STEMscopes Georgia Math Teacher Guide Grade 8 by acceleratelearning

8 Georgia Math Teacher Guide

Grade 8 Teacher Guide

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

ISBN: 979-8-88826-667-0

8 GEORGIA

MATH G8

GEORGIA

Teacher Guide: Grade 8 ISBN: 979-8-88826-667-0 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

Square Roots and Cube Roots ............................................................................. 18

SCOPE 2

Irrational Numbers.............................................................................................. 36

SCOPE 3

Integer Exponents .............................................................................................. 60

SCOPE 4

Scientiﬁc Notation .............................................................................................. 78

SCOPE 5

Operations with Scientiﬁc Notation ..................................................................... 94

SCOPE 6

Solve Equations ................................................................................................ 110

SCOPE 7

Solve Inequalities ............................................................................................. 140

SCOPE 8

Create Non-proportional Relationships from Proportional Relationships ............. 164

SCOPE 9

Functions ......................................................................................................... 176

TABLE OF CONTENTS

Table of Contents

SCOPE 10 Rate of Change and Initial Value ........................................................................ 196 SCOPE 11 Linear Forms .................................................................................................... 218 SCOPE 12 Bivariate Data ................................................................................................... 240 SCOPE 13 Parallel and Perpendicular Lines ....................................................................... 262 SCOPE 14 Solving Pairs of Linear Equations ...................................................................... 284 SCOPE 15 Pythagorean Theorem ...................................................................................... 308 SCOPE 16 Volume ............................................................................................................ 330

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

Square Roots and Cube Roots Scope Introduction SCOPE SUMMARY

Student Expectations

8.NR.2.2 Use square root and cube root symbols to represent solutions to equations. Recognize that x² = p (where p is a positive rational number and |x| ≤ 25) has two solutions and x³ = p (where p is a negative or positive rational number and |x| ≤ 10) has one solution. Evaluate square roots of perfect square ≤ 625 and cube roots of perfect cubes to ≥ –1000 and ≤ 1000.

Up to this point, students have been exposed to squared and cubed numbers as they began to solve equations and used the order of operations. Students will now have ample opportunities to work systematically with square root and cube root symbols through writing and ﬁnding solutions to equations. They will be writing with both square and cube root symbols in order to solve given expressions. Students will also be exploring the different ways to write solutions to square roots and cube roots with the possibility of a positive and negative outcome.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In Grade 5, students began denoting the whole number powers of 10 with exponential notation. In Grade 6, students were writing and evaluating expressions that contained whole-number exponents. The students have encountered situations where they need to expand whole-number exponents in order to complete the solving of a numerical expression.

In high school, students will expand their knowledge of radicals as they begin to expand and simplify radicals within equations. Students will be focusing on properties of rational exponents to ﬁnd roots of various expressions through polynomials, quadratic equations, Pythagorean theorem, and complex area and volume questions. They will also apply knowledge of radicals to graph various parent functions (square root and cube root).

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

write and evaluate numerical expressions involving wholenumber exponents.

•

match numbered cards with lettered cards posted around the room.

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

Hook

Accessing Prior Knowledge

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

determine the volume of a cube based on the area.

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. 18

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Square Roots and Perfect Squares In this exploration, groups of tasked with solving a realworld scenario where students are asked to help ﬁnd the area of different frames so the cardboard backs can be created. Students will: •

ﬁnd patterns that recognize that solutions to perfect squares can never be negative.

•

apply their knowledge of squaring numbers being inverses to square roots.

•

calculate solutions to problems containing either a perfect square or a square root.

Explore 2

Explore 1

EXPLORE ACTIVITIES Cube Roots and Perfect Cubes In this exploration, groups of students will need to ﬁnd the side lengths and volume for memory boxes to be sold at a craft fair. Students will: •

ﬁnd patterns to recognize that solutions to positive perfect cubes can never be negative.

•

apply their knowledge of cubing being the inverse to taking cube roots.

•

calculate solutions to problems containing either a perfect cube or a cube root.

SQUARE ROOTS AND CUBE ROOTS

Home

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.

SQUARE ROOTS AND CUBE ROOTS

Square Roots and Cube Roots 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.PAR.6.1 Write and evaluate numerical expressions involving rational bases and whole-number exponents.

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.

SQUARE ROOTS AND CUBE ROOTS

Home

Procedure and Facilitation Points 1. 2.

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

Card 1 matches with Card B.

Card 2 matches with Card A.

Card 3 matches with Card C.

FACILITATION TIP For an extra challenge, have students ﬁnd a third equivalent expression for each match. Cards 1 and B are equal to 36. Cards 2 and A are equal to 12. Cards 3 and C are equal to 4 x 4 x 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. FACILITATION TIP

Identifying Misconceptions •

Students may confuse an exponent with a factor and multiply the integer by the exponent

Reiterate that an exponent tells how many times the base number is multiplied by itself. Point out the base number and the exponent on cards 3, A, and B.

SQUARE ROOTS AND CUBE ROOTS

Square Roots and Cube Roots Hook – Thinking Outside the Box ACTIVITY PREPARATION Students will determine the volume of a cube based on the area of the top face.

Materials

Preparation

Printed •

• •

1 Thinking Outside the Box (per class)

Reusable •

•

1 Phenomena Video (per class)

Plan to show the video. Prepare to project the Thinking Outside the Box 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 FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) Do you have a box that holds items that are special to you? What does the box look like?; 2) What would be important to consider if you wanted to make a box in which to hold special items?; 3) If you could make a special box, what materials would you use and how would you decorate it? FACILITATION TIP Make sure students take note that the situations described the box as a cube. Ask the students what they remember about the attributes of a cube. They should mention that the length, width, and height are all equal.

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: Prisha is taking an art class after school. She has begun a project to create colorful decorated jewelry boxes to sell for the upcoming holiday of Diwali. She is starting by crafting a test box. It is a hollow box made with extremely thin sheets of wood in the shape of a cube with an almost invisible seam hiding a removable top. She will paint the outside of the box in bright colors. She will cover the top face with decorative silver contact paper and then beautiful glass jewels. She also needs to determine the volume of the jewelry box so she can tell customers how much room they will have for jewelry. 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 Prisha is working with both area and volume. I wonder what the dimensions of the box she is working on are. What sizes of boxes will Prisha be selling? I can use math to determine the area and the volume of a cube. Project Thinking Outside the Box. Notes

Engage

Explore

Explain

Elaborate

Evaluate

Acceleration

Explain to students that Prisha knows the area of the top of her ﬁrst wooden jewelry box cube, so she can cut out the correct size of silver contact paper to use on it. She will use that information about the area of the top to determine the volume of the box. Discuss the following questions: a.

DOK-1 If the box is a cube, what two-dimensional shape is the top of the box? A square

DOK-1 What is the formula to determine the area of a square? A = s2

DOK-1 What is the area of the top face of the box? 64 square inches

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Intervention

Show the Phenomena Video again, and restate the problem. Refer to the Thinking Outside the Box Slide and discuss the following questions: a.

DOK-1 How can knowing the area of the top face help Prisha to determine the lengths of the sides of the top square? Because the top face of the cube is a square, it means that both sides are the same length. The formula for the area of a square is A = s2. Therefore, the length of a side is the square root of the area.

DOK-1 What is the √64? 8

DOK-1 If the length of each side of the top face of the cube is 8 inches, what does that mean regarding the lengths of all of the edges of the cube? Explain. Each edge of the cube is 8 inches. This is because in a cube, all edges are the same length. The side of the square on top is also one of the edges of the cube.

DOK-1 What is the formula for the volume of a cube ? V = s3, which is the same as the volume formula for a rectangular prism (because a cube is a rectangular prism), V = lwh. Because the length, width, and height are all the same in a cube, it is the same as side × side × side or s3.

e. DOK-1 What is the volume of the jewelry box? The volume of the jewelry box is 512 cubic inches (512 in3) because 8 × 8 × 8 = 512. f.

DOK-1 How could you check to see whether the volume of 512 is correct? Find the cube root of 512. The 3√512 = 8, so the answer is correct.

FACILITATION TIP Ask the students if this gives enough information to be able to solve the problem. It does, but they do not have to solve it yet. You could allow students to try to solve the problem, save their answer, then check their work in the Post-Explore section below.

SQUARE ROOTS AND CUBE ROOTS

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FACILITATION TIP Let students make the connection between cubic units and the fact that they just cubed the side length (or multiplied it by itself 3 times). Ask the students if they think this could be why volume is measured in cubic units. Extend this thinking to square units.

SQUARE ROOTS AND CUBE ROOTS

Square Roots and Cube Roots Explore 1 – Square Roots and Perfect Squares ACTIVITY PREPARATION Students will ﬁnd patterns to recognize that solutions to perfect squares can never be negative. Students will apply their knowledge of squaring numbers as the inverse to square roots in order to calculate solutions to problems containing either a perfect square or a square root.

Standards for Mathematical Practice • •

MP.2 Reason abstractly and quantitatively. MP.5 Use appropriate tools strategically.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Craft Cards (per group) 1 Exit Ticket (per 2 students)

Reusable • •

•

16 Linking cubes (per student) 1 Resealable bag (per group)

• •

Plan to separate the class into groups of 2 or 3 to complete this activity. Print a Student Journal for each student. Print one set of Craft 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. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student has one. Gather 16 linking cubes per student. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Linking Cubes)

PROCEDURE AND FACILITATION Part I FACILITATION TIP

After reading the scenario, ask the class 1) What is a “perfect square”?; 2) What is the formula for ﬁnding the area of a perfect square?; 3) If the area of a perfect square is known but the side lengths are unknown, how can you use the area to ﬁnd the lengths of the sides?

2. 3.

FACILITATION TIP

Read the following scenario to the class: Your family decided to enter a craft fair this year! You are determined to help your Nana create beautiful frames to sell at the craft fair. Nana says the back of each frame has a cardboard base to hold the picture in place. Nana is making many frames in different sizes, but all of the frames will be perfect squares. Help Nana determine the area of each size frame so the cardboard backs can be created. Give 16 linking cubes to each student. Ask students the following questions: a.

DOK-1 The ﬁrst square frame Nana is making has a side length of 1 unit. How can we model this frame with the linking cubes? We can take out 1 linking cube to show a square with a side length of 1 unit.

FACILITATION TIP

Reiterate that s = s x s. To ﬁnd the area of a square, we need to multiply the side length times itself.

DOK-1 What is the measure of each side of the square frame? Each side of this square measures 1 unit in length.

DOK-1 What formula can you use to determine the area of a square frame? To ﬁnd the area of a square, we can use the formula A = s2.

DOK-1 What is the area of the square frame with a side length of 1 unit? Using the formula A = s2, I solve 12, which is 1. The area of a square frame with a side length of 1 unit is 1 square unit.

Alternatively, you could use graph paper and have students draw the squares. 2

4. 24

7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

Explain to students that they will collaborate with their groups and use the linking cubes to determine the area of square frames with side lengths of 2 units and 3 units. Then, have students determine the side length of a square frame with an area of 16 square units. Instruct students to use the patterns they ﬁnd to complete the table in Part I of their Student Journals. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 What do you know about the side lengths of a square? All of the side lengths of a square are the same measure.

DOK-2 What can you do to determine the side length of a square when given the area of the square? When given the area, I can determine what number times itself gives me the area measure. The number times itself that gives me the area measurement is the length of the side.

DOK-2 How can I determine the side length if I don’t know what number squared gives me the area? I can use a guess-and-check method to square numbers until I reach the value of the area.

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

Math Chat DOK-2 Describe the relationship between the side length of a square and its area. The area of each square frame is the value of the side length squared. • DOK-2 When given the area, how can you determine the side length of a square? I can determine what number squared equals the area. •

Intervention

Acceleration

FACILITATION TIP Extension activity: Have students additionally identify an equivalent equation in each row in the table. The format of the last column would then look like A = 32 = 3 x 3.

SQUARE ROOTS AND CUBE ROOTS

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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.

Explain the following to the class: Mathematicians call this ﬁnding the square root. The square root of a number is the inverse operation of squaring the number. (Show students what the square root symbol looks like.) •

DOK-2 What if the square root wasn’t about area; would there possibly be more solutions? Yes, the area could be negative too because a negative multiplied by another negative has a positive product.

Part II 1.

2. 3. 4.

Read the following scenario to the class: Nana has also decided she would like to make square knitted pot holders to sell at the craft fair. She has determined 4 different styles of knitted pot holders and needs your help determining their measurements. Help Nana find the missing measurements for each of the knitted pot holders. Give one set of Craft Cards to each group. Explain to students that they will collaborate with their groups to use the Craft Cards to find the missing measurements of each square knitted pot holder. Have students find the option 2 card and read the information with their groups. Ask students the following questions: a.

DOK-1 What do you notice about the area for this knitted pot holder? The area for this card is a fraction.

DOK-2 How can I determine the side length of the square pot holder when given the area? I can ﬁnd the square root of the area to ﬁnd the side length since side squared gives the area.

DOK-2 What strategy could I use to determine the square root of a fraction? First, I can ﬁnd the square root of the numerator. Then, I can ﬁnd the square root of the denominator. Once I have the square root for each part of the fraction, I can write the new fraction.

FACILITATION TIP After reading the scenario, ask the class 1) How can you ﬁnd the missing measurements of a square?

FACILITATION TIP You could expand on this if students are readily understanding and show them the ____ ____

√ (m)

____ ___ . rule √ (__ n) =

√ (n)

SQUARE ROOTS AND CUBE ROOTS

Square Roots and Cube Roots Explore 1 – Square Roots and Perfect Squares d.

e. DOK-1 What is the square root of 121? The square root of 121 is 11 because 11 times 11 equals 121. f. FACILITATION TIP Project idea: Allow students to create their own set of craft cards to trade with another group. Students should create cards similar to the craft cards provided using blank index cards and calculate the answers on a separate paper. Once groups have traded cards and completed the activity, allow the groups to check each others’ work.

FACILITATION TIP Extension Activity: Provide students with index cards to make ﬂash cards of the perfect squares and the square roots.

7. 8.

FACILITATION TIP Additional math chat questions to further develop this line of thought: Do all square roots have a positive and negative solution? Are there some situations in which you don’t need to worry about the negative solution?

•

• • •

FACILITATION TIP Depending on the time of the school year that this Exit Ticket is given, take time to clarify expectations about writing names, showing work, and labeling any answers.

DOK-1 What is the square root of ____ ? The square root of ____ will be ___ , 11 121 121 2 2 because 9 = 81 and 11 = 121.

Explain that Nana has provided the area as a square root on each of the Craft Cards. Students will work with their groups to ﬁnd the missing measurements on the remaining cards. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 What do you know about the side lengths of a square? All of the side lengths of a square are the same measure.

DOK-2 How can I determine the perfect square? I can start with the greatest perfect square I know and then use a guess-and-check method to square numbers until I reach the value of the area.

Allow students time to complete Part II and 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 square root of 81? The square root of 81 is 9 because 9 times 9 is 81.

DOK-2 What is the relationship between perfect squares and square roots? Give an example to explain. Squaring and taking square roots are inverse operations. For example, if I am given the square root of 100, I can think of what number squared (or number times itself) will equal 100. The square root of 100 is 10. DOK-2 Two students are given the equation x2 = 81. Johans says that x = 9 because 9(9) = 81. Shiloh says that x = −9 because (−9)(−9) = 81. Who is correct? Explain your answer. Both students are correct because 9 times 9 is 81 and −9 times −9 is 81. DOK-2 Is (√25)2 = 52? Explain. Yes, (√25)2 = 52 because when you square the square root, you are using inverse operations. So (√25)2 is 25 and 52 is 25. DOK-2 Do you think all solutions to whole-number square roots are integers? No, not all whole-number square roots have integer solutions. DOK-2 How many solutions do square roots have? How do you determine whether the answer is valid? All square roots have two answers, a positive and negative solution. In cases of area or length, there is only one positive answer. In all other cases, such as solving for a variable in an equation, there are two solutions. By plugging in the solution to the equation, I can double-check the validity of the solution.

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

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SQUARE ROOTS AND CUBE ROOTS

Square Roots and Cube Roots Explore 2 – Cube Roots and Perfect Cubes ACTIVITY PREPARATION Students will ﬁnd patterns to recognize that solutions to positive perfect cubes can never be negative. Students will apply their knowledge of cubing being the inverse to taking cube roots to calculate solutions to problems containing either a perfect cube or a cube root.

Standards for Mathematical Practice • •

MP.2 Reason abstractly and quantitatively. MP.5 Use appropriate tools strategically.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Gift Box Cards (per group) 1 Exit Ticket (per 2 students)

Reusable • •

•

27 Linking cubes (per student) 1 Resealable bag (per pair)

• •

Plan to have students work in groups of 2 or 3 to complete this activity. Print a Student Journal for each student. Print one set of Gift Box 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. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student will have one. Gather 27 linking cubes per student. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. (Linking Cubes)

PROCEDURE AND FACILITATION Part I FACILITATION TIP After reading the scenario, ask the class 1) What is a “perfect cube”?; 2) What does volume measure?; 3) What formula do you use to ﬁnd volume of a cube?

2. 3.

FACILITATION TIP You could list all the ways the students know how to ﬁnd the volume of a cube on the board. This would include V = l x w x h, where l = w = h; and V = s x s x s. 4. 28

Read the following scenario to the class: Your aunt Jeanne has created beautiful memory boxes to sell at the craft fair. She is making many memory boxes in different sizes, but all of the frames will be perfect cubes. Help Jeanne determine the volume of each memory box. Give 27 linking cubes to each student. Ask students the following questions: a.

DOK-1 The ﬁrst cube memory box aunt Jeanne is making has a side length of 1 unit. How can we model this cube with the linking cubes? We can take out 1 linking cube to show a cube with a side length of 1 unit.

DOK-1 What is the measure of each side of the cube-shaped box? Each side of this box measures 1 unit in length.

DOK-1 What formula can you use to determine the volume of a cube? To ﬁnd the volume of a cube, we can use the formula V = s3.

DOK-1 What is the volume of the cube-shaped memory box with a side length of 1 unit? Using the formula V = s3, I know 13 is 1. So the volume of the cube-shaped memory box with a side length of 1 unit is 1 unit cubed.

Explore

Explain

Elaborate

Evaluate

Explain to students that they will collaborate with their groups to use the linking cubes to determine the volume of cube-shaped memory boxes with side lengths of 2 units and 3 units. Students will use the patterns they ﬁnd to complete the table in Part I on their Student Journals. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

7. 8.

Engage

DOK-2 What do you know about the side lengths of a cube? All the side lengths of a cube are the same measure.

DOK-2 What can you do to determine the side length of a cube when given the volume of the cube? When given the volume, I can determine what number cubed will give me the same values as the volume. The number that, when cubed, gives me the volume will be the length of each side of the memory box.

DOK-2 How can I determine the side length if I don’t know what number cubed gives me the volume? I can use a guess-and-check method to cube numbers until I reach the value of the volume.

Intervention

Acceleration

FACILITATION TIP Student could write the equation both as a perfect cube and as a product of the side length multiplied by itself 3 times in the equation column on the Student Journal. FACILITATION TIP Go more in depth here and review past concepts by asking questions such as, “What is the length? What is the width? What is the height? What is the shape of each face of a cube? How do we ﬁnd the area of a face? Now, how can we ﬁnd the volume using that area?”

SQUARE ROOTS AND CUBE ROOTS

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

Math Chat • •

DOK-2 Describe the relationship between the side length of a cube and its volume. The side length cubed is equal to the volume of the cube. DOK-2 When given the volume, how can you determine the side length of a cube? I can ﬁnd what number cubed will give me the value of the volume.

Explain the following to the class: Mathematicians call this ﬁnding the cube root. The cube root of a number is the inverse operation of cubing the number. (Show students what the cube root symbol looks like.) •

DOK-2 Explain the difference between square root and cube root solutions. Square roots have two solutions because the negative and positive factors can be possible solutions. In cube roots, however, the product will have the same sign as the root.

FACILITATION TIP If calculators are permitted in the classroom, this could be a good opportunity to demonstrate how to use the cube root button on the calculator.

Part II 1.

2. 3. 4.

Read the following scenario to the class: Jeanne has also decided that she would like to make decorative cube gift boxes to sell at the craft fair. She has determined 4 different styles and needs your help determining their side lengths. Help Jeanne find the missing side lengths for each of the gift boxes. Give one set of Gift Box Cards to each group. Explain to students that they will collaborate with their groups to use the Gift Box Cards to find the missing side lengths of each cubed decorative gift box. Have students find the option 1 card and read the information with their groups. Ask students the following questions: a.

DOK-1 What do you notice about the cube root for this gift box? The cube root for this card is a fraction.

DOK-2 What strategy could I use to determine the cube root of a fraction? First, I can ﬁnd the cube root of the numerator. Then, I can ﬁnd the cube root of the denominator. Once I have the cube root for each part of the fraction, I can write the fractional side length of the gift box.

DOK-1 What is the cube root of 64? The cube root of 64 is 4 because 4 times 4 times 4 equals 64.

DOK-1 What is the cube root of 216? The cube root of 216 is 6 because 63 = 216.

FACILITATION TIP After reading the scenario, ask the class 1) How can you ﬁnd missing side lengths of perfect cubes?

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.

SQUARE ROOTS AND CUBE ROOTS

Square Roots and Cube Roots Explore 2 – Cube Roots and Perfect Cubes 64

e. DOK-1 What is the cube root of ____ ? The cube root of ____ will be __6 or __3 , 216 216 3 3 because 4 = 64 and 6 = 216. 5. FACILITATION TIP Extra practice: Allow students to make their own ﬂash cards using index cards. They should write the cubed or cube root expression on one side and the solution on the other.

7. 8.

Explain that Jeanne has provided the volume of each cube-shaped gift box as a cube root on each Gift Box Card. Students will work with their groups to ﬁnd the missing side lengths on the remaining cards. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 What do you know about the side lengths of a cube? All the side lengths of a cube are the same measure.

DOK-2 What can you do to determine the side length of a cube when given the volume of the cube? When given the volume, I can determine what cubed number gives me the value of the volume. The number cubed that gives me the volume measurement is the length of one side of the cube.

DOK-2 How can I determine the perfect cube? I can start with the greatest perfect cube I know and then use a guess-and-check method to cube other numbers until I reach the value of the volume.

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

Math Chat FACILITATION TIP Review what is meant by the term inverse operations. Remind students that multiplication and division are inverse operations, and addition and subtraction are inverse operations. Inverse here just means opposite. FACILITATION TIP Ask students to compare and contrast (–4)2 and (–4)3.

DOK-2 What is the relationship between perfect cubes and cube roots? Give an example to explain. Cubing and taking cube roots are inverse operations of each other. For example, if I am given the cube root of 1,000, I can think of what number cubed (or number times itself times itself) will equal 1,000. The cube root of 1,000 is 10. • DOK-2 Two students are given the equation x3 = 64. Jacob says that x = 4 because (4)(4)(4) = 64. Sam says that x = −4 because (−4)(−4)(−4) = 64. Who is correct? Explain your answer. Jacob is correct. Sam is not correct because a negative times a negative gives a positive solution, but then you have to multiply that by another negative. A positive times a negative is negative; therefore, x cannot equal −4. • DOK-2 Is (3√27)3 = 33? Explain. Yes, (3√27)3 = 33 because when you cube the cube root, you are using inverse operations. So (3√27)3 is 27, and 33 is 27. • DOK-2 Do you think all solutions to whole-number cube roots are integers? No, not all whole-number cube roots have integer solutions. •

Post-Explore FACILITATION TIP

Depending on the time of the school year that this Exit Ticket is given, take time to clarify expectations about writing names, showing work, and labeling any answers.

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

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SQUARE ROOTS AND CUBE ROOTS

Square Roots and Cube Roots 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

Square Roots and Perfect Squares 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

Cube Roots and Perfect Cubes 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

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

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

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

Intervention and Assessment SMALL-GROUP PLANNER 1

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

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

SQUARE ROOTS AND CUBE ROOTS

Square Roots and Cube Roots

3 34

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 ﬁnd patterns within a list of square numbers.

What prompts will be used?

What does mastery look like?

SQUARE ROOTS AND CUBE ROOTS

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I can ﬁnd patterns within a list of cube numbers.

I can recognize that squaring a number is the inverse operation of taking the square root of a number.

I can recognize that cubing a number is the inverse operation of taking the cube root of a number.

I can evaluate square roots of perfect squares.

I can evaluate cube roots of perfect cubes.

SCOPE 1

Irrational Numbers Scope Introduction SCOPE SUMMARY Students are exposed to multiple opportunities to identify and compare rational and irrational numbers. They show the difference between the deﬁnitions of a rational number and an irrational number through number approximations and decimal makeup. In this scope, students identify the differences between rational numbers and irrational numbers in order to compare number values. Students estimate the numerical values of irrational numbers in order to locate them on a number line in comparison to given rational numbers. Student Expectations

8.NR.1.1 Distinguish between rational and irrational numbers using decimal expansion. Convert a decimal expansion which repeats eventually into a rational number. 8.NR.1.2 Approximate irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

Students encounter rational numbers in sixth grade when they understand rational numbers as points on a number line. They work with fractions and decimals to identify their appropriate locations on any given number line. When using division in seventh grade, students are exposed to inﬁnitely 1 repeating decimals when converting fractions like __3 to decimals. Students work to classify the decimals as terminating or repeating using the repetitive pattern that may or may not occur during the division process.

As students enter high school, they will be applying their knowledge of irrational numbers in order to study functions that take on irrational values, like exponential, logarithmic, and power functions. Students will study how rational and irrational numbers work together in the real number system. Students will explore how operations between irrational and rational values can produce irrational answers even when one value is rational. Students will also apply their knowledge of irrational numbers to complete computation involving complex numbers, such as computations with the quadratic equation and imaginary numbers.

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

convert a rational number to a decimal using long division.

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know the decimal form of a rational number terminates in 0s or repeats.

•

read responses and decide if they agree or disagree.

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 tasked with solving a real-world scenario. Through completing the Hook, students will demonstrate their ability to: •

convert a square root to an approximate decimal.

•

identify the approximate location of a decimal on a number line.

•

compare and order numbers to determine the greatest value.

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. 36

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Irrational Numbers vs. Rational Numbers In this exploration, groups of students are tasked with creating a graphic organizer for a job interview. Students will: •

create a graphic organizer to classify real numbers.

•

classify numbers as rational or irrational.

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justify their reasoning.

Explore 2

Explore 1

EXPLORE ACTIVITIES

Locate and Compare Irrational Numbers on a Number Line In this exploration, groups of students will help a building company repair a measurement tape. Students will: •

ﬁnd rational approximations of irrational numbers.

•

plot irrational numbers on a number line.

•

compare rational and irrational numbers using decimal approximations for irrational numbers.

In this exploration, groups of students will be tasked with helping a building company purchase materials. Students will: •

write fractions as repeating decimals and repeating decimals as fractions.

•

show that the decimal expansion of a rational number terminates or repeats.

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

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

Decimal Expansion

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Estimate and Compare Irrational Number Expressions In the ﬁnal exploration, groups of students will help a building company gather their ﬁnal construction needs to get started on their projects. Students will: •

estimate and compare irrational number expressions.

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.

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Irrational 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 read different student responses to a posed question on the prior standard, decide if they agree or disagree with the student, and explain their reasoning. 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.

Materials

Preparation

Printed •

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Print one Agree or Disagree for each student.

1 Agree or Disagree (per student)

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

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

Disagree with Roberto

Disagree with Sydney

Agree with Zia

FACILITATION TIP 1 __

Give the example of 9 to show that it will keep repeating and never terminate.

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 Demonstrate the examples and nonexamples in numbers 2 and 3 from the handout to show students how these answers were found.

Identifying Misconceptions • • •

Students may not recognize that adding terminal zeros does not change the value of a number in the decimal position. Emphasize to students that dividends ending in decimal values cannot have quotients with remainders. Students may think that a repeating decimal will only repeat the same value (ex. 0.33333).

FACILITATION TIP Show an example of a repeating decimal that 5 repeats a string of numbers. For example: ___ . 11

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Irrational Numbers Hook – Win, Place, or Show ACTIVITY PREPARATION Students will convert a square root to a decimal approximation. They will then identify the approximate location of the decimal on a number line with other numbers, allowing its value to be compared and ordered to determine the number with the greatest value.

Materials

Preparation

Printed •

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1 Win, Place, or Show (per class)

Reusable •

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1 Phenomena Video (per class)

Plan to show the video. Prepare to project Win, Place, or Show 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 game do you like to play with your friends?; 2) How do you determine who wins?

FACILITATION TIP

Ask students to think of other ways or models to compare and order numbers.

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: Jaxon was playing a new video game with his friends Armstrong and Sammy. They decided to make it a competition. Each boy would play one game, determine his score to the hundredths place, and plot it on a number line. After the last boy finished his turn, they would compare and order the scores and award prizes like a horse race — “win” for first place, “place” for second place, and “show” for third place. Jaxon went last and his score was a square root that he needed to convert to a decimal approximation to see whether he would win, place, or show. 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 boys are using a number line to compare and order numbers. I wonder who will win, place, and show and what the scores will be. How will Jaxon find the decimal approximation of a square root? I can use math to determine Jaxon’s score and compare and order the scores. Project Win, Place, or Show. Notes

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Explain to students that Jaxon must determine a decimal approximation for √15 to the hundredths place and plot it on the number line the boys created. Discuss the following questions: a.

DOK-1 What is a square root? A factor of a number that gives the number when multiplied by itself.

DOK-1 What is √16? 4

DOK-1 What is √9? 3

DOK-1 What do you know about the value of √15 ? Its value will be greater than 3 but less than 4.

Complete the Explore activities.

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FACILITATION TIP Ask students to think about whether it would be closer to 3 or 4. Why do they think that?

Part II: Post-Explore 1. 2.

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

DOK-1 Is Jaxon’s score greater than or less than Sammy’s score? Explain. Jaxon’s score is greater than Sammy’s score. Sammy’s score is 3, which is the √9. 15 is greater than 9, so √15 is greater than 3.

DOK-2 What is the decimal approximation of Jaxon’s score? How do you ﬁgure it out? The decimal approximation of Jaxon’s score is an estimation. Because 15 is only 1 away from 16 but 6 away from 9, √15 is much closer to 4 than it is to 3. I estimate it will be 3.85.

DOK-1 Is Jaxon’s score greater than Armstrong’s score? Yes, it is greater because 3.85 is greater than 3.25.

DOK-1 What are the ﬁnal results of the competition? Jaxon wins. Armstrong places. Sammy shows.

FACILITATION TIP Extension activity: Allow students to come up with 3 new square root expressions, trade with a partner to order them, then check the answers.

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Irrational Numbers Explore 1 – Irrational Numbers vs. Rational Numbers ACTIVITY PREPARATION Students will create a graphic organizer to classify real numbers. Students will classify numbers as rational or irrational and justify their reasoning.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.5 Use appropriate tools strategically. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • • • •

1 Student Journal (per student) 1 Set of Real Numbers Student Classiﬁcation Cards (per group) 1 Set of Real Numbers Classiﬁcation Explanation Cards (per class) 1 Real Numbers Venn Diagram (per class) 1 Exit Ticket (per student)

Reusable • •

1 Projector or document camera (per class) 1 Resealable bag (per group)

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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 a set of Real Numbers Student Classiﬁcation Cards for each group. Cut them 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 one Real Numbers Venn Diagram for the class. If desired, print it on card stock, and laminate it for future use. Print one set of Real Numbers Classiﬁcation Explanation Cards for the class. These cards are a key for the teacher and do not need to be cut out. If desired, print them on card stock, and laminate them for future use. Have a projector or document camera ready to show the Real Numbers Venn Diagram to the class.

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) What is an internship?; 2) What steps are typically involved when someone is interested in getting a job?; 3) If math skills were required for a job you were interested in, how might you prove you have good math skills?

FACILITATION TIP Allow students to write examples of these numbers on the board or a chart.

FACILITATION TIP Project idea: Have students create their own Venn diagram. It should be colored neatly and include at least 3 examples of each type of number. 42

3. 4.

Read the following scenario to the class: You are looking for an internship to improve your job skills. You’ve applied to a local builder, Team Builders, to plan and build youth sporting areas throughout the city. The hiring manager has called you in for an interview. She feels you have great potential, but you will need to prove you can learn new math skills and apply your new knowledge before she can offer you the job. Help Team Builders sort their numbers by their classiﬁcations. Discuss the following questions with the class: a.

DOK-1 What types of numbers have we discovered so far? Accept all reasonable answers. Fractions, decimals, whole numbers, negative numbers

DOK-1 What is an example of a whole number? 2; 100; 75; 1,000,000,000

DOK-1 What is an example of an integer? −15, 88

DOK-1 What is an example of a rational number? 8, __2 , 75.50

Display the Real Numbers Venn Diagram. Discuss the following with the class: This is called a “Venn diagram.” Venn diagrams help to show information based on relationships. Ask the class the following questions: a.

5. 6. 7.

10. 11.

Explore

Explain

Elaborate

Evaluate

DOK-1 What set of numbers is within the integers set? Integers include the set of whole and natural numbers. Whole numbers and natural numbers are called “subsets” of integers.

DOK-1 What are the names of the type of numbers that include integers as subsets? Integers are a subset of rational numbers and real numbers.

Give one set of Real Number Student Classification Cards to each group. Explain to students that they will collaborate with their groups to match the classification name, definition, and example for each of the sets of real numbers. Have students work cooperatively to use their prior knowledge to determine which classification name card matches the definition and example given for each section of the Real Numbers Venn Diagram. Students may use the Venn diagram displayed for guidance as they discuss with their groups. Monitor and assess students as they work. Use the Real Numbers Classification Explanation Cards to provide hints and guidance as appropriate for each group. Ask students the following guiding questions: a.

Engage

DOK-2 What are some key words on this classiﬁcation card that were helpful? How did the key word help? Answers will vary depending on the deﬁnition card students are working with. The words numbers greater than or equal to zero with no fractional or decimal parts tell me that these numbers are a subset of whole numbers.

DOK-2 What are some key numbers in the examples on this classiﬁcation card that were helpful? How did the key numbers help? Answers will vary. The fractions and decimals as examples let me know that these numbers were not whole numbers. The negative signs in front of the numbers let me know that these were integers.

DOK-3 How can the Venn diagram help you? The diagram shows sets and subsets. I can use my prior knowledge and the subsets to determine the classiﬁcation for each card.

Direct the class to share their answers and justify their choices. Use the Real Numbers Classification Explanation Cards to ensure that all students have the correct understanding of the classification of numbers. Give a Student Journal to each student. Allow time for students to complete the Venn diagram on the first page of their Student Journals using the Real Numbers Student Classification Cards as a guide.

Part II 1.

Read the following scenario to the class: The hiring manager for Team Builders was impressed with your ability to sort each classiﬁcation of real numbers! But can you apply this knowledge? Help determine if each number is rational or irrational. Then, show proof of your decision. Explain to students that they will collaborate with their groups to determine if each of the numbers provided is a rational number or an irrational number. Then, they will need to use the Venn diagram to help prove their decisions are correct. Students will collaborate to apply their knowledge of real-number classiﬁcation to a variety of numbers provided. Students should use the Venn diagram completed in Part I to help prove their thinking is correct. Allow students to use a calculator to assist in determining if a number is a repeating decimal or not. Monitor and assess students as they work. Ask guided questions as needed:

Intervention

Acceleration

FACILITATION TIP Game idea: Students may play a matching game or a Go Fish type game.

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 Do a quick refresher on what types of numbers are considered rational. FACILITATION TIP After reading the scenario, ask the class 1) What is a rational number?; 2) What is an irrational number?; 3) What are examples of rational and irrational numbers? FACILITATION TIP For early ﬁnishers, provide a list of additional rational and irrational numbers for students to decide on and give reasoning. They may use a separate sheet of paper. FACILITATION TIP

DOK-1 Why do you think 0.5 is rational? 0.5 is rational because it can be written as a fraction.

If students need a refresher, review

DOK-2 Can a decimal that does not repeat be written as a fraction? No, nonrepeating decimals like pi cannot be written as fractions.

examples of common repeating decimals

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what repeating decimals are. Give a few 1 2

. written as fractions: __3 , __9 , ___ 11 43

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Irrational Numbers Explore 1 – Irrational Numbers vs. Rational Numbers 5.

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

Math Chat •

DOK-2 Based on the Real Numbers Venn Diagram, is zero a rational or irrational number? Zero is a whole number. Whole numbers are a subset of rational numbers. Zero is a rational number.

DOK-2 Is a terminating decimal rational or irrational? Explain using an example. A terminating decimal is a rational number because it can be written as a fraction. 27 Examples may vary. 5.27 is a terminating decimal. It can be written as 5 ____ . 100 • DOK-2 How would you classify the sum of a rational and irrational number? The sum of a rational and irrational number would be irrational because you cannot write that sum as a fraction.

• FACILITATION TIP Ask students to use a calculator to add 2 + pi. What number do they get?

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

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Irrational Numbers Explore 2 – Decimal Expansion ACTIVITY PREPARATION Students will write fractions as repeating decimals and repeating decimals as fractions. Students will show that the decimal expansion of a rational number terminates or repeats.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.5 Use appropriate tools strategically. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

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1 Student Journal (per student) 1 Set of Land Survey 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 Land Survey Cards per 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 Part I: Converting Fractions to Decimals with Decimal Expansion FACILITATION TIP

Before reading the scenario, ask the class 1) If you were involved in ﬁnding areas to have youth sports, what types of things would you look for? After reading the scenario, ask the class 2) Why might the land measurements need to be recorded in the same form?

2. 3.

FACILITATION TIP Direct the students to talk about examples of fractions that are best suited for each method. For example, for a fraction with a denominator of 3 it might be best to use method 1, but for a fraction with a denominator of 5 it might be best to use method 2.

4. 5.

FACILITATION TIP For fun extra practice, call two students to the board at a time. Give the students a fraction, and let them race to ﬁnd the decimal form then state whether it is a terminating or repeating decimal. 6. 46

Read the following scenario to the class: Congratulations! You got a new job with Team Builders! Your company is in the process of planning youth sporting areas throughout the city. The company sent out employees to get land measurements from several locations. Unfortunately, some of the measurements were recorded in decimal form, and some were recorded in fraction form. Help Team Builders determine the decimal forms for each measurement. Give a Student Journal to each student. Explain the following to the class: Your boss has provided you with two examples of how to determine the decimal form of a fraction. Discuss with your groups what you notice about each method. Allow students time to discuss with their groups. Then, have the following class discussion: a.

DOK-1 How can I make a fraction into a decimal number? I can use long division to make a fraction into a decimal number, or I can ﬁnd an equivalent fraction with a denominator of 10 or 100.

DOK-1 Which number is a terminating decimal? How do you know? 0.4 is a terminating decimal because the digits after the decimal point stop.

DOK-1 Which number is a repeating decimal? How do you know? 0.333... is a repeating decimal because if I continue dividing, I will always have one left over. We would bring down another zero to get 10, and 3 goes into 10 three times again.

DOK-1 What is another way to show that a digit is repeating? We can use bar notation to write repeating decimals. We would put a bar over the digit 3 to show that it continues to repeat.

Engage

Explore

Explain

Elaborate

Evaluate

Acceleration

Explain to students that they will collaborate with their groups to use the Land Survey Cards to change the fractional measurements into their decimal expansions. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 How can you determine by looking at the fraction if you will need to perform long division? If the fraction cannot be converted into a fraction with a denominator of 2 or 5, then we will need to perform long division to ﬁnd the decimal expansion.

DOK-2 How did you know when you could stop dividing? Answers will vary. I knew I could stop dividing when the remainder kept repeating.

DOK-3 How can you determine the decimal expansion for 3 __8 ? Answers

FACILITATION TIP

may vary. I performed long division on the fraction __8 , which will be

0.625. Then, I added the whole number 3 to my decimal and found that 5

9. 10.

Intervention

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3 __8 in decimal expansion is 3.625.

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

Review that a mixed number is composed 5

of a whole number plus a fraction. 3 __8 is 5

the same as 3 + __8 . This may help students

see the fraction part a little easier for the decimal expansion.

