions and equations are new concepts for most students. Consider using the Skill Review and Practice (four pages) under Intervention as needed.

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UNDERSTAND SLOPE

Understand Slope Explore 2 – Determine the Rate of Change 5. 6. 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.

Math Chat •

•

DOK-2 Does a proportional relationship have a constant rate of change? Explain. Yes, in proportional relationships, k (constant of proportionality) is equivalent to m (constant rate of change). DOK-2 What is an example of rate of change used in a real-world situation? Buying ice cream at a set price and paying additional money for toppings

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

UNDERSTAND SLOPE

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UNDERSTAND SLOPE

Understand Slope Explore 3 – Slope ACTIVITY PREPARATION Students will derive the slope-intercept form of an equation by using ordered pairs to identify m and b in an equation. Students will compare proportional relationships represented in different ways.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. 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 Slope Scenario Cards (per group) 1 Exit Ticket (per student)

• • •

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 a set of Slope Scenario Cards for each group. Cut out and place each set of cards in a resealable bag. If desired, print them on card stock, and laminate them for future use.

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION Part I 1.

2. 3. 4. FACILITATION TIP Use Picture Vocabulary and Skill Review and Practice (four pages) under Intervention as needed. Student knowledge of y-intercept, and the abstracted forms of m and b may be limited.

FACILITATION TIP

Read the following scenario to the class: Isabella and her friends are recording the amount of money spent on renting bikes in New York. They will record their information on graphs. Help Isabella and her friends find out how much money is spent on bike rentals in New York. Give a Student Journal to each student. Give a set of Slope Scenario Cards to each group. Explain to students that they will collaborate with their groups on how to derive the slope-intercept form of the equation and will understand when slopes represent a proportional relationship . Have students use the information on the Slope Scenario Cards to find m, which is the ratio of the change in y to the change in x.. They will also find b and write an equation that represents the situation. Monitor and assess students as they are working by asking the following guiding questions: a.

Struggling students may need to be directed to create an input output table from a graph in order to write an equation.

DOK-2 What do the ratios of the change in y to the change in x represent? Student responses will vary. The ratios represent the slope, which is the ratio of the change in y to the change in x between any two points. DOK-2 What method can you use to ﬁnd the slope? Student responses Change in Y

will vary. I can use the formula __________. c.

7. 128

Change in X

DOK-2 How can you identify the y-intercept? Student responses will vary. I can identify the y-intercept on the graph because it is the point where the line crosses the y-axis.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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

Math Chat DOK-2 Why is y = mx + b called the slope-intercept form of the equation of a line? In the equation of a straight line (when the equation is written as y = mx + b), the slope is the number, m, that is multiplied by x, and b is the y-intercept (the point where the line crosses the vertical y-axis). • DOK-2 How do you know whether a graph shows a proportional relationship? A graph shows a proportional relationship when the line goes through the origin, (0, 0). • DOK-2 What is a real-world example of slope? Renting a bike and charging $10 per hour represents a proportional relationship. •

FACILITATION TIP Slow down and identify all of the parts of this standard y = mx + b equation for students. It is a key concept for later success with Algebra.

UNDERSTAND SLOPE

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Part II 1.

4. 5.

Read the following scenario to the class: Isabella and her friends are curious about doing some different excursions in New York. She contacted a few different companies to compare prices. Help Isabella and her friends ﬁnd out which company offers the best price for each excursion. Have students ﬁnd the slope and unit rate of the tables and graphs. Once they’ve created an equation, they can compare the information to determine which excursion has the better price. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-2 How can you compare a graph to a table? Once you ﬁnd the unit rate, you can record that in an equation so they’re both written in the same form.

DOK-2 How do you ﬁgure out if a table has the point (0, 0)? You can count backward by the same amount being counted forward in order to see if the table would reach the point (0, 0).

FACILITATION TIP Depending on your students, consider using Part II tables and graphs to demonstrate step by step how to calculate unit rate and slope. In addition, model how to write a linear equation from a table or graphs.

Math Chat •

•

DOK-2 How does the slope of the line compare to the unit rate on the graph? You ﬁnd the slope of the line and the unit rate the same way. You ﬁnd the change in y and divide it by the change in x. DOK-2 How do you know whether a table is proportional? A table is proportional when it has the point (0, 0) on it. You may need to calculate whether the table will reach the point (0, 0).

FACILITATION TIP Unit rates are a focus in the next scope; take time to clarify.

Post-Explore 1. 2. 3. 4.

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UNDERSTAND SLOPE

Understand Slope 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

Similar Triangles 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

Determine the Rate of Change 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

Slope 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

130

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

Rate of Change and Slope

UNDERSTAND SLOPE

Home

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

Interactive Practice First Contact A game to practice the skills established by the standards in the scope

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UNDERSTAND SLOPE

Understand Slope 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

3 132

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

UNDERSTAND SLOPE

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What does mastery look like?

I can explain why the slope is the same between any two distinct points using proportional reasoning.

I can demonstrate that two slopes are the same by using similar triangles.

I can represent proportional relationships using graphical reasoning.

I can determine the unit rate when given the slope of a graph.

I can compare different representations of proportional relationships.

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SCOPE 1

Ratios, Rates, and Percents Scope Introduction SCOPE SUMMARY In this scope, students will be able to determine the unit rate of fractional ratios pertaining to real-world situations. They will be able to solve multistep ratio and percent problems through the use of proportional relationships. By the end of the scope, students will understand simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, and percent error. Student Expectations

7.PAR.4.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units presented in realistic problems. 7.PAR.4.9 Use proportional relationships to solve multistep ratio and percent problems presented in applicable situations.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In Grade 6, students learned about how ratios and rates can help to solve real-world and mathematical problems. They used tape diagrams, number line diagrams, and tables to determine the unit rate associated with a ratio. Students learned the correct language and terms to be able to write and understand ratio word problems.

In an upcoming scope in 7th grade, students will continue to work with multistep ratios and percents to ﬁnd tax, tip, commission, markups and markdowns, simple interest, and percent error. Students will continue to work with proportional relationships such as these through 8th grade and high school. They will dive deeper into proportional relationships using tables, graphs, equations, and diagrams throughout 7th grade and beyond. Proportional relationship skills will continue to develop as students work with percents as they discover statistics. They will use the idea of ratios to determine the probability that something will occur.

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

use and ﬁnd ratios, rates, and percents.

•

reason and solve problems.

•

examine a series of tables and determine which option does not belong with the group.

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 ratios, rates, and percentages.

•

use multiple models.

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. 134

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Unit Rates with Ratios of Fractions In this exploration, groups of students will solve a realworld scenario where they help Travis, a production assistant on a reality show, decipher baker’s recipes to determine how much ﬂour each baker will need for their cupcakes. Students will: •

compute unit rates associated with ratios of fractions in like or different units.

•

analyze graphs, tables, equations, and diagrams to see various ways to discover unit rates.

Explore 2

Explore 1

EXPLORE ACTIVITIES

•

apply their knowledge of ratios and percents.

•

solve multistep word problems.

Explore 4

Explore 3

In this exploration, students will help Travis save money by ﬁnding the best prices at stores for various baking items needed for the show. Then, determine the constant of proportionality and the equations needed to calculate the total camp fees he will be collecting. Students will:

In this exploration, students will solve an extension from the scenario in the ﬁrst exploration. Here, students help bakers in the competition convert their ideas into measurements for determining the exact size of each cake they are baking. Students will: •

use ratios to solve proportions to ﬁnd length or width.

•

use the dimensions to calculate area.

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. Multistep Ratio Problems

Ratios of Length and Area

RATIOS, RATES, AND PERCENTS

Home

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

Solve Problems – Percents In this exploration, students will be guided through various concepts involving ratios, rates, and percents; and, analyze proportional relationships to solve problems. Students will: •

compute unit rates related to ratios of fractions and other units.

•

use proportional relationships to solve problems that are multistep and involve ratios and percents.

Students conclude the exploration with an Exit Ticket and revisit the Hook to solve using their learning from the explorations.

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RATIOS, RATES, AND PERCENTS

Ratios, Rates, and Percents 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 examine a series of tables and determine which option does not belong with the group. This element is designed to uncover student misconceptions, and it should not be used as a summative assessment. 6.NR.4.3 Solve problems involving proportions using a variety of student-selected strategies. 6.NR.4.6 Calculate a percent of a quantity as a rate per 100 and solve everyday problems.

Materials

Preparation

Printed •

•

1 Does Not Belong (per student or per group)

•

RATIOS, RATES, AND PERCENTS

Home

If not assigning the APK digitally, print one Does Not Belong for each student. Plan to assign students to groups of two or three, optionally.

Procedure and Facilitation Points 1. 2. 3.

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 there is 1 red flower for every 2 yellow flowers, not 1 yellow flower for 2 red flowers.

FACILITATION TIP Consider completing at least one part of this APK together as a class. Use Set 1 to review the three ways to set up and view ratios. Have students copy the original ratio in Set 1: red to 6 yellow (___ , 6 to 12, and 6:12) before they look 12 for equivalent ratios.

b. Set 2: D does not belong because the cost of 2 slices of pizza is not $4.00; it is $5.00. c.

Set 3: B does not belong because there are not 0.02 of an ounce in 1% of the bag; there are 0.2 of an ounce in 1% of the bag.

d. Set 4: A does not belong because there is not 1 bee for every grasshopper; there is 1 grasshopper for every two bees. 4. 5.

Identifying Misconceptions • • • •

FACILITATION TIP This APK provides excellent visual mathematical tools. For some students’ using tape diagrams, double number lines, and tables is essential for success with this scope. Take time to identify which students are able to represent ratios with these tools.

Students may reverse the terms within a ratio. Students may not understand how to ﬁnd the unit rate per 1%. Students may not distinguish between part-to-part and part-to-whole ratio relationships. Students may not know how to make use of a table, tape diagram, or double number line when representing ratio relationships.

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RATIOS, RATES, AND PERCENTS

Ratios, Rates, and Percents Hook ACTIVITY PREPARATION Students will determine ratios, rates, and percents by using multiple models.

Materials

Preparation

Reusable •

• •

1 Phenomena Video (per class)

Plan to show the video. Prepare to introduce the scenario and to encourage students to think about how to represent 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) How are cupcakes made?; 2) Do all types of cupcakes use the same ingredients?; 3) What would need to be done to a cupcake recipe to make different serving sizes of that cupcake?

FACILITATION TIP In addition to presenting the video, consider showing students a set of measuring cups with labels. Perhaps tell a story about a time when you miscalculated when doubling or halving a recipe.

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: Baking involves many different measurements. The Bake Shop wants to produce different portion options of different baked goods that they sell. The ratio of ingredients will need to stay 1 consistent so that the taste stays the same. Emmi ﬁnds that the recipe calls for __2 1 __ of a cup of ﬂour and 4 of a cup of sugar.

Ask students the following fol questions: What do you notice? What do you wonder? Where can you see math in this situation? Allow students to share all ideas. Answers will vary. I notice that there are different cupcakes that will require using different recipes. I wonder how the recipe would change based on the serving size. Explain to students that when you are baking, you have to use the correct measurements that are required for the recipe. Discuss the following questions: a.

DOK-1 Where do you see the math in baking? Allow students to share all ideas. Answers will vary. Baking requires measuring the ingredients and calculating the portions of ingredients to use based on the serving size.

DOK-1 Where might you ﬁnd ratios in baking? Ingredients are deﬁned in parts that are relative to each other in quantity. By using ratios, we can easily scale baked goods to the desired quantity, which means adjusting the ingredients based on the serving size.

FACILITATION TIP Project a copy of a recipe so students can see the fractions, mixed numbers, and whole numbers. Challenge students to consider how they might make half, double, or triple the recipe. Accept all ideas and return to this recipe after the Explore activities. FACILITATION TIP As a follow-up question, consider asking students to demonstrate or explain how serving sizes are related to ratios.

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Complete the Explore activities.

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and repeat the scenario. Refer to the video, and discuss the following questions: a.

DOK-1 Do portions and serving sizes make more sense after the Explore activities? Yes, this is called using ratios.

DOK-1 How can you represent ratios? Ratios represent quantities of the same unit. I can use double number lines, graphs, tables, and tape diagrams.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How much ﬂour and sugar would you need to make four times a recipe? You would need 2 cups of ﬂour and 1 cup of sugar.

e. DOK-1 Do you feel that you have a strong understanding of ratios, rates, and percents? Answers will vary based on students’ success during the activity and their conﬁdence levels.

Intervention

Acceleration

FACILITATION TIP For question 2e, allow students to use hand signals to share with you how strong their understanding of ratios, rates, and percents measures currently. Thumbs up (I’ve got this!), thumbs sideways (not sure), or thumbs down (still need help).

Notes

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Ratios, Rates, and Percents Explore 1 – Unit Rates with Ratios of Fractions ACTIVITY PREPARATION Students will compute unit rates associated with ratios of fractions in like or different units. They will analyze graphs, tables, equations, and diagrams to see various ways to discover unit rates.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.5 Use appropriate tools strategically.

Materials

Preparation

Printed • • •

• • •

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

Plan to divide the class into groups of 3 or 4 to complete this activity. Print a Student Journal and an Exit Ticket per student. Print a set of Recipe Cards for each group. Cut out the Recipe Cards, and place each set in a resealable bag. If desired, print them on card stock, and laminate them for future use.

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION 1. FACILITATION TIP Before reading the scenario, ask the class 1) What happens during a reality show?; 2) What ingredients are needed to make cupcakes? After reading the scenario, ask the class 3) Why is using the right amount of ﬂour important when making cupcakes?

Read the following scenario to the class: Travis is a production assistant on the reality competition show Ready, Set, Bake. In this week’s episode, the contestants will be making cupcakes. The baker with the most delicious cupcake will win this round. Each baker has requested very speciﬁc ingredients that Travis is responsible for supplying. Your job is to help Travis decipher each baker’s recipe to determine how much ﬂour each baker will need for a batch of cupcakes. Review fractions, rates, and ratios with students. a.

DOK-2 How is a fraction similar to a ratio? How is it different? Both fractions and ratios show amounts and use a fraction bar. A fraction has a numerator and denominator and usually shows a part-to-whole relationship. A ratio can be written like a fraction. It can show a partto-whole relationship and a part-to-part relationship. It’s important to deﬁne what each part of the ratio represents.

DOK-1 What is a unit rate? A rate that compares two different things, like how far you travel in a certain amount of time (miles per hour). One of the numbers in the comparison has to be 1.

FACILITATION TIP This review of the meanings of fraction, ratio, and unit rate can be copied on the Student Journal. Remind students that a unit rate is also called a constant of proportionality. FACILITATION TIP Students may notice that these recipe cards include very few ingredients. Note that most recipes have several ingredients, but in this case each baker is only requesting ﬂour.

3. 4.

FACILITATION TIP

Take time to complete page one all together as suggested in Step 5. Select some examples from students to show how they doubled both values (Be sure to continue to review the words double, triple, quadruple, halve, etc in this scope). 140

Give a Student Journal to each student. Give a set of Recipe Cards to each group. Explain to students that they will be working with their small groups to determine the amount of ﬂour that each baker will need for one batch of cupcakes. Students will use the various prompts in their Student Journals to ﬁnd ratios and unit rates. Complete page one of the Student Journal as a class. Guide the students with the following questions: a. b.

DOK-1 According to the Recipe Card, how much ﬂour is Kiana 1 1 requesting? She is requesting __8 cup of ﬂour per __4 batch of cupcakes. 1 1

DOK-1 How can this s be written as a ratio? __8 : __4

9. 10.

Engage

Explore

Explain

Elaborate

Evaluate 2

Intervention

Acceleration

DOK-1 If Kiana doubles both values, what will the new values be? __8 or __4 2 1 cups of ﬂour per __4 or __2 batch of cupcakes

DOK-1 How can you write rite these new values as a ratio? __4 : __2

1 1

Guide students to continue the double number line by extending the pattern created in the first two sections of their Student Journals. Encourage students to continue the pattern until they find the number of cups of flour needed for one batch of cupcakes. Assist as needed. Allow students to work cooperatively with their small groups to complete the remainder of the Recipe Cards and their Student Journals, including the reflection questions. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 How do you determine when to increase your values or decrease your values when ﬁnding the unit rate? First, determine the starting value of the unit for which you want to ﬁnd the unit rate. If it is less than one, you must increase both values until you get to one. If it is greater than one, you must decrease both values until you get to one.

DOK-1 What operation does a fraction represent? A fraction represents a division of two numbers.

DOK-3 How do you solve complex fractions? Divide the fractions by rewriting the fraction as the numerator multiplied by the reciprocal of the denominator.

DOK-2 How do you know which value to place on the denominator of the complex fraction? Whichever value needs to be the singular unit rate will be the denominator. The goal is to end with a denominator of 1.

FACILITATION TIP Ensure that students can use the double number line and create a table. Viewing the mathematical relationships creates concrete understanding that applies to many real-world ratio problems.

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FACILITATION TIP Project questions 8b and 8c to guide students’ collaboration discussions. Direct them to be ready to share their answers during the Math Chat.

Allow time for students to complete their Student Journals, including the reﬂection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat • •

•

DOK-1 List three ways to find a unit rate. Answers may include tables, double line diagrams, graphs, equations, and number line models. DOK-1 How can tables and graphs help to solve problems of unit rate with ratios of different units? Graphs and tables show comparisons of two different things. When you have two different units, tables and graphs show a good representation of how one unit compares to the other. DOK-2 Why does it make sense to simplify a fraction when computing unit rate? Relate your answer to the context of the problem. It is easier to compare unit rates in simplest form. In terms of the context of the problem, it would make more sense to use one scoop of a one-half cup rather than use four scoops of a one-eighth cup when baking. DOK-2 When dealing with the division of fractions, describe why solving a mathematical expression is easier than drawing a diagram. Sometimes creating double number lines or tables can be difficult if the numbers you have don’t multiply or divide into easy mathematical numbers. Dividing fractions gives you an answer without determining all of the in-between values. DOK-2 What are the benefits of creating tables? They allow you to easily see and compare all ratios, not just the unit rate.

FACILITATION TIP Use real measuring cups to demonstrate how it makes more sense to use simpliﬁed fractions when baking (and for other realworld activities as well). FACILITATION TIP These last two discussion points are critical to helping students learn to use appropriate tools and strategies (MP.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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Ratios, Rates, and Percents Explore 2 – Ratios of Length and Area ACTIVITY PREPARATION Students will use ratios to solve proportions to ﬁnd length or width. They will use the dimensions to calculate area.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.5 Use appropriate tools strategically.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Baking Notes (per group) 1 Exit Ticket (per student)

Plan to have students work in groups of 2 or 3 to complete this activity. Print a Student Journal and an Exit Ticket per student. Print one set of Baking Notes per group. Cut out the Baking Notes, and place each set in a resealable bag. If desired, print them on card stock, and laminate them for future use.

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What would make a cake “impressive”?; 2) Other than round cakes, what other shapes and sizes of cakes have you seen?; 3) How do you think cake bakers take their ideas for a cake and make them a reality?

FACILITATION TIP

Procedure and Facilitation Points 1.

2. 3. 4.

Determine whether students will need calculators for this Explore activity and Exit Ticket. FACILITATION TIP Depending upon student understanding, provide time for students to copy notes about deﬁnitions and examples for ratio and proportion on their Student Journal. Be sure to include a review of the formula for Area of a Rectangle. FACILITATION TIP Reading the ratio out loud or using letters to represent the values on their Student Journals will help students prevent these errors. Note that the Student Journal Answer Key has width/length written out under ratio on the Baking Notes. 142

5. 6.

Read the following scenario to the class: In this week’s episode of Ready, Set, Bake, the contestants will be making beautiful cakes. The baker with the most impressive cake will win the round. Each baker has worked out their design and has some notes to help them remember the sizing that will work best with their design. Your job is to help the bakers convert their ideas into measurements to determine the exact size for each cake. Give a Student Journal to each student. Give a set of Baking Notes to each group. Review ratios and proportions with students. a.

DOK-1 What is a ratio? Student responses will vary. A ratio can show a part-to-whole relationship and a part-to-part relationship.

DOK-1 How can you write a ratio? Student responses will vary. You can use “per,” “to,” a colon, or /.

DOK-1 What is a proportion? Student responses will vary. A proportion is a comparison of two equal ratios.

DOK-1 How do you set up a proportion? Student responses will vary. Set up the preferred ratio and the desired length as a proportion.

Explain to students that they will be working cooperatively with their small groups to help the bakers turn their notes into measurements for their cakes. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 Why is it important to represent what each number represents using words when you write a ratio? Student responses will vary. If you do not represent what each number in the ratio represents correctly, you could make an error when solving the proportion. © Accelerate Learning Inc. - All Rights Reserved

7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

DOK-1 Does it make a difference if you are solving proportions that include complex fractions? Student responses will vary. No, it does not make a difference. You will still set up and solve proportions the same way you would if the proportions included whole numbers.

Allow time for students to complete their Student Journals, including the reﬂection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 How did you solve for the missing value in the proportion? I found the scale factor using the values that were given (if given the length, solved for the width; if given the width, solved for the length) and multiplied the scale factor times the given value to solve for the missing value. • DOK-2 Compare and contrast ratios and proportions. Ratios and proportions both compare things. A ratio is a comparison of two things that shows their size in relation to each other. A proportion is a comparison of two ratios. The ratios are equal. • DOK-2 Which baker will create the longest cake? Tai’s cake is 36 inches long, Valerie’s cake is 15 inches long, Kiana’s cake is 21 inches long, and Neri’s cake is 45 inches long. Neri has the longest cake. • DOK-2 How can you determine the length-to-area ratio for Kiana’s cake? After you have solved for the missing dimensions, you can use the area of a rectangle formula to solve for the area since each cake is shaped like a rectangle. Once you solve for the area, then include the length : area ratio. Kiana’s cake’s length:area ratio is 21 : 294.

FACILITATION TIP Rather than having students write detailed answers for these reﬂection questions, use Think, Pair, Share as a strategy and ask for volunteers to explain to the whole class.

•

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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.

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 After reviewing how to ﬁnd the area of a rectangle, post the formula on the board for students to refer to during the Exit Ticket.

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Ratios, Rates, and Percents Explore 3 – Multistep Ratio Problems ACTIVITY PREPARATION Students will apply their knowledge of ratios and percents to solve multistep word problems.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.5 Use appropriate tools strategically.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Price Guide (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 Price Guide per group. Cut out each set of cards, and place them in a resealable bag. If desired, print it on card stock, and laminate it for future use.

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) How do you try to get the most reasonable price when buying something you want?; 2) Why do you think the cost of the same type of item varies? After reading the scenario, ask the class 3) Why would Travis want to save money for the show by ﬁnding the best prices at the store? FACILITATION TIP This scenario provides students some excellent real-world practice about how understanding ratios and percents can save them money. Be prepared with some other real-world examples that they can relate to. People save money when they are able to calculate costs quickly and correctly. FACILITATION TIP To guide students’ thinking, read and post questions 5a and 5b before collaboration. Students’ understanding that consumers need to know dollars or cents per item is important in most real-world problems.

2. 3. 4.

Read the following scenario to the class: Travis, the production assistant on Ready, Set, Bake, has the task of stocking the show’s pantry for baking. Travis has written down how much he needs and the prices from a local store on a Price Guide. Your job is to help Travis save money by ﬁnding the best prices at the store. Give a Student Journal to each student. Give a Price Guide to each group. Explain to students that they will be working cooperatively with their small groups to help Travis ﬁnd the best discounts and prices. Students will use the various prompts in their Student Journals for guidance. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 When you set up your ratio, should you find the price per item or item per price? You should find the price per item because you are not trying to find the item for a price; you are trying to determine the price for an item.

DOK-1 How can you be sure you are ﬁnding the ratio you want? You can put the values in this order: price ﬁrst and then quantity. You should also represent the ratios using words that describe each value.

DOK-1 What strategy can you use to ﬁnd the possible savings on different amounts of sugar? Answers may vary. In the table, I found the savings for one pound of sugar for option 1 and option 2 and then evaluated the lowest price per pound of sugar.

DOK-2 How did you compare butter prices using proportions? I set up the proportion using price per pound to determine the price for 1 pound.

e. DOK-2 How did you compare sugar prices using unit rates? For option 1, I divided the price of the 5-pound bag by 5 to get the price for one pound. Option 2 included the price per pound. 144

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

Allow time for students to complete their Student Journals, including the reﬂection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 Is it easier to ﬁnd the discount price when the discount is a percentage off or when the discount is a certain dollar (or cents) amount off? It’s easier when it’s a certain dollar amount off because you don’t have to calculate the percentage; you just subtract. • DOK-2 Why is unit rate helpful? A unit rate allows you to easily compare things because it shows you the amount per 1 unit. • DOK-2 Is a dollar amount off coupon better than a percent off coupon? It depends on the dollar amount, the percent amount, and the price. If you have a coupon for $1 off, that is probably a big percent of something that doesn’t cost a lot, like a candy bar, but not a great discount on something expensive, like a new phone. •

Intervention

Acceleration

FACILITATION TIP Ask for student volunteers to provide some real-world examples of when they or their families had to compare prices. Be prepared with some of your own ideas (pizza coupons, extra lives in video games, gym visits, group discounts). FACILITATION TIP Challenge students to consider why different companies and vendors might use different discount strategies. Consider asking students why it is important for consumers to be able to ﬁnd prices on their own.

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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.

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Ratios, Rates, and Percents Explore 4 – Solve Problems – Percents ACTIVITY PREPARATION Students will calculate percentages using real-world problems. They will also use percent proportions to calculate the part, whole, and percent.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.5 Use appropriate tools strategically.

Materials

Preparation

Printed • • •

•

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

• •

Reusable •

Plan to divide the class into groups of two or three to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print one set of Appliance Cards per group. Cut out each set of cards, and place them in a resealable bag. If desired, print them on card stock, and laminate them for future use.

1 Resealable bag (per group)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What types of kitchen appliances are used to make smoothies? After reading the scenario, ask the class 2) Why would stores give discounts to a reality competition show?; 3) What steps are needed to determine the best price for each small kitchen appliance when stores are offering discounts? FACILITATION TIP In addition to reading the scenario aloud, post it for students to view while you read it and ask for volunteers to read it or parts of it aloud as well. FACILITATION TIP Depending on students’ prior knowledge, review and record the deﬁnition of percent. Some students may know that “cent” means 100 and “per” means each. Include a clear deﬁnition and examples on the scope anchor chart or word wall.

2. 3. 4.

Read the following scenario to the class: Liam, the production assistant on the reality show Healthy Cooking for Healthy Living, has one more job before ﬁlming can start. Liam must purchase all of the small appliances needed for the cooks to make their smoothies. He has made some pricing notes from two stores to help him decide where to buy each item. The production company also gets discounts from each store, so Liam has some work to do to make sure he gets the best price. Your job is to help Liam determine the prices for each small kitchen appliance. Give a Student Journal to each student. Distribute sets of Appliance Cards to student groups. Explain to the students that they will be working cooperatively with their groups to help Liam ﬁnd the actual dollar amount allocated for each item based on a given budget and given percent. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 What is a percent? Responses will vary. A percent is a ratio that has a denominator of 100.

DOK-1 How can models be used to solve percent problems? Responses will vary. Models can create visuals to help with solving percent problems by allowing you to see the part, whole, and percent you are solving for in the problem. They help to make sure that as you are solving the problem, you are working toward a reasonable answer.

Notes __________________________________________________________________________________________________________________________________________________

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Engage

Explore

6. 7.

Elaborate

Evaluate

DOK-1 What is an example of a percent problem that can be set up with a tape diagram to ﬁnd the part, whole, or percent? (Note that tape diagrams are scaffolds that help students keep their percents organized so they can set up a proportion. Quantities go on one side of the bar (part and whole). The other side of the bar will include the percents (the given percent or the variable that shows you are solving for the percent Part Whole 75 x be used to ﬁnd 75% of 60. ___ = ____ 60 100

Explain

Percent

Intervention

Acceleration

FACILITATION TIP Create an example problem about an appliance or complete some of the Appliance Cards together. Have students record a clear tape diagram with labels on top and bottom. Model ﬁnding 50%, or 1

and 100%). Responses will vary. ______ = _______ . The tape diagram could 100

half, and then half of that (25% or __4). Use

x = 45

These models of commonly used percents will support students as they complete the Student Journal.

DOK-2 How can we determine the percent of any given number? Responses will vary. We can use percent proportions to determine the percent of any given number.

Allow time for students to complete their Student Journals, including the reﬂection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

. another one to model ﬁnding 10% or ___ 10

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FACILITATION TIP On question 5d, it is important to take time to ensure that each student can identify strategies to ﬁnd the percent of a number. Students will have varying tools and methods, but before moving on, ensure that students can demonstrate this real-world skill.

Math Chat •

•

DOK-2 How do proportions relate to percents? The relationship between a proportion and a percent is that when a proportion is multiplied by 100, it gives the percent of parts taken; in other words, Parts/Whole X 100= percent. Similarly, when a percent is multiplied by total, it gives the number of parts taken; in other words, parts = percent × whole. DOK-2 If Liam is given a new budget and $2,000 represents 40% of the budget, 2,000 ____ 40 what amount represents the new budget? _____ = 100 x

2,000 ÷ 40 = 50; 40 × 50 = 2,000 100 × 50 = 5,000 x = 5,000 DOK-3 What are some examples of percents in the real world? Examples include grades, sales, and taxes. Grades—the percent of items that are correct; sales—the percent off of the original price; taxes—the percent of a retail sale price that is added to a total cost of an item that is paid to the government.

FACILITATION TIP Additionally, point out that the word “of” when placed between two values often means multiplication. For example, 40% “of” the budget can be written as “.40 multiplied by the budget.” In this scenario, “.40 times 2000.”

Post-Explore 1. 2. 3. 4.

FACILITATION TIP Note that the Exit Ticket includes an appliance that may be unfamiliar to some students; an immersion mixer. Be prepared to explain or edit the Exit Ticket.

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Ratios, Rates, and 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

Unit Rates with Ratios of 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

Ratios of Length and Area 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 Ratio Problems

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

Solve Problems – Percents

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

Ratios, Rates, and Percents 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 Ratios and Rates in Various Representations 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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Ratios, Rates, and Percents

3 150

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 unit rate problems in relation to everyday situations.

What prompts will be used?

What does mastery look like?

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I can use the ratios of lengths to compute unit rates.

I can use the ratios of areas to compute unit rates.

I can solve multistep ratio problems using proportional relationships.

I can solve multistep percent problems using proportional relationships.

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SCOPE 1

Percent Application Scope Introduction SCOPE SUMMARY In this scope, students will use their prior knowledge of proportional relationships to solve mathematical problems involving percentages. By the end of the scope, students will feel comfortable solving many different types of multistep ratio problems such as interest, tax, gratuities, and commissions. Student Expectations

VERTICAL ALIGNMENT

7.PAR.4.9 Use proportional relationships to solve multistep ratio and percent problems presented in applicable situations.

Background Knowledge

Future Expectations

In sixth grade, students learned how to recognize proportional relationships and use ratios and unit rates to solve mathematical problems. In the prior scope, students learned how to solve multistep ratio problems and began working with percents. This knowledge of proportions will be further explored as they discover the importance of proportional relationships within percent problems.

Students will continue to use proportional relationships as they begin to learn about functions in 8th grade. Throughout high school, percentages and the importance of their relationships will help students to better gain insight into ﬁnding the probability that an event might occur.

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

ﬁnd a percent of a quantity as a rate per 100.

•

solve relating problems.

•

identify two truths and a lie by reading statements about the prior standard.

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 selling price when the cost to make and the percent markup are known.

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.

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In this exploration, groups of students will be tasked with solving a scenario about Anthony, a diner owner, and the need to help calculate the sales tax for new supplies, and determine how much he spent plus the sales tax for his new supplies. Students will:

Percent Change In this exploration, students will be tasked with determining the amount of change and percent change for the employees that will be working at the diner; as well as, calculating each customer’s discount on their meal. Students will:

solve problems that involve sales tax using proportional reasoning.

•

solve problems involving percent increase and percent decrease.

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

•

calculate markups and markdowns in prices using proportional reasoning.

•

Explore 3

Explore 2

Tax

Tips and Commissions In this exploration, students will solve a scenario about the diner and help calculate the amount of tips left by 4 different customers, and the amount of commission a salesperson earns from selling appliances. 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.

Explore 4

Explore 1

EXPLORE ACTIVITIES

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solve problems solving tips and commissions using proportional reasoning.

Simple Interest In this exploration, learning will occur through students collaborating with peers to solve another extension from the diner scenario. Here, students help the diner manager calculate simple interest on loan options for a diner expansion. 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.

solve problems involving simple interest using proportional reasoning.

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

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Percent Application Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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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. 6.NR.4.6 Calculate a percent of a quantity as a rate per 100 and solve everyday problems.

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.

FACILITATION TIP

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 a 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.

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Statement C is the lie because 32.4 is 40% of 81, not 37.4.

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.

To streamline this APK, display the Two Truths and Lie for students to use Think, Pair, Share. Have students think (individually), pair (discuss with partner), and share (with the class). Students can vote with hand signals and/or you can ask for volunteers to explain their thinking. FACILITATION TIP In addition to the Foundation Builder, provide students with the structure of translating these words into equations. For example “75.25 is 35% of 215” can be written as “75.35 = .35 times 215”. Replacing “is” with “=” and “of” with “times” may support students’ review of prior standards.

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Percent Application Hook – Piece of Cake ACTIVITY PREPARATION Students will determine the selling price when the cost to make and the percent markup are known.

Materials

Preparation

Printed •

• •

1 Piece of Cake (per class)

Reusable •

•

1 Phenomena Video (per class)

Plan to show the video. Prepare to project the Piece of 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.

PROCEDURE AND FACILITATION FACILITATION TIP

Part I: Pre-Explore

Before showing the video and reading the scenario, ask the class 1) What is your favorite types of cake or pastry?; 2) What do you think a pastry shop owner considers when he or she sets the price for your favorite cake or pastry?

FACILITATION TIP Print and project the scenario so students can read along with you, or have volunteers read parts of the situation to the class. Be prepared to preteach any words you think students may need to learn or review: patisserie, steep markup, high-end, production costs, proﬁt. FACILITATION TIP Write Claudette’s formula out step by step for students if they cannot see it on the Piece of Cake slide.

4. 5.

FACILITATION TIP Allow students to contribute real-world connections to item markups in their own lives (clothing, collectibles, favorite snacks, games, etc.)

DOK-1 What is the total cost of making the cake? $25 The sum of $15 for ingredients and $10 for employee pay is $25.

DOK-1 What is a markup? It is the retail price of an item minus what it cost the company to make it.

DOK-1 Why do companies mark up items? If they did not mark up items, they would make no proﬁt.

Complete the Explore activities. Notes

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Part II: Post-Explore 1. 2.

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

DOK-1 How can you determine the retail price of an item? Find the sum of the cost to make the item and the amount of the markup.

DOK-1 How do you ﬁnd the dollar amount of the markup? Find the product of the total cost of making the cake and the percent of the markup.

DOK-1 What is the amount of the markup on Claudette’s cake? $20 because 25. × 0.8 = 20

DOK-1 How do you ﬁnd the retail price of the cake? Find the sum of the cost to make the cake and the amount of the markup.

e. DOK-1 What is the retail price of Claudette’s strawberry vanilla creme cake? $45 because 25 + 20 = 45

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FACILITATION TIP Review translating words into mathematical formulas with students. For example, “80% of 25 is ?” is the same as “.8 times 25 = x” (“Of” often becomes “multiplied by” or “times” and “is” becomes “equal” or = ).

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Percent Application Explore 1 – Tax ACTIVITY PREPARATION Students will solve problems involving sales tax using proportional reasoning.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.5 Use appropriate tools strategically.

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 FACILITATION TIP

Part I: Calculating Sales Tax

Before reading the scenario, ask the class 1) What do you think a new restaurant owner needs to do to prepare a restaurant for opening day?; 2) What types of equipment would need to be purchased to stock a new kitchen restaurant?; 3) What is “sales tax”?

FACILITATION TIP

2. 3.

Project this scenario in print as you read it with students. FACILITATION TIP

Before distributing the Student Journal to each student, assess for ﬂuency with converting percents to decimals with a few examples. Have students copy some common percents as decimals on their Student Journals as reference. For example: 10%= .1; 5% = .05 FACILITATION TIP Continue to review the skill of translating words into math expressions and equations. “Percent of a number” can be written as “% times y.” 5. 6. 158

Read the following scenario to the class: Anthony is opening a new restaurant, the Yellow Rose Diner. There is a lot of planning and preparation before the restaurant’s upcoming grand opening. He begins stocking his kitchen by buying pots and pans, cake pans, silverware, and measuring utensils. Anthony will pay a sales tax on his new supplies for his kitchen. The sales tax rate will vary because it depends on the state that Anthony and his staff purchase the items from. Let’s help Anthony calculate the amount of sales tax for his new supplies for his kitchen. Give the Student Journal to each student. Have students work in pairs to determine the tax amount Anthony and his restaurant staff will pay on the supplies for the restaurant. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 How do you convert a percent to a decimal? We move the decimal point two places to the left. For example, 25% is equivalent to 0.25.

DOK-2 How can you determine the percent of a number? We can ﬁrst convert the percent to a decimal and then multiply this decimal value by the number using this equation: tax amount = tax rate (as a decimal) times the cost of the item.

DOK-2 How would you determine the amount of sales tax on an item if the sales tax rate is 8%? What about 8.5%? I would convert 8% to 0.08 and then use the equation tax amount = tax rate (as a decimal) times the cost of the item. If the sales tax rate was 8.5%, I would convert this to 0.085 and then use the equation tax amount = tax rate (as a decimal) times the cost of the item.

Allow students enough time to record all their work for Part I on their Student Journals. 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

•

• • •

Acceleration

FACILITATION TIP

Math Chat •

Intervention

DOK-1 What is sales tax? Sales tax is a fee added to the cost of purchases and services. That fee is then given to the government. DOK-2 How do you calculate the amount of sales tax on a purchase? First, convert the percent to a decimal by moving the decimal point two places to the left. Next, multiply this decimal by the cost of the item. DOK-2 Why do you think we pay sales tax on the purchase of goods and services? The government is funded by tax dollars. Without paying a sales tax, we wouldn’t be afforded some of the public services the government provides. DOK-1 What is the formula for sales tax? Tax amount = tax rate (as a decimal) times the cost of the item DOK-1 Do all US states have the same sales tax rate? No, each state has its own sales tax rate. DOK-1 What is the sales tax rate in your state? (Allow students time to look up this information online if they do not already know the sales tax for their state.) Examples include the following: Texas = 6.25% California = 7.25% New York = 4% Arizona = 5.6% Florida = 6%

Prior to the Math Chat, select three of these questions to post. Allow students some thinking and partner time before the whole group Math Chat.

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FACILITATION TIP

Part II: Calculating Total Cost 1.