Math Chat DOK-2 What two types of decimals did you get when you converted the fractions to decimals? When we converted fractions to decimals, we got either terminating or repeating decimals. • DOK-2 What is an example of a fraction that you know the decimal for without 7 having to do any work? I know that ___ is written as 0.7 as a decimal. 10 • DOK-3 What is the decimal for √16? How do you know? The decimal is 4.0 because 42 is 16. •

Part II: Converting Decimals to Fractions 1.

2. 3.

Read the following scenario to the class: Some of the materials Team Builders will be purchasing are ordered in fractional measurements. Your boss has provided you with a method that Team Builders uses to convert repeating decimals into fractions. Use this method to help Team Builders determine the fractions for each measurement. Direct students to analyze the method the boss provided on their Student Journals. Allow students time to discuss the method with their groups. Have the following class discussion: a.

DOK-2 Why do you think your boss multiplied by 10? Answers will vary. The boss wanted to get a number greater than 1, so they multiplied by 10.

DOK-2 What other numbers can you multiply by? You could multiply by any power of 10, such as 100 or 1,000.

Explain to students that they will use the Land Survey Cards and the method provided by their boss to determine the fractional representation of each of the given repeating decimals. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 Does it matter if you multiply by 10, 100, or 1,000? No, you can multiply by any power of 10.

DOK-2 Does this method work if you have a decimal number greater than one where only the decimal repeats? Yes, I can use this same method to ﬁnd the fractional representation for the decimal and then add the whole number to it.

FACILITATION TIP Before reading the scenario, ask the class 1) How do you convert decimals into fractions? After reading the scenario, ask the class 2) What are “repeating decimals”?; 3) Do you think the process for converting decimals to fractions is different when the decimals are repeating? Why or why not?

FACILITATION TIP Allow students to try working one of the problems using 100 or 1,000. Discuss the steps used and how the solution was the same or different than working the problem using 10.

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Irrational Numbers Explore 2 – Decimal Expansion 6. 7. FACILITATION TIP

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

Math Chat DOK-3 Why do you use powers of ten when converting a repeating decimal to a fraction? You can’t write all the digits of a repeating decimal because they repeat forever. Using powers of ten allows you to convert a repeating decimal. • DOK-3 Can you use this method to determine the fractional representation for a number that has only part of the decimal repeating, like 0.123? 0.123 Yes, we can use this same method for any type of repeating decimal. • DOK-3 Why can’t you write a nonterminating, nonrepeating decimal as a fraction? Because the numbers don’t repeat, you can’t set it up using powers of ten. • DOK-3 What is an example of a repeating decimal that you would want to multiply by 100 instead of 10? I could multiply a repeating decimal that has 2 different digits repeating by 100 to help when I subtract out the original equation. •

Provide early finishers with a list of repeating decimals, and ask them to find the fractional representation of each. Challenge the students to find patterns 2 - for example .22222... is __9 and .33333... 3 is __9 . Could they conclude that when a single digit is repeated, the fractional representation is that digit in the numerator with a 9 in the denominator? Have them try other examples to see if they fit the patterns found.

Post-Explore

FACILITATION TIP

Have students think of examples of these decimals and think of all the ways they know write them. Examples include ____how to ____ pi, √ (2) , and √ (3) .

2. 3.

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

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Irrational Numbers Explore 3 – Locate and Compare Irrational Numbers on a Number Line ACTIVITY PREPARATION Students will ﬁnd rational approximations of irrational numbers and plot them on a number line. Students will compare rational and irrational numbers using decimal approximations for irrational numbers.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.5 Use appropriate tools strategically. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Team Builders Measuring Tape (per group) 1 Set of Special Measurements Cards (per 5 groups) 1 Exit Ticket (per student)

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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 Special Measurements Cards per 5 groups. Cut Special Measurements Cards Part I into rows for each group. Cut Special Measurements Cards Part III into rows for each group. Print one Team Builders Measuring Tape per group.

Reusable • •

1 Pair of scissors (per group) 1 Glue stick (per group)

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) If you needed to measure a very large object, what tool would you use?; 2) How are measuring tapes arranged?; 3) If some measurements on a measuring tape were missing, how might you determine the missing parts?

FACILITATION TIP Review the ﬁrst 5 perfect squares with the class. Students may write in the values above the Team Builders Measuring Tape to help them place the cards on their tape measure. FACILITATION TIP If students need to, they may write the decimal values of each number on the front or back of their cards.

Read the following scenario to the class: Team Builders uses a special measuring tape to take their measurements. This measuring tape has been used out in the sunlight a lot, and now part of the numbers have faded away. Help Team Builders place the measurements on the special measuring tape again. Give one row of Special Measurement Cards Part I, one Team Builders Measuring Tape, one pair of scissors, and one glue stick to each group. Explain to the students that they will work with their groups to cut out each of the special measurements and glue the cards into the correct locations on the measuring tape. Students will collaborate to determine the value of each of the square roots on the Special Measurement Cards. Then, students will determine the location of each square root on the measuring tape. Monitor and assess student understanding as each group collaborates by asking the following questions: a.

DOK-2 What number squared will give you 16? I know that 42 = 16, so ___ √ 16 = 4.

DOK-2 Where would you place__√ 4 on the measuring tape? The √ 4 has a value of 2, so I should place √ 4 above 2 on the measuring tape.

DOK-2 What are these special measurements called? These special measurements are square roots. Each of these square roots is from a perfect square.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Allow students time to locate and glue each of the special measurements onto the Team Builders Measuring Tape.

Part II 1.

2. 3.

Read the following scenario to the class: Your boss at Team Builders needs to ﬁnd the approximate location of other square root measurements. The boss knows these measurements are all irrational numbers but tells you to determine the approximate location of each of these square roots on the special measuring tape. Help Team Builders determine an approximate decimal value for each of these irrational number measurements. Give a Student Journal to each student. Explain to students that they will use the Team Builders Measuring Tape to help determine the approximate decimal value of some of the measurements that have been recorded. Students should determine the perfect square below the square root and the perfect square above the square root to help determine an approximate decimal value of the irrational number. Monitor and assess student understanding as each group collaborates by asking the following questions: √ 3 fall between? Looking at the measuring DOK-1__What square roots __ will __ __ __ √ 3 falls between √ 1 and √ 4 . Since √ 1 = 1 and √ 4 = 2, the value of tape, __ √ 3 must be between 1 and 2.

DOK-2 How can the approximate decimal value for √ 3 ? __ __ you determine __ Since I know √ 1 = 1 and √ 4 = 2, the__value of √ 3 must be between 1 and √ 2. 3 is closer to 4, so the value of __ 3 will be closer to 2. I can estimate the decimal approximation for √ 3 to be 1.6.

DOK-2 What two measurements on your measuring tape would 12 fall ___ __ ___ between? I know √ 12 falls between √ 9 and √ 16 because 12 is between 9 and 16.

___

After reading the scenario, ask the class 1) How might you ﬁnd the approximate location of a square root on a measuring tape?; 2) How can you determine the approximate decimal value of a square root? FACILITATION TIP Return to the Venn diagram from Explore 1, and show where various square roots fall on the Venn diagram. Which ones are rational (perfect squares) and which are irrational?

√ d. DOK-2 How can you determine the ___ __ ___approximate decimal value for 12 ? Since √ 12 falls between √ 9 and √ 16 squares ___, I can use their perfect __ ___to know what two whole numbers √ 12 will be between. √ 9 = 3 and √ 16 = 4, so 12 must be a decimal number between 3 and 4. 12 is closer to 9, so I can start with a___ decimal less than 3.5 to determine the decimal approximation for √ 12 .

FACILITATION TIP

IRRATIONAL NUMBERS

Home

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

Math Chat DOK-2 How does a number line help to determine the approximate decimal value of a square root that is not a perfect square? A number line can help determine the approximate value of a square root that is not a perfect square because you can determine which 2 perfect squares your square root would be between and what whole number the square root would be closer to. ___ • DOK-2 Estimate the value of √ 42 . 62 = 36, and 72 = 49, so the square root of 42 is between 6 and 7. • DOK-2 Would the square root of 42 be closer to 6 or closer to 7? Explain your thinking. I think it would be closer to 6 because 42 is 6 away from 36 and 7 away from 49. __ __ • DOK-3 What would the value of −√ 4 be? I know that √ 4 is 2, so the opposite of 2 would be −2. •

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.

FACILITATION TIP Ask students who ﬁnish early to ﬁnd the approximate decimal values and the perfect squares that the _____ following _____ _____square _____ roots fall between: √ (40) , √ (50) , √ (75) , √ (90) . FACILITATION TIP Go deeper: Ask students, “How would you place other irrational numbers, pi for example, on a number line?”

IRRATIONAL NUMBERS

Irrational Numbers Explore 3 – Locate and Compare Irrational Numbers on a Number Line Part III 1. FACILITATION TIP Before reading the scenario, ask the class 1) What process do you go through when comparing numbers?; 2) What do you do when you want to compare numbers but they are in different forms?

2. 3.

FACILITATION TIP As an alternative, you could write these values on the board then have students write them on the measuring tape instead of cutting and pasting.

Read the following scenario to the class: Team Builders wants to compare the measurements they have been provided for different locations where they could build. All of the measurements have been given to you in different forms. Help determine the approximate value of each measurement to compare their sizes. Give one row of Special Measurement Cards Part III to each group. Explain to students that they will use the Team Builders Measuring Tape to help determine the approximate decimal value of some of the measurements that have been given to them. Students should determine the perfect square below and above each of the given measurements to help determine an approximate decimal value. Monitor and assess student understanding as each group collaborates by asking the following questions: DOK-1 What two square roots will √__2 fall between? Looking at __ the __ __ __ √ 2 falls between √ 1 and √ 4 . Since √ 1 = 1 and √ 4 = 2, measuring tape, __ the value of √ 2 must be between 1 and 2 but closer to 1.

DOK-2 How can the approximate decimal value for 2? __ __ you determine __ Since I know √ 1 = 1 and √ 4 = 2, the__value of √ 2 must be between 1 and √ 2. 2 is closer to 1, so the value of __ 2 will be closer to 1. I can estimate the decimal approximation for √ 2 to be 1.3.

DOK-2 What two measurements on your measuring tape would π___ fall __ between? I know π is about 3.14, so π would be between √ 9 and √ 16 because 3.14 is between 3 and 4.

DOK-2 Which is greater, √ 18 or 4.36? √ 18 will be between √ 16 and √ 25 , which is between 4 and 5 on the number ___ line. I can square each number to determine which one is greater. (√ 18 )2 = 18 and 4.362 = 19.01, so 4.36 is greater.

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 For an additional exercise, have the students put all of the numbers from Part III in order from least to greatest. Have students try this without using the measuring tape.

5. 6.

___

DOK-2 How does a number line help to compare an irrational number to a rational number? A number line can help determine the approximate value of an irrational number by determining what two whole numbers the irrational number will be between. Then, you locate both numbers on the number line to compare. ___

___

DOK-2 Which is greater, 7.2 or √ 70 ? √ 70 is greater because it will be located ___ ___ between √ 64 and √ 81 , or between 8 and 9. • DOK-3 What other strategies could you use to compare irrational numbers with rational numbers? Give an example. I can find the square of both numbers to compare because I know that ___the square of a square root ___ will be that number. For 2 √ example, when comparing √ 12 to 3.5, I can ___ first find ( 12 ) , which is 12. Then, I can find 3.52, which is 12.25; therefore, √ 12 is less than 3.5. Post-Explore 1. 2. 3.

___

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

•

As students answer this question, have them use their suggested strategy in an example. You could work the problem on the board, while having the students call out the steps or allow students to show their work on the board.

___

Math Chat •

FACILITATION TIP

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

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

IRRATIONAL NUMBERS

Home

IRRATIONAL NUMBERS

Irrational Numbers Explore 4 – Estimate and Compare Irrational Number Expressions ACTIVITY PREPARATION Students will estimate and compare irrational number expressions.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.5 Use appropriate tools strategically. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Set of Construction Cards (per group) 1 Building Card (per class) 1 Exit Ticket (per student)

•

Reusable • •

1 Resealable bag (per group) 1 Projector or document camera (per class)

• •

Plan to divide the class into groups of 2 or 3 to complete this activity. Print a Student Journal for each student. Print one set of Construction 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 durability. Print a Building Card for the class. If desired, print it on card stock, and laminate it for durability. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student will have one. Be prepared to project the Building Card for the class to see.

PROCEDURE AND FACILITATION FACILITATION TIP After reading the scenario, ask the class 1) Why is estimating numbers rather than determining the actual numbers suﬃcient in this scenario?; 2) What process is required to estimate irrational numbers to their closest decimal approximation?; 3) Why do you think it is beneﬁcial to convert irrational numbers into decimal approximations when solving expressions?

2. 3.

Read the following scenario to the class: Team Builders is gathering their ﬁnal construction needs to get started on their projects! All of the measurements given to your team need to be estimated to their closest decimal approximation. Help Team Builders determine each of the decimal approximations. Give a Student Journal to each student. Display the Building Card for students to view.

Explain the following to students: The perimeter of a square portion of the ____

concrete slab for the new____ youth activity center measures √ 410 feet. Team Builders

√ 410

is using the expression ____ to calculate the side lengths of the square slab. Ask 4 students the following questions: a.

What is unique about the side lengths of a square? All of the sides of a square are the same length.

How does the expression ____ relate to the side length of the square 4

____

slab? To determine the side length of one side given the perimeter, we

FACILITATION TIP Remind students that we chose to approximate sqrt(410) as 20.2 because it is closer to 400 than to 212 (441).

can divide the perimeter by 4. c.

What step should we take ﬁrst to solve for the side length of____ the slab? √ 410 . We First, we should ﬁnd the rational number approximation for ____ know that 202 = 400, so we can approximate √ 410 to be 20.2. Then, we can divide by 4 since there are 4 equal sides.

What does one side of the square slab measure rounded to the nearest tenth? One side of the slab will measure 5.1 feet in length.

FACILITATION TIP This could be a good opportunity to review the perfect squares that students should have learned in another scope. Review with ﬂash cards, making a list, or quizzing students orally. 54

√ 410

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

Give one set of Construction Cards to each group. Explain to students that they will collaborate with their groups to determine decimal approximations for irrational number expressions for different measurements at the youth activity center. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 Why does the expression for the new length of the garden include a 2 before the square root? We need to double the length of the garden, so we must multiply the length by 2 to ﬁnd the new length of the garden.

DOK-2 What is another way to write____ the expression for the basketball

Intervention

Acceleration

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.

IRRATIONAL NUMBERS

Home

√ 250

8. 9.

court problem? We can also write ____ . 2

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

Post-Explore 1. 2. 3. 4.

FACILITATION TIP During the Math Chat, review inverse operations. Have students list other operations that are inverses – addition and subtraction, multiplication and division. Demonstrate what happens when you apply inverse operations.

IRRATIONAL NUMBERS

Irrational 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

Irrational Numbers vs. Rational Numbers Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

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

Decimal Expansion 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

Locate and Compare Irrational Numbers on a Number Line

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

Estimate and Compare Irrational Number Expressions

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

Representation of Real Numbers

IRRATIONAL NUMBERS

Home

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

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

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

IRRATIONAL NUMBERS

Irrational Numbers

3 58

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

IRRATIONAL NUMBERS

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What does mastery look like?

I can classify real numbers as rational or irrational numbers.

I can convert a repeating decimal into a rational number.

I can use visual models and numerical reasoning to approximate irrational numbers.

I can locate irrational numbers on a number line and estimate the value of expressions.

SCOPE 1

Integer Exponents Scope Introduction SCOPE SUMMARY

Student Expectations

Students are exposed to the different properties that are tied to operations involving integer exponents. They will understand the process of applying integer operations to same base exponents to develop equivalent exponent expressions. In this scope, students will learn to evaluate exponents by expanding exponents in a quicker, more eﬃcient way by including integer operation rules. Students should be supported to develop strategies to multiply and divide exponent expressions as well as raise an exponent to a given power. They will develop an understanding of negative exponents and how they relate to their positive counterparts. Students will apply their understanding of exponent and integer operations to develop equivalent exponents expressions.

8.NR.2.1 Apply the properties of integer exponents to generate equivalent numerical expressions.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

Students are introduced to the concepts of exponents and integers in Grades 6 and 7. In Grade 6, students are taught to write and evaluate numerical expressions involving whole-number exponents. Students in Grade 7 understand the basic operation rules when working with signed numbers also known as integers. When working with integers, students are exposed to the differences between addition and subtraction rules and multiplication and division rules. Students are challenged to further expand their knowledge of integers to develop the concept of integer exponents using the principles of integer operations and exponent properties to develop equivalent expressions.

In high school algebra, students will be exploring rational exponents by extending their knowledge of integer exponents. Students will apply their knowledge of rational exponents to solve problems involving scientiﬁc notation. They will be expanding their integer exponent rules allowing for a notation for radicals in terms of rational exponents. Students will also use their knowledge of the properties of integer exponents to rewrite expressions involving radicals and rational exponents.

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

multiply and divide rational numbers.

•

examine a set of multiplication and division expressions.

•

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: •

evaluate an expression to determine which person successfully created an equivalent expression.

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. 60

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Properties of Integer Exponents In this exploration, groups of students will be tasked with matching property cards about exponents that have examples and nonexamples of the different properties. Students will: •

Explore 2

Explore 1

EXPLORE ACTIVITIES

generalize the properties of integer exponents.

Dividing with Exponents In this exploration, groups of students will determine whether division problems with exponents are true or false by matching. Students will: •

In this exploration, students will match equivalent multiplication expressions and show how one expression can be simpliﬁed to generate another. Students will: •

match equivalent multiplication expressions.

•

show how one expression can be simpliﬁed to generate another.

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

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

Multiplying with Exponents

INTEGER EXPONENTS

Home

Exponential Powers In this exploration, groups of students will solve a scenario about preparing for a test. Students will: •

determine if division problems with exponents are true or false.

ﬁnd equivalent expressions when given expressions with exponential terms raised to a power.

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.

INTEGER EXPONENTS

Integer Exponents 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 set of multiplication and division expressions 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.

INTEGER EXPONENTS

Home

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

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.

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.

Page 1: Letter C does not belong; A, B, and D have the same solution of 18, and C is −18.

Page 2: Letter B does not belong; A, C, and D have the same solution of −24, and B equals 9.6.

• •

Remind students that division problems can be written with the division symbol like in box A or by writing the numbers as a fraction like in box B.

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

Identifying Misconceptions •

FACILITATION TIP

Students may confuse the rules for multiplying and dividing positive and negative rational numbers with adding and subtracting rational numbers. Students may confuse the rules for multiplying and dividing a variety of rational numbers. Students may make a calculation error and not evaluate the expressions correctly, which would cause them to choose the wrong letter.

Create a chart with “positive” and “negative” along the left side and across the top. Have the students help you ﬁll it in to show what happens when you multiply or divide a positive by a positive, a positive by a negative, a negative by a positive, and a negative by a negative.

INTEGER EXPONENTS

Integer Exponents Hook – Game On! ACTIVITY PREPARATION Students will evaluate an expression and the steps taken by two different people to determine which person successfully created an equivalent expression.

Materials

Preparation

Printed •

• • •

1 Game On! (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Game On! 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 students have completed the Explore activities.

PROCEDURE AND FACILITATION Part I: Pre-Explore 1.

FACILITATION TIP As you read the scenario, slowly write out (under a document camera) the tiebreaker expression with exponents. Consider keeping the twins’ solutions covered up until students have had some time to make observations about the expression.

FACILITATION TIP

4. 5.

Print and post these guiding questions to lead your discussion. Provide students some independent think time and shoulder partner time before leading the class discussion. FACILITATION TIP Check for understanding regarding the vocabulary for base and exponent before moving on to the Explore activities.

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: The Rodriguez family was holding a family game night on Friday night. They were playing a trivia game. After it ended in a tie, Mr. Rodriguez suggested a tiebreaker question for twin 8th graders, Alejandro and Paloma. He wrote out an expression with exponents, asked the twins to evaluate the expression, and find an equivalent expression. He told them they had to show their work. Whoever got it right and finished first won the tiebreaker, and that person’s team won game night. 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 the twins are evaluating expressions. I wonder what operations are involved—addition, subtraction, multiplication, or division. How are exponents being used? I can use math to determine whether Alejandro and Paloma correctly evaluated the expression and found an equivalent expression by checking their steps and their solutions. Project Game On! for students. Explain to students that only one twin found an equivalent expression. The other twin made an error during one step of the process. Discuss the following questions: a.

DOK-1 What is an exponent? It is a raised number to the right of a base number that tells how many times to multiply the base number by itself. It is a way to indicate repeated multiplication.

DOK-1 What operations are being used in this expression? I can see addition, subtraction, multiplication, and division.

DOK-1 What number is the base of the exponents? 10

Complete the Explore activities.

Part II: Post-Explore 1. 2.

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

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

FACILITATION TIP

DOK-1 Which twin has an equivalent expression and the correct answer? Alejandro

DOK-1 What mistake did Paloma make? When dividing 10 by 10 , she divided the exponents instead of subtracting them. Her process was 8 ÷ 4 = 2, but the correct process is 8 – 4 = 4. The expression should have been 104 inside the parentheses.

DOK-1 What was the equivalent expression that Alejandro found? 1020

FACILITATION TIP

e. DOK-2 How would the expressions change if they were raised to the negative 5th power instead of positive 5th power? The answers would be the reciprocal.

Consider challenging students to create and solve expressions together in partners. Provide some speciﬁc constraints to limit the complexity if needed. Perhaps it can only include 3 operations, one set of parentheses, or negative exponents. Carefully select some expressions to share with the class and solve.

Take time to clarify how dividing exponents with the same base works. Conﬁrm that students understand that the bases must match and demonstrate why we use subtraction to simplify.

INTEGER EXPONENTS

Home

INTEGER EXPONENTS

Integer Exponents Explore 1 – Properties of Integer Exponents ACTIVITY PREPARATION Students will use reasoning to work with integer exponents to generalize the properties of integer exponents.

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 Set of Property Deﬁnition Cards (per student) 1 Exit Ticket (per student)

Plan to divide the class into groups of two to complete this activity. Print a Student Journal, a set of Property Deﬁnition Cards, and an Exit Ticket for each student.

Reusable • •

1 Pair of scissors (per pair) 1 Glue stick (per pair)

PROCEDURE AND FACILITATION Part I: Finding the Pattern FACILITATION TIP

After reading the scenario, ask the class 1) What do exponents tell us?; 2) How does knowing the properties of exponents help us when solving equations?; 3) How can having examples and nonexamples of different properties help determine which property of exponents is being used?

2. 3.

FACILITATION TIP Keep an eye out for students who see the negative exponent and ﬁll in the chart with a negative number. Students may need a reminder of the meaning of a negative exponent.

a. DOK-2 What patterns do you notice when moving to the left in the Expanded row? In the Expanded row, there is one less 4 until you get to 1 0. Then, there is an additional __4 each time after 0. (Numbers in student responses are dependent on which table students are working on.)

FACILITATION TIP Challenge students to ﬁgure out what values will come next in the Expanded and Evaluate rows in the table. 66

Read the following scenario to the class: Your best friend just finished studying exponents in her class. She’s about to take her test but found that her notes got all messed up. She has a chart on exponents and a chart of examples and nonexamples for the different properties of exponents but is missing her notes for the properties themselves. Look over her notes, help her fill in what she’s missing, and match the examples to the correct properties. Give a Student Journal to each student. Student partnerships should analyze the tables on their Student Journals. Students will fill in the missing sections, looking for patterns to help them complete the chart. Monitor and assess student understanding as each group collaborates by asking the following guiding questions:

DOK-2 What patterns do you notice when moving to the left in the Evaluate row? Answers will depend on which table students are looking at. The number is divided by 2 to get the next number to the right.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Math Chat • •

DOK-1 Why is 41 equal to 4? 41 means there is 1 four, which is equal to 4. DOK-2 What can you determine about any number raised to the power of 1? Any number raised to the power of 1 will be itself.

•

__? 4−1 equals __ because each section is __ of the section DOK-1 Why is 4−1 equal to __ 4 4 4

•

DOK-2 What pattern do you see that helps you ﬁll in the missing sections? In the

1 to the left. 40 is 1, and of that is __4.

INTEGER EXPONENTS

Home

ﬁrst table, when looking at the Evaluate row, each column is __2 of the column to 1

the left. In the second table, each column of the Evaluate row is __4 of the column

to the left. In the third table, each column of the Evaluate row is of the column to

•

the left. DOK-2 Looking at the chart, what could a negative exponent mean for the integer? A negative exponent means the base number is the denominator of a fraction.

Part II: Finding the Property 1. 2. 3. 4.

Give a set of Property Definition Cards to each student and a pair of scissors and a glue stick to each pair of students. Have students cut out the Property Definition Cards. Explain that they will use the If and But sections on the tables of their Student Journals to match the correct Property Definition Card to the correct section. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 What is happening with the exponents in the ﬁrst If section? The exponents seem to be added together.

DOK-1 What do you notice about the integers in the If section? They are the same number.

DOK-1 What is changing in the If section? What is staying the same? Answers will vary. The exponents are changing, but the base numbers are staying the same.

DOK-2 Looking at the equation 42 • 43 = 45, how can you tell it is a true statement? 42 is (4 • 4), and 43 is (4 • 4 • 4). This is the same as 45, which is (4 • 4 • 4 • 4 • 4).

e. DOK-2 How is the If section different from the But section? Answers will vary. In the If section, the base numbers were the same and exponents were subtracted. In the But section, the integers were different and the exponents were not subtracted. 5.

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

Math Chat DOK-2 Why is it important to know the properties of integer exponents? Knowing the properties of integer exponents helps to create equivalent expressions, which makes solving equations easier. • DOK-2 How did the But section help you identify the rule? The But section identiﬁed a nonexample that helped me narrow down what the If section was trying to show. •

FACILITATION TIP An alternative answer students could give would be that the negative exponent tells how many times to divide 1 by the base number (versus a positive exponent tells how many times to multiply the base number). FACILITATION TIP Students may write the property directly into the space if these materials are limited or unavailable.

FACILITATION TIP Keep an eye out for students who struggle ﬁnding the difference between these two sections. The difference is subtle (different base numbers, therefore the rule cannot be applied). FACILITATION TIP For extra practice, have the students write each law in their own words with an example on a separate sheet of paper. This could be done in a math journal, in class notes, as a pop quiz, or to turn in for a classwork grade.

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 Create clear criteria for student written responses to these questions.

INTEGER EXPONENTS

Integer Exponents Explore 2 – Multiplying With Exponents ACTIVITY PREPARATION Students will match equivalent multiplication expressions and show how one expression can be simpliﬁed to generate another.

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 Set of Matching Cards (per pair) 1 Exit Ticket (per 2 students)

Reusable •

•

1 Resealable bag (per pair)

Plan to divide the class into groups of two to complete this activity. Print a Student Journal for each student. Print a set of Matching Cards for each partnership. Cut out and place the 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 will have one.

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) How do you study for tests?; 2) How can ﬂash cards help you memorize information? FACILITATION TIP Watch for students who try to multiply the exponents instead of adding them. Show examples with numbers to prove that the exponents are added.

FACILITATION TIP Alternatively, students could work in groups of 3–4 and deal 3 cards to each player. The remaining cards would become the “draw” pile. Students would take turns asking another player for a card with a value equivalent to the value of one of their cards (in the style of “Go Fish”). If the player they ask does have the card, they must hand it over, and a pair can be made. If not, he or she must draw a card. The goal is to make matches. Students will record the expressions for all matches on their Student Journals. 68

2. 3. 4.

Read the following scenario to the class: A Plus School Supplies creates ﬂash cards to help students study for tests. There was a mistake at the factory, and a whole set of cards got mixed up on the ﬂoor. Help the workers pick up and reorganize the cards. Give a Student Journal to each student. Give a set of Matching Cards to each pair. Review the property of exponents when multiplying with common bases and exponents. Ask the following questions: a.

DOK-1 What happens to the exponents when you multiply numbers with like bases? When you multiply with like bases, the base stays and you can add the exponents together.

DOK-1 What happens to the exponents when you multiply numbers with unlike bases but common exponents? If the bases are different but the exponents are the same, the exponent can stay and you can multiply the bases together.

Explain to students that they will spread out their Matching Cards and take turns ﬁnding matches. When either person ﬁnds a match, they both will record each expression in their Student Journals. They will use the Generate column to show how one expression equals the other. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How do you know when to add the exponents together? If the base numbers are the same, you can add the exponents together.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 What would you do if neither the exponents nor the bases are the same? If neither are the same, the expression is already simpliﬁed.

DOK-2 What would you do if both the exponents and the bases are the same? If both are the same, you can choose whether to add the exponents or multiply the bases and keep the exponent.

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

Intervention

Acceleration

FACILITATION TIP Show an example to the class. 33 x 33 = 36 (adding the exponents)= 93 (multiplying the base and keeping the exponent) = 729 (Both options give the same result.)

INTEGER EXPONENTS

Home

Math Chat DOK-2 Why is it important to be able to generate equivalent expressions? It is important because it simpliﬁes the expression and should make it easier to solve. • DOK-1 Given the expression 32 • 32, generate an equivalent expression. 32 • 32 = 92 = 34 • DOK-2 For the expression 6(64 • 64), 6 has no exponent. Explain how you would get 69 and not 68. 6 has a power of 1. Instead of 6(64 • 64), it could be written as 61(64 • 64). • DOK-2 Explain what occurs with the exponent when the bases are different, like we saw in the expression (4 • 3)2. Each base is being raised to the exponent. It is 42 • 32. •

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.

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.

INTEGER EXPONENTS

Integer Exponents Explore 3 – Dividing with Exponents ACTIVITY PREPARATION Students will determine whether division problems with exponents are true or false.

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 Set of True or False Cards (per group) 1 Exit Ticket (per 2 students)

• •

•

Plan to divide the class into groups of three or four to complete this activity. Print a Student Journal for each student. Print a set of True or False Cards for each group. Cut them out, and place them in random order around the room. 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 will have one.

PROCEDURE AND FACILITATION FACILITATION TIP After reading the scenario, ask the class 1) What strategies can you use to determine whether the division expressions are correct?; 2) What do you know about dividing exponents? FACILITATION TIP Alternatively, you could project each card onto the board instead of having the students search for the cards. FACILITATION TIP

2. 3.

This is a good opportunity to review the properties of exponents that students have learned so far. Speciﬁcally, the ones used in this lesson – negative property, quotient property, power of a quotient – should be covered. FACILITATION TIP The discussion questions asked above will help students answer the reﬂection questions on their Student Journal. Let them discuss these questions together in their groups or discuss the questions together as a class. 70

5. 6.

Read the following scenario to the class: The review for the exponents test did not go very well. Your teacher took all the division expressions you answered and placed them on cards around the room. It’s your job to ﬁnd the cards and determine whether they were answered correctly. Find the cards, and decide whether they are true or false. Use what you know about dividing exponents to prove your answer. Give a Student Journal to each student. Explain to students that they will collaborate with their groups to use the properties of exponents to determine whether the cards around the room are true or false. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What should you be looking for to simplify when dividing exponents? You should look for common base numbers or common exponents.

DOK-1 What happens when the bases are the same when dividing exponents? When the bases are the same, you can subtract the exponents from each other.

DOK-1 What happens when the exponents are the same when dividing exponents? When the exponents are the same, you can divide the base numbers and keep the exponent the same.

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

Intervention

Acceleration

Math Chat DOK-2 How is creating an equivalent expression when dividing numbers with like bases different from when dividing numbers with like exponents? When the bases are the same, you subtract the exponents. When the exponents are the same, you divide the bases and keep the exponent the same. • DOK-1 What happens when neither the bases nor the exponents are the same? When neither the bases nor the exponents are the same, you have to calculate each exponent and then divide normally. • DOK-2 How is dividing numbers with like bases different from multiplying numbers with like bases? When you divide exponents with like bases, you subtract the exponents. When you multiply exponents with like bases, you add the exponents. • DOK-2 How is dividing numbers with unlike bases different from multiplying numbers with unlike bases? When you divide exponents with unlike bases, you can simplify the fraction before distributing the exponent. When you multiply exponents with unlike bases, you can multiply the bases ﬁrst as well as simplify. When dividing exponents with unlike bases, however, you cannot always simplify the bases ﬁrst. •

STEMscopes Tip

INTEGER EXPONENTS

Home

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

Post-Explore 1. 2. 3.

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

INTEGER EXPONENTS

Integer Exponents Explore 4 – Exponential Powers ACTIVITY PREPARATION Students will ﬁnd equivalent expressions when given expressions with exponential terms raised to a power.

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 Exit Ticket (per student)

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

PROCEDURE AND FACILITATION FACILITATION TIP

After reading the scenario, ask the class 1) What is an exponential term raised to a power?; 2) What steps are necessary when creating equivalent expressions? FACILITATION TIP Clarify the deﬁnition of equivalent expression. Explain that students are to create an expression that speciﬁcally can be derived by using the exponent properties that have been learned in this scope.

2. 3.

FACILITATION TIP Before beginning the activity, you could review the exponent properties, speciﬁcally the ones used in this activity: Power of a Power, Power of a Product, Power of a Quotient, Product Property, Quotient Property.

Read the following scenario to the class: It’s the day before the test, and Ms. Taylor has provided one ﬁnal review activity for you. She has given you six expressions that include an exponential term raised to a power. Not only do you have to create an equivalent expression, but you also need to explain the steps you take to make that expression. Show Ms. Taylor that you’re ready for the test by completing her test review. Give a Student Journal to each student. Explain to students that they will collaborate with their groups to use the properties of exponents to determine equivalent expressions they can use for each of the given expressions in the table. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What do the properties of integer exponents teach us about when an exponential term is raised to a power? When an exponential term is raised to a power, you can multiply the exponents together.

DOK-1 What happens to the exponents when you multiply numbers with like bases? When you multiply with like bases, the base stays and you can add the exponents together.

DOK-1 What happens when the bases are the same when dividing exponents? When the bases are the same, you can subtract the exponents from each other.

STEMscopes Tip A Parent Letter, located in the Home section, provides parents with a breakdown of the concepts being learned in school, as well as a choice board of related activities that students can complete at home. Sending home the Parent Letter at the start of each scope strengthens the family-school connection by keeping parents informed and included in the learning process. 72

e. DOK-1 What happens when the exponents are the same when dividing exponents? When the exponents are the same, you can divide the base numbers and keep the exponent the same. 5.

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

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Math Chat DOK-1 What does it mean when an exponential term has an exponent? It means the exponential term is multiplied that many times. • DOK-2 What strategies could you use to check your work with equivalent expressions? I could ﬁnd the value of each expression and determine if they are equal to each other. • DOK-2 Why is it important to know the properties of integer exponents? Knowing the properties of integer exponents helps to create equivalent expressions, which makes solving equations easier. •

Post-Explore 1. 2. 3. 4.

FACILITATION TIP This is a good place to reiterate with an example showing the exponential term multiplied that many times.

INTEGER EXPONENTS

Home

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.

INTEGER EXPONENTS

Integer Exponents 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

Properties of Integer Exponents 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

Multiplying with Exponents 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

Dividing with Exponents

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

Exponential Powers

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

Spiraled Review

Fluency Builder

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

Integer Exponents

Can be done independently

INTEGER EXPONENTS

Home

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

INTEGER EXPONENTS

Integer Exponents 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 76

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

INTEGER EXPONENTS

Home

What does mastery look like?

I can use numerical reasoning to identify patterns associated with properties of integer exponents.

I can generate equivalent numerical expressions with integer exponents by using the product rule, quotient rule, power rule, power of product rule, power of a quotient rule, zero exponent rule, and negative exponent rule.

SCOPE 1

Scientific Notation Scope Introduction SCOPE SUMMARY Students will be working to convert extremely large and extremely small numbers into a format that allows them to be easier to calculate with. They will be converting numbers from standard form to scientific notation as well as scientific notation to standard form. Students will be focusing on comparing values that are written in both standard form and scientific notation to determine which has a better value overall. Student Expectations

8.NR.2.3 Use numbers expressed in scientiﬁc notation to estimate very large or very small quantities, and to express how many times as much one is than the other.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

Students begin to understand the place value system beginning in Grade 5 when they work to recognize that a digit in one place of a multidigit number represents 10 times as much as it 1 represents in the place to its right as well as ___ of 10 the value of the place to the left. They also explore the patterns in the placement of a decimal point when a decimal is multiplied or divided by 10. Students will also have experience with denoting whole numbers as powers of 10 in exponential form.

Students will be using their knowledge of scientific notation to complete multistep operations with numbers in scientific notation. They will be converting numbers between standard form and scientific notation as well as performing operations like addition, subtraction, multiplication, and division with those numbers.

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

explain patterns in the number of zeros of the product when multiplying a number by powers of 10.

•

explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.

•

use whole-number exponents to denote powers of 10.

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: •

represent numbers in scientiﬁc notation.

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. 78

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Writing in Scientiﬁc Notation In this exploration, groups of students will be tasked with helping a stop-motion animator write the number of frames it takes to create a project. Students will: •

Explore 2

Explore 1

EXPLORE ACTIVITIES

write small and large numbers in scientiﬁc notation.

In this exploration, students will be tasked with creating reasonable estimates of the large numbers of movie ticket sales for a local news report. 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 3

Estimating Numbers and Scientiﬁc Notation

SCIENTIFIC NOTATION

Home

estimate large and small numbers using scientiﬁc notation.

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

Comparing Numbers in Scientiﬁc Notation In this exploration, students will compare two values written in scientiﬁc notation for numerical facts. Students will: •

compare two values written in scientiﬁc notation for numerical facts.

•

determine how many times greater or smaller one value is when compared to the other.

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

SCIENTIFIC NOTATION

Scientiﬁc Notation 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 listen to prompts 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.

SCIENTIFIC NOTATION

Home

6.PAR.6.1 Write and evaluate numerical expressions involving rational bases and whole-number exponents.

Materials

Preparation

Printed •

•

1 Fact or Fiction (per class)

•

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, and allow students to move to different sides of the room. Have students discuss their reasonings amongst peers. Before reading the next prompt, allow students to move back to their starting points. Repeat with another prompt. Provided below is an answer key: a.

Prompt 1 is ﬁction. The correct answer is 100,000.

Prompt 2 is a fact.

Prompt 3 is ﬁction. The correct answer is 794.5.

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 Instead of having students shuﬄe around, you could have them hold up whiteboards and write “fact” or “ﬁction” in response to each prompt.

FACILITATION TIP Show the students 10 , 103, and 104. Ask them if they see the pattern. Since the exponent for the 10 is a 5, we know there are 5 zeroes after the 1. The exponent tells how many zeroes there will be. 2

Identifying Misconceptions • •

•

Students may have diﬃculty relating a power of 10 to its number in standard form. Students may confuse the direction that a decimal point should move when multiplying or dividing a decimal number by a power of 10. When multiplying by a power of 10, the decimal point moves to the right. When dividing by a power of 10, the decimal point moves left. Students may not understand that the exponent denotes the number of places the decimal point should move when multiplying or dividing a decimal number by a power of 10.

FACILITATION TIP Remind students that when multiplying, the number gets bigger, so the decimal point moves right. However, when dividing, they are breaking a number into smaller parts, so the decimal point moves left.

SCIENTIFIC NOTATION

Scientific Notation Hook – Scientific Notation ACTIVITY PREPARATION Students will represent numbers in scientific notation.

Materials

Preparation

Printed •

• •

1 Scientiﬁc Notation (per class)

Reusable •

•

1 Phenomena Video (per class)

Plan to show the video. Prepare to project the Scientiﬁc Notation slide for the whole class to view. Prepare to introduce the scenario and 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) If you were going to make a short movie, what would it be about?; 2) What would you need to do before ﬁlming the movie?

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.

4. 5.

6. 82

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: Saanvi is working on a new movie for the Independent Film Festival. She plans on filming a 2-hour-and-15-minute movie about living in a small town. As she makes the movie she notices that 24 frames are needed for 1 second of film in a typical 2-hour movie. Saanvi’s goal is to find the most efficient way to represent the total number of frames needed for her 2-hour-and-15-minute movie called Life in a Small Town. 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 there are many frames in a storyboard that is used to make a movie. I wonder what type of math can be used to find the most efficient way to represent the total number of frames. Project the Scientific Notation slide. Explain to students that finding the most efficient way to represent numbers can help us represent really large and really small numbers. Discuss the following questions: a.

DOK-1 Where do you see math in movies? Allow students to share all ideas. Student answers will vary. Calculating the amount of minutes it takes to make the movie and the number of frames per second.

DOK-1 Using your prior learning of representing numbers in different ways, how could we represent a really large number? Allow students to share all ideas. Student answers will vary. Since the movie will be 2 hours and 15 minutes and there are 24 frames per second, we can take the larger number and get a better understanding of the place value of the different numbers we are working with.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II: Post-Explore 1. 2.

Show the phenomena video again and restate the problem. Refer to the Scientiﬁc Notation slide and discuss the following questions: a.

DOK-1 Does representing the total frames per second using this method make more sense after the Explore activities? Yes, this method is called scientific notation. The total frames per second represent large numbers and an efficient way to write these numbers is by using scientific notation.

DOK-1 How do you convert a number from standard notation to scientiﬁc notation? I would place or move the decimal point of a number until the number is greater than 1 and less than 10.

DOK-1 How do you convert a number from scientiﬁc notation to standard notation? I would move the decimal point the number of places indicated by the power of 10.

DOK-1 Do you feel that you have a strong understanding of scientiﬁc notation? Answers will vary based on students’ success during the activity and on their conﬁdence level.

SCIENTIFIC NOTATION

Home

FACILITATION TIP Make sure students can identify where the numbers in the multiplication problem came from. The 24 is from the 24 frames per second. The 60 is from 60 seconds per minute. FACILITATION TIP How many minutes are in 2 hours and 15 minutes? If students need a review of time, this could be a good opportunity.

SCIENTIFIC NOTATION

Scientific Notation Explore 1 – Writing in Scientific Notation ACTIVITY PREPARATION Students will write small and large numbers in scientific notation.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.3 Construct viable arguments and critique the reasoning to others. 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 two to complete this activity. Print a Student Journal and an Exit Ticket for each student.

PROCEDURE AND FACILITATION Part I: Writing Large Numbers in Scientiﬁc Notation FACILITATION TIP Before reading the scenario, ask the class 1) What do you think stop-motion animation is? After reading the scenario, ask the class 2) How many frames do you think it takes to create a stop-motion animation video?; 3) In what ways can really large numbers be written? FACILITATION TIP This could be a good opportunity to review place value through the hundred thousands place. Draw a place value chart, and have students tell the value of each digit, as well as the name of that place.

2. 3.

FACILITATION TIP Watch out for students who forget to write the 10 in scientiﬁc notation and are placing the exponent immediately following the base number. Remind students that they are not multiplying that base number by itself, but they are multiplying that base number by a power of ten.

DOK-2 How do you write a large number in standard form from scientiﬁc notation by moving the decimal? Responses will vary. We move the decimal place in the factor (or after the whole number) to the right the number of spaces in the exponent.

DOK-1 What does an exponent represent? Responses will vary. The exponent represents the number of times we multiply the base by. For example, 105 is 10 × 10 × 10 × 10 × 10.

DOK-2 What is 103? Responses will vary. 10 × 10 × 10 = 1,000

d. DOK-2 How would you convert 60,000 from standard form to scientiﬁc notation? 60,000 can be rewritten as 6 × 10⁴ because we move the decimal (after the last zero) 4 spaces to the left to identify our factor of 6.

FACILITATION TIP Review opportunity: Have students name this number (one thousand). Show other powers of ten, for example 104 and 105, and have students name those numbers as well.

Read the following scenario to the class: Penelope is a stop-motion animator. She creates her ﬁlms using clay characters and backgrounds. Each of her movies is created by slightly moving each character, taking a picture of the character and background, and then repeating this process. The images, or frames, are then assembled together to form an animated scene. Penelope notices the number of frames it takes to create a project is a pretty large number. Let’s learn about a new way for Penelope to write these large numbers. Give a Student Journal to each student. Direct students’ attention to Part I of their Student Journals. Students will write large numbers that represent the frame/total number of frames in either standard form or in scientiﬁc notation using positive exponents. Monitor and assess student understanding as each group collaborates by asking the following guiding questions:

5. 6.

Allow time for students to complete Part I of their Student Journals, including the reflection questions. Explain to students that mathematicians write very large numbers in scientific notation. These numbers expressed in scientific notation are written as a factor multiplied by a power of 10 with a positive exponent. The factor must be greater than 1 and less than 10. © Accelerate Learning Inc. - All Rights Reserved

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 How can you write 4.5 × 109 in standard form? 4,500,000,000 DOK-2 How can you write 81,000,000,000,000 in scientific notation? 8.1 × 1013 DOK-1 Does writing a number in scientific notation change its value? No, writing a number in scientific notation is just another way of writing a number.

SCIENTIFIC NOTATION

Home

Part II: Writing Small Numbers in Scientiﬁc Notation 1.

Read the following scenario to the class: Penelope wonders what part of the entire project one frame is. She notices this number is really small. Let’s now learn about a new way for Penelope to write these really small numbers. Direct students’ attention to Part II of their Student Journals. Students will write small numbers that represent the frame/total number of frames in standard form as a ratio expressed as a decimal. They will then write these small numbers in scientiﬁc notation using negative exponents. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

4. 5.

DOK-2 How do you write a small number in standard form from scientiﬁc notation by moving the decimal? Responses will vary. We move the decimal place in the factor (or after the whole number) to the left the number of spaces in the exponent.

DOK-3 What is 10−3? Responses will vary. 1 ÷ 10 ÷ 10 ÷ 10 = 0.001

DOK-2 How are 0.006 and 6 × 10−3 equivalent values? Responses will vary. 10−3 = 0.001, and 6 × 0.001 = 0.006. Both numbers are equal to each other. One is written in standard form, and the other is written in scientiﬁc notation.

Allow time for students 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.

FACILITATION TIP After reading the scenario, ask the class 1) In what ways can really small numbers be written? FACILITATION TIP As students write the small numbers in scientiﬁc notation, watch out for students who leave out the negative sign or place the negative sign in the wrong place. FACILITATION TIP This is a good opportunity to review negative exponents. Ask the students for the meaning of a negative exponent. You could have students solve simple negative exponent problems on the board, such as 3−2 or 10−3. FACILITATION TIP Ask the students if they know other ways to write 0.006. They should be able to name 6 3 the fraction _____ , or ____ if they simplify. 1000 500

Math Chat • • •

DOK-2 How can you write 3.3 × 10−10 in standard form? 0.00000000033 DOK-2 How can you write 0.000000000054 in scientiﬁc notation? 5.4 × 10−11 DOK-2 How can you tell whether a number is really large or really small when given a value written in scientiﬁc notation? If the power of 10 is positive, then the number is large. If the power of 10 is negative, then the number is a small decimal.

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.

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.

SCIENTIFIC NOTATION

Scientific Notation Explore 2 – Estimating Numbers and Scientific Notation ACTIVITY PREPARATION Students will estimate large and small numbers using scientific notation.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.3 Construct viable arguments and critique the reasoning of others. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

• • • •

1 Student Journal (per student) 1 Estimation Card Match (per pair) 1 Exit Ticket (per student)

Reusable • •

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 a set of Estimation Card Match cards for each pair. Gather enough pairs of scissors and glue sticks for each student to have one of each.

1 Pair of scissors (per student) 1 Glue stick (per student)

PROCEDURE AND FACILITATION FACILITATION TIP

Before reading the scenario, ask the class 1) How do you think news organizations determine which movies are the top movies of the weekend? After reading the scenario, ask the class 2) Why is it suﬃcient for Pierre to develop a reasonable estimate rather than exact number of ticket sales?

2. 3.

FACILITATION TIP Extension activity: Practice rounding each number on the Student Journal to the nearest 10. Then, practice rounding to the nearest 100, nearest 1,000, and so on.

Read the following scenario to the class: Every Monday, the local news station creates a report of the top movies from the weekend based on ticket sales. Pierre is developing the report; however, he believes that using the actual amount of money each movie made isn’t necessary to the viewer. Pierre decides to develop a reasonable estimate of each movie’s ticket sales. Let’s help Pierre create reasonable estimates of these large numbers. Give a Student Journal to each student. Give a set of Estimation Card Match cards to each pair. Give a pair of scissors and a glue stick to each student. Review the purpose of estimation with students to ensure they understand why we would need to estimate numbers using scientiﬁc notation: a.

DOK-1 Why should we estimate numbers? Responses will vary. Estimating can help us determine approximately how big or how small a number is. Estimation gives us a relatively good idea of the size of the number.

DOK-1 How does estimation help with scientiﬁc notation? Responses will vary. Estimation and scientiﬁc notation are useful when working with really big and really small quantities.

DOK-1 How do you estimate large numbers using scientiﬁc notation? Responses will vary. Write each number in scientiﬁc notation, and then round each decimal number to the nearest whole number.

DOK-1 How do you estimate small numbers using scientiﬁc notation? Responses will vary. Write each number in scientiﬁc notation, and then round each decimal number to the nearest whole number. © Accelerate Learning Inc. - All Rights Reserved

7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

Explain to students that they will estimate numbers using scientiﬁc notation. Each student will then cut their own set of Estimation Card Match cards. Students will use their set of matching cards to determine each movie’s reasonable estimates according to the actual values. (Note that not all cards will be used.) Students will check each other’s work before gluing the appropriate cards to their Student Journals. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 Is there only one way to estimate a large or small number? Responses will vary. We can approximate the estimated value by evaluating the digit in the 2nd-farthest decimal place. For example, if I needed to estimate 793,312, I would look at the 3 in the thousands place. Since 3 is less than 5, I would estimate 793,312 to 790,000. 800,000 would even be a reasonable estimate, depending on how I approximated the actual value. The same would apply to a much smaller number. I would estimate 0.000032 to 0.00003.

DOK-3 What makes an estimation reasonable? Responses will vary. A reasonable estimation should be pretty close to the actual number. The closer we approximate or round to the actual number, the more reasonable it becomes. For example, if I were to estimate 21,567,678, a good estimation might be 22,000,000 or 21,500,000.

DOK-1 Does estimating a number change the actual value of the number? Responses will vary. No, estimating a number gives a solution near the actual value but does not change the actual value of a 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.

Math Chat DOK-2 Estimate 81,768,478,348. Write your estimation in scientific notation. If I approximate 81,768,478,348 to 82,000,000,000, the estimation would be written as 8.2 × 1010. • DOK-2 Estimate 0.00000079. Write your estimation in scientific notation. If I approximate 0.00000079 to 0.0000008, the estimation would be written as 8 × 10−7. • DOK-2 What are several ways you can estimate 7,549,687? Write each response in scientific notation. I can estimate the number to be 7.5 × 106 (7,500,000) or 8 × 106 (8,000,000). I can also estimate the number to be 7.55 × 106 (7,550,000). All are valid depending on how we approximate the actual value. •

Intervention

Acceleration

FACILITATION TIP If materials are not available, students may write in the values on the Student Journal instead of cutting and pasting. FACILITATION TIP For early ﬁnishers: On the backs of cards that did not get used, have the students write a number that could use that card as an estimate. Allow students to trade within their group or with another group to check answers.

SCIENTIFIC NOTATION

Home

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.

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.

SCIENTIFIC NOTATION

Scientific Notation Explore 3 – Comparing Numbers in Scientific Notation ACTIVITY PREPARATION Students will compare two values written in scientific notation. Students will also determine how many times greater or smaller one value is when compared to the other.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.3 Construct viable arguments and critique the reasoning of others. MP.6 Attend to precision.

Materials

Preparation

Printed •

•

1 Student Journal (per student) 1 Set of Comparing Trivia Task 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 Comparing Trivia Task Cards for each group of students. Gather enough pairs of scissors for each group to have one.

Reusable •

1 Pair of scissors (per group)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What is a trivia contest? After reading the scenario, ask the class 2) What is “scientiﬁc notation”?; 3) How is having numbers written in scientiﬁc notation helpful when comparing them? FACILITATION TIP

2. 3. 4.

An alternative activity would be to have the students put all the task cards in order from least to greatest or greatest to least. Students could perform this task as individuals or in small groups. FACILITATION TIP After ﬁlling out the Student Journal, students will have one pair of task cards remaining. Students who ﬁnish early can compare those remaining two cards. If they have more time, students can restart the task, drawing and comparing different pairs of cards.

Read the following scenario to the class: Daphne hosts a weekly trivia contest. This week’s theme is animated films. Daphne is gathering numerical facts to use for the trivia contest and notices the values can also be written in scientific notation. Daphne also wonders how much bigger or smaller some of these numerical facts are compared to the other facts gathered. Let’s help Daphne compare numbers written in scientific notation! Give a Student Journal to each student. Give a set of Comparing Trivia Task Cards and a pair of scissors to each group of students. Explain to students that they will work in their groups and select one teammate to cut out the 8 Comparing Trivia Task Cards. A teammate will shuffle the cards and place them face down in a pile. Students will then take turns selecting one card from the top of the pile. After a student selects the first card, the group will use the table provided in their Student Journals to record the card letter and the value on that card for card 1. Students will repeat this process 5 more times. 3 pairs of values (cards 1 and 2, cards 3 and 4, and cards 5 and 6) will be created from the 6 cards drawn. Students will follow each step in the table to determine the greater or lesser value between the two cards. They will use the formula provided in the table to determine how much bigger or smaller the two cards are when compared to each other. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 How can you determine how much larger one number is compared to another? Responses will vary. I can divide the larger value by the smaller value to determine how much bigger the larger value is compared to the smaller value. © Accelerate Learning Inc. - All Rights Reserved

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 What is one way you can compare two numbers written in scientiﬁc notation? Responses will vary. I can rewrite these numbers in standard form and then compare the values.

DOK-3 Write a number, in scientiﬁc notation, that is greater than 3 × 10−4. Answers will vary. 3 × 10−3 is greater than 3 × 10−4.

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 •

smaller value

2. 3. 4.

Specify if you want students to provide answers with a negative or positive exponent. You can also ask students for a number, in scientiﬁc notation, that is less than the given number. Before students answer, ask them if it’s possible to have an answer with a positive exponent.

−4

−5

4.5 × 10 ________ 5

9 × 10 4

DOK-2 Which value is smaller: a number written in scientific notation with a positive exponent or a number written in scientific notation with a negative exponent? Why? Between a number written in scientific notation with a positive exponent and one with a negative exponent, the negative exponent will be smaller than the number with the positive exponent. A decimal less than 1 will be smaller than a whole number greater than 1.

Post-Explore 1.

FACILITATION TIP

9 ×________ 10 = 20 4.5 x 10

DOK-2 Which value is smaller: 9 × 104 or 4.5 × 105? By how much? 9 × 104 is 5 times smaller than 4.5 × 105.

larger value ___________

•

Acceleration

DOK-2 Which value is larger: 4.5 × 10−5 or 9 × 10−4? By how much? 9 × 10−4 is 20 times larger than 4.5 × 10−5. larger value ___________

•

Intervention

SCIENTIFIC NOTATION

Home

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

SCIENTIFIC NOTATION

Scientiﬁc Notation 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

Writing in Scientiﬁc Notation 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

Estimating Numbers and Scientiﬁc Notation 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

Comparing Numbers in Scientiﬁc Notation 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

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

Scientiﬁc Notation

Can be done independently

SCIENTIFIC NOTATION

Home

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

SCIENTIFIC NOTATION

Scientiﬁc Notation 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 92

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

SCIENTIFIC NOTATION

Home

What does mastery look like?

I can estimate and express very large or very small numbers using scientiﬁc notations.

I can use the magnitude of quantities to compare numbers written in scientiﬁc notation to determine how many times larger (or smaller) one number written in scientiﬁc notation is than another.

SCOPE 1

Operations with Scientiﬁc Notation Scope Introduction SCOPE SUMMARY

Student Expectations

8.NR.2.4 Add, subtract, multiply and divide numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Interpret scientific notation that has been generated by technology (e.g., calculators or online technology tools).

Students will have ample opportunities to perform basic operations containing numbers in scientific notation. They will use addition, subtraction, multiplication, and division to evaluate real-life and mathematical problems containing values written in scientific notation. Students will be manipulating the decimal locations to establish the same exponential values when adding and subtracting numbers in scientific notation. They will be applying the laws of exponents when multiplying and dividing numbers in scientific notation.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

Students have interacted with exponents in Grade 6 when they began writing and evaluating numerical expressions containing whole-number exponents. They explored how whole-number exponents represent the repetitive multiplication of a given number. Students began to complete whole-number exponents and multistep real-life mathematical problems containing rational numbers in Grade 7. With this skill, students evaluate expressions containing numbers in various forms combined with basic operations. In the prior scientiﬁc notation scope in Grade 8, students have learned how to write, estimate, and compare numbers in scientiﬁc notation.

In high school, students will be working with various scales on all numerical levels including very large whole numbers and very small decimals. Students will be using numerical conversions to change the values in order to understand and guide the completion of multistep real-life or mathematical problems.

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

use numbers expressed in single digit times an integer power of 10 to estimate large or small quantities.

•

express how much one is than the other.

•

identify two truths and a lie by reading statements.

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 how many times greater one value is than another.

•

use values expressed in scientiﬁc notation.

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. 94

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Adding and Subtracting with Scientiﬁc Notation

Explore 2

Explore 1

EXPLORE ACTIVITIES

In this exploration, groups of students will be tasked with helping an electronics company complete orders by determining the sizes of the combined data and how much free space will be on each hard drive after the data is combined. Students will: •

In this exploration, students will be tasked with assisting a landscaping company determine the total area of each of the golf courses they have as customers. Students will: •

practice adding and subtracting numbers written in scientiﬁc notation both with and without common powers.

practice multiplying numbers written in scientiﬁc notation.

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

Multiplying with Scientiﬁc Notation

OPERATIONS WITH SCIENTIFIC NOTATION

Home

Dividing with Scientiﬁc Notation In this exploration, students will help compare numbers written in scientiﬁc notation about different numerical facts for a trivia contest. Students will: •

practice dividing numbers written in scientiﬁc notation.

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

OPERATIONS WITH SCIENTIFIC NOTATIONS

Operations with Scientiﬁc Notations 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 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. 8.NR.2.3 Use numbers expressed in scientiﬁc notation to estimate very large or very small quantities, and to exprress how many times as much one is than the other.

Materials

Preparation

Printed •

•

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

•

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

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

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

A is the lie because 6 × 104 is equal to 6(10 × 10 × 10 × 10), 6 times 10,000, which is 60,000.

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.

OPERATIONS WITH SCIENTIFIC NOTATIONS

Home

FACILITATION TIP Have students correct the lie statement before turning in their paper. They can do this by either changing the exponent to a 3 or changing the 6,000 to 60,000. FACILITATION TIP Challenge students to create their own set of Two Truths and a Lie. They can share with another group to identify the lie.

OPERATIONS WITH SCIENTIFIC NOTATIONS

Operations with Scientiﬁc Notations Hook – Won’t You Be My Neighbor? ACTIVITY PREPARATION Students will determine how many times greater one value is than another by using values expressed in scientiﬁc notation.

Materials

Preparation

Printed •

• •

1 Won’t You Be My Neighbor? (per class)

Reusable •

•

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Won’t You Be My Neighbor? 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 do you learn about when you study astronomy?; 2) What do you know about the distance of the planets from the Sun?; 3) What unit of measure is used to identify these large distances?

FACILITATION TIP Challenge activity: Challenge students to see what other units they could express these distances in (examples: meters, centimeters). How would they be written in scientiﬁc notation? In standard notation?

4. 5.

6. 98

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: Archie is working on a project in his astronomy class. He wants to figure out how far Earth is from the Sun compared with how far Earth is from the farthest planet from the Sun, Neptune. He has decided to research the distances and record them in kilometers, which requires him to use scientific notation so the numbers are not too large and awkward. He wonders how many times farther away Earth is from its farther solar system neighbor than it is from its closer neighbor. 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 Archie is using scientific notation. I wonder how large the numbers are that are being described with scientific notation. How will Archie compare the two numbers by using scientific notation? I can use math to determine how many times farther away Neptune is from Earth than the Sun is. Project Won’t You Be My Neighbor? Explain to students that Archie has researched the distances (in kilometers) from Earth for both the Sun and Neptune. Now he wants to know not only the difference in distances but also how many times farther Neptune is from Earth than the Sun is. Discuss the following questions: a.

DOK-1 Which celestial body is farther from Earth? Neptune

DOK-1 How far is the Sun from Earth? 1.49 x 108 km

DOK-1 How far is Neptune from Earth? 4.41 x 109 km

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and restate the problem. Refer to Won’t You Be My Neighbor? and discuss the following questions: a.

DOK-2 What is the process to determine how many times farther from Earth Neptune is than the Sun? First set up a ratio comparing the distance from Earth to Neptune to the distance from Earth to the Sun. Divide the decimal numbers between one and ten first and find the quotient. Then, divide the powers of ten and find the quotient. Then, find the product of both quotients.

DOK-1 What is the quotient of the decimal numbers between one and ten? 4.41 ÷ 1.49 = 2.96

DOK-1 What is the quotient of the powers of ten? 109 ÷ 108 = 101 = 10

DOK-1 What is the solution? How many times farther away is Neptune from Earth than the Sun is from Earth? Neptune is 29.6 times farther from Earth than the Sun is.

FACILITATION TIP Check for understanding: Do you need to review math terms? Do students know what a “quotient” is? Do you need to deﬁne “powers of ten”?

OPERATIONS WITH SCIENTIFIC NOTATIONS

Home

OPERATIONS WITH SCIENTIFIC NOTATIONS

Operations with Scientific Notations Explore 1 – Adding and Subtracting with Scientific Notation ACTIVITY PREPARATION Students will practice adding and subtracting numbers written in scientific notation both with and without common powers.

Standards for Mathematical Practice • •

MP.5 Use appropriate tools strategically. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

•

1 Student Journal (per student) 1 Set of Work Order Cards (per class) 1 Exit Ticket (per 2 students)

• •

•

Plan to divide the class into groups of three or four to complete this activity. Print a Student Journal for each student. Print a set of Work Order Cards for the class. Cut out the cards along the dashed lines (HDD 1, HDD 2, and New HDD will be on each card), and place the cards in a random order around the room. 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.

PROCEDURE AND FACILITATION FACILITATION TIP

Before reading the scenario, ask the class 1) What do people do when they need more storage space on their computers? After reading the scenario, ask the class 2) Why would Plus/Minus Electronics need to know both the amount of combined data and the free space remaining on the hard drive after combining data?

2. 3.

FACILITATION TIP Alternatively, you could print out a copy for each group and have students work at their desks. FACILITATION TIP This is a good opportunity to review combining like terms. Let students problem-solve to ﬁgure out how to modify the numbers to make them be like terms so they can be combined. FACILITATION TIP Look out for students who are not following instructions and adding all three numbers. They should be adding the numbers from HDD 1 and HDD 2 together, then subtracting that total from the new HDD value.

100

4. 5. 6.

Read the following scenario to the class: Plus/Minus Electronics has a special service for people who need more storage space on their computers or external hard drives. They will combine the hard drives of people’s computers and external hard drives so they have enough room to store all their data. They record all data sizes in bytes. Some of their new work orders are placed around the room. Help Plus/Minus know the sizes of the combined data as well as how much free space will be on the hard drive after the data is combined. Give a Student Journal to each student. Explain to students that they will collaborate with their groups to add hard drive 1 and hard drive 2 to ﬁnd the total amount of hard-drive space used. Then, students will subtract the total space used in hard drives 1 and 2 from the new hard-drive space. Explain to students how much time they will have at each Work Order Card and how they will know when to rotate to the next card. Allow students to rotate around the room, completing the work orders with their groups and recording the results on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How do you know if a number is written in correct scientiﬁc notation? Correct scientiﬁc notation starts with a number greater than or equal to 1 and less than 10 that is raised to a power of 10.

DOK-1 What happens when you add or subtract numbers in scientiﬁc notation that are raised to the same power? When adding or subtracting numbers raised to the same power, you can just add or subtract the ﬁrst number and keep the power the same.

DOK-1 What happens when you add or subtract numbers in scientiﬁc notation that are raised to different powers? When adding or subtracting numbers raised to different powers, you will need to change the exponent so they are raised to the same power. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

DOK-1 If numbers are raised to different powers, what happens to the ﬁrst number or coeﬃcient when you change the power? If the exponent is getting smaller, you will shift the decimal that many places to the right. If the exponent is getting larger, you will shift the decimal that many places to the left.

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

Math Chat DOK-1 If you’re trying to add two numbers that are raised to a different power of ten, how can you make it so you can add those numbers together? You need to change the exponent so they are both raised to the same power of ten. • DOK-2 Do you prefer making the powers of ten equal by making the exponent larger or smaller? Explain. I prefer making the exponent larger because that means I’m adding a decimal to a whole number. • DOK-1 What happens when you add or subtract numbers written in scientific notation but the answer is not written in correct scientific notation? You need to write it in correct scientific notation. You do this by moving the decimal to make the coefficient a number greater than or equal to 1 and less than 10 and then changing the exponent accordingly. If the decimal moves to the right, the exponent gets smaller. If the decimal moves to the left, the exponent gets larger. • DOK-2 Why is it important to be able to add and subtract numbers written in scientific notation? Numbers written in scientific notation are either very large or very small. Being able to add or subtract them written this way saves time in doing calculations as you don’t need to write all the zeros. •

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.

OPERATIONS WITH SCIENTIFIC NOTATIONS

Home

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.

101

OPERATIONS WITH SCIENTIFIC NOTATIONS

Operations with Scientific Notations Explore 2 – Multiplying with Scientific Notation ACTIVITY PREPARATION Students will practice multiplying numbers written in scientific notation.

Standards for Mathematical Practice • •

MP.5 Use appropriate tools strategically. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

•

1 Student Journal (per student) 1 Set of Course Model Cards (per class) 1 Exit Ticket (per student)

• •

Plan to put 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 Course Model Cards for the class. Place the cards in a random order around the room. Depending on class size, you may need to print out 2 sets of cards. If desired, print them on card stock, and laminate them for future use.

PROCEDURE AND FACILITATION FACILITATION TIP

After reading the scenario, ask the class 1) How might knowing the sizes of lawns Greyson’s Lawn Service cuts be useful to them?; 2) What is area?; 3) What formula is used to ﬁnd that area of a location?

2. 3.

FACILITATION TIP Students may need a quick review on area versus perimeter. Review the formulas, and ensure understanding of why the area formula applies in this situation. FACILITATION TIP Watch out for students who are struggling with multiplying the decimal portion. You may need to review multiplying decimals.

4. 5. 6.

FACILITATION TIP Instead of having the students rotate around the room, you could hand each group one card. After the time limit, have the students pass their card to the next group.

7. 102

Read the following scenario to the class: Greyson’s Lawn Service is putting together a list of their clients and the sizes of lawns that they cut. The majority of their services are performed on golf courses. Help Greyson determine the total area of each golf course they have as customers. Give a Student Journal to each student. Explain to students that they will collaborate with their groups to multiply numbers written in scientific notation. Some numbers provided are not in scientific notation, but students will need to convert all numbers to scientific notation to solve. Explain to students how much time they will have at each golf-course question and how they will know when to rotate to the next card. Allow students to rotate around the room, finding the area of each golf course with their groups and recording the results on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What do you do with the numbers written in decimal notation? The numbers written in decimal notation will need to be changed to scientiﬁc notation so they can be multiplied together.

DOK-1 Scientiﬁc notation always has a base of ten. Based on the properties of exponents, what does that mean we do with the exponents when multiplying? The properties of exponents tell us that when we multiply with like bases, we can add the exponents together.

DOK-1 How do you know when a number is written in scientific notation? To be in scientific notation, the coefficient needs to be greater than or equal to 1 and less than 10.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Math Chat DOK-1 How do you correct any answer that is not written in the correct scientific notation? You move the decimal to make the coefficient a number greater than or equal to 1 and less than 10. Then, change the exponent according to how many places you moved the decimal point. If the decimal moves to the right, the exponent gets smaller. If the decimal moves to the left, the exponent gets larger. • DOK-1 Why do you need to change the numbers written in decimal notation to be in scientific notation? The numbers should be in the same form so they can be multiplied effectively. • DOK-2 Why is it important to be able to multiply in scientific notation? Numbers written in scientific notation are either very large or very small. Being able to multiply them written this way saves time in doing calculations, as you don’t need to write all the zeroes. • DOK-3 Where might you see numbers written in scientific notation being multiplied in the real world? Finding the area of farmland •

STEMscopes Tip Transition students into the current concept by meeting them at their level with the Hook activity, found in the Engage section. These real-world scenario-based activities frame the overall learning throughout the scope and serve as both an introduction and concluding aspect of each concept. The Hook fosters personal growth.

Post-Explore 1. 2. 3.

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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.

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OPERATIONS WITH SCIENTIFIC NOTATIONS

Operations with Scientific Notations Explore 3 – Dividing with Scientific Notation ACTIVITY PREPARATION Students will practice dividing numbers written in scientific notation.

Standards for Mathematical Practice • •

MP.5 Use appropriate tools strategically. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

•

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

• •

Reusable •

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 Building Volume Cards for each group of students. Cut out and place cards in a resealable bag for each group. 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 is a trivia contest? After reading the scenario, ask the class 2) How do you write numbers in scientiﬁc notation?; 3) How is having numbers written in scientiﬁc notation helpful when comparing them? FACILITATION TIP For students who may need an easier division problem, consider using 2.0 x 102, or 200, as the divisor. This will allow students to practice the same concepts, but with simpler numbers. FACILITATION TIP Check if students recall how to divide the power of ten portion. You may need to review dividing exponents with the same base.

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2. 3. 4.

5. 6.

Read the following scenario to the class: Daphne hosts a weekly trivia contest. This week’s theme is big buildings. Daphne is gathering numerical facts to use for the trivia contest and notices the values can also be written in scientific notation. Daphne also wonders how much bigger or smaller some of these numerical facts are compared to the other facts gathered. Help Daphne compare numbers written in scientific notation! Give a Student Journal to each student. Give a set of Building Volume Cards to each group. Explain to students that they will collaborate with their groups to compare sizes of different buildings or rooms. One teammate will shuffle the Building Volume Cards and place them face down in a pile. Students will draw two cards from the pile and record the name of each building or room drawn. Then, students will divide their values written in scientific notation to compare the two buildings/ rooms. Students will repeat this process 3 more times to create 4 pairs of values. Explain that any coefficient that has a large decimal place can be rounded to the closest thousandth. Students will use the ratio provided to divide the numbers written in scientific notation. Some numbers provided are not in scientific notation, but students will need to convert all numbers to scientific notation to solve. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 How do you know when a number is written in scientific notation? To be in scientific notation, the coefficient needs to be greater than or equal to 1 and less than 10.

DOK-1 How do you convert a number written in decimal notation into scientific notation? Give an example. You start with a coefficient greater than or equal to 1 and less than 10 and count how many places you move the decimal. For example, for the number 1,234, you would start with 1.234 as the coefficient. Since you are moving the decimal 3 places to the left, your exponent would be 103. © Accelerate Learning Inc. - All Rights Reserved

8. 9.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How can you determine how much larger one number is compared to another? Student responses will vary. I can divide the larger value by the smaller value to determine how much bigger the larger value is compared to the smaller value.

DOK-2 How is dividing numbers in scientific notation different from dividing numbers in decimal notation? It’s similar, but scientific notation can be easier. You divide the coefficients and the exponents separately. To divide the exponents, since the bases are the same, you can just subtract the exponents.

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

DOK-2 Do you think it’s easier to divide numbers written in decimal notation or scientific notation? I prefer dividing numbers written in scientific notation, as I only have to worry about dividing a few digits, and then I just subtract the exponents. • DOK-1 Why do you need to change the numbers written in decimal notation to be in scientific notation? The numbers should be in the same form so they can be divided effectively. • DOK-2 Why is it important to be able to divide in scientific notation? Numbers written in scientific notation are either very large or very small. Being able to divide them written this way saves time in doing calculations as you don’t need to write all the zeroes. • DOK-3 Where might you find numbers written in scientific notation being divided in the real world? Comparing large distances such as the distance between stars •

Post-Explore

2. 3. 4.

Acceleration

FACILITATION TIP Students who ﬁnish early or have extra time can place all the cards in order from least to greatest or greatest to least. FACILITATION TIP

Math Chat

Intervention

Challenge students to attempt the division using the smaller number divided by the larger number. How does this quotient compare to the quotient they found during the activity? Does the scientiﬁc notation have a positive or negative exponent?

OPERATIONS WITH SCIENTIFIC NOTATIONS

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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.

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OPERATIONS WITH SCIENTIFIC NOTATION

Operations with Scientiﬁc Notation 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

Adding and Subtracting with Scientiﬁc Notation 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

Multiplying with Scientiﬁc Notation 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.

Dividing with Scientiﬁc Notation 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

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

Notes __________________________________________________________________________________________________________________________________________________

OPERATIONS WITH SCIENTIFIC NOTATION

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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

OPERATIONS WITH SCIENTIFIC NOTATION

Operations with Scientiﬁc Notation

3 108

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

What does mastery look like?

OPERATIONS WITH SCIENTIFIC NOTATION

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I can subtract numbers expressed in scientiﬁc notation.

I can multiply numbers expressed in scientiﬁc notation.

I can divide numbers expressed in scientiﬁc notation.

I can use place value reasoning to multiply by a power of 10.

I can interpret scientiﬁc notation generated by technology.

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SCOPE 1

Solve Equations Scope Introduction SCOPE SUMMARY

Student Expectations

8.PAR.3.1 Interpret expressions and parts of an expression, in context, by utilizing formulas or expressions with multiple terms and/or factors. 8.PAR.3.2 Describe and solve linear equations in one variable with one solution (x = a), inﬁnitely many solutions (a = a), or no solutions (a = b). Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). 8.PAR.3.3 Create and solve linear equations and inequalities in one variable within a relevant application.

In this scope, students should continue to develop their ability to apply properties of operations to solve multistep equations. Students should be able to use models, construct arguments for equality, and explain why an equation or model is equivalent when performing operations to isolate a variable. They should be able to determine and explain constraints for equations that represent a contextual situation, including the reasonableness of a solution. Students should make connections between solving equations with numerical coefficients and solving equations with variable coefficients. They should be able to rearrange formulas, including finding the inverse, to highlight a variable and explain the steps used. Students should make connections to solving equations and rearranging formulas. They will use a conceptual approach to inverses and verify they are inverses using substitution.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grade levels, students created and solved one variable linear equations, learned and applied the properties of operations, and reasoned with inequalities. Students used a variety of formulas, including but not limited to simple interest, temperature conversions, and perimeter. They represented and solved problems that involved proportional relationships. This included ﬁnding the constant of proportionality. Students also used the distributive property to make equivalent expressions and solve equations with variables on both sides.

Solving equations and ﬁnding inverses is a central topic across all math courses. In Algebra I, students will solve and explore quadratic expressions, quadratic equations, exponential equations, and others. In Algebra II, students will solve and explore inverses of cubic functions, rational functions, trigonometric functions, exponential and logarithmic functions, and others.

8.PAR.3.4 Using algebraic properties and the properties of real numbers, justify the steps of a one-solution equation or inequality.

8.PAR.3.6 Use algebraic reasoning to ﬂuently manipulate linear and literal equations expressed in various forms to solve relevant, mathematical problems.

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

apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coeﬃcients.

•

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

Hook

Accessing Prior Knowledge

8.PAR.3.5 Solve linear equations and inequalities in one variable with coeﬃcients represented by letters and explain the solution based on the contextual, mathematical situation.

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 value of a variable in an equation.

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

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

One-Variable Equations In this exploration, students will be tasked with helping a research company evaluate and solve equations. Students will: •

Explore 2

Explore 1

EXPLORE ACTIVITIES

solve linear equations with one variable and determine if the equations have one solution, inﬁnite solutions, or no solutions.

•

determine if given values are valid solutions to equations.

•

predict if an equation will have one value for x,, many values for x, or no values for x.

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Write, Model, and Solve Two-Step Equations In this exploration, students will write, model, and solve one- and two-step equations involving variables on one side of the equations. Students will: •

write and solve equations.

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

Write, Model, and Solve Multistep Equations In this exploration, students will write, model, and solve multistep equations, including equations with variables on both sides. 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.

In this exploration, groups of students will analyze the information provided to determine how much two brothers charged for walking dogs. Students will:

•

analyze each scenario to identify the variable.

•

write, model, and solve multistep equations.

•

write an equation.

•

analyze strategies for solving.

•

model the problem using the balanced hanger to identify the 3-digit combination of the new vault.

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

Solve Multistep Equations

Solve Literal Equations In this exploration, students will be tasked with escaping the Algebra Dungeon to solve the puzzle dials at each level using their knowledge of solving algebraic equations. Students will: •

solve literal equations for a given variable.

After solving the scenario, students discuss learning with the class, complete an exit ticket for assessment; then, revitist the hook to solve.

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SOLVE EQUATIONS

Solve Equations 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 expressions 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.

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7.PAR.2.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coeﬃcients.

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.

Page 1: Choice C does not belong. Choices A, B, and D are all equivalent, but C equals 2x + 24.

Page 2: Choice B does not belong. A, C, and D are all equivalent, but choice B would equal 3x + 63.

FACILITATION TIP As students work, look out for students ignoring the parentheses. Remind students of the correct order of operations. FACILITATION TIP Some students may choose A because it’s the only one without parentheses. Explain that they need to solve the problems to ﬁnd which one isn’t like the others.

Identifying Misconceptions • • •

Students may compute their math in the wrong order. They must be familiar with the order of operations. Students may work from left to right instead of following the order of operations. Students may forget that expressions within parentheses must be computed ﬁrst.

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Solve Equations Hook – Let Them Eat Cupcakes ACTIVITY PREPARATION Students will determine the value of a variable in an equation.

Materials

Preparation

Printed •

• •

1 Let Them Eat Cupcakes (per class)

Reusable •

•

1 Phenomena Video (per class)

Plan to show the video. Prepare to project the Let Them Eat Cupcakes 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 are food prices set?; 2) How is the proﬁt a food item makes calculated?; 3) Do you think discounted food makes the same proﬁt as when it was on sale for full price? Explain your reasoning.

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: Alisa works for her mom’s cupcake company during the summer and on breaks from school. The day after a holiday, fancy jumbo cupcakes that did not sell go on sale at a 50% discount. Alisa’s mom wants Alisa to figure out what the profit is for a dozen of those special cupcakes after the discount. She gives Alisa the initial equation including the variable y and asks Alisa to solve it for the profit. 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 Alisa is finding profit. I wonder what the y stands for in the equation. Will there be more than one variable in the equation? I can use math to solve the equation and find the profit from selling a dozen cupcakes at a 50% discount. Project Let Them Eat Cupcakes. Notes

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Explain to students that Alisa has made a mistake solving the equation and trying to find the profit of the dozen discounted cupcakes. Her mom has asked her to go back through her work, find where she made her mistake, and correct the mistake to find the real profit. Discuss the following questions: a.

DOK-1 How many different variables do you see in the equation? There is only one variable, y.

DOK-1 What else do you notice about the given equation? Answers may vary. There are both decimals and fractions in it.

SOLVE EQUATIONS

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Complete the Explore activities.

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and restate the problem. Refer to Let Them Eat Cupcakes, and discuss the following questions: a.

3 DOK-1 What is the ﬁrst step Alisa performs? She changes the fraction __4

DOK-1 What is the next step Alisa takes? She multiplies both sides of the equation by 100 to get rid of the decimals.

DOK-1 Can you identify Alisa’s mistake? What did she do wrong? Yes, she made a mistake when she subtracted 400 from each side. She was supposed to add 400 to both sides to start isolating the variable y.

DOK-1 What is the incorrect value of y?? What is the correct value of y? The incorrect value was 36. The correct value is 68.

to a decimal of 0.75, so the units will be the same.

e. DOK-1 What does that value represent? It represents the proﬁt made off of one dozen fancy jumbo cupcakes discounted by 50% on the day after a holiday. f.