Read the following scenario to the class: Anthony is getting ready to purchase the supplies for his kitchen. In order to determine the total amount Anthony pays, the cashier or the customer service rep that places the orders that are placed in different states will add the cost of the pots and pans, cake pans, silverware, and measuring utensils and the amount of sales tax in order to determine the total cost of each item. How much do you think Anthony spent for the new supplies for the kitchen, including the sales tax? Students will work in their pairs and will use the tax amounts found in Part I to determine the total cost of the items that will be purchased by Anthony and his staff. They will represent the total cost of each item using a tape diagram and an equation to show how to solve for the total cost of each item. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 Why do you think a tape diagram is used to help us understand the total cost of an item? The total cost of an item consists of the cost of the item and the amount of tax on the item. The length of the tape diagram represents the total cost of the item.

DOK-2 Given a state’s sales tax rate, what operations are used to calculate the total cost of an item? Multiplication to ﬁnd the amount of sales tax and then addition to ﬁnd the total cost

c. 4. 5.

DOK-2 Create a formula that could be used to determine the total cost of an item. Cost of an item + amount of sales tax = total cost of the item

Allow students enough time to record all their work for Part II on their Student Journals. After Part II, invite the class to a Math Chat to share their observations and learning.

After reading the scenario, ask the class 1) What steps will the cashier or customer service rep take to determine the total cost for the kitchen items?; 2) Would you rather see the total cost or the cost of the individual items and sales tax to make your decision about your purchase decisions? Why? FACILITATION TIP Model a tape diagram for total cost with tax for students.

FACILITATION TIP Encourage students to write the tax amounts as percents and as decimals on the Student Journal. Some students may need to continue to practice translating percents into decimals.

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Percent Application Explore 1 – Tax Math Chat DOK-2 Including sales tax, how can you determine the total cost of a purchase? I would add the amount of sales tax to the cost of the item to determine the item’s total cost. • DOK-1 How is the tape diagram helpful in understanding the total cost of a purchase, including sales tax? The entire length of the tape represents the total cost of an item. The total cost of an item is made up of two parts: the cost of the item and the sales tax on the item. • DOK-1 When you go grocery shopping, is the price listed on each item the actual amount you pay for the item? Explain. No, these prices do not include the sales tax. You will need to pay an additional amount on these items when you checkout. • DOK-2 How does the sales tax rate affect the total cost of an item? The greater the sales tax rate, the greater the amount of sales tax on an item. This would cause an increased cost in the total price of the item. •

FACILITATION TIP Before assigning the Exit Ticket, decide whether to allow calculators. In order to assess misconceptions, encourage students to record percents as decimals, set up equations, and show steps in the workspace.

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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Percent Application Explore 2 – Percent Change ACTIVITY PREPARATION Students will solve problems involving percent increase and percent decrease that will also include understanding how to calculate markups and markdowns in prices using proportional reasoning.

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.

Materials

Preparation

Printed • • •

• • • •

1 Student Journal (per student) 1 Discount Spinner (per pair) 1 Exit Ticket (per student)

Reusable • •

Plan to divide the class into pairs to complete this activity. Print a Student Journal and an Exit Ticket for each student. Provide each pair of students with one paper clip and one pencil. Print the Discount Spinner for each pair of students. If desired, print the Discount Spinner on card stock, and laminate it for future use.

1 Paper clip (per pair) 1 Pencil (per pair)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What is a restaurant manager responsible for to make sure a restaurant runs smoothly on opening day?; 2) What are the consequences of not being prepared for a grand opening? After reading the scenario, ask the class 3) What is meant by “amount of change” and “percent change”?

FACILITATION TIP Demonstrate a tape diagram and a double number line with an example problem or do some of the problems together to model what students are to record in the work spaces on the Student Journal

FACILITATION TIP For struggling students, break this process into three clear steps. Demonstrate with an example. 162

Part I: Amount of Change and Percent Change 1.

2. 3.

Read the following scenario to the class: The Yellow Rose Diner is getting ready to open its doors to its ﬁrst customers, but before it does, diner manager Lacey begins working with the head chef to make sure the employees are trained and their work schedule hours are set so the diner is ready for the grand opening. Let’s help Lacey determine the amount of change and the percent change for the employees that will be working at the diner. Give a Student Journal to each student. Explain to students that they will use models, proportions, and equations to determine the amount of change in the number of hours worked at the diner. They will also determine whether the scenario represents a percent increase or a percent decrease. Have students work in pairs and use models, a tape diagram, or double number line to represent the amount of change in hours worked by employees at the diner. Then, have students apply their understanding of percents by identifying whether the scenario represents a percent increase or percent decrease in hours worked at the diner. Students will calculate the number of hours using a proportion or equation to determine the amount of change in hours worked by the employees at the diner. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 How can you calculate a percent increase or percent decrease? To find the percent increase or percent decrease, find the amount of change by subtracting the lesser value from the greater value (amount of change = greater value – lesser value). Then, you divide the amount of change by the original number, and multiply by 100 to find the percent increase or percent decrease. © Accelerate Learning Inc. - All Rights Reserved

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How can you determine the amount of change? Find the amount of change by subtracting the lesser value from the greater value (amount of change = greater value – lesser value).

Allow time for students to complete Part I of their Student Journals, including the reﬂection questions. After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 What is percent change? A comparison between two values DOK-2 How does a tape diagram help model and calculate percent change? Tape diagrams help us see the connection between the percents and the amount. • DOK-3 There were 20 employees who completed training at the diner during week 1. During week 2, there were 6 employees who completed training. Was there a percent increase or a percent decrease from week 1 to week 2? Since the number of employees was at 20 during week 1 and 6 during week 2, there was a decrease by 14. • •

14 ___ 20

= ____ 100

Intervention

Acceleration

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FACILITATION TIP Project the DOK-3 question as students ﬁnish up Part I. Alert students to be ready to share their solution and thinking. Be sure that they are prepared to explain the amount of increase or decrease using a percent.

Part II: Markups and Markdown 1.

2. 3.

Read the following scenario to the class: Lacey is getting excited for the Yellow Rose Diner’s upcoming grand opening. To spread the word, she decides to send out mailers to local residents. The mailer includes a coupon. Let’s use the Discount Spinner to help calculate each customer’s discount on their meal at the Yellow Rose Diner. Give the Discount Spinner, a paper clip, and a pencil to each group. For 1 and 2, explain to students that they will be introduced to markup and markdown as examples of a percent increase and percent decrease. They will apply their understanding of markups and markdowns using the word bank provided in their Student Journals. They will use the table and the markup percent to calculate the selling price and the proﬁt that will be made to prepare and cook each item. For 3 and 4, have students work in their pairs and ﬁll out the tape diagram using the words and phrases provided in the word bank. They will use the information in the scenarios to complete each table for markup and markdown. For markdown, they will use the Discount Spinner, paper clip, and pencil and spin the spinner to get the markdown percent. Students will use the markdown percent to calculate the discount and total cost of each meal after the discount is applied. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 Why is understanding percents and markup important for a business? A company needs to price the item just right so customers buy the item but not low enough where the company doesn’t make money from the sale.

DOK-1 What happens when a markdown is applied to the cost of an item? The markdown is subtracted from the cost of an item.

DOK-2 What formula could be used to determine the amount a customer pays for an item after a markdown is applied? Original cost of item – markdown = total cost of item

DOK-2 How do you know whether a markdown increases or decreases the total cost of an item? Examples of markdowns include coupons, sales, and discounts, and these examples all cause the original price of the item to be reduced.

Allow time for students to complete Part II of their Student Journals, including the reﬂection questions.

FACILITATION TIP Consider spinning the Discount Spinner for the whole class to streamline Part II. You can have volunteers come up front to spin or hand the spinner to a student at their desk to spin for the discount for 3 and 4. FACILITATION TIP Encourage students to continue to record translation of percents into decimals in the Student Journal as they work. Requiring this step will support student success with calculations and will allow you to monitor and assess.

FACILITATION TIP Bring in some local coupons or ﬂyers to show students some real-world markdowns (pizza coupons or car wash group sales). If time allows, use the formula in part 4c. to ﬁnd total costs of some real-world items. 163

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Percent Application Explore 2 – Percent Change 6.

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

Math Chat •

•

• FACILITATION TIP Display the word form and equation form for Lacey’s proﬁt. Point out to students a pattern. In this example, “80% of $5.25 is ?” translates to “.8 times 5.25 equals.” “Of” can be “times” and “is” can be “equals.”

• •

•

FACILITATION TIP Encourage students to look for discounts and coupons to bring in to share. Find a place to post them in your classroom.

•

FACILITATION TIP In order to accurately assess student understanding, ask them to show their evidence and thinking in the work space. Encourage them to label their answers.

DOK-1 Do you think all businesses apply a markup to the products they sell? Yes, I feel all businesses apply a markup when selling products. Companies need to stay in business and in order for a company to do so, it must charge a markup and make a profit. DOK-2 What do you think happens when a company applies too big of a markup to their products? If a company applies too big of a markup, the product might not sell because it’s too expensive for customers to buy. DOK-2 What does it mean for a company to mark up their products by 0%? A markup of 0% means the company sells the product for exactly the same amount the product cost to make. DOK-2 How much profit will Lacey make after selling a cheeseburger that cost $5.25 to make with an 80% markup? $5.25 · 0.8 = $4.20 DOK-2 Why would a business create a markdown for customers? People love markdowns because it saves them money. Markdowns help attract customers to stores. DOK-4 What are some other examples of markdowns you’ve seen recently when shopping? Sales and discounts are other examples of markdowns I’ve seen when shopping. DOK-2 How are markdowns and markups similar? How are they different? Markdowns and markups are similar in that they both involve finding the percent of the cost of an item. They are different because markups increase the cost of an item, while markdowns decrease the cost of an item. DOK-2 What is the total cost of a $13.00 fried chicken dinner with a 25% off coupon? $13.00 · 0.25 = $3.25 $13.00 – $3.25 = $9.75

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Percent Application Explore 3 – Tips and Commissions ACTIVITY PREPARATION Students will solve problems involving tips and commissions using proportional reasoning.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.5 Use appropriate tools strategically.

Materials

Preparation

Printed • • • • •

• • • •

1 Student Journal (per student) 1 Commission Spinner (per pair) 1 Set of Tip Task Cards (per pair) 1 Bob’s Appliance Store Sales Ad (per pair) 1 Exit Ticket (per student)

•

Reusable • • •

•

1 Paper clip (per pair) 1 Pencil (per pair) 1 Reusable bag (per pair)

Plan to divide the class into pairs to complete this activity. Print a Student Journal and an Exit Ticket for each student. Provide each pair with one paper clip and one pencil. Print and cut out a set of Tip Task Cards for each pair. Place each set of Tip Task Cards in a resealable bag, and label the bag “Part 1.” If desired, print them on card stock, and laminate them for durability. Print one Commission Spinner per pair. If desired, print it on card stock, and laminate it for future use. Print one Bob’s Appliance Store Sales Ad per pair. If desired, print it on card stock, and laminate it for future use.

PROCEDURE AND FACILITATION Part I: Tips FACILITATION TIP Before reading the scenario, ask the class 1) Other than a salary from a restaurant, what other income do waiters receive for serving customers?; 2) How do customers determine how much to tip waiters? FACILITATION TIP Complete one of the customer problems together to demonstrate how students are to show their thinking. In Part II, the Student Journal answer key shows the percents written as decimals. Consider requiring students to show the percents as decimals in Part I also.

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Read the following scenario to the class: The Yellow Rose Diner is oﬃcially open! Will is a waiter at the diner, and in addition to receiving an hourly wage, he also receives the tips left by customers after eating their meals. The tips left by his customers are percentages of their meals. Let’s help Will calculate the amount of tips left by 4 different customers. Give a set of Tip Task Cards to each pair. Give a Student Journal to each student. Direct students’ attention to Part I of the Student Journal. Students will work with their partners to create a tape diagram for each Tip Task Card. Students will then use the Tip Task Cards to apply their understanding of tips by calculating the tip on a meal and the total cost of the meal after the tip is applied. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 What operation(s) do you use when calculating a tip, and how do you use it/them? We use multiplication. We begin by calculating the tip by changing the percent of tip to a decimal and then multiplying by the cost of the meal.

FACILITATION TIP

Students may not be as aware of tipping etiquette as adults, so this scenario is a good opportunity to explain this common real-world application of percents.

DOK-2 What formula could be used to determine the total cost of a meal after a tip is applied? Cost of meal + tip = total cost of meal

c. DOK-2 Why do you think some customers left a bigger percent tip than others? The percent of tip is based on the service you receive from the waitstaff. Typically, the better the service is, the greater the percent of tip. © Accelerate Learning Inc. - All Rights Reserved

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Allow students enough time to record all their work for 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 How does a tip affect the total cost of a meal? The greater the tip left by a customer, the greater the total cost of a meal. DOK-2 Will earns an hourly wage in addition to the tips left by customers. If Will is planning his budget, which is easier to predict, his hourly wage or the amount of tips he receives? The hourly wage is easier to predict because he can multiply the number of hours he works by his hourly wage. Different customers can leave different tips even if the cost of their meal is the exact same. It would be easier to predict his hourly wage. DOK-2 What is the total cost of a $25.50 meal with an 18% tip? $25.50 · 0.18 = $4.59 $25.50 + $4.59 = $30.09 DOK-4 The waitstaff at the Yellow Rose Diner receives tips left by their customers. Do you think all restaurants encourage tipping? Usually all sit-down restaurants encourage tipping since you are being waited on by a waiter or waitress. You typically don’t leave a tip when visiting fast food restaurants.

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FACILITATION TIP Consider projecting two of the Math Chat questions as students ﬁnish up Part I to give them some think time with their groups. Students (or you) may note that in some real-world instances. tips are pooled and shared between the wait staff.

Part II: Commissions 1.

2. 3.

Read the following scenario to the class: Great news! The diner is a huge success, so much so that diner co-owner Troy wants to expand this business. He visits the local appliance store in search of an additional oven. Brooke is a salesperson at the local appliance store and helps Troy ﬁnd a new oven for the diner. Brooke is paid a commission for selling appliances. The more Brooke sells, the more money she earns. Commission is calculated as a percent of the total sale of the appliance. Let’s use the Commission Spinner to help calculate the amount of commission Brooke earns from the sale of different appliances. Give a Commission Spinner, Bob’s Appliance Store Sales Ad, paper clip, and pencil to each group. Students will use the Commission Spinner, paper clip, and pencil to apply their understanding of commission and how to determine the amount earned from the sale of an item. They will use the Bob’s Appliance Store Sales Ad to get the prices of each appliance. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

5. 6.

DOK-2 Does commission affect the total cost the customer pays for an item? Why or why not? No, the customer pays the full price of the item. Commission only affects how much money the salesperson receives from the sale of an item.

DOK-2 What operation(s) do you use when calculating the amount of commission on a sale? Explain how you would use this operation. We use multiplication. We would ﬁrst convert the commission rate from a percent to a decimal and then multiply the decimal value by the price of an item.

DOK-3 Why might a salesperson prefer to get paid based on commission? Getting paid based on commission gives you the opportunity to earn a lot of money depending on how many items you sell. Some salespeople might work a few hours a week but still get paid a large amount for selling a large amount of products.

Allow students enough time to record all their work for Part II on their Student Journals. After Part II, invite the class to a Math Chat to share their observations and learning.

FACILITATION TIP Before reading the scenario, ask the class 1) Salespeople are sometimes on commission. What do you think “being on commission” means?; 2) What are the advantages of being on commission rather than on a set salary?; 3) What are the disadvantages of being on commission rather than on a set salary? FACILITATION TIP Commission may be a new concept for many students. Before sharing the scenario; check for students’ prior knowledge and teach accordingly. Some families may be more familiar with commissions than others. See Picture Vocabulary in Explain. FACILITATION TIP Alternatively, to simplify the procedure, spin the Commission Spinner for the class for each appliance before students begin their group work. You could have a student volunteer come spin for each appliance and record the commissions on the board. Have students copy the commission percents and equivalent decimals on their Student Journal work spaces. FACILITATION TIP Using the same commission for the whole class for each appliance will make monitoring and assessing more eﬃcient as students collaborate.

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PERCENT APPLICATION

Percent Application Explore 3 – Tips and Commissions Math Chat DOK-1 Describe how to determine the amount of commission earned. I ﬁrst change the commission percent to a decimal by moving the decimal point two places to the left. I then multiply this decimal by the sale price to determine the amount of commission the salesperson earns. • DOK-3 Would you prefer to get paid via an hourly wage or by commissions? Why? I would prefer to get paid via an hourly wage because it would be easier to budget my money. I could easily predict how much I am going to make instead of getting paid by commissions, where I don’t know exactly how much income I’ll receive per month. • DOK-1 What are some other professions that get paid based on commissions? Car salespeople and real estate agents also get paid based on commissions. •

FACILITATION TIP Allow students to vote on which pay structure they would prefer.

Post-Explore FACILITATION TIP

Challenge students to display their thinking with words and equations on the Exit Ticket.

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

PERCENT APPLICATION

Home

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PERCENT APPLICATION

Percent Application Explore 4 – Simple Interest ACTIVITY PREPARATION Students will solve problems involving simple interest using proportional reasoning.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.5 Use appropriate tools strategically.

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. Gather enough number cubes for each partnership to have one.

Reusable •

1 Number cube (per pair)

PROCEDURE AND FACILITATION FACILITATION TIP

Before reading the scenario, ask the class 1) Lending institutions charge consumers getting loans interest. What is “interest”?; 2) Why do lending institutions charge interest?; 3) Do you think the amount of interest charged is always the same? Why or why not?

2. 3.

FACILITATION TIP Print and project the scenario. Have volunteers read it aloud to the class. Before giving the Student Journal to each student, explain simple interest. Explain that many adults borrow money for larger purchases, and that it costs money to borrow money.

Read the following scenario to the class: The Yellow Rose Diner is such a big success that the owners and managers of the diner have decided to open a second location. In order to build a new restaurant, the owners and manager will need to apply for a business loan at the local bank. Diner manager Sienna will ﬁll out an application for a $10,000 loan. The bank provides Sienna with four loan options. Let’s help Sienna calculate the simple interest on each of these options. Give a Student Journal to each student. Have students read the background information in the box at the top of page 1 of their Student Journals with their partners. Instruct students to practice using the simple interest rate formula in order to determine the amount of interest. Students will be asked to solve for the interest for four different loan options. Students will roll a number cube to determine each loan’s interest rate and length of time. Monitor and assess student understanding as each group collaborates by asking the following guiding questions:

FACILITATION TIP

Display the box with the variables and formula for simple interest to the class and practice some examples with students.

DOK-1 What operation(s) is/are used to calculate the amount of simple interest? We use multiplication. We multiply the principal by the rate by the time.

DOK-2 How do you determine the total amount paid back on a loan by a customer to a bank? We would ﬁrst determine the amount of simple interest on the loan using the simple interest rate formula. We would then need to add the principal to this amount.

DOK-3 Does order matter when multiplying the principal, rate, and time? Why or why not? No, because of the commutative property, order doesn’t matter when multiplying.

FACILITATION TIP To begin, rather than having the students roll a number cube to determine the rate and time, consider rolling these variables as a class (or ahead of time) so you can monitor and assess student solutions more eﬃciently.

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5. 6.

Allow time for students to complete their Student Journals, including the reﬂection questions. 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

•

DOK-1 What does the amount of simple interest paid on a loan represent? The amount of simple interest on a loan is the amount a bank charges you for borrowing money. This is income to the bank. DOK-1 What does the amount of simple interest earned on a savings account represent? The amount of simple interest on a savings account is the amount the bank provides you for investing money in the bank. This is income to you. DOK-2 What formula would help you calculate the amount of simple interest you would pay on a $5,200 loan with an interest rate of 7% for 4 years? I = prt I = $5,200 · 0.07 · 4 DOK-2 How much simple interest would you pay on a $5,200 loan with an interest rate of 7% for 4 years? I = $5,200 · 0.07 · 4 I = $1,456 DOK-4 If you were to borrow money from a bank, what would the ideal interest rate and length of loan look like to you? Why? I would prefer to have a low interest rate and a small length of time since this would cause the amount of interest I pay to the bank to be minimal. The longer the time and the higher the rate, the greater the interest I would have to pay the bank.

2. 3. 4.

Take time and use the variables in the box to teach the deﬁnitions of principal, rate, time, and interest. Have students take notes in the box. FACILITATION TIP Address that interest can be both paid and earned before or during the Math Chat. Note that the practice problems are mostly related to interest paid on loans.

FACILITATION TIP

Post-Explore 1.

Acceleration

FACILITATION TIP

Math Chat •

Intervention

PERCENT APPLICATION

Home

As a support, you can provide the standard interest formula by writing it on the board or allowing students to copy it on their Exit Ticket. Some students may want to challenge themselves by memorizing the formula and/or not using any calculators on the Exit Ticket.

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PERCENT APPLICATION

Percent Application 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

Tax 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

Percent Change 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

Tips and Commissions

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

Simple Interest

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

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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

Applications of Percents

PERCENT APPLICATION

Home

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

Fluency Builder Calculate Simple Interest Independent and partner games and other activities that provide students with an engaging way to practice the new concept

173

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

PERCENT APPLICATION

Percent Application

3 174

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

PERCENT APPLICATION

Home

What does mastery look like?

I can use strategies to represent a percent as a decimal.

I can identify the part, percent, and whole of proportional relationships to determine simple interest, tax, markups and markdowns, and gratuities and commissions.

I can use proportional relationships to solve problems that involve adding a percent tax to a total.

I can use proportional relationships to solve simple interest problems.

I can use proportional relationships to solve percent increase and percent decrease.

I can use proportional relationships to solve for tips and commissions.

I can solve multistep percent problems using proportional relationships.

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SCOPE 1

Expressions Scope Introduction SCOPE SUMMARY In this scope, students will use operational properties along with basic operations in order to expand linear equations with rational coeﬃcients. They will then be able to ﬁnd relationships between quantities and terms in the problem through rewriting expressions in different ways. Using a variable, they will solve algebraic equations of real-world and mathematical problems. Student Expectations

7.PAR.2.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coeﬃcients. 7.PAR.2.2 Rewrite an expression in different forms from a contextual problem to clarify the problem and show how the quantities in it are related.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In Grade 6, students learned the order of operations, along with the operational properties and identities. They understood how to write and read expressions with variables and how to evaluate these expressions. Students have prior knowledge of equality and understand that either side of an equal sign represents an equal quantity. They are able to represent variables in real-world problems through solving equations.

Solving real-world problems using algebraic expressions and equations is the foundation of algebra as students move into 8th grade as well as high school. In 8th grade, students will expand expressions through the use of the distributive property to combine like terms in order to simplify radical expressions. Students will continue their work with linear expressions, as they use these skills to solve nonlinear expressions in further grades.

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

read different student responses to a posed question on the prior standard.

•

decide whether they agree or disagree with the student.

•

explain their reasoning.

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 whether two different equations are equivalent.

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.

176

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Combining Like Terms with Rational Coeﬃcients In this exploration, groups of students will help solve a scenario that involves students being tasked with helping Josiah write expressions and equivalent expressions for companies to determine proﬁt opportunities for a new app he has created. Students will: •

write expressions and equivalent expressions.

•

apply properties of operations to combine like terms in expressions with rational coeﬃcients.

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.

EXPRESSIONS

Home

Distributive Property In this exploration, students will be presented with an extension scenario involving Josiah. However, this time students are tasked with helping Josiah and his brother. Students will: •

determine which expression is correct by using area models to ﬁnd equivalent expressions.

•

identify equivalent expressions.

•

apply the distributive property to determine equivalent expressions involving rational numbers.

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

Finding Equivalent Expressions Using Properties In the ﬁnal exploration of this scope, students will learn through solving a scenario involving Josiah and helping him write expressions and determine if the expressions are equivalent to aid in his app creations. Students will: •

explain in writing their reasoning regarding how to evaluate and generate equivalent expressions involving word problems.

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

177

EXPRESSIONS

Expressions Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

178

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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE

EXPRESSIONS

Home

Students will read different student responses to posed questions on the prior standards, decide whether they agree or disagree with each student, and explain their reasoning. 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 from the variables, including expressions that arise in everyday situations. 6.PAR.6.5 Apply the properties of operations to identify and generate equivalent expressions.

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.

Agree with Jay

Disagree with Kit

Disagree with Austin

Disagree with Justin

•

Encourage students to make notes on the Agree or Disagree as they work independently. Have them take pencils with them while they complete the “stand up, hand up” activity and collaborate to correct and make more notes with their partners as they discuss. FACILITATION TIP Use a timer to signal when students are to rotate to ﬁnd a new partner.

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.

Kit’s expression includes a variable without a coeﬃcient that is being subtracted (–y); Watch for common errors when combining like terms like these. Be prepared to remind students what is implied if there is no coeﬃcient with the variable. FACILITATION TIP

Identifying Misconceptions •

FACILITATION TIP

Students may forget to multiply the number outside the parentheses by both numbers inside the parentheses when using the distributive property. Students may not pay attention to the signs when combining like terms.

These procedural errors are very common. Provide some simple review and practice on whiteboards to refresh the calculation habits from prior standards.

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EXPRESSIONS

Expressions Hook – Even Steven? ACTIVITY PREPARATION Students will determine whether two different equations are equivalent.

Materials

Preparation

Printed •

• • •

1 Even Steven? (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Even Steven? 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) What video games do you like to play with your friends?; 2) How do you know who wins the game?; 3) What do you do if your scores are tied?

FACILITATION TIP Before asking the questions in Step 3, consider posting Even Steven? 4.

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: Steven is playing a video game against his best friend Jamaal. He is player 1 and Jamaal is player 2. At the end of the game, both of their scores were posted as expressions. Steven’s little sister Avery came in, looked at the scores, and asked who won. Steven and Jamaal need to decide whether the expressions are equivalent or not. If they are equivalent, it’s a tie. If not, the player with the greater score wins! 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 Steven and Jamaal have scores that are displayed as expressions. I wonder whether the scores have variables or not. Will the scores be equivalent or will one score be a greater value than the other and allow its player to win? I can use math to evaluate and compare each expression. Project Even Steven? Notes

180

Explore

Explain

Elaborate

Evaluate

Explain to students that the scores are in different formats and will have to be converted to a common format to be reliably compared. Discuss the following questions: a.

DOK-1 What do you notice about the expression that is the score for player 1? Answers will vary. It has all positive numbers. It uses the distributive property. It includes fractions.

DOK-1 What do you notice about the expression that is the score for player 2? Answers will vary. It has all negative numbers. It uses the distributive property. It includes fractions and decimals.

c. 6.

Engage

DOK-1 Can you tell which score is greater through just looking at the expressions? No, it is diﬃcult to tell even if I try to estimate.

Acceleration

FACILITATION TIP Challenge students who calculate or think they’ve calculated the correct solutions to save their notes and answers for Part II: Post-Explore. They can also turn them in to you for you to examine after class. FACILITATION TIP Offer students the chance to share whether they like to determine values of expressions with fractions or with decimals. They could vote: Decimals, Fractions, or Both.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Intervention

EXPRESSIONS

Home

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

DOK-1 How can you determine the value of an expression? Evaluate it by simplifying it and combining like terms.

DOK-2 What should you do if one score has terms that are fractions and one score has terms that are decimals? I can pick one type of term and convert both scores to that format.

DOK-1 What is player 1’s score when simpliﬁed? 7.5x + 18

DOK-1 What is player 2’s score when simpliﬁed? 7.5x + 18

FACILITATION TIP As an extension, depending on students’ experiences with video games, ask “What could “x” possibly represent in a game?” (gems gathered, planets targeted, challenges completed...).

e. DOK-1 Who won the video game and how can you tell? It was a tie. I can tell because the expressions are equivalent. They both equal 7.5x + 18. f.

DOK-1 What would the scores be if x = 1,000? 7,518 (7.5 × 1000 + 18 = 7500 + 18 = 7,518) Notes

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EXPRESSIONS

Expressions Explore 1 – Combining Like Terms with Rational Coeﬃcients ACTIVITY PREPARATION Students will apply properties of operations to combine like terms in expressions with rational coeﬃcients.

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 • •

1 Set of algebra tiles (per group) 1 Resealable bag (per group)

• •

Plan to divide the class into groups of four to complete this activity. Print a Student Journal for each student. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so that each student has one. Print a set of Expressions Cards for each group. Cut out the cards, and place each set in a resealable bag. If desired, print them on card stock, and laminate them for future use. Gather algebra tiles 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. (Algebra Tiles)

PROCEDURE AND FACILITATION FACILITATION TIP

Part I

Before reading the scenario, ask the class 1) When app developers create new apps, what things do they consider when deciding on a company to help them market their apps?; 2) How is the proﬁt that app developers make on their apps determined?

FACILITATION TIP As you review these tips for how to use algebra tiles, post them somewhere that students can refer to as they work. Complete 2–3 expressions modeling how to use the tiles before distributing tiles and Expression Cards to students.

182

What does a green rectangle represent? A green rectangle represents the value of one x.

What is x? This is a variable. An x represents an unknown amount.

What does a green square represent? A green square represents one.

How can you represent 2? You can use 2 green squares.

e. How could you represent negatives using the tiles? The rectangles always represent x, and the squares always represent one. If the rectangles and squares are green, the value is positive. Negative values for x and one are represented with red rectangles and squares.

FACILITATION TIP Take time to demonstrate how students should record on the Student Journal. Consider if you want to require color or labels in their drawn models.

Read the following scenario to the class: Josiah has created a new app! He decides that he would like to sell his app to earn extra money but isn’t sure which company to work with. Josiah looks at the first five companies he sees listed to sell apps through to determine which has the best opportunity. Help Josiah write expressions for each company to determine profit opportunities. Review how to use the algebra tiles with students by asking the following questions:

f. 3. 4.

What is the purpose of the algebra tiles? You use the tiles to represent values.

Explore

Explain

Elaborate

Evaluate

Explain to the class that each Expression Card matches one company’s pay plan. Have students collaborate with their groups to ﬁnd each company’s expression, model the expression with algebra tiles, and rewrite the expression to help determine proﬁtability with the company. Monitor and assess students as they are working by asking the following guiding questions: a.

7. 8.

Engage

DOK-1 How can you model Extraordinary Apps’s expression with algebra tiles? Student responses may vary. Use three green squares to represent positive three, and two red squares to represent negative two. Model −x with one red rectangle and +7x with seven green rectangles.

DOK-2 Can you combine negative variables with negative constants? No, you can only combine like terms, so variables can only be combined with variables, and constants can only be combined with other constants.

DOK-2 Is it possible to combine 7xx and 2 2yy? No, you cannot combine different variables together. They must be the same variable to combine them together.

Allow students time to complete Part I of their Student Journals. After Part I, invite the class to a Math Chat to share their observations and learning.

Intervention

Acceleration

FACILITATION TIP Support struggling students by reading the scenarios together. Model underlining, circling, or highlighting the important parts of the word problem. Encourage students to persevere through translating from story to expression to solution.

EXPRESSIONS

Home

FACILITATION TIP Post question 6a where students can see it while they collaborate. Look for volunteers to share their model with the class, or select a student sample yourself to display.

FACILITATION TIP Find students who can remind the class the reason you cannot combine unlike terms. Students may have creative ways for remembering this procedure from prior standards.

Math Chat DOK-2 DOK-2 Which company offers the best plan? Extraordinary Apps is the best company to work with. • DOK-2 If you have 12 apps ready to sell, how can you determine how much you will proﬁt with the company Extraordinary Apps? To determine how much you will make with Extraordinary Apps, you can substitute 12 in for x. 6(12) + 1 = $73 •

Part II 1.

5. 6.

Read the following scenario to the class: Josiah is not sure if Extraordinary Apps really is the best. He decides to look at more companies to make sure he selects the right one to work with. Help Josiah ﬁnd equivalent expressions for the new companies he is reviewing. Explain to students that they will work with the expressions provided on their Student Journals. Students should collaborate with their groups to determine equivalent expressions for each of the new companies. Have students evaluate the ﬁve expressions on their Student Journals to determine whether they are equivalent. After writing their answers and explanations, once again, give students time to discuss their answers and explanations as small groups. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 What are equivalent expressions? Expressions that have the same value

DOK-1 How can you generate an equivalent expression? There are many different strategies. Using knowledge about order of operations and properties of operations can help.

Allow students time to complete their Student Journals and the reﬂection questions. After Part II, invite the class to a Math Chat to share their observations and learning.

FACILITATION TIP After reading the scenario, ask the class 1) What are some reasons Josiah might be questioning whether Extraordinary Apps is the best?; 2) How would finding equivalent expressions for the new companies Josiah is reviewing help him make a decision on which company to choose to market his app? FACILITATION TIP Before students collaborate, determine if you will accept all equivalent expressions or if you want students to create a simplified expression. Be prepared to address expressions that include 1 as a coefficient, that have variables in different order, and/or include different methods for subtraction (adding the opposite). FACILITATION TIP Before this Explore activity, post or repost the Order of Operations and Rules for Integers in your classroom.

183

EXPRESSIONS

Expressions Explore 1 – Combining Like Terms with Rational Coefficients Math Chat DOK-1 What are some strategies you can use to generate equivalent expressions for a given expression? I can use the commutative and associative properties to rearrange the order of each term to solve. Then, I can combine like terms to determine a new equivalent expression. • DOK-2 How does the knowledge of how to combine like terms help you understand equivalent expressions? It allows me to simplify expressions. It shows me different ways to view expressions and understand equivalence. • DOK-2 What is a mathematical term, and why is it important to understand how to combine like terms properly? Give an example. A mathematical term can be a constant, a variable, or a variable with a coefficient. It cannot include addition or subtraction. An example is 4x. • DOK-2 Give an occasion in the real world when people evaluate expressions to see if they are equivalent. People look at compound interest rates to see which loan best fits their situation and goals. •

FACILITATION TIP This review from prior standards about terms is important content for students to be able to simplify expressions. If needed, generate a list of examples of terms (like and unlike) for students to copy on their Student Journal. FACILITATION TIP Before assigning the Exit Ticket, demonstrate ways for students to show their thinking or steps in the work space so you can evaluate misconceptions. Also, clarify whether the equivalent expression must have all like terms combined (simpliﬁed) or just be equivalent.

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

184

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Explain

Elaborate

Evaluate

Intervention

Acceleration

Notes

EXPRESSIONS

Home

185

EXPRESSIONS

Expressions Explore 2 – Distributive Property ACTIVITY PREPARATION Students will apply the distributive property to determine equivalent expressions involving rational numbers.

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 Exit Ticket (per student)

Plan to divide the class into groups of 4. Print a Student Journal and an Exit Ticket for each student.

PROCEDURE AND FACILITATION FACILITATION TIP After reading the scenario, ask the class 1) What would be involved in creating an expression to determine earnings?; 2) How can an area model be used to ﬁnd equivalent expressions? FACILITATION TIP Before giving the Student Journal to students, ask students about their prior experience with area models.

Part I: Using Area Models to Determine Equivalent Expressions 1.

2. 3.

Read the following scenario to the class: Joshia is eagerly wanting to determine how much he is earning for the apps he has created. He created an expression to determine his earnings. His brother, Jakobi, says he doesn’t think he has the correct expression. Help the brothers determine which expression is correct by using an area model to ﬁnd equivalent expressions. Give a Student Journal to each student. Direct students’ attention to the area models on their Student Journals. Have students look at the area model started for Joshia. Discuss the following questions with the class: a.

DOK-1 Joshia has started his area model by decomposing his expression. What step should Joshia take next? Joshia should multiply 4 times 2x.

DOK-1 What is the value of 4 times 2x? 4 times 2x equals 8x.

DOK-1 What step should Joshia take now? Joshia needs to multiply 4 times −5.

DOK-1 What is the value of 4 times −5? 4 times −5 equals −20.

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.

FACILITATION TIP

Remind students to pay close attention to the signs of integers in expressions.

186

e. DOK-1 How can Joshia write his equivalent expression? Joshia’s equivalent expression is 8x – 20. Direct students’ attention to the area model for Jakobi’s expression. Explain to students that they should work with their groups to determine the equivalent expression for Jakobi’s expression using the area model. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 Will a negative number times a negative number result in a negative or positive product? A negative number times a negative number will be a positive number.

DOK-2 What are some common mistakes you or others have made while determining if expressions are equivalent? Student responses will vary. We have to remember that we are multiplying the terms together and not adding or subtracting them to get the numbers in our area model. © Accelerate Learning Inc. - All Rights Reserved

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

EXPRESSIONS

Home

Acceleration

Allow students time to complete Part I of their Student Journals, including the reﬂection question. After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 What similarities do you notice about the brothers’ expressions? The brothers’ expressions are equivalent because all of the same numbers and variables are included. The equivalent expressions found using the area model are the same expression for both brothers. • DOK-2 What differences do you notice about the brothers’ expressions? The terms in Jakobi’s expression are all opposites of the terms in Joshia’s expression. • DOK-2 Are the expressions written by Joshia and Jakobi different? Explain. Joshia’s and Jakobi’s expressions may look different, but they are equivalent expressions. I know they are equivalent because when I used the area models, I found the same expression for each brother.

FACILITATION TIP

•

To facilitate the Math Chat, create a two-column chart to quickly list student comments about similarities and differences between the two expressions.

Part II: Using Distributive Property to Determine Equivalent Expressions 1.

Read the following scenario to the class: Joshia wants to find out the cost and profit he can make for different apps he has created. The company he chose to sell the apps through has a unique expression for each app, depending on its level of difficulty. Joshia has created apps at two different levels. Help Joshia determine equivalent expressions where he can see which app levels are earning and costing the most. Explain to students that they will collaborate with their groups to use the distributive property to determine equivalent expressions Joshia can use for each of the given expressions in the table. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 What strategies can you use to help generate equivalent expressions? Student responses will vary. I can use an area model to help generate equivalent expressions.

DOK-1 What is the product of −2 times −3x? −3 The value of −2 times −3x is 6x.

DOK-1 How can you determine the value of -13 times −15x? −15 I can ﬁnd __3

a negative will result in a positive value, the product of –__3 • –15x will be 5x.

Allow students time to complete Part II of their Student Journals, including the reﬂection questions. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 What are some strategies you can use to generate equivalent expressions for a given expression? I can use factoring, the distributive property, the commutative property, and the identity property of multiplication. I could also use knowledge about the relationship between multiplication and division. • DOK-2 Can you determine whether two expressions are equivalent without knowing the numerical value of a variable? Explain. Yes. One way I can determine if two expressions are equivalent is to simplify both expressions and see if they are the same. •

Before reading the scenario, ask the class 1) What costs are involved when making an app?; 2) Why might the costs when developing and selling apps vary? After reading the scenario, ask the class 3) How can equivalent expressions help Joshia see which app levels are earning and costing the most? FACILITATION TIP Demonstrate the distributive property step by step for students. Model distributing the signs for integers outside of the parentheses. As a support, some students can draw arrows from the factor outside of the parentheses to each factor inside as they multiply to be sure they distribute to all factors inside the group. FACILITATION TIP

of __5 by multiplying __3 • ___ , which equals ___ , or 5. Since a negative times 1 3

FACILITATION TIP

Remind students that just like with percents, multiplication can be expressed 1

as “of” with fractions. In this example, __3 1

of 15 is the same as __3 times 15. Using the word “of” in place of “times” may support some students to successfully determine the value.

FACILITATION TIP Use the Picture Vocabulary in Explain to visually represent some strategies for students. FACILITATION TIP If needed, take time to have students practice simplifying expressions with variables on individual whiteboards.