DOK-1 What can you tell about the price of the cupcakes? The cupcakes are expensive.

FACILITATION TIP Let half the class solve the problem by converting to a decimal, but have the other half solve by converting to a fraction. Have them compare answers.

FACILITATION TIP Ask students to calculate how much proﬁt they make on one dozen cupcakes when they are not on sale. The costs should remain the same. Only the sale price is changing by 50%, so the proﬁt should be double, which would be $136.

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Solve Equations Explore 1 – One- Variable Equations ACTIVITY PREPARATION Students will solve linear equations with one variable and determine if the equations have one solution, inﬁnite solutions, or no solutions.

Standards for Mathematical Practice • • • •

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

Materials

Preparation

Printed • •

•

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

•

Plan to separate the class into groups of 3 or 4 students to complete the 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) Where would math be used in a research company? After reading the scenario, ask the class 2) What is a solution to an equation?; 3) How can you determine if a solution is valid? FACILITATION TIP Some students may not be familiar with the term “solution” to an equation. This might be a good time to remind them that the solution to an equation is the answer, or the value of x in an equation.

2. 3.

Read the following scenario to the class: For week two of summer savings, Erika and Tammi decided to work for Tammi’s aunt. Tammi’s aunt owns a research company. This week’s project is determining solutions to equations. Help the girls determine if given values are valid solutions to given equations. Give a Student Journal to each student. Explain to students that they will need to substitute the values of 0, 5, and 10 for x in each equation Tammi and Erika are working with. Then, students will shade in the boxes of the equations that are true for each value of x. Monitor and assess students as they are working by asking the following guiding questions:

FACILITATION TIP Students may use a highlighter or may just circle the boxes that are true.

FACILITATION TIP For early ﬁnishers, have them solve each of the equations. For the ﬁrst one, show them how to denote the solution as “all real numbers.” For the last one, show them the notation for “no solution.”

5. 6.

DOK-1 What operation did you solve ﬁrst? After substituting the value for x, I multiplied the coeﬃcient by the value I substituted.

DOK-1 What does it mean to substitute x = 0? It means to replace the x with 0. Then, we will solve the equation to see if it is true.

DOK-1 What does 6 = −4 mean when you substitute 0 for x in the equation x + 6 = 3x – 4? It means the value substituted for x is not a solution to this equation. 0 cannot be a solution to the equation.

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

Math Chat •

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DOK-2 What did you notice about the equations? In the ﬁrst equation, all of the values for x were solutions to the equation. In the second equation only x = 5 was a possible solution to the equation. None of the values were solutions for the third equation. DOK-2 Does an equation have to have only one solution? No, an equation can have more than one solution. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

DOK-2 How can you tell if an equation has more than one solution? The terms on both sides of the equation are the same. • DOK-2 Do you think an equation can have no solution to it? Yes, I think this is possible. •

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Part II 1.

4. 5.

Read the following scenario to the class: Tammi’s aunt wants the girls to help develop an app to determine how many solutions there are for different equations. Help the girls solve different equations to predict if the equation will have only one value for x, many values for x, or no values for x. Explain to students that they will collaborate with their groups to solve each equation for x.. Then, each group will discuss to make a prediction if the equation will have only one solution for x,, many solutions for x,, or no solutions for x. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 Why do you think 4 4xx – 6 = 4 4xx has no solution? When we solved, we got 0 = 6, which is not true. Therefore, we think there will be no solutions for this equation.

DOK-1 Why do you think −12x −12x + 15 = 3x – 30 will have only one solution for x? When we solved this equation, we got that x = 3. Therefore, we think 3 will be the only solution for this equation.

DOK-1 Why do you think x + 6 + 2x = 3x + 6 will have many solutions? When we solved this problem, we got that x = x. Therefore, we believe there are many different values we could substitute for x in this equation.

Allow students time to complete Part II and the reﬂection questions on their Student Journals. 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 does the x in an equation represent?; 2) What can you do to help you predict whether an equation will have one value, many values, or no values for x?; 3) When would an app that determines how many solutions there are for different equations be useful? FACILITATION TIP To ensure each group member is working on the problems, you could assign each student a problem to present to their group. FACILITATION TIP Challenge early finishers to come up with another equation that will have no solution. Let them explain why it has no solution. FACILITATION TIP Challenge early finishers to come up with another equation that will have infinite solutions. Ask them what the equation to simplify to.

Math Chat •

•

DOK-2 Were there clues that helped you see what type of equation it was before solving it? Yes, if we could see that both sides of the equation were the same or the terms could be combined to be the same on both sides, then there would be many solutions. If both sides of the equation showed an untrue statement (like 4x – 6 = 4x), then there would be no solution. DOK-2 How could you tell if there will be no solutions? There will be no solutions if when you solve you get an untrue statement like 6 = 0.

Explain the following to the class: If a number equals a different number, then mathematicians say there are no solutions to the equation. •

DOK-2 How could you tell if there will be only one solution? There will be only one solution when you solve and you get x equals a value, like x = 12.

Explain the following to the class: If x equals one number, then there is only one solution to the equation. •

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

DOK-2 How could you tell if there will be many solutions? There will be many solutions when you solve the equation and you get x = x.

Explain the following to the class: If x = x, then there are inﬁnitely many solutions to the equation. Mathematicians categorize all equations as having either one solution, inﬁnitely many solutions, or no solutions. All equations will fall under one of these three types.

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Solve Equations Explore 1 – One- Variable Equations Post-Explore 1. 2. 3.

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

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

SOLVE EQUATIONS

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SOLVE EQUATIONS

Solve Equations Explore 2 – Write, Model, and Solve Two-Step Equations ACTIVITY PREPARATION Students will write, model, and solve one- and two-step equations involving variables on one side of the equations.

Standards for Mathematical Practice • • • •

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

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Algebra Equation Mat (per group) 1 Exit Ticket (per 2 students)

• • •

Reusable • •

1 Set of algebra tiles (per group) 1 Set of colored pencils (per group)

•

Separate the class into groups of 2 or 3 students. Print a Student Journal for each student. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student has one. Print an Algebra Equation Mat for each group. Organize a set of algebra tiles and colored pencils for each group. Prepare an Algebra Equation Mat and set of algebra tiles to display. Virtual algebra tiles or traditional algebra tiles can be displayed. 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 Part I FACILITATION TIP Project this scenario. Give students time to read it alone, and then read it with a partner. Coach students to ﬁnd the essential phrases and values. FACILITATION TIP Project these questions and record or display the answers as you review the algebra tiles.

2. 3.

Read the following scenario to the class: Lily is having a 5-day bake sale at her school to raise funds for a children’s health organization. She decided to sell different items on each day of the fundraiser. She decided to track how much she spent on supplies and how much proﬁt she earned each day. Help Lily determine how much of each baked good she sold each day of the fundraiser. Display an Algebra Equation Mat and a set of algebra tiles, or display the virtual algebra tiles. Brieﬂy review how to use the algebra tiles with students by asking the following questions: a.

DOK-1 What does the rectangle represent? A rectangle represents the value of one x.

DOK-1 What is x? This is a variable. An x represents an unknown amount.

DOK-1 What does a small square represent? A small square represents the value of one.

DOK-1 How can you represent 2? You can use 2 small squares.

e. DOK-1 How can you represent 2x? 2 You can use 2 rectangles. 120

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 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 not shaded, the value is positive. Negative values for x and one are represented with shaded rectangles and squares.

DOK-1 What happens mathematically when you combine an unshaded and a shaded tile of the same size (for example, an unshaded and a shaded rectangle or unshaded and shaded square)? You create a zero pair, which means the value is equal to zero.

h. DOK-1 What is the purpose of the algebra tiles? You use the tiles to represent the values given and to help solve the problem visually. i. DOK-1 What is the purpose of the Algebra Equation 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. 4. 5. 6.

Give a Student Journal to each student. Give a set of Algebra Tiles, an Algebra Equation Mat, and a set of colored pencils to each group. Explain to students that they will work with their groups to write and solve the equations found on their Student Journals using the algebra tiles and Algebra Equation Mat. Students will record their results on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What should you do ﬁrst when analyzing each problem? First, you have to read the problem carefully. Then, you write an equation to solve the problem.

DOK-2 Would we model subtracting 12 from both sides with shaded or unshaded tiles? Why? We would use shaded tiles to model subtracting 12 from both sides because shaded tiles represent negative numbers.

DOK-2 Would we model adding 8 on both sides with shaded or unshaded tiles? Why? We would use unshaded tiles to add 8 on both sides because unshaded tiles represent positive numbers.

DOK-3 What equation did you write for the brownie sales? Justify why you think the equation you wrote is correct. Answers may vary. The scenario says that brownies are one dollar each, so I knew the variable and coeﬃcient would be 1b. It says she received a tip. That would increase the money she made, so I knew that should be +4. Since she made $12 that day, I knew 12 would be the output.

e. DOK-2 How do you show your solution? First, solve the equation using the algebra tiles and Algebra Equation Mat. Then, show your solution by drawing out what you did on the mat or using numerical representations. 8. 9.

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

Intervention

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FACILITATION TIP Reassure students that being ﬂuent with the algebra tiles will support their problemsolving skills in Algebra and beyond. Students may be resistant to hands-on and visual models. Remind them that even if they don’t need them for this particular set of problems, the physical representations provide support for visual thinking. FACILITATION TIP Clarify expectations for how students are to draw their models, show their thinking, and record solutions on the Student Journal. FACILITATION TIP The Student Journal includes ﬁve pages. Consider giving students pages 1–4 and then projecting page 5 as a guide for a class discussion. FACILITATION TIP If needed, model how to carefully read a problem. Highlighting important values or circling/underlining key phrases can guide students to writing accurate equations.

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.

Math Chat • • •

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 Equation Mat, or you can use an algebraic representation on paper.

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Solve Equations Explore 2 – Write, Model, and Solve Two-Step Equations DOK-3 Which way—algebra tiles or algebraic equations—did you prefer using to solve the equations? Why? 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. • FACILITATION TIP Consider allowing students to vote for their preferred method. However, consistently remind them that fluency with models contributes to powerful cognitive skills.

Part II: Property Management

FACILITATION TIP

Project this scenario in print and conduct a careful read aloud with the class. Model highlighting the essential phrases and values.

2. 3.

5. 6.

Read the following scenario to the class: Lily wrote and solved an equation to represent her sales during days 4 and 5 of the bake sale. Help Lily ﬁnalize her fundraising by analyzing her work and verifying her solutions for days 4 and 5. Students should still have the algebra tiles, the Algebra Equation Mats, and their Student Journals. Explain to students that they will work with their groups to analyze Lily’s equations and strategies for solving her bake sales for days 4 and 5 and record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 How did Lily isolate 1.50xx in step 2 of her day 4 sales? Lily added 25.75 to both sides of the equation to isolate 1.50x.

DOK-2 How did Lily get 66 as her answer for her day 5 sales? First, Lily subtracted 5.50 on both sides, then she multiplied both sides by 5, and lastly, she divided both sides by 3 to get the ﬁnal answer.

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

Math Chat FACILITATION TIP

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Remind students that they are basically “undoing” the order of operations. •

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FACILITATION TIP This question provides another chance to show students that they are doing the order of operations in reverse.

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DOK-2 What are the rules of solving a two-step equation? To isolate the variable on one side of the equation to determine its value, we first add or subtract on both sides of the equation and then multiply or divide on both sides to get the final solution to the two-step equation. DOK-2 What operation will you use to isolate a variable if 2 is being added to it? I will use the inverse operation of addition; that is subtraction. I will subtract 2 from both sides to isolate the variable. DOK-2 What operation will you use to isolate a variable if it is being multiplied by 4? I will use the inverse operation of multiplication; that is division. I will divide both sides by 4 to isolate the variable. DOK-2 If there are addition and division going on in a two-step equation, which operation will you solve first to isolate the variable? I will use the inverse operation to solve addition. I will subtract from both sides and then use the inverse operation of division to solve for multiplication. DOK-2 If there are multiplication and subtraction going on in a two-step equation, which operation will you solve first to isolate the variable? I will use the inverse operation to solve subtraction. I will add on both sides and then use the inverse operation of multiplication to solve for division.

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Clarify criteria for success on this Exit Ticket. Be clear about how students are to write the solution and its veriﬁcation.

2. 3.

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

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Acceleration

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Solve Equations Explore 3 – Write, Model, and Solve Multistep Equations ACTIVITY PREPARATION Students will write, model, and solve multistep equations, including equations with variables on both sides.

Standards for Mathematical Practice • • • •

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

Materials

Preparation

Printed • •

• •

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

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

PROCEDURE AND FACILITATION

FACILITATION TIP Project a print version of this scenario. Provide students time to read it individually, and then with a partner. Have a student volunteer read it aloud and model how to locate the essential math phrases and values. FACILITATION TIP Depending on your students’ prior experience or recent success with hanger models, take time to review. Encourage students to continue to use these models even when they can intuit solutions. Fluency with physical models contributes to growth in cognitive skills needed for more complex algebra later.

Part I 1.

2. 3.

Read the following scenario to the class: William, Henry, and Ethan work at Greengotts bank. They have been assigned a task to store all of the bank’s valuables, records, and documents in a new vault, as the ﬁrst vault is almost full. Unfortunately, they have forgotten the 3-digit combination that opens the new vault. William remembers that the way the bags of coins and the bands of money are organized in the original vault is a clue to the 3-digit combination needed to open the new vault. The employees decide to check each wall of the vault to solve for a digit to the combination lock. Give a Student Journal to each student. Explain to students that they will work with their groups to analyze each scenario to identify the variable, write an equation, and model the problem using the balanced hanger to identify the 3-digit combination of the new vault. Students will record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What should you do ﬁrst when analyzing each problem? First, we should read the problem carefully. Then, we should write an equation to solve the problem.

DOK-2 What does one rectangle represent in the key? One rectangle represents the value of a bag of gold coins.

DOK-3 How can you represent 5b 5 using the rectangles? I can use ﬁve rectangles.

DOK-2 What does one square represent in the key? One square represents one band of $10 bills.

e. DOK-3 How can you represent 6 bands of $10 bills using the squares? I can use six squares. f.

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DOK-2 Would we model adding 8 bands of $10 bills on both sides with rectangles or squares? Why? I would use squares to add 8 bands of $10 bills on both sides because each square represents a band of $10 bills. © Accelerate Learning Inc. - All Rights Reserved

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Explain

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Intervention

Acceleration

DOK-3 How do you know if the hanger is balanced? If we have equal weights on the ends of a hanger, the hanger will be in balance. If there is more weight on one side than the other, the hanger will tilt to the heavier side.

h. DOK-2 How do you show your solution? First, model the equation using the balanced hanger model. Then, show our solution by using an algebraic representation. 5. 6.

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

Math Chat 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 hanger model, or you can use numbers. • DOK-3 Which way— hanger models or numbers—did you prefer using to solve the equations? I preferred using hanger models 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 hanger model was the most difficult because I don’t have a lot of experience with hanger models. •

FACILITATION TIP Consider allowing students to vote for their preferred method. However, consistently remind them that ﬂuency with models contributes to powerful cognitive skills.

Part II 1.

2. 3. 4.

5. 6.

Read the following scenario to the class: William, Henry, and Ethan have successfully deciphered the 3-digit combination of the new vault. They have to arrange the valuables in such a way that the left side wall and the right side wall of this new vault have the same monetary value for safekeeping. Students should still have their Student Journals. Explain to students that they will work with their groups to analyze the hanger model, answer the questions, and record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-3 How will you determine the equation for the hanger model? I will use the key and count the number of rectangles and squares drawn on both sides of the hanger.

DOK-3 How will you solve the equation to ﬁnd the value of one bag of gold coins that is represented by one green rectangle? I will get the variable on one side of the equation and then solve to ﬁnd its value.

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

Math Chat

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.

DOK-1 Which two methods did you learn today to solve multistep equations? I learned how to solve multistep equations using a balanced hanger and an equation. • DOK-2 How do you know if the hanger is balanced? If we have equal weights on the ends of the hanger, the hanger will be in balance. If there is more weight on one side than the other, the hanger will tilt to the heavier side. • DOK-3 Why would you do the same operation on both sides while balancing a hanger? If we have a balanced hanger and add or remove the same amount of weight from each side, the result will still be in balance. •

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Solve Equations Explore 3 – Write, Model, and Solve Multistep Equations • FACILITATION TIP This is an important ﬁrst step for solving multistep equations. Take time to repeat the rule and have students record it on their Student Journal or in notebooks. Consider providing extra practice on individual whiteboards with some whole number multistep equations.

•

DOK-3 How useful is the balancing hanger method in solving multistep equations? We can use a balanced hanger to think about steps to finding an unknown amount in an associated equation. DOK-2 What are the rules of solving a multistep equation with variables on both sides? The first step in solving an equation with a variable on both sides is to get the variables on one side. We solve the variables on one side, and then to determine the variable’s value, we first add or subtract on both sides of the equation and then multiply or divide on both sides to get the final solution to the multistep equation. DOK-2 What operation would you use to isolate a variable if x is being added on one side and 2x 2x on the other? I would subtract x from both sides to get the variable on one side and then solve to find its value. DOK-2 If I remove 2 bands of $10 bills from one side of the balanced hanger, what should I do on the other side? You have to remove 2 bands of $10 bills from the other side of the balanced hanger as well; otherwise, the hanger will not be in balance. DOK-2 What operation would you use to isolate a variable if it is being multiplied by 3? I would use the inverse operation of multiplication; that is division. I would divide both sides by 3 to isolate the variable. DOK-2 If there were addition and division in a multistep equation, which operation would you solve first to isolate the variable? I would use the inverse operation to solve addition. I would subtract from both sides and then use the inverse operation of division to solve for multiplication. DOK-2 If there were multiplication and subtraction in a multistep equation, which operation would you solve first to isolate the variable? I would use the inverse operation to solve subtraction. I would add on both sides and then use the inverse operation of multiplication to solve for division. DOK-2 How will you verify your answer to check if it makes the equation true? I will substitute my answer back into the equation and solve it. If the answer is true, it will have the same value on both 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.

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Intervention

Acceleration

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Solve Equations Explore 4 – Solve Multistep Equations ACTIVITY PREPARATION Students will write, model, and solve multistep equations. Students will analyze strategies for solving to identify the most eﬃcient practices.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.3 Make sense of problems and persevere in solving them. MP.4 Model with mathematics.

Preparation

Materials Printed • •

• • •

1 Student Journal (per student) 1 Exit Ticket (per 2 students)

Reusable •

•

1 Set of algebra tiles (per group, optional)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) Who has ever wanted to buy something but didn’t have enough money?; 2) How did you get the money to buy what you wanted?; 3) How long did it take you to get enough money? FACILITATION TIP As students sketch models, make sure they sketch terms correctly. If they use algebra tiles, they should model terms with the correct shapes, shading tiles they will subtract. If they make a quick sketch, they should use sticks for x’s, plus signs for positive constants, and minus signs for negative constants.

Part I 1.

2. 3.

Read the following scenario to the class: Brothers Diego and Andrew are saving the money they earn in the summer to put toward a new car to share. For the ﬁrst few weeks, they decided to walk dogs in their neighborhood. At the end of each week, the brothers met up to check their earnings. Even though the brothers worked separately, they continuously brought home the same amount of money at the end of each week. Give a Student Journal to each student. Explain to students that they will work with their groups to analyze the information given for each week and determine how much Diego and Andrew charged for each dog walked. Students will model, explain, and write out the algebraic steps on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 Using algebra tiles is one way to model solving equations. What are other models you could use? I could make a quick sketch with sticks for x’s, plus signs for positive integers, and minus signs for negative integers.

DOK-1 In week 2, Diego walked 4 dogs and spent $2 on treats on each of two days. How is this represented in an equation? Why? This is represented by 2(4x – 2). The 2 in front of the parentheses means 2 groups of 4x – 2.

DOK-2 What does your solution represent in terms of the situation? Each solution represents the amount of money the brothers charged per dog.

DOK-2 How do you know which operations to use to isolate the variable? Reverse the operations starting with anything outside the parentheses. When isolating the variable, always reverse addition and subtraction before reversing the multiplication or division.

FACILITATION TIP Be sure students know to use a linear equation for each brother’s earnings for each week. Make sure students understand that the number of dogs a brother walks in a week represents the coeﬃcient in the corresponding earnings equation. FACILITATION TIP As students work through their Student Journal, make sure they understand that the term “tips” correlates with addition while the term “spent” correlates with subtraction. 128

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Explain

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Evaluate

e. DOK-2 How can you conﬁrm your solution is correct? I can substitute my solution back into the original equation. If the solution makes the equation true, then it is correct. 5.

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

Math Chat •

•

DOK-2 Did everyone in your group solve each equation using the exact same strategies? Explain. Some people used more steps than others. Some people moved integers first, and some people moved x terms first. Some used the distributive property, and some divided. DOK-2 If you used different strategies, did you get different solutions? Explain. No, we all got the same solution. As long as you are using correct strategies, your solution will be correct. DOK-1 What can you do to be more efficient when solving equations? I can try to simplify or combine like terms first. I can scan the problem and determine on which side to subtract x terms so the coefficient of x stays positive.

Part II 1.

2. 3.

Read the following scenario to the class: Diego and Andrew grew tired of walking dogs in the heat, so they decided to try some other professions to earn money. They realized how convenient it was that they kept earning the same amount of money each week and decided to continue this trend. For weeks 4 and 5, the brothers wrote and solved equations when they were calculating their weekly earnings. Diego and Andrew kept arguing that one was taking longer to solve or their equations didn’t look the same. Help the brothers settle their arguments by analyzing their work. Explain to students that they will work with their groups to read the weekly scenarios and analyze each brother’s equations and strategies for solving. As students are collaborating on their work, monitor their understanding by asking the following guiding questions: a.

DOK-2 What do you notice about the structure of the equations? Diego used fractions, while Andrew used a combination of fractions and decimals.

DOK-2 Is it best to use fractions or decimals when solving equations? They are both useful in different situations. I can look at the other numbers in the equation to help me know which would be more eﬃcient to use. I can clear fractions by multiplying by a multiple of the denominators, which helps with eﬃciency.

DOK-2 Would multiplying both sides of the equation from week 4 by the LCD to clear the equation of fractions be more eﬃcient than the method Diego used? It is still 4 steps, so it isn’t more eﬃcient. It might be a preferred method for students who don’t like dealing with fractions.

DOK-2 How can you tell when to combine terms containing the same variable or when to use inverse operations? If the like terms are on the same side of the equal sign, we should combine the terms as written. If the like terms are on opposite sides of the equal sign, we should use inverse operations to “move” one of the terms to the other side. This is done using a zero pair—adding or subtracting the zero pair on each side of the equation will cancel on one side and “move” it to the other side.

e. DOK-1 What was Diego’s ﬁrst step in solving for week 5? What property did he use? Diego used the distributive property as his ﬁrst step in solving. f.

DOK-1 Is this the only ﬁrst step that will work? Why or why not? No, most equations can be solved in a variety of orders and still result in the correct solution. Diego could have combined like terms ﬁrst.

Intervention

Acceleration

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.

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FACILITATION TIP Before reading the scenario, ask the class 1) What jobs have you had that you got paid for doing?; 2) Which of those jobs did you really not like to do?; 3) Did you keep working or ﬁnd other jobs to replace the jobs you didn’t like? FACILITATION TIP Some students may have forgotten or be unfamiliar with the term additive inverse, which is mentioned in Question 1. Before they start Part II, explain to the class what an additive inverse is, and show them a simple example. FACILITATION TIP After they answer the question, ask the class for examples of when it would be better to use a fraction to solve an equation. If they are stumped, present scenarios such as one involving cooking measurements.

STEMscopes Tip The Explain section, located along the scope menu, has a variety of elements designed to solidify students’ understanding of the content presented in the Explore section. Each scope’s Explain section includes a Picture Vocabulary, independent practice assignments, anchor charts, journal prompts, and interactive notebook activities.

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Solve Equations Explore 4 – Solve Multistep Equations 4. 5. FACILITATION TIP After the Facilitation Tip for Question 3b ask the class for examples of when it would be better to use a decimal to solve an equation. If they are stumped, present scenarios such as one involving taxed goods.

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

Math Chat DOK-2 How can you prove that a solution is correct for an equation? I can substitute my solution back into the original equation. If the solution makes the equation true, then it is correct. • DOK-2 Describe the properties of equality Diego used to solve the equation for week 4. Diego used the multiplication property of equality and the subtraction property of equality. •

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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Solve Equations Explore 5 – Solve Literal Equations ACTIVITY PREPARATION Students will solve literal equations for a given variable and use the same reasoning as that used in solving algebraic equations.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.3 Make sense of problems and persevere in solving them. MP.4 Model with mathematics.

Preparation

Materials Printed • • • •

• • •

1 Student Journal (per student) 1 Set of Code Cards (per group) 1 Set of Puzzle Dials (per group) 1 Exit Ticket (per 2 students)

•

Reusable •

2 Resealable bags (per group) •

Separate the class into groups of 2 or 3 students. Print a Student Journal for each student. Print a set of Code Cards for each group of students. Cut out the level 1 cards, and place them in a resealable bag labeled “Level 1.” Cut out the level 2 cards, and place them in a resealable bag labeled “Level 2.” If desired, print them on card stock, and laminate them for future use. Print one set of Puzzle Dials for each group. You may print double sided to save paper. 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.

PROCEDURE AND FACILITATION

FACILITATION TIP Before reading the scenario, ask the class 1) Have you ever participated in an Escape Room?; 2) What was the Escape Room’s theme?; 3) What did you have to do to escape?

Part I 1.

2. 3.

FACILITATION TIP After they answer the question, ask the class if they can spot any similarities between the equations on the Code Cards and typical algebraic equations. They should notice that the operators are the same. As they work through Part I, ask them if they think they could still solve the puzzles if the equal signs were substituted with inequality signs.

4. 5. 6.

FACILITATION TIP Encourage students to take their time drawing shapes so that they draw them clearly. Have students check each other’s codes to make sure they are actually sketching the expressions they are stating. Be ready to resolve disagreements over what is sketched. 132

Read the following scenario to the class: Your group has been locked in the Algebra Dungeon, and the only way to escape is to use your knowledge of solving equations to solve the puzzle dials at each level. Give a set of level 1 Code Cards to each group. Instruct students to spread out the Code Cards. Ask the following question:

DOK-2 What do you notice about the equations on the level 1 Code Cards? I noticed that there are no numbers or variables in these equations, only shapes.

Give a Student Journal to each student. Give the level 1 Puzzle Dial to each group. Explain to students that they will work with their groups to solve each puzzle by isolating the . Students will apply their knowledge of solving equations to solving the puzzles in Part I on their Student Journals. Once students solve a puzzle, they will sketch the code in the last column of the table on page 1. Groups will place the corresponding Code Card onto the level 1 Puzzle Dial. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 How can you apply your knowledge of solving equations to solving these puzzles? I can treat each shape like a variable. Each shape is a different term. Then, I can apply the properties of equality to solve.

9. 10.

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Intervention

Acceleration

DOK-2 How can this thinking help you with puzzle D? Since there are 2 circles, I thought of these as like terms. So when I had + , that equaled 2 .

•

• •

After Part I, invite the class to a Math Chat to share their observations and learning. Groups must submit their completed level 1 Puzzle Dial to the teacher in order to move on to level 2.

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Math Chat DOK-2 What are the similarities and differences between solving these puzzles and solving algebraic equations? The processes for solving both were exactly the same. The difference was that these puzzles used shapes instead of variables and numbers. • DOK-2 Could you have solved for a different shape besides the circle? Explain. Yes, I could have solved for any shape, but the instructions were to solve for the circle. • DOK-2 What were some of the strategies your group used to solve these puzzles? I replaced each shape with a different variable. Then, it was easier for me to solve algebraically. At the end, I translated the variables in my solution back to shapes. • DOK-1 What would it look like if you replaced puzzle C with variables? Puzzle C would be a(b + c) = c. •

FACILITATION TIP Student Journal, Page 1, ﬁrst column: After they answer the question, have students use cell B to solve for the square. Then, have students use cell D to solve for the triangle.

Explain the following to the class: When you have an equation that consists of mostly variables, it is called a literal equation. You can solve literal equations for a given variable or quantity of interest using the same processes as in solving algebraic equations. •

DOK-2 Brainstorm some literal equations you have used in school. A = πr2, A = lw, r P = 4s, d = _t , ...

Part II 1.

2. 3.

6. 7. 8.

Read the following scenario to the class: Your group made it through level 1, but level 2 is deﬁnitely more challenging. The puzzles in level 2 are literal equations from the math world. Can your group escape by cracking the code for level 2? Give a set of level 2 Code Cards and a level 2 Puzzle Dial to each group. Students should still have their Student Journals from Part I. Explain to students that they will work with their groups to solve each literal equation for the speciﬁed variable. A workspace is provided on their Student Journals. Once students solve a puzzle, they will write the code in the last column of the table on page 3. Groups will place the corresponding Code Card onto the level 2 Puzzle Dial. As students are collaborating on their work, monitor their understanding by asking the following guiding questions: a.

DOK-2 For puzzle E, how did you decide your ﬁrst step? I thought about what I would do if this were an algebraic equation. I knew I should remove the 2l term before dividing by the coeﬃcient of w.

DOK-2 Was that the only ﬁrst step? Explain. No, I could have divided both sides by 2, but that would be less eﬃcient.

Allow students enough time to complete Part II and answer the reﬂection questions that follow. Groups must submit their completed level 2 Puzzle Dial to the teacher in order to successfully “crack the code.” 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) If you did participate in an Escape Room, were the problems more difficult to solve the longer you were in the room?; 2) Were you successful at escaping the room? FACILITATION TIP There are significantly more variables at play for literal equations in Part II than in Part I. As students solve equations, watch out for them making more algebraic errors, such as mixing up terms and/or leaving terms out. FACILITATION TIP Make sure students write the codes legibly. Have students check each other’s codes to make sure they are actually writing the expressions they are stating. Be ready to resolve disagreements over what is written. FACILITATION TIP After students answer Part c of the first reflection question on Page 4, have them manipulate the equation for perimeter so they can solve for width. Then, give them a value of your choice for perimeter and length, and have them solve for the width.

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Solve Equations Explore 5 – Solve Literal Equations Math Chat DOK-2 Can you check your solution for a literal equation by substituting back into the original equation? Explain. Yes I can, but it is not very efficient and gets very messy since there are mostly variables in these equations. • DOK-2 Describe the properties of equality your group used to solve puzzle G. My group used the multiplication property of equality and the division property of equality. • DOK-3 Why would it be beneficial to know how to manipulate literal equations? It would be beneficial because I may not always be given a value for the isolated variable. But if I have other values, I can manipulate the equation to be more useful. • 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.

Post-Explore 1. 2. 3. 4.

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Explain

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Intervention

Acceleration

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Solve Equations Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

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

One-Variable 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

Write, Model, and Solve Two-Step Equations Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 3

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Write, Model, and Solve Multistep Equations Independent practice assignment that gives students an opportunity to demonstrate their learning

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

Show What You Know, Part 5 Solve Literal Equations Independent practice assignment that gives students an opportunity to demonstrate their learning

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Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

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

Equations with Variables on Both Sides

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

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Solve Equations 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

3 138

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

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What does mastery look like?

I can interpret parts of an expression and extend my understanding to more complex expressions.

I can describe and solve linear equations in one variable with one solution.

I can describe and solve linear equations in one variable with many solutions.

I can describe and solve linear equations in one variable with no solutions.

I can use algebraic properties and the properties of real numbers to justify the steps of a one-solution equation.

I can solve linear equations in one variable with coeﬃcients and can also explain the solution based on the situation.

I can use algebraic reasoning to ﬂuently manipulate linear and literal equations expressed in various forms to solve mathematical problems.

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Solve Inequalities Scope Introduction SCOPE SUMMARY Students will be able to model and solve one-variable, two-step inequalities with variables on both sides of the inequality. They will write one-variable inequalities and corresponding real-world problems when given a one-variable inequality with variables on both sides.

Student Expectations

VERTICAL ALIGNMENT

8.PAR.3.3 Create and solve linear equations and inequalities in one variable within a relevant application. 8.PAR.3.4 Using algebraic properties and the properties of real numbers, justify the steps of a one-solution equation or inequality. 8.PAR.3.5 Solve linear equations and inequalities in one variable with coeﬃcients represented by letters and explain the solution based on the contextual, mathematical situation.

Background Knowledge

Future Expectations

Students in previous grades wrote, modeled, and solved one-variable, one-step equations and inequalities and one-variable, two-step equations and inequalities. They also wrote corresponding real-world problems when given one-variable, onestep equations and inequalities and one-variable, two-step equations and inequalities.

Students will continue to build upon this foundation and will extend their knowledge of writing and solving equations. In high school algebra, students will solve linear equations with variables on both sides utilizing the distributive property. They will also write and solve systems of linear equations.

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

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES

determine how that amount changes when the proportional relationship is changed to a nonproportional relationship.

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

discuss math involved in a scenario about solving inequalities.

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

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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

In this exploration, students will write inequalities and scenarios that describe inequalities with variables on both sides. Students will:

Explore 3

•

Explore 2

Write Inequalities

determine which inequality symbol to use based on different scenarios.

Write, Model, and Solve Inequalities with Variables on Both Sides In this exploration, students will write, model, and solve inequalities with variables on both sides of the inequality symbol. Students will: •

compare how much money per hour people earned.

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

Solve Inequalities with Variables on Both Sides

Solve Inequalities with the Distributive Property and Fractions

In this exploration, groups of, students will help determine options to purchase video games with a speciﬁc amount of money and have money left over. Groups are also asked to weigh the pros and cons of each gaming option with each option represented by an advertisement with an inequality and a related graph. Students will: •

use inequalities to compare.

•

solve inequalities.

•

interpret inequality solutions.

Explore 4

Explore 1

EXPLORE ACTIVITIES

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In this exploration, students will be tasked with exploring averaging grades and using inequalities to make predictions. Students will: •

model and solve inequalities.

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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Solve Inequalities Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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Acceleration

ACCESSING PRIOR KNOWLEDGE Students will examine a series of descriptions 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.

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7.PAR.3.2 Construct algebraic inequalities to solve problems, leading to inequalities of the form px ± q > r, px ± q < r, px ± q ≤ r, or 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.

Materials

Preparation

Printed •

• •

1 Does Not Belong (per student or per group)

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

Procedure and Facilitation Points 1. 2. 3.

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

Set 1: Letter B does not belong because A, C, and D are all equivalent with solutions of x > 6. B represents all solutions where x is greater than or equal to 6.

b. Set 2: Letter C does not belong because A, B, and D are all equivalent with solution of y > − 7. C represents all solutions where y is less than − 7. 4. 5.

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

Identifying Misconceptions • •

• •

Students may incorrectly combine unlike terms, such as 7x 7x and 4. Students may not accurately identify the operation and, therefore, select an incorrect inverse operation. For example, students may think that the operation in 5xx is addition, rather than multiplication, and use subtraction, rather than division, as the inverse. Students may forget to change the direction of the inequality sign when dividing by a negative. Students may not recall the speciﬁcs for graphing an inequality. For example, students may confuse an open circle and a closed circle, as well as, the direction for shading the line graph.

FACILITATION TIP To facilitate a cohesive whole class discussion, consider projecting the pages and/or options one at a time. Ask students to make observations about what they notice and wonder. FACILITATION TIP Provide some time (allow for note taking) for students to quietly think, then pair with a shoulder partner, then share their ideas with the whole class. FACILITATION TIP This Foundation Builder provides several good examples that may be used to reteach reading inequalities, translating scenarios into inequalities, and ﬂuency with the symbols.

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Solve Inequalities Hook – Solve Inequalities ACTIVITY PREPARATION Students will relate solving inequalities to a real-world situation.

Materials

Preparation

Reusable • •

• •

1 Phenomena Video (per class) 1 Solve Inequalities (per class)

•

Plan to show the video. Prepare to project Solve Inequalities 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 or reading the scenario, ask the class 1) Has anyone ever been involved in a fundraising event?; 2) If so, what was the event raising money for?; 3) What did you do? FACILITATION TIP Be sure students interpret this phrase correctly. Since there are 24 hours in a day, some may see this as the amount of time in a day instead of a rate of joggers dropping out of the Jog-a-thon.

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. Read the following scenario while showing the video behind you: Marquez County is hosting its annual Jog-a-thon benefitting local charities. The Jog-a-thon is a jogging competition, where competitors jog for as long as they can. When they cannot jog anymore and have to stop, they are out of the competition. The Jog-a-thon gives $2,000 to each jogger’s charity when there are between 20 and 40 joggers left on the course. There are 215 competitors at the beginning of the Jog-a-thon, and joggers are dropping out at a rate of 24 joggers per hour. You have a goal of earning the $2,000 cash prize for an autism awareness charity. 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. Here are sample student answers: I notice that the prize is awarded when there are between 20 and 40 joggers left on the course. I wonder if there are additional prizes awarded for joggers who finish first, second, third, etc. I notice that if 24 joggers drop out per hour, you would only have to run for about 9 hours to be the last jogger standing (225 ÷ 25 = 9). I wonder how many hours it takes to win the prize. I wonder if you jog too long, do you miss out on the prize? Project Solve Inequalities. Notes

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Explain

Elaborate

Evaluate

Intervention

Acceleration

Explain to students that the Jog-a-thon team suggested that this math sentence could be used to calculate the number of hours a jogger would need to stay in the competition in order to receive the prize for their charity. Discuss the following questions: a.

Engage

DOK-1 Why would this math sentence have two inequality symbols? Allow students to share all ideas. Student answers will vary. Sample student answer: Since the prize is awarded when there are between 20 and 40 joggers left, the expression goes in between two inequality symbols. DOK-1 What does the “215 – 24 24x” part of this math sentence represent? 215 represents the number of joggers at the beginning of the Jog-athon, and – 24x represents the rate at which joggers are dropping out of the competition (24 joggers per hour).

Complete the Explore activities.

FACILITATION TIP

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After students answer the question, build on it by asking why the Jog-a-thon team wouldn’t just use the inequality 215 – 24x ≤ 40. A possible answer is the team is set on awarding at least 20 different charities. If students struggle to come up with a reasonable answer, ask them the question again when they return to the Hook after the Explore activities.

Part II: Post-Explore 1. 2.

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

DOK-1 Does this inequality make more sense after the Explore activities? Yes, this is called a compound inequality.

DOK-1 What strategies would you use to solve this compound inequality? I would split the compound inequality into two pieces and solve each piece.

DOK-1 Does this inequality represent conjunction and/or disjunction? How would the graph of this inequality be shaded? This inequality represents conjunction. The graph would have shading in between two values.

DOK-1 Do you feel that you have a strong understanding of solving inequalities? Answers will vary based on students’ success during the activity and conﬁdence level.

FACILITATION TIP In addition to this question 2d., ask students to pretend that they are explaining solving inequalities to a ﬁfth grader or a grandparent. What are the important things to look for? Where do they recommend to start? FACILITATION TIP Be prepared for some students to say that they still struggle to understand how to solve inequalities. Ask students where they are confused in the process, and review inequalities from their Student Journal with them.

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Solve Inequalities Explore 1 – Write Inequalities ACTIVITY PREPARATION Students will write inequalities and scenarios that describe inequalities with variables on both sides.

Standards for Mathematical Practice • • •

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

Materials

Preparation

Printed •

• • •

1 Student Journal (per student) 1 Exit Ticket (per 2 students)

Plan to separate the class into groups of 2 or 3 students. Print a Student Journal for each student. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student has one.

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Project this scenario and read it aloud together with the class. FACILITATION TIP Before distributing the Student Journal, consider modeling Day 1. Show students how to translate from words to inequalities. When you move on to Part II, model Day 2. Show students how to translate from inequalities to words.

2. 3.

Read the following scenario to the class: Devi and Tamara are students at Lake High School. They decided they would work this summer to earn money for an upcoming trip they want to take. Each day, Devi and Tamara earn different amounts of money to prepare for the trip. Help Devi and Tamara write inequalities to track their earnings for the trip. Give a Student Journal to each student. Explain to students that they should work with their groups to determine how to write Devi’s earnings and how to write Tamara’s earnings. Then, students can determine which inequality symbol to use based on the scenario. As students collaborate, monitor their work and use the following guiding questions to assess student understanding:

FACILITATION TIP If needed, take time to review each inequality symbol and encourage students to say the symbol meanings out loud (greater than, greater than or equal to, less than, less than or equal to).

DOK-1 What expression can you write for Tamara’s earnings on day 1? I can write 15.25x – 2 because she earned $15.25 per hour and spent $2 on a slushie.

DOK-1 What expression can you write for Devi’s earnings on day 1? I can write 14.75x + 4 because she earned $14.75 an hour and received $4 in tips.

DOK-1 What inequality symbol should we use for day 1? Why? I will use the greater than symbol because Tamara earned more than Devi on day 1.

Allow students enough time to complete Part I.

Part II 1.

2. 146

Read the following scenario to the class: Tamara and Devi have written inequalities about their earnings on days 4 through 6. Help the girls by writing scenarios that match their inequalities. Explain to students that they will work with their groups to create scenarios that match the inequalities given on their Student Journals. © Accelerate Learning Inc. - All Rights Reserved

4. 5.

Engage

Explore

Explain

Elaborate

Evaluate

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

DOK-1 What scenario could you write for the day 4 inequality? Student scenarios may vary. Tamara earned $10 an hour and spent $6 for lunch. Devi earned $12 an hour and spent $4 on lunch. At the end of the day, Tamara had earned less than or equal to the amount Devi had. Let x represent the number of hours worked.

DOK-2 What do the inequality symbols in the inequalities mean in these scenarios? The inequality symbols tell us if Tamara earned more than, less than, more than or equal to, or less than or equal to the amount Devi earned.

Intervention

Acceleration

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.

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

Math Chat DOK-2 How are these inequalities different from ones we have seen previously? These inequalities are different because there are variables on both sides of the inequality symbol. • DOK-2 How can you determine the inequality from a scenario? You can determine the inequality by finding the amount of money earned per hour. This will give the coefficient to go with the variable. Then, if tips were earned, you will add the amount of tips, and if money was spent, you will subtract what was spent. Last, you will determine the inequality symbol by finding out if more was earned, less was earned, or more than or equal to or less than or equal to was earned that day. • DOK-2 How did you write a scenario from an inequality? The coefficient was the amount earned per hour. Then, if it was adding, tips were earned, and if it was subtracting, something was purchased. • DOK-3 Where in the real world might you see inequalities? You could see inequalities when you are at an amusement park and need to pay a certain number of tickets per ride or attraction. •

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 Encourage students to use these speciﬁc terms (coeﬃcient, variable). Review as needed.

FACILITATION TIP Be prepared with some additional relevant real-world applications of inequalities to coach students (speed limits, age restrictions for movie tickets, height restrictions for amusement rides). FACILITATION TIP On this Exit Ticket, struggling students may need some read aloud support and guidance to locate the essential phrases and values.

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Solve Inequalities Explore 2 – Write, Model, and Solve Inequalities with Variables on Both Sides ACTIVITY PREPARATION Students will write, model, and solve inequalities with variables on both sides of the inequality symbol.

Standards for Mathematical Practice • • • •

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

Materials

Preparation

Printed • •

• • • •

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

Reusable • •

1 Set of algebra tiles (per group) 1 Projector or document camera (per teacher)

•

Plan to separate the class into groups of 2 or 3 students. Print a Student Journal and an Exit Ticket for each student. Gather one set of algebra tiles for each group. Have a projector or document camera prepared to show students how to model solving equations with variables on both sides using algebra tiles. 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 Part I FACILITATION TIP Project a printed version of this scenario and read it aloud together as a class. Consider providing students time to think independently, pair chat, and share their observations with the whole class. FACILITATION TIP To help students focus on the text, distribute the algebra tiles Step 6.

FACILITATION TIP Consider breaking this process into actionable steps that you can post or project. For example, 1) Identify and label what the parts of the inequality represent. 2) Identify what the variable represents. 3) Explain why the two sides are not equal. 4) Model both side with tiles. 5) Sketch the model. 6) Explain the steps. 7) Write the algebraic solution of the inequality. 148

2. 3. 4.

Read the following scenario to the class: Erika and Tammi wanted to earn money for the upcoming trip with friends at Lake High School. They decide to work for different relatives this summer to earn money for their trip. Each day, the girls earn different amounts of money to prepare for the trip. Help Erika and Tammi determine how much money per hour they earned on day 1. Give a Student Journal to each student. Give a set of algebra tiles to each group. Direct students to look at the day 1 scenario on their Student Journals. Read day 1 together: Erika worked for 3 hours and spent $1 on a soda. Tammi worked for 2 hours and earned $2 in tips. At the end of the day, Tammi had more money than Erika. Let x represent the amount of money per hour each girl was paid on day 1. How much money did Tammi get paid per hour on day 1? Discuss the following questions with the class: a.

DOK-1 The inequality 3x 3x – 1 < 2x + 2 represents this situation. Which part of the inequality represents Erika’s earnings, and which part of the inequality represents Tammi’s earnings? Erika’s earnings are represented by 3x – 1, and Tammi’s earnings are represented by 2x + 2.

DOK-1 What does the variable x represent in this scenario? The variable x represents the amount of money each girl was paid per hour.

DOK-2 Why are the two expressions not equal to each other? Tammi earned more than Erika. This is shown by Erika’s expression being less than Tammi’s expression, as shown by the inequality symbol. © Accelerate Learning Inc. - All Rights Reserved

Engage

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Explain

Elaborate

Evaluate

Intervention

Acceleration

DOK-1 How can you model Erika’s expression using algebra tiles? I can use 3 green rectangles for 3x and 1 red square for −1.

e. DOK-1 How can you model Tammi’s expression using algebra tiles? I can use 2 green rectangles for 2x and 2 yellow squares for positive 2. f.

DOK-1 How can you show that the girls’ expressions are not equal? I can draw a less than sign to show that Erika’s expression is less than Tammi’s expression.

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Explain to students that they should model the inequalities with their algebra tiles and then sketch the model on their Student Journals. Model drawing the algebra tiles for students. Discuss and model solving by using algebra tiles with students. a.

DOK-1 What is different about this inequality versus all of the inequalities you have solved in the past? This inequality has variables on both sides of the less than sign.

DOK-2 When solving, what strategies do you use to isolate the variable? I perform inverse operations to get x by itself. What I do to one side of the equation, I must do to the other side.

DOK-2 How can you manipulate the inequality so you only have x terms on one side of the equal sign? I can remove 2 x’s from both sides of the equation.

Model removing 2xx tiles from each side as well as crossing through 2x tiles on the sketch. Write the explanation “remove 2x 2x from both sides” in the corresponding column. Lastly, model algebraically, subtracting 2x 2 from both sides.

STEMscopes Tip Students take notes, express ideas, and/or process the information presented in class using the Interactive Notebook element, located in the Explain section of each scope. These cut-and-glue activities provide an interactive way for students to showcase the concepts and skills learned in the Explore activities and can be added to a notebook for future reference.

e. DOK-1 What is remaining on the left side of the inequality? x – 1 f. 7.

8. 9.

DOK-1 What is remaining on the right side of the inequality? 2

Explain to students that now the inequality looks like the inequalities they have solved previously. Complete the model, explanation, and algebraic steps to determine the amount of money each of the girls were paid per hour. 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.

Math Chat DOK-2 How are these inequalities different than ones that we have seen previously? These inequalities are different because there are variables on both sides of the less than sign. • DOK-2 Does this change how you solve for x?? Explain. No, we still have to isolate the variable by using inverse operations to solve for x. •

Part II 1.

Read the following scenario to the class: Erika and Tammi are helping Erika’s aunt with her food truck order this week! Each day, the girls work different hours and earn different amounts of money. Help Erika and Tammi determine how much money per hour they earned on days 2 through 5. Explain to students that they will work with their groups to determine the inequality for each of the remaining days Erika and Tammi worked this week. Students will model using algebra tiles, draw their models on their Student Journals, explain, and write out the algebraic steps on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

FACILITATION TIP Project the seven steps (or create your own) listed above to guide students through the process required.

DOK-1 Using algebra tiles is one way to model solving inequalities. What are other models that you could use? I could make a quick sketch with sticks for x’s, plus signs for positive integers, and minus signs for negative integers.

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Solve Inequalities Explore 2 – Write, Model, and Solve Inequalities with Variables on Both Sides

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. FACILITATION TIP Create and post a list of some common phrases of real-world inequalities (maximum, at most, at least, fewer than, under, exceeds...) to coach students to recognize more real-world examples. FACILITATION TIP Depending on your success criteria, consider allowing students to merely show the algebraic steps and explanation for solving. Drawing the model may be limiting for some students.

4. 5.

DOK-2 How do you know which operations to use to isolate the variable? We use inverse operations. When there is addition, we use subtraction, and when there is subtraction, we use addition.

DOK-2 How can you conﬁrm your solution is correct? I can substitute my solution back into the original equation. If the solution makes the equation true, it is correct.

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

Math Chat DOK-2 How can you prove a solution is correct for an inequality? I can substitute my solution back into the original equation. If the solution makes the inequality true, then it is correct. • DOK-2 What does your solution represent in terms of the situation? Each solution represents how much money the girls earned per hour that day. • DOK-3 Where else in the real world would you use inequalities? Comparing the cost of cell phone plans •

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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Solve Inequalities Explore 3 – Solve Inequalities with Variables on Both Sides ACTIVITY PREPARATION Students will use inequalities to compare the costs of different gaming options. Students will solve the inequalities and interpret their solutions in the context of the situations presented.

Standards for Mathematical Practice • • •

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

Materials

Preparation

Printed • • •

1 Student Journal (per student) 2 Sets of Advertisement Stations (per class) 1 Exit Ticket (per 2 students)

• • • •

•

Separate the class into 8 groups to complete this activity. Print a Student Journal for each student. Print 2 single-sided sets of Advertisement Stations. If desired, print them on card stock, and laminate them for future use. Create 8 stations around the room with one advertisement at each station. (Each advertisement is 2 pages: one with the pictures and one with the scenario.) 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 Before reading the scenario, ask the class 1) Does anyone buy video games?; 2) Approximately how much is a new video game?; 3) Where do you buy your video games from?

Part I 1.

FACILITATION TIP Discuss with the class if there is any information within the scenario that is unneeded to solve the problem. Eliminating arbitrary information comes in handy for instances such as long word problems and large data sets.

2. 3. 4. 5. 6.

FACILITATION TIP Make sure students understand why only whole numbers make sense as solutions in this scenario. They should remember that solutions represent the number of purchases and not the amount of money spent. 152

Read the following scenario to the class: Zahra’s stepbrother has decided to give her his gaming system as he is packing up and getting ready to leave for college. Zahra wants to purchase new video games but also needs to save money to go on a trip with friends from Lake High. Zahra decides to take out only $24 from her savings to spend on video games, but she isn’t sure if she wants to spend all of the money. New games are $25, but she found a bargain bin that has games for $4. Zahra wants to make sure she has $5 remaining after making her video game purchases. Help Zahra determine her options for the number of bargain bin games she can afford. Give a Student Journal to each student. Continue reading to students: Zahra created the inequality to calculate the number of bargain bin games she could purchase. Students will record their work on their Student Journals. Encourage students to make connections between the current work and onevariable equations and inequalities that they have previously solved. Monitor students as they collaborate on their work. Use the following guiding questions to assess student understanding: a.

DOK-2 What types of solutions make sense in Zahra’s context? Although the inequality has solutions along the real number line, only whole numbers make sense for possible video game purchases.

DOK-2 What does each term in the inequality represent? The 24 represents the amount of money Zahra has to spend. The –4x represents the amount spent on the $4 bargain games. The greater than or equal to 5 part of the inequality represents that Zahra must have at least 5 dollars left. © Accelerate Learning Inc. - All Rights Reserved

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DOK-2 Why does an inequality best represent this situation? We are considering a situation that has a range of solutions. We are not looking for one speciﬁc point; we are looking for all the values that make the inequality true. DOK-2 Why is a closed circle used to graph the solution set? A closed circle shows that that number is included. The inequality symbol has the equal-to bar at the bottom that shows it should include that number.

e. DOK-2 What did you notice was different about solving the inequality when you did not move the linear term of –4 –4xx to the other side? I needed to ﬂip the direction of the inequality or the answer would not have been correct. 7.

Allow students enough time to complete Part I, and then invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-2 Did you prefer adding 4 4xx to both sides or subtracting 24, and why? I preferred adding the 4x because I did not have to divide by a negative coeﬃcient. • DOK-2 Why does the direction of an inequality change when you multiply both sides by a negative number? Two inequalities are said to have the opposite sense if the signs of inequality point in the opposite direction. Multiplying by a negative number changes what side of zero the number is on. For example, if we have 3 < 7 and multiply both sides by −1 to get −3 and −7, they are still the same distance from zero; now −7 is further to the left than −3 on a number line, so the sense of the inequality must change. We would end up with the inequality −3 > −7. • DOK-3 The addition property of inequality states that we can add any number to both sides of an inequality to produce an equivalent inequality. Based on your work in Part I, what do you think the multiplication property of inequality states? The multiplication property of inequality states that we can multiply any positive number to both sides of an inequality to produce an equivalent inequality. If multiplying any negative number by both sides of an inequality, an inequality of the opposite sense is produced. An inequality of the opposite sense means the original inequality and the resulting inequality point in the opposite direction. •

Part II 1.

2. 3.

Read the following scenario to the class: As a newbie to the gaming world, Zahra had no idea how many gaming options there were. To help weigh the pros and cons of each gaming option, she decided to represent the situation with an inequality and a related number line. The advertisements are posted around the room. Visit each Advertisement Station to help Zahra weigh the pros and cons of each gaming option. Students should still have their Student Journals. Have students look around the room to notice the locations of the Advertisement Stations. Explain to students that there are two sets of the same Advertisement Stations posted on the walls so no station gets too congested. Instruct students to only complete one set of stations (A–D). Explain to students that they will work with their groups to help Zahra weigh the pros and cons by examining each advertisement and analyzing the information. Students should use the space provided on their Student Journals to organize the information from the advertisements. Monitor students as they collaborate on their work. Use the following guiding questions to assess student understanding: a.

DOK-2 The inequality shows 62 + 8x 8x is greater. What does 62 + 8x represent? Why would the inequality show that 62 + 8x 8x is greater? 62 + 8x represents the cost of bargain games at Retro Games. The question is asking when Game Shop is cheaper, which means Retro Games is the larger quantity. (This question can be adjusted and asked of any inequality in Part II.)

Intervention

Acceleration

FACILITATION TIP After students answer the question, ask them for other instances where an inequality would best represent a real-life scenario. They may start with situations that are incomplete as far as representing an inequality. Guide them with questions and statements to ﬁll in the gaps.

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FACILITATION TIP After students complete Part I, ask them for real-life scenarios where an inequality with a fraction or decimal value would make sense. If they are stumped, present examples pertaining to situations such as baking measurements or money saved.

FACILITATION TIP Before reading the scenario, ask the class 1) What are other ways you could get video games without paying full price?; 2) What are the pros of buying used video games?; 3) What are the cons of buying used video games? FACILITATION TIP To compare gaming options, students will organize information from the ads using inequalities for each ad station. Encourage students to focus on one station at a time so they don’t get confused with or overwhelmed by the number of ad stations or gaming options. FACILITATION TIP The expression is used in more than one context in Part II of the Student Journal. Make sure students understand that Question 5a. applies to Advertisement A as opposed to Advertisement B, where the expression is set up as the smaller quantity. 153

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Solve Inequalities Explore 3 – Solve Inequalities with Variables on Both Sides

FACILITATION TIP Be sure students clearly match the right solution set with the right condition. When the number sentence is true, the solution includes all real numbers; when the number sentence is false, there is no solution.

FACILITATION TIP Students may be thrown off when solving for the inequality for Advertisement C. The solution has no variable, and the solution for Advertisement B is false. If students are confused, read the solution to the inequality out loud to them and remind them that a mathematical statement can be true with or without a variable.

6. 7.

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FACILITATION TIP

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DOK-1 What words let you know that an inequality you are creating should have an equal-to piece attached instead of being strictly greater than or less than? Phrases like at least and no more than indicate that values that make the quantities equal should be included in the solution set.

DOK-3 How do you know if the inequality has no solutions or if all real numbers are solutions? Inequalities that yield these solution sets occur when solving removes all of the variables and we are left with a number sentence that is either true or false.

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

Math Chat

STEMscopes Tip

Depending on how students solve the inequality, they may divide by a negative coeﬃcient. Continue to watch out for students leaving the inequality sign the same after dividing by a negative coeﬃcient.

DOK-2 How did you decide which numbers from the ﬂyer were constants and which should have a variable attached in the inequality you created? The values that were one-time payments were constants, but the values that represented a cost per month or per bargain game purchased that would repeat had a variable attached in the inequality. (This question can be adjusted and asked of any inequality in Part II.)

e. DOK-3 Why was a test point needed to solve the related equation for advertisement A, but a test point wasn’t needed to solve the inequality for advertisement B? When solving an inequality, it helps to test a point to verify what side to shade or where the solutions exist. For advertisement B, there were no solutions, so we did not have to determine which side of the number line required shading.

•

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.

DOK-2 How do you know if an inequality has no solutions? As I solve the inequality, I end up with a statement that is never true regardless of x, such as 32 < 15. DOK-2 How would you know when the solution to an inequality is all real numbers? Note that you should avoid calling this scenario infinite solutions because all open-ended inequality solutions have an infinite number of values. As I solve the inequality, I end up with a statement that is always true regardless of x, such as 70 < 190. DOK-2 Can negative coefficients be avoided when solving an inequality with variables on both sides? Explain. Yes, they can be avoided if the variables are moved and combined strategically. For example, if you have 5x on one side and 3x and on the other, subtracting 3x on each side would avoid a negative coefficient. If you subtract 5x on each side, you would create a negative coefficient and have to switch the direction of the inequality when you divide by −2. DOK-1 When graphing the solution set, when is an open circle used, and when is a closed circle used? An open circle is used when the boundary point is not included. A closed circle is used when the boundary point is included. DOK-2 How is solving inequalities with variables on both sides similar to solving equations with variables on both sides? If a related equation is used, it is exactly the same process until the end, when you need a test point. If you’re not using a related equation, it is the same unless you have a negative coefficient. If there is a negative coefficient, the inequality will change direction when multiplying or dividing by a negative.

Post-Explore 1. 2. 3.

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Solve Inequalities Explore 4 – Solve Inequalities with the Distributive Property and Fractions ACTIVITY PREPARATION Students will explore averaging grades and using inequalities to make predictions. Students will model and solve inequalities including the distributive property and fractions.

Standards for Mathematical Practice • • •

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

Preparation

Materials Printed • • •

• • •

1 Student Journal (per student) 1 Set of Gym Grades Cards (per group) 1 Exit Ticket (per 2 students)

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Separate the class into groups of 3 or 4 students. Print a Student Journal for each student. Print a set of Gym Grades Cards for each group. Cut out the cards. 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.

PROCEDURE AND FACILITATION 1. 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.

Begin the Explore activity with the whole class invited to a Math Chat to discuss calculating averages.

Math Chat DOK-1 How do you ﬁnd the average of three numbers? Add the three numbers, and divide the sum by three. • DOK-1 How do you know to divide by three? I know to divide by three because there are three numbers. • DOK-2 If your grades are 100, 100, and 100, what is the average? How do you know? The average would be 100. 300 divided by 3 is 100. When averaging three numbers that are the same, the average will also be the same. • DOK-2 If your grades are 100, 50, and 75, what is the average? How do you know? The average would be 75. 225 divided by 3 is 75. 50 is 25 less than 75, and 100 is 25 more than 75. Therefore, 75 is the center number, or average. •

Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Do you like participating in gym class?; 2) What are your gym grades based on?; 3) How can you earn a high grade in gym? 2. 3. 4.

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Read the following scenario to the class: Pari, Elizabeth, and Tomas are students at Lake High School. They are wanting to go on a trip with their friends but must get an A in gym class. They are trying to ﬁgure out if it is even possible for each of them to get an A in gym class. At LHS, an A is a 90 or higher. Gym grades at LHS are determined by averaging the students’ weekly grades. Help them ﬁgure out if they can all earn A’s. Give a Student Journal to each student. Give Pari’s Gym Grades Card to each group. Explain to students that they will work with their groups to analyze Pari’s Gym Grades Card. They will use her grades given on the card to complete Part I on their Student Journals. Monitor students as they collaborate on their work. Use the following guiding questions to assess student understanding: © Accelerate Learning Inc. - All Rights Reserved

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DOK-1 How did you know to divide by 4 to ﬁnd Pari’s current average in gym class? I knew to divide by 4 because there are 4 numbers.

DOK-1 Could the expression for finding Pari’s current average be written with distribution? What number would be used for the coefficient? Yes, 1 the coefficient would be _4_.

DOK-2 How would the expression change if we wanted to know the average of 5 grades? We would add 5 numbers together and divide the sum by 5.

DOK-1 How did you know to divide by 5 to determine her 5-week average? I knew to divide by 5 because there are 5 numbers.

e. DOK-1 Could the expression be written using distribution? What number 1 would be used for the coeﬃcient? Yes, the coeﬃcient would be _5_. f.

DOK-2 I see you wrote out each grade in the expression to determine the 5-week average. How could that be made simpler so there is less you have to write each time? I could add up her grades for weeks 1 through 4 because those will stay the same each time. I would use 350 instead of 100 + 100 + 100 + 50.

DOK-2 How did you know the variable should be added to the numerator and not the denominator? The number added to 350 changed each time. That is the value that was varying, so I knew to use the variable in the numerator, where the value was changing each time.

h. DOK-2 When solving the equation for x,, what was your first step? How did you know to multiply by 5 before subtracting 350? The first step is to multiply by 5. When there is a constant in the denominator, it is like having parentheses around the numerator, so we have to eliminate the coefficient before we consider any operations inside the parentheses (in this case, the numerator). i. DOK-3 Are there other ways to solve this equation? Is there a more advantageous method? We could distribute the coefficient. This is not the most advantageous method because it takes more steps to distribute to both terms and to later undo the multiplication to isolate the variable. It is more advantageous to use multiplying by a reciprocal to eliminate the coefficient. j. DOK-2 How did you know what inequality symbol to use? How did you know you should use an inclusive inequality symbol? The problem said “90 or better.” This language told me that 90 is included in the solutions, so I knew I should use an inclusive inequality that has the equal-to part. 6.

Allow students enough time to complete Part I, and then invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 Could the expression be written using distribution? What number would be 1 used for the coeﬃcient? Yes, the coeﬃcient would be _5_. • DOK-2 Do you think writing the expression using a fraction and distribution will make it easier or harder to solve? Explain. I think it would make it harder to solve because I think fractions are hard. I think it would be the same steps to solve because we would just multiply by 5, the same way we did for the one we already solved. • DOK-1 How could we represent this solution on a number line? What values would be shaded? Would you shade 100? We would shade all the numbers 100 and greater. We would use a closed (shaded) circle for 100. •

Intervention

Acceleration

STEMscopes Tip Spiraled Review, located in the Elaborate section, provides students with a contextual scenario used to solve four different problems. This activity helps students maintain essential knowledge, see how mathematical skills connect from one topic to the next, and experience real-world applications of previously learned skills.

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FACILITATION TIP Go over the answer with the whole class. If it goes unmentioned, add that the denominator, 5, is constant, meaning no variable should go in that part of the fraction. FACILITATION TIP Some students may have a different first step. For instance, they may have split the 1 fraction into two fractions or factored __5 out of the fraction. Be sure students apply algebraic principles correctly throughout their solution regardless of the first step they choose. Alternative first steps may provide a segue into Question 5i.

FACILITATION TIP After they finish Part I, ask the class to find the average of the weekly grades if Pari gets a perfect score plus ten points of extra credit for Week 5. FACILITATION TIP Watch out for students answering that the 1 coefficient is 5 instead of __5. Remind them that the “5” is part of a fraction with an understood “1” as the numerator.

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Solve Inequalities Explore 4 – Solve Inequalities with the Distributive Property and Fractions DOK-2 What are the reasonable solutions for this situation? Is 250 reasonable? Are decimals reasonable? Explain. The only reasonable solution would be 100. 100 is the highest grade possible. Reasonable solutions would be 100 and higher. 250 doesn’t seem reasonable because that would be extra credit more than a regular class grade. Reasonable solutions would be from 100 to 150 because 50 points of extra credit is reasonable. Decimals would be reasonable because some teachers give grades that are decimals. • DOK-2 How did you decide on the direction of the inequality? The problem said “90 or better.” That’s another way of saying 90 or higher. To indicate we want something higher than 90, we would use an inequality that points to 90. • DOK-3 How would the scenario need to be different for the inequality to point in the other direction? The scenario would need to be looking for grades of 90 or lower. •

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.

Part II 1. 2. 3.

FACILITATION TIP

Remind students that this option involves ﬂipping the inequality sign if and only if they perform the inverse operation of a negative coeﬃcient.

Distribute the remaining Gym Grades Cards to each group. Each group should still have Pari’s Gym Grades Card from Part I. Explain to students that they will work with their groups to use the Gym Grades Cards to calculate the grade information for Elizabeth and Tomas. Point out to the class that solving inequalities is similar to solving equations. They can always solve the equation and use test points to determine the appropriate solution. Another option is to solve the inequality if that is their preferred method. Students will then work together using the Gym Grades Cards to model and determine the answer to the scenario and record their work on their Student Journals. Monitor students as they collaborate on their work. Use the following guiding questions to assess student understanding: a.

DOK-1 Do you predict Elizabeth will need a higher or lower grade than Pari in week five in order to earn an A? Why? I believe Elizabeth will not need to earn as high of a grade in gym class as Pari in week five because she performed better than Pari during the first four weeks.

FACILITATION TIP

After students answer the question, have them look back in Part I at what Pari needs in Week 5 to earn a 90. Ask them, “Would it be reasonable to think Elizabeth could earn a 90 if she had earned fewer points than Pari so far?”

DOK-2 How will the inequality for Elizabeth’s gym grade compare to the inequality for Pari’s grade? The setup will be similar, except Elizabeth has earned more points so far, so the constant in the numerator will be larger.

DOK-1 How can you create an equivalent inequality for Elizabeth’s grade that does not contain any fractions? Multiply both sides by 5.

DOK-1 How does multiplying both sides by 5 use inverse operations to help solve the inequality? The expression 370 + x is being divided by 5, so we have to use the inverse operation and divide by 5 to get 370 + x by itself.

FACILITATION TIP Students have a 50-50 chance of answering correctly. After the majority of students answer the question correctly, check their understanding by having them justify their answer.

e. DOK-1 Do you predict Tomas will need a higher or lower grade than Elizabeth and Pari in week ﬁve of gym class in order to earn an A? I believe Tomas can earn a lower grade in week ﬁve than Elizabeth and Pari and still earn an A. f.

DOK-1 What needs to be different about the inequality for Tomas’s ﬁnal English grade? There are ten weeks in the term, so the denominator is 10. Tomas is aiming for a B instead of an A, so the fraction should be greater than or equal to 80 instead of 90.

g. DOK-2 Do you think this type of inequality would work to determine what grades a student would need no matter how many weeks have gone by? The inequalities we created could help us ﬁnd what grade or average grade we need at any point throughout the term. We would just have to adjust the values in our fraction and change the target grade we want to earn. 158

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

Math Chat DOK-3 Compare and contrast solving equations with fractions with solving inequalities involving fractions. All steps are the same unless there is a negative coefficient. If there is a negative coefficient, you either solve the inequality as an equation and use a test point to determine the direction of the inequality or remember to change the inequality when multiplying or dividing by a negative. • DOK-2 How were you able to write an expression for the average after 5 weeks even though you did not have five grade values yet? I wrote the fifth-week value as the variable x and added it to determine the five-week sum. •

•

3 + 5x 5

DOK-2 What would be your ﬁrst step to solve the inequality _____ < 4 + x, and 17 why? I would multiply both sides of the inequality by 17 to create an equivalent inequality that does not have a fraction. 1

4 ) < 4 + x, DOK-2 What would be your first step to solve the inequality –__5(12 – 4x and why? I would multiply both sides of the inequality by –5 to create an 1 inequality without fractions and not have to distribute the –__5. I would have to flip the inequality sign if I performed this operation. • DOK-3 What is another real-world scenario where an inequality would make more sense to use than an equation? An inequality would make more sense to use if someone was trying to figure out how to make a profit of at least $300. They would want to make at least $300, not exactly $300, so an inequality would be more appropriate than an equation. •

Post-Explore 1. 2. 3. 4.

Intervention

Acceleration

FACILITATION TIP After students answer the question, ask them if the same logic would apply to decimals. Give them space to justify their reasoning, and present them with a simple inequality to solve that has a decimal coeﬃcient or constant.

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FACILITATION TIP Some students may incorrectly separate the Week 5 grade from the previous four grades. They may take the fourweek average and simply add x for the expression for the five-week average. Remind them that an average is the sum of values divided by the number of values. Then, ask: How do you find the sum of the values? What is the total number of values after 5 weeks? FACILITATION TIP After students answer the question, ask them for another real-world scenario where finding an average would be helpful. If they are stumped, present an example such as predicting what a basketball player will score based on his scoring performance in previous games. FACILITATION TIP After the class completes the Exit Ticket, present them with a few fraction and decimal values of your choice. Some should be less than 13 and some should be greater than 13. Ask them if any of the values are solutions to the inequality in the Exit Ticket, and have them explain their answers.

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Solve 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

Write Inequalities 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

Write, Model, and Solve Inequalities with Variables on Both Sides 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

Solve Inequalities with Variables on Both Sides

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 Inequalities with the Distributive Property and Fractions

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

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ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

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

Inequalities with Variables on Both Sides

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

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Solve Inequalities 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

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What prompts will be used?

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What does mastery look like?

I can use algebraic properties and the properties of real numbers to justify the steps of a one-solution inequality.

I can create and solve inequalities in one variable.

I can solve inequalities in one variable with coeﬃcients represented by letters and explain the solution based on the situation.

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Create Non-Proportional Relationships from Proportional Relationships Scope Introduction SCOPE SUMMARY Students will be able to take a proportional relationship and determine what would cause it to make it a non-proportional relationship. They will learn distinguishing characteristics that will allow them to identify proportional and non-proportional linear relationships using tables, graphs, and equations. This develops the foundational concepts of functions.

Student Expectations

8.PAR.4.1 Use the equation y = mx (proportional) for a line through the origin to derive the equation y = mx + b (non-proportional) for a line intersecting the vertical axis at b. 8.PAR.4.2 Show and explain that the graph of an equation representing an applicable situation in two variables is the set of all its solutions plotted in the coordinate plane.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

Students in previous grade levels represented realworld situations using verbal descriptions, tables, graphs, and equations in the form y = kx and y = mx + b.

Students will continue to build upon this foundation and will extend their knowledge of working with proportional and non-proportional linear relationships. As students move into high school algebra, they work more in depth with functions.

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

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES

compare two different proportional relationships represented in different ways.

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: •

distinguish between proportional and non-proportional situations using tables and graphs.

•

represent situations using equations.

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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Explore 1

EXPLORE ACTIVITIES Create Non-Proportional Relationships In this exploration, students will create non-proportional relationships by using proportional relationships shown on tables, graphs, and equations. Students will: •

determine how an amount changes when the proportional relationship is changed to a non-proportional relationship.

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

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Create Non-Proportional Relationships from Proportional Relationships Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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ACCESSING PRIOR KNOWLEDGE Students will listen to prompts about the prior standard, decide whether each prompt is fact or ﬁction, and communicate their decisions by walking to the designated sides of the classroom. This element is designed to uncover student misconceptions; it should not be taken for a grade. 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.

Materials

Preparation

Printed •

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1 Set of Fact or Fiction Prompts (per class)

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Print one set of Fact or Fiction Prompts to read aloud to students. Another option is to project the prompts using a digital projector.

Procedure and Facilitation Points 1.

2. 3. 4. 5.

Designate one side of your room as the Fact side of the room and the other side as Fiction. Explain to students that they will decide whether they think each prompt is fact or ﬁction and then move to the corresponding side of the room. Read the prompt, and allow students to move to different sides of the room. 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 ﬁction.

Prompt 2 is fact.

Prompt 3 is fact.

•

Project each prompt one at a time and allow students time at their seats to think and make notes independently in silence, then share their ideas with a shoulder partner. If space for movement is limited, have students just vote fact or ﬁction. They can secretly vote with a ﬁst or open hand on their chest facing the teacher. FACILITATION TIP

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

Consider distributing each prompt to students so they can take notes, create tables or graphs, and write on each one. Be prepared for a wide variety of reasoning. FACILITATION TIP Take time to explain the three different account names mentioned on Prompt 1 (bank, checking and investment savings).

Identifying Misconceptions •

FACILITATION TIP

Students may struggle with ﬁnding an equation that represents the situation presented in a word problem or scenario. Students may forget how to determine the unit rate for various representations and need assistance comparing the same rates.

CREATE NON-PROPORTIONAL RELATIONSHIPS FROM PROPORTIONAL RELATIONSHIPS

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FACILITATION TIP For Prompt 2, consider that some students will ask for labels on the x- and y-axes. Clarify that Go Cart A is modeled by the graph and Go Cart B is modeled by the equation.

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CREATE NON-PROPORTIONAL RELATIONSHIPS FROM PROPORTIONAL RELATIONSHIPS

Create Non-Proportional Relationships from Proportional Relationships Hook – Family Bowling Night ACTIVITY PREPARATION Students will distinguish between proportional and non-proportional situations using tables and graphs and will represent the situations using equations.

Materials

Preparation

Printed •

• • •

1 Family Bowling Night (per class)

Reusable •

1 Phenomena Video (per class)

PROCEDURE AND FACILITATION Part I: Pre-Explore 1.

FACILITATION TIP

Print and project this scenario. Read it aloud together as a class. Guide students to identify the essential phrases and values.

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.

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Plan to show the video. Prepare to project Family Bowling Night 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 students have completed the Explore activities.

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: The Carlos family is planning a family bowling night. Some of the Carlos family members regularly bowl and own their bowling shoes. They have researched prices at Awesome Strikes Bowling Alley and Magic Strikes Bowling Alley and have recorded the results in a table and a graph. The cost of renting bowling shoes is included in the cost at Awesome Strikes Bowling Alley. Magic Strikes Bowling Alley charges an extra $3.00 to rent bowling shoes. The Carlos family will compare prices and determine which bowling alley offers the best options for their family members. 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 the Carlos family is planning a family bowling night. I wonder which bowling alley will offer the best prices. How will I compare data in a graph and a table? I can use math to determine the equations that can be used to represent the cost of bowling at each bowling alley. Project Family Bowling Night for students. Explain to students that both the graph and the table show linear relationships. They can compare the rates and discuss the following questions: a.

DOK-2 How can you determine whether a situation is proportional? Answers may vary. A situation is proportional if it has an ordered pair of (0, 0), the line passes through the origin, and the equation is written in the form y = kx.

DOK-2 How can you determine whether a situation is non-proportional? Answers may vary. A situation is non-proportional if the equation can be written in the form y = mx + b when b doesn’t equal 0.

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 Family Bowling Night, and discuss the following questions: a.

DOK-1 Which bowling alley represents a proportional situation? Awesome Strikes Bowling Alley

DOK-1 Which bowling alley represents a non-proportional situation? Magic Strikes Bowling Alley

DOK-2 Write the equation that can be used to determine the cost of the bowling games at Awesome Strikes Bowling Alley. y = 8x

DOK-2 Write the equation that can be used to determine the cost of the bowling games at Magic Strikes Bowling Alley. y = 5x + 3

e. DOK-2 How much would it cost for 6 members of the Carlos family to go bowling at Awesome Strikes Bowling Alley? y = 8x; y = 8(6) = 48 It would cost $48 for the Carlos family to bowl at Awesome Strikes Bowling Alley. f.

DOK-2 How much would it cost for 6 members of the Carlos family to go bowling at Magic Strikes Bowling Alley? y = 5x + 3; y = 5(6) + 3 = 33 It would cost $33 for the Carlos family to bowl at Magic Strikes Bowling Alley.

DOK-2 If Awesome Strikes Bowling Alley decided to mimic Magic Strikes Bowling Alley and charge $3 for shoes, how would the equation change? It would no longer be proportional. It would become non-proportional because it would have the y-intercept at 3. The equation would change to y = 8x + 3.

Depending on your students, these questions may contain vocabulary and concepts that require review. The last experience with proportional relationships and slope may have been Scopes 4 and 5 in 7th grade.

FACILITATION TIP Consider creating a list of common proportional vs. non-proportional situations for students to reference. Many cell phone plans, gym memberships, rental agreements, and subscriptions can provide examples.

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CREATE NON-PROPORTIONAL RELATIONSHIPS FROM PROPORTIONAL RELATIONSHIPS

Create Non-Proportional Relationships from Proportional Relationships Explore 1 – Create Non-Proportional Relationships ACTIVITY PREPARATION Students will create non-proportional relationships by using proportional relationships shown on tables, graphs, and equations.

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 Job Listing 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 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.

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION Part I FACILITATION TIP Project a printed version of the scenario and allow students to note down the important values and phrases. FACILITATION TIP This Student Journal is ﬁve pages. Print pages 1 and 2 back to back in one color and pages 3 and 4 back to back in a different color. Keep page 5 to use as a projected visual to guide the small group and whole class discussions.

2. 3.

FACILITATION TIP Before giving out the Student Journal project these directions and read them together with students. Be prepared to clarify proportional relationship vs. nonproportional relationship for students.

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 Student Journal to each student. Explain the following to the class: We are going to use the tables provided to determine the amount of money Jake will make. Then, we are going to determine how that amount changes when the proportional relationship is changed to a nonproportional relationship. 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? -axis? The x-axis represents the number of hours.

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

DOK-2 How does adding a signing bonus affect the graph? The signing bonus means the graph no longer goes through the origin.

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. 170

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II 1.

2. 3.

4. 5.

Read the following scenario to the class: Jake encouraged his friend Jania to apply for assistant coaching jobs, too! She started looking up job listings online. The job listings she saw showed the pay rate with and without signing bonuses. Help Jania graph the pay rate and determine the amount she’ll be able to earn. Give a set of Job Listing Cards to each group. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How did you determine where to start your graph? If the equation was in the form y = mx, I started at the origin (0, 0). If the equation was in the form y = mx + b, I started at b (the y-intercept).

DOK-1 How did you determine how much Jania’s salary increased each hour? By using the rate of the equation, I was able to determine the slope of the line on the graph.

DOK-2 How do you differentiate between proportional and nonproportional relationships in scenarios? Scenarios that have a proportional relationship just have an amount per hour. Scenarios that have a non-proportional relationship have the amount per hour and the signing bonus.

DOK-2 What do you notice about all of the points on the line? All of the points on the line make the equation true.

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 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. DOK-2 If all of the other solutions to the equations were graphed, where would they appear? All of the solutions would appear along the line on the graph. DOK-2 What did you notice about the scenarios of the offers with proportional relationships? The scenarios with proportional relationships give the amount earned per hour with no signing bonus. DOK-2 When the x-axis is 0, what does the point on the yy-axis -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 To quickly engage students, brieﬂy project the Job Listing Cards and have students vote to predict which one they think is the best deal.

STEMscopes Tip The Skills Quiz, located in the Evaluate section, is a short standardsbased assessment where students demonstrate their computational ﬂuency. These assessments include a variety of question types and can be used to formatively evaluate students’ knowledge about topics covered in the scope or to review the content.

CREATE NON-PROPORTIONAL RELATIONSHIPS FROM PROPORTIONAL RELATIONSHIPS

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Post-Explore 1. 2. 3. 4.