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EXPRESSIONS

Expressions Explore 2 – Distributive Property DOK-1 What can be challenging about working with negative numbers in expressions? When using the distributive property of multiplication, it is sometimes diﬃcult to remember to distribute the negative sign when multiplying the numbers. • DOK-2 Where might you use equivalent expressions in the real world? I can use equivalent expressions to determine costs of cell phone plans. •

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.

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

Notes

EXPRESSIONS

Home

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EXPRESSIONS

Expressions Explore 3 – Finding Equivalent Expressions Using Properties ACTIVITY PREPARATION Students will explain in writing their reasoning regarding how to evaluate and generate equivalent expressions involving word problems.

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 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 Part I FACILITATION TIP Before reading the scenario, ask the class 1) What would be important for app developers to know when considering improving the apps they created? After reading the scenario, ask the class 2) Why is it important that Joshia ﬁnd the total cost for his app creation?; 3) What information will ﬁnding equivalent expressions give him? FACILITATION TIP

2. 3.

Before students collaborate, direct them to the Order of Operations and Rules for Integers if they are posted in your classroom.

Read the following scenario to the class: Joshia is looking at improving some of his apps and purchasing more products to use for his app creation. He has gathered information and is ready to determine expressions he can use to ﬁnd the total cost for an unknown amount of apps. Help Joshia write expressions and determine if the expressions are equivalent. Give a Student Journal to each student. Have students collaborate with their groups to read each scenario provided in Part I. Then, have students use the properties of operations on the two expressions given to determine whether the expressions are equivalent and provide an explanation. Monitor and assess students as they are working by asking the following guiding questions:

FACILITATION TIP Remind students that they cannot solve for the variable in these expressions but that they can ﬁnd equivalent expressions by simplifying using the Order of Operations and combining like terms.

Part II

FACILITATION TIP

Alternatively, to save time, move on to the Math Chat rather than have students create equivalent expressions in the second column. 190

DOK-1 How were you able to determine whether two given expressions were equivalent? We simpliﬁed both expressions, and if they were the same when simpliﬁed, the expressions were equivalent.

DOK-2 What differences did you notice in some sets of equivalent expressions? Answers will vary. The most common difference I noticed was that some expressions added the cost of the parts together and then multiplied the sum by 4. Other equivalent expressions multiplied each part by 4 and then found the sum of the parts.

Have students solve the four expressions provided in Part II using the properties of operations. Instruct students to collaborate with their groups to determine an equivalent expression for each of the provided expressions. Have students solve their equivalent expressions to prove their equivalency to the given expressions. © Accelerate Learning Inc. - All Rights Reserved

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Monitor and assess students as they are working by asking the following guiding question: a.

4. 5.

Engage

EXPRESSIONS

Home

DOK-2 How can you determine an equivalent expression for the given expression? Answers will vary. I can use another property of operations to ﬁnd an equivalent expression for each given expression.

Allow students time to complete the reﬂection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat •

•

DOK-2 What are some different ways to generate an expression that shows a discount of 20%? Which method do you prefer? Why? One way is to solve for the total cost (or number) of something and then subtract the product of the total cost and 20% (or 0.20). Another way is to ﬁnd the product of 80% (or 0.80) and the total cost. Both solve for the discounted cost. I prefer the second method because the ﬁrst way requires more steps than the second method. DOK-2 Why do you think writing an explanation of equivalent expressions is important in math? When you have to write an explanation, you have to take the time to evaluate how the expression works to solve the word problem. It helps me learn better, and then sharing my writing with my group helps others see and learn things they may not have seen or understood. The same thing happens to me when I learn about how other students solved problems. DOK-2 When you generated an equivalent expression, did you have the same expression as all of your group members every time? Why? No, there are many different ways to generate an equivalent expression that will solve for the same answer, so it is unlikely that we would all solve all the problems the same exact way.

FACILITATION TIP As students finish Part II, post the first question and challenge early finishers to generate as many ways to solve for a discount of 20%. Have them record notes on their Student Journal. Ask for volunteers to share their methods.

Post-Explore 1. 2. 3. 4.

FACILITATION TIP Prompt students to show their thinking in steps so you can assess their progress toward new skills before you return to the Hook.

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EXPRESSIONS

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

Combining Like Terms with Rational Coeﬃcients 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

Distributive Property 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 Equivalent Expressions Using Properties 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

Spiraled Review

Fluency Builder

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

Expressions

EXPRESSIONS

Home

Can be done independently

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

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EXPRESSIONS

Expressions 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

3 194

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER

EXPRESSIONS

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 the order of operations to rewrite expressions.

I can apply properties of operations to add, subtract, factor, and expand linear expressions with rational coeﬃcients.

I can show how quantities are related by rewriting expressions in different forms.

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SCOPE 1

Solve Equations and Inequalities Scope Introduction SCOPE SUMMARY In this scope, students will use their knowledge of rational numbers and operations in order to solve algebraic equations. They will use variables to represent these quantities as a real-world or mathematical problem. Through reasoning, they will solve these problems by creating simple equations and inequalities. They will use mathematical techniques such as the additive inverse and the multiplicative inverse in order to recognize the similarities between algebraic solutions and arithmetic solutions. Student Expectations

7.PAR.3.1 Construct algebraic equations to solve practical problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are speciﬁc rational numbers. Interpret the solution based on the situation. 7.PAR.3.2 Construct algebraic inequalities to solve problems, leading to inequalities of besides the form px ± q > r, px ± q < r, px ± q ≤ r, px ± q ≥ r , where p, q, and r are speciﬁc rational numbers. Graph and interpret the solution based on the realistic situation that the inequalities represent.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In Grade 6, students expanded their knowledge of expressions with variables as equations and inequalities. They began to solve these problems using one-step equations and mathematical reasoning. These arithmetic solutions lead to understanding the concepts behind equations and how to ﬁnd the values that will make each equation true.

The skills to accurately solve simple and two-step algebraic equations are the foundation of algebra. In Grade 8, equations will evolve to include radical and exponent properties. Students will continue to expand their knowledge of algebraic equations throughout high school as they relate them to graphs and real-world scenarios.

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

understand inequalities.

•

listen to prompts about the prior standard and communicate whether they believe the prompts are fact or ﬁction.

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 an unknown value by using clues and a model to create and solve an equation.

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.

196

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Construct Equations In this exploration, groups of students will be tasked with solving a scenario about helping their school’s game booth volunteers solve problems that involve setting up and running the game booth at their school’s fundraiser. Students will: •

draw pictures to model word problems.

•

match drawings to models.

•

write equations to represent the word problems.

Explore 2

Explore 1

EXPLORE ACTIVITIES Solve and Compare Equations In this exploration, students will help the craft booth volunteers solve problems related to crafts and their sales. Students will: •

write, model, solve, and compare equations using algebra tiles.

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

Construct Inequalities In this exploration, students will help the Culinary Club solve problems as they prepare for and run a fundraiser. 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.

create models of inequalities using algebra tiles and an Algebra Inequality Mat.

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

SOLVE EQUATIONS AND INEQUALITIES

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Solve and Graph Inequalities In this exploration, students will help the Math Club solve various problems as they prepare for a fundraiser. Students will: •

write inequalities from word problems.

•

solve the inequalities.

•

graph the solution sets.

Students conclude the exploration with an Exit Ticket and revisit the Hook to solve using their learning from the explorations.

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SOLVE EQUATIONS AND INEQUALITIES

Solve Equations and Inequalities 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 listen to 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. 6.PAR.7.4 Recognize and generate inequalities of the form x > c, x ≥ c, x < c, x ≤ c to explain situations that have inﬁnitely many solutions; represent solutions of such inequalities on a number line.

MaterialsPrinted •

Preparation

1 Sheet of Fact or Fiction Prompts

• •

Print one Fact or Fiction Prompts sheet to read aloud to your students. Another option is to project the prompts by using a digital projector.

SOLVE EQUATIONS AND INEQUALITIES

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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 false.

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.

•

If space is an issue, you can have students use hand signals to indicate their thinking. FACILITATION TIP Students can use table partners to share their reasoning.

FACILITATION TIP Students may not remember the symbol and number line representation for “or equal to.”

Identifying Misconceptions •

FACILITATION TIP

Students may not make the connection between solving equations and solving inequalities. Students may forget what the less-than (<) and greater-than (>) signs represent.

FACILITATION TIP Complete a quick review of greater and less than signs before completing this APK.

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SOLVE EQUATIONS AND INEQUALITIES

Solve Equations and Inequalities Hook – Scorer’s Table ACTIVITY PREPARATION Students will determine an unknown value by using clues and a model to create and solve an equation.

Materials

Preparation

Printed •

• •

1 Scorer’s Table (per class)

Reusable •

•

1 Phenomena Video (per class)

Plan to show the video. Prepare to project the Scorer’s Table 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) How is a basketball game played?; 2) What does the amount of points players can score when making baskets depend on?; 3) How many points does a player earn if they make a basket from the free throw line?

FACILITATION TIP Print and project this scenario so students can read it to themselves, along with you, or volunteer to read it to the class. FACILITATION TIP Be sure to add variable and the greater and less than symbols to a word wall if you have one in the classroom.

4. 5.

6. 200

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: Fabio and Daniel play on the same basketball team. In their ﬁrst game of the season, Fabio scored a number of points. Daniel scored three times the number of points Fabio scored plus some free throws. Together the two boys scored half the points their team scored during the game. They celebrated when their team won the game by a score of 56–51. How many points did Fabio score? How many points did Daniel score? 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 both Fabio and Daniel scored points toward their team score of 56 points. I wonder how many free throws Daniel made. What was the total number of points Fabio and Daniel scored together? I can use math to create an equation from the data and the model, and then I can solve the equation to ﬁnd how many points each scored. Project Scorer’s Table. Explain to students that Daniel added 4 points to his total with free throws he made! The score keeper showed Fabio and Daniel a model so they could determine how many points they each scored. Discuss the following questions: a.

DOK-1 What is a variable? A variable is a letter or symbol that stands for a value that is not known yet.

DOK-1 What does the variable p represent in the model? The number of points scored in the game by Fabio.

DOK-1 How could the number of points scored by Daniel be represented? 3p + 4

Engage

Explore

Explain

Elaborate

Evaluate

Show the Phenomena Video again, and restate the problem. Refer to the Scorer’s Table, and discuss the following questions: a.

DOK-1 Whose points should you solve for ﬁrst? Why? I should solve for Fabio’s points ﬁrst because his score is represented by p. Then, I can use his score to determine Daniel’s score that is represented by 3p + 4. I have to know Fabio’s points to be able to determine Daniel’s points.

DOK-1 What is the equation that solves for p? p + 3p + 4 = __2(56)

DOK-1 Solve the equation. How many points did Fabio score? 6 points

1 p + 3p + 4 = _2_(56)

÷ 4 = 24 ÷ 4

4p + 4 = 28

4p + 4 – 4 = 28 – 4

4p = 24

Consider covering up the equation model before the discussion to encourage students to think independently before seeing the solution.

STEMscopes Tip 4p

p = 6.

DOK-1 How many points did Daniel score? 22 points 3(6) + 4 18 + 4 = 22 3p + 4

e. DOK-2 Solve the scenario arithmetically. 56 ÷ 2 = 28 28 – 4 = 24 24 ÷ 4 = 6 f.

Acceleration

FACILITATION TIP

Part II: Post-Explore 1. 2.

Intervention

DOK- 2 Compare the steps to solve arithmetically with the steps to solve algebraically. Explanations may vary. In both, I would use division and subtraction. First, I would divide by two. Next, I would subtract four. Therefore, I would divide by 4.

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.

SOLVE EQUATIONS AND INEQUALITIES

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Solve Equations and Inequalities Explore 1 – Construct Equations ACTIVITY PREPARATION Students will draw pictures to model word problems. Students will match their drawings to models and write equations to represent the word problems.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Game Booth 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 Game Booth Cards for each group. Cut out and place each set of cards in a resealable bag. If desired, print the cards on card stock, and laminate them for future use.

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What types of fundraisers have you participated in that raised money for your school or an organization?; 2) What might be some problems a group would encounter when setting up and running a game booth at a fundraiser?

FACILITATION TIP

What is an equation? An equation is a math sentence with an equal sign.

Print and post questions 2a–2c. so students can view them as you discuss. Review how an expression is different from an equation.

What does the equal sign tell you? The sides of the equation are equal to each other.

How do you represent an unknown amount in an equation? An unknown amount can be represented with a variable, usually a lowercase letter. Often the variable used is x.

What problem solving strategies can be used to solve problems? Answers may vary but might include guess and check, make a table or chart, draw a picture or diagram, act out the problem, ﬁnd a pattern, make a list, work backward, or solve a simpler problem.

e. How do you choose a problem solving strategy? Answers may vary. Choose the one that makes the most sense to you. Choose one that goes best with the problem. Use your math experience to help you choose the best strategy.

FACILITATION TIP Point out to students the blank space at the bottom of each card where they can create a diagram to model the situation. Expect most models to match the one already shown on the Student Journal unless you specify. 202

Read the following scenario to the class: Your school is hosting a fundraiser to beneﬁt some areas around campus that need improvements. Several school clubs have volunteered to host booths to help. The Athletics Club will host a game booth to help raise money. Students will pay a fee to play games and win prizes. Help the game booth volunteers solve some problems that involve setting up and running the game booth. Review some key points about equations with students by asking the following questions:

Explain that students will work cooperatively on the Game Booth Cards to analyze several math problems and create a diagram or a drawing to model the situation. © Accelerate Learning Inc. - All Rights Reserved

5. 6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

Give one set of Game Booth Cards to each group. Allow groups time to talk through each problem and create a diagram or sketch in the blank space at the bottom of each card. When students have completed the Game Booth Cards, explain that students will now use their Game Booth Cards to complete their Student Journals. Give a Student Journal to each student. Monitor and talk with students as they work. Check for understanding by using the following guiding questions: a.

DOK-1 Which diagram looks the most like your drawing? Answers will vary.

DOK-2 What makes you think you have selected the correct diagram? Answers will vary but should include reasoning that reﬂects the nature of the math problem.

DOK-3 Do you think your equation is correct? Why? Answers will vary. Students should justify their equations. Provide feedback to ensure the equations are written correctly.

DOK-1 How do you know what letter to use as a variable? The problem tells you what variable to use.

Intervention

Acceleration

FACILITATION TIP Note that the bean bag toss card includes the word perimeter. Review as needed.

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.

SOLVE EQUATIONS AND INEQUALITIES

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

Math Chat •

•

DOK-2 How can you decide on which side of the equal sign you should put the variable? It doesn’t matter which side the variable is on because both sides are equal. DOK-2 How can you evaluate your equation to make sure it makes sense? I can use what I already know about solving math problems to help me. I can actually ﬁnd the answer using my math skills and then make sure my equation will give me the answer I am looking for. I can draw or model the situation and use the drawing or model to write an equation. DOK-2 What should you do if a problem does not tell you what letter to use as a variable? Use whatever lowercase letter you want. You may want to avoid letters that can be confused with numbers, like the letter l. DOK-3 Is there more than one way to write the equation correctly? Yes. You can put the variable on either side. Or you might write an equation as an addition problem, but someone else may use the inverse operation, subtraction. For example, 4 + 2 = 6 and 6 – 2 = 4 can mean the same thing. Both can be correct if written the right way. DOK-3 What was the most challenging part of this activity? I thought reading the problem and creating my own diagram was the hardest. I thought communicating with my group and letting every voice be heard was challenging. I thought selecting an equation was hard because the equation presented didn’t always match the one that made the most sense to me.

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 Select three of the Math Chat questions to print and post for students to read before sharing their observations and learning.

FACILITATION TIP Before the Exit Ticket, prepare students by explaining that the written equation is the important part of the assessment. Students might only answer or solve how many points Ming earns rather than writing an equation with the variable p on one side.

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Solve Equations and Inequalities Explore 2 – Solve and Compare Equations ACTIVITY PREPARATION Students will write, model, solve, and compare equations using algebra tiles.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Algebra Equations Mat (per group) 1 Exit Ticket (per student)

•

Reusable • • •

1 Set of algebra tiles (per group) 1 Set of colored pencils (per group) 1 Projector or document camera (per class)

• •

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 Algebra Equations Mat for each group. If desired, print it on card stock, and laminate it for future use. Gather enough sets of algebra tiles and sets of colored pencils for each group to have one set of each. Be prepared to display an Algebra Equations Mat and set of algebra tiles, or display the virtual algebra tiles for the class to see. 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 FACILITATION TIP After reading the scenario, ask the class 1) What types of materials might the Art Club need in their craft booth?; 2) What types of problems related to the crafts and sales in their booth might the Art Club encounter? FACILITATION TIP If you can, locate a balance scale to show to students. Take time to demonstrate how it balances and what it looks like unbalanced. Many students will not have experience with this tool. FACILITATION TIP Before you you ask questions 3a–3e, prepare a list of algebra tile rules to show to students as you go over these representations.

2. 3.

Read the following scenario to the class: The Art Club has volunteered to host a booth for the school fundraiser. The Art Club will host a craft booth with several crafts for sale. Help the craft booth volunteers solve some problems related to crafts and sales in their booth. Display an Algebra Equations Mat and a set of algebra tiles, or display the virtual algebra tiles. Analyze the algebra tiles and Algebra Equations Mat with the students by asking the following questions: a.

What does a green rectangle represent? A green rectangle represents the value of one x.

In algebra, what is the purpose of x? An x represents an unknown amount.

What does a yellow square represent? A yellow square represents one.

How can you represent 4? You can use 4 yellow squares.

e. How could you represent negatives using the tiles? The rectangles always represent x, and the squares always represent one. If the rectangles are green and the squares are yellow, the value is positive. Negative values for x and one are represented with red rectangles and squares. f.

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4. 5.

Explore

Explain

Elaborate

Evaluate

What is the purpose of the Algebra Equations Mat? The mat helps you set up problems that involve variables. The scale reminds you that both sides of the equations are equal. As you solve, you must keep both sides equal.

DOK-1 What should you do ﬁrst when analyzing each problem? First, you should read the problem carefully. Then, you write an equation to solve the problem.

DOK-2 How can you represent 2x 2x using algebra tiles? You can use two green rectangles.

DOK-3 What equation did you write for the balloon animals booth? Justify why you think the equation you wrote is correct. Answers will vary. Students should justify their equations. Provide feedback to ensure the equations are written correctly.

DOK-2 How do you show your solution? First, solve the equation using the algebra tiles and Algebra Equations Mat. Then, show your solution by drawing out what you did on the mat or using numerical representations.

• • •

•

DOK-2 Which side of the scale should you put the variable on? It doesn’t matter. Both sides are equal. DOK-2 Why is it important to identify the variable? The variable tells you what you are solving for. DOK-2 What are two ways to solve for the missing variable? You can use the algebra tiles and Algebra Equations Mat, or you can use numbers. DOK-3 Which way, algebra tiles or numbers, did you prefer using to solve the equations? I preferred using algebra tiles because it was very visual and easy to understand. I preferred using numbers because it was faster. DOK-3 What was the most challenging part of this activity? I thought figuring out how to write the equation was the most difficult because you really had to read carefully and think things through. I thought using the algebra tiles was the most difficult because I don’t have a lot of experience with the algebra tiles. DOK-3 Analyze how a negative coefficient would affect how you would isolate a variable. It would not change the process for solving for the variable. The only change that would occur is that the final sign will be the opposite because you have to divide by a negative to solve.

Post-Explore 1. 2. 3.

Acceleration

FACILITATION TIP Consider providing only red, green, and yellow colored writing utensils to students.

FACILITATION TIP Remind students that it is good practice to read the problem more than once. Some students ﬁnd it helpful to underline, circle, or highlight key words or values. FACILITATION TIP To ensure the equations are written correctly, do the ﬁrst one together to demonstrate the expected form, steps, and how to show the solution.

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

Math Chat •

Intervention

Give a Student Journal to each student. Give a set of algebra tiles, an Algebra Equations Mat, and a set of colored pencils to each group. Optionally, direct students to utilize the algebra tiles virtual manipulatives. Students will work cooperatively to write and solve the equations found in their Student Journals using the algebra tiles and Algebra Equations Mat. Students will record their results in the workspace on their Student Journals and answer the reﬂection questions. Monitor and talk with students as they work. Check for understanding by using the following guiding questions: a.

Engage

SOLVE EQUATIONS AND INEQUALITIES

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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.

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.

FACILITATION TIP Be aware that some students may use alternative methods to solve equations (inverse operations). Encourage students to show their thinking so you can assess if they can meet the criteria for writing and solving equations. 205

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Solve Equations and Inequalities Explore 3 – Construct Inequalities ACTIVITY PREPARATION Students will create models of inequalities using algebra tiles and an Algebra Inequality Mat.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.6 Attend to precision.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Algebra Inequality Mat (per group) 1 Set of Cupcake Booth Cards (per group) 1 Exit Ticket (per student)

Reusable • • •

• • • •

• • •

1 Set of algebra tiles (per group) 1 Resealable bag (per group) 1 Set of colored pencils (per group)

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 two-sided Algebra Inequality Mat for each group. If desired, print it on card stock, and laminate it for future use. Print one set of Cupcake Booth Cards for each group. Cut out and place each set of cards in a resealable bag. If desired, print the cards on card stock, and laminate them for future use. Gather enough sets of algebra tiles for each group to have one. Gather enough sets of colored pencils 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. (Algebra Tiles)

PROCEDURE AND FACILITATION FACILITATION TIP

After reading the scenario, ask the class 1) What does a Culinary Club do?; 2) What supplies would the Culinary Club need to host a cupcake booth at the fundraiser?; 3) What problems might the Culinary Club encounter when preparing for and setting up their cupcake booth?

FACILITATION TIP Note that culinary may be an unfamiliar word for some students. FACILITATION TIP Provide some examples of simple numbers only inequalities ( 4 < y or y < 9 ) for students to copy down, read out loud and explain when you review 2a. “What is an inequality?”

Read the following scenario to the class: The Culinary Club has volunteered to host a cupcake booth for the school fundraiser. The Culinary Club has several ideas planned. You are going to help the club solve some problems as they prepare for and run the fundraiser. Review some key points about inequalities with the students by asking the following questions: a.

What is an inequality? An inequality is a mathematical number sentence where one side is greater than the other.

How is an inequality different from an equation? An equation has two equal sides and an equal sign. An inequality has one side greater than the other side and has a greater than or less than sign.

Why might you need an inequality? Sometimes you may want to ﬁnd an answer that is the minimum or maximum acceptable. For example, speed limits tell you that you can go less than or equal to a certain speed on a road.

What do the symbols <, ≤, >, and ≥ mean? They mean “less than,” “less than or equal to,” “greater than,” and “greater than or equal to.”

e. Display an Algebra Inequality Mat. What is the purpose of the Algebra Inequality Mat? The mat provides a place for you to solve inequalities using algebra tiles. The uneven balance scale reminds you that one side is greater than the other. The greater (heavier) side will be lower than the lesser (lighter) side. 206

f. 3. 4. 5.

Explore

Explain

Elaborate

Evaluate

Why is the Algebra Inequality Mat two sided? The mat has a “greater than” side and a “less than” side.

Give one set of Cupcake Booth Cards, one set of algebra tiles, one set of color pencils, and one Algebra Inequality Mat to each group. Give a Student Journal to each student. Explain that students will work cooperatively to read each of the word problems on the Cupcake Booth Cards, model the problems on the Algebra Inequality Mat, and record their thinking on their Student Journals. Allow groups time to complete the task, including the reﬂection questions. Monitor and talk with students as they work. Check for understanding by using the following guiding questions: a.

DOK-1 What are some key words that let you know you are working with an inequality? Some key words include at least, more than, no more than, and less than.

DOK-2 How can you alter the mat to show “greater than or equal to?” You can add a straight line under the “greater than” sign to change it to “greater than or equal to.”

Engage

DOK-3 Do you think your inequality is correct? Why? Answers will vary. Students should justify their inequality. Provide feedback to ensure the inequalities are written correctly.

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

Intervention

Acceleration

FACILITATION TIP In addition to asking why you might need an inequality, ask for examples for when you might need one. Be prepared with some real-world examples of your own (A teacher may decide that any student that scores greater or equal to 80% on a test does not have to retake it. A coach may decide that only players with at least a GPA of 2.5 may join the high school team...). FACILITATION TIP Find a place to post these symbols and their meanings in the classroom and allow students to make notes on their Student Journal.

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FACILITATION TIP Pose and post the ﬁrst question about key words before students begin collaboration. Alert them to be on the lookout for some key words and be prepared to share them as they work and when they ﬁnish. These words can be added to a word wall.

Math Chats •

• •

•

DOK-2 How do you decide which Algebra Inequality Mat to use? You have to read very carefully to ﬁnd out if you are looking for greater than or less than a certain amount. DOK-1 What words help you know if you are using > or ≥? > means “greater than.” ≥ means “greater than or equal to.” DOK-2 Why are the scales on the Algebra Inequality Mat at different levels? Inequalities aren’t equal, so the scales shouldn’t be equal, either. The greater value should be lower because on a balance scale the side with the most weight on it is lower. DOK-3 Is it possible for one student to use the greater than side of the Algebra Inequality Mat and another student to use the less than side of the Algebra Inequality Mat and both get the correct answer? Explain. Yes. When writing an equation, you can put each side on either side of the scale because each side of an equation is equal. For an inequality, you have to put the larger amount on the lower side of the scale. As long as you put the larger amount on the lower side of the scale, you can use either scale. For example, x < 3 means the same as 3 > x. DOK-4 Name some real-life situations where inequalities are used. If you are grocery shopping on a budget, you will want to spend less than or equal to your budget. If you are competing in a race, you would want your race time to be < everyone else’s race time. If you are cooking meat, you want the internal temperature to be > or equal to a certain amount to avoid getting sick.

FACILITATION TIP If you have a balance scale in the room, use it to demonstrate this concept. It may sound confusing for students to hear, “the greater value should be lower” when they can’t visualize a balance scale.

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 Students may need the scenario read and explained to them. Not all students may be familiar with cupcake liners and batches.

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Solve Equations and Inequalities Explore 4 – Solve and Graph Inequalities ACTIVITY PREPARATION Students will write inequalities from word problems, solve the inequalities, and graph the solution sets.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.6 Attend to precision.

Materials

Preparation

Printed • •

• • • •

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

Reusable •

1 Set of colored pencils (per group)

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. Gather enough sets of colored pencils 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. (Algebra Tiles)

PROCEDURE AND FACILITATION FACILITATION TIP

Before reading the scenario, ask the class 1) What type of booth might the Math Club host for the fundraiser? After reading the scenario, ask the class 2) How might a rubber duck booth be set up?; 3)What types of problems might the Math Club encounter when preparing for and running the lucky duck booth?

Read the following scenario to the class: The Math Club has volunteered to host a booth that allows players to select a rubber duck for a random chance to win a prize. They hope their lucky duck booth will raise a lot of money for the school fundraiser. You are going to help the club solve some problems as they prepare for and run the fundraiser. Review some key points about inequalities with the students by asking the following questions: a.

The solution to an inequality is known as a solution set. Why do you think it is a set? An inequality shows you the largest or smallest possible answer. If the speed limit is 30 mph, you are not allowed to go over 30 mph. You are obeying the speed limit if your speed is 30, 29, 28, etc.

Using the example of a speed limit of 40 mph, how could you show all the speeds allowed using a number line? You can start at 40 on a number line and then include all numbers below 40 to 0 using an arrow on the number line.

Using the same example, how could you indicate that 40 would be included in the solution set? If you include 40, you can mark 40 with a circle that is ﬁlled in. If you don’t include 40, you can mark it with an empty circle on 40.

Using the same speed limit example, would you include negative numbers on your graph? No, you wouldn’t because it doesn’t make sense to have a negative speed.

FACILITATION TIP To facilitate this review, provide students with some printed number lines to practice graphing some inequalities. For example, post a simple inequality like “s > 30” or “s is greater than or = to 30”. Monitor and assess before moving on to graphing inequalities from word problems. Alternatively, using individual whiteboards would be eﬃcient if students can quickly draw number lines of their own.

FACILITATION TIP To simplify this activity, you might have students only graph the inequalities (or only model them) depending on time and standards. If you choose to only graph the equations, you won’t need colored pencils. 208

3. 4. 5.

Give a Student Journal to each student. Give a set of colored pencils to each group. Explain that students will work cooperatively to analyze four word problems, model the problems, and write inequalities to represent the problems. Groups will also solve the problems, graph the solution sets, and answer the reﬂection questions. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Monitor and talk with students as they work. Check for understanding by using the following guiding questions: a. DOK-1 Recall how a negative coefficient affects solving an equation. It would not change the process for solving for the variable. The only change that would occur is that the final sign will be the opposite because you have to divide by a negative to solve. b. DOK-2 Predict whether the same will be true for inequalities and explain. Answers will vary. I think it won’t change because it didn’t change for equations, and solving is the same process. I think it will change because if the sign is flipped, then the inequality will be affected.

FACILITATION TIP Pose and post these two essential questions about open and closed circles before students begin collaborating.

c. DOK-2 Let’s test your theory. Solve −4 −4xx < 8. What is your solution? Test a value to ensure you have solved it correctly. What do you ﬁnd? Answers will vary. I found that x < −2. When I tested a value that was less than −2, −5, I got a false statement. −4(−5) is 20, and that is not less than 8.

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d. DOK-2 What are some key words that indicate you should use an open circle when you graph your solution set? Words like less than or greater than tell you to use an open circle. An open circle means the number is not included in the solution set. e. DOK-2 What are some key words that indicate you should use a closed circle when you graph your solution set? Words like less than or equal to, greater than or equal to, and equal to tell you to use a closed circle. A closed circle indicates the number is included in the solution set. f. DOK-2 How can you model

1 __

2 2x

? You can make the rectangle half as large as

the usual rectangle. You can label the rectangle explains the model means

1 __

. You can create a key that

g. DOK-3 Justify why you think your inequality is correct. Answers will vary. Students should justify their inequalities. Provide feedback to ensure the inequalities are written correctly. 7.

• •

•

Note that question 6f is referring to Kai’s 1 budget (__2 b). This representation with algebra tiles may be a new challenge for some students.

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

Math Chat •

FACILITATION TIP

DOK-1 Why is it necessary to show your solution set with a ray? You don’t know the exact answer when you solve an inequality. When you solve, you might find the highest possible answer, and all of the numbers that are lower than the highest possible answer are also in the solution set. DOK-2 What did you learn when you analyzed Rasul’s and David’s inequalities? It is possible to write inequalities that mean the same thing but look different. DOK-3 What are some ways to write inequalities differently that mean the same thing? You might combine terms or use the distributive property. You might swap the < for a > and swap the location of either side of the inequality. DOK-4 Can you think of a real-world example that could include a negative solution set? If someone is asking how much money you have, the solution set could include negative numbers if you are in debt. A question about winter temperatures might have a solution set with both positive and negative numbers. A question about elevation might have a solution set with negative numbers if the elevations are below sea level. DOK-2 What conclusion can you draw about inequalities and negatives? If the coefficient is negative, then the sign will need to flip.

Post-Explore 1. Have students complete the Exit Ticket to formatively assess their understanding of the concept. 2. Complete the Anchor Chart as a class. 3. Have each student complete their Interactive Notebook. 4. Return to the Hook and instruct students to use their newly acquired skills to successfully complete the activity. © Accelerate Learning Inc. - All Rights Reserved

STEMscopes Tip The Communicate Math – Questioning page, found under the Communicate Math tab of the Teacher Toolbox, includes questioning strategies teachers can use to help challenge and stimulate students’ ability to clarify and extend their mathematical thinking. Examples of possible questioning types are provided.

FACILITATION TIP If you choose to adjust the Explore activity to only include graphing or algebra tile/scale model, edit the Exit Ticket accordingly. You could also have students complete it after they have experience with both models. 209

SOLVE EQUATIONS AND INEQUALITIES

Solve 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

Construct 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

Solve and Compare 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

Construct Inequalities

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

Solve and Graph Inequalities

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

210

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

Two-Step Equations and Inequalities Independent and partner games and other activities that provide students with an engaging way to practice the new concept

Notes

SOLVE 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

SOLVE EQUATIONS AND INEQUALITIES

Solve 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 represent relationships with equations involving variables and positive and negative rational numbers.

What prompts will be used?

What does mastery look like?

SOLVE EQUATIONS AND INEQUALITIES

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I can explain the meaning of the solution of a mathematical situation. I can compare an algebraic solution to an arithmetic solution. I can identify the sequence of operations used in algebraic solutions. I can identify the sequence of operations used in arithmetic solutions. I can solve equations in the form px + q = r and p(x + q) = r. I can solve for the value of a variable in equations and inequalities by using the properties of equality. I can represent relationships with inequalities involving variables and positive and negative rational numbers. I can solve inequalities of the form px ± q > r, px ± q < r, px ± q ≤ r, px ± q ≥ r. I can model an inequality on a graph to interpret the solution.

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SCOPE 1

Scaling Scope Introduction SCOPE SUMMARY In this scope, students will solve problems that involve the scale drawings of different geometric ﬁgures. This involves computing the lengths and areas of a scale drawing. Students will be able to explore how a scale factor affects the area of a ﬁgure by squaring the scale factor. They will also be able to calculate the scale and reproduce a scale drawing using another scale. Student Expectations

7.PAR.4.6 Solve everyday problems involving scale drawings of geometric ﬁgures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In Grade 6, students determined the area of special quadrilaterals and parallelograms through the use of shape composition and decomposition skills. The students learned how to compute areas with given lengths as well as ﬁnding lengths from a given area. This background knowledge will be the framework that allows students to reproduce scale drawings at a different scale than the original.

Computing scale drawings and the ratios of the scales will be important as the students begin rigid motions and congruences in 8th grade. Determining if two figures are the same size and shape based on scaling will help reinforce the concept of determining congruent versus similar figures. As the students enter high school, these ideas will be used to prove theorems and definitions.

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

ratio reason to convert measurement units.

•

manipulate and transform units when multiplying and dividing.

•

identify two truths and a lie by reading statements about the prior standard.

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 scale factor when given the original.

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.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

SCALING

Home

Scaling Drawings In this exploration, groups of students will help solve a real-world scenario involving a park and the need for students to help determine which signs belong to which group of signs so that they can be repainted; and, the need for students to also create a new sign for each type to ensure each sign is scaled correctly. Students will: •

pair scale drawings of rectangles.

•

determine scale through the use of ratios.

•

create additional rectangles by enlarging or reducing a scale model.

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.

Perimeter and Area In this exploration, students will pretend they are volunteering at a national park and are tasked with ﬁnding the scale factors used to update the parks maps and determine how much the sizes of the new maps have been enlarged or reduced from the original maps. Students will: •

determine a scale factor when given original dimensions.

•

ﬁnd the area and perimeter of rectangular locations by using proportions and formulas for area and perimeter of a rectangle.

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

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SCALING

Scaling Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

216

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

SCALING

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. 6.NR.4.7 Use ratios to convert within measurement systems (customary and metric) to solve authentic problems that exist in everyday life.

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. FACILITATION TIP

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

Students should select the third choice as the lie because when converting from gallons to ounces, the number of gallons is multiplied by 128, not divided.

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.

Rather than printing one per student or group, post the prompt for students to see and read along with you. Pace the thinking time by uncovering each statement one at a time. FACILITATION TIP Provide students time to collaborate and calculate each statement with a shoulder partner. Consider posting a standard measurement table for students to view in the classroom. FACILITATION TIP In addition to completing the Foundations Builder, print a standard measurement table for students to refer to as needed.

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SCALING

Scaling Hook – Scaling the Stars ACTIVITY PREPARATION Students will determine the scale factor when given the original size.

Materials

Preparation

Printed •

• • •

1 Scaling the Stars (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Scaling the Stars 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) Have you ever been to an art museum? Describe your experience.; 2) Art museums usually have gift shops that sell prints of famous artworks. What art print would you like to buy?

FACILITATION TIP To engage students, ﬁnd a visual of the original Starry Night by Vincent Van Gogh to project for students to view. FACILITATION TIP In order to support students’ answers to question 3, project Scaling the Stars before asking what they notice, wonder, and see.

4. 5.

FACILITATION TIP Post these questions for students to see as you discuss them. Be prepared to share some real-world examples for when accurate scaling is required. 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: One summer, the Drawert family went on a vacation to New York City. They visited the Museum of Modern Art and viewed the famous painting Starry Starry Night by Vincent Van Gogh. The children were very taken with it, so Mrs. Drawert decided to buy them a print to hang on the wall in their bedroom at home. The print was not the same size as the painting hanging in the museum, but it was proportional. She wondered what the scale factor of the print was when compared with the original painting. 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 dimensions of the painting are proportional. I wonder whether the print is a reduction or an expansion of the original painting. What will the scale factor be? I can use math to determine the scale factor of the print compared with the original painting if I have the dimensions. Project Scaling the Stars. Explain to students that Mrs. Drawert measured the print when she got home and found its length was 12 inches and its width was 10 inches. Then, she looked up the size of the original painting and discovered its length is 36 inches and its width is 30 inches. Discuss the following questions: a.

DOK-1 Is the print smaller or larger than the original painting? It is smaller.

DOK-1 Do the painting and the print need to be proportional to be considered scaled? Yes, they must be proportional.

DOK-1 What is scaling? It means multiplying each dimension of a shape by the same value.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

SCALING

Home

Part II: Post-Explore 1. 2.

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

DOK-1 What is a scale factor? A scale factor is the ratio of the length of a side of one ﬁgure to the length of the corresponding side of the other ﬁgure.

DOK-1 Does each ratio (for both length and width) have to have the same ratio for the ﬁgures to be proportional and to be considered scaled? Yes, both ratios will need to be the same to be considered proportional.

DOK-1 What is the scale factor of the print of Scaling the Stars to the 1

original painting? Show your work. The scale factor is __3. 12

= __; width = ___ = __. Length = ___ 36 3 30 3

FACILITATION TIP These Post-Explore questions are a good time to review and reinforce the terms scale factor and ratio. Ask for volunteers and random students to state the deﬁnitions and be sure the terms are included in their notes, on a word wall, or on their Student Journal.

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SCALING

Scaling Explore 1 – Scale Drawings ACTIVITY PREPARATION Students will pair rectangles that are scale drawings of each other and determine the scale through use of ratios. Based on the ratios, students will create additional rectangles by enlarging or reducing a scale model.

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 National Park Sign Cards (per group) 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. Print a set of National Park Sign Cards for each group. Cut out and place each set in a resealable bag. If desired, print them on card stock, and laminate them for future use.

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION FACILITATION TIP

Before reading the scenario, ask the class 1) What types of activities are there to do at a national park?; 2) What types of information are on hiking trail signs in a park?; 3) What might happen if signs around the park are missing or unreadable? FACILITATION TIP Provide some background information about National Parks. Depending on your students, some may not have experience with National Parks and the signs. Consider showing some images. FACILITATION TIP Prepare students by explaining whether they should use decimals or fractions (or both) when ﬁnding enlargements or reductions. FACILITATION TIP Complete page 2 of the Student Journal “Signs for Trail directions and distances” together so you can model “between ratios,” “within ratios,” and how to simplify. 220

2. 3. 4. 5.