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CREATE NON-PROPORTIONAL RELATIONSHIPS FROM PROPORTIONAL RELATIONSHIPS

Create Non-Proportional Relationships from 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

Create Non-Proportional Relationships Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Interactive Notebook

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

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

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

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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

Distinguish between Proportional and Non-Proportional Relationships Independent and partner games and other activities that provide students with an engaging way to practice the new concept

CREATE NON-PROPORTIONAL RELATIONSHIPS FROM PROPORTIONAL RELATIONSHIPS

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Intervention and Assessment SMALL-GROUP PLANNER 1

Students who are approaching mastery and need review

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

CREATE NON-PROPORTIONAL RELATIONSHIPS FROM PROPORTIONAL RELATIONSHIPS

Create Non-Proportional Relationships from Proportional Relationships

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Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

I can identify two lines with the same slope but different intercepts as translations of each other.

I can explain that the slope of a line is the rate of change.

I can use algebraic reasoning to show and explain that the graph of an equation represents the set of all of its solutions.

I can identify the parts of the equation y = mx + b, where m is the slope and b is the y-intercept.

What prompts will be used?

What does mastery look like?

CREATE NON-PROPORTIONAL RELATIONSHIPS FROM PROPORTIONAL RELATIONSHIPS

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I can identify proportional relationships by using the idea that one variable is conditioned on another.

I can relate tables and graphical representations to contextual and mathematical situations on the coordinate plane.

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SCOPE 1

Functions Scope Introduction SCOPE SUMMARY

Student Expectations

In this scope, students will understand that in a function, a rule is created that assigns every input to exactly one output. Students will use the inputs and outputs as ordered pairs in order to graph functions. They will understand the precise language that is used when pertaining to functions, such as range and domain, and will be able to state the differences and relationships between them. Students will be able to recognize functions as graphs, tables, and ordered pairs. They will be able to describe the functional relationship between two quantities through the analysis of a graph. In contrast, they will also be able to sketch a graph based on a verbal description of the features of a function. Students will determine whether a function is increasing or decreasing based on the provided slope. They will be able to distinguish given functions as linear or nonlinear from the context using appropriate vocabulary.

8.FGR.5.1 Show and explain that a function is a rule that assigns to each input exactly one output. 8.FGR.5.2 Within realistic situations, identify and describe examples of functions that are linear or nonlinear. Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grades, students learned how to plot points on a graph. They learned about the relationship between the x-- and yy-coordinates -coordinates and how to analyze tables and graphs. In 7th grade, they discovered proportional relationships between quantities. All of these concepts will tie together in order for students to understand the basics of functions. In previous grades, students gained a solid understanding of the relationships between tables and graphs. They were able to plot various points on a graph based on the given data in a problem. Students found regularities between different graphs and tables, preparing them for ﬁnding correspondences between functions.

In the coming years, students will continue their work with linear and nonlinear functions. They will expand this thinking into quadratic, exponential, logarithmic, and other types of functions. Students will learn how to express the inputs and outputs of functions in function notation as well as how to build and interpret all types of functions and their rules. They will continue their work with functions as they begin to write in formal function notation. Students will describe the features of a function such as increasing, decreasing, domain, and range according to the interval notation. They will evaluate the output values according to the individual inputs and create equations and function rules from this data. Students will become familiar with different types of nonlinear functions including quadratic, exponential, and trigonometric.

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

use variables to represent two quantities in a problem that change in relationship.

•

use variables to represent two quantities in a problem that change in relationship.

•

determine misconceptions about analyzing relationships between dependent and independent variables using graphs and tables, and.

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: •

analyze information from graphs, tables, and diagrams.

•

look at models and decide which is not a function.

•

explain reasonings.

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.

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

FUNCTIONS

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In this exploration, students will help analyze monthly and quarterly graphs of deposits and revenues for a boutique. Students will: •

determine which graphs are functions and which are not by looking at them.

•

analyze graphs.

•

Explore 2

Understand Functions on a Graph

Understand Functions on a Table In this exploration, students will be tasked with solving an extension scenario involving the boutique. Here, students are tasked with taking inventory by style, color, size, and cost for the boutique. Students will: •

ﬁnd patterns to recognize that solutions to positive perfect cubes can never be negative.

compare the graphs of functions and nonfunctions.

•

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

apply their knowledge of cubing being the inverse to taking cube roots.

•

calculate solutions to problems containing either a perfect cube or a cube root.

Analyzing Graphs In this exploration, groups of students will betasked with solving a real-world scenario, where students are tasked with analyzing and using data from a data tracking software program to analyze graphs about speed, distance, and time traveled. 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 3

Explore 1

EXPLORE ACTIVITIES

analyze the graph and describe it as linear or nonlinear and increasing or decreasing.

Sketching Graphs In this exploration, students will be tasked with solving an extension scenario involving the previous exploration. Here, students are tasked with using data from a data tracking software program to analyze graphs about speed, distance, and time traveled. 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.

sketch a graph that shows the qualitative features of a function that can be described as linear or nonlinear and increasing or decreasing.

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

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FUNCTIONS

Functions 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

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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. 8.PAR.4.1 Use the equation y = mx (proportional) for a line through the origin to derive the equation y = mx + b (non-proportional) for a line intersecting the vertical axis at b. 8.PAR.4.2 Show and explain that the graph of an equation representing an applicable situation in two variables is the set of all its solutions plotted in the coordinate plane.

Materials

Preparation

Printed •

• •

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

Print 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.

The second statement is a lie. The equation should read y = 30x + 75.

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

FACILITATION TIP Instead of distributing copies to each student or group, project one copy for the whole class to view. FACILITATION TIP Have the students identify the independent and dependent variables in each statement. This will help the justiﬁcation for this particular lie.

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Functions Hook – Function or Malfunction? ACTIVITY PREPARATION Students will look at four models, decide which model does not show a function, and explain their reasoning.

Materials

Preparation

Printed •

• •

1 Function or Malfunction? (per class)

Reusable •

•

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Function or Malfunction? 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 wanted to start a business, what type of business would you start?; 2) What expenses would be involved in starting your new business?; 3) How would you get the money needed to start your business?

STEMscopes Tip Supplemental Aids, located in the Intervention section, provide materials that will meet the needs of diverse learners. These materials include graphic organizers, handouts, and manipulatives that can further support students.

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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: Kamea’s older sister Hiapo is starting a cupcake business. First, she baked in her own kitchen for small events. People loved her cupcakes so much that she has saved enough money to buy a food truck! She has collected all sorts of data on ﬂavors, prices, locations, times, expenses, ingredients, advertising, and other considerations. She has put a lot of information into graphs, tables, and diagrams to make the information easy to interpret. Hiapo believes that all of her displays of information show functions. Kamea says they do not. Which sister 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. Student answers will vary. I notice that Hiapo is using graphs, tables, and other diagrams to display data. I wonder whether all of Hiapo’s displayed data are functions. Will I be able to look and see whether Hiapo has used functions? I can use math to determine whether something is a function or not. Project Function or Malfunction? Explain to students that Kamea is looking at four representations of data and says that some are not functions. She can tell by looking at it. Discuss the following: a.

DOK-1 What do you think a function is? Accept all reasonable answers.

DOK-1 How did Hiapo display her data? Tables, graphs, and diagrams

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II: Post-Explore 1. 2.

FUNCTIONS

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Show the Phenomena Video again, and restate the problem. Refer to Function or Malfunction? and discuss the following questions: a.

DOK-1 What is a function? A special relationship in which each input has a single output

DOK-1 If something is a function, how many y values exist for each x value? Only one

DOK-1 How can you determine whether something is NOT a function? It will have more than one y value for an x value.

DOK-1 Can something be a function if a y value has more than one x value? Yes

e. DOK-1 Look at square number one. Is there more than one y value for each x value? No, each x value has only one y value. Is it a function? Yes, it is a function. f.

DOK-1 Look at square number two. Is there more than one y value for each x value? Is it a function? No, each x value has only one y value. Is it a function? Yes, it is a function. DOK-1 Look at square number three. Is it a function? No, it is not, because there are two y values for the x value of 12.

h. DOK-1 Look at square number four. Is it a function? No, it is not a function because the x value of 2 has a y value of both 25 and 30, so it is not a function.

FACILITATION TIP You could also review previously learned terms such as proportional relationship or linear relationship. Have students identify these relationships in the representations in the tables or graph. FACILITATION TIP After this activity is completed, allow students to come up with their own example and nonexample of a function. Students may use a graph, table, or diagram to represent their data. FACILITATION TIP Point out that if there is more than one y value for an x value, connecting them will form a vertical line. Ask students if they recall the slope of a vertical line. They should remember that it is undeﬁned.

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Functions Explore 1 – Understand Functions on a Graph ACTIVITY PREPARATION Students will determine which graphs are functions and which are not by looking at them. They will be able to compare the graphs of functions and nonfunctions to one another.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

1 Student Journal (per student) 1 Set of Monthly Deposits Cards (per group) 1 Exit Ticket (per 2 students)

Reusable • •

1 Gallon-sized resealable bag (per group) 1 Quart-sized resealable bag (per group)

• • • •

•

Plan to have the class work in groups of 4 to complete this activity. Print a Student Journal for each student. Print a set of Monthly Deposits Cards for each group. If desired, print them on card stock, and laminate them for future use. Cut out the Monthly Deposits Cards. Place the Part I cards in a gallonsized resealable bag labeled “Part I” for each group. Place the Part II cards in a quart-sized resealable bag labeled “Part II” for each group. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student has one.

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) What types of information do business owners need to document?; 2) What types of information do you think business owners need to include on their taxes? After reading the scenario, ask the class 3) When are graphs functions or not functions?

2. 3. 4.

Read the following scenario to the class: Rosie is the sole owner of Rosie’s Boutique. As owner, she has the job of making sure all of the store’s ﬁnances are well documented. It is almost tax season, and her accountant has asked her to gather together these important documents and send them over to him. Before she can do that, she must create monthly and quarterly graphs of her deposits and revenues. Help Rosie analyze her graphs to determine whether they are functions or not. Give a Student Journal to each student. Give a set of Monthly Deposits Cards to each group. Explain to the class that they will use the Monthly Deposits Cards Part I to graph the coordinates of the monthly deposits and analyze each graph. Discuss the following questions with the class:

FACILITATION TIP

Some students may struggle with labeling the axes themselves. Encourage students to count how many gridlines there are to determine what scale to use.

DOK-1 What could you label the x-axis? Student responses may vary. I can label the x-axis with the months in the ﬁrst half of the year.

DOK-1 What could you label the y-axis? y Student responses may vary. I can label the y-axis as “Deposit (dollars),” starting at 200 and counting by 200s to 2,000.

FACILITATION TIP

DOK-1 Would it make sense to connect the points with a line? Student responses may vary. No, it does not make sense to connect the points with a line.

Have students highlight or circle with a different color any points on the graph that indicate that this is not a function. The ﬁrst graph is a function, but the second graph is not.

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Students will collaborate with their groups to graph each monthly deposit on the coordinate plane on their Student Journals. Then, students will analyze their graphs to answer the questions that follow.

7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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

DOK-1 When plotting coordinates on a graph, which direction does the x value go? The x value moves left or right on the graph.

DOK-2 In diagram 1, why are three different arrows pointing to $1,550? Three inputs all have the same output.

DOK-2 In diagram 2, why are there two different arrows coming from November? November has two different outputs.

DOK-1 Why do you think Rosie deposited money twice in the month of December? Answers may vary. She is making more money because of the holidays.

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

Math Chat DOK-1 What was the input for the graphs in Part I? The input was each month Rosie earned money. • DOK-1 What was the output for the graphs in Part I? The output for the graphs was the deposit or deposits Rosie made each month. • DOK-2 How many outputs were there for each input in diagram 1? Each input has exactly one output in diagram 1. •

FUNCTIONS

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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.

Explain the following to the class: Mathematicians call this type of diagram or graph a function. A function is when each input has exactly one output. DOK-2 How many outputs were there for each input in diagram 2? In diagram 2, July, August, September, and October each had exactly one output, but November and December each had two outputs. • DOK-2 Is diagram 2 (2nd Half of the Year graph) showing a function? No, the 2nd Half of the Year graph does not show a function because there are two inputs that have more than one output. •

Part II 1.

4. 5.

Read the following scenario to the class: At the end of each week, Rosie counts all the revenue from the store and writes it down. On particularly busy weeks, she may count the money more than once. She then creates a monthly graph to compare her weekly revenue. Help decipher Rosie’s graphs to help answer questions for her accountant. Explain to students that they will be analyzing the graphs and maps of each month to determine whether they are functions or not. They will be discussing the different inputs and outputs and how they can be arranged to make functions. Monitor and assess students as they are working by asking the following guiding questions: a.

DOK-1 How do you know when there is more than one output? There is more than one output for one input if there are multiple y values for the same x value.

DOK-1 Which value determines whether the graph is a function, the input or the output? The input determines if the graph is a function.

DOK-1 How can inputs and outputs be written as coordinates on a graph? Inputs are the x values, and outputs are the y values.

FACILITATION TIP Before reading the scenario, ask the class 1) How often do you think business owners record their revenue? After reading the scenario, ask the class 2) Why might Rosie count the money more than once on busy weeks?; 3) What types of information would analyzing Rosie’s monthly graphs give her accountant? FACILITATION TIP If students are still struggling with determining if a graph is a function, have them do the pencil test. They will hold their pencil vertically on the graph, then slide it along the x-axis. If it comes to any two points that are both on that same vertical line created by the pencil, the graph is not a function.

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

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Functions Explore 1 – Understand Functions on a Graph Math Chat DOK-2 When Rosie counts her money more than once a week in Part II, does this represent a function? Why or why not? It does not represent a function. When she counts her money more than once, it creates two different outputs for the same input. • DOK-1 Deﬁne a function in your own words. A function is when one x value does not have multiple y values. • DOK-1 Do duplicate output values with different input values affect whether a graph is a function or not? Why or why not? No, they do not. A function only focuses on the uniqueness of the input. •

Post-Explore FACILITATION TIP

On this Exit Ticket, consider asking students to explain their responses rather than just replying yes or no.

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

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Functions Explore 2 – Understand Functions on a Table ACTIVITY PREPARATION Students will determine which tables are functions and which are not by looking at them. They will be able to compare the tables of functions and nonfunctions to one another.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

• • •

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

Plan to have the class work in groups of 4 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Weekly Count 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 is considered “inventory” in a business?; 2) Why do business owners keep track of their inventory? FACILITATION TIP Print and project this scenario for students to read along with you. Provide some time for them to make observations about key math phrases and terms.

2. 3. 4.

FACILITATION TIP Instead of giving out cards to each group, you could post cards at stations around the room and have the students rotate between the stations.

Read the following scenario to the class: Every week, Rosie takes an inventory of her products. She needs to know how many items she has sold so she can order more from the warehouse. To ensure she has the correct items in stock, she counts her inventory many different ways. These include by style, color, size, or cost. On busy weeks, she takes inventory twice. Your job is to help Rosie analyze her inventory lists to determine whether they create functions. Give a Student Journal to each student. Give a set of Weekly Count Cards to each group. Explain to students that they will be analyzing tables and written information to determine whether the inventory lists create functions. They will be discussing the different inputs and outputs and how they can be arranged to make functions. Monitor and assess students as they are working by asking the following guiding questions:

FACILITATION TIP Look out for students who reverse the criteria for a function. Students may see that a y value, or output, repeats and decide that is not a function; however, they should look to see if the x values repeat. FACILITATION TIP When students ﬁnish, have them identify whether or not any of the functions represent linear relationships. This will tie in to a previous scope 186

DOK-1 How do you know when a table is a function or not? If there are no repeating values in the input column, the table is a function.

DOK-1 How can you create a table from coordinates? The x values go into the input column, and the y values go into the output column.

DOK-1 What happens if you have repeating outputs? Repeating outputs do not affect whether a relationship is a function.

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

Math Chat •

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

DOK-2 What is the signiﬁcance of inputs when determining a function? The inputs cannot repeat. If they repeat, it is not a function. • DOK-2 Does it matter where the inputs and outputs are placed in the table? Why or why not? Yes, inputs determine the outputs. • DOK-2 Which coordinate does the input represent? Which coordinate does the output represent? Inputs represent x values. Outputs represent y values. •

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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.

FACILITATION TIP On this Exit Ticket, consider asking students to explain their responses rather than just replying yes or no.

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Functions Explore 3 – Analyzing Graphs ACTIVITY PREPARATION Students will analyze the graph and describe it as linear or nonlinear and increasing or decreasing.

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 Road Trip Cards (per group) 1 Analyzing Data on a Graph Card (per group) 1 Exit Ticket (per student)

• • •

•

Reusable • • • •

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

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 Road Trip Cards for each group. Cut them 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 Analyzing Data on a Graph Card for each group. Place it inside a clear sheet protector for each group. 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) Has anyone ever been on a road trip before? Where did you go?; 2) What kinds of information might data tracking software provide?

2. 3.

FACILITATION TIP

Have students describe what else they know about the graphs. What is the slope, y-intercept, independent variable, dependent variable, etc? If a graph is linear, is it a proportional relationship?

FACILITATION TIP Let students make the connection that when a graph is increasing it has a positive slope, and when it is decreasing it has a negative slope.

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Read the following scenario to the class: Lashawn and Diego are going on a road trip during their summer break. They will use the analytics from their data tracking software program to keep track of the speed, distance, and time traveled using graphs. Help Lashawn and Diego analyze the graphs from their car’s analytic reports. Give a Student Journal to each student. Give a set of Road Trip Cards, an Analyzing Data on a Graph Card, and a dry-erase marker to each group. Explain to students that they will analyze each graph and describe the graph as linear or nonlinear and increasing or decreasing by using the Analyzing Data on a Graph Card. Have students use the Analyzing Data on a Graph Card to analyze the graph and describe it as linear or nonlinear and increasing or decreasing. (Note that students should understand how a graph looks when it is described as linear, nonlinear, increasing, or decreasing.) Project the Analyzing Data on a Graph Card. (Note that when showing graph 3, you should show students where the function is increasing and decreasing.) Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 How can you determine whether a function is linear? Which function(s) are linear? Student responses will vary. A linear function is a function whose graph is a straight line. The functions that are linear are 1 and 2. © Accelerate Learning Inc. - All Rights Reserved

8. 9.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How can you determine whether a function is nonlinear? Which function(s) are nonlinear? Student responses will vary. The graph of a nonlinear function is not a straight line. It is a curved line. The function that is nonlinear is 3.

DOK-1 How can you determine whether a function is increasing? Which function(s) are increasing? Student responses will vary. A function is increasing when the y value increases as the x value increases. The function that is increasing is 2.

DOK-1 How can you determine whether a function is decreasing? Which function(s) are decreasing? Student responses will vary. A function is decreasing when the y value decreases as the x value increases. The functions that are decreasing are 1 and 3.

Explain to students that they will analyze each graph from Lashawn’s and Diego’s road trip and will describe the graph as linear or nonlinear and increasing or decreasing on their Student Journals. Have students use the graphs to describe the graph as linear or nonlinear and increasing or decreasing. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 How can you describe the relationship the graph shows? Student responses will vary. The distance from the zoo entrance graph shows a linear relationship that is increasing.

DOK-2 Which graphs show a linear relationship? The distance from the zoo entrance graph and the distance from home graph show a linear relationship. (Note that students should notice that the distance to the hotel and the distance to the museum include intervals that show a linear relationship.)

Intervention

Acceleration

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FACILITATION TIP Allow students who finish early to flip through a math textbook or search online to research what type of function the nonlinear graphs represent. Have them discuss with a partner or group before presenting their findings.

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 Compare and contrast graphs that show linear relationships and graphs that show nonlinear relationships. When the graph shows a straight line, it represents a linear relationship, and when the graph shows a line that is not straight, the graph shows a nonlinear relationship. Both linear and nonlinear graphs can show relationships that are increasing and decreasing. • DOK-2 What is an example of a real-world situation where the function is linear? Driving home at a speed of 65 miles per hour is an example of a real-world situation where the function is linear. • DOK-2 What is an example of a real-world situation where the function is nonlinear? A rocket launched in the sky increasing in height and then decreasing in height is an example of a real-world situation where the function is nonlinear. •

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 To create another real-world connection, many middle school physics lessons include graphs about speed/distance and rate of change. Consider using these images or class experiment results in your discussion. FACILITATION TIP Before this Math Chat, find some additional real-world engaging examples (race cars, local companies profits, how long it takes to finish an assembly job). FACILITATION TIP For this Exit Ticket, provide success criteria for the written explanation. If needed, provide sentence starters for students.

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Functions Explore 4 – Sketching Graphs ACTIVITY PREPARATION Students will sketch graphs that show the qualitative features of a function that can be described as linear or nonlinear and increasing or decreasing.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.6 Attend to precision.

Preparation

Materials Printed • • •

• • •

1 Student Journal (per student) 1 Set of Beach 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 Beach Cards for each group. Cut them 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 Before reading the scenario, ask the class 1) Has anyone ever vacationed at a beach? Where did you go?; 2) If you had a software tracking program that graphed the speed, distance, and time traveled to get to your vacation spot, what types of information could you glean by analyzing the graphs? FACILITATION TIP Have a beach day! Set up stations around the room with beach toys (sand bucket, beach ball, sunglasses, etc) and post one scenario card at each station. Have the students take their own “road trip” to visit each station.

2. 3. 4. 5.

Read the following scenario to the class: Lashawn and Diego are going to the beach for another vacation. They will use their software tracking program to keep track of the speed, distance, and time traveled using graphs. Help Lashawn and Diego sketch and analyze the graphs from their software tracking program. Give a Student Journal to each student. Give a set of Beach Cards to each group. Explain to students that they will sketch graphs and describe the graphs as linear or nonlinear and increasing or decreasing. Have students sketch graphs and describe the graphs as linear or nonlinear and increasing or decreasing. (Note that students should be able to create a story based on the graph and understand why time is the independent variable.) Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-2 How can you sketch a graph based on a story? Student responses will vary. When sketching a graph based on a story, you need to identify what the independent variable and dependent variable are in the story. You also need to understand what words describe the qualitative features that the graph should have.

DOK-2 How can you distinguish between a linear and a nonlinear relationship on a graph? Student responses will vary. A linear function has a constant rate of change. A nonlinear function does not have a constant rate of change.

FACILITATION TIP Struggling students may need a scaffold or lined paper to successfully sketch the graphs. Provide clear constraints for what needs to be included/not included on each graph. FACILITATION TIP Make sure students understand that they are just doing a rough sketch. They don’t need their graphs to be at speciﬁc values, because the scenarios do not list any values. They are only showing a sketch of the line on the graphs. 190

6. 7.

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

Math Chat

Intervention

Acceleration

FACILITATION TIP

DOK-2 How did you determine whether the verbal description was describing a linear relationship? If the Beach Cards describe the relationship as increasing, decreasing, or constant, then the relationship is linear because the rate of change is constant with respect to time. • DOK-2 How did you determine whether the verbal description was describing a nonlinear relationship? If the Beach Cards describe the relationship as gradually increasing, gradually decreasing, or slowing down, then the relationship is nonlinear because the rate of change is not constant with respect to time. • DOK-2 What is an example of a real-world situation where the function is both linear and nonlinear? An example could be 2 people running a race. One person’s representation on a graph could be linear because that racer has a constant rate of speed, and the other person’s representation on the graph could be nonlinear because that racer is not running at a constant rate of speed. •

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If students are struggling with the ﬁrst reﬂection question, ask them to think about how what is happening in the verbal description is changing. Is it changing at a constant rate or is it changing at a gradual rate?

FACILITATION TIP Take time to research some engaging local examples of real-world linear/nonlinear functions.

Post-Explore 1. 2. 3. 4.

FACILITATION TIP For this Exit Ticket, provide success criteria for the written explanation. If needed, provide sentence starters for students.

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Functions 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

Understand Functions on a Graph 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

Understand Functions on a Table 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

Analyzing Graphs

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

Sketching Graphs

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

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

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

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

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

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Functions 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

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What prompts will be used?

What does mastery look like?

I can use algebraic reasoning to identify and explain that a relationship is a function or not a function.

I can describe the graph of a function as the set of ordered pairs consisting of an input and the corresponding output.

I can use graphs to model practical situations and interpret them based on the situations.

I can use mathematical language to explain the difference between linear and nonlinear functions.

I can analyze a graph by determining whether the function is increasing or decreasing and whether the function is linear or nonlinear.

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Rate of Change and Initial Value Scope Introduction SCOPE SUMMARY Students will construct functions to model linear relationships between two quantities. They will determine the rate of change (slope) as well as the initial value (y-intercept) of the function from a given description, graph, equation, or table. Students will understand that the slope is found from the ratio created by the change in the y values and the change in the x values. They will become familiar with ﬁnding the y-intercept at the point when the x value is equal to zero. Student Expectations

8.FGR.5.3 Relate the domain of a linear function to its graph and where applicable to the quantitative relationship it describes.

8.FGR.5.8 Explain the meaning of the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 8.FGR.7.1 Interpret and solve relevantmathematical problems leading to two linear equations in two variables.

Future Expectations

In previous grades, students were introduced to problems with a single variable. They gained experience dissecting tables and graphs, determining any regularity between them. Students learned how to ﬁnd proportionality and multiplicative relationships between values. All of these ideas will be key concepts for students to break down a function into its various pieces.

Students will continue to use slope in mathematics as they enter high school. As they learn more about functions, they will begin to dive deeper into evaluating speciﬁc pieces of what makes up a function. They will analyze the graphs of functions as well as interpret and recognize function notation. Students will also gain an understanding of how to determine the slope between certain segments of nonlinear functions.

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

use proportional relationships to solve multistep ratio and percent problems.

•

identify two truths and a lie by reading statements.

Hook

8.FGR.5.7 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph.

Background Knowledge

Accessing Prior Knowledge

8.FGR.5.4 Compare properties (rate of change and initial value) of two functions used to model an authentic situation each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

VERTICAL ALIGNMENT

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

make predictions based on a rate of change based on linear functions.

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

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Determine the y-intercept and Rate of Change In this exploration, groups of students will be tasked with determining the number of hours that people volunteer at a hospital. Students will: •

Explore 2

Explore 1

EXPLORE ACTIVITIES

identify the yy-intercept -intercept from a graph, table, equation, or verbal description.

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.

Comparing Key Features In this exploration, groups of students will determine which graph represents the ideal allowance representation.Students will: •

analyze a representation of a function.

•

compare functions.

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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Create Equations In this exploration, students will create equations for scenarios that are given as descriptions, graphs, or tables. Students will: •

match the correct equation with the verbal description found in an advertisement.

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

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RATE OF CHANGE AND INITIAL VALUE

Rate of Change and Initial Value 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 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. 7.PAR.4.9 Use proportional relationships to solve multistep ratio and percent problems presented in applicable situations.

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.

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

Read the prompt aloud to the class. Allow 2 minutes of thinking time for the students to read the three statements and determine which two statements are truths and which one is a 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 choice for the lie. a.

The ﬁrst statement is the lie because two loaves should cost a total of $6.

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 Provide students with dry erase boards for scratch work. They will need to compute 50% of $4, then add that to $4. They may not be able to compute this using mental math. FACILITATION TIP Partners that ﬁnish their discussion early could make a table showing the cost of a variety of quantities of bread loaves. The number of bread loaves would be the independent variable. The cost would be the dependent variable.

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RATE OF CHANGE AND INITIAL VALUE

Rate of Change and Initial Value Hook – Rate of Change and Initial Value ACTIVITY PREPARATION Students will make predictions based on linear functions.

Materials

Preparation

Printed •

• •

1 Rate of Change and Initial Value (per class)

Reusable •

•

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Rate of Change and Initial Value 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 or reading the scenario, ask the class 1) Has anyone seen, been on, or read about military ships?; 2) What do military ships do?; 3) What types of machinery do military ships have on them?

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.

2. 3.

4. 5.

DOK-1 What information would you need to answer the question? Allow students to share all ideas. Student answers will vary. We could use the slopes, y-intercepts, and equations of each line to help us answer the question.

DOK-1 How could the skills used for answering this question connect to the satellite and laser? The satellite is far away, and we would need to be able to program the military ship to shoot the laser on the correct line or trajectory to hit the satellite. We can see the initial path of each line, but cannot see the graph for large x values.

DOK-2 Why is graphing an ineﬃcient method for answering this question? We would have to extend each of the graphs very far to verify which line passes through (510, 341).

Explain to students that we will use different forms of lines throughout the scopes and that they should spend time deciding which form is most helpful in each new scenario.

e. Complete the Explore activities. 200

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and restate the problem. Refer to Rate of Change and Initial Value, and discuss the following questions: a.

DOK-1 What skills can you use to answer this question? I could ﬁnd the slope of each line using two points and then write linear equations in point-slope or slope-intercept form.

DOK-1 Which line passes through (510, 341)? The blue line.

DOK-1 Write the equation for the correct line in two different ways. 2 2 y = ___ + 1 and y − 341 = __3(x – 510). 3x

DOK-1 Do you feel that you have a strong understanding of linear functions? Answers will vary based on students’ success during the activity and conﬁdence level.

FACILITATION TIP After students answer the question, have them write an equation for each of the other lines in the form of their choice. Then, ask, by a show of hands, who chose slopeintercept form and who chose point-slope form. Next, show an accurate equation for both forms for both lines, and solve for y for both lines when x = 510 with the class. Remember, point-slope form has inﬁnite correct equations.

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RATE OF CHANGE AND INITIAL VALUE

Rate of Change and Initial Value Explore 1 – Determine the yy-intercept -intercept and Rate of Change ACTIVITY PREPARATION Students will identify the yy-intercept -intercept and rate of change from a graph, table, equation, or verbal description.

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 Bike Rental Cards (per group) 1 yy-intercept -intercept Work Mat (per group) 1 Equation of a Straight Line Card (per teacher) 1 Exit Ticket (per student)

•

Reusable • • • •

•

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

• •

Plan to separate 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 Bike Rental 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 cards on card stock, and laminate them for future use. Print a yy-intercept -intercept Work Mat for each group. Place it inside a clear sheet protector for each group. If desired, print it on card stock, and laminate it for future use. Print an Equation of a Straight Line Card for each teacher. If desired, print it on card stock, and laminate it for future use. Be prepared to project the Equation of a Straight Line Card for the class to see. Gather enough dry-erase markers for each group to have one.

PROCEDURE AND FACILITATION FACILITATION TIP Project this scenario and allow time for students to read it silently, then chat with a shoulder partner and note down essential phrases and values. Read it all together as a class and coach students to answer, “What do we know? What do we need to ﬁnd out?” FACILITATION TIP Depending on your students, consider reviewing key vocabulary before giving out the Student Journals (ordered pair, y-intercept, dependent and independent variable). FACILITATION TIP

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Part I: Determine the y-intercept and Rate of Change in Tables and Graphs 1.

2. 3. 4. 5.

Read the following scenario to the class: Javier’s family is taking a family vacation and plans to rent bikes. He has been put in charge of determining the best place to rent bikes, and he is meeting with you to get some help. Javier needs help determining the initial cost or y value when the x value is 0 when using tables or graphs. He also needs to ﬁnd the cost per hour for each rental. Give a Student Journal to each student. Give a set of Bike Rental Cards Part I, a yy-intercept -intercept Work Mat, and a dry-erase marker to each group. Explain to students that they will analyze each Bike Rental Card carefully and determine the y value and cost per value for each rental. Have students use the tables and the graphs to determine the y value when the x value is 0. Monitor and assess student understanding as each group collaborates by asking the following guiding questions:

Clarify “the y value” and “cost per value” for students. Check for understanding before they begin collaboration.

DOK-1 What is the dependent variable? Total cost spent to ride bikes

FACILITATION TIP

Print and post these guiding questions to support and guide student collaboration.

DOK-1 What are the domain and range? The domain is the set of all the input values, and the range is the set of all the output values.

DOK-1 What does the y value represent in the table when the x value is 0? 60

a. DOK-1 What is the independent variable? Number of hours spent riding bikes

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

e. DOK-1 What does the y value represent in the graph when the x value is 0? 25 f. 6.

7. 8.

DOK-1 What is the ordered pair when x is 0? (0, 25)

Explain the following to the class: When you determined the y value when the x value was 0, you determined the y-intercept. The y-intercept is the point on a graph of an equation where the line crosses the y-axis. When you determined how much the bike rental cost per hour, you determined the rate of change. The rate of change determines the slope of the line on a graph. 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 How do you determine the yy-intercept -intercept from a table? The y-intercept can be found when the x value is equal to 0. If you are given a table that does not start at 0, you need to identify the pattern in the table and then use the pattern to count backward to 0. • DOK-2 How do you determine the rate of a change from a table? The rate of change can be found by locating y value when x is one or by dividing the change in the y values by the change in the x values. • DOK-2 How do you determine the yy-intercept -intercept from a graph? We can determine the y-intercept by looking at the graph and identifying the point where the line intersects the y-axis. • DOK-2 How do you determine the rate of change from a graph? We can look at a graph and determine the y value when x is one or by counting how much the y value increases when the x value increases by 1. •

Part II: Determine the y-intercept in Equations and Verbal Descriptions 1.

2. 3.

Read the following scenario to the class: Now that Javier has reviewed the bike rental costs in the tables and graphs, he needs your help looking at equations and verbal descriptions and identifying the y-intercept and rate of change. Explain to students that they will be looking at equations and verbal descriptions to identify the y-intercept. y-intercept. Project the Equation of a Straight Line Card and introduce students to the equation of a line, y = mx + b. b. (Note that students will only need to understand b as the yy-intercept -intercept and m as the rate of change for this Explore activity.) Have students use the yy-intercept -intercept Work Mat as needed to determine the yy-intercept -intercept for each situation. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 What is the equation of a line? y = mx + b

DOK-1 What does b represent? b is the y-intercept.

DOK-1 What does m represent? m is the rate of change.

DOK-2 How can you use the equation of a line to ﬁnd the y-intercept? y You can substitute 0 for x and solve the equation for y to get the y-intercept.

e. DOK-2 How do you determine what m and b are from a verbal description? m is the rate of change, so you can look for key words such as per hour or in a day. The b is the initial value; it will be a value that only occurs once in the scenario. 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.

FACILITATION TIP Repeatedly reinforce this explanation with students. They will need to hear and say the deﬁnition of y-intercept several times. Keep asking, “What is y when x is zero?” Practice ﬁnding the y-intercept will support future success in Algebra. FACILITATION TIP As with the y-intercept, repeatedly reinforce rate of change (or linear function). Find a way to help students see how it is related to rise over run/slope.

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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.

FACILITATION TIP If students have trouble ﬁnding the y-intercept from the verbal description, suggest that they write an equation to represent that description. Then, it should be easier for them to ﬁnd the y-intercept. FACILITATION TIP This printout can be saved and posted on your wall as a reference for students. Laminate or insert it in a sheet protector to make it more durable. FACILITATION TIP Print out these essential questions and have student take notes on their Student Journal or in a notebook. Have them label the parts of the standard equation. Take time to check for understanding by calling on students, partner pairs, and whole groups to answer verbally.

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Rate of Change and Initial Value Explore 1 – Determine the yy-intercept -intercept and Rate of Change Math Chat DOK-2 How can you describe the yy-intercept -intercept when given a verbal description? The y-intercept is the starting amount or the amount that does not change depending on the amount of time. • DOK-2 What is an example of a yy-intercept -intercept used in a real-world situation? The amount someone pays to rent a car before adding in the price per mile driven • DOK-2 How can you describe the rate of change when given a verbal description? The rate of change is the consistent change in the ratio of the y value to the x value. In a verbal description, it is the amount per one value. • DOK-2 What is an example of rate of change used in a real-world situation? The price per pound a grocery store charges for apples •

Post-Explore FACILITATION TIP This Exit Ticket could be used as a pre- and post-assessment.

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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Rate of Change and Initial Value Explore 2 – Comparing Key Features ACTIVITY PREPARATION Students will determine the domain for the function in relation to the situation. Students will determine which function has the greater rate of change when given one as a table and one as an expression. Students, when giving functions in scenarios, compare and describe slope and yy-intercept. -intercept.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Set of Allowance Options Graphs (per group) 1 Set of Allowance Options Cards (per group) 1 Exit Ticket (per student)

• • •

•

Reusable •

Separate the class into groups of 3 or 4 students. Print a Student Journal and an Exit Ticket for each student. Print a set of Allowance Options Graphs for each group of students. Cut the graphs apart, and place them in a resealable bag labeled “Part I.” If desired, print them on card stock, and laminate them for future use. Print a set of Allowance Options Cards for each group of students. Cut the cards apart, and place them in a resealable bag labeled “Part II.” If desired, print them on card stock, and laminate them for future use.

2 Resealable bags (per group)

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) Do you get an allowance?; 2) Does your allowance amount stay the same, or does it vary?; 3) What do you do with your allowance? FACILITATION TIP Part I, Question 1: Some students may be confused initially when trying to ﬁll out the second column of the table. Let them know that the title of the column refers to characteristics of a graph, and ask them to name characteristics of a graph.

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Read the following scenario to the class: Your family is looking to start you on an allowance before your family biking trip, and they’ve given you the freedom to pick which setup you want! The catch is that they have given you the options as representations of algebraic functions! Start by classifying the graphs to see which representations show allowances that are the most ideal. Give a Student Journal to each student. Give a set of Allowance Options Graphs to each group. Explain to students that they will work with their groups to sort the graphs based on certain characteristics. Where appropriate, students should list the letters in order. (Graphs with shared characteristics can be shown with a slash between them.) As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How do you identify the domain? The domain is the values for x in a function.

DOK-1 How do you determine an appropriate domain? Are all real numbers appropriate in all situations? An appropriate domain is one that makes sense in a scenario. Not all real numbers would make sense in all situations if a scenario wouldn’t use both negative and positive values. © Accelerate Learning Inc. - All Rights Reserved

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How are you finding the slope on the graphs? You can find two points and identify the rise/run between them. You can find two points and find the slope by hand using the formula. You can find two points and plug them into the linear regression in the calculator. DOK-1 How are you finding the intercepts on the graphs? The intercepts are where the lines cross the axis. So the y-intercept crosses the y-axis, and the x-intercept crosses the x-axis.

Intervention

Acceleration

FACILITATION TIP Students should recognize that all of the graphs are lines. Before asking this question, ask students if the slopes of the graphs are constant and how they know.

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

Math Chat •

•

DOK-1 What are different ways we can find the slope on a graph? You can find two points and identify the rise/run between them. You can find two points and find the slope by hand using the formula. You can find two points and plug them into the linear regression in the calculator. DOK-1 When you look at a graph with a positive slope, is it increasing or decreasing? What about a negative slope? Positive slope = increasing graph Negative slope = decreasing graph DOK-2 When considering an allowance option that has a negative slope, what does that mean about the allowance? Is this a good option? Why or why not? The negative slope means I would be paying someone, so that is not a good option for an allowance. Paying someone is the opposite of an allowance. DOK-2 When considering an allowance option, which graphs did you eliminate? Explain. I eliminated the graphs with negative slopes because I don’t want to give my money away. I want money to be given to me. DOK-3 Consider both the appropriate domain and slope. Conclude how they are related in this scenario. The domains for positive slopes are all positive values. When the slopes are negative, there is a limited amount of positive values for the appropriate domain.

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FACILITATION TIP Connect this scenario to the students. After students answer the question, ask them if they have a personal scenario where they have to pay someone repeatedly out of money they’ve been given or earned. Ask them if they are receiving money over time while they are making payments over time.

Part II 1.

2. 3. 4.

Read the following scenario to the class: Since no one wants to lose money, we are going to only look at the graphs that show allowance options with a positive slope! However, your parents give you some more options for your allowance using different representations. You will answer some questions about the remaining cards to ﬁgure out which allowance option you might want! Give a set of Allowance Options Cards to each group. Students should still have their Student Journals and the Allowance Options Graphs with positive slopes. Explain to students that they will work with their groups to answer the questions in Part II regarding the different representations. It could be valuable for students to identify the slope and yy-intercepts -intercepts on all of the cards ﬁrst before answering the questions. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 How can the slope or yy-intercept be determined from the verbal description? The starting amount is the y-intercept, and the slope is the weekly or monthly allowance or rate of change.

DOK-2 How can the slope or yy-intercept -intercept be determined from the table? The slope can be found using the slope formula, the change in y divided by the change in x. The y-intercept might be shown in the table, or the equation can be written to determine the y-intercept.

DOK-2 For the equations in standard form, are you able to identify the slope immediately? No, we need to isolate the y ﬁrst, and then we can ﬁnd the slope.

FACILITATION TIP After reading the scenario, ask the class 1) What other representations could be used to show allowance options?

FACILITATION TIP Students should know that they can also ﬁnd the y-intercept when they isolate y by plugging in 0 for x. After they answer the question, ask them if they can ﬁnd the x-intercept if they isolate x and plug in 0 for y. Have them explain their reasoning. 207

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Rate of Change and Initial Value Explore 2 – Comparing Key Features 6. 7.

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

Math Chat FACILITATION TIP After students answer the question, name each linear representation and have students raise their hands when you name the one that is easiest for them to understand. Then, ask the class for real-world scenarios for each linear representation.

• •

•

FACILITATION TIP Some students may forget that negative values for f(x) do not necessarily indicate a negative slope for the graph. They may also forget that the dependent variable in a linear equation should be isolated to ﬁnd the slope. Encourage students to complete the Exit Ticket attentively.

DOK-1 When comparing different representations of linear functions, what key features are helpful to identify? Slope, y-intercept, and domain DOK-1 How do we ﬁnd the slope in linear representations (a graph, a table, a word problem, an equation)? Graph: Identify rise over run from two points on the line of a graph. Table: Change in y divided by change in x Word problem: The value that is changing per unit of input Equation: Once y is isolated, the slope is the coeﬃcient of x. DOK-2 Which key feature on the cards we examined had a greater effect on the amount that is earned? Explain. Slope has a greater effect than the y-intercept because it changes how quickly money is earned or lost.

Post-Explore 1. 2. 3.

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

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Rate of Change and Initial Value Explore 3 – Create Equations ACTIVITY PREPARATION Students will create equations for scenarios that are given as descriptions, graphs, or tables.

Standards for Mathematical Practice • • • •

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Set of Bike Shop Flyer Match Cards (per group) 1 Set of Bike Shop Cards (per group) 1 Exit Ticket (per student)

• • •

•

Reusable •

Separate the class into groups of 3 or 4 students. Print a Student Journal and an Exit Ticket for each student. Print a set of Bike Shop Flyer Match Cards for each group. Cut the cards apart, and place them in a resealable bag labeled “Part I.” If desired, print them on card stock, and laminate them for future use. Print a set of Bike Shop Cards for each group. Cut the cards apart, and place them in a resealable bag labeled “Part II.” If desired, print them on card stock, and laminate them for future use.

2 Resealable bags (per group)

PROCEDURE AND FACILITATION

FACILITATION TIP Print and project this scenario for students. Provide independent time to read and take notes, shoulder partner check in and then read it aloud as a class.

Part I 1.

FACILITATION TIP

2. 3.

Depending on your students’ skill levels, consider projecting the guiding questions ﬁrst to review these processes.

FACILITATION TIP If needed, you could guide students through a few of the equations and verbal descriptions by projecting them. Coach students using questions 4a–4c as you problem solve.

5. 210

Read the following scenario to the class: Before your family goes on vacation, you get the information from the bike shops about their rental prices at their other chains. You apply your knowledge of algebraic equations to write functions for each of the advertisements to make it easier to compare and ﬁnd the best rates. Later that day, you ﬁnd that your notes got separated and you need to rematch the equation with the correct advertisement. Give a Student Journal to each student. Explain to students that they will work with their groups to match the correct equation with the verbal description found in the advertisement and answer the questions that follow. As students collaborate, monitor their work, and use the following guiding questions to assess student understanding: a.

DOK-2 What form are the equations in? What does each part of the equation represent? They are in slope-intercept form, y = mx + b. The x and y represent values of the equation, m is the slope, and b is the y-intercept.

DOK-2 How do you determine which value is the slope? What does the slope represent in this scenario? The value that is showing a consistent change is the slope. In these descriptions, it is the hourly or daily charge.

DOK-2 How do you determine which value is the yy-intercept? -intercept? What does the yy-intercept -intercept represent in this scenario? The y-intercept is a value that occurs one time. In these descriptions, it is the cost of equipment.

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 How did you determine if the values should be positive or negative? All the daily/hourly charges were money being paid, so I knew they were positive. When thinking about the y-intercepts, I knew if it was a discount, it was negative, but if it was money being paid for the rental, it was positive. • DOK-2 How was the process similar for writing all the equations? I had to determine which value was the constant change to identify the slope and what value was the one-time charge to identify the y-intercept. • DOK-2 To determine which is the best equation/shop for the rental, would you just look at the yy-intercept? -intercept? Why or why not? No, slope has to be considered as well. Slope is going to have a greater impact because it will affect how quickly the cost increases. •

Part II: Property Management 1.

2. 3. 4.

Read the following scenario to the class: Your family decides to bring some bikes on vacation with them and then rent the others. The bikes your family are bringing need tune-ups, though. Luckily, the shops all offer tune-up services. Create the equations for each shop from their tables or graphs to determine which shop has the best tune-up rates for your family. Give a set of Bike Shop Cards to each group of students. Explain to students that they will work with their groups to create equations in slope-intercept form that reﬂect the tables and graphs shown for each shop. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

5. 6.

DOK-1 How can you determine the yy-intercept on a table? If a value is given for (0, y), that y value is the y-intercept. If that value is not given. I can determine the slope and work backward until I determine the point for (0, y).

DOK-1 How can you determine the yy-intercept -intercept on a graph? Where the graph intersects the y-axis is the y-intercept.

DOK-1 How can you determine the slope on a table? I need to calculate the difference of the y values divided by the difference of the x values.

DOK-1 How can you determine the slope on a graph? I need to determine the rise and run. I can ﬁnd two points and complete the same process as I would for a table, or I can count the rise and run and then divide the answers.

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

Math Chat DOK-1 Which table or graph did not have an obvious yy-intercept? Explain how you calculated the yy-intercept -intercept for it. Two Wheels Only did not have a point for (0, y) given. I ﬁrst found the slope was 53. I then took the smallest y value of 89 and subtracted 53 twice because it was the y value for when x was 2. I found then when x is 0, y is −17. • DOK-2 Your friend wanted to argue that the slope for Bikes N’ More should be 60. Where is the error in their thinking? My friend would be looking at the point and thinking of unit rates because x is 1 and y is 60. However, the y-intercept isn’t 0, so that can’t be the case. •

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FACILITATION TIP Print and project the scenario. Read it aloud as a class with student volunteers. FACILITATION TIP There are only four bike shop cards. The page can be distributed whole to small groups or individuals without cutting. Alternatively, you could project them one at a time and guide students through the solutions if needed. FACILITATION TIP Project questions 4a–4d and have students record the steps for each process. Review and repeat as needed. Assess student understanding by calling on select students’ to respond to each question in their own words.

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.

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Rate of Change and Initial Value Explore 3 – Create Equations •

DOK-2 Your friend argues that the slope for Two Wheels Only is 106. What is the error in their thinking? They subtracted the y values but ignored that the x values are increasing by 2. This means 106 has to be divided by 2, making the slope 53.

Post-Explore FACILITATION TIP This Exit Ticket could be used as a pre- and post-assessment.

1. 2. 3. 4.

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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Rate of Change and Initial Value 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

Determine the y-intercept and Rate of Change 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

Comparing Key Features 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

Create Equations 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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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

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

Notes

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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

RATE OF CHANGE AND INITIAL VALUE

Rate of Change and Initial Value

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Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

What does mastery look like?

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I can compare the properties of two functions represented algebraically.

I can construct a function to model a linear relationship between two quantities.

I can determine the rate of change and initial value of the function from a description of a relationship or from two values.

I can explain the meaning of the rate of change and the initial value of a linear function when it is represented as a situation, a table, or a graph.

I can interpret and solve mathematical problems involving two linear equations in two variables.

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SCOPE 1

Linear Forms Scope Introduction SCOPE SUMMARY In this scope, students will expand their understanding of a function and properties of functions to identify key features and represent constraints of linear models. They will expand their understanding of solving systems graphically to solve or estimate the solutions of complex equations. Students will expand their understanding of rearranging and rewriting equations to translate linear equations between forms (standard, point-slope, and slopeintercept). They should be able to expand on their experience constructing a linear function to use the structure of each form to identify the key attributes of the linear function it represents. Students will make connections between linear functions and arithmetic sequences. Student Expectations

8.FGR.5.5 Write and explain the equations y = mx + b (slope-intercept form), Ax + By = C (standard form), and (y - y�) = m(x - x�) (point-slope form) as deﬁning a linear function whose graph is a straight line to reveal and explain different properties of the function. 8.FGR.5.6 Write a linear function deﬁned by an expression in different but equivalent forms to reveal and explain different properties of the function. 8.FGR.5.9 Graph and analyze linear functions expressed in various algebraic forms and show key characteristics of the graph to describe applicable situations.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grade levels, students have determined whether an equation or a set of ordered pairs were linear and determined if a graph was increasing, decreasing, linear, or nonlinear. Students have distinguished between proportional and nonproportional relationships using tables and graphs. They have created, solved, and interpreted a linear function based on the context given a table, graph, description, or algebraic model. Students have constructed a linear function given a description, two ordered pairs, a table, a graph, or a scatter plot. They have generated and veriﬁed equivalent expressions, rewritten expressions to understand how the quantities relate to the context, and solved equations involving distribution and rearranging like terms. They have compared properties of functions represented in different ways. Students have used similar triangles to compare slope and understand that a function is a rule that assigns to each input exactly one output. Students have solved systems of two linear equations graphically and understand that the point of intersection is the solution.

Students will apply linear skills to nonlinear functions throughout Algebra I. They will also examine linear factors of quadratics. Students will transform quadratic and exponential functions later in Algebra I. In Algebra II, students will continue manipulating linear and nonlinear functions. In future courses, students will use transformations to analyze many function families. When working with trigonometric functions, transformations are particularly important to analyze amplitude and period.

8.FGR.7.1 Interpret and solve relevant mathematical problems leading to two linear equations in two variables.

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

model a linear relationship between two quantities.

•

determine the rate of change and initial value of a function from a description of a relationship or from two ((x, yy) values.

•

interpret the rate of change and initial value of a linear function.

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: •

make predictions based on linear functions.

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

Point-Slope Form In this exploration, students will help treasure hunters identify which jewel they would pick up if they traveled from a certain starting point with a given slope. Groups will then write an equation to automate a digging machine. Students will: •

Explore 2

Explore 1

EXPLORE ACTIVITIES

In this exploration, students will explore transformations of linear functions through comparisons of their representations. Students will: identify which jewel they would pick up if they traveled from a certain starting point with a given slope.

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

Standard Form

•

understand transformations of linear functions through comparisons of their representations.

LINEAR FORMS

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Compare Equation Forms In this exploration, students will use various forms of linear equations to analyze characteristics of the equations in scenarios. Students will: •

compare the linear equations to determine the different properties of the given functions

•

determine when to use slope-intercept, point-slope, and standard forms.

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

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Linear Forms 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

LINEAR FORMS

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Students will listen to prompts about the prior standard and communicate if 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. 8.FGR.5.7 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or graph.

Materials

Preparation

Printed •

•

1 Fact or Fiction Prompts (per class)

•

Print one Fact or Fiction Prompts to read aloud to students. Another option is to project the prompts by using a digital projector.

Procedure and Facilitation Points 1.

2. 3. 4. 5.

Designate one side of your room as the Fact side and the other side as Fiction. Instruct students to decide whether each prompt is fact or ﬁction and then move to the corresponding side of the room. Read the ﬁrst 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 the remaining prompts.

FACILITATION TIP Consider making other prompts based on the given graphs and reading them out loud to see which side of the room students go to. For instance, go back to Prompt 1 and say “The y-intercept of this line is (0,3)” or “This line is steeper than the line y = x.”

Prompt 1 is fact.

FACILITATION TIP

Prompt 2 is fact.

Prompt 3 is ﬁction.

State whether a prompt is fact or ﬁction before going to the next prompt. If you make additional prompts, avoid making them all fact or ﬁction so the students stay engaged.

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

FACILITATION TIP

Identifying Misconceptions •

Students may struggle to identify whether a graph is proportional or nonproportional. Have students focus on the origin and whether the graph passes through the origin.

This Foundation Builder includes Math Match Cards with images, equations, tables and scenarios that can be used to provide effective practice and review for all students.

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Linear Forms Hook – Laser Beam Lines ACTIVITY PREPARATION Students will make predictions based on linear functions.

Materials

Preparation

Printed •

• • •

1 Laser Beam Lines (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Laser Beam Lines 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 students have completed the Explore activities.

PROCEDURE AND FACILITATION Part I: Pre-Explore 1.

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2. 3.

4. 5.

FACILITATION TIP

Connect key features of lines to real life. After students answer the question, ask the class for other real-world scenarios where they could use slopes, y-intercepts, and equations of lines to make observations.

DOK-1 What information would you need to answer the question? Allow students to share all ideas. Answers will vary. We could use the slopes, y-intercepts, and equations of each line to help us answer the question.

DOK-1 How could the skills you used to answer this question connect to the satellite and laser? The satellite is far away, and we would need to be able to program the military ship to shoot the laser on the correct line or trajectory to hit the satellite. We can see the initial path of each line but cannot see the graph for large x values.

FACILITATION TIP

After students answer the question, ask the class for other real-world scenarios where graphing would be an ineﬃcient method to make observations. They may mention scenarios with coordinates that have relatively large positive or negative values.

DOK-2 Why is graphing an ineﬃcient method for answering this question? We would have to extend each of the graphs much farther to verify which line passes through (510, 341).

Explain to students that we will use different forms of lines throughout the scope and that they should spend time deciding which form is most helpful in each new scenario.

Engage

Explore

Explain

Elaborate

Evaluate

Part II: Post-Explore 1. 2.

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

DOK-1 What skills can you use to answer this question? I can ﬁnd the slope of each line using two points and then write linear equations in point-slope form or slope-intercept form.

DOK-1 Which line passes through (510, 341)? The blue line

DOK-1 Write the equation for the correct line in two different ways. 2 2 y = __ + 1 and y – 341 = __3(x - 510) 3x

DOK-1 Do you feel you have a strong understanding of different forms for the same line? Answers will vary based on students’ success during the activity and their conﬁdence level.

Intervention

Acceleration

FACILITATION TIP After students answer the question, have them write an equation for each of the other lines in the form of their choice. Then, ask by a show of hands who chose slope-intercept form and who chose point-slope form. Next, show an accurate equation for both forms for both lines, and solve for y for both lines when x = 510 with the class. Remember, point-slope form has inﬁnite correct equations.

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Linear Forms Explore 1 – Point-Slope Form ACTIVITY PREPARATION Students will explore transformations of linear functions through comparisons of their representations.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

1 Student Journal (per student) 1 Jewel Island Treasure Map (per group) 1 Exit Ticket (per 2 students)

Reusable • • •

• • • •

1 Dry-erase marker (per group) 1 Sheet protector (per group) 1 Chenille stem or straightedge (per group)

Plan to have students work in groups of 2 or 3 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 each student has one. Print a Jewel Island Treasure Map, on card stock for durability, for each group of students. Place it in a sheet protector to create an erasable surface.

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever gone on a treasure hunt?; 2) If so, what were you trying to ﬁnd?; 3) Was the treasure buried? FACILITATION TIP Print and project the scenario for students to read along with you.

2. 3. 4.

FACILITATION TIP Students need to keep up with signs as they calculate slopes on their Student Journal. Remind students that a positive slope coincides with a line that increases in value while a negative slope coincides with a line that decreases in value.

Read the following scenario to the class: You have been given a treasure map with different treasures and places posted throughout. You have a special machine that can effectively dig for treasure without destroying it, but it can only move in a straight line once it starts digging. Give a Student Journal to each student. Give a Jewel Island Treasure Map, dry-erase marker, and chenille stem to each group. Explain to students that they will work with their groups to identify which jewel they would pick up if they traveled from a certain starting point with a given slope. Explain to students that they should record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How do we ﬁnd the slope between two points? We ﬁnd it y2 – y1

using rise over run on a graph or with the formula m = ______ x – x using the

coordinate points. b. 6. 224

DOK-2 What is true about all of the points on a line? The slope between any two points on the line must always be the same.

Engage

Explore

Explain

Elaborate

Evaluate

Part II 1.

2. 3.

Read the following scenario to the class: You are tired of directing the digging machine on your own and want to automate the digging. We still have a few jewels we have not been able to reach yet, and we want to be able to input an equation for our machine to use to dig. Students should still have their Jewel Island Treasure Maps, dry-erase markers, chenille stems, and Student Journals. Explain to students that they will work with their groups to graph each line in Part II and make determinations about other points on the line. Instruct them to record their answers on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 Can we use the slope formula without having numerical coordinate points? Yes, we can use variables or numerical coordinates.

DOK-2 How can you tell if a new point will be on the line between the yellow triangle and blue teardrop? The new point also has to have a 3 slope of −__5 between it and the yellow triangle.

DOK-2 Where does the slope show up in your ﬁnal equation? The slope is the coeﬃcient in front of the expression with x.

DOK-2 How does the coordinate point of the yellow triangle show up in the equation? The opposite of the coordinate point is next to the respective variable.

e. DOK-2 How can you determine if a point lies on the graph of any equation? Substitute in the coordinate point and see if it makes the equation true. f. DOK-2 How are the lines represented by the equations in questions 9 and 10 related? The two equations represent the same line. They have the same slope and just start or anchor at a different point along the line. 5. 6.

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

Math Chat DOK-2 How can two different equations represent the same line? They are equivalent equations that are expressed differently. • DOK-2 This form of linear equations is called point-slope form. Why do you think this form got that name? The equation clearly shows the coordinate of one point on the line and the slope of the line. • DOK-2 Why was it easier and more eﬃcient to create an equation in point-slope form than in slope-intercept form? We already had a point on the line and did not have to solve for the y-intercept. • DOK-2 Suppose you were given an equation in slope-intercept form and needed to write it in point-slope form. How would you go about rewriting the equation y = 3x + 5? I would ﬁrst notate that the slope is 3, and then I would write the y-intercept as a point (0, 5). Then, I would write it in point-slope form as y – 5 = 3(x – 0).

Intervention

Acceleration

FACILITATION TIP Before reading the scenario, ask the class 1) What is an example of a machine you have worked with?; 2) What did the machine do?; 3) How do you think the machine could be redesigned to be more eﬃcient?

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FACILITATION TIP The slope for this scenario clearly indicates if students understand and remember the principle of rise over run. Watch out for students ﬂipping the numerator and denominator and/or forgetting the negative sign for the slope.

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.

•

Part III 1.

Read the following scenario to the class: We still have a few jewels we have not been able to reach yet. You and your partner will enter an equation into the GPS to reach the jewels. Students should still have their Jewel Island Treasure Maps, dry-erase markers, chenille stems, and Student Journals.

FACILITATION TIP Compare point-slope form with slopeintercept form further. Ask students when it is easier and more eﬃcient to create an equation in slope-intercept form than in point-slope form. FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever used GPS?; 2) Where were you going?; 3) Did the GPS accurately locate the location you were going? FACILITATION TIP Print and project the scenario for students to read along with you. 225

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Linear Forms Explore 1 – Point-Slope Form 3.

Explain to students that they will work with their groups to graph each line in Part III, identify key features of each equation, and identify which jewels they would pick up on that route. Instruct them to record their answers on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding:

FACILITATION TIP After students answer the question, have them write the given equation and convert it to slope-intercept form. Have them compare the constant and coeﬃcient values of the two equations.

FACILITATION TIP Ask students why there are not infinite ways to write a simplified equation in slopeintercept form. Then, note to the class that there are infinite ways to write an equation in slope-intercept form when terms are not simplified. Next, show several equivalent equations in slope-intercept form that are not simplified. FACILITATION TIP Students may not remember terms like additive inverse, which is used in the answer key. Give students space to clearly answer the question in their own words. FACILITATION TIP After they complete the Exit Ticket, have students write what the equation would be in point-slope form if the signs for the given coordinate values were changed. Then, have them compare the new equation with the equation they selected from the Exit Ticket.

5. 6.

DOK-1 When the equation y – 4 = −2( −2(xx + 7) is graphed, how is the equation related to the point (−7, 4) on the line? The point exists on the line, and the equation uses the opposite of those same numbers next to their corresponding variable.

DOK-1 When given two jewels, what information do you need to find first? We need to find the slope between their two coordinate points.

DOK-2 Why is it easy to substitute the starting point into the point-slope equation? The coordinates create an additive inverse and make each side of the equation equal 0.

Allow students enough time to complete Part III 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-3 Why are there infinitely many ways to write the equation of a single line in point-slope form? You can use any of the infinite points on the line along with the slope to create an equation in the form y – k = m(x – h). • DOK-2 Are two points enough information to write the equation of a line? Why or why not? Yes, because there is only one line between those two points, and you can find the slope between them and then use either one to write an equation in point-slope form. • DOK-2 Why do the values in a point-slope equation look like the opposite of the coordinate point the equation passes through? When you plug in the coordinate point, each side of the equation becomes 0, so the values in the equations must be the additive inverse. •

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

Notes __________________________________________________________________________________________________________________________________________________

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Linear Forms Explore 2 – Standard Form ACTIVITY PREPARATION Students will explore transformations of linear functions through comparisons of their representations.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

•

1 Student Journal (per student) 1 Jewel Island Treasure Map (per group) 1 Exit Ticket (per 2 students)

Reusable • • •

• • •

1 Dry-erase marker (per group) 1 Sheet protector (per group) 1 Chenille stem or straightedge (per group)

Separate the class into groups of 2 or 3 students 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 each student has one. Print a Jewel Island Treasure Map, on card stock for durability, for each group of students. Place it inside a sheet protector to create an erasable surface. (This is the same map used in Explore 1.)

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Print and project the scenario for students to read along with you. FACILITATION TIP This Student Journal is ﬁve pages. Consider printing the different Parts (I, II and III) in different colors. Distribute each part as needed rather than all at once. Save the ﬁfth page as a guide for class discussion rather than printing for all of the students. FACILITATION TIP Print the Jewel Island Treasure Map in color.

2. 3. 4.

Read the following scenario to the class: You continue your search for treasure. Your initial success allowed you to upgrade your treasure-hunting machinery. You can now place points on the intercepts and be told where the treasure is between the points. Give a Student Journal to each student. Give a Jewel Island Treasure Map, dry-erase marker, and chenille stem or straightedge to each group. Explain to students that they will work with their groups to identify which jewel they would pick up if they traveled from a certain starting point with a given slope. Explain to students that they should record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How do we ﬁnd the slope between two points? We ﬁnd it y2 – y1 using rise over run on a graph or with the formula m = ______ x2 – x1 using the coordinate points.

DOK-2 What is true about all of the points on a line? The slope between any two points on the line must always be the same.

DOK-1 What is an intercept? It is a point that is plotted exactly on the x-axis or y-axis.

DOK-1 Plot 3 points on the x-axis. Write down all the points. What do you notice? All the points have a value in the x-coordinate location and a 0 in the y-coordinate location.

FACILITATION TIP Print and project questions 5a, 5b, and 5c to guide students as they collaborate. Consider using the questions to coach a quick review before students begin working.

228

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

e. DOK-1 Plot 3 points on the yy-axis. -axis. Write down all the points. What do you notice? All the points have a value in the y-coordinate location and a 0 in the x-coordinate location. 6.

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Allow students enough time to complete Part I and answer the questions that follow.

Part II 1.

2. 3.

Read the following scenario to the class: After taking some time with your new machine in the ﬁeld, you notice that you can input an equation. However, both variables are on one side of the equation and the constant is alone on the other. You decide to rework what you know about equations to make your work with your machine more eﬃcient. Students should still have their Jewel Island Treasure Maps, dry-erase markers, chenille stems or straightedges, and Student Journals. Explain to students that they will work with their groups to graph each line in Part II and make determinations about other points on the line. Instruct them to record their answers on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What is point-slope form? y – k = m(x – h)

DOK-2 When rearranging your equation for question 4, how many terms will be on the left side, and how do you know? There will be two terms because x and y are different variables and not like terms.

DOK-2 When the equation is in slope intercept form, which part of y = –2xx + 7 is the y-intercept? y-intercept? 7 is the y-intercept because it is in the location of b.

Print and project the scenario for students to read along with you.

FACILITATION TIP Post question 4a. and its answer for students to record on their Student Journals or in notebooks. FACILITATION TIP

DOK-2 How do you write the yy-intercept -intercept as a point? What does that indicate about the variables in the equation? 7 as a y-intercept would create the point (0, 7). This means that when x is 0, y is 7.

Questions 4c and 4d are critical. Take time to call on some select students to respond verbally to help you do a quick assessment. Project the questions and record clear answers for students to record in notes.

e. DOK-2 How do you write the estimated x-intercept as a point? What does that indicate about the variables in the equation? 3.5 as an x-intercept would create the point (3.5, 0). This means that when x is 3.5, y is 0.

STEMscopes Tip

FACILITATION TIP

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

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.

Part III 1.

Read the following scenario to the class: We still have a few jewels we have not been able to reach yet. You and your group will enter an equation in standard form into the GPS to reach the jewels. Students should still have their Jewel Island Treasure Maps, dry-erase markers, chenille stems or straightedges, and Student Journals.

FACILITATION TIP Print and project the scenario for Part III for students to read along with you.

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Linear Forms Explore 2 – Standard Form 3.

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.

5. 6.

DOK-1 When the equation –x –x + y = –8 is graphed, how is the equation related to the point (0, –8) on the line? The point exists on the line; it is the y-intercept.

DOK-1 When given two jewels, what information do you need to find first? We need to find the slope between their two coordinate points.

DOK-2 Why is it easy to substitute the starting point into the point-slope equation and change the equation into standard form? It allows me to easily input the information I have and rewrite the equation into the new form.

Allow students enough time to complete Part III 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-3 Why are there multiple ways you can create your equation in standard form? The information I have determines which way I will create an equation in standard form. I can start from point-slope form or slope intercept form depending on the information I have, and I will have the same resulting equation. • DOK-2 Are two points enough information to write the equation of a line? Why or why not? Yes, because there is only one line between those two points, and you can ﬁnd the slope between them and then use either one to write an equation in point-slope form. • DOK-2 Why does the yy-intercept appear to be the constant in some equations in standard form and not all? If there is no coeﬃcient in front of the y-coordinate, then the constant value is the y-intercept. •

FACILITATION TIP

Post-Explore

This Exit Ticket provides one multiplechoice question. Consider using this Exit Ticket to have students create a table and a graph after they have selected the correct equation.

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 __________________________________________________________________________________________________________________________________________________

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Linear Forms Explore 3 – Compare Equation Forms ACTIVITY PREPARATION Students will use various forms of linear equations to analyze characteristics of the equations in scenarios. Students will compare the linear equations to determine the different properties of the given functions in the scenarios.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials Printed • •

1 Student Journal (per student) 1 Exit Ticket (per 2 students)

Preparation • • •

Reusable •

Plan to have students work in groups of 3 or 4 students 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 each student has one.

1 Graphing calculator (per group)

PROCEDURE AND FACILITATION

FACILITATION TIP Print and project the scenario for students to read along with you. FACILITATION TIP Print the Student Journal in two different colors. Distribute Part I ﬁrst, and then when students are ready, distribute Part II.

Part I 1.

2. 3. 4.

FACILITATION TIP Before passing out the Student Journal and/or reading the scenario, consider conducting a mini-lesson on how to use the graphing calculators. If students have their own, be prepared for a variety of different user manuals.

Read the following scenario to the class: After all your work in the treasurecollecting ﬁeld, you want to analyze the data you collected from your machines. The challenge you are ﬁnding is in determining which linear form is best with the data you have collected at different times and situations. Give a Student Journal to each student. Give a graphing calculator to each group. Explain to students that they will work with their groups to determine when to use slope-intercept, point-slope, and standard forms. They will record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 In the equations given for the ﬁrst problem, which variable is the independent variable? The independent variable is t.

DOK-2 In the equations given, which variable is the dependent variable? The dependent variable is s.

DOK-2 How do you calculate the rate of change? I calculate the change in the dependent variable s and divide it by the change in the independent variable t over some interval.

DOK-2 How do you choose the points to calculate the rate of change? I can choose any two points because the rate of change is constant.

FACILITATION TIP Print and project some of the guiding questions to coach students as they collaborate.

e. DOK-2 In the equations, do you see the same values that you calculated from the tables for the slopes? If so, where? Yes, the value is multiplying the variable t in each equation. 232

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 From the scenario in the second problem, do you expect the rate of change (slope) to be positive or negative? I expect it to be negative because it is cooling, so the temperature is decreasing as time is increasing. DOK-2 In the equation, do you see the same value that you calculated from the graph for the slope? If so, where? Yes, the value is multiplying the variable t again in the equation.

h. DOK-2 When you are being asked to find the starting temperature, what are you being asked to find? Explain what that point means in regard to the scenario. I need to find the y-intercept because I need to know when 0 minutes have passed (x) the temperature of the machine (y). i. DOK-2 When you are being asked to find the time to return to the original temperature, what are you being asked to find? How do you expect that to look as a point? I need to find the x-intercept because I need to know, when the temperature of the machine (y) has returned to 0, how many minutes have passed (x). 6. 7.

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

Math Chat •

•

DOK-1 On page 1, the equation was given in slope-intercept form y = mx + b. On page 2, the equation was given in a different form. How could you still identify the slope? Since m is the slope or rate of change of y with respect to x when y is in slope-intercept form, I can isolate y to determine the slope for any other equation. DOK-2 Both scenarios were given in a table. Why wouldn’t you use the same linear form for both tables? The ﬁrst table had the data given in points, and one was the y-intercept. Once I determined slope because I wanted to compare slope and y-intercept, slope-intercept form made the most sense. The second table gave a point that was not the y-intercept and the slope; this made point-slope form the easiest form to input the information into. DOK-3 Connect how the name of the linear form helps you know which form to use. Slope-intercept form tells me I know the slope and y-intercept. Point-slope form tells me I know a point and the slope. Standard form lets me know I don’t know a point or slope directly.

Intervention

Acceleration

STEMscopes Tip

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The Standards list is located along the menu bar. Here, a keyword can be entered to locate each standard. The search will result in a list of standards and direct links to the scopes where those standards appear. The standards are organized by grade level as well. Clicking on a standard within a grade level will also provide direct links to the scopes.

FACILITATION TIP The reﬂection questions provide some excellent assessment questions. Consider using page 5 as a quick check before the Exit Ticket.

FACILITATION TIP Create a quick three-column chart to list the properties of each linear form.

Part II 1.

2. 3.

Read the following scenario to the class: All these forms have left your head spinning. You want to sit down for a minute and jot down all you know to keep it straight as you work on your machines in the future. Work through all the linear forms to create a reference note card to help yourself as you dig for treasure in the future. Students should still have their Student Journals. Explain to students that they will work with their groups to summarize all the linear forms and determine which form would be the best in a given scenario. They will record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 In y = mx + b,, what do the variables indicate? What name would be that form? m is the slope, and b is the y-intercept. I am given slope and the intercept, so it is slope-intercept form. DOK-2 When you have an equation in point-slope form, what point are you usually not given that leads you to choose that form? Why? It is usually a point on the line that is not the y-intercept because otherwise, when given the slope and y-intercept, the slope-intercept form is chosen.

FACILITATION TIP Print and project the scenario for Part II for students to read along with you.

FACILITATION TIP If y = mx + b is not already posted with labels in the classroom, have students copy it on their Student Journal and label each variable.

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FACILITATION TIP Use questions 4c and 4d to quickly check student comprehension. Ask select students to respond in their own words and use different open-ended formats.

5. 6.

DOK-1 When the scenario details a starting temperature, what feature is being described? The y-intercept

DOK-1 When the scenario details how quickly something is occurring, what feature is being described? The slope

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

Math Chat DOK-2 For a linear equation, is the rate of change constant? Yes, for a linear equation the rate of change of y with respect to x is constant and equal to the slope of the line corresponding to the equation. • DOK-2 How can the slope be identiﬁed from an equation in any form? The slope is always the coeﬃcient of the independent variable once y is isolated, so we can convert any equation into slope-intercept form. • DOK-2 Will your key features change when you change the linear form? Why or why not? The key features of slope and y-intercept will stay the same because the variables and constants are being rearranged as to their location in the equation; their values are not being changed. •

Post-Explore 1. 2. 3. 4.

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Evaluate

Intervention

Acceleration

Notes __________________________________________________________________________________________________________________________________________________

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Linear Forms 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

Point-Slope Form 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

Standard Form 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

Compare Equation Forms 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

Linear Forms

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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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Linear Forms 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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Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER

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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 write linear equations in standard, slope-intercept, and point-slope forms.

I can write a linear function in equivalent forms and explain properties of a linear function.

I can graph and analyze a linear function expressed as a verbal description, a table, and a graph with and without technology.

I can show key characteristics of the linear function graph to describe situations.

I can describe graphs using the concepts of rate of change (slope), intercepts, and end behavior, and using the vocabulary terms strictly increasing, strictly decreasing, positive, and negative.

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Bivariate Data Scope Introduction SCOPE SUMMARY In this scope, students will investigate patterns of association between two quantities to construct and interpret scatterplots of bivariate data. They will model the relationships between two variables using a straight line (the line of best ﬁt) and assess the closeness of the data points. Students will interpret slope and intercept from equations of linear models.

Student Expectations

8.FGR.6.1 Show that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, visually fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line of best fit. 8.FGR.6.2 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercepts.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In Grade 7, students learned the concept of statistics and the relationship between different data. Students discovered the measures of center and variability that arose from samples of different populations. They made informal inferences about these populations based on the data sets, gaining a basic understanding of bivariate measurement data.

Students gain a basic understanding of bivariate data and scatterplots in 8th grade. As they progress to high school, they will learn to create an exact line of best ﬁt with the use of technology. This will lead them to discover linear regressions and correlation coeﬃcients. Through extensive interpretations of frequency tables, they will be able to determine possible data trends and associations.

8.FGR.6.3 Explain the meaning of the predicted slope (rate of change) and the predicted intercept (constant term) of a linear model in the context of the data.

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

understand how statistics can be used for information about populations by examining a sample of the population.

•

make generalizations about population from a sample.

•

describe how random sampling tends to produce representative samples and support valid inferences.

Hook

ENGAGE ACTIVITIES Accessing Prior Knowledge

8.FGR.6.4 Use appropriate graphical displays from data distributions involving lines of best ﬁt to draw informal inferences and answer the statistical investigative question posed in an unbiased statistical study.

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

look at a set of data on a graph and ﬁnd a line of best ﬁt.

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

Bivariate Data In this exploration, groups of students will solve realworld scenarios that are about analyzing data about students like interests, attendance, ages, eye color, and other data to help a visitor from another planet develop a school on their home planet. Students will:

Explore 2

Explore 1

EXPLORE ACTIVITIES

In this exploration, students will solve real-world scenarios that are about assisting a hospital to analyze the relationship between the length and width of the hand to help determine the missing sizes for a glove order. Students will:

construct and interpret bivariate data.

•

model linear relationships on a graph.

•

determine and describe associations that are positive, negative, clustering, outliers, no association, linear, and nonlinear.

•

construct straight lines to informally ﬁt data in a scatterplot.

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

In this exploration, groups of students will be tasked with assisting different grade levels make predictions based on data to ensure each grade level meets their fundraising goal. Students will: •

Explore 4

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

Lines of Best Fit

•

Linear Equations

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create a linear equation based on that data using the slope and y-intercept. y-intercept.

Interpret Data In this exploration, students will analyze bivariate data to draw informal inferences and answer statistical investigative questions posed in an unbiased statistical study. 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.

analyze the data on the graphs and use the line of best ﬁt.

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

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Bivariate Data Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

242

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE

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Students will dialogue with classmates about their understanding of the prior standard through a Four Corners discussion. This element is designed to uncover student misconceptions; it should not be taken for a grade. 7.PAR.4.11 Analyze sampling methods and conclude that random sampling produces and supports valid inferences.

Materials

Preparation

Printed •

• •

1 Set of Four Corner Slides (per class)

•

Print one set of the Four Corners Slides. Hang the scenario slide at the front of the room, easily visible to all students. Hang the remaining four 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 population best deﬁnes a valid sample population. Allow 2 minutes of thinking time. Ask students to move to the corner sample population they chose. Ask each group to discuss why they chose the sample population with each other. Allow 2–5 minutes of discussion at the corner sample populations. After students have discussed why they chose their answers, 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 •

•

Slide 1: Students who select this slide may believe that as long as a portion of the population is asked then the sample is valid. This sample may not be large enough or diverse enough to be representative of the population. Slide 2: Students who select this slide may believe that as long as a portion of the population is asked then the sample is valid. This sample may not be large enough or diverse enough to be representative of the population. Slide 3: Students who select this slide may not understand that asking just Student Council oﬃcers is taking a very small sample and not representative of the population. Slide 4: This is the correct slide.

FACILITATION TIP Ask students to brainstorm another sample they could take that would be representative of all the teachers. This could include things like putting a survey in every other teacher box in the oﬃce or standing outside the school and asking the ﬁrst 39 teachers that arrive at school. FACILITATION TIP This Foundation Builder includes six true/ false scenarios that could provide effective review and reteaching discussions for all students. FACILITATION TIP As students are discussing, if there are students who don’t understand why this slide is correct, explain that in the lunch period, all types of teachers are represented (males, females, math teachers, English teachers, science teachers, etc.). In the other choices, only one group of teachers is represented at a time.

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Bivariate Data Hook – Scatterplot ACTIVITY PREPARATION Students will look at a set of data on a graph and ﬁnd a line of best ﬁt.

Materials

Preparation

Printed •

• • •

1 Scatterplot (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project the Scatterplot 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) Why do people buy stock in companies?; 2) If you wanted to buy stock in a company, which company would you choose?; 3) When would it be wise to buy the comapny’s stock? FACILITATION TIP Print and project the text of this scenario for students to read independently and then with a shoulder partner. Read it all together as a class. Coach students to identify math phrases. Guide them to identify, “’What do we know?” and “What do we want to ﬁnd out?”

4. 5.

FACILITATION TIP

You might need to explain that a trend means a pattern in the data. She is looking to see if the price of the stock generally continues to go up or down.

DOK-1 Why might this line be a better way to predict future behavior of the stock? The price of the stock moves up and down so randomly, but this line shows more of an average of the stock price.

DOK-1 Does the line Jane created have a positive or negative slope? The line has a positive slope.

DOK-1 Are there any points that are far away from the rest of the data? There are a couple of points that are not near the rest of the data.

6. 244

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: Jane has collected a lot of data on the price of a stock she has been following in the stock market. The stock has been called “volatile,” which she figured out means the price changes many times a day. After following the price of the stock for the last month, she thinks she has enough data to see a trend. She wants to try to predict what the stock will do next week to decide whether she should buy more or sell what she already has. 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 Jane is finding a line of best fit or a trend line. I can use math to find the equation of the line. Project the Scatterplot. Explain to students that Jane plans to create a line that will best approximate all the data she collected instead of connecting the dots one by one. Discuss the following questions:

Engage

Explore

Explain

Elaborate

Evaluate

Part II: Post-Explore 1. 2.

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

DOK-1 How can you find the equation of the best fit line? Choose two points on the line, find the slope, and then find the y-intercept. Plug the slope and y-intercept into the slope-intercept form of the linear equation.

DOK-1 What is the equation of the line? Using the points (0, 6) and (30, 9 60) I found the equation to be y = __ + 6. 5x

DOK-1 What do the faraway points mean? Those points are outliers. They can be discarded from the data.

Intervention

Acceleration

FACILITATION TIP Watch out for students who may be overwhelmed by the amount of data shown on the graph. Tell them not to think about all the points, and just to look at the line. They know how to ﬁnd the equation for a line of the form y = mx + b.

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FACILITATION TIP Note the vocabulary term outlier. Have the students add this term to their notes. An outlier is a number in a set of data that is much larger or smaller than other numbers in the set.