Read the following scenario to the class: Your group is volunteering at a national park. The park has three types of signs: “Hiking Only Trail” signs; signs that give information about the direction and the distances of trails, and “Take Only Pictures, Leave Only Footprints” signs to remind people to care for the park. All signs of a particular type are scaled models of each other. The paint has faded, so your group’s job will be to determine which signs belong to which groups so rangers can repaint messages on them. Because the park loses signs to harsh weather in the winter, your group will also need to make a new sign for each category, making sure each sign is scaled correctly for its category. Give a set of National Park Sign Cards to each group, as well as a Student Journal to each student. Have students work together in their groups to ﬁnd three pairs of signs representing the three different sign groups. After students ﬁnd their preliminary pairs, have them draw a record of their sign groups for the park rangers and check their pairings with ratios. Explain to students that because some of the park signs have been destroyed due to harsh winter weather, they should create a new sign for each category of signs, paying close attention to requirements of enlargement or reduction of signs. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 How do you ﬁnd rectangles that look like they are scale drawings of each other? Student responses will vary. A rectangle usually has a shorter side and a longer side, so we look at the rectangles, and if they look like the short and long sides are about the same proportion, we pair them up. © Accelerate Learning Inc. - All Rights Reserved

7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How did your group check to see if the rectangles were really scale drawings of each other? Student responses will vary. We determined the ratios and made sure they were equal.

DOK-2 What are corresponding sides on two given rectangular signs in the same group? The long sides on both rectangles and then the short sides on both rectangles

Allow time for students to complete their Student Journals, including the reﬂection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat •

•

DOK-2 What is the ratio of any two corresponding sides? If two polygons are similar, then the ratio of the lengths of any two corresponding sides is called the “scale factor.” This means the ratio of all parts of a polygon is the same as the ratio of the sides. DOK-1 How did you find the “between” ratios? I used a pair of corresponding sides and made it into a fraction. I chose another pair of corresponding sides and made sure it was equal to the other pair. DOK-1 How did you find the “within” ratios? I used the length and the width of each rectangle, created a fraction, and checked whether they were equal to each other. DOK-2 What are similarities and differences between “between” ratios and “within” ratios? Both types of ratios help determine whether rectangles are scaled drawings of each other. “Between” ratios compare corresponding parts between two rectangles. “Within” ratios compare the lengths and widths of the same rectangles. DOK-2 What did you notice when you simplified each ratio? Each simplified ratio was equal to each other. Both “between” ratios were equal, and then both “within” ratios were equal. DOK-2 What operations did you use to draw new rectangles that were scaled drawings of given rectangles? I used division when I made a ratio in fraction form and when I simplified it. I used multiplication or division to find the length or width of the new rectangle that was a scaled drawing of a given rectangle. I used multiplication when I enlarged the rectangle and division when I reduced it. DOK-3 When might you have to create scaled drawings in real life? When an architect is drawing plans for a room in the house and wants to design furniture to fit exactly in the room

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

SCALING

Home

FACILITATION TIP Model drawing a variety of shapes with corresponding sides and labeling them to support student learning. Students could draw along with you on their Student Journal. FACILITATION TIP Allow time for students to draw a few small sketches on the reﬂection questions to solidify their understanding of length, width, corresponding sides, and where the two values in each different ratio come from. FACILITATION TIP Remind students that other polygons besides rectangles and squares can be scaled using ratios. As a challenge, consider asking students about polyhedrons. Is it possible to scale a polyhedron? How would you scale a polyhedron if you could?

FACILITATION TIP Be prepared with some visual examples of real-world applications for scaled drawings: maps, landscape plans, plumbing plans, etc. As an extension, ask students if they know of some models that are diﬃcult to display to scale (atoms, molecules, distances between planets, etc.) FACILITATION TIP To scaffold this Exit Ticket, have some students create more than one sign (enlargement and reduction). Other students may need clear, explicit directions to either enlarge or reduce the sign on the Exit Ticket.

221

SCALING

Scaling Explore 2 – Perimeter and Area ACTIVITY PREPARATION Students will determine a scale factor when given original dimensions and revised dimensions for a rectangular ﬁgure. Students will ﬁnd the area and perimeter of real-life rectangular locations by employing proportions and formulas for the area and perimeter of a rectangle.

Standards for Mathematical Practice • •

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

Materials

Preparation

Printed • • •

1 Student Journal (per student) 1 Set of National Park Task Cards (per group) 1 Exit Ticket (per student)

• • •

Reusable •

2 Resealable bags (per group)

Plan to divide the class into groups of four to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of the National Park Task Cards for each group. Cut out and place each set of National Park Task Cards Part I in a resealable bag labeled “Part I.” Cut out and place each set of National Park Task Cards Part II in a resealable bag labeled “Part II.” If desired, print the cards on card stock, and laminate them for future use.

PROCEDURE AND FACILITATION Part I: Finding the Scale Factor FACILITATION TIP Before reading the scenario, ask the class 1) Why might maps handed out at national parks need to be updated periodically?; 2) What makes maps “user friendly”? After reading the scenario, ask the class 3) What are “scale factors”? FACILITATION TIP Provide some background information about National Parks. Depending on your students, some may not have experience with National Parks and the signs. Consider showing some images.

2. 3. 4. 5.

FACILITATION TIP Model one of the National Park Task Cards for students; point out that the simplified fraction is the scale factor. Consider having students write “scale factor” above “simplified” on the table. FACILITATION TIP Reflection question number 5 is essential for student learning. Emphasize the order of new before old. The Exit Ticket requires independent fluency with this skill. 222

6. 7.

Read the following scenario to the class: Your group is volunteering at a national park. The rangers have plans to update their existing park maps to make them more user friendly for visitors. Your group will work as volunteers to ﬁnd the scale factors used to update the park’s maps. Your job is to determine how much the sizes of the new maps have been enlarged or reduced from the original maps. Give a Student Journal to each student. Give Part I of the National Park Task Cards to each group. Have students identify the scale factor of each set of maps. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 What operation is your group using to ﬁnd the scale factor? We are using division.

DOK-1 Can you use addition or subtraction to ﬁnd the scale factor? No, we cannot because we cannot add or subtract the same number to each dimension and still get a similar ﬁgure.

DOK-2 How do you make sure you are ﬁnding the scale factor showing how the new map has changed in size from the old map? Student responses will vary. When setting up the fractions to compare sizes, the dimension of the new map needs to be the numerator and the dimension of the original map needs to be the denominator. Also, the fraction must compare corresponding sides.

Allow time for students to complete Part I of their Student Journals, including the 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

• •

DOK-1 What kind of numbers are the scale factors when the figures get smaller? The scale factors are fractions or decimals greater than 0 but less than 1. DOK-1 What kind of numbers are the scale factors when the figures get bigger? The scale factors are positive numbers greater than 1. DOK-2 Why is there a specific way to set up a fraction to find the scale factor? You must divide the new dimension by the old dimension. They are not reversible because the order matters, as it is a difference between making the new figure smaller or bigger. Also, you must make sure to use corresponding sides of the rectangle.

Part II: Finding Area and Perimeter of Rectangles Using Proportions 1.

Read the following scenario to the class: Your group is still volunteering at a national park. The park rangers have asked your group to create maps of certain areas of the park. After you create the maps, the rangers want you to provide them with information based on the perimeter and area of the newly mapped locations. Distribute Part II of the National Park Task Cards. Have students ﬁnd the perimeter and/ or area of mapped locations to help determine data that park rangers need to share information with visitors or purchase supplies. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a. DOK-1 What information do you need to be able to determine the perimeter or area of a rectangular area? Student responses will vary. We need to know the length and the width. b. DOK-1 How do you ﬁnd the length and width of the actual area when given the scale and dimensions on the map? Student responses will vary. We have to set up a proportion that has the scale on one side of the equation. On the other side of the equation, the dimension on the map is the numerator and the dimension in the real world is the denominator, marked unknown. Then, we solve for the unknown.

4. 5.

Allow time for students to complete Part II of their Student Journals, including the reﬂection questions. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat •

• • • •

DOK-2 When given a scale and measurements on a map, how do you build a proportion to find measurements that are the actual length or width of something? Set up an equation. Use the scale as the fraction on one side of the equation. Include the measurement units. On the other side of the equation, use length or width on the map as the numerator. Use x as the denominator. Then, solve for x to find the corresponding length or width in the real world. DOK-2 Can you apply a scale factor to only one dimension and not the other? No. The scale factor has to be applied to all dimensions to create rectangles that are scaled to each other. DOK-1 What is the formula for the perimeter of a rectangle? P = 2l + 2w DOK-1 What is the formula for the area of a rectangle? A = l · w DOK-2 Can you think of a time you had to find the life-sized area or perimeter of something that was represented on a smaller scale? Explain. If you are looking at a blueprint of a house, you would need to have the dimensions to determine how much tile or hardwood to put in your house for flooring.

Post-Explore 1. 2. 3. 4.

Acceleration

FACILITATION TIP

Math Chat •

Intervention

SCALING

Home

If students are struggling to explain what kind of numbers the scale factors are when enlarging or reducing, direct them to simplified ratios on the chart. Create a T-chart with enlarging numbers and reducing numbers so they can see the similarities. FACILITATION TIP Consider asking students what would happen if they did not follow the specific way and accidentally reversed the numbers. Perhaps come up with some real-world examples or humorous “what ifs.” FACILITATION TIP After reading the scenario, ask the class 1) Why would maps focused on certain areas in a park be useful to visitors?; 2) What information does finding the perimeter of a location provide?; 3) What information does finding the area of a location provide? FACILITATION TIP Post this scenario for students to read along with you. Provide vocabulary support for perimeter and area. A common error is for students to confuse these two terms repeatedly. Help them find a unique way to memorize their meanings and uses. FACILITATION TIP Before students begin collaborating, prepare them for the question about the fencing around the pool. Address whether space will be required between the pool and the fence for any reason before they begin. FACILITATION TIP Point out the unusual ratio of metric and standard on the first card. Centimeters and feet create a unique ratio for construction. FACILITATION TIP Depending upon student experience with setting up and solving these equations, consider modeling several examples. FACILITATION TIP Post these formulas for perimeter and area for different polygons in the classroom, then have students copy them on their Student Journal, or post on a word wall if you have one.

223

SCALING

Scaling 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

Scale Drawings 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

Perimeter and Area 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

224

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

SCALING

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

Similar Figures Using Scale Factors Independent and partner games and other activities that provide students with an engaging way to practice the new concept

225

SCALING

Scaling 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

3 226

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

SCALING

Home

What prompts will be used?

What does mastery look like?

I can compute unknown lengths through the use of proportions shown in tables.

I can compute unknown lengths through the use of proportions shown in equations.

I can use a given scale to determine the dimensions of ﬁgures.

I can identify the impact that a scale has on the length or area of a ﬁgure.

I can identify the scale factor of two ﬁgures.

I can make connections between scale drawings and similarity.

I can create a scale drawing at a different scale.

227

SCOPE 1

Angles Scope Introduction SCOPE SUMMARY In this scope, students will be able to measure and ﬁnd the unknown angles in a ﬁgure. By understanding types of angles and angle relationships, students will be able to use algebra in solving these geometric problems.

VERTICAL ALIGNMENT

Student Expectations

7.GSR.5.1 Measure angles in whole nonstandard units. 7.GSR.5.2 Measure angles in whole number degrees using a protractor.

Background Knowledge

Future Expectations

In fourth grade, students were introduced to terms such as lines, segments, rays, and angles. They discovered perpendicular and parallel lines and their deﬁnitions based on these terms. Students used fractions of a circle to determine angles and angle measurements. They gained the understanding of angle measurements that will be used to measure angles both with nonstandard and standard units of measure. The knowledge of these basics will be expanded on within this scope.

The knowledge of angles will play a role as students progress with geometry. In 8th grade, students begin studying the congruency of shapes and angles and how they relate to different series of rigid motions. Their work with transformations will continue through geometry in high school as they begin to analyze the difference between congruence and similarity. Angle rules and patterns will be the groundwork for creating new theorems in geometry.

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

draw right, acute, and obtuse angles based on the relationship of the angle measure to 90 degrees.

•

determine an angle’s measure in relation to the 360 degrees in a circle through division or as a missing factor problem.

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 angles in a real-world problem.

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.

228

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ANGLES

Home

Measuring in Nonstandard Units In this exploration, students will use parts of a circle to measure angles. Students will: •

determine the angle measures of the different furniture.

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.

Measuring Angles In this exploration, students will use protractors to measure angles in whole-number degrees. Students will: •

use the protractors to ﬁnd the angle measure of each piece.

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

Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________

229

ANGLES

Angles Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

230

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ANGLES

Home

ACCESSING PRIOR KNOWLEDGE Students will dialogue with classmates about their understanding of angles through a Four Corners discussion. This element is designed to uncover student misconceptions; it should not be used as a summative assessment. 4.GSR.7.1 Recognize angles as geometric shapes formed when two rays share a common endpoint. Draw right, acute, and obtuse angles based on the relationship of the angle measure to 90 degrees. 4.GSR.7.2 Measure angles in reference to a circle with the center at the common endpoint of two rays. Determine an angle’s measure in relation to the 360 degrees in a circle through division or as a missing factor problem.

Materials

Preparation

Printed •

•

1 Set of Four Corners Slides (per class)

•

If not assigning the APK digitally, 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 measuring angles. 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.

Identifying Misconceptions • •

•

Students who choose slide 1 think that the half circle is the whole, instead of 180°. They don’t understand the part shown is a quarter of 360°, or 90 degrees. Students who choose slide 2 think because the circle is divided into eighths they must multiply the total angle measure of 8 to solve. They don’t understand that they are needing to solve 8x = 360, or use dividing, in order to ﬁnd the missing angle measure. Students who choose slide 3 think that they can align an angle anywhere on the protractor. They do not recognize that the angle is less than 90° and would be classiﬁed as an acute angle. Slide 4 is the correct slide.

FACILITATION TIP To give students time to think independently, project each Four Corners slide individually and allow students to take notes. FACILITATION TIP Provide sentence frames to help students focus their discussions: “I chose this corner because I see...” or “I think this corner has the best explanation about angles because..” FACILITATION TIP This Foundation Builder includes some interesting images. A simple classroom discussion would provide an effective review for students. It would also provide a quick check for you to note student knowledge.

231

ANGLES

Angles Hook – Raining on Playground ACTIVITY PREPARATION Students will determine the angles in a real-world problem.

Materials

Preparation

Printed •

• •

1 Raining on Playground (per class)

Reusable •

•

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Raining on Playground 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 Part I: Pre-Explore 1. FACILITATION TIP

Print and project this scenario and read it along with students. Coach students to note the essential words and phrases (obtuse, no way, easily, disagrees). 3.

FACILITATION TIP

4. 5.

Be prepared for students to wonder how Monica can physically measure using the circle and protractor. FACILITATION TIP These discussion questions create a great opportunity to review angles. Students may not have recent experience with the vocabulary. Take time to use the Picture Vocabulary (under Explain). Students should record visual examples and deﬁnitions in their notebooks. Depending on your students, try using some physical responses to model acute, obtuse, and right angles with arms or hands. 232

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: Monica is disappointed that she can’t enjoy going outside to relax because of the rain. She is bored and watching to see if the rain and wind will move the swings. Suddenly the wind picks up and the swings begin to sway. Her classmate tells her there is no way that the wind will be strong enough to cause an obtuse angle. She disagrees and believes that the swing by going back and forth will easily make an obtuse angle. They decide to watch and see who is correct. 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. Answers will vary. I notice that Monica and her classmate are wanting to measure angles. I wonder if she will try to use a protractor or circle measure to determine the angle measures. I wonder how strong the wind is blowing. What is the most common angle that people swing on a swing? I can use math to determine the angle of the swing. Project Raining on Playground. Explain to students that Monica measured using a circle and a protractor. She measured when the swing went from the neutral position to the top for the ﬁrst half of the swing, and it was the same measure from the neutral position to the back of the swing. Discuss the following questions: a.

DOK-1 In what units will the angles be measured? Degrees or circle parts

DOK-1 What type of angle is created with each swing? How can you tell? I can visualize a part of a circle and determine if it is more or less than 90 degrees. If it is more than 90 degrees, it is obtuse. If it is less than 90 degrees, it is acute.

DOK-1 How can you ﬁnd the kind of angle? I will measure each angle, and then I will double the measurement to ﬁnd the total angle of that swing.

Engage

Explore

Explain

Elaborate

Evaluate

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and repeat the scenario. Refer to Raining on Playground, and discuss the following questions: a.

DOK-1 What would be the circle part of the ﬁrst swing? What kind of 1 2 1 angle is it? __8 , so together the whole swing is __8 or __4 of 360 degrees. This is 90 degrees or a right angle.

DOK-1 What would be the degree measure of the second swing? What kind of angle is it? It measures 50°, and doubling that makes 100°. That is an obtuse angle.

DOK-2 Who was correct, Monica or her classmate? What likely error was made? Monica was correct because the ﬁrst angle was a right angle and the second was obtuse. The classmate might not have doubled the angle and only considered the center to the top of the swing, or they may have forgotten what is classiﬁed as an acute angle.

Intervention

Acceleration

ANGLES

Home

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.

233

ANGLES

Angles Explore 1 – Measuring in Nonstandard Units ACTIVITY PREPARATION Students will use parts of a circle to measure angles.

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 Circle Units (per student) 1 Set of Furniture Cards (per group) 1 Exit Ticket (per student)

Divide students into groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Circle Units on transparency ﬁlm for each student. Print a set of Furniture Cards for each group. Cut out the cards, and place each set in a resealable bag for each group. If desired, print the cards on card stock, and laminate them for future use.

Consumable •

1 Sheet of transparency ﬁlm (per student)

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION FACILITATION TIP

Project this scenario for students to read along with you. Guide students to note the essential math vocabulary and phrases (shapes, measurement of angles, ﬁt, etc.) FACILITATION TIP Prior to asking students to use the Circle Units to measure, assess their prior knowledge of circles, angle measures in circles, and estimating angle measures. FACILITATION TIP Consider providing students with some protractors and time to explore the angles on the Circle Units page.

234

2. 3.

Read the following scenario to the class: Melanie is a design drafter. She takes people’s ideas for the design of their home and creates a technical drawing so they can see their ideas come to life. She uses shapes to represent different types of furniture and the measurement of angles to determine whether items will fit where they’re supposed to be placed. Help her find the measurement of the items so she’ll know whether or not they will fit into each room. Give a bag with the Furniture Cards to each group. Explain to students that they will be working in their groups to determine the angle measures of the different furniture in the bag. Tell students that to find the measurement of each angle, they will need to use their Circle Units to find out how many sections the angle takes up on the circle. Monitor and assess students as they are working by asking the following guiding questions: a.

FACILITATION TIP

DOK-1 How many units does it take to create an acute angle? Anything less than 4 units is an acute angle.

Remind students or coach them to make the connections between ratios, fractions, analog clocks, and scaling when using the Circle Units to measure these angles.

DOK-1 How many units does it take to create an obtuse angle? Anything more than 4 units is an obtuse angle.

DOK-1 How many units does it take to create a right angle? A right angle is exactly 4 units.

DOK-2 What happens when the shape doesn’t line up with the units on the circle? When the shape doesn’t line up, you estimate by going with the closest unit. © Accelerate Learning Inc. - All Rights Reserved

5. 6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

Give students enough time to draw each shape and label its interior angles. Allow time for students to complete their Student Journals, including the reﬂection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

DOK-2 How do you use the circle to determine the number of units in the angle? When you’ve chosen an angle to measure, you line up one side of the shape with one of the lines on the circle. Then, you ﬁnd the closest line that matches the other line in the angle. However many units you count is the measure of the angle. • DOK-2 What does this technique tell us about what it means to measure angles? When you’re measuring angles, you’re really telling what part of a circle the angles are. • DOK-3 In the real world, in what other situations would you need to measure angles? For example, builders need to build a roof at the proper angle to allow for snow and rain runoff. •

Post-Explore

2. 3.

Acceleration

FACILITATION TIP Provide clearly posted expectations or constraints for the drawings and labels. Consider modeling using one of the six shapes. FACILITATION TIP

Math Chat

Intervention

ANGLES

Home

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.

If students are still struggling to use the circle, label each section on a circle and have the students do the same on their own copy. Post a large example on your classroom wall with angles measured and labeled. FACILITATION TIP Take time to ﬁnd additional engaging realworld examples for when accurate angle measurements are helpful. Consider space exploration, sports plays, bridge construction, or favorite video game landscapes. FACILITATION TIP Before assigning this Exit Ticket, consider allowing students access to a transparent protractor or a transparency copy of the Circle Units.

235

ANGLES

Angles Explore 2 – Measuring Angles ACTIVITY PREPARATION Students will use protractors to measure angles in whole-number degrees.

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 Protractor (per student, optional) 1 Set of Puzzle Pair Cards (per pair) 1 Exit Ticket (per student)

•

Consumable •

1 Sheet of transparency ﬁlm (per student, optional)

Divide students into pairs. Print a Student Journal and an Exit Ticket for each student. Print a Protractor on transparency ﬁlm for each student, or gather protractors for each student. Print a set of Puzzle Pair Cards for each pair. Cut out the cards, and place each set in a resealable bag for each pair. Cards should be completely cut so that they have to be put together like a puzzle. If desired, print the cards on card stock, and laminate them for future use.

Reusable • •

1 Protractor (per student) 1 Resealable bag (per pair)

PROCEDURE AND FACILITATION FACILITATION TIP If protractors were not used in Explore 1, take time to model how to use them effectively. Students may not have used this skill since ﬁfth grade. If using the protractor printed on transparencies, have them precut. Show students how to read the measures for acute and obtuse angles. Students need to slow down and carefully place the center of the protractor on the center of the angle. Sometimes they need to be shown how to extend the lines to read the angle. FACILITATION TIP Use these guiding questions (4a–4e) to model accurate measuring before students begin collaborating. FACILITATION TIP Consider helping students mark the vertices on the Puzzle Pair Cards angles with a dot before they begin using the protractors. 236

2. 3.

Read the following scenario to the class: Penny’s Puzzles is a place in town where people go to relax, hang out, drink coffee, and do puzzles. One of Penny’s customers is having some trouble putting their puzzle together. Use your knowledge of angles to help find a match for each puzzle piece. Give a bag with the Puzzle Pair Cards to each group. Explain to students that they will be working in their pairs to find a match to each puzzle piece. One student will pick up all the number cards, and another student will pick up all the letter cards. They will use the protractors to find the angle measure of each piece and work together to match them all up. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 What unit do you use to measure angles? Angles are measured in degrees.

DOK-2 How do you use a protractor to ﬁnd the measure of an angle? Place the vertex of the angle at the center of the protractor. Line up one side of the angle with the bottom line of the protractor. Find the measurement closest to the other line that touches the protractor to ﬁnd the angle measurement.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How do you know if you should read the top set of numbers or the bottom set of numbers on a protractor? When an angle opens to the right side of the protractor you read the bottom numbers. If an angle opens to the left side of the protractor, you read the top numbers.

DOK-1 How many degrees does it take to create a straight angle? A straight angle is exactly 180 degrees.

e. DOK-2 What happens when the shape doesn’t line up with the degrees on the protractor? When the shape doesn’t line up, you estimate by going with the closest mark on the protractor. 5. 6.

Allow time for students to complete their Student Journals, including the reﬂection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 Why are there two sets of numbers on a protractor? There are two sets of numbers on a protractor so you can measure angles from either direction. • DOK-2 How does using a protractor compare to using the circle units from the last Explore? Using a protractor is very similar to using the circle units. The main difference is that you can be more precise with the protractor and know the actual degrees of the angle. • DOK-2 What do you need to do to get an accurate measurement when using a protractor? We need to make sure the vertex is on the correct spot on the protractor. We need to line one side up with the zero line. We need to make sure we are using the correct number scale. •

Intervention

Acceleration

ANGLES

Home

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.

Post-Explore 1. 2. 3. 4.

FACILITATION TIP In addition to using this Exit Ticket as a formative assessment, it could be used as a pretest for this scope to see how many students have protractor skills.

237

ANGLES

Angles 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

Measuring in Non-Standard Units 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

Measuring Angles 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

238

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ANGLES

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

Measure Angles

Can be done independently

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

239

ANGLES

Angles 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

3 240

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ANGLES

Home

What prompts will be used?

What does mastery look like?

I can recognize that angles are formed by the endpoint of two rays.

I can draw an angle by using a protractor.

I can measure an angle by using a protractor.

I can determine an angle’s measure by relating it to the 360 degrees in a circle.

I can connect nonstandard units to standard units of angle measurement (degrees).

241

SCOPE 1

Angle Relationships Scope Introduction SCOPE SUMMARY In this scope, students will be able to write and solve simple and multistep equations in order to ﬁnd an unknown angle in a ﬁgure. They will identify supplementary, complementary, vertical, and adjacent angles in order to determine these measurements. By understanding angle relationships and theories, students will be able to use algebra in solving these geometric problems. Student Expectations

7.GSR.5.3 Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve equations for an unknown angle in a ﬁgure.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous years, students were introduced to terms such as lines, segments, rays, and angles. They discovered perpendicular and parallel lines and their deﬁnitions based on these terms. They gained the understanding of angle measurements that will be used to solve supplementary, complementary, vertical, and adjacent angle problems. The knowledge of these basics will be expanded on within this scope.

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

measure angles in whole nonstandard units.

•

measure angles in whole number degrees using a protractor.

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 which angles are complementary, supplementary, vertical, or adjacent.

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.

242

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Supplementary and Complementary Angles In this exploration, groups of students will help solve a scenario which involves helping two people review a bike trail proposal to see if it meets their Parks Department’s requirements for its Rustic Oak Park improvements. Students will: •

measure angles.

•

calculate the sum of supplementary and complementary angles.

•

compare and contrast supplementary and complementary angles.

Explore 2

Explore 1

EXPLORE ACTIVITIES

In this exploration, students will help solve a scenario that is extended from the ﬁrst exploration. Here, students are tasked with examining a proposal for a park to look for mistakes and determine whether it meets the Parks Department guidelines before they submit it. Students will: •

locate, measure, compare, and contrast vertical and adjacent angles.

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

Vertical and Adjacent Angles

ANGLE RELATIONSHIPS

Home

Multistep Angle Problems In this exploration, students will deal with other Parks Department scenarios and ensuring a proposal meets the park’s guidelines. Students will: •

apply their knowledge of supplementary, complementary, vertical, and adjacent angles to solve multistep problems.

Students conclude the exploration with an Exit Ticket and revisit the Hook to solve using their learning from the explorations.

243

ANGLE RELATIONSHIPS

Angle Relationships Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

244

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will read different student responses to posed questions on the prior standards, decide whether they agree or disagree with each student, and explain their reasoning. 7.GSR.5.1 Measure angles in whole non-standard units.

ANGLE RELATIONSHIPS

Home

7.GSR.5.2 Measure angles in whole number degrees using a protractor.

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 To save time, project the Agree or Disagree statements and read through them one at a time with students. After reading each one, have students stand up, put their hand up, and discuss with a partner. Move on to the next statement after students have shared. Students can return to their seats between statements or ﬁnd a new partner in the room for each statement.

a. Disagree with Isabella FACILITATION TIP

b. Disagree with Ryan c. 8.

Agree with Nicole

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 forget how to measure with circle measurements and the total measures for a circle. Students may forget how to measure with a protractor and how to use proper placement to measure an angle. Students may forget the vocabulary of acute and obtuse angles.

Consider assessing student measurement skills after Scope 11 (Angles) with protractors . Have a variety of sizes and types of protractors available for students to use. Some may need smaller and transparent protractors, and others may prefer larger opaque tools. FACILITATION TIP Angle measurement vocabulary from prior standards is essential. In addition to clarifying verbally with students, post visual examples and deﬁnitions for students to refer to on a word wall. Students can later record notes on their Student Journal.

245

ANGLE RELATIONSHIPS

Angle Relationships Hook – Yarn Art ACTIVITY PREPARATION Students will determine which angles are complementary, supplementary, vertical, and adjacent in a plan for artwork.

Materials

Preparation

Printed •

• • •

1 Yarn Art (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Yarn Art 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) What is your favorite type of artwork (for example: paintings, sculptures, sketches, etc.)?; 2) What do you know about modern art?; 3) What types of materials and media do artists use when creating their art?

FACILITATION TIP To help students visualize the situation, project Yarn Art before asking them to make observations.

4. 5.

6. 246

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: Indira is working to create a new medium for art. She wants to create modern art with mixed media, such as yarn, paint, and grease pencil. She will draw out her plan by using thick lines of colored grease pencil. It will feature colorful lines of yarn intersecting to form angles. Under the angles will be textured paint. Indira wonders how the angles she creates will be related to one another. 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 Indira is using straight lines and creating angles. I wonder what types of angles Indira is creating. What angle relationships will be created by Indira’s design. I can use math to determine what types of angles are created in Indira’s artwork. Project Yarn Art. Explain to students that Indira has created the plan for her new unique artwork. She has used three colorful straight lines to create many angles. Discuss the following: a.

DOK-1 How many angles did Indira create with the yarn? There are ﬁve labeled angles (A, B, C, D, and E), but there are more angles if they are combined in different ways.

DOK-1 What are complementary angles? Two angles whose sum is 90°.

DOK-1 What are supplementary angles? Two angles whose sum is 180°.

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 Yarn Art, and discuss the following questions: a.

DOK-1 What angles are complementary angles?

A and

DOK-1 What angles are supplementary angles? C

A and

B and

DOK-1 What are vertical angles? The pairs of opposite angles created by two intersecting lines

DOK-1 What angles are vertical angles?

A and

B and

e. DOK-1 What are adjacent angles? Two angles that share a common side and vertex and do not overlap f.

DOK-1 What angles are adjacent angles? and D, D and E, and A and E

A and

B and

DOK-1 Which angles are congruent? How do you know? A and congruent. I know because vertical angles are congruent.

If examples and deﬁnitions for complementary and supplementary are not already posted in the classroom or on the Student Journal, take time to review and post for students to reference throughout the scope.

ANGLE RELATIONSHIPS

Home

FACILITATION TIP This scope is very vocabulary heavy. Consider supporting the vocabulary development in various ways with ﬂashcards, visual note taking, partner quiz practice, and charades/picture drawing type games with whiteboards.

C are

247

ANGLE RELATIONSHIPS

Angle Relationships Explore 1 – Supplementary and Complementary Angles ACTIVITY PREPARATION Students will measure angles and calculate the sum of supplementary and complementary angles. Students will compare and contrast supplementary and complementary angles.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Rustic Oak Park Map (per group, optional) 1 Rustic Oak Park Map (per class) 1 Exit Ticket (per student)

Reusable • •

• • • • •

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. Be prepared to project the Rustic Oak Park Map for the class to see. Optionally, print one Rustic Oak Park Map per group to provide a larger workspace for students. Gather enough protractors for each group or each student to have one.

1 Protractor (per group or per student) 1 Projector or document camera (per teacher)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What activities are typically available in a city park?; 2) What types of improvements might a city make to its parks?; 3) Who might the city ask for input about making park improvements? FACILITATION TIP The Rustic Oak Park Map is the current map (no upgrades made yet) of the park. Be sure to project it and discuss with students for questions 3a–3c before projecting or distributing the Student Journal to each student.

FACILITATION TIP Be prepared to make color copies and enlarge the Rustic Oak Park Map in order for students to be able to read and draw accurately. 248

Read the following scenario to the class: The City Parks Department is planning an improvement to Rustic Oak Park. They have invited the community to provide suggestions for upgrading the park. Jaime and Aliyah would like to see the city upgrade the park with a bike trail. They have created a proposal to present to the Parks Department. Jaime and Aliyah would like your help to look over their bike trail proposal and see if it meets the Parks Department’s requirements for Rustic Oak Park improvements. Display the Rustic Oak Park Map on the board. Explain to students that this is the current map of the park. The bike trail proposal will create changes to the current park. Brieﬂy review the features of the Rustic Oak Park Map by asking the following questions: a.

DOK-1 How can you determine cardinal directions? Use the compass rose to ﬁnd north, south, east, and west.

DOK-1 What are the names of the current trails? Sunset, Happy, Main, Duck, and Shady Trails

DOK-2 Is Sunset Trail connected to Duck Trail? If so, how? Yes, according to the map, Sunset Trail connects directly with Duck Trail, past the fountain.

Give a Student Journal to each student. Give one protractor either to each student or to each group. Optionally, also give one Rustic Oak Park Map to each group. The larger format will make it easier for some students to complete the activity. If students use the larger map, instruct them to record their thinking in their Student Journals. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Have students work cooperatively to read Jaime and Aliyah’s bike trail proposal that is found at the bottom of their Student Journals. Instruct students to use a protractor, including the protractor’s straight edge, to sketch Jaime and Aliyah’s plan onto the current park map on their Student Journals. After adding the proposals to the map, students will measure the angles created by the trails and determine if the bike trail proposal meets the Parks Department’s guidelines. Monitor and talk with students as they work. Check for understanding by using the following guiding questions: a.

DOK-3 What do you notice about the trails in the park? The park has only straight trails.

DOK-2 How can you tell which set of numbers you should read on the protractor? If the angle is acute, read the smaller number. If the angle is obtuse, read the larger number. If an angle is pointed to the right, read the bottom number. If an angle is pointed to the left, read the top number.

DOK-1 What is the angle measurement of a right angle? 90 degrees

DOK-1 What is the angle measurement of a straight line? 180 degrees

e. DOK-1 According to the City Park Improvement Guidelines for Rustic Oak Park, what is a complementary angle? Complementary angles are pairs of angles with a sum of 90°. f.

7. 8.

DOK-1 According to the City Park Improvement Guidelines for Rustic Oak Park, what is a supplementary angle? Supplementary angles are pairs of angles with a sum of 180°.

Students will use the completed Rustic Oak Park Map to complete the remainder of their Student Journals. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat •

•

• • •

DOK-4 Design a change to the bike trail proposal to meet the guidelines. Add a trail that would bisect a right angle. Connect the intersection of Main and Jaime Trails with Happy Trail. Connect the intersection of Main and Jaime Trails with Duck Trail. DOK-3 Compare and contrast supplementary and complementary angles. Supplementary angles and complementary angles are pairs of angles that have speciﬁc sums. Supplementary angles have a sum of 180 degrees, and complementary angles have a sum of 90 degrees. DOK-1 Which angle was the largest? What was its measurement? C was the largest. It was 145°. DOK-1 Which angles were formed by adding Jaime’s trail? B, E, and F. B is 55°. E and F are both 90°. DOK-3 How do you know the line you drew for Jaime Trail is perpendicular to Main Street? To make Jaime’s Trail perpendicular, I measured E or F to make the line, and I drew a right angle. To make the trail perpendicular, I placed my protractor on the intersection of Sunset Trail and Happy Trail and marked where 90° was located on Main Trail. Then, I drew a line with my straightedge. I knew it was right by measuring the angles after I drew the trail.

Intervention

Acceleration

FACILITATION TIP The Student Journal version of the park map includes some upgrades already. If you want students to draw and create their own upgrades with protractors and straight edges, consider using the editable ﬁles.

ANGLE RELATIONSHIPS

Home

FACILITATION TIP Depending on students’ prior or recent experience with measuring with protractors, consider providing some larger simple black and white practice angles. They can measure and record with partners or independently before trying to read the map, decipher colors, and incorporate City Park Guidelines. 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.

FACILITATION TIP Select and post two or three essential Math Chat questions for students to view and discuss as they collaborate. Direct students to be prepared to answer these prompts during the upcoming Math Chat. FACILITATION TIP Avoid misconceptions by drawing students’ attention to the geometrical deﬁnition of “right” and common usage of the word “right.” If they say, “I knew it was ‘right’...” challenge them to clarify.

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 As an extension, offer students the time to label acute, obtuse, and right angles on the Exit Ticket.

249

ANGLE RELATIONSHIPS

Angle Relationships Explore 2 – Vertical and Adjacent Angles ACTIVITY PREPARATION Students will locate and measure vertical and adjacent angles. Students will compare and contrast vertical angles.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Tall Pines Park Map (per group, optional) 1 Tall Pines Park Map (per class) 1 Exit Ticket (per student)

Reusable • •

• • •

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. Be prepared to project Tall Pines Park Map for the class to see. •

•

Optionally, print one Tall Pines Park Map per group to provide a larger workspace for students.

Gather enough protractors for each group or each student to have one.

1 Protractor (per group or per student) 1 Projector or document camera (per teacher)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) How might hiking trails be upgraded in a park?; 2) How can hiking trails be designed to be accessable to all people? After reading the scenario, ask the class 3) What types of information and documents are needed in Min and Xavier’s proposal to the Parks Department?

FACILITATION TIP To help students focus on the features of the park in questions 3a–3c, give each student a color copy of the Tall Pines Park Map before distributing the Student Journal with the proposal details. Help them locate and note the essential features (compass rose, parking lot, lakes, trails). Allow them to make marks on this map. 250

Read the following scenario to the class: The City Parks Department has asked the public to provide suggestions for upgrading one of the city parks. Min and Xavier would like to see an upgrade to Tall Pines Park that improves hiking opportunities in the park. Xavier walks in the park with his grandmother. He would also like to add benches to the trails to make the park more accessible to people of all ﬁtness levels. Min and Xavier have prepared a proposal to present to the Parks Department. They have asked you to examine their proposal to look for mistakes and determine whether it meets the Parks Department’s guidelines before they submit their proposal. Display the Tall Pines Park Map on the board using a projector. Explain that this is a map of the current park. The hiking trail proposal will create changes to the current park. Brieﬂy review the features of the Tall Pines Park Map by asking the following questions: a.

DOK-1 Where is the parking lot located? It is at the north end of the park.

DOK-1 Is Big Lake east or west of Pine Lake? West

DOK-1 How can you go from the parking lot to the swings? Answers may vary. You can take Big Lake Trail south until it intersects Back Trail, and then head west.

Give a Student Journal to each student. Give one protractor to either each student or each group. Optionally, also distribute one Tall Pines Park Map to each group. The larger format will make it easier for some students to complete the activity. If students use the larger map, instruct them to record their thinking in their Student Journals. © Accelerate Learning Inc. - All Rights Reserved

7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

Have students work cooperatively to read Min and Xavier’s hiking trail proposal that is found on their Student Journals. Instruct students to use a protractor, including the protractor’s straight edge, to sketch the plan onto the current park map on their Student Journals. After adding all of the plans to the map, students take measurements and determine whether the hiking trail proposal meets the Parks Department’s guidelines. Monitor and talk with students as they work. Check for understanding by using guiding questions. a.

DOK-1 How can you add perfectly straight trails to the map? Use the straight edge of the protractor.

DOK-2 If you add symbols to the map, what can you include to make it easier to read? You can add a map key.

DOK-1 According to the City Park Improvement Guidelines for Tall Pines Park, what are adjacent angles? Adjacent angles are angles that have the same vertex and a common side.

DOK-1 According to the City Park Improvement Guidelines for Tall Pines Park, what are vertical angles? Vertical angles are angles that are opposite of each other when two lines cross. They share a vertex.

Students will use the completed park map to complete the remainder 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 Design a change to the hiking trail proposal to meet the guidelines. Relocate the rest areas at the intersection of West Trail and Back Trail to make them vertical. Move the rest area from R to S. DOK-3 Compare and contrast adjacent and vertical angles. Adjacent angles and vertical angles share a vertex. Adjacent angles also share one side, but vertical angles do not share a side. Vertical angles are opposite of each other. DOK-3 What do you notice about the vertical angles on the map? The vertical angles are congruent. There are many vertical angles in Tall Pines Park. DOK-3 What do you notice about adjacent angles on the map? The adjacent angles on the map are supplementary. The sum of their angles is 180°. DOK-3 Why are the adjacent angles supplementary on this map? All of the trails are drawn with straight lines. If you bisect a straight line to form two angles, the sum will be equal to the measurement of a straight line, which is 180°.

2. 3.

Acceleration

FACILITATION TIP Project the detailed proposal and read it together with students or have volunteers read it to the class. Clarify the details. Students may think they must widen the parking spaces for mobility aid users or think they need to eliminate other spaces somehow. They also may ask for information on constraints regarding widening Big Lake Trail. FACILITATION TIP Be sure to monitor for accurate measuring skills with protractors. Recent Explore activities provided some practice, but students may need consistent reminders for how to place the protractor and read and record the measurements. FACILITATION TIP Project questions 6c. and 6d. for students and prompt them to be prepared to draw and deﬁne adjacent and vertical angles. Before or after the activity, post examples and deﬁnitions on a word wall or have students draw examples on the Student Journal for later reference. FACILITATION TIP To help students identify similarities and differences between adjacent and vertical angles, create a T-chart and note student observations. FACILITATION TIP As an alternative extension, challenge students to consider: Are all or only some adjacent angles supplementary? Are all or only some adjacent angles complementary? Could they draw two vertical angles that are not equivalent? FACILITATION TIP

Post-Explore 1.

Intervention

ANGLE RELATIONSHIPS

Home

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.

To scaffold this Exit Ticket for a variety of learners, consider telling students the number of possible answers. Some students may ﬁnd only one solution, while others may be challenged to ﬁnd all of the possible answers.

251

ANGLE RELATIONSHIPS

Angle Relationships Explore 3 – Multistep Angle Problems ACTIVITY PREPARATION Students will apply knowledge of supplementary, complementary, vertical, and adjacent angles to solve multistep problems.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • • •

•

1 Student Journal (per student) 1 Redwood Park Map (per group, optional) 1 Redwood Park Map (per class) 1 Exit Ticket (per student)

• •

•

Reusable • •

1 Straightedge (per student) 1 Projector or document camera (per teacher)

Plan to divide the class into groups of 2 or 3 students to complete this activity. Print a Student Journal and an Exit Ticket for each student. Be prepared to project Redwood Park Map for the class to see.

•

Optionally, print one Redwood Park Map per group to provide a larger workspace for students.

Gather enough straightedges for each student to have one.

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What are examples of natural features in parks?; 2) In what ways could natural features in a park be upgraded? After reading the scenario, ask the class 3) What types of guidelines might the Park Department have regarding improvements to the park?

2. 3.

FACILITATION TIP If you haven’t already, help students make a real-world connection with a nearby park or your school campus by projecting a local map. FACILITATION TIP Be prepared with color copies of Redwood Park Map for each student to view up close in order to facilitate steps 1–5. FACILITATION TIP

252

Consider projecting Gabriella and Jasmine’s proposal and read it together as a class before distributing the Student Journal. Allow time for clarifying questions from students. Possible questions could include: “Can we remove trees? Is the overlook also a water feature? What are natural enhancements?”

Read the following scenario to the class: Gabriella and Jasmine have decided to submit a plan to the City Parks Department to upgrade one of the city parks. They love nature and would like to upgrade Redwood Park to highlight natural features within the park. Gabriella loves wildflowers and gardening. Jasmine loves bees and butterflies. Gabriella and Jasmine have prepared a proposal and would like you to help them be sure their proposal meets the Parks Department’s guidelines. Project the Redwood Park Map onto the board. Explain that this is a map of the current park. The nature trail proposal will create changes to the current park. Briefly review the features of the Redwood Map by asking the following questions: a.

DOK-1 What are the current features of Redwood Park? Redwood Park has a parking lot, a ranger station, a rest area, a hydration station, and a scenic overlook.

DOK-2 What kind of things might be found at the ranger station? Park rangers, maps, and information about the park are usually found at ranger stations.

Give a Student Journal and a straightedge to each student. Optionally, also give one Redwood Park Map to each group. The larger format will make it easier for some students to complete the activity. If students use the larger map, instruct them to record their thinking on their Student Journals. Have students work cooperatively to read Gabriella and Jasmine’s nature trail proposal that is found on their Student Journals. Instruct students to sketch the plan onto the current park map on their Student Journals. Students will not use protractors to determine the measurement of angles in this activity. After adding all of the plans to the map, students will use the measurements provided and their knowledge of complementary, supplementary, vertical, and adjacent angles to determine whether the nature trail proposal meets the Parks Department’s guidelines. © Accelerate Learning Inc. - All Rights Reserved

Explore

Explain

Elaborate

Evaluate

Monitor and talk with students as they work. Check for understanding by using the following guiding questions: a.

DOK-1 Can you locate a pair of supplementary angles? How do you know they are supplementary? Answers may vary. Answers should identify two angles that have a sum of 180°, such as A and C or L and N. I know they are supplementary because together they make a straight line.

DOK-1 Can you locate a pair of complementary angles? How do you know they are complementary? P and Q are complementary. I know they are complementary because together they make a right angle.

DOK-1 Can you identify a pair of adjacent angles? How do you know they are adjacent? Answers may vary. Answers should identify two angles that have the same vertex and a common side, such as A and B or W and X. Adjacent angles are angles that have the same vertex and a common side.

Engage

DOK-1 Can you identify a pair of vertical angles? How do you know they are vertical? Answers may vary. Answers should identify two angles that have the same vertex and are opposite of each other, such as X and Z or K and N. Vertical angles are angles that are opposite of each other when two lines cross. They share a vertex.

Instruct students to use the completed park map to complete the remainder 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 Design a change to the nature trail proposal to meet the guidelines. Bisect any right angle on the trail to create a pair of complementary angles. Extend Buckwheat Trail to the parking lot to bisect right angle T. • DOK-3 Why is it important to understand adjacent, vertical, supplementary, and complementary angles? Understanding the properties of angles helps you determine the measurements of missing angles. • DOK-3 Which property of angles is easiest to identify visually? Supplementary angles are easiest to identify because they form a straight line. Adjacent angles are easiest to identify because they share a vertex and a common side. Vertical angles are easiest to identify because they share a vertex and are opposite of each other. • DOK-3 Of the three park improvement projects (bike trail, hiking trail, and nature trail), which design did you like the best? Why? I liked the bike trail because I enjoy riding bikes. I liked the hiking trail because it provided hiking opportunities for people of many mobility levels. I preferred the nature trail because I love wildlife.

Intervention

Acceleration

FACILITATION TIP Students may be surprised that they will not need protractors for these measurements. Consider engaging students by telling them that you “lost” the protractors or loaned them to another teacher. How can they complete the evidence collection without the tool? Allow some time for students to consider and discover how they can determine the measurements without protractors.

ANGLE RELATIONSHIPS

Home

FACILITATION TIP Print and project these guiding questions for students. Have students read them out loud as you monitor their collaborations. FACILITATION TIP Be prepared to support students when they need to record equations on the Student Journal. The Exit Ticket includes equation solutions as well. Students may need a step-by-step explanation of where the terms and the format of the equations come from. FACILITATION TIP Remind students how to bisect an angle.

•

Post-Explore 1. 2. 3. 4.

FACILITATION TIP Some of the Math Chat questions provide great opportunities for students to vote and express their opinions. Use hand signals or other physical responses to let students vote. FACILITATION TIP Some of the Math Chat questions provide great opportunities for students to vote and express their opinions. Use hand signals or other physical responses to let students vote. FACILITATION TIP On the Exit Ticket, support struggling students by projecting the image of the angles and allowing students time to make observations and ask clarifying questions before working independently. You could use color markers to help them identify vertical angles. Be prepared to respond to some student questions about the variable x as well.

253

ANGLE RELATIONSHIPS

Angle Relationships 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

Supplementary and Complementary Angles 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

Vertical and Adjacent Angles 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 Angle Problems 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

254

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

Angle Relationships

Can be done independently

ANGLE RELATIONSHIPS

Home

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

255

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

ANGLE RELATIONSHIPS

Angle Relationships

3 256

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

ANGLE RELATIONSHIPS

Home

What does mastery look like?

I can write and solve equations by drawing and measuring angles with a protractor.

I can identify supplementary angles.

I can identify complementary angles.

I can identify vertical angles.

I can identify adjacent angles.

I can write and solve multistep equations using angle relationships.

257

SCOPE 1

Circles Scope Introduction SCOPE SUMMARY In this scope, students use the relationship between the circumference and diameter of a circle to discover the formula for circumference. This helps students to gain a deeper understanding of the formula. Students understand the relationship between the circumference and area of a circle and use them to solve problems.

Student Expectations

7.GSR.5.4 Explore and describe the relationship between pi, radius, diameter, circumference, and area of a circle to derive the formulas for the circumference and area of a circle. 7.GSR.5.5 Given the formula for the area and circumference of a circle, solve problems that exist in everyday life.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In elementary school, students ﬁnd the area of shapes such as triangles, quadrilaterals, and polygons. They use composition and decomposition to create shapes within shapes in order to solve mathematical problems.

Students will continue to ﬁnd the area and circumference of circles throughout 8th grade and beyond. They will learn to manipulate the formulas to ﬁnd geometric pieces such as arcs, sectors, and radian measures. The understanding of new circle concepts will be expanded in geometry classes in high school.

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

ﬁnd the area of various shapes.

•

compose shapes into rectangles or decompose them into other shapes.

•

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 area and circumference of circles.

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.

listen to prompts in relation to the prior standard and communicate if they feel the prompts are fact or ﬁction.

258

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

CIRCLES

Home

Discovering Circumference In this exploration, groups of students will be tasked with solving a scenario involving a pizzeria and helping the owner, Diego. Students are tasked with identifying parts of a circle, understanding relationships between parts so each pizza is the perfect shape and size; and, measuring the diameter and circumference of each pizza to determine what size each should be. Students will: •

identify and label the radius, diameter, and circumference of a circle.

•

determine relationships.

Explore 2

Explore 1

EXPLORE ACTIVITIES

•

approximate the formula for the area of a circle.

•

use the formula to solve real-world problems.

Explore 4

Explore 3

In this exploration, students will help Diego calculate the area of a circle to ensure each pizza has the perfect amount of dough. Students will also be asked to use the area of a circle formula to calculate the area of a small, medium, large, and mega pizza in order to use the correct amount of pizza dough. Students will:

In this exploration, students will be tasked with helping calculate the circumference of a circle and different pizzas offered so that correct pans can be ordered. Students will: •

use their knowledge of pi.

•

calculate the circumference of a circle.

•

solve real-world problems.

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. Area of a Circle

Circumference

Area and Circumference Problem Solving In this exploration, students will be tasked with helping Diego determine when they should ﬁnd the circumference of a circle and when they should ﬁnd its area. Students will: •

determine whether to calculate the circumference or area of a circle to solve a problem.

•

differentiate between area and circumference.

Students conclude the exploration with an Exit Ticket and revisit the Hook to solve using their learning from the explorations.

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

259

CIRCLES

Circles Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

260

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

CIRCLES

Home

ACCESSING PRIOR KNOWLEDGE Students will listen to prompts in relation to 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 used as a summative assessment. 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.

Materials

Preparation

Printed •

•

1 Fact or Fiction (per teacher, per student, or per group)

•

If not assigning the APK digitally, print one Fact or Fiction to read aloud to your students. Another option is to project Fact or Fiction 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 the side of the room based on what they think the prompt is: fact or ﬁction. Read the prompt, show it to your students, and allow students to move to different sides of the room. Have students discuss their reasonings among their peers. Before reading the next prompt, allow students to move back to their starting point. Repeat with another prompt. An answer key is provided below: a.

Prompt 1 is false because the height is always perpendicular to the base.

Prompt 2 is true.

Prompt 3 is false because the formula for the area of a trapezoid is A = (b1 + b2)(h), which is not the same formula used to ﬁnd the area of a rectangle.

• •

Before reading the three prompts, consider taking time to review the terms parallelogram and trapezoid in order to assess what speciﬁc knowledge gaps (related to base, height, and area) exist.

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 not understand that the height is always opposite and perpendicular to the base. Students may not understand that any triangle can be duplicated to form a parallelogram, and that is why the formula is half of the base times the height. Students may not understand how to decompose the trapezoid into a square and a rectangle.

FACILITATION TIP In addition to the Foundation Builder, have students draw triangles in a parallelogram, and decompose (with pencil) a trapezoid on paper. It may also be helpful to have them draw some rectangles and squares halved into triangles to warm-up.

261

CIRCLES

Circles Hook – Pizza ACTIVITY PREPARATION Students will determine the area and circumference of circles.

Materials

Preparation

Printed •

• • •

1 Pizza (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project the Pizza 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 Part I: Pre-Explore 1. FACILITATION TIP

Before showing the video and reading the scenario, ask the class 1) What size pizza does your family get when they order pizza?; 2) What other sizes of pizzas are there?; 3) What toppings do you like to get on your pizza? FACILITATION TIP Project the measurements (or have students write them down on paper) of the current pizza sizes for students to read along with you and refer to as you discuss the situation. Personal: 4, Small: 6, Medium: 10, Large: 12, XL: 15 FACILITATION TIP

4. 5.

To prevent students from being overwhelmed by the formulas, consider covering up the three formulas (use sticky notes), before you project the Pizza Slide. Reveal them one at a time and discuss the two questions.

6. 262

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: Pantastic Pizza Shop is looking at the size of each of their pizzas. The owner, Pam, wants to know how much sauce is used to cover the pizza dough and the amount of butter that is used on the crust of each pizza. They currently sell round pizzas in five different sizes: personalsized, which measures 4 inches from side to side; small, which measures 6 inches from side to side; medium, which measures 10 inches from side to side; large, which measures 12 inches from side to side; and extra large, which measures 15 inches from side to side. Ask students, “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 each pizza has a different measurement. I wonder why the personal and small pizzas have measurements that are closer together than the small and medium pizzas. Maybe these measurements will help us to find the amounts of sauce and butter that are used. Project the Pizza slide. Explain to students that Pam has two equations that we can use to find how much sauce and butter will be needed for each pizza. Discuss the following questions: a.

DOK-1 What do you think the equations mean? Allow students to share all ideas. Student answers will vary. Since we need to know how much sauce to use for one pizza, one equation must be ﬁnding the part that should have sauce.

DOK-1 Why do you think that it might be important to know the amount of sauce needed to cover the pizza? It is important to know the amount of sauce to cover each pizza because they will need to have enough sauce to make and sell pizza.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

CIRCLES

Home

Part II: Post-Explore 1. 2.

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

DOK-1 Do these equations make more sense after the Explore activities? Yes, these equations are used to ﬁnd circumference and area of circles.

DOK-1 What is the difference in the two circumference formulas? The formula C = πd uses the diameter of a circle and the formula C = 2πr uses the radius of a circle to ﬁnd circumference.

DOK-1 How can you ﬁnd what amount of sauce will be needed to cover one small pizza? I can use the area formula for a circle to determine the amount of sauce needed for a small pizza. Since a small pizza has a diameter of 6 inches, I will ﬁnd the radius by dividing the diameter in half, which will give a radius of 3 inches.

DOK-1 Do you feel that you have a strong understanding of solving for area and circumference of circles? Answers will vary based on students’ success during the activity and on their conﬁdence level.

FACILITATION TIP Before asking question 2a., ask a more open ended question. For example, “What can you tell me about these equations now that we’ve completed the Explore activities?” FACILITATION TIP In addition to asking for the difference in the two circumference formulas, ask what is the same about them. FACILITATION TIP To follow up on question 2d, create a list of real-world applications for using area and circumference of circles. Be prepared with some of your own examples to engage students.

263

CIRCLES

Circles Explore 1 – Discovering Circumference ACTIVITY PREPARATION The students will identify and label the radius, diameter, and circumference of a circle and determine the relationships between them.

Standards for Mathematical Practice • • • •

MP.5 Use appropriate tools strategically. MP.6 Attend to precision. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • • •

• •

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

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.

Part I • •

Reusable • • • • • • •

1 Gallon-size resealable bag (per group) 1 Resealable bag (per group) 1 Piece of string (per group) 1 Ruler (per group) 1 Red colored pencil (per group) 1 Blue colored pencil (per group) 1 Green colored pencil (per group)

•

Label a resealable bag “Part I” for each group. Print one set of Deﬁnition Cards for each group. Cut out the cards, and place each set in the resealable bag. If desired, print them on card stock, and laminate them for future use. Gather enough red, blue, and green colored pencils for each group to use one of each color. Add each set of colored pencils to the resealable bag of Deﬁnition Cards.

Part II • •

•

Label a gallon-size resealable bag “Part II” for each group. Print one set of Pizza Cards for each group. Cut out the Pizza Cards along the dashed lines. Place each set in the gallon-size resealable bag labeled “Part II.” If desired, print them on card stock, and laminate them for future use. Cut a piece of string at least the length of the circumference of the largest pizza. Optionally, cut strings the length of the circumference of each pizza. Place the string in the gallon-size resealable bag labeled “Part II.” Gather enough rulers for each group to have one.

PROCEDURE AND FACILITATION

264

FACILITATION TIP

Part I: Parts of a Circle

Before reading the scenario, ask the class 1) What would pizzeria employees need to know when making pizzas?; 2) What shape is a typical pizza? After reading the scenario, ask the class 3) What do you know about the parts of a circle and their relationship to each other?

2. 3. 4.

Read the following scenario to the class: Diego recently decided to open a pizzeria in his community. As the owner of the pizzeria, it is his responsibility to teach his employees how to make pizza. Diego’s pizzeria is making pizzas shaped like circles, and he needs your help to identify the parts of a circle and to understand the relationship between them so each pizza is the perfect shape and size. Give 1 set of Deﬁnition Cards to each group. Give a Student Journal to each student. Have students work in their groups to identify and label the parts of a circle based on the provided deﬁnitions. Then, students will trace them on the circle using the colored pencils. © Accelerate Learning Inc. - All Rights Reserved

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

Monitor and talk with students as needed to check for understanding by using the following guiding questions: a.

DOK-1 What are the different parts of a circle? Responses will vary. The four parts of a circle are the center, radius, diameter, and circumference.

DOK-2 If the radius of a circle is 3 inches, what is the circle’s diameter? Responses will vary. The diameter would be 6 inches, since the diameter is twice the length of the radius.

DOK-2 Why is the center a key feature of a circle? Responses will vary. One of the endpoints of the radius is the center, while the diameter must pass through the center.

•

DOK-1 In your own words, what are the different parts of a circle? The center is the most inside point of a circle. There is an equal distance from the center to any point on the circle. The radius is the length from the center to any point on the circle. The diameter is the distance from one point on the circle through the center to the opposite side point. DOK-2 If the length of a circle’s radius is given, how can you determine the length of the diameter? We can multiply the length of the radius by 2 to get the length of the diameter. DOK-2 If the length of a circle’s diameter is given, how can you determine the length of the radius? We can divide the length of the diameter by 2 to get the length of the radius. DOK-3 Outside of the math classroom and not using the pizza example, where else might you see a diameter or radius? A real-life example of a radius would be a spoke on a bicycle tire or the distance from the center of a bicycle wheel to the tire. A real-life example of a diameter would be the distance from one Ferris wheel pod to the pod on the opposite side of the Ferris wheel.

Part II: Discovering Pi as a Constant 1.

2. 3.

Acceleration

FACILITATION TIP To avoid confusion between radius and diameter, ensure that students’ labels clearly show that the diameter stretches all the way across. On the Student Journal, there are several ways to label the radius and the diameter. Be sure to clarify for students.

Allow time for students to complete Part I of their Student Journals, including the reﬂection questions. After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat •

Intervention

CIRCLES

Home

Read the following scenario to the class: Diego’s Pizzeria will make and sell 4 different-sized pizzas: small, medium, large, and the mega pizza. Help Diego and his pizza makers measure the diameter and circumference of each pizza to determine what size each pizza should be. Record the measurements, and complete the table in your Student Journal. Give a bag of Pizza Cards and string and a ruler to each group. Have students work in their groups to measure and record the diameter of each pizza using the provided ruler. Students will use the provided string to measure the circumference of each circle by laying the string along the edge of each circle. Students will measure this distance around the circle using the ruler and record the length of the circumference. Students will record this data in their Student Journals. Students will then use the provided formula (circumference/ diameter) to discover the value of pi (π). Monitor and talk with students as needed to check for understanding by using the following guiding questions:

FACILITATION TIP Be sure to include circumference in the list of parts of a circle. Consider using physical responses (using arms and hands) and encourage all students to draw in the air the different parts of a circle as you review this question.

FACILITATION TIP Consider your students’ interests when coming up with real-world examples for diameter and radius examples (skateboard wheels, planets, moons, vinyl records, jewelry, roundabouts, etc.). FACILITATION TIP After reading the scenario, ask the class 1) How do you measure the diameter of a circle?; 2) How do you measure the circumference of a circle?; 3) How do you think knowing the diameter and circumference of a circle will help to determine what size each pizza should be? FACILITATION TIP Be prepared to provide students with some tips on how to manage measuring the circumference of each circle with a string. Demonstrate by yourself and then perhaps with another student or two helping you. Using a ﬂexible measuring tape might help.

a. DOK-1 If pi is the ratio of the circumference to the diameter, what whole number is pi (π) ( closest to? Student responses will vary. Pi (π) is around 3.

265

CIRCLES

Circles Explore 1 – Discovering Circumference b.

FACILITATION TIP Emphasize to students the constancy of pi. When measured accurately, every perfect circle creates this constant ratio (For example: the eraser on their pencil end, the circle of the clock on the wall, a basketball, Earth’s equator, etc.) In addition, students may be fascinated by the non-repeating nature of pi. Ask for students who may have already memorized several digits. FACILITATION TIP Consider providing ﬂexible measuring tapes for students to use on the Exit Ticket. Some students may struggle to measure with lightweight string. Be prepared for some students to just write pi or 3.14... rather than calculate their own ratio under Decimal Form.

5. 6.

DOK-2 Do you think the ratio of the circumference to its diameter would be the same as the other size pizzas if there was a pizza with a diameter smaller than the small pizza? Responses will vary. Yes, I would expect to see the same value, around 3.14.

Allow time for students to complete Part II of their Student Journals, including the reﬂection questions. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 Pi (π (π) is the ratio of a circle’s circumference to its diameter. Using the last column in the table, what is the approximate value of pi? Depending how we round, the value is approximately 3.14 or a little over 3. • DOK-3 Outside of the math classroom, where else might you see an example of circumference in use? An example of everyday circumference is the distance a bicycle travels after one of the bicycle wheels makes one rotation. •

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.

266

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

CIRCLES

Home

267

CIRCLES

Circles Explore 2 – Circumference ACTIVITY PREPARATION Students will use their knowledge of pi (π)) to determine how to calculate the circumference of a circle. They will determine the circumference of circles to solve mathematical and real-world problems.

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.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Pizza Cutout (per group) 1 Set of Pizza Circumference Cards (per group) 1 Exit Ticket (per student)

1 Gallon-size resealable bag (per group) 1 Resealable bag (per group) 1 Bundle of yarn or string (per teacher) 1 Pair of scissors (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.

Part I •

•

Reusable • • • •

• •

•

Print a Pizza Cutout for each group. Cut out the pizza along the dashed lines. Place each cutout into a gallon-size resealable bag labeled “Part I.” If desired, print it on card stock, and laminate it for future use. Cut around 2 ft. of yarn or string for each group, and place the string into the gallon-size resealable bag labeled “Part I.” Gather enough pairs of scissors for each group to have one.

Part II •

Print and cut out a set of Pizza Circumference Cards for each group. Place the cards in a resealable bag labeled “Part II.” If desired, print them on card stock, and laminate them for future use.

PROCEDURE AND FACILITATION Part I: Connecting to the Formulas FACILITATION TIP Before reading the scenario, ask the class 1) What information does a pizzeria need to have to be able to purchase the correct sized pizza pans to cook their pizzas?; 2) What is the name for the distance around a circle? FACILITATION TIP Depending on time and supplies, consider modeling/demonstrating Part I for students using a projector. Student volunteers can assist as you measure and cut in front of the class.

268

2. 3. 4.

Read the following scenario to the class: Now that Diego has trained his team on the ins and outs of a circle, it is time to order pizza pans to cook the pizzas to perfection. Diego and his team of pizza makers must work together to determine how to calculate the circumference of a circle so he can order the correct pans. Distribute the bags containing the Pizza Cutouts and the string. Give a Student Journal to each student. Have students work in their groups using the string and the Pizza Cutout to measure the distance around the pizza. The distance around the pizza is called the circumference. Note that students will wrap the string around the circumference of the Pizza Cutout and will cut the string at exactly one circumference of the Pizza Cutout.

Engage

Explore

Explain

Elaborate

Evaluate

Continue to have students work in their groups using the string and the Pizza Cutout to determine how many diameters it takes to go around the pizza. Note that students will take their string circumference and stretch it across the diameter of the Pizza Cutout. They will cut as many string diameters from their string circumference as they can.

Intervention

Acceleration

CIRCLES

Home

FACILITATION TIP To support accuracy, carefully select the string you use as some can be very stretchy. Also, check your color copies to be sure the printed version matches the digital version before printing multiples.

STEMscopes Tip

7. 8.

Monitor and talk with students as needed to check for understanding by using the following guiding questions: a.

DOK-1 How can we model the circumference of a circle using a piece of string? Student responses will vary. The circumference is the distance around a circle. We could take a piece of string and lay it along the perimeter of the circle, making sure both ends of the string connect.

DOK-1 What is the relationship between the circle’s diameter and the circumference? Student responses will vary. It takes a little over three diameters to equal the same length of a circle’s circumference.

Allow time for students to complete Part I of their Student Journals, including the reﬂection questions. After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 What do you think the value of the number of diameters needed to go around the circumference represents? Pi (π) or 3.14 is the same as the circumference divided by the diameter. • DOK-2 How can you calculate the circumference of an object? Since pi (π) or 3.14 can be calculated by dividing the circumference by the diameter, we can multiply pi times the diameter to calculate the circumference. • DOK-1 How many different ways do you think there are to calculate the circumference of a circle? There are two different ways to calculate the circumference of a circle: one for if you have the radius of a circle and the other for if you are given the diameter. The formulas are C = 2πr and C = πd. • DOK-3 Not including the pizza example, what is another everyday example of circumference? The distance a Ferris wheel travels in one rotation is an example of circumference. •

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.

FACILITATION TIP Struggling students will need support to 1 determine that __7 of the string remained after cutting the string diameters. Some school scissors may not cut thin string very precisely, therefore making it diﬃcult for students to measure accurately.

Part II: Calculating Circumference 1.

2. 3.

Read the following scenario to the class: Now that you and your team know how to calculate the circumference of a circle, you and your team must work together to determine the circumference of each of your pizzas to place your pizza pan order. Distribute the bags of Pizza Circumference Cards to student groups. Have students work in their groups to match each pizza image with its appropriate circumference equations. Students will use that information to calculate the circumference of each pizza and record it in Part II of their Student Journals. Monitor and talk with students as needed to check for understanding by using the following guiding questions:

FACILITATION TIP After reading the scenario, ask the class 1) How is the circumference of a circular pizza calculated?; 2) What is the relationship between the radius and diameter of a circular pizza?

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Circles Explore 2 – Circumference a.

DOK-1 What is the difference between a circumference formula that uses a circle’s radius and a circumference formula that uses a circle’s diameter? Student responses will vary. Because it takes two radii to equal the length of a circle’s diameter, the circumference formula, when using the radius, should include a 2.

b. DOK-2 How can rounding the value of pi (π (π) be used to check your answer when solving for the circumference? Responses will vary. We can round 3.14 to 3 and use the appropriate circumference formula. Our answer should be very close to the circumference when using 3.14 for pi. 5. FACILITATION TIP As students ﬁnish up their reﬂection questions, project the three Math Chat questions. As you monitor their collaboration, draw attention to these essential questions and encourage students to be prepared to discuss. FACILITATION TIP As you conduct the Math Chat, take time to post the circle formulas somewhere in the room for students to reference. The vocabulary can be recorded on a word wall and on their Student Journal. FACILITATION TIP

Math Chat DOK-1 Why does one circumference formula use 2, while the other does not? It takes two radii to equal the same length of one diameter. The circumference of a circle can be found by multiplying the diameter by pi (π) or 3.14 or by multiplying the radius by 2 and then multiplying by 3.14. • DOK-1 What does the circumference formula look like for a circle with a radius of 4 inches? If r = 4 and C = 2πr, then C = 2π4. • DOK-3 Which circle, do you think, has a greater circumference: a circle with a radius of 6 inches or a circle with a diameter of 12 inches? Why? The circumference is the same for both circles since a diameter of 12 inches is the same as 2 radii of 6 inches each. •

• • • • • • •

As a follow up to the question about the circle with a radius of 4, provide some extra examples of radii for students to substitute in and perhaps calculate.

FACILITATION TIP Before assigning the Exit Ticket, preview the color graphic. If students have the circumference formulas memorized, take time to cover them up if they are posted. Also, be prepared to allow calculators for some students.

Allow time for students to complete Part II of their Student Journals, including the reﬂection questions. After Part II, invite the class to a Math Chat to share their observations and learning.

Diameter = 2 · radius C = 3.14 · d C = 3.14 · 12 inches C = 37.68 inches C = 3.14 · 2 · r C = 3.14 · 2 · 6 inches C = 37.68 inches

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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Circles Explore 3 – Area of a Circle ACTIVITY PREPARATION Students will approximate the formula for the area of a circle. They will use the formula for the area of a circle to solve real-world problems.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.6 Attend to precision.

Materials

Preparation

Printed • • • • •

1 Student Journal (per student) 1 Decomposing a Circle (per group) 1 Area of a Circle Work Mat (per group) 1 Set of Pizza Area Cards (per group) 1 Exit Ticket (per student)

Part I • • •

Reusable • •

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.

• •

1 Resealable bag (per group) 1 Pair of scissors (per group)

Print a Decomposing a Circle for each group. Print an Area of a Circle Work Mat for each group. If desired, print it on card stock, and laminate it for future use. Gather enough pairs of scissors for each group to have one pair.

Part II •

Print and cut out a set of the Pizza Area Cards for each group. Place each set in a resealable bag labeled “Part II.” If desired, print them on card stock, and laminate them for future use.

PROCEDURE AND FACILITATION Part I: Connecting to the Area Formula FACILITATION TIP Before reading the scenario, ask the class 1) What ingredients are needed to make a pizza?; 2) How can knowing the area of a pizza help when purchasing pizza toppings? FACILITATION TIP If time and materials are limited, you can demonstrate Part I for students using large cardboard circle models with 4 and 14 slices to show how to create the parallelogram. FACILITATION TIP Ensure that students understand the mathematical deﬁnition of decomposing as you begin this activity.

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Read the following scenario to class: Diego and his team of pizza makers are ready to make some pizza. First, he will need to know how much dough is needed to make each pizza. Diego needs your help calculating the area of a circle so you can make sure each pizza has the perfect amount of dough. Give a Decomposing a Circle and an Area of a Circle Work Mat to each group. Give a Student Journal to each student. Have students work in their groups using Decomposing a Circle to decompose the circle by cutting out the sectors (parts) of the circle and creating a parallelogram to demonstrate how the base and height of the parallelogram relate to the parts of the circle. Monitor and discuss with students as needed to check for understanding by using the following guiding questions: a.

DOK-2 Why do you think decomposing a circle helps us better understand the area of a circle formula? Student responses will vary. Decomposing a circle helps us see that a circle can be divided into slices and then rearranged. This rearrangement closely resembles a parallelogram. It’s easier for us to ﬁnd the area of a parallelogram since the formula is base times height (b · h). We can apply the same idea to the decomposed circle with our understanding of a circle’s parts. © Accelerate Learning Inc. - All Rights Reserved

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 What did you notice when you decomposed the circle with 4 wedges and the circle with 14 wedges? Student responses will vary. The smaller the wedges were, the more the shape looked like a rectangle.

Allow time for students to complete Part I of their Student Journals, including the reﬂection questions. After Part I, invite the class to a Math Chat to share their observations and learning.

Intervention

Acceleration

CIRCLES

Home

FACILITATION TIP Be prepared to address possible questions about the rounded pizza crusts on the top and bottom of the parallelogram.

Math Chat • •

DOK-1 What is another way of writing radius × radius (r (r · rr)? Another way of writing radius · radius is “radius squared,” or r2. DOK-2 Does the area of the circle change between keeping the circle intact or decomposing the circle to form a parallelogram? No, the area does not change since after you decompose the circle, you are not taking any pieces away or changing the sizes of the pieces.

FACILITATION TIP This Math Chat question provides a great opportunity to review the Order of Operations, particularly when it involves exponents.

Part II: Calculating the Area of a Circle 1.

2. 3.

5. 6.

Read the following scenario to the class: Now it is time to make the pizza dough. You and your team of pizza makers need to use the area of a circle formula to calculate the area of a small, medium, large, and mega pizza in order to use the correct amount of pizza dough. Distribute the Pizza Area Cards to each group. Have students work in their groups to calculate the area of each pizza and record their work on their Student Journals. Have students work in their teams using the Pizza Area Cards to match each pizza to the corresponding area formula. Not all cards will be used. Once students have matched the cards, students will calculate the area of each pizza on their Student Journals. Explain the following to the class: Work with your team to use the Pizza Area Cards to match each pizza with the equation that can be used to calculate its area. Not all cards will be used. Once you have matched the cards, calculate the area of each pizza in your Student Journal. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a.

DOK-1 How do you determine the area of a circle when given the diameter? Responses will vary. Since the area of a circle formula is A = π • r2, we ﬁrst need to divide the diameter by 2 in order to get the radius.

DOK-1 What does it mean when a number or variable is squared? Responses will vary. When a number or variable is squared, it means the number or variable is multiplied by itself.

DOK-2 Which circle has a larger area, a circle with a radius of 5 inches or a circle with a diameter of 10 inches? Responses will vary. Both circles have the same area since the radius of a circle with a diameter of 10 inches is 5 inches.

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-2 How would you determine the area of a circle if you are given the diameter? Because the area of a circle formula uses the radius, you would ﬁrst need to divide the diameter by 2 in order to get the measurement of the radius. • DOK-3 Outside of this pizza example, where else might you need to be able to calculate the area of a circle? If you are decorating your bedroom, it might be helpful to know how much ﬂoorspace a new circular rug takes. •

FACILITATION TIP After reading the scenario, ask the class 1) What information do you need to know to ﬁnd the area of a circle?; 2) How does knowing the area of the pizzas help when ordering pizza boxes? FACILITATION TIP Break the directions into smaller steps for students and project the directions. Step 1. Match each pizza to a formula (not all formulas will be used). Step 2. Calculate the areas of each pizza.

FACILITATION TIP Page 4 (Specialty Pizzas and Mega Pizza problems) of the Student Journal might be an excellent extension activity for students who ﬁnish early. The reﬂection questions on page 5 can also be posted for Math Chat rather than individually copied for each student. FACILITATION TIP To engage students, be prepared with some more real-world applications of circles. (For example: the rotation in graphics to create 3-D images, locating the epicenter of an earthquake, elements of design and architecture)

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Circles Explore 3 – Area of a Circle 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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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Circles Explore 4 – Area and Circumference Problem Solving ACTIVITY PREPARATION Students will determine whether to calculate the circumference or the area of a circle to solve a problem in a real-world situation. Students will differentiate between area and circumference.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.6 Attend to precision.

Materials

Preparation

Printed • •

• •

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

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.

PROCEDURE AND FACILITATION FACILITATION TIP

After reading the scenario, ask the class 1) What reasons would pizza makers have to determine the area of pizzas?; 2) What reasons would pizza makers have to determine the circumference of pizzas?

FACILITATION TIP Before breaking the students into groups, read the situations together as a class without allowing any discussion. Be certain students are clear on the critical difference between circumference and area. Perimeter and area are often diﬃcult concepts for students to differentiate.

Read the following scenario to the class: Every pizza that Diego’s pizza makers make must be made to each customer’s satisfaction. It is important that the pizza makers use the correct measurements to make every pizza. Help Diego’s pizza makers determine when they should ﬁnd the circumference of a circle and when they should ﬁnd its area. Have students work in their groups to determine which situations represent the area of a circle and which represent the circumference of a circle. Then, have students complete their Student Journals. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a.

DOK-2 How do you know when to calculate the circumference of a circle and when to calculate the area of a circle? Responses will vary. The circumference of a circle is the perimeter of a circle, so if a question is asking us to ﬁnd the distance around a circle, then we are looking for the circumference. If we are trying to ﬁnd the amount of space within a circle, then we are looking for the area of a circle.

DOK-2 What are some of the similarities and differences between the area of a circle and circumference formulas? Responses will vary. Both formulas involve multiplying by 3.14 (pi). The area of a circle formula only uses the radius when calculating, whereas we can use the diameter or radius when calculating the circumference of a circle.

FACILITATION TIP As students try to determine when to use circumference or area, point out some clue words frequently used, like “cover,” “around,” or “length.” FACILITATION TIP

276

As students try to visualize or sketch the pizza situations, note that some of the questions on the Student Journal may need clariﬁcation regarding speciﬁc measurements and ingredients provided. For example, some students may wonder, “Is 3 cm the radius of the pepperoni or the pizza? Are these red peppers small circles or long red peppers? Is 30 cm the diameter of the pizza or the pepper? How do you measure melted butter in inches?”

4. 5.

Allow time for students to complete their Student Journals, including the reﬂection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

CIRCLES

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Math Chat DOK-1 If a circle had a radius of 3 cm and you are finding the circumference and area of a circle, how would the circumference and area of a circle be affected if the radius doubled? The circumference of a circle with a radius of 3 cm would be 18.84 cm. If we double the radius, the circumference would also double and would be 37.68 cm. The area of a circle with a radius of 3 cm would be 28.26 cm2 and if we double the radius, the area would increase by 4 times the original area of a circle and would be 113.04 cm2. • DOK-2 Compare using the area and circumference formulas when given the radius or diameter. If you are finding the circumference of a circle, you can use either the radius or the diameter to find the circumference using the formula. If you are finding the area of a circle, you will need to use the radius for the formula. If you are asked to find the area of a circle and are given the diameter, you would need to get half of the diameter, the radius, to use the formula. • DOK-2 What are examples of how the area of a circle and circumference can be used in the real world? An example of how circumference is used in the real world is when you need to determine the length of the border for a circular garden. An example of how area is used in the real world is finding the area of a circular flower bed in a garden to get an idea of how many flowers can be planted in the garden. •

FACILITATION TIP Struggling students may need the second question broken up into smaller chunks. If so, project these questions:. 1) How do you use the area formula when given the radius? 2) How do you use the area formula when given the circumference? 3) How do you use the circumference formula when given the radius? 4) How do you use the circumference formula when given the diameter?

Post-Explore 1. 2. 3. 4.

FACILITATION TIP Before students work independently on the Exit Ticket, read the questions together so you can clarify words like “across the pizza” and “inside the crust.” If you print the Exit Ticket, the color graphic may be affected.

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Circles 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 Circumference 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

Circumference 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

Area of a Circle

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

Area and Circumference Problem Solving

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

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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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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

Circles

Can be done independently

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

Interactive Practice Planet Scanner A game to practice the skills established by the standards in the scope

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Circles 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

3 280

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

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Home

What prompts will be used?