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Bivariate Data Explore 1 – Bivariate Data ACTIVITY PREPARATION Students will construct and interpret bivariate data. They will determine and describe associations that are positive, negative, clustering, outliers, no association, linear, and nonlinear.

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 Data Collection Sheets (per class) 1 Set of Blank Scatterplots (per class) 1 Exit Ticket (per student)

Reusable •

• • • •

Plan to divide the class into 6 groups to complete this activity. Number the groups 1–6. Print a Student Journal and an Exit Ticket for each student. Print the Data Collection Sheets for each class. Print the Blank Scatterplots for each class. Tape the Blank Scatterplots across the front wall. Provide a measuring tape for Group 2.

1 Measuring tape (per class)

Consumable •

1 Roll of tape (per teacher)

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) What do you think aliens would want to know about humans?; 2) What questions might aliens ask about students? 2. FACILITATION TIP

Students will collect data from their peers or from students in nearby classrooms. There is space in each data table to collect up to 10 data points, but at least 5 data points should be collected.

4. 5.

FACILITATION TIP Consider printing the Student Journal in two different colors to correlate with Part I and Part II. Distribute each part as needed.

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Read the following scenario to the class: General Zane has come to planet Earth with a mission of learning more about humans. He wants to open a school on his home planet but ﬁrst wants to explore the relationships between different human variables. General Zane has asked some weird questions, and it is up to your group to do some research and give him the answers he needs. Collect and analyze data to complete General Zane’s report. Give a corresponding Data Collection Sheet to each group and a measuring tape to Group 2. Explain to students that they will work in their groups to collect at least 10 pieces of data for their assigned survey topic. Groups may need to collaborate with other groups to get enough data points. Next, have groups plot their data on the corresponding Blank Scatterplot taped on the front wall. Once the scatterplots are completed, give a Student Journal to each student. Explain to students that they will quickly sketch each group’s scatterplot on their Student Journals. Students will work with their groups to analyze the scatterplots and answer General Zane’s questions. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What would the data points on a scatterplot with a positive relationship look like? The points would be increasing across the graph.

7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 Do you see any scatterplots that are not increasing or decreasing? Would you consider these positive or negative? Graphs 1 and 6 have data points that are scattered around the grid. They are not increasing or decreasing. I would consider these neither positive nor negative.

Intervention

Acceleration

FACILITATION TIP Reiterate that this is because there is no correlation between the two variables presented in these graphs. The points are all random.

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DOK-2 What does it mean for data to be linear? Are there any scatterplots that appear linear? Linear means that the data is in a straight line. Graphs 2, 3, 4, and 5 appear to be linear.

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

Math Chat Explain the following to the class: Each group compared 2 different variables to analyze the relationship between them. Any time mathematicians are comparing 2 variables to each other, this is called bivariate data.

FACILITATION TIP Post a clear explanation and deﬁnition of bivariate data. See Picture Vocabulary.

DOK-2 Think about the terms bicycle bicycle, bifocal bifocal, and bipartisan. Why do you think mathematicians use the term bivariate data? All of those words start with the preﬁx bi-, which means 2. So bivariate must mean to compare 2 sets of data or 2 variables. • DOK-2 How would you describe the relationship between the height and shoe size of a human to General Zane? I would tell General Zane that as a human grows taller, their shoe size increases. There is a positive relationship between these two variables. • DOK-2 How would you explain the relationship between the amount of money spent and the amount of money saved to General Zane? I would tell General Zane that as a human spends more money, they save less money. There is a negative relationship between these two variables. •

Explain the following to the class: When mathematicians are describing the relationship between two variables, they call it association. Association between two variables can be positive or negative, or there can be no association. •

DOK-1 Did you see any scatterplots that were perfectly linear? Explain. No, there were scatterplots that looked close to a perfectly straight line, but nothing with the exact same slope in between every data point.

Explain the following to the class: Variables can also have linear association and nonlinear association. If data points closely resemble a line, the relationship is described as linear. Because of human error in data collection, it is often diﬃcult to achieve a perfectly linear set of data.

FACILITATION TIP Print this explanation and clarify the term association for students. Consider having them note down the deﬁnition, some examples, and non-examples. FACILITATION TIP Print and project the text of this explanation. Provide students time to take notes and ask clarifying questions.

Part II 1.

2. 3.

Read the following scenario to the class: General Zane has returned to Earth with Principal Zed to analyze one more set of bivariate data. They surveyed a school to ﬁnd how absences affect test scores among students. Use your knowledge of bivariate data to analyze the scatterplot and complete Principal Zed’s report. Explain to students that they will analyze the scatterplot Absences vs. Test Scores and work with their groups to complete Principal Zed’s report. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 Describe what you think is meant by the term outlier outlier.. I think an outlier is a data point that doesn’t really ﬁt in with the rest of the data.

DOK-2 Would an outlier be a good representation of the relationship between the variables? Explain. No, because an outlier doesn’t ﬁt in with the rest of the data, it would not be a good representation.

FACILITATION TIP After reading the scenario, ask the class 1) What is bivariate data?; 2) What is your prediction about how absences affect test scores?; 3) What is a scatterplot? FACILITATION TIP Print and project the scenario and conduct a read aloud with students. FACILITATION TIP Write the term outlier and a clear deﬁnition for students. Provide a list of examples and nonexamples. See the Math Chat to clarify.

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Bivariate Data Explore 1 – Bivariate Data

FACILITATION TIP If students ﬁnish early, ask them to make a list of other relationships that might have a negative relationship like this one. For example, if the supply of new game systems (independent variable) is really low, the demand for them (dependent variable) might be very high.

4. 5.

DOK-2 Describe what you think is meant by the term cluster cluster. I think a cluster is a bunch of data points that are clumped or close together.

DOK-2 The data points are not in a perfectly straight line. Does this mean the relationship is nonlinear? Explain. The relationship is still linear even though the data is not perfectly straight. Linear association occurs when the data points are close to linear.

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

Math Chat •

DOK-1 How did you determine whether the association was positive, negative, or neither? As I moved across the graph from left to right, the data points were decreasing. This means the association was negative.

Explain the following to the class: An outlier is a data point that is much larger or much smaller than the rest of the data points. •

DOK-2 How can you identify outliers on a scatterplot? I can look at the trend of the data points. There was a data point at (13, 64) that did not ﬁt in with the rest of the data. This is an outlier.

Explain the following to the class: A cluster is a group of data that are close together. •

DOK-2 What could be the reason for the cluster that appears on the Absences vs. Test Scores scatterplot? Maybe there were more students with fewer absences, so there were more students with high test scores.

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.

248

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Notes __________________________________________________________________________________________________________________________________________________

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Bivariate Data Explore 2 – Lines of Best Fit ACTIVITY PREPARATION Students will model linear relationships on a graph and construct straight lines to informally ﬁt data in a scatterplot. Students will justify each of these lines as a good or bad ﬁt.

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 Class Data Table (per class) 1 Exit Ticket (per student)

Reusable • • •

• •

1 Measuring tape (per group) 1 Hard spaghetti noodle (per student) 1 Projector or document camera (per teacher)

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. Be prepared to project the Class Data Table onto the front board. Make sure the surface being projected on may be written on. Optionally, print one copy for each class. Gather enough measuring tapes for each group to have one. Gather enough hard spaghetti noodles for each student to have one.

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What do hospital staff do to prevent the spread of germs in a hospital?; 2) What information would you need to determine what size of latex gloves to order for the hospital staff to use to prevent the spread of germs? FACILITATION TIP Print and project the scenario so that students can read it along with you. FACILITATION TIP If you want students to have more data points, make this a two-day exercise. Collect data from all of your classes on the ﬁrst day. On the second day, students will make their graphs using the data from all the classes. FACILITATION TIP Instead of a spaghetti noodle, you could use a piece of yarn or string, a clear straw, or anything else that is thin and can be made straight. 250

2. 3.

4. 5.

Read the following scenario to the class: Due to a viral outbreak, increased safety measures are being imposed in Woodville Hospital. Doctors are not happy, as the latex gloves in a new shipment are all the wrong size. The staff at Woodville Hospital need to know each doctor’s hand length and hand width to ensure the glove order gets corrected. The hospital needs you to analyze the relationship between the length and width of the hand to help determine the missing dimensions for the glove order. Project the Class Data Table onto the front board, or hang a printed Class Data Table on the front board. Explain to students that they will work in their groups to measure each member’s hand in centimeters. Groups will record each person’s data on the Class Data Table on the front board. Give a Student Journal and hard spaghetti noodle to each student. Once all data is collected for the class, have students create a scatterplot of the data on their Student Journals. Instruct students to work with their groups to analyze the scatterplot to complete the glove order for the hospital. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 How would you describe the association between the two variables? Use the terms positive positive, negative, linear linear, or nonlinear. The association between the two variables is positive and linear.

DOK-2 Why would it be useful to draw a line through the data points? The line could help me predict the width or length of a hand for people that are not in our class data. © Accelerate Learning Inc. - All Rights Reserved

7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How do you determine if your line ﬁts the data well? Answers will vary. I make sure to have the same number of data points on either side of the spaghetti noodle. I try to make my spaghetti noodle go through the most data points. DOK-2 Will outliers affect the line that best represents the data? Explain. Outliers are data points that are much larger or much smaller than the rest, so they are not good representations of the relationship. I will ignore the outliers when I am making my line.

Intervention

Acceleration

FACILITATION TIP Have the students note any clusters of data. Discuss what this means. In this situation, it means that many of the students are about the same size.

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Allow students enough time to complete the Explore activity 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 you determine where to place your spaghetti noodle to best represent the data? I tried to have the line go through the most data points. I tried to have the same amount of data points on either side of the line.

Explain the following to the class: The straight line that best represents the data on a scatterplot is called the line of ﬁt. •

•

DOK-2 What would it mean if all of the data points are close to the line of best fit? When the data points are close to the line, it represents data that fits the prediction well. DOK-2 What would it mean if the data points are farther from the line of best fit? When the data points are farther from the line, it represents data that does not fit the prediction well. DOK-3 Can you draw a line of best fit when there is no correlation? Some scatterplots have no correlation. This means you cannot draw a line of best fit.

FACILITATION TIP Have students record the definitions of scatterplot and line of fit (or line of best fit).

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 On this Exit Ticket, consider asking students to record their reasoning for their choices.

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Bivariate Data Explore 3 – Linear Equations ACTIVITY PREPARATION Students will create a scatterplot to informally ﬁt a line through data. They will create a linear equation based on that data using the slope and y-intercept. y-intercept.

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 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. Gather enough rulers to be used as straightedges for each student to have one.

1 Ruler (per student)

PROCEDURE AND FACILITATION

FACILITATION TIP Before reading the scenario, ask the class 1) Have you ever been involved in a fundraiser? Explain who you were raising funds for.; 2) How could funds be raised for people in need?

Part I 1.

FACILITATION TIP Print the Student Journal for Part I in one color (page 1) and Part II (pages 2 and 3) in another color. Distribute them one at a time as needed.

2. 3.

FACILITATION TIP Students may struggle with drawing a line of best ﬁt. Their slope and y-intercept may not be exactly what the answer key has.

FACILITATION TIP Print and project guiding questions 4a–4c. Post responses and check for student understanding. Ask several selected students to respond verbally in their own words.

Read the following scenario to the class: The Tanaka High School Student Council decided to do a massive 20-week fundraiser to raise money for hurricane relief for a neighboring county. Each grade level chose a different fundraiser and set the goal of raising at least $500 per class. Some of the classes started off with big donations from parents, while the others started at $0. The 9th-grade class chose to sell cookies during lunch periods, and the data for the first 8 weeks has been collected. Based on the data, are they on track to meet the goal of at least $500? Give a Student Journal and a ruler to each student. Explain to students that they will work in their groups to create a scatterplot to represent the funds raised by the 9th-grade class over the first 8 weeks of the fundraiser. Instruct them to use their rulers as straightedges to draw the line of best fit. Have students work with their groups to answer the questions to determine whether the 9th-grade class is on track to meet their fundraising goal. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How do you determine the slope of a line? I pick two points on the line and calculate the change in y divided by the change in x.

DOK-1 How do you identify the y-intercept? y The y-intercept is the point at which the line crosses the y-axis.

DOK-2 How can you write an equation once you know the slope and y-intercept? I know that linear equations can be written in the form of y = y-intercept? mx + b, where m is the slope and b is the y-intercept, so I just substitute my calculated slope and y-intercept into the equation.

DOK-2 How can you use the equation to determine the predicted funds raised at 20 weeks? I know the variable x stands for the number of weeks, so I can substitute 20 into the equation for x and simplify.

FACILITATION TIP If you want everyone to be using the same equation, do a check in, and have students update their line of best ﬁt and equation. Do not mark students incorrect if their equations were slightly “off” from the key. 252

Engage

Explore

Explain

Elaborate

Evaluate

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

Math Chat •

DOK-1 What pieces of information do you need from a graph to be able to write an equation? I need the slope and y-intercept.

DOK-2 How do you determine what the slope represents in the scenario? I know that slope is the change in y values divided by the change in x values, so I think 25 25 dollars about the variables in the same way. The slope is ___ or ________. This tells me 1 1 week that the slope represents the amount of funds raised per week. • DOK-2 Could there have been a different equation for the line of best ﬁt than the one your group created? Explain. Yes, everyone’s line of best ﬁt could be a little different and could have different slopes and y-intercepts. The equations of different groups would likely be a little different but still close to our values.

•

Intervention

Acceleration

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Part II: Property Management 1.

Read the following scenario to the class: The 10th, 11th, and 12th graders are also curious about whether they are on track to hit the $500 goal by the end of the 20 weeks. Use your knowledge of bivariate data and lines of fit to predict each class’s 20-week profit total. Explain to students that they will work with their groups to analyze the data from each class’s funds raised over the first 8 weeks of the fundraiser. Students will write an equation to represent each line of best fit and use those equations to predict the funds raised by each class after 20 weeks. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

4. 5.

DOK-2 What does it mean in the scenario when the yy-intercept is 0? This means that those classes did not receive any donations from parents. They started the fundraiser with an initial value of $0. DOK-2 For the 12th-grade fundraiser data, I see the line has a rise of 3 3 and a run of 2. Does this mean the slope is __2? You have to consider the scale when calculating slope. The line rises by 3 tick marks, but since the scale of the y-axis is counting by 25s, the actual rise is 75. The run is 2. Since the scale of the x-axis is counting by ones, the run matches up 75 with the amount of tick marks. The slope is ___ = 37.5. 2

FACILITATION TIP Project the text of this scenario and have student volunteers read it aloud to the class while you coach students to note essential math phrases and terms. FACILITATION TIP Since the lines are given in this part, students’ equations should match the answer key exactly. The students do not have to estimate and draw a line of best ﬁt in this part of the Explore activity. FACILITATION TIP Follow up question 3a by challenging students to write an equation for when the y-intercept is 0.

Allow students enough time to complete the Explore activity 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 When given an equation in the form y = mx + b, b how can you identify which number represents the slope and which number represents the y-intercept? y In slope-intercept form, the slope is always the number that is multiplied by the x value. The y-intercept is the constant that is added or subtracted. • DOK-3 Why would it be useful to write a linear equation to represent the line of best ﬁt of a scatterplot? The equation of a line of best ﬁt can be used to predict an output for any given input. This means that with just a few data points, I could make predictions about the relationship between the variables. •

FACILITATION TIP Post and label this equation form in your classroom. Ensure that students have it labeled and recorded in their notebooks as well. Take time to call on several select students to identify the variables in the equation.

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BIVARIATE DATA

Bivariate Data Explore 3 – Linear Equations Post-Explore 1. 2. 3.

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

254

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Notes __________________________________________________________________________________________________________________________________________________

BIVARIATE DATA

Home

255

BIVARIATE DATA

Bivariate Data Explore 4 – Interpret Data ACTIVITY PREPARATION Students will analyze bivariate data to draw informal inferences and answer statistical investigative questions posed in an unbiased statistical study.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.7 Look for and make use of structure.

Preparation

Materials 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 Print and project the scenario for students. Be prepared for students to need time to share opinions about the data being collected. If time allows, this might be a good time to address data collection ethics.

FACILITATION TIP

2. 3. 4.

Consider posting and discussing questions 4a–4c before distributing the Student Journal.

FACILITATION TIP Since there are only four graphs to analyze, consider projecting them one at a time and conduct a whole class analysis to coach and guide students to make inferences.

5. 6.

256

Read the following scenario to the class: The principal at Spring Meadows Middle School is very interested in what variables have an effect on students’ test scores and grades. She’s collected data from random students to ﬁnd their sleeping habits, age, study habits, and extracurricular activities. Look at the data she’s collected, and answer questions so that she can create teaching strategies to help her students get the best scores possible. Give a Student Journal to each student. Explain to students that they will analyze the data on the graphs and use the line of best ﬁt to answer questions. Ask students the following questions to assess their understanding of bivariate data and statistical questions: a.

DOK-2 What is bivariate data? Bivariate data is data for two variables.

DOK-2 How do we use bivariate data? We can look at the association between the two variables.

DOK-2 How does a line of best ﬁt help when we interpret bivariate data? A line of best ﬁt gives us an estimated rate of change and y-intercept to use when making inferences about the data.

Have students use the graphs with lines of best ﬁt to answer questions, including the statistical question for each. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 How does the line of best ﬁt help us make inferences with the data? Student responses will vary. You can use the line of best ﬁt to make an inference for places on the graph where there is no data.

DOK-2 How can we see evidence of an association in a graph? By visually looking at the shape of the data we can see if there is any kind of association between the two variables.

Allow students enough time to complete the Explore activity and answer the reﬂection questions.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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

Math Chat DOK-2 Is there an association between the extracurricular activities and the study habits data? The more time spent on extracurricular activities, the less time there is to spend studying. • DOK-2 How can you use a line of best fit to make predictions? The line of best fit shows you the trend in the data. You can follow the line to show where the data should be. The line of best fit shows you the trend in the data. You can follow the line to show where the data should be. • DOK-2 What is an example of how bivariate data is used in the real world? Determining how many 7th and 8th graders want to go to the museum or zoo for the end-of-year field trip

BIVARIATE DATA

Home

•

Post-Explore 1. 2. 3. 4.

FACILITATION TIP Before this Math Chat, gather some additional real-world examples when bivariate data is used. For example, “How does temperature affect ice cream truck sales? How does spending on ads in videos increase proﬁt for companies?”

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BIVARIATE DATA

Bivariate Data Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

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

Bivariate 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

Lines of Best Fit 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

Linear 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

Interpret Data

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

258

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

Bivariate Data

BIVARIATE DATA

Home

Can be done independently

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

259

BIVARIATE DATA

Bivariate Data Intervention and Assessment SMALL-GROUP PLANNER 1

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

Students who are still acquiring the concept and need remediation

How to Use the Review

3 260

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER

BIVARIATE DATA

Home

What prompts will be used?

What does mastery look like?

I can show that straight lines can be used to model relationships between two quantitative variables.

I can discover the line of best ﬁt as the one that comes closest to most of the data points in a scatterplot.

I can solve practical, linear problems involving situations using bivariate quantitative data.

I can explain the meaning of a predicted slope (rate of change) and intercept (constant term) of a linear model.

I can use graphical displays from data distributions involving lines of best ﬁt to draw inferences and answer statistical questions.

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SCOPE 1

Parallel and Perpendicular Lines Scope Introduction SCOPE SUMMARY In this scope, students expand on their knowledge of slope to distinguish between parallel and perpendicular slopes of lines. Students are able to write an equation of a line that contains a given point and is either parallel or perpendicular to a given line. They are also able to write an equation of a line that is parallel or perpendicular to the x-axis or y-axis and determine whether the slope of the line is zero or undeﬁned. Student Expectations

8.FGR.7.5 Create and compare the equations of two lines that are either parallel to each other, perpendicular to each other, or neither parallel nor perpendicular.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In 6th grade, students are able to plot coordinates in all four quadrants. In 7th grade, students work on graphing proportional relationships. Previously in 8th grade, students explore slope-intercept form and identify the slope and yy-intercept. -intercept. Students are already familiar with fractions and recognizing reciprocals.

In Geometry, students will examine tangent lines to circles that are perpendicular to the radius, as well as parallel lines that are cut by transversals.

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

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES

use data from a table or graph to determine the rate of change or slope and y-intercept. y

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

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.

relate parallel and perpendicular lines to a real-world situation.

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 _____________________________________________________________________________________________________________ _____________________________________________________________________________________________________________ _____________________________________________________________________________________________________________

262

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Parallel and Perpendicular Slopes In this exploration, students will explore with graphs that are parallel and perpendicular to discover characteristics of the slopes of parallel and perpendicular lines. Students will: •

•

Explore 2

Explore 1

EXPLORE ACTIVITIES

determine that parallel lines have the same slopes and perpendicular lines have negative reciprocal slopes. analyze graphs and investigate the lines to help create perfect boxes.

In this exploration, students will determine that parallel lines have the same slopes. Students will: •

write the equation of a line that is parallel to another line given a point.

•

use point-slope form.

•

create parallel lines to the x- and y-axes y-axes.

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

Parallel Lines

PARALLEL AND PERPENDICULAR LINES

Home

Perpendicular Lines In this exploration, students will determine that perpendicular lines have slopes that are opposite reciprocals of each other. Students will: •

write the equation of a line that is perpendicular to another line given a point.

•

use slope-intercept form.

•

use point-slope form.

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

263

PARALLEL AND PERPENDICULAR LINES

Parallel and Perpendicular Lines Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

264

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will read different student responses to a posed question on the prior standard, decide if they agree or disagree with the student, and explain their reasoning. This element is designed to uncover student misconceptions; it should not be taken for a grade. 8.FGR.5.5 Write and explain the equations y = mx + b (slope-intercept form), Ax + By = C (standard form), and (y – y1) = m(x – x1) (point-slope form) as deﬁning a linear function whose graph is a straight line to reveal and explain different properties of the function. Use data from a table or graph to determine the rate of change or slope and y-intercept in mathematical and real-world problems.

Materials

Preparation

Printed •

•

Print an Agree or Disagree for each student.

1 Agree or Disagree (per student)

PARALLEL AND PERPENDICULAR LINES

Home

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

Instruct students to complete 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 hands raised in a high-five position. On your instruction, students will stop and high five the closest person. This will be their partner. Give students a couple of minutes to discuss their answers and justifications together. Repeat steps 3–5 as many times as you want with different partners. Discuss the responses as a class. Allow students to explain their reasoning for each problem. a.

Disagree with Jack

Agree with Gabby

Disagree with Nakia

If students are struggling to complete this task, move on to 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 struggle to ﬁnd key information from word problems that will help them set up the equation they need to write. Students may struggle identifying what information from the problem the variables represent. Students may need to be reminded about the forms of linear equations and where slope and yy-intercept -intercept are represented. Students may confuse slope and yy-intercept with information presented in the problem when a form other than slope-intercept form is used. Students may struggle with understanding how to transfer the information from a word problem to a graph in order to show the problem using multiple representations.

FACILITATION TIP Consider encouraging students to try to create a graph or a table for each scenario while they work independently.

FACILITATION TIP While students are walking around, monitor for clear explanations/reasoning. Have successful students share aloud. If students have created clarifying tables or graphs in Step 1, have them project them under a document camera. FACILITATION TIP This Foundation Builder has several true/ false statements that would generate an excellent review activity for partners or a whole class discussion.

265

PARALLEL AND PERPENDICULAR LINES

Parallel and Perpendicular Lines Hook – Parallel and Perpendicular Lines ACTIVITY PREPARATION Students will relate parallel and perpendicular lines to a real-world situation.

Materials

Preparation

Printed •

• • •

1 Parallel and Perpendicular Lines (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project Parallel and Perpendicular Lines 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 Engage students by connecting with the students that you know play or watch a lot of baseball/softball. Ask them to share some information about the differences and significance of the infield, outfield, foul territory, and the base lines. 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.

4. 5.

6. 266

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: The school’s baseball field is being set up for a playoff game. After the groundskeeper mows the outfield and rakes the infield, it is time to put down the bases and draw the chalk foul lines. 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 groundskeeper has to create multiple lines. I wonder what shape the bases make. I wonder what tools the groundskeeper needs to use in order to complete the job and make sure measurements and lines are perfect. Project Parallel and Perpendicular Lines. Explain to students that the first base line has been given an equation, and the picture of the field is mapped over a coordinate plane. Discuss the following questions: a.

DOK-1 How will the lines connecting the base paths be similar and different? Allow students to share all ideas. Student answers will vary. The two foul lines are perpendicular. The lines between home and third as well as ﬁrst and second should be parallel. The lines between second and third and ﬁrst and second should intersect on second base.

DOK-1 Is the point (14, −6) on the foul line? How do you know? The line does pass through (14, −6) because substituting that point in for x and y in the equation makes it true.

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 Parallel and Perpendicular Lines, and discuss the following questions: a.

DOK-1 Does creating these lines make more sense after the Explore activities? Yes, the lines all have parallel or perpendicular slopes.

DOK-1 What strategies would you use to create the equations? I would use the same slope to create a parallel line to the one given and use the opposite reciprocal slopes to create a perpendicular line.

DOK-1 What form would be easiest to use to create each of the lines? I would use point-slope form for the line connecting ﬁrst and second base and I would use slope-intercept form for the line connecting home plate to third base.

DOK-1 Do you feel that you have a strong understanding of parallel and perpendicular lines? Answers will vary based on students’ success during the activity and conﬁdence level.

Project question 5a and encourage students to jot down similarities and differences on a T-chart. Collect their responses on a whole class chart. Listen to see if anyone uses the terms perpendicular, parallel, right angles, and/or intersect. Determine if you want to coach them with a word bank or wait until after the Explore activities. FACILITATION TIP Before asking question 5b, consider asking, “What part of the field would be the x-axis, y-axis, and, origin?” It may not be clear to some students where this specific foul line is. Some may ask if it is the right field or left field foul line. Clarify that students should visualize a four quadrant coordinate plane.

PARALLEL AND PERPENDICULAR LINES

Home

267

PARALLEL AND PERPENDICULAR LINES

Parallel and Perpendicular Lines Explore 1 – Parallel and Perpendicular Slopes ACTIVITY PREPARATION Students will explore with graphs that are parallel and perpendicular to discover characteristics of the slopes of parallel and perpendicular lines. Students will discover that parallel lines have the same slopes and perpendicular lines have negative reciprocal slopes.

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. 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 Exit Ticket (per student)

Divide the class into groups of 2 or 3 students. Print a Student Journal for each student. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student has one.

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Project this scenario and have student volunteers read it along with you. Coach students to jot down essential math phrases as you model careful reading. FACILITATION TIP Consider distributing the Student Journal in two parts as needed.

2. 3.

FACILITATION TIP Before students begin collaborating, create a word bank or word wall that includes the mathematical vocabulary they should be using during this scope. (horizontal, vertical, slope, x-axis, y-axis, ordered pairs, points). FACILITATION TIP Project some of these guiding questions to support student collaboration as they work.

Read the following scenario to the class: Brandon’s robotics class has partnered with a local box factory to help them make boxes for the upcoming holiday rush. The company has sent over their plans for the class to study before programming the machines they will loan them. They know that squares and rectangles have pairs of parallel lines. They know that opposite sides are equal in length and parallel. They need to determine what slopes will create parallel lines to create perfect boxes. Give a Student Journal to each student. Explain to students that they will work with their groups to analyze the graphs and investigate the lines to help create the perfect boxes. They will record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How are horizontal and vertical line equations written? Horizontal lines are y = some value, and vertical lines are x = some value.

DOK-1 For question 1b, does the line cross the x- or y-axis? y-axis? y-axis

DOK-1 Where does it cross the y-axis? y At (0, 5)

DOK-1 What is the equation of a horizontal line that passes through (0, 5)? y = 5

e. DOK-1 For question 1c, what are the ordered pairs for the corners of the top of the box? (0, 5), (5, 5) f. g. 268

DOK-1 When you plug those points into the slope formula, what is the 0 fraction? __5

DOK-1 What happens when a fraction has 0 as the numerator? The fraction simpliﬁes to 0.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

h. DOK-1 What is the slope for any horizontal line? m = 0 i. DOK-1 What is the slope for any vertical line? m is undeﬁned. j. DOK-1 For question 1g, what are the ordered pairs for the corners of the left side of the box? (0, 0), (0, 5) k.

DOK-1 When you plug those points into the slope formula, what is the 5 fraction? __0

l. DOK-1 What happens when a fraction has 0 as the denominator? The answer is undeﬁned. m.

DOK-1 What is the formula for calculating slope? The change in the y’s divided by the change in the x’s

n. DOK-1 How can you determine the slope of a line on a graph without calculating the slope? Count the change in the y values and the x values between two points, and simplify the fraction. 5. 6.

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

PARALLEL AND PERPENDICULAR LINES

Home

Math Chat DOK-1 If line f and line g are parallel, what must be true about their slopes? The slopes must be equal. • DOK-2 Explain in your own words why equal slopes would create lines that don’t intersect. If the slopes are the same, then both lines have the same rise and run. When one line goes up one unit, the other does, as well. If they do this to inﬁnity, they will never touch. •

Part II 1.

2. 3. 4. 5.

Read the following scenario to the class: Brandon’s robotics class continues to practice writing plans for their projects. They know that squares and rectangles have 4 right angles and perpendicular lines. They know that opposite sides are equal in length and parallel. They need to determine what slopes will create perpendicular lines to create perfect boxes. Explain to students that they will work with their groups to determine equations of the missing sides. Point out to the class that using a vertex and point-slope formula is an alternate way to ﬁnd or check their equations. Students will then work together to analyze equations and record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 For box 3, what did you notice about the slopes of the lines? The slopes are opposite reciprocals.

DOK-1 Did your generalization and conclusion about parallel lines remain true? Yes, parallel lines have the same slope.

DOK-1 What do you notice about the numerators and denominators of the fractions of the corners? The numerators become the denominators, and the denominators become the numerators.

DOK-1 What is it called when the numerators and denominators change positions? Reciprocals

e. DOK-1 When numbers are the same distance from 0 on either side of the number line, what are they referred to as? Opposites f.

FACILITATION TIP Project the text of this scenario and have student volunteers read it aloud. Have students identify the key vocabulary as it is read aloud. FACILITATION TIP Review vertex and point-slope formula as needed.

FACILITATION TIP Post questions 5c–5h before students begin collaborating. Encourage them to focus their discussions on observations about these guiding questions.

DOK-1 What makes two numbers reciprocals? The product of the numbers is 1.

269

PARALLEL AND PERPENDICULAR LINES

Parallel and Perpendicular Lines Explore 1 – Parallel and Perpendicular Slopes g.

h. DOK-1 What do you notice about the sign of the slopes? They are opposites.

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.

DOK-1 What do you notice about the product of the slopes? Their product is a negative one.

i. DOK-1 You have all the vocabulary you need to make your generalization. What might you call slopes that are the same distance from zero in opposite directions? Opposite reciprocals j. DOK-1 In slope-intercept form, what value must you identify to know the slope? The m or coeﬃcient of the x. k. DOK-1 How do you determine the slope of a line? Using two points, the slope is the quotient of the change in y values and the change in x values. 6. 7.

Allow students enough time to complete Part II 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 Just by looking at the equations, how can you tell if linear equations are parallel? The slopes are the same, and the y-intercepts are different. • DOK-2 Just by looking at the equations, how can you tell if linear equations are perpendicular? The slopes are opposite reciprocals. • DOK-2 Just by looking at the equations, how can you tell if linear equations are neither parallel nor perpendicular? The equations would either be identical or they would not have the same or opposite reciprocal slope in order to be neither parallel nor perpendicular. •

Post-Explore FACILITATION TIP For this Exit Ticket, consider providing a word bank or sentence starters for struggling students.

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.

270

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

PARALLEL AND PERPENDICULAR LINES

Home

271

PARALLEL AND PERPENDICULAR LINES

Parallel and Perpendicular Lines Explore 2 – Parallel Lines ACTIVITY PREPARATION Students will explore that parallel lines have the same slopes. Students will be able to write the equation of a line that is parallel to another line given a point. If the yy-intercept -intercept is given, then students will use slope-intercept form as the most eﬃcient method. Otherwise, students will use point-slope form. Parallel lines to the x- and yy-axes will also be created.

Standards for Mathematical Practice • • • • •

Materials

Preparation

Printed • • •

1 Student Journal (per student) 1 Set of Ripped Notes Cards (per group) 1 Exit Ticket (per 2 students)

Reusable •

1 Resealable bag (per group)

• • • •

Separate the class into groups of 3 or 4 students 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 each student has one. Print a set of Ripped Notes Cards, on card stock for durability, for each group of students. Cut the cards apart, and place them in a resealable bag. Label the bag “Part I.”

PROCEDURE AND FACILITATION Part I: Matching Parallel Equations FACILITATION TIP Project the scenario for Part I. Have students read it silently and jot down as many math terms as they can ﬁnd with a timer set. Next, have them share the words with a partner.

Distribute each part of the Student Journal only as it is needed.

2. 3. 4.

FACILITATION TIP

Some students may need graph paper or blank tables to help them visualize the equations they are comparing. FACILITATION TIP Print and project some of the most essential guiding questions. Consider reading them with students before they begin collaborating.

Read the following scenario to the class: The robotics class has been challenged by the box company to help them finish the plans for their latest boxes. Saanvi and Brandon are partnered up to write equations of parallel sides. Brandon finds the notes left with the machine to program the parallel sides of boxes. Somehow, the notes were ripped apart before Saanvi and Brandon arrived. Find the parallel matches for the first phase of programming. Give a Student Journal to each student. Distribute the Ripped Notes Cards to each group. Explain to students that they will work with their groups to find four pairs of parallel lines and record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How can you identify if equations are parallel? The slopes are the same.

DOK-1 How can you rewrite an equation given in standard form to determine the slope? Solve the equation for y. This will rewrite the equation in slope-intercept form, and m can be identiﬁed for the slope.

DOK-1 When the x-coordinates -coordinates are the same in multiple points, what kind of line does that indicate? The line is a vertical line.

DOK-1 What is the slope of a vertical line? Undeﬁned

e. DOK-1 When the slope is 0, what kind of line is it? A horizontal line 272

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How could the slope-intercept equation with 0x 0x be simpliﬁed? y=4

DOK-1 What are some differences in equations of horizontal and vertical lines? Horizontal equations are y = some value, the slopes are 0, and the y-coordinates are all the same values. Vertical equations are x = some value, the slopes are undeﬁned, and the x-coordinates are all the same values.

Intervention

Acceleration

h. DOK-1 What form do equations need to be in to compare slopes eﬃciently? Slope-intercept or point-slope form 6. 7.

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

Math Chat DOK-1 How did you determine if the equations were parallel? I put them in the same form and compared the slopes. • DOK-2 Suppose you are given an equation and a point the parallel equation must go through. How might you go about determining the equation of the parallel equation? I could use the point-slope form. By substituting the point and the same slope, I can create a parallel equation. •

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Part II: Creating Parallel Equations 1.

2. 3. 4. 5.

Read the following scenario to the class: The factory decides to have the students help design flower boxes since their homecoming dance is coming up. The factory sends over the starts of some plans for the students to practice and work on. Use the given side to create a parallel side that fits the specifications given. Students should still have their Student Journals. Explain to students that they will work with their groups to create equations in various forms that create parallel sides. Students will then work together to create parallel equations and record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a. b. c.

DOK-1 What is the slope for the graphed equation? m = __4

DOK-1 Will the slope change when you are creating a parallel equation? No, the slope will remain the same. DOK-1 What is the formula for point-slope form? How will you use it to write parallel equations? The formula for point-slope is y – y1 = m(x – x1). I will keep the slope the same and substitute the point the line needs to pass through as the x- and y-coordinates. DOK-1 How do you rewrite an equation in point-slope form into slopeintercept form? Complete the distributive property by multiplying the slope by the variable x and substituting the x-coordinate. Solve for y by adding or subtracting.

FACILITATION TIP Post the formulas for point-slope form, standard form, and slope-intercept form with variables labeled. If needed, have students copy them again on their Student Journal.

e. DOK-1 What is the formula for slope-intercept form? How will you use it to write parallel equations? The formula for slope-intercept form is y = mx + b. I will keep the slope the same and substitute the y-intercept for b. f.

DOK-1 What is the formula for standard form? The formula for standard form is Ax + By = C.

DOK-1 Can you identify the slope in standard form, or do you need to rewrite it? It needs to be rewritten in slope-intercept form.

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Parallel and Perpendicular Lines Explore 2 – Parallel Lines 6. 7. FACILITATION TIP On the Student Journal, page 3 question 3 provides a great opportunity to demonstrate or review several methods to create new parallel lines. Use this question to model or guide students to model each method.

Allow students enough time to complete Part II 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-1 If two lines are parallel, what must be true about their slopes? Both lines have equal slopes. • DOK-1 Which form (slope-intercept, point-slope, or standard) was your preferred equation? What was your least preferred? I preferred point-slope because I could input any point and the slope. I least liked standard form because I had to change it into slope-intercept form, and that was not eﬃcient. • DOK-2 Justify why the slope must stay the same in all the parallel equations. The slope must stay the same so the lines are at the same angle to keep them parallel and never intersecting. •

Post-Explore FACILITATION TIP

On this Exit Ticket, some students may need the support of additional graph paper and blank tables.

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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Explain

Elaborate

Evaluate

Intervention

Acceleration

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Parallel and Perpendicular Lines Explore 3 – Perpendicular Lines ACTIVITY PREPARATION Students will discover that perpendicular lines have slopes that are opposite reciprocals of each other. Students will be able to write the equation of a line that is perpendicular to another line given a point. If the yy-intercept -intercept is given, then students will use slopeintercept form as the most eﬃcient method. Otherwise, students will use point-slope form. Lines perpendicular to the x- and y-axes y will also be created.

Standards for Mathematical Practice • • • • •

Materials

Preparation

Printed • •

• • •

1 Student Journal (per student) 1 Exit Ticket (per 2 students)

PROCEDURE AND FACILITATION Part I 1.

2. 3. FACILITATION TIP Depending on your students, consider having them complete page 1 with small groups and then guiding them through page 2 as a whole class.

Read the following scenario to the class: The robotics class is looking to speed up the process of making boxes. They’ve learned that they can use a program that allows a cut like stairs to be made, cutting the top and a side repeatedly. Use your knowledge of perpendicular slope to determine the equations needed to use the stair program shortcut and speed up their progress. Give a Student Journal to each student. Explain to students that they will work with their groups to ﬁnd the equations for the missing lines and record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What do you notice about all of the coordinates of line A? All of the x values are the same, −2.

DOK-1 When the x-coordinates -coordinates are the same in multiple points, what kind of line does that indicate? The line is a vertical line; the x’s will always equal −2.

DOK-1 What is the slope of a vertical line? Undeﬁned

DOK-1 Put it together: How do we write vertical lines with an undeﬁned slope? x = −2

e. DOK-1 Pick two points on line B. What is the slope? 0

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DOK-1 What do you notice about all of the coordinates of the line? All of the y values are the same, 6.

Engage

Explore

Explain

Elaborate

Evaluate

h. DOK-1 What are some differences in equations of horizontal and vertical lines? Horizontal equations are y = a value, the slopes are 0, and the y-coordinates are all the same values. Vertical equations are x = a value, the slopes are undeﬁned, and the x-coordinates are all the same values.

Intervention

Acceleration

FACILITATION TIP Use question 4h to create a T-chart and compare horizontal and vertical lines.

i. DOK-1 What form is most eﬃcient for comparing their slopes? Slopeintercept or point-slope form j. DOK-1 What is each form? Slope-intercept form is y = mx + b; point-slope form is y – y1 = m(x – x1). k.

DOK-1 How can you determine the slope using two points? Divide the difference of the y-coordinates and the difference of the x-coordinates.

l. DOK-1 What makes slopes perpendicular? The slopes are opposite reciprocals and have a product of −1. 5. 6.

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

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Math Chat DOK-1 Why will horizontal and vertical slopes always be perpendicular? The slopes are opposite reciprocals because horizontal lines always have 0 as the numerator, and vertical lines always have 0 as the denominator. • DOK-1 What is the generalization about perpendicular slopes? Does