What does mastery look like?

I can explain the relationship between the diameter and the circumference of a circle by using proportional reasoning.

I can explain the relationship between the diameter and the circumference of a circle and that the unit rate is pi.

I can derive the formulas for circumference and area of a circle.

I can deﬁne the symbol π by both its name and its meaning.

I can explain the relationship between the radius and the diameter of a circle.

I can determine the circumference and the area of a circle by using given formulas.

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SCOPE 1

Surface Area Scope Introduction SCOPE SUMMARY In this scope, students will find the surface area of different two and three-dimensional figures. They will understand that slicing three-dimensional figures, creates two-dimensional figures. This can help students to decide which formulas to use on which figures. They will extend this thinking to solve triangles, quadrilaterals, polygons, cubes, and right prisms within real-world and mathematical situations. Student Expectations

7.GSR.5.6 Solve realistic problems involving surface area of right prisms and cylinders. 7.GSR.5.7 Describe the two-dimensional ﬁgures (cross sections) that result from slicing three-dimensional ﬁgures, as in the plane sections of right rectangular prisms, right rectangular pyramids, cones, cylinders, and spheres.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In elementary school, students were introduced to geometric shapes and their classiﬁcations. In Grade 6, students began to calculate the areas of shapes through the composition and decomposition of other shapes. They focused on the areas of triangles and quadrilaterals along with the discovery of volume among more simplistic three-dimensional shapes. Students are able to view smaller shapes within a larger shape. This background knowledge will help students understand how surface area formulas are derived in this scope.

As students move on to 8th grade and high school, they will use the concepts of area, surface area, and volume in order to derive new formulas and prove theorems for more in-depth three-dimensional figures. In geometry, they will learn to rotate, reflect, and translate two- and three-dimensional figures, using these ideas in order to prove similarity and congruence between shapes.

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

ﬁnd the area of various shapes.

•

compose shapes into rectangles or decompose them into other shapes.

•

listen to prompts in relation to the prior standard and communicate if they feel the prompts are fact or ﬁction.

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 the surface area of a composite three-dimensional ﬁgure.

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.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Slicing 3-D Figures In this exploration, groups of students will be tasked with solving a real-world scenario involving Maria and helping her predict and discover what two-dimensional figures the plane sections of clay will be so she can paint and fire the two-dimensional clay figures to make coasters and plates. Students will: •

slice right rectangular prisms, right pyramids, cylinders, cones and spheres.

•

discover what two-dimensional ﬁgures are revealed as plane sections.

Explore 2

Explore 1

EXPLORE ACTIVITIES

SURFACE AREA

Home

Surface Area In this exploration, students will be tasked with helping Maria ﬁnd the surface area of each shipping box Mrs. Lopez sells. Students will: •

ﬁnd the surface area of cubes, right prisms, and complex three-dimensional ﬁgures.

•

solve single-step and multistep problems involving surface area.

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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Surface Area Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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Evaluate

Intervention

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ACCESSING PRIOR KNOWLEDGE

SURFACE AREA

Home

Students will examine a series of descriptions and determine which option does not belong with each group. This element is designed to uncover student misconceptions; it should not be taken for a grade. 6.GSR.5.2 Given the net of a three-dimensional ﬁgure with rectangular and triangular faces, determine the surface area of these ﬁgures.

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.

4. 5.

FACILITATION TIP

Give a Does Not Belong to each student or group. Explain that each description 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.

On the ﬁrst card, B does not belong because it describes surface area and the rest describe volume.

On the second card, C does not belong because it describes volume and the rest describe surface area.

On the third card, D does not belong because it describes volume and the rest describe surface area.

To engage students and perhaps help them see “ﬁlling” and “wrapping,” you can demonstrate without explanation. Without speaking, take an empty smaller box and “ﬁll” it with something (dumping in beads, styrofoam pieces, crumpled paper spheres, wooden cubes) and empty it. Next, “wrap” it with paper quickly. Follow up with the Does Not Belong. FACILITATION TIP After passing out the Does Not Belong to each student, structure some time for students to think silently and independently, then turn to a shoulder partner and pair up to think together. Finally, they should be prepared to share and explain their thinking to the whole class. FACILITATION TIP

Identifying Misconceptions • •

Students may not know how to use a formula to calculate volume. Students may confuse the measurements of solid ﬁgures when breaking them apart into 2-D net shapes.

Be sure to use the sentence frame on the Does Not Belong to support student discussion. FACILITATION TIP The vocabulary related to geometry is essential for future standards and needs explicit review as students progress through these scopes. Take time to frequently support students by posting images on the walls, repeatedly reviewing terms, and restating deﬁnitions related to perimeter, area, and volume.

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Surface Area Hook – Show-Off Shelves ACTIVITY PREPARATION Students will determine the surface area of a composite three-dimensional ﬁgure to solve a real-world problem.

Materials

Preparation

Printed •

• • •

1 Show-Off Shelves (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Show-Off Shelves 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 Part I: Pre-Explore 1. FACILITATION TIP

Before showing the video and reading the scenario, ask the class 1) Why might people want to refinish furniture?; 2) Do you think you would like to refinish a piece of furniture? Why or why not?; 3) What steps are involved in refinishing furniture? FACILITATION TIP Project the essential details of the problem for students to read with you. Ask students, “What are we trying to find?”

FACILITATION TIP Think about projecting the Show-Off Shelves image before step 3 (it may look more like stairs than shelves to some students). Students can then make more productive observations about what they notice, wonder, and see.

4. 5.

a. DOK-1 In what unit will the surface area of the furniture and area of the shelves be measured? Square inches

FACILITATION TIP Students will need some help understanding and viewing the 3-D figure of the shelves. Take time to illustrate and explain how it is labeled. If you have a way to model the shelf with blocks or pieces, it might illustrate for students how three-dimensional figures are frequently represented on a flat surface. 6. 286

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: Sheba found a small piece of antique furniture at a flea market. She decided to make it a display for her trophies from different sporting, music, and art contests in which she participates. She sanded it and wiped it down. Next, she will seal the piece to protect the wood. Finally, she will decorate the shelves with decorative self-adhesive paper. She needs to determine the surface area of the whole piece to find how much sealant to purchase, and she needs to find the area of the top of each shelf so she knows how much decorative paper 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. Answers will vary. I notice that Sheba is finding surface area and area. I wonder how much wood sealant and decorative paper Sheba will need to buy. I wonder how much her supplies will cost. What are the dimensions of the piece of furniture? I can use math to determine the surface area of the piece of furniture and the area of the shelves. Project Show-Off Shelves. Explain to students that Sheba has measured the dimensions of her new furniture piece and recorded them in inches. Discuss the following questions:

DOK-1 Is the area of all the shelves the same size? How can you tell? No, all of the shelves are the same length, but the top shelf is one inch wider. I can tell this because the width/depth of the piece is 10 inches. The bottom two shelves are each 3 inches wide; 10 – 3 – 3 = 4. The top shelf is 4 inches wide.

DOK-1 How can you find the surface area of the whole piece of furniture? I have to find the area of each face of the furniture and find the sum of all of the areas.

Engage

Explore

Explain

Elaborate

Evaluate

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and repeat the scenario. Refer to Show-Off Shelves, and discuss the following questions: a. DOK-1 What is the total area of the shelves so Sheba knows how much self-adhesive decorative paper to buy? The total area of all three shelves is 40 square inches. 3 × 4 = 12; 3 × 4 = 12; 4 × 4 = 16 12 + 12 + 16 = 40 b.

Intervention

Acceleration

FACILITATION TIP As before, project the essential details of the problem for students to read with you. Ask students, “What are we trying to ﬁnd?”

SURFACE AREA

Home

DOK-2 What is the surface area for the whole piece of furniture so Sheba knows how much wood sealant to buy? 356 square inches Shelves = 40 3 × 4 = 12; 3 × 4 = 12; 4 × 4 = 16 12 + 12 + 16 = 40 Backs of shelves = 48 3 × 4 = 12; 4 × 4 = 16; 5 × 4 = 20 12 + 16 + 20 = 48 Back = 48 4 × 12 = 48 Bottom = 40 10 × 4 = 40 15 + 27 + 48 = 90 90 Sides = 180 3 × 5 = 15 3 × 9 = 27 4 × 12 = 48 × 2 = 180 Total surface area = 356 square inches 40 + 48 + 48 + 40 + 180 = 356

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Surface Area Explore 1 – Slicing 3-D Figures ACTIVITY PREPARATION Students will slice right rectangular prisms, right pyramids, cylinders, cones, and spheres to discover what two-dimensional ﬁgures are revealed as plane sections.

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) 1 Set of Slicing 3-D Figures Scenario Cards (per pair) 1 Exit Ticket (per student)

Reusable • • •

• • •

•

Plan to divide the class into pairs to complete the activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Slicing 3-D Figures Scenario Cards for each group. Cut out the cards. Place each set of cards in a resealable bag for each group. If desired, print the cards on card stock, and laminate them for future use. Obtain modeling clay, divide it into 1 lb. lumps, and wrap each pound in plastic for each set of partners.

1 Pair of scissors (per teacher) 1 Plastic knife or craft stick (per pair) 1 Resealable bag (per pair)

Consumable • •

1 Pound of modeling clay (per pair) 1 Roll of plastic wrap (per teacher)

PROCEDURE AND FACILITATION FACILITATION TIP

After reading the scenario, ask the class 1) What is a right rectangular prism?; 2) What is a right pyramid?; 3) In what ways could Maria slice the right rectangular prism and right pyramid to make plane sections? FACILITATION TIP Take time to review the similarities and differences between pyramids and prisms. A set of 3-D solids will help. If time allows or students need it, plan time for them to explore hands on sets individually or in partners before moving into the abstract images included in these scopes. FACILITATION TIP Depending on the type of clay you use, success with forming accurate ﬁgures and slicing precisely will vary. Be sure to model for your students how this process should look in your classroom. You might want trays for students to work on to contain the clay. 288

2. 3.

Read the following scenario to the class: One day, Maria looked through the storage room at her mom’s shop. She found old containers in the figures of right rectangular prisms and right pyramids. Mrs. Lopez said they were reusable shipping containers she no longer used. Maria decided to use them for an art project. She wanted to put clay inside each container to mold it to the shape of the container. Then, she would remove the clay from inside the container and cut plane sections of the clay, either vertically or horizontally. Your job is to help Maria predict and discover what two-dimensional figures the plane sections of the clay will be. Then, Maria will paint and fire the two-dimensional clay figures to make colorful and unique coasters and plates. Give a Student Journal to each student and a bag of Slicing 3-D Figures Scenario Cards, a lump of clay, and a plastic knife or craft stick to each pair. Explain to students that they will work with their partners to determine the 2-D figures of the plane sections sliced from the various right rectangular prisms and right pyramids.They will accomplish this by first building each 3-D figure from the Slicing 3-D Figures Scenario Cards and then following instructions to slice the figure correctly. Then, they will draw the 2-D figure that is revealed.

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-2 Is there more than one way to slice a right pyramid? Yes, it can be sliced vertically or horizontally.

DOK-1 What two-dimensional ﬁgure is revealed with each way the right pyramid is sliced? When it is sliced vertically, a triangle is revealed. When it is sliced horizontally, a square is revealed.

DOK-2 Is there more than one way to slice a right rectangular prism? Yes, it can also be sliced vertically or horizontally.

DOK-1 What two-dimensional ﬁgure is revealed with each way the right rectangular prism is sliced? Whether it is sliced vertically or horizontally, the two-dimensional ﬁgure that is revealed is a rectangle. (This also includes squares, which are special types of rectangles.)

Allow time for students to complete their Student Journals, including the reﬂection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Intervention

Acceleration

FACILITATION TIP Be sure to deﬁne and model how to “slice” cross sections. Clarify beforehand that they might need to create equivalent slices or maintain the integrity of the original 3-D shape as they slice.

SURFACE AREA

Home

FACILITATION TIP When you preview the scenario cards, look for vocabulary that you need to review or reteach before this activity (horizontal, vertical, apex, base). Determine if color or black and white copies work best. FACILITATION TIP Include some follow-up questions about the many different ways a 3-D ﬁgure could be sliced if the constraints were different. Could the slices be angled?

Math Chat DOK-2 When you slice a right pyramid horizontally in several different places, why STEMscopes Tip does it always produce a square, and why are they different sizes? Slicing a right Supplemental Aids, located in the pyramid horizontally always produces a square because the cut is parallel to the Intervention section, provide materials base, which is a square. The squares are different sizes because a right pyramid that will meet the needs of diverse narrows to a point at the apex (top) and widens to its widest square at the base. learners. These materials include If you slice close to the top, it will be a smaller square than at the bottom, where graphic organizers, handouts, and it will be larger. manipulatives that can further support • DOK-2 Why does a right rectangular prism only have one type of 2-D figure students. (rectangle) created when it is sliced, while a right pyramid has options of both triangle and square? A right rectangular prism only produces rectangles (including squares) when sliced because all of its faces are rectangles (including squares). The rectangles may have different dimensions, depending on which way the prism is sliced. A right pyramid can have a square or a triangle because it has both triangular and square faces. • DOK-2 Did you notice anything special about the triangle that was revealed when the right pyramid was sliced vertically? Explain. We noticed it was an isosceles triangle. This is because the right prism was sliced from the apex to the base FACILITATION TIP and all of the triangular faces and their edges are the same length, so it creates a Be prepared with some images of sliced triangle with same-length sides. transparent 3-D shapes for students to • DOK-2 In the real world, when might people slice three-dimensional figures to use view. the 2-D figures created? We do this when we make cookies. We make a 3-D figure (like a cylinder or a right rectangular prism) of cookie dough and then slice it in quarter-inch slices to bake cookies in the shapes of circles, squares, or rectangles. • DOK-2 Why is a rectangle created when the cylinder is sliced vertically but a circle is created when the sphere is sliced vertically? The sphere is curved all around and does not contain any flat sides, which means a circle will be created no matter which way the sphere is sliced. • DOK-2 What did you notice about the shapes created when the right pyramid and cone were sliced vertically? They both created isosceles triangles. This is because both 3-D figures were sliced from apex to base, creating triangles with samelength sides. •

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 Before assigning the Exit Ticket, explain stackable sculpture using a similar but not congruent ﬁgure.

289

SURFACE AREA

Surface Area Explore 2 – Surface Area ACTIVITY PREPARATION Students will ﬁnd the surface areas of cubes, right prisms, and complex three-dimensional ﬁgures. They will solve single-step and multistep real-world problems involving surface area.

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) 1 Set of Shipping Box Cards (per group) 1 Exit Ticket (per student)

• •

Reusable •

Plan to divide the class into student groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Shipping Box Cards for each group. Cut out the cards, and place a set of cards in a resealable bag for each group. Optionally, print the cards on card stock, and laminate them for future use.

1 Resealable bag (per group)

PROCEDURE AND FACILITATION FACILITATION TIP

After reading the scenario, ask the class 1) What is surface area?; 2) How can we ﬁnd the surface area of a rectangular prism?; 3) How can we calculate the surface areas of the boxes and the quantities of boxes sold monthly to help determine the amount of waterprooﬁng spray needed? FACILITATION TIP Project the critical information from the scenario for students to see and refer to as you read it together. Ask students, “What are we trying to solve?”

2. 3. 4.

FACILITATION TIP If you are printing the Shipping Box Cards, be sure students can see the 3-D shadowing shown on the digital color version. If not, take time to model how to create a drawing of a 3-D ﬁgure with dotted lines and solid lines. FACILITATION TIP Be prepared for students to offer alternative slicing methods (angled, rather than horizontal or vertical).

290

Read the following scenario to the class: Mrs. Lopez received customer feedback that some of her boxes ripped and fell apart if they got wet during shipping. To protect her boxes and her customers’ property, she created a spray-on product to waterproof her boxes. One bottle covers 10,000 square inches of surface area. Mrs. Lopez asked Maria to help her determine the surface area of the six different sizes of shipping boxes she sells. Mrs. Lopez can then track the quantities of the boxes she sells monthly to know how many bottles of waterproofing spray she needs to make monthly. Your job is to help Maria find the surface area of each shipping box Mrs. Lopez sells. Give a Student Journal to each student. Give a set of Shipping Box Cards to each group. Explain to students that they will be working in their small groups to determine the surface areas of the six different shipping boxes Mrs. Lopez sells. Tell students that to find the surface area of each box, they will need to sketch each box face, label the dimensions, and then find the areas. Students will find the total surface area of each box by combining the areas of each face. Monitor and assess students as they are working by asking the following guiding questions: a. DOK-1 How does slicing affect the rectangular prism? No matter how it is sliced, a rectangular prism will always show a rectangle (which could also be a square, which is a special rectangle). b.

DOK-1 What is important to remember about the base of any triangular prism? The base is always a triangle.

DOK-2 Do the faces of a triangular prism have the same area? Not necessarily. The only time the area of each of the three faces would be the same would be if the base of the prism was an equilateral triangle. © Accelerate Learning Inc. - All Rights Reserved

6. 7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

Give students enough time to draw each type of box face, calculate its area, and determine the surface area of each box. Allow time for students to complete their Student Journals, including the reﬂection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

DOK-1 What were the surface area formulas you used for rectangular prisms and cubes? . • DOK-2 What are the similarities and differences of the surface area formulas for a rectangular prism and a cube? Both formulas ﬁnd the sum of the area of all 6 faces. Since all edges and faces of a cube are congruent, to ﬁnd the surface area, just square the length of an edge and multiply it by six. • DOK-1 What were the surface area formulas you used for cylinders? The formula for the surface area of a cylinder is the area of the two bases, A = 2πr², added to the area of the face, A = 2πrh. • DOK-1 What were the surface area formulas you used for triangular prisms? The formula for the surface area is the area of two bases (always use the triangle in a 1 triangular prism), A = __2bh, added to the area of each of the three faces, A = lw. •

DOK-2 In the real world, when might people need to determine the surface area of complex three-dimensional ﬁgures? When someone needs to know how much paint to buy to paint three-dimensional playground equipment, such as climbing cubes or rectangular prism tunnels, they would need to know how to determine surface area.

Post-Explore 1. 2. 3. 4.

Acceleration

FACILITATION TIP Review the terms face and base with students when you distribute the Student Journal. Take time to ensure that the pages of the Student Journal allow enough room for students to create relevant sketches. FACILITATION TIP

Math Chat

•

Intervention

SURFACE AREA

Home

Most shipping containers are simple rectangular prisms. Note that Box 5 and Box 6 are not commonly used in shipping, but they still provide good practice for calculating surface area.

FACILITATION TIP Engage your students with some more relevant real-world examples (For example: painting the inside of an outdoor pool in the neighborhood, covering a propane tank with fresh paint, cereal box companies trying to save on packaging, or video games with map components). FACILITATION TIP On the Exit Ticket, highlight the image for students (if not in color) so they can visualize the ﬁgure accurately. There may be more spaces for faces on the Exit Ticket than students need.

291

SURFACE AREA

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

Slicing 3-D Figures 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

Surface Area 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

292

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

293

SURFACE AREA

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

3 294

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER

SURFACE AREA

Home

What prompts will be used?

What does mastery look like?

I can solve for the surface area of right prisms and cylinders.

I can use the area of triangles, rectangles, and other polygons to solve problems involving surface area.

I can solve surface area problems using geometric and spatial reasoning.

I can solve multistep problems involving the surface area of right prisms and cylinders.

I can determine which planes can be created by slicing a prism vertically or horizontally.

295

SCOPE 1

Volume Scope Introduction SCOPE SUMMARY In this scope, students will find the volume of different three-dimensional figures: rectangular prisms, triangular prisms, and cylinders. They will understand that volume formulas are derived from finding the area of the base multiplied with the height. This can help students to decide which formulas to use on which figures. They will extend this thinking to calculate volume within real-world and mathematical situations. Student Expectations

7.GSR.5.8 Explore volume as a measurable attribute of cylinders and right prisms. Find the volume of these geometric ﬁgures using concrete problems.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

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

ﬁnd the area of various shapes.

•

compose shapes into rectangles or decompose them into other shapes.

•

listen to prompts in relation to the prior standard and communicate if they feel the prompts are fact or ﬁction.

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 a composite three-dimensional ﬁgure to solve a real-world problem.

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

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

VOLUME

Home

Volume of Rectangular Prisms In this exploration, students will beMaria determine the volume of each shipping box Mrs. Lopez sells in her store; and, solve packing and shipping problems involving volumes of boxes and packing peanuts. Students will: •

ﬁnd the volume of cubes, right prisms, and complex three-dimensional ﬁgures.

•

solve single-step and multistep problems involving volume.

Explore 2

Explore 1

EXPLORE ACTIVITIES Volume of Triangular Prisms In this exploration, students will ﬁnd the volume of triangular prisms and will solve real-world problems involving volume. Students will: •

determine the volumes of the six different shipping boxes.

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

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment. Volume of Cylinders In this exploration, students will ﬁnd the volume of cylinders and will solve real-world problems involving volume. Students will: •

determine the volumes of the six different columns.

297

VOLUME

Volume Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

298

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

VOLUME

Home

ACCESSING PRIOR KNOWLEDGE Students will examine a series of descriptions and determine which option does not belong with each group. This element is designed to uncover student misconceptions; it should not be taken for a grade. 6.GAR.5.3 Calculate the volume of right rectangular prisms with fractional edge lengths by applying the formula, V = (area of base) × (height).

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.

4. 5.

On the ﬁrst card, B does not belong because it describes surface area and the rest describe volume.

On the second card, D does not belong because it describes the surface area of a triangular prism and the rest describe volume.

On the third card, C does not belong because it describes surface area and the rest describe volume.

FACILITATION TIP If you have not had time to demonstrate “ﬁlling” and “wrapping,” try showing students now. Without speaking, take an empty smaller box and “ﬁll” it with something (dumping in beads, styrofoam pieces, crumpled paper spheres, wooden cubes) and empty it. Next, “wrap” it with paper quickly. Follow up with the Does Not Belong. FACILITATION TIP After passing out the Does Not Belong to each student, structure some time for students to think silently and independently, then turn to a shoulder partner and pair up to think together. Finally, they should be prepared to share and explain their thinking to the whole class. FACILITATION TIP

Identifying Misconceptions • •

Students may not know how to use a formula to calculate volume. Students may confuse the measurements of solid ﬁgures when breaking them apart into 2-D net shapes.

Be sure to use the sentence frames on the Does Not Belong to support student discussion.

299

VOLUME

Volume Hook – Swim Meet ACTIVITY PREPARATION Students will determine the volume of a composite three-dimensional ﬁgure to solve a real-world problem.

Materials

Preparation

Printed •

• • •

1 Swim Meet (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Swim Meet 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 Part I: Pre-Explore 1. FACILITATION TIP Project a printed version of this scenario and read it aloud with students. Have students read it more than once and coach them to locate key phrases and math terms. Students may be curious about why people would complain about the absence of a strong smell. Explain that the right amount of chlorine keeps pools safe for large amounts of swimmers (too much chlorine is very dangerous).

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: Justin is cheering on his sister at a state swim meet. His sister, the teammates, and the fans have complained that there doesn’t seem to be a strong smell of chlorine in the pool. He notices the pool is not a perfect rectangular prism, and is much larger than their school pool. He knows the amount for chemicals is determined by the size of the pool and wonders how the pool’s volume is determined when it isn’t an exact rectangular prism. He wants to calculate the volume to check that the correct amount of chemicals are being used and that the team and his sister are safe. 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 Justin is ﬁnding the volume. I wonder how many shapes Justin will need to construct or deconstruct to calculate the volume. I wonder how much chlorine is used in a pool. What are the dimensions of the pool? I can use math to determine the volume of the pool in several ways. Project Swim Meet. Explain to students that Justin has measured the dimensions of the pool and recorded them in feet. Discuss the following questions:

FACILITATION TIP

Review the difference between linear, square, and cubic measurements.

DOK-1 In what unit will the volume of the pool be measured? The volume of the pool is measured in cubic feet.

DOK-1 What shape was removed to create this pool from a rectangular prism? How can you tell? The pool has a wedge the shape of a triangular prism that was removed. I can sketch the current pool’s missing sides to recreate a rectangular prism, and the missing shape is a triangular prism.

FACILITATION TIP Consider demonstrating this “removal” with a 3-D model if you can (perhaps a shoe box or a brick of cheese with a wedge removed).

Complete the Explore activities.

Part II: Post-Explore 1. 2.

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

300

DOK-1 What would be the volume of the pool if it didn’t have the wedge cut out and was a perfect rectangular prism? The total volume would be 43,050 cubic feet. 7 × 75 × 82 = 43,050 © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How can you find the volume of the pool as it is? I will find the volume of the pool as if it was a rectangular prism. Then, I will find the volume of the triangular prism wedge that was removed. I will subtract the two volumes to find the volume of the current pool. DOK-2 What is the volume of the pool that Justin needs to discuss with the person in charge of the pool? 33,825 cubic feet Pool as a rectangular prism = 43,050 7 × 75 × 82 = 43,050 Triangular prism wedge that’s removed = 9,225 0.5 × 3 × 75 × 82 = 9,225 Difference to determine actual pool = 33,825 43,050 − 9,225 = 33,825

Intervention

Acceleration

VOLUME

Home

FACILITATION TIP Allow students to work in pairs. Consider providing a scaffold for them to show their thinking steps. Step 1: Record the measures and volume of the rectangular prism. Step 2: Record the measures and volume of the triangular prism. Step 3: Find the difference.

301

VOLUME

Volume Explore 1 – Volume of Rectangular Prisms ACTIVITY PREPARATION Students will ﬁnd the volumes of cubes, right prisms, and complex three-dimensional ﬁgures and will solve single-step and multistep real-world problems involving volume.

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) 1 Set of Shipping Box Cards (per group) 1 Exit Ticket (per student)

Plan to divide the class into groups of 4. Print a Student Journal and an Exit Ticket for each student. Print a set of Shipping Box Cards for each group. Cut out the cards, and place a set of cards in a resealable bag for each group. If desired, print the cards on card stock, and laminate them for future use.

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION FACILITATION TIP After reading the scenario, ask the class 1) What is volume?; 2) How can we find the volume of a rectangular prism?; 3) How can we calculate the volumes of the boxes and the quantities of boxes sold monthly to help determine the amount of packing peanuts needed? FACILITATION TIP Engage students by showing a shipping box and filling it with packing peanuts. This simple visual demonstration of how much it takes to fill the box might help some students to connect to the scenario.

Part I: Box Volume 1.

2. 3.

FACILITATION TIP Students may comment that these Shipping Box Cards look the same as ones from an earlier scope. You can alert them ahead of time or be prepared to address the comments.

Read the following scenario to the class: To keep the contents of packages from being damaged, Mrs. Lopez fills her shipping boxes with environmentally friendly, biodegradable packing peanuts. For ordering purposes, Mrs. Lopez needs to know how many packing peanuts she uses monthly. She can determine that by finding the volume of each of her six shipping boxes and applying that data to how many boxes she sells and ships monthly. Your job today is to help Maria determine the volume of each shipping box Mrs. Lopez sells in her store. Give a Student Journal to each student and a bag of Shipping Box Cards to each group. Explain to students that they will be working in their groups to determine the volumes of the six different shipping boxes Mrs. Lopez sells. Tell students that to find the volume of each shape, they will need to sketch each box, decomposing boxes that are complex three-dimensional figures into cubes or rectangular prisms. Students will find the total volume, combining volumes of the complex three-dimensional figures when necessary. They may decide as a group how best to accomplish this task or try multiple methods and check their answers. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-2 Are there multiple ways to ﬁnd the volume of cube-shaped shipping boxes? No, s · s · s is exactly the same as l · w · h since all the edges are the same.

DOK-1 Do you ﬁnd it easier to ﬁnd the volume of rectangular prisms or cubes? Why? A cube is a type of rectangular prism. We think it is equally challenging because no matter what, we are multiplying 3 numbers, either l · w · h or s · s · s.

FACILITATION TIP For these complex calculation tasks, it might be important to provide some silent work time so students can focus independently. Set a timer (or closely monitor) and then allow groups to collaborate about methods and answers. 302

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 Which box or boxes did you find to be the most challenging to calculate volume? Why? We thought boxes 5 and 6 were the most challenging because they were complex 3-D figures. We had to break them into two 3-D figures and find the volumes of both figures. Then, we had to find the sum of both volumes. It took twice as long because we had to do double the number of steps.

Intervention

Acceleration

VOLUME

Home

FACILITATION TIP Before asking question 4a., ask “What are some ways to ﬁnd the volume of a cube?”

Give students enough time to draw each box; label its length, width, and height; and determine its volume. Allow time for students to complete Part I of their Student Journals.

Part II: Customer Service 1.

Read the following scenario to the class: When customers come into the shop, Mrs. Lopez often has to problem solve to ﬁnd the best way to pack items. She must determine what size box to use and the volume of packing peanuts required. Your job today will be to help Mrs. Lopez and Maria solve packing and shipping problems involving volumes of boxes and volumes of packing peanuts. Explain to students that they will be working with their small groups to determine solutions to complex problems involving dimensions and volumes of packing and shipping boxes. They will use common sense, volume formulas, the four operations, and strategies to best solve real-world customer requests. Monitor and assess students as they are working by asking the following guiding questions: a.

5. 6.

DOK-2 Why is ﬁnding the volume of a packing and shipping box not equivalent to the volume of packing peanuts needed for that box? The packing peanuts will not ﬁll the whole box because whatever is being shipped will take up some of the space inside the box.

DOK-1 What strategy did you use? We found the total volume of the shipping box. We found the volume of the item being shipped. We subtracted the volume of the item being shipped from the total volume of the box. The difference was the solution, the volume of packing peanuts needed.

DOK-3 When might it be hard to ﬁgure out what volume of packing peanuts are needed to ship something in the real world? Why? Answers will vary. It would be hard to ﬁgure out the necessary volume of packing peanuts when the volume of the item being shipped cannot be determined (like clothes in a soft pile).

Give students enough time to draw each box, find the volume of each, and find the difference between the volume of the packing/shipping box and the volume of the item being shipped. Allow time for students to complete Part II of their Student Journals and its reflection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat

FACILITATION TIP If you have the packing box and peanuts, you can quickly demonstrate adding an item to the box and how it changes the volume of peanuts required. FACILITATION TIP Before reading the scenario, ask the class 1) What process do you use to ﬁnd the best way to pack items in boxes?; 2) How can you use ﬁnding volume to help determine the best boxes for items?; 3) Besides boxes, what other materials are needed to pack and ship items? FACILITATION TIP Project or post question 3a and 3c. before students begin working. Draw their attention to these guiding questions as they collaborate. Encourage them to be prepared to share their thoughts during the Math Chat.

FACILITATION TIP Most shipping containers are simple rectangular prisms. Note that Box 5 and Box 6 are not commonly used in shipping, but they still provide good practice for calculating volume.

DOK-1 What were the volume formulas you used for rectangular prisms and cubes? Rectangular prism: l · w · h Cube: s3 or s · s · s or l · w · h • DOK-2 Maria tells her mom that figuring out the volume of a box does not really explain the volume of peanuts a box will use when something is being shipped inside the box. Is Maria correct? Explain what she means. Maria is correct. What she means is that when something is being shipped inside the box, it takes up some of the space/volume that would have been filled with packing peanuts, so fewer packing peanuts are necessary than would fill the total volume of the box. •

303

VOLUME

Volume Explore 1 – Volume of Rectangular Prisms •

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.

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DOK-3 How would you find the volume of packing peanuts needed for a large box in the shape of a rectangular prism containing another, smaller box shaped like a cube? First, I would find the volume of the larger shipping box. l · w · h gives me the volume of a box in the shape of a rectangular prism. Next, I would find the volume of the box being placed inside the shipping box by using s3 or s · s · s. Finally, I would subtract the volume of the box being shipped from the total volume of the shipping box. The difference is the volume of packing peanuts needed to fill the packing box and protect the contents of the box during shipping. DOK-3 In the real world, why might people need to know the volume of rectangular prisms or cubes? People need to know the volume of a pool that is in the shape of a rectangular prism so they know how many cubic feet of water it holds. They might need to know this to know how long it will take to fill the pool or how much it will cost to fill it.

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Volume Explore 2 – Volume of Triangular Prisms ACTIVITY PREPARATION Students will ﬁnd the volume of triangular prisms and will solve real-world problems involving volume.

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) 1 Triangular Prism Net (per group) 1 Set of Shipping Box Cards (per group) 1 Exit Ticket (per student)

Divide students into groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a Triangular Prism Net on card stock for each group. Print a set of Shipping Box Cards for each group. Cut out the cards, and place each set in a resealable bag for each group. If desired, print the cards on card stock, and laminate them for future use.

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION FACILITATION TIP

Project the essential question for students to read. Coach students to consider the connection to the original Hook about the pool volume.

FACILITATION TIP Project the Shipping Box Cards (in color) before distributing them to groups. Help students see the 3-D features of each image and then pass out copies for students to make observations. If you have any 3-D models, use them to illustrate the prisms.

a. 3.

FACILITATION TIP If giving groups a Triangular Prism Net, consider having them cut and glued or taped ahead of time (use an adult volunteer or older student to help).

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Read the following scenario to the class: Mrs. Lopez runs a successful packing and shipping business because she puts customer service first. She likes to serve her customers in a friendly and efficient manner. She knows time is precious during Fiesta season. Therefore, she builds some of her colorful triangular prism boxes ahead of time and fills them with colorful recycled material filled with air bubbles. Packing her special edition boxes ahead of time helps her know when to restock her Fiesta packing material since she can tell when they are running low. In order to know when to reorder, she wants to know the volume of packing material required by each of her special-edition Fiesta boxes in the shapes of triangular prisms. Your job is to use the given dimensions for each box and calculate the volume in cubic inches. Give a bag with the Shipping Box Cards to each group.

Optionally, give one Triangular Prism Net to each group to reference as needed.

Explain to students that they will be working in their groups to determine the volumes of the six different shipping boxes Mrs. Lopez uses to ship items during Fiesta. Tell students that to ﬁnd the volume of each ﬁgure, they will need to sketch each package in their Student Journals and then solve real-world volume problems to help Mrs. Lopez. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 What happens to the volume of a triangular prism as its height increases when its base stays the same size? Its volume increases.

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DOK-1 How do you ﬁnd the volume of a triangular prism? We use the volume formula for a triangular prism (Bh, with B as the area of the base) and plug in its dimensions.

Give students enough time to draw each box, label its dimensions, and determine its volume. Allow time for students to complete their Student Journals, including the reﬂection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 What were the volume formulas you used for triangular prisms? Triangular 1 Prism: Bh or (__2 bh)h • DOK-2 What is something you find confusing with the volume formula for the triangular prism? I find it confusing that there are two bases and two heights to work with. There is the base and the height of the triangle that is the base of the 3-D figure (either triangular prism or triangular pyramid) and there is the base and the height of the entire 3-D figure (either triangular prism or triangular pyramid). • DOK-1 What is the height of a triangular prism? It is the distance between the two bases. • DOK-3 In the real world, why might people need to know the volume of triangular prisms? People may want to know how much storage space an attic under a roof has.

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FACILITATION TIP Have students record the speciﬁc formula for a triangular prism: Volume = Base • height. FACILITATION TIP Create clear criteria for success for the drawings of each box. Some students will struggle to create images that communicate the 3-D elements of the boxes. Also, clarify if labels are required. If needed, model how to successfully sketch one of the boxes.

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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.

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Volume Explore 3 – Volume of Cylinders ACTIVITY PREPARATION Students will ﬁnd the volume of cylinders and will solve real-world problems involving volume.

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) 1 Cylinder Net (per group) 1 Set of Column Cards (per group) 1 Exit Ticket (per student)

Divide students into groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a Cylinder Net on card stock for each group. Print a set of Column Cards for each group. Cut out the cards, and place each set in a resealable bag for each group. If desired, print the cards on card stock, and laminate them for future use.

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading this scenario, project some images of columns on construction projects. Consider ancient historical buildings from students’ social studies texts and familiar newer local architecture. FACILITATION TIP

Depending on your students’ recent experience with circles, consider reviewing circle vocabulary as needed (radius, circumference, diameter, z, and pi).

FACILITATION TIP

Alternatively, give students access to cylindrical models and allow them to make observations. You can show them several cylinders or distribute different 3-D shapes (A toilet paper roll or paper towel roll vs a petri dish would provide good contrast for how height affects volume). FACILITATION TIP Create clear criteria for success for the drawings of each cylinder. Some students will struggle to create images that communicate the 3-D elements of the columns. Also, clarify if labels are required. If needed, model how to successfully sketch one of the shapes. 308

Read the following scenario to the class: Henry Johnson owns a remodeling business. The popular thing this year is adding columns. He builds his columns using molds and then ﬁlls them with a material that forms a solid column such as concrete for large columns or resin for smaller columns. He has a lot of upcoming jobs, and he needs to know how much material to order. Help him ﬁnd the volume of his upcoming project columns so he knows how much to order. Give a bag with the Column Cards to each group.

5. 6. 7.

Optionally, give one Cylinder Net to each group to reference as needed.

Explain to students that they will be working in their groups to determine the volumes of the six different columns Henry Johnson will need to build. Tell students they will need to sketch each package before ﬁnding the volume. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 What happens to the volume of a cylinder as its height increases when its base stays the same size? Its volume increases.

DOK-1 What happens to the volume of a cylinder when the area of its base increases and its height stays the same? Its volume increases.

DOK-1 How do you ﬁnd the volume of a cylinder? We use the volume formula for a cylinder (Bh, with B as the area of the base) and plug in its dimensions.

Give students enough time to draw each column, label its dimensions, and determine its volume. Allow time for students to complete their Student Journals, including the reﬂection questions. After the Explore activity, invite the class to a Math Chat to share their observations and learning. © Accelerate Learning Inc. - All Rights Reserved

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Math Chat • • •

DOK-1 What were the volume formulas you used for cylinders? Bh or πr2h DOK-2 What do you do when you are given the diameter as opposed to the radius of a cylinder? I divide the diameter by 2 to get the radius. DOK-3 In the real world, why might people need to know the volume of cylinders? People might need to know the volume of a cylindrical ﬁsh tank so they know how much water it holds.

FACILITATION TIP Preview the images on the Exit Ticket with students and allow time for clarifying questions about the decorative parts of the columns if needed.

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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

Volume of Rectangular Prisms 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

Volume of Triangular Prisms 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

Volume of Cylinders 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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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

Volume – Cylinders

Can be done independently

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

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

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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

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What prompts will be used?

What does mastery look like?

I can determine the base of a cylinder using cross sections.

I can determine the volume of a cylinder using the area of a circle.

I can ﬁnd the volume of a cylinder by using the area of the base and the height of the cylinder.

I can use my knowledge of the volume of right rectangular prisms to determine the volume of cylinders and other solids composed of cubes and right prisms.

I can ﬁnd the volume of a right prism.

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SCOPE 1

Probability Scope Introduction SCOPE SUMMARY In this scope, students will be introduced to the probability of chance. Students will design and conduct simple probability experiments to determine the likelihood of events, and they will reason that the probability of an outcome depends on its long-run relative frequency. They will calculate the number of outcomes in a given sample space to determine the probability of an event as a number between 0 (impossible) and 1 (certain). Student Expectations

7.PR.6.3 Develop a probability model and use it to ﬁnd probabilities of simple events. Compare experimental and theoretical probabilities of events. If the probabilities are not close, explain possible sources of the discrepancy. 7.PR.6.4 Develop a uniform probability model by assigning equal probability to all outcomes and use the model to determine probabilities of events. 7.PR.6.5 Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

Background Knowledge

Future Expectations

In previous grades, students determined equivalences involving fractions, decimals, and percents, and they performed all four operations with rational numbers. Students have collected, represented, and interpreted data. In sixth grade, students studied ratios, rates, and percents. Sixth graders were also introduced to statistics through the investigation of data distribution and measures of center and spread. Students have not encountered the study of probability prior to this grade level.

Students will continue to build on probability and statistics in future years. In eighth grade, students will 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.

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

Hook

7.PR.6.2 Approximate the probability of a chance event by collecting data on an event and observing its long-run relative frequency will approach the theoretical probability.

VERTICAL ALIGNMENT

Accessing Prior Knowledge

7.PR.6.1 Represent the probability of a chance event as a number between 0 and 1 that expresses the likelihood of the event occurring. Describe that a probability near 0 indicates an unlikely event, a probability around __1 2 indicates an event that is neither unlikely or likely, and a probability near 1 indicates a likely event.

listen to prompts about the prior standard and communicate whether they feel the prompts are fact or ﬁction.

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: •

compare theoretical probability versus experimental probability.

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explain why they may not be equal.

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. Notes _____________________________________________________________________________________________________________ _____________________________________________________________________________________________________________

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Probability In the ﬁrst exploration, students will help solve a realworld scenario where students pretend they are helping a game manufacturer assess the probability of selecting a color marble from a mystery bag; and, make decisions about which spinners to use for games based on the probability and likelihood associated with each spinner. Students will: •

investigate chance events.

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ﬁnd the probability of events.

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evaluate the likelihood of events.

Explore 2

Explore 1

EXPLORE ACTIVITIES Predicting Probability In the second exploration, students will solve an extension scenario from the previous scenario where they are tasked with helping ﬁnd the difference between theoretical and experimental probability in action through ﬂipping coins and recording data. Students will: •

predict the probability of an outcome.

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collect data by conducting repeated trials.

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understand the similarities and differences between theoretical and experimental probability.

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

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Probability Models In ﬁnal exploration of this scope, students will be tasked with solving another game manufacturing scenario where they help demonstrate their understanding of probability models in order to participate in future games for the company. Students will: •

apply their knowledge of experimental and theoretical probability.

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design and conduct experiments.

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compare probability results to make predictions.

Students conclude the exploration with an Exit Ticket and revisit the Hook to solve using their learning from the explorations.

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Probability Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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ACCESSING PRIOR KNOWLEDGE

PROBABILITY

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Students will listen to 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. 6.NR.4.6 Calculate a percent of a quantity as a rate per 100 and solve everyday problems given a percent.

Materials

Preparation

Printed •

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1 Set of Fact or Fiction Prompts

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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.

FACILITATION TIP

Prompt 1 is true.

Prompt 2 is false.

Prompt 3 is false.

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.

If space is an issue in the room, students can use Think, Pair, Share by taking time to sit and think independently, pair with a partner, and then be prepared to share with the class from their desks or tables. Students can also use hand signals. FACILITATION TIP If you decide to project the prompts, be aware that students will need time and space to tally the color words on Prompt 3. Also, alert students that the images in probability problems are not always accurate to the scenario. For example, students may try to count the coins in Prompt 1.

Identifying Misconceptions •

Students may struggle with making the connection between probability and percent. Notes

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Probability Hook – He’s Lost His Marbles! ACTIVITY PREPARATION Students will compare theoretical probability versus experimental probability and explain why they may not be equal.

Materials

Preparation

Printed •

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1 He’s Lost His Marbles! (per class)

Reusable •

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1 Phenomena Video (per class)

Plan to show the video. Prepare to project He’s Lost His Marbles! 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) If you had a bag full of red, green, and blue marbles, which color do you predict you will pull out the of the bag the most often?; 2) How would you test your prediction? FACILITATION TIP

Project this scenario so students can read it themselves along with you. Take time to read through it more than once so students understand the details (marbles are put back, there are 20 draws). FACILITATION TIP Make and record observations when you project He’s Lost His Marbles. Help students count and record the total number of marbles. There are green, blue, spotted, yellow, and red marbles.

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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: One rainy day, Damon and Maka were playing with marbles. They were playing a game to compare theoretical probability with experimental probability. They were taking turns. One person would give a condition, and then the other person would draw a marble out, record it, put it back into the bag, and draw again. They would draw a total of 20 times, and then they would switch roles. Damon and Maka were investigating whether theoretical probability was equivalent to experimental probability. 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 Damon and Makani are testing probability. I wonder why they are putting the marbles back in the bag after each draw. Will theoretical probability be equivalent to experimental probability? I can use math to determine theoretical probability. I can use math to determine experimental probability. I can use math to compare theoretical probability and experimental probability. Project He’s Lost His Marbles! Explain to students that Damon recorded the results for his experimental probability for three different turns. He and Makani are trying to determine whether the theoretical and experimental probabilities are equivalent. Discuss the following: a.

DOK-1 What do you think theoretical probability is? Student answers will vary. Accept all reasonable answers. It is how mathematically likely something is to occur.

DOK-1 What do you think experimental probability is? Student answers will vary. Accept all reasonable answers. This is what actually happens when you conduct the experiment.

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Part II: Post-Explore 1. 2.

DOK-2 Should theoretical and experimental probability be equivalent? Answers will vary. In a perfect world, yes, but in reality it doesn’t always happen. DOK-1 What is the theoretical probability for each of Damon’s turns? 8

Spotted marbles: ___ or __5. Green or blue marbles: ___ or __2. Yellow 20 20

2 1 or ___ . marbles: ___ 20 10

DOK-1 What was the experimental probability for each of Damon’s 10

turns? Spotted marbles: ___ or __2 Green or blue marbles: ___ or __2 . Yellow 20 20 0

or 0. marbles: ___ 20

Acceleration

FACILITATION TIP

Show the Phenomena Video again, and restate the problem. Refer to He’s Lost His Marbles! and discuss the following questions: a.

Intervention

DOK-2 Were the theoretical and experimental probabilities ever equivalent? In only one out of the three turns, they were equivalent.

Be ready to explain to students what a “turn” means in this case. In this scenario a turn means “20 draws.”

PROBABILITY

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STEMscopes Tip 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.

Drawing out green or blue marbles had a theoretical probability of __2 and

1 also an experimental probability of __2.

FACILITATION TIP

e. DOK-2 Why are theoretical and experimental probabilities not always equivalent? Theoretical probability is what should happen, but that is not what happens in reality. f.

DOK-2 What practice could help bring theoretical and experimental probabilities closer to the same values? The more times you repeat an experiment, the more likely it is to be close to the theoretical probability.

Being able to answer questions 2e and 2f is essential for student learning. Project these questions and record student responses. Clear up any misconceptions during the Explore activities and before assessing this scope.

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Probability Explore 1 – Probability ACTIVITY PREPARATION Students will investigate chance events, ﬁnd the probability of events, and evaluate the likelihood of events.

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 Probability Task Cards (per group) 1 Exit Ticket (per student)

Reusable • •

• • •

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1 Resealable bag (per group) 1 Set of marbles (per teacher) • • • •

Plan to divide the class into student groups of 4. Print a Student Journal and an Exit Ticket for each student. Print a set of Probability Task Cards for each group. Cut out and place each set of Task Cards in a resealable bag. If desired, print them on card stock, and laminate them for future use. Gather 20 marbles (10 red marbles, 5 blue marbles, 4 green marbles, and 1 yellow marble), and place them in a brown bag. This is the teacher’s mystery bag.

10 Red marbles 5 Blue marbles 4 Green marbles 1 Yellow marble

Consumable •

1 Brown paper bag (per teacher)

PROCEDURE AND FACILITATION FACILITATION TIP

Part I: Discovering Likelihood

Before reading the scenario, ask the class 1) How do you think game companies test their games before selling them? After reading the scenario, ask the class 2) How is “likelihood” related to selecting colored marbles from a Mystery Bag?

FACILITATION TIP

2. 3.

In addition to this scenario, demonstrate a real-life probability experiment with your class using the Mystery Bag. You could draw the marbles yourself or have student volunteers draw the marbles. Create engagement with prizes like mini erasers, pencils, or whatever you have on hand. Have the class calculate various probabilities of “winning” an item (Maybe the yellow marble means the student gets a bigger prize and the red marble is a sticky note.).

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Read the following scenario to the class: Your group is volunteering at a family game manufacturer called Famfun. For today, your group will be a test group for the Mystery Bag division of the company. You will help the research department decide what color marbles will be used in the games. Before you begin, Famfun wants to make sure you understand likelihood, so the manager wants you to assess selecting a color from the mystery bag. Give a Student Journal to each student. Explain to students that probability is a measure of how likely it is an event will happen and is represented by a number between 0 and 1. Direct the students’ attention to the mystery bag with marbles and the mystery bag model on the Student Journal. While pulling different marbles out of the bag, ask the following questions: a.

DOK-1 What are all of the possible outcomes when a contestant picks a marble out of the mystery bag? The contestant might pick a red, blue, green, or yellow marble from the bag.

DOK-2 If I picked a handful of marbles out of the bag, how likely is it that one of the marbles will be orange? Impossible

6. 7.

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Have students work with their group members to complete the table about the likelihood of selecting different color marbles from the mystery bag. As students complete the table, monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 Which color is most likely to be picked from the bag? Red is most likely to be picked because there are so many red marbles.

DOK-2 How likely is it to pick a yellow marble from the bag? Unlikely

DOK-1 Are there any marbles that are impossible to pick? Why? Purple is impossible because there are no purple marbles included in the mystery bag.

Allow time for students to complete Part I of their Student Journals, including the reﬂection questions. After Part I, invite the class to a Math Chat to share their observations and learning.

Intervention

Acceleration

FACILITATION TIP When you review probability, have students copy a deﬁnition on their Student Journal.

PROBABILITY

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FACILITATION TIP During step 4, have students record the answers on their Student Journal as you proceed. Provide students an opportunity to vote on question 5c. FACILITATION TIP Students may need to be review terms before being able to participate in the Math Chat. Preview vocabulary before Part I: likelihood, likely, unlikely, certain, and range of values.

Math Chat •

•

DOK-1 What is the range of likelihood for different events? The range is 0–1. 1 Zero means an event will never happen. 1 means an event will always happen. __2 means an event will happen half the time. DOK-2 What are the numerical values of probability associated with each category of likeliness? Explain. •

Impossible: 0 because it has no chance of happening

•

Unlikely: between 0 and __2 because it should happen less than half the time

•

Equally likely: __2 because it should happen half the time

•

Likely: between __2 and 1 because it should happen more than half the time

•

Certain: 1 because it will happen every single time

DOK-2 Do you think events always happen according to likelihood? Give an example to support your answer. Sometimes they do. If an event is impossible or certain, that will happen according to likelihood. Unlikely, equally likely, and likely are unpredictable in one instance. For example, it is equally likely I would ﬂip a coin and get heads or tails. So if I ﬂip a coin 8 times, it makes sense that I would get 4 heads and 4 tails. But one time, I got 8 tails in a row. • DOK-3 Which likelihoods in scenarios would probably create the best games? Why? I think unlikely, equally likely, and likely are the scenarios that would probably create the best games because even if the likelihood of something happening is known, what is expected does not always happen in individual cases. That makes it exciting, fun, and unpredictable. Impossible and certain would be boring because everyone would know the outcome before it happened. •

Part II: Assessing Probability and Likelihood 1.

2. 3.

Read the following scenario to the class: Your group is still volunteering at Famfun, but for today, your group will work for the spinner game division of the company. You will help the research department make decisions about which spinners to use for games based on the probability and likelihood associated with each spinner. Give a set of Probability Task Cards to each group. Have students take turns drawing a card. On their Student Journals, have students record the number of the card, the probability, and the likelihood of the scenario on each card. Monitor and assess students as they are working by asking the following guiding questions: a. DOK-1 What information do you need to be able to determine the probability of an outcome? We need to know how many possible outcomes there are and how many possibilities there are for a speciﬁc outcome.

FACILITATION TIP In addition to asking how to create the “best game,” challenge students to think about games of chance in the real world. Students may be aware that the creators of those games may want the likelihood of winning big to be unlikely; however, the players of the game may prefer a likely win. Be aware that students may have strong opinions and values about such games. FACILITATION TIP Before Part II, demonstrate a variety of spinners (from commercial board games) to the class. As with the Mystery Bag, you can increase engagement with these spinners by creating your own class or student outcomes. FACILITATION TIP Before reading the scenario, ask the class 1) What types of random selection tools are used in games?; 2) How are spinners used in games? After reading the scenario, ask the class 3) How is “probability” related to spinning spinners? FACILITATION TIP If you are short on time, you can do Part II as a group with the teacher drawing the cards or having student volunteers come up front. The cards are clearly numbered, so they could be done in order from 1–8 FACILITATION TIP Choose two or three guiding questions to project as students collaborate to facilitate their discussions. As you monitor, direct students to these questions. 321

PROBABILITY

Probability Explore 1 – Probability

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.

DOK-1 How do you determine probability? We set up a fraction. The numerator is the number of possibilities for a speciﬁc outcome. The denominator is the total number of possible outcomes. We simplify the fraction.

DOK-2 How does the probability affect the likelihood of an event? The probability can be plugged into the likelihood scale we did in Part I to determine the likelihood.

DOK-1 What scenarios were the easiest to label with a likelihood description, and why? Those that were impossible or certain were the easiest because it was easy to see and understand that they would never or always happen.

e. DOK-1 What is an example of something that is equally likely? Explain why. Examples will vary. Getting heads or tails when ﬂipping a coin and rolling an odd or even number on a standard die are equally likely because the probability of the event happening is always one-half. 5. 6.

Allow time for students to complete Part II of their Student Journals, including the reﬂection questions. After Part II, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 Why does the likelihood scale not exceed 1? There cannot be more specific outcomes than possible outcomes. For example, you cannot flip a coin 5 times and get heads 7 times. The most times you could get heads would be 5 5 times, which would be __5 , which simplifies to 1. • DOK-2 Why does the likelihood scale not go below 0? You cannot get a specific outcome less than 0 times. If you flip a coin 5 times, the least amount of times 0 you could get heads would be zero times. __5 simplifies to 0. The number cannot be negative. •

•

FACILITATION TIP Create a real-world list of equally likely scenarios. Students may be familiar with the phrase, “50/50 chance”.

•

DOK-3 Create an example of a scenario with a probability of __2 and an equally likely likelihood. Imogene was rolling a die with the numbers 1, 2, 3, 4, 5, and 6 on the six faces. What is the probability she will roll an even number?

•

DOK-2 Can you think of a time that knowing the probability helped you make a decision? One time I was supposed to go on a picnic, but a meteorologist said 3 there was a probability of __4 that it would rain, so we went to the movies instead. I was happy because it did rain.

FACILITATION TIP Before students complete the Exit Ticket, display a ten-sided die if you have one and consider reviewing vocabulary: multiples, even, and odd. Determine if students will be allowed to use calculators for the percents.

DOK-2 What does it mean if the probability of a spinner landing on black is __4 and 3 your spin lands on white? It just means you did an unlikely spin. If __4 of the time 1 the spinner lands on black, __4 of the time it lands on a different color.

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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Probability Explore 2 – Predicting Probability ACTIVITY PREPARATION Students will predict the probability of an outcome and collect data by conducting repeated trials, understanding the similarities and differences between theoretical and experimental probability.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. 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 2. Print a Student Journal and an Exit Ticket for each student. Gather enough coins for each pair to have one.

Reusable •

1 Coin (per pair)

PROCEDURE AND FACILITATION Part I: Flip a Coin FACILITATION TIP

Before reading the scenario, ask the class 1) Do you think a coin will land on heads or tails most often when flipped? After reading the scenario, ask the class 2) What is “theoretical probability”?; 3) What is “experimental probability”? 2. 3. 4. FACILITATION TIP Demonstrate flipping a coin with your expectations. For example, the coin should not go too high. Anticipate what you want students to do if coins roll away or get dropped. Allow students to choose how they take turns; some may want to alternate each flip rather than waiting for their partner to flip it 10 times.

Read the following scenario to the class: Your group continues to volunteer at a family game manufacturer called Famfun. For today, your group will be a test group for the money game division of the company. You will help the research department decide what games involving money are worth pursuing and field testing. Many of the games involve flipping coins to decide paths of action. Famfun wants to see the difference between theoretical probability and experimental probability in action, so the manager wants you to assess this by flipping coins and recording data. Give a Student Journal to each student. Give a coin to each pair of students. Have students pull out the coin and predict how many heads and tails they will toss if they flip the coin 20 times. Have students flip the coin 20 times. (Suggest that students take turns flipping the coin halfway through so both students get a chance to flip the coin 10 times each and record the tallies on their Student Journals.) Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 What is theoretical probability? Give an example. It is the ratio of the outcome you expect from an event. When you ﬂip a coin 10 times, you expect to get 5 heads and 5 tails.

DOK-1 What is experimental probability? It is the ratio of a speciﬁc outcome to the number of trials. When you ﬂip a coin ten times, you might actually get 7 heads and 3 tails, even though that is not what you expect to get.

DOK-2 Do you think you will get exactly 10 heads and 10 tails? Why? No, I don’t because I don’t think things always work out exactly like theoretical probability when I am only doing 10 coin ﬂips.

FACILITATION TIP For 6c, many students like to make predictions. Challenge students (optionally) to predict and record their guess before they begin ﬂipping coins. 324

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Allow time for students to complete Part I of their Student Journals, including the reﬂection questions. After Part I, invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 What is the theoretical probability of getting heads when you flip a coin? How many times should heads come up if you flip the coin 20 times? The 1 theoretical probability of getting heads when you flip a coin is __2 . Theoretically, if I flip a coin 20 times, I should get 10 heads, and the other 10 times I should get tails. • DOK-2 Did you get exactly 10 heads and 10 tails? Why? Answers will vary, but will usually be no. Experimental probability can be different from theoretical probability. I was close to 10 heads and 10 tails, but it was not exact. Instead, I got 8 heads and 12 tails. • DOK-2 Are your results reliable enough that you could predict how many heads and tails you will get if you flipped the same coin 1,200 times instead of 20 times? No, theoretical probability says I would get 600 heads and 600 tails. If I took my results from the trial, I would multiply my results by 60 and get 480 heads and 720 tails. It is unlikely that either of those results would be exactly what I would get if I actually flipped the coin 1,200 times, but it is likely that I would get something close to 600 heads and 600 tails. • DOK-2 When might theoretical probability and experimental probability be close to equal? The more times I flip the coin, the closer to theoretical probability I will be. •

Part II: Flip a Coin (Long-Run Relative Frequency) 1.

Read the following scenario to the class: Your group continues to volunteer at a family game manufacturer called Famfun. For today, your group will continue investigating probability for the money game division of the company. You will help the research department learn about tossing heads and tails on a coin. The manager wants you to assess the probability of tossing heads or tails over multiple trials. This will aid them in understanding long-term frequency and making important decisions about game rules. Have students ﬂip a coin 10 times and record their predictions and their data collected from ﬂipping a coin on their Student Journals. Students will use this data to calculate and record the frequency for getting heads and tails. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 How do you determine the frequency of tossing heads and tails? You can determine the frequency of tossing heads or tails by counting the total number of heads and tails that were tossed during the total number of trials. DOK-1 How can you represent your probability as a fraction? The numerator will include the total number of times the coin was tossed heads or tails, and the denominator includes the total number of trials.

Have students use the data they collected from the 10 coin tosses to calculate and record the relative frequency for getting heads and tails on the data table on page 4 of their Student Journals. (When students record the relative frequency, they will need to represent the relative frequency as a fraction and a decimal, rounded to the nearest hundredth.) Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 Why do you need to do more than one trial? Gathering more data makes it more accurate and should make the data closer to the theoretical probability.

Intervention

Acceleration

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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 Understanding the difference between theoretical and experimental probability is essential knowledge for this scope. Have students copy and recite the deﬁnitions. Include them on a word wall if you have one. FACILITATION TIP After reading the scenario, ask the class 1) How many trials would be reasonable to asses the probability of tossing heads or tails on a coin?; 2) How could using the data they gather help them develop game rules? FACILITATION TIP Depending on your students, you may want to conduct this experiment, record data, and calculate the frequency together as a class. If time allows, you could do the experiment together as a class and then have students do 10 tosses with partners. FACILITATION TIP If students are completing this activity with a partner, model and post the steps for calculating the relative frequency before they begin tossing. You could also break this into steps. Students could collect all their data with partners and then you could model the frequency calculation steps. Finally, students could calculate their own relative frequencies and round them. FACILITATION TIP Before students begin collaborating, post questions 5a–5c and read them to students. Direct students to these questions as they collaborate. 325

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Probability Explore 2 – Predicting Probability

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.

FACILITATION TIP Be prepared with some of your own real-world examples of long-run relative frequency. Students may need prompting to come up with examples. FACILITATION TIP When you preview the Exit Ticket, note that the answer spaces for the Probability Prediction are to the left of the possible outcomes. Clarify this placement for students.

6. 7.

DOK-2 What is relative frequency? How often something happens divided by all outcomes

DOK-2 After calculating the relative frequencies when adding each relative frequency for each outcome, what is the result? The sum equals one.

Allow time for students to complete Part II of their Student Journals, including the reﬂection questions. 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 frequency and relative frequency? Frequency is the total number of times a given result came up in an experiment. Relative frequency is a fraction of how many times a result occurs over the total number of tries. • DOK-2 What do you notice in the table about the changes in the relative frequency of the number of tails as the number of tosses increases? The relative frequency changes as the number of tosses increases. • DOK-1 What is long-run relative frequency, and why does it usually come closer to theoretical probability than a one-time short experiment? Long-run relative frequency is a ratio of the number of speciﬁc outcomes that occur to the total number of trials when there are a large number of trials. The more trials there are, the closer to theoretical probability the results are. This is because it is then an average of all test results. • DOK-3 Give an example of a trial or event where you would use long-run relative frequency. It would have to include a larger number of trials and/or testing an event many times, such as ﬂipping a coin 300 times, spinning a spinner 1,000 times, etc. •

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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Probability Explore 3 – Probability Models ACTIVITY PREPARATION Students will apply their knowledge of experimental and theoretical probability by designing and conducting experiments. Students will also compare probabilities and use their results to make predictions.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Set of Probability Model Cards (per group) 1 Set of Spinners (per 2 groups) 1 Exit Ticket (per student)

• • •

•

Reusable • • •

2 Resealable bags (per group) 1 Brown paper bag (per group) 1 Set of 20 color counters (per group) • •

• •

•

Plan to divide the class into groups of 4. Print a Student Journal and an Exit Ticket for each student. Print a set of Probability Model Cards for each group. Cut out and place each set of cards in a resealable bag for each group. If desired, print the cards on card stock, and laminate them for future use. Print a set of Spinners for every 2 groups. (There are 2 spinners on a page, enough for two groups.) Cut out the spinners, and place one in a resealable bag with a paper clip and pencil for each group. If desired, print them on card stock, and laminate them for future use. Gather the color counters and brown paper bags for each group. Place 10 red counters and 10 yellow counters in a bag.

10 yellow counters 10 red counters

1 Paper clip (per group) 1 Pencil (per group)

PROCEDURE AND FACILITATION FACILITATION TIP After reading the scenario, ask the class 1) What are probability models?; 2) What are examples of probability models?; 3) How might you be able to prove you understand probability models?

FACILITATION TIP To break this activity into smaller steps, you might do each scenario or card one at a time. For example project Card 1, set a timer, and allow groups to record their answers. Then, project Card 2, set a timer, and allow groups to complete that card. Monitor and continue as needed. Consider keeping Card 6 for the next math session. 328

2. 3. 4.

Read the following scenario to the class: Your group is ﬁnishing up its last week of training at Famfun, the company that creates and manufactures family games. Your job this week is to prove that you understand probability models. If your group does well at this task, next week you get to test and tweak current games set for production to make sure they are exciting, fun, and surprising rather than boring and predictable. Give a Student Journal to each student. Give a set of Probability Cards, a spinner, and a brown bag with 20 color counters to each group. Explain to students that they will apply their understanding of theoretical probability and experimental probability to ﬁnd the probability of events. They will also use probability models to compare and predict probabilities. Inform students that they will model two of the events with their groups and will select a probability model their groups will choose to develop. (If students select the brown bag with color counters, they do not have to use all 20 color counters; they can choose the number of counters they would like to use for their models.) When students develop their models, they need to create a table that includes a frequency column and a column that shows the frequency written as a fraction. © Accelerate Learning Inc. - All Rights Reserved

Engage

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Explain

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Intervention

Acceleration

Monitor and assess students as they are working by asking the following guiding questions: a.

PROBABILITY

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DOK-2 How do you calculate the theoretical probability for the brown bag with the color counters? I count the number of red and yellow Number of Yellow Counters Total Counters

counters we had in the brown bag. I use ______________________ to ﬁnd the theoretical probability. b.

DOK-2 How do you calculate the experimental probability for the brown bag and the number of yellow counters that are pulled from the brown Number of times a yellow counter is pulled

bag? I use __________________________________ to ﬁnd the experimental probability.

Total Number of trials

FACILITATION TIP DOK-1 Can you use the same method to ﬁnd the theoretical probability and experimental probability of landing on the color blue on the Spinner? Present or post the terms theoretical and Yes, the same process can be used to calculate both the experimental experimental probability as students work. probability and the theoretical probability. As you monitor and assess, encourage students to use the terms in their Allow time for students to complete their Student Journals, including the discussions. In addition, place these words reﬂection questions. on a word wall in your classroom. After the activity, invite the class to a Math Chat to share their observations and c.

7. 8.

learning.

Math Chat DOK-2 How do probability models help with comparing probabilities? Instead of just guessing about how likely something is to happen, probability models help us use theoretical probability and experimental probability to calculate the outcomes of an event. • DOK-2 What is an example of a real-world probability model you could design? We included how many of each ﬂavor of fruit chew was in the bag. We made a prediction about the probability of pulling out a lemon fruit chew. Then, we drew out a fruit chew 100 times. Every time, we put the fruit chew back in the bag and recorded what ﬂavor we drew in a data table. Then, we compared our prediction to our actual results. • DOK-2 What did you learn about setting up probability experiments? I learned that we need more trials to get experimental probability closer to theoretical probability.

•

FACILITATION TIP In addition to the ﬁrst question, follow up with asking how probability models help us make informed decisions in the real world. FACILITATION TIP As an extension, allow students to bring their own appropriate items to design a probability model (dice, spinners, items for brown bag).

Post-Explore 1. 2. 3. 4.

FACILITATION TIP Note that this Exit Ticket has color in the prompt. If color printing is not available, project a color version, have students note the colors on their copies, or edit the Exit Ticket before printing for students.

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Probability 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

Probability 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

Predicting Probability 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

Probability Models 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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ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

PROBABILITY

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Can be done independently

Spiraled Review

Fluency Builder

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

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

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

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Probability 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

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Acceleration Acceleration

ASSESSMENT PLANNER

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What prompts will be used?

What does mastery look like?

I can represent probability as a fraction, decimal, or percentage.

I can make predictions about relative frequency using theoretical probability.

I can determine probabilities using various random generation devices.

I can collect data that can be used to determine probability.

I can determine whether an event is impossible, unlikely, equally likely (or unlikely), likely, or certain to occur.

I can develop probability models through observations in frequencies of data.

I can develop probability models to ﬁnd the probability of simple events.

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SCOPE 1

Informal Inferences Scope Introduction SCOPE SUMMARY In this scope, students will expand their knowledge of statistics to explore where data comes from. They will understand that random sampling can create probabilities about a general population that is too large to gain complete data from. They will create estimates about the generalizations of both the population and sample data based on the collected data. Students will understand that this statistical data is just an estimate and that discrepancy between the sample and the population will be present. Student Expectations

7.PR.6.6 Use appropriate graphical displays and numerical summaries from data distributions with categorical or quantitative (numerical) variables as probability models to draw informal inferences about two samples or populations. 7.PAR.4.10 Predict characteristics of a population by examining the characteristics of a representative sample. Recognize the potential limitations and scope of the sample to the population. 7.PAR.4.11 Analyze sampling methods and conclude that random sampling produces and supports valid inferences.

VERTICAL ALIGNMENT Background Knowledge In previous grades, students were introduced to categorical and measurable data. They were introduced to important vocabulary such as mean, median, mode, range, etc. Students are able to describe sets of data using measures of center and variability. They learn that the median may be a more useful measure to represent the center of data rather than the mean. Students learn to make observations based on data sets in relation to their context and numerical values.

Future Expectations As students continue on in statistics, they will utilize informal inferences on a larger scale. They will expand this thinking to represent measures of center in terms of correlation, coeﬃcients, and strong and weak associations. They will expand their thinking to ﬁnd trends and patterns. In high school, students will begin measuring normal distribution along with sampling error to solve problems of expected value.

7.PAR.4.12 Use data from repeated random samples to evaluate how much a sample mean is expected to vary from a population mean. Simulate multiple samples of the same size.

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

identify two truths and a lie by reading statements about statistical questions.

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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: •

use data collected in a survey and represented in a data table to make inferences.

•

evaluate if a population surveyed is a fair representation of the population and the data collected is valid.

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.

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Variability in Data In this exploration, groups of students will help solve a scenario to help a survey company create a survey for data to be used in research with identifying which questions are statistical and which are non-statistical. Students will: •

determine the difference between a statistical and a nonstatistical question.

•

explain that both types of questions require data.

•

determine that a statistical question has data that varies.

Explore 2

Explore 1

EXPLORE ACTIVITIES

•

decide whether conclusions drawn from a sample of a population are valid for determining an outcome for a population.

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

Explore 4

Explore 3

In this exploration, students will solve a real-world scenario involving a movie theater and moviegoers. Students will analyze data to make predictions about who purchased snacks and use the data to make positive changes to bring customers back. Students will:

In this exploration, groups of students will be tasked with solving a real-world scenario about a middle school and helping the assistant principal read sample population cards to decide whether it is a valid sample to use for ordering pizzas. 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 Data to Make Inferences

Valid Generalizations

INFORMAL INFERENCES

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use data collected from a random sample to make predictions about an entire population.

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

Compare Data In this exploration, students will be tasked with solving real-world scenarios where students help Dr. Miranda determines the answer to this question: Across the world, men are statistically taller than women by a global average of 7%, can the same be said of 7th-grade students? Students will also make predictions about midJuly temperatures for two cities. Students will: •

visually compare data from two populations.

•

determine whether there is a visible difference between two populations based on sample data from each population.

Explore 5

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment. Centers and Measures of Variability In the ﬁnal exploration, students will be tasked with helping oﬃcials use data to decide which school should represent Columbus City in a basketball-shooting contest. Students will: •

calculate measures of center and variability from random samples.

•

draw informal inferences from two data sets.

Students conclude the exploration with an Exit Ticket and revisit the Hook to solve using their learning from the explorations.

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INFORMAL INFERENCES

Informal Inferences Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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Acceleration

ACCESSING PRIOR KNOWLEDGE Students will identify two truths and a lie by reading statements about statistical questions. This element is designed to uncover student misconceptions; it should not be taken for a grade. 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.

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.

FACILITATION TIP

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 their 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.

INFORMAL INFERENCES

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The third statement is the lie because symmetrical means the data matches on either side of the center.

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.

To save time and paper, project the prompts rather than print them. Have students record notes about their thinking in a journal or notebook. FACILITATION TIP Engage students by telling them you will collect student responses to the two statistical questions after step 5. Prompt them to focus on determining the truth and lie ﬁrst. FACILITATION TIP As you monitor partner collaborations, consider highlighting/emphasizing key words in the statements as needed (statistical, favorite). FACILITATION TIP This would be a good time to allow students to vote anonymously on the truths and the lie. They could use hand signals, put heads down, or turn in small ballots.

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INFORMAL INFERENCES

Informal Inferences Hook – Lunch Bunch ACTIVITY PREPARATION Students will use data collected in a survey and represented in a data table to make inferences while also evaluating whether the population surveyed is a fair representation of the population and the data collected is valid.

Materials

Preparation

Printed •

• • •

1 Lunch Bunch (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Lunch Bunch 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) Do you feel like you have plenty of room to eat in our cafeteria? Why or why not?; 2) How do you think our cafeteria could be redesigned to make it a better place to have lunch? FACILITATION TIP Before discussing questions 5a–5e, project or highlight the important details of the scenario for students to read with you. Ask students, “What do we know? or What data do we have? What is the principal trying to ﬁgure out?”

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. Gutierrez is working to solve the overcrowding issue in the cafeteria at Jefferson Middle School. He has brainstormed some potential solutions and wants students to take a survey to see how they vote on the options. He will make a data table to record the results. Then, Mr. Gutierrez will decide whether the survey data is valid and will also survey the Jefferson Middle School teachers. 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. Gutierrez is collecting data through a survey. I wonder how many students attend Jefferson Middle School. How many students will be surveyed? I can use math to compare data, and I can use data to make inferences about the most popular solution for the problem. Project Lunch Bunch. Notes

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Acceleration

Inform students that approximately 1,000 students attend Jefferson Middle School. Explain that Principal Gutierrez has come up with four solutions for the problem with cafeteria overcrowding. Mr. Gutierrez knew both the girls’ and the boys’ basketball, swim, and cross country teams practiced their sport during 4th period and then showered and got ready for class at the beginning of their 5th-period lunch. He waited for them as they rushed from the locker rooms to the cafeteria to grab food before 5th-period lunch was over. He surveyed 50 girls and 50 boys in less than ﬁve minutes and was very happy regarding the eﬃciency of his survey. Discuss the following questions: a.

DOK-1 What is an inference? Reaching a conclusion based on the evidence

DOK-1 How many people did Principal Gutierrez survey? 100. He surveyed 50 girls and 50 boys.

DOK-1 What percentage of the student population was surveyed? 100 1 10% because _____ = ___= 10% 1,000 10

DOK-1 What is the most popular answer? Option 2: Add 10 minutes to lunches (4th and 5th period) and assign students study hall rooms to go to when they are ﬁnished eating.

e. DOK-1 What is the least popular answer? Option 4: Assign students a classroom for lunch and make everyone have lunch during 5th period. 6.

Intervention

Complete the Explore activities.

INFORMAL INFERENCES

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FACILITATION TIP Consider that some students may question the strategy of surveying boys and girls as speciﬁc binary groups. FACILITATION TIP Questions 5c, 5d, and 5e are essential questions. Project them for students to see and think about (either independently or with a partner) before the class discussion. This is a good review of prior standards regarding reading charts and graphs.

Part II: Post-Explore 1. 2.

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

DOK-1 What is an inference you can make about all 1,000 students based on the data represented in the table? Answers will vary. No students want to have lunch in an assigned classroom during 5th period. Most students want to have an extra 10 minutes added to their lunch period and have study hall available.

DOK-1 Do you think the group of students surveyed accurately represents the student population of the school? Why or why not? No, I do not think the group of students surveyed accurately represents the student population. Although an equal number of girls and boys was surveyed, all of the students were athletes. All of the students surveyed were rushing from their sport to lunch with not enough time because of having to use the ﬁrst part of lunch to shower and get ready for class after sports. It makes sense they would want more time for lunch. However, the vast majority of students are not coming from a sport and already have enough time to eat. They might prefer another option.

FACILITATION TIP Ask for student volunteers to restate the problem.

FACILITATION TIP Be sure to post question b. and allow enough time for discussion. You may need to provide some follow up guiding questions to help students reach these conclusions, or you can leave it unresolved and return to it after the Explore activities.

DOK-1 How could Principal Gutierrez find a random sample of students that would accurately reflect the student population? He could find at least 100 students, an equal number of boys and girls, an equal number of students from each grade level, and try to find students from all different interest groups. He could get this group of students from lunches in the cafeteria asking only one member per group of students.

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INFORMAL INFERENCES

Informal Inferences Explore 1 – Variability in Data ACTIVITY PREPARATION Students will determine the difference between a statistical and a nonstatistical question and explain that both require data, but a statistical question has data that varies.

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.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Survey Cards (per class) 1 Exit Ticket (per student)

•

Reusable • •

6 Quart-sized resealable bags (per class) 1 Permanent marker (per class)

•

Plan to divide the class into 6 groups to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print one set of Survey Cards. Cards can be printed on card stock for durability, if desired. Cut off the questions at the bottom of each Survey Card, and place them in a quart-sized resealable bag for each station. Use a permanent marker to label each resealable bag with the station those question cards belong to: “Survey 1,” “Survey 2,” “Survey 3,” etc. Create 6 stations around the room, each with a Survey Card and its corresponding bag of questions.

PROCEDURE AND FACILITATION Part I: Understanding a Survey FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever taken a survey?; 2) If so, who was the survey for?; 3) What are some examples of questions you had to answer? FACILITATION TIP Take some time beforehand to create some constraints for the types of questions students can create in your classes. Think about: How would you deﬁne a “good” question? What might be considered too personal, not appropriate for school, not inclusive enough, etc?

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2. 3.

Read the following scenario to the class: You have been asked by a company called Survey Tiger to create a survey on a question of your choice. A survey is a data collection tool based on a question or a set of questions that is used to gather information about individuals. With this question, you will receive feedback and collect data from the responses of your classmates. Survey Tiger would then like to use this data you collected to aid them in the research they are doing regarding the question you asked. They just have some steps you need to follow when submitting your survey data. Give a Student Journal to each student. Students will work collaboratively with their groups to complete the steps in Part I of their Student Journals. Groups must work together to create a question to ask the class and then collect data by asking their fellow classmates that question and recording their responses as data. Brainstorm with the class what possible questions they could conduct a survey on that would require them to collect data. Create a list together, and display this list of questions so students can refer to it during the activity. Guide students to create statistical questions to ask their classmates in which their data will be numerical. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Give groups time to create their questions and conduct their surveys of the class. Students will then use the data collected to answer their original questions from step 1. Monitor and assess each group’s understanding of the task by asking the following guiding questions: a.

DOK-1 What survey question will you be asking your classmates? Answers will vary.

DOK-2 What does it mean when it says your question will be answered with data? Answers may vary. Data is information collected from individuals.

Intervention

Acceleration

FACILITATION TIP After supporting the creation of appropriate and relevant statistical questions, create a step-by-step plan for how students will collect the data from their classmates. Will the teacher conduct classwide surveys for each question and have students record data? Will students create an online survey for peers or families to respond to? If each group is creating one question, who conducts the survey?

INFORMAL INFERENCES

Home

FACILITATION TIP Explain the following to the class: The information collected can be numerical or categorical. Numerical data are responses that are numbers. Take time to write down, project, and Categorical data include responses such as favorite pizza toppings. clarify the deﬁnitions of numerical and categorical. Provide opportunities for d. DOK-3 How can you organize the data collected on the table provided students to practice saying the words on the survey document? Answers may vary depending on the survey question. I can list the classmate responses on the left side of the table out loud and look for root words that aid comprehension. Refer to the Visual and mark a tally for each classmate who responds with that response Glossary or Picture Vocabulary to provide on the right side of the table. examples and clear deﬁnitions for students e. DOK-3 How can you use the data collected to create a dot plot? Answers to copy down and refer to. may vary. We can split the dot plot into equal parts for each response FACILITATION TIP and label each line with the student response. Then, for each tally represented in the table, we can place a dot above that response on the Although students should be competent in dot plot. If there are a lot of responses, we can create a key that shows creating a dot plot, you might display some that each dot on the dot plot equals 2, 3, 4, 5, etc. responses. examples from prior scopes or create your c.

7. 8.

DOK-1 Using the data provided by your classmates, what is the answer to your survey question? Answers will vary based on data.

Students will then answer the Part I reﬂection questions on their Student Journals. Quickly discuss the student responses to the Part I reﬂection questions to see what students learned from this activity. a.

DOK-1 Was there only one response to the question, or did the responses vary? The responses to the question varied depending on the individual asked.

DOK-2 What is the difference between numerical data and categorical data? Both numerical and categorical data can result in numbers, but categorical data is data that is divided into groups. Categorical data can also be nonnumerical, like animal categories.

Explain to students that they just created and responded to a statistical question. A statistical question is a question that can be answered with data, and the data, or information, they collected will vary. When data varies, that means there can be more than one response.

Part II: Understanding Statistical Questions 1.

Read the following scenario to the class: Survey Tiger also has a business that contacts small and big businesses to ask them questions and track the feedback they receive to these questions. Some of these questions are statistical questions, and their data and responses can vary. Other questions are nonstatistical questions, and their data and responses do not vary. Survey Tiger needs your help determining which questions they asked are statistical and nonstatistical. Quickly review the deﬁnition of a statistical question by asking the following questions: a.

own. Model for students how to choose the numbers that go on the line (intervals), how to neatly stack the dots, and how to use scale appropriately. STEMscopes Tip In the Teacher Toolbox, the Communicate Math – Representations page under the Communicate Math tab features methods to help teachers show students how to select and use representations and to make connections between representations and what is being represented. A variety of possible representations is provided.

FACILITATION TIP Before reading the scenario, ask the class 1) Why do you think companies send out surveys?; 2) What do you think the responses they receive are used for?

DOK-1 What is a statistical question? A statistical question is a question that is answered with data, and the data collected needs to vary.

341

INFORMAL INFERENCES

Informal Inferences Explore 1 – Variability in Data

FACILITATION TIP

Ensure that these vocabulary words and deﬁnitions (statistical, non-statistical, numerical, categorical, data, survey) are included on a word wall, anchor chart, and/ or Student Journal. Encourage students to use these speciﬁc words as they discuss during their collaboration and Math Chat.

4. 5.

7. 8. 9.

DOK-2 What are some examples of data that can have various categories? Eye color, hair color, favorite subjects in school, favorite hobbies, favorite seasons, etc.

DOK-2 What are some examples of data that can have various numbers as a response? Weight, height, value or cost of items, number of siblings, etc.

DOK-2 Does this question require a response with one answer or various answers? Answers will vary depending on the question.

DOK-1 What do we call a question that requires a response with various data? A statistical question

DOK-1 What do we call a question that requires a response with only one piece of data? A nonstatistical question

DOK-2 Can a statistical question only be answered with various numbers? Explain. No, a statistical question can also be answered with various categories, such as animal colors.

Allow students enough time to talk about and sort the questions at each station and then record their responses and explanations on their Student Journals. Once the groups have completed all six stations, have students 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 • •

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DOK-1 Describe how data can vary. Data can vary when the response to a question has more than one answer.

e. DOK-1 What does it mean to find the average of something? To find the average of something, we must collect various amounts of data to find the center of that data.

FACILITATION TIP

Encourage students to be as speciﬁc as possible in their explanations. Think about creating some sentence frames that support detailed responses (For example, “B is a statistical question because the _________ will vary.”)

Assign each group to a Survey Card around the room. Instruct students to read each scenario and the questions that belong with that scenario. Encourage each group to collaborate in deciding how to sort each question and whether the question is a statistical question and has a response with various data or a nonstatistical question and has a response of only one answer. Monitor and check each group for understanding by asking the following questions:

Prior to asking an either/or question, ask students, “What type of response is required for this question?” Post these guiding questions for students to read as they collaborate to help them focus.

FACILITATION TIP

DOK-1 What is a survey question? A survey question is a question that we use to collect data to answer a statistical question.

Explain to the class that a statistical question can be answered with numerical data or categorical data, meaning the data can vary with numbers or different categories as a response to the question.

FACILITATION TIP

When asking about average, it might be a good time to review the most common measures of center (mean, median, mode).

FACILITATION TIP

•

Be prepared to discuss some of the current real-world affects of data collection; the ethics, the pervasiveness, and expansion. Data collection can be an engaging, relevant topic for students.

• •

DOK-2 What should you consider when writing a statistical question? A statistical question is answered by collecting data and must have varying responses. DOK-3 How would you describe the difference between a statistical and nonstatistical question? Both questions require data, but a statistical question requires various data responses, while a nonstatistical question only has one response. DOK-3 How can collecting data and answering questions be helpful in a real-world situation? Collecting data and answering questions can be helpful because you can determine what a group of individuals prefers or doesn’t prefer in various situations. DOK-3 Give one example of a statistical question. What is the average height of a sixth grader? DOK-3 Give one example of a nonstatistical question. Do you like the color blue?

Engage

Explore

Explain

Elaborate

Evaluate

2. 3.

Acceleration

FACILITATION TIP

Post-Explore 1.

Intervention

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.

This Exit Ticket requires students to be able to recognize statistical and non-statistical questions and be able to explain their reasoning. Part II of this scope directly supports success with this skill. Therefore, if needed, Part I could be used as a challenge or follow-up activity when time allows.

INFORMAL INFERENCES

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INFORMAL INFERENCES

Informal Inferences Explore 2 – Valid Generalizations ACTIVITY PREPARATION Students will decide whether conclusions drawn from a sample of the population are valid in determining the outcome for an entire population.

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.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 Sample Population Cards (per group) 1 Set of Sample Data Cards (per group) 1 Exit Ticket (per student)

Reusable •

• • •

2 Resealable bags (per group)

Print a set of Sample Data Cards for each group. Cut out and place cards in a resealable bag labeled “Part II.” If desired, print them on card stock, and laminate them for future use. Plan to divide the class into groups of three or four to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Sample Population Cards for each group. Cut out and place cards in a resealable bag labeled “Part I.” If desired, print them on card stock, and laminate them for future use.

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) When might we want to know what a large group of people want or like?; 2) What can we do to get a general idea of what a large group of people want or like without asking each person?; 3) Is taking a sample of what a large population of people want or like always accurate? Why or why not? FACILITATION TIP

Project the scenario and help students determine the essential information as you read it together.

FACILITATION TIP

When you clarify the meaning of valid and invalid, post the deﬁnitions and examples in the classroom for students to refer to during the scope. Encourage students to use these words in their collaborations.

344

Part I Read the following scenario to the class: The 7th-grade students at North County Middle School won a pizza party for donating the most canned goods at the Thanksgiving Festival. The assistant principal would like to ask some students what pizza toppings they prefer to get an idea of what types of pizzas she should order for the 200 students in the 7th grade. Read each Sample Population Card, and help the assistant principal decide whether it is a valid sample to use for ordering the pizzas. Give a Student Journal to each student. Give a set of Sample Population Cards to each group. If necessary, clarify the meaning of the words valid and invalid by explaining that valid conclusions are logical, based on reasonable results. Conclusions can be invalid for a variety of reasons, and these conclusions are not reliable. Read and discuss the ﬁrst Sample Population Card with the whole class: The assistant principal asks one student which type of pizza they like best. a.

DOK-1 Would this be a good sample of the entire 7th grade to base the pizza order on? No, not enough students were asked.

DOK-2 Why might this sample produce inaccurate results? The one student selected could prefer a type of pizza that many students out of the 200 do not like, such as pineapple. Then, the only pizza the assistant principal will order is that one type of pizza.

d. 5.

6. 7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

DOK-1 Would you classify this sample population as valid or invalid? This is not a valid sample population.

Explain to students that as they read each Sample Population Card, they should think about how this sample population of students compares to the entire 7th grade. Students will be collaborating with their groups to determine if the sample population on each card will result in valid or invalid conclusions about the entire 7th-grade class. Encourage each group to discuss these questions together before making a ﬁnal decision for each Sample Population Card. Explain that after they have made a decision about each Sample Population Card, students should ﬁll out the table in their Student Journals. While students are working together, monitor groups and ask the following questions to check for understanding: a.

DOK-2 I see you have categorized ____ sample population as valid/ invalid. Can you share the reasons your group had to justify that decision? Students should provide justiﬁcation based on the speciﬁc Sample Population Card and their group discussion.

DOK-3 For the cards you categorized as invalid, what change might you make to the Sample Population Card wording so the card would be a valid sample population? Students should provide justiﬁcation based on the speciﬁc card and their group discussion. The necessary change should be made so the student sample population is proportionally representative of the entire student population.

FACILITATION TIP Be prepared for students to be eager to share their own opinions about favorite pizzas and for them to be passionate about the validity of this (or any) survey. For example, some middle school students may say, “What if they don’t like pizza? What if they’re allergic? Consider collecting your class’ data regarding pizza before, during, or after this Explore activity.

INFORMAL INFERENCES

Home

FACILITATION TIP Select and project a few key questions from this Explore activity to encourage students to focus their collaboration discussions and be prepared for the Math Chat. As you monitor, direct students’ attention to the focus questions.

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

Math Chat DOK-1 If a sample population is not valid, should you use it to make decisions for the entire population? No, a group of people could have been left out, or another group might have been overrepresented. The decisions made would not match what the entire population wanted/preferred. • DOK-2 What would you say are the characteristics of a good sample population? Fair representation of all subgroups within a population, a large enough sample to reasonably represent a cross section of the entire population • DOK-3 What is an example of a good sample population? Why do you think that would be a good sample? A good sample would satisfy the characteristics from the question above. • DOK-3 What is an example of a bad sample population? Why do you think that would be a bad sample? A bad sample would only include a speciﬁc portion of the entire population or would exclude some people groups. It may also include too few people to make any generalizations for the group. •

Part II 1. 2.

Give a set of Sample Data Cards to each group. Explain that each card shows the results of different surveys taken by the assistant principal. The results shown are from valid sample populations of the 200 7th-grade students at North County Middle School. Groups will read each card and make three generalizations, inferences, conclusions, or predictions about the entire 7th-grade population based on the results of the assistant principal’s collected data. Read and discuss the ﬁrst card in the set with the whole class: The assistant principal asked 50 7th-grade students which type of pizza they preferred. a.

DOK-1 Do you believe this sample population is large enough to represent the entire 7th grade? Yes, 50 out of 200 is one-fourth of the entire 7th grade.

FACILITATION TIP Be prepared to follow up with discussions about when students have felt unrepresented and times that the use of invalid data impacted them. FACILITATION TIP As you discuss good and bad data, consider using terms like relevant, accurate, complete, timely etc to help students determine the validity of the samples. FACILITATION TIP Before passing out all four data cards, project each one individually and coach students to note the labels and different ways to represent data. This is a good time to review prior standards regarding tally marks, bar graphs, and averages (mean). FACILITATION TIP If time allows, create two or three data cards that reﬂect the interests of your current group(s) of students. Consider adding in a “survey” on video games, YouTubers, after-school hobbies besides sports, favorite ways to relax, favorite class, or favorite season.

345

INFORMAL INFERENCES

Informal Inferences Explore 2 – Valid Generalizations DOK-2 What recommendations would you give to the assistant principal before she orders 25 pizzas for the entire 7th-grade class? Answers may vary. About half of the pizzas she orders should be pepperoni.

DOK-3 Based on the results shown in the table, what can you assume about all 200 7th-grade students at North County Middle School? Answers may vary. None of the students like sausage on their pizza.

DOK-3 Can you make any inferences about the whole group of 7thgrade students based on the results from the assistant principal’s data? Answers may vary. None of the students like sausage on their pizza.

e. DOK-3 How could you use this data to make decisions for all the 7th graders? Answers may vary. The assistant principal should still order pineapple and veggie pizzas because some students prefer them, even though they are not a majority.

Explain that students will read each of the remaining Sample Data Cards as a group. Students will write three statements about the entire population based on the data provided in their Student Journals. While students are working together, monitor groups and ask the following questions to check for understanding: a.

DOK-1 Do you believe this sample population is large enough to represent the entire 7th grade? Yes, I believe it is large enough because it has (40/50/25) students.

DOK-2 How could that conclusion be used to make a decision for the entire population? We can assume that since approximately half of the students surveyed prefer ____, then about half, or 100 out of the 200 students, would also prefer ____.

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

Math Chat DOK-1 How can the sample data be useful when making decisions for the whole group? Look at the data, and see which categories are selected most often and least often. Then, you will know how the larger population might respond to the same questions. • DOK-2 When might someone in the real world use data from a sample population? It is easier to ask a smaller number of people from the entire population than to ask every single member. A store can easily ask a sample of the customer population which brand of soda they prefer so they can order the right varieties for their stock. • DOK-3 If you had to make a prediction, how many of the 200 7th-grade students would you expect to select ______? Can you explain the process you used to get that answer? I used a proportion of x over 200 and value over total sample size, and then I found the relationship between the two denominators and applied it to the numerators to ﬁnd x, or I found the relationship between the numerator and denominator and applied it to the other fraction in my proportion. •

FACILITATION TIP Consider having some real-world (local) examples prepared for when data from a sample population has been used to make decisions.

Post-Explore

346

FACILITATION TIP

For this Exit Ticket, consider providing some sentence frames or starters for students.

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

INFORMAL INFERENCES

Home

347

INFORMAL INFERENCES

Informal Inferences Explore 3 – Use Data to Make Inferences ACTIVITY PREPARATION Students will use data collected from a random sample to make predictions about an entire population.

Standards for Mathematical Practice • • • • •

Materials

Preparation

Printed • • •

•

1 Student Journal (per student) 1 Set of Theater Survey Result Data Cards (per group) 1 Exit Ticket (per student)

• •

Plan to divide the class into groups of three or four to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Theater Survey Result Data Cards for each group. Cut out and place the cards in a resealable bag. If desired, print them on card stock, and laminate them for future use.

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) How does information collected from customers beneﬁt businesses?; 2) What methods do businesses use to collect information from their customers? After reading the scenario, ask the class 3) How is the sampling size related to the accuracy of the responses received? FACILITATION TIP Project the scenario and allow students to read through it silently, and then together. Guide students to locate the essential information. Ask, “What is the problem Ms. Harshan is trying to solve?” FACILITATION TIP

2. 3.

Read the following scenario to the class: Ms. Harshan has just been hired as the new manager of Cinema Star 16, the movie theater in her town. She wants to increase concession-stand purchases and ticket purchases for the theater, so she decides to conduct a survey of some movie-theater guests. The survey is sent to 250 random customers who have purchased at least one movie ticket at Cinema Star 16 in the past 12 months. She will use the results of the survey to make changes that will improve the customer experience at Cinema Star 16. With these changes, she is hopeful that customers will return to the theater more frequently in the following year and spend more money at the concession stand. Give a Student Journal to each student. Give a set of Theater Survey Result Data Cards to each group. Share the ﬁrst Theater Survey Result Data Card, “Days of the Week,” with the whole class. Explain the following to the class: We will be using the data from this card to make predictions about the population of all moviegoers. 2,000 people visited Cinema Star 16 in the past week. a.

This would be a good time to review proportion (Stemscopes visual glossary).

, and for the whole population total visitors. For the survey, it will be ____ 250 x

348

DOK-1 What can we use to compare the survey data to the whole population data? We can set up a proportion for Saturday visitors over . of visitors, it will be _____ 2,000

DOK-2 How can we use the proportion to determine how many people visited the movie theater on Saturday of the past week? We can use our 80 x proportion, ____ = _____. There is a relationship between 250 and 2,000. 250 2,000 Because 250 times 8 is 2,000, we can also multiply 80 by 8 to ﬁnd x, which is 640. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 Is there another way we can calculate the number of visitors who likely visited the theater on Saturday last week? i. Since we know that 80 out of 250 people surveyed visited on a 80

2,000

to ___ and then multiply by _____ . It Saturday, we can simplify ____ 1 250 25 16,000

, which is 640. would be ______ 25

Intervention

Acceleration

FACILITATION TIP Question 3c is important. Encourage students to keep thinking of other ways to solve this proportion (some may remember double number lines with percents).

ii. Since we know that 80 out of 250 people surveyed visited on a 80 Saturday, we can ﬁnd an equivalent decimal for ____ , which is 0.32. 250 We can then multiply 2,000 by 0.32, which is 640. d.

INFORMAL INFERENCES

Home

DOK-3 If you were the movie theater manager, why might you want to know this information? It might be helpful to compare this data to the actual ticket sales for Saturday to see if her results are accurate. She can see if the theater is selling more tickets after her improvements have been made.

Explain the following to the class: Variability is how far apart the actual data points are from the expected value or mean. Students should understand that sample data will give a reasonable approximation of a predicted value but may not be exactly accurate every time. To get a truer picture of the overall population, it is sometimes helpful to collect multiple samples and compare the results as a range of possible reasonable values. a.

DOK-1 How will the predictions for a population be affected if the sample size is low? The variability in a prediction will increase with a smaller sample size. There may be a wider gap between the survey results and the actual results.

DOK-2 What method can be used to decrease variability? Larger data samples are better for more accurate results, but if it is not possible to get one large sample, multiple data samples of equal size should be collected and compared, and variability will decrease with a larger sample size.

Share the last Theater Survey Result Data Card, “Snack Time!,” with the whole class. Explain the following to the class: We will be using the data from this card to make predictions about the population of people who purchase snacks from the concession stand. a.

DOK-1 Which concession worker had the lowest percent of guests that preferred chocolate candy? Neysa

DOK-1 What percent of Neysa’s guests preferred chocolate candy? Her guests preferred chocolate candy 32% of the time.

DOK-1 Which concession worker had the highest percent of guests that preferred chocolate candy? Eliana

DOK-1 What percent of Eliana’s guests preferred chocolate candy? Her guests preferred chocolate candy 35% of the time.

FACILITATION TIP Slow down and take time to carefully explain sample data and extrapolation to the students. Have students take notes on their Student Journal about variability, range, reasonable, mean, bias, and other terms as needed. Consider mentioning some real-world examples.

e. DOK-2 What generalization can you make about all concession stand guests who like chocolate candy best? Somewhere between 32% and 35% will select chocolate candy as their favorite. f.

DOK-3 Currently, 25% of the snacks ordered by the concession-stand manager is chocolate candy. What recommendation would you make to the concession-stand manager when she places future snack orders? i. She should increase the amount of chocolate candy ordered since about a third of the guests said that was their favorite, but only onefourth of the snacks ordered are chocolate candy. ii. On her next snack order, she should select chocolate candy for 32%–35% of the total order. If there is some candy left over, she can sell it the next month, since those snacks do not go bad quickly.

349

INFORMAL INFERENCES

Informal Inferences Explore 3 – Use Data to Make Inferences 6.

Explain to students that they will use the four Theater Survey Result Data Cards to ﬁnd the sample size for each data set. Then, students will work collaboratively through each problem in their Student Journals. While students are working together, monitor groups and ask the following questions to check for understanding:

FACILITATION TIP Project question 7b. as a guiding question during student collaboration. Encourage students to focus their discussion on methods and reasoning. 8.

DOK-1 Which method did you choose to find the predicted value for this chart? Student answers may vary and could include setting up a proportion, finding the percent and then multiplying by the total, finding the fraction and then multiplying by the total, creating a percent bar/ strip diagram, or drawing out the situation.

DOK-2 Is it always necessary to get an exact answer when making a prediction? It depends on what you are using the predictions for. If you just want to see how many people might want to see a certain movie, it might be OK to get a general idea. But if you are ordering food or making sure you have enough supplies/food for everyone it might be more important to be as accurate as you can.

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

Math Chat • •

• FACILITATION TIP Biased data is a critical concept for students. Have student volunteers deﬁne it in their own words and check for student understanding. Have students note the deﬁnition on their Student Journal and post it in your classroom.

•

DOK-1 Why do we use surveys instead of collecting data from the entire population? It might take too long or be too diﬃcult to collect data from an entire population. DOK-1 When using a proportion to make predictions about a population, how do you determine where the values go? The fractions should be set up as part over whole.Total sample size, total population values, or 100% will be represented in the denominator. The desired group of the sample size, the desired part of the population, or the other percentage will be represented in the numerator. DOK-2 Does it make sense to base inferences/predictions on biased data? No, biased data will not give us a true picture of what the entire population would select and could be unfair to a group of people. DOK-2 Why might the predicted result based on a survey differ from the actual population? There is always some variability when you base a result on a random sample. As long as the sample size is reasonable and representative, it is OK to have variability. DOK-3 Is it always reasonable to use a sample population to make predictions or decisions for an entire population? Can you think of an example? No, some predictions/decisions need to be made as a whole population. Examples include the following: • • •

•

DOK-3 What are some other situations where people might make predictions based on sample data from a larger population? • •

•

350

US census—every person in every region needs to be counted. Local, state, and national elections—every voter’s voice counts. A TV game show where every contestant has a vote to kick someone off the island.

Polls taken before big elections help politicians see how they are doing before the ﬁnal votes are counted. Media companies do surveys about the types of television programming people prefer so they can create and produce the types of shows people will want to watch more often. The school district asks parents and students what kind of extracurricular school activities they may want in the future so they can plan their budget to include the new programs that are most popular.

Engage

Explore

Explain

Elaborate

Evaluate

2. 3.

Acceleration

FACILITATION TIP

Post-Explore 1.

Intervention

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.

Read through the Exit Ticket with students before providing independent work time. Allow for relevant questions about size of theaters, how many people were surveyed, etc. Be sure students note the details in the scenario.

INFORMAL INFERENCES

Home

351

INFORMAL INFERENCES

Informal Inferences Explore 4 – Compare Data ACTIVITY PREPARATION Students will visually compare the data from two populations and determine whether there is a visible difference between them based on sample data from each population.

Standards for Mathematical Practice • • • • •

Materials

Preparation

Printed • • • •

• •

1 Student Journal (per student) 1 Set of Class Heights (per class) 1 Set of Dot Plot Data Cards (per group) 1 Exit Ticket (per student)

Part I • • •

Reusable • •

Plan to divide the class into groups of four to complete this activity. Print a Student Journal and an Exit Ticket for each student.

1 Measuring tape (per group) 1 Projector or document camera (per class)

•

Print a set of Class Heights for the class. Have a measuring tape for each group to measure each student’s height. Optionally, aﬃx multiple measuring tapes to the wall for students to measure each group member’s height. Have a document camera or projector ready to project the Class Heights on the board.

Part II •

Print a set of Dot Plot Data Cards for each group. Cut out the cards. If desired, print them on card stock, and laminate them for future use.

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) Do you think men are statistically taller than women around the world? Why or why not? After reading the scenario, ask the class 2) How can we determine the difference between the height of 7th-grade girls and boys?; 3) What visual tools can we use to compare the height data collected? FACILITATION TIP Consider breaking this Explore activity into smaller steps. Do Part I one day and Part II the next. Also, collecting the height of students can be done the day before or with adult help ahead of time. However, some middle-school students may feel self-conscious about their height. Perhaps use a sample data set (you could create one from another class section or from a previous year) to avoid having one or two students revealed as the outliers. 352

Part I 1.

3. 4. 5. 6.

Read the following scenario to the class: Dr. Miranda from Prime Pediatrics needs our help. Across the world, men are statistically taller than women by a global average of 7%. Can the same be said of 7th-grade students? Help Dr. Miranda ﬁnd the answer to this question. Explain the following to the class: Think about this question, and then come up with a hypothesis. Will there be a visible difference between the height of the girls and the height of the boys when displayed in a dot plot? If so, which gender do you think will be taller? Give the Student Journal to students. After allowing students time to record their hypothesis statements, students will collaborate with their groups to measure each member’s height. Project the Class Heights on the board. Have each student record their height on the lists for girls’ heights or list for boys’ heights, as appropriate. Explain to students that they will collaborate with their groups to record the collected data on their Student Journals to determine the ﬁve-number summary and interquartile range to create a box plot for heights of girls and a box plot for the heights of boys.

9. 10.

Engage

Explore

Explain

Elaborate

Evaluate

Explain to students that they should look at each box plot as it compares to the other box plot. Allow groups time to review the data and discuss their observations. Monitor and assess students learning by asking the following guiding questions: a.

DOK-3 Would it be reasonable for a teacher to include their height on the appropriate gender dot plot? No, we are comparing the heights of 7th-grade students, and since teachers are not 7th-grade students, the teacher’s height would make the data invalid if it were to be included in the data set.

DOK-2 What do you notice about the median of the girls’ data versus the boys’ data? Responses will depend on class data. The medians are very close together, but the girls’/boys’ is slightly higher.

DOK-3 Compare the interquartile range for each set of data, and share some observations with your group. Responses will depend on class data. The interquartile range for the boys is higher than the girls, which means the data for this group is more spread out and the individual data points are not as close to the mean as the other group.

DOK-3 Revisit your hypothesis statement. What changes or additions would you make to your statement now that you have seen the data displayed and learned the median and interquartile range? Answers will vary. Although our original statement was factually accurate, there is more that can be understood from the data when exploring the difference in the centers and variability.

Intervention

Acceleration

FACILITATION TIP The binary categories of boys and girls may not work for some classes.

INFORMAL INFERENCES

Home

Allow time for students to complete the questions and reﬂection 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 are some similarities or differences between these two sets of data? The boys’ data set seems to be more spread out than the girls’ data set. DOK-3 How can median and interquartile range help you compare data in box plots? By knowing the median, you can get a representation of the center of the data set. Using the interquartile range, you can determine if the median is an accurate representation of the data set.

Part II 1.

3. 4.

FACILITATION TIP

Read the following scenario to the class: Both Los Angeles, California, and Little Rock, Arkansas, lie near the 34th parallel north—that is, 34°N latitude—above the equator. Both cities receive just over 14 hours of sunlight per day for the ﬁrst three and a half weeks in July. What predictions would you make about the mid-July temperatures for these two cities? Students should discuss with their groups to create their predictions. Then, students will write their predictions about mid-July temperatures for these two cities on their Student Journals. Give the July High Temperatures Dot Plot Data Card to each group. Students will collaborate with their groups to investigate the data from the July High Temperatures Dot Plot Data Card. Then, students will complete the questions about the July High Temperatures Dot Plot Data Card on their Student Journals. Monitor and assess students’ learning by asking the following guiding questions: a.

DOK-1 What is the difference in the means for these two data sets, and what does this tell us? The difference in means is 7.6°. This tells us that Little Rock typically has higher temperatures in July than Los Angeles does.

Depending on your students, consider completing one of the scenarios (July High Temperatures or Sodium Content) together as a class so you can model accurate, precise investigation of the data. FACILITATION TIP Before reading the scenario, ask the class 1) Do the hours of sunlight an area gets affect its temperatures?; 2) What other factors can affect an area’s temperature? FACILITATION TIP Use a map or globe to illustrate and review latitude and longitude for students. FACILITATION TIP Project the guiding questions for students to see as you monitor and assess during their collaboration. 353

INFORMAL INFERENCES

Informal Inferences Explore 4 – Compare Data DOK-1 What is the difference in the mean absolute deviations (MADs) for these two data sets? 0.22

DOK-2 Based on these data sets, do you think latitude is a good way to determine if two areas will have similar weather? No, it may be helpful in general, but not for speciﬁc planning purposes. You might know that the temperatures will be warm instead of cold, but you cannot tell how warm or hot it will be just by knowing the latitude.

DOK-3 What do the means and MADs tell you about the midsummer temperatures in these two cities? While Little Rock, AR, is on average about seven and a half degrees warmer than Los Angeles during the same time period in July, the variability between the July temperatures for these two cities is almost exactly the same. The temperature ﬂuctuates about the same amount in both cities during this time period.

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 reading the scenario, ask the class 1) What is your favorite type of soda?; 2) Do you believe diet sodas are healthier than regular sodas? Why or why not?; 3) What other reasons, besides their sugar content, might cause people to believe sodas are not a healthy drink? FACILITATION TIP If you can, bring in a few cans of regular and diet soda to compare sodium content on real-world sodas.

e. DOK-3 What other factors besides latitude might affect the temperatures of a city or area? There are different climates; southern California is in the dry subtropical or Mediterranean climate, while central Arkansas is part of the humid subtropical climate. Los Angeles is very close to the Paciﬁc Ocean, which has colder water temperatures. This could cause the air from the ocean to have a cooling effect on the city as wind brings lower temperature air onto the land. 6.

7. 8. 9. 10.

FACILITATION TIP

Read the following scenario to the class: Diet soda was invented in the 1950s as a choice for people with diabetes who could not eat sugar but was later promoted as a healthier choice for people trying to eat healthy or lose weight. However, sugar and sweeteners are not the only ingredients in various sodas. Sodium has been identiﬁed as a cause of increased heart rate and blood pressure, which can lead to heart attacks. Let’s investigate the amount of sodium in regular sodas and diet sodas. Discuss with your group how you think diet soda will compare to regular soda on sodium content. Allow students time to discuss their predictions with their groups. Then, students will write down their predictions on their Student Journals. Give a Sodium Content in Soda Dot Plot Data Card to each group. Students will collaborate with their groups to investigate the data shown in the dot plots and write down their observations on their Student Journals. Monitor and assess students’ learning by asking the following guiding questions: a.

DOK-1 What is the difference in the means for these two data sets? 0.56

DOK-1 What is the difference in the MADs for these two data sets? 4.94

DOK-2 How do the MADs of these two data sets help us understand the data more thoroughly? The MADs tell us there is a lot more variability in the data for the diet soda. The data values are much more spread out.

DOK-3 What do the means and MADs tell you about the sodium content in regular soda versus diet soda? Although the averages are almost exactly the same, the spread in the data is not the same or even close. The MAD for diet soda is about 1.5 times the MAD of regular soda.

e. DOK-3 What suggestions would you give to someone trying to make a healthy drink choice? It is important to look at the ingredients for each of the drinks you want to compare. Just because something says “diet” doesn’t mean it is the healthier choice for someone trying to reduce the amount of sodium they are consuming.

Students may be curious about their own favorite drink choices. As an extension and connection, encourage students to bring in data about their favorite drinks. 11. 12.

354

Allow time for students to complete the questions on their Student Journals, including the reﬂection questions. After Part II, invite the class to a Math Chat to share their observations and learning.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 Are the mean and the MAD the same? No, the mean is the average. The MAD shows how far apart each data point is from the mean and shows how closely the mean represents the data. • DOK-2 Does knowing the variability (MAD) of a set of data give you enough information to make conclusions about the data? No, some data can have the same MAD, but their means are very different. You would need also to know the mean or see the data to make any conclusions. • DOK-2 Does a variability of about 2 mean a large spread or a smaller spread? It will depend on the data. Two is not a large number when looking at temperatures in the 80s and 90s. Two degrees won’t make much of a difference. If this were different data, 2 might be a bigger deal, such as looking at shoe sizes. Since shoe sizes are usually 5–10, a difference of 2 shoe sizes would be a bigger change. • DOK-3 How can mean and MAD help you compare data in dot plots? By knowing the mean (average), you have a general value that represents the group, but by looking at the MAD as well, you will be able to tell whether the mean is really the best representation of a set of data. Then, you can use that information to help make good decisions or ask for more information. •

Post-Explore

2. 3.

Acceleration

FACILITATION TIP

Math Chat

Intervention

Before asking this question, ask “How are mean and mean absolute deviation the same?” Then, ask, How are they different? FACILITATION TIP Consider asking, “How does knowing the variability help you make decisions? What other data can you use to make reliable decisions?”

INFORMAL INFERENCES

Home

FACILITATION TIP In addition, ask, “What does a variability of 2 tell us about the data?” FACILITATION TIP Project this question for students to see and think about individually. Have them discuss it with a partner and be prepared to share with the class.

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.

355

INFORMAL INFERENCES

Informal Inferences Explore 5 – Centers and Measures of Variability ACTIVITY PREPARATION Students will calculate measures of center and variability from random samples and draw informal inferences from two data sets.

Standards for Mathematical Practice • • • • •

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 2 Sets of Basketball 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 and cut out 2 sets of Basketball Cards for the class. If desired, print them on card stock, and laminate them for future use. Create 8 stations around the classroom. You will be running 2 sets of 4 stations in class. At each station, place 1 Basketball Card.

PROCEDURE AND FACILITATION FACILITATION TIP After reading the scenario, ask the class 1) What types of information can the district get from the data they collect?; 2) How will the district use the data to decide which schools should be represented in the competition?; 3) What could the district do to increase the accuracy of the data they collect?

Part I: Measures of Center 1.

FACILITATION TIP Before distributing the Student Journals, project (as well as read aloud) the scenario to help students note the critical pieces of information.

2. 3.

FACILITATION TIP If space is an issue, you can rotate the Basketball Cards rather than have students rotate. Consider printing each Basketball Card on a different color. FACILITATION TIP

Read the following scenario to the class: The Columbus City School District has decided to enter a statewide basketball-shooting contest. The district is allowed to choose up to two schools, one boys’ team and one girls’ team, to represent them in the competition. To help the district choose, each player has to shoot 100 shots and record the number of shots they make. To make it fair, the district took a random sampling of ten students from each of the four junior high schools. Help the officials use that data to decide which schools should represent Columbus City in the competition. Give a Student Journal to each student. Explain to students that they will rotate around the room two times. The first time, they will use the data on the Basketball Cards to find the measures of center. Provide students with enough time at each station to find the median and mean of each data set. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What are some measures of center? Measures of center are the mean, median, and mode of a data set.

DOK-1 How do you ﬁnd the mean of a set of data? You add all the data points and divide them by how many data points there are.

DOK-1 How do you find the median of a set of data? You arrange all the data points in order from least to greatest and find the middle number. If there are two middle numbers, you find the mean of those two numbers.

Use one of the Basketball Cards to model for students how to show ﬁnding the median and mean. Use a student work sample to share strategies. 356

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How do you decide which measure of center is the best one to use? You can look at the measure of center and compare it to the data. If there are not a lot of outliers, it is good to use the mean to ﬁnd the measure of center. If there are enough outliers, the mean will give you a number that doesn’t accurately describe the data because it’s higher or lower than most of the numbers.

Allow students enough time to complete Part I and answer the questions that follow.

Acceleration

FACILITATION TIP Print and project the guiding questions for students to view as they collaborate. If students a need review of the three most common measures of center, have them take notes on their Student Journal and post examples and deﬁnitions in your classroom. FACILITATION TIP

Part II 1.

Intervention

INFORMAL INFERENCES

Home

Explain to students that they will rotate around the room a second time. This time, they will use the data on the Basketball Cards to ﬁnd the measures of variability. Provide students enough time at each station to ﬁnd the measures of variability of each data set. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How do you ﬁnd the range? You subtract the smallest data point from the largest data point.

DOK-1 How do you find the interquartile range? You find the median of the data set. Then, you find the upper and lower quartiles and find the range of those two numbers.

DOK-1 How do you ﬁnd the mean absolute deviation (MAD)? You ﬁnd the absolute distance between data points and the mean and then divide that number by the number of data points.

DOK-2 Both the girls’ teams have the same interquartile range. Does that mean they have the same variability? No, the Columbus City School has a larger range and a larger MAD; therefore, it has a larger variability.

Post a word bank with some words students can include in their reasoning for the reﬂection questions (range, spread, outlier, deviation, etc.). FACILITATION TIP Choose one of the Basketball Cards to demonstrate how to ﬁnd and show work for range, interquartile range, and mean absolute deviation. FACILITATION TIP If you have time, create a visual number line showing the interquartile range from one of the data sets. Even though this comes from a prior standard, students may need to see the Q1, Q2, and Q3 markers to make a connection.

e. DOK-2 Why can’t you just use the mean to decide which teams should be chosen? The means for the four teams are very similar. We need to look at all the data to decide which team would be the best choice. f.

3. 4.

DOK-2 How does the MAD help you decide in this situation? The MAD tells us how far each data point is from our mean. This tells us how consistent the shooters should be in the competition.

Allow students enough time to complete 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 • •

•

DOK-1 Looking at all 4 schools, which school had the greatest variability? Hawkeye Junior High School had the greatest variability with a range of 65. DOK-2 How did you know which measure of center to use when choosing between the median and the mean? I looked at the measure of center and compared it to the data. If there are not many outliers, it is good to use the mean to ﬁnd the measure of center. If there are enough outliers, the mean will give us a number that doesn’t accurately describe the data because it’s higher or lower than most of the numbers. DOK-2 Why was it important also to look at the measures of variability when choosing the best team? The measures of variability tell us how spread out our data is. In this case, it helped us see how consistent the shooters would be for each individual school.

FACILITATION TIP Engage students by making some realworld connections to sports analytics in college and professional sports.

357

INFORMAL INFERENCES

Informal Inferences Explore 5 – Centers and Measures of Variability DOK-2 Instead of putting the scores in order and taking the sample, the school district used the ﬁrst column of the Columbus City Girls for the sample mean. Describe the possible implications, and gauge how far off the sample is from the actual mean. Using the ﬁrst column results in a sample mean of 74. This is much higher than using every other score from the scores put in numerical order. Instead of the schools appearing similar, this could lead to inferences that Columbus City Girls have a greater mean. • DOK-2 What could the school district do to increase the accuracy of their prediction? Explain. The school district could take more than 10 data points from each school. The more data points you have, the more accurate your predictions will be. •

Post-Explore FACILITATION TIP

To support student explanations on the Exit Ticket, post a word bank that they can use. Some struggling students may beneﬁt from sentence frames on this Exit Ticket.

2. 3. 4.

358

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

INFORMAL INFERENCES

Home

359

INFORMAL INFERENCES

Informal Inferences 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

Variability in Data 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

Valid Generalizations 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

Use Data to Make Inferences

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

Compare Data

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

Show What You Know, Part 5 Centers and Measures of Variability Independent practice assignment that gives students an opportunity to demonstrate their learning

360

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

Compare Data

Can be done independently

INFORMAL INFERENCES

Home

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

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

361

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

INFORMAL INFERENCES

Informal Inferences

3 362

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

INFORMAL INFERENCES

Home

What does mastery look like?

I can compare data through observations of bar graphs.

I can compare data of various samples and populations by using box plots and dot plots.

I can draw inferences by using measures of central tendency and variability from samples.

I can describe a distribution by using measures of central tendency and variability.

I can gather data to create and answer a statistical investigative question.

I can determine strategies for gathering data that will help to answer statistical investigative questions.

I can draw inferences based on various collections of sample data.

363

Kindergarten ten Published by Acceler ve, Suite 800, Houston, on, TX 77056. Copyright © 2023, by Acceler Accelerate Learning Inc. All rights reserved. ved. No par partt of this publication may be repr reproduced or distributed in any form or by any means, or stored in a database or retriev retrieval system, without prior written consent of Accelerate Learning Inc., including, but not limited to, in any network or other electronic onic storage or transmission, tr T

7 Georgia Math Teacher Guide

Grade 7 Teacher Guide

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

ISBN: 979-8-88826-666-3

7 GEORGIA

MATH G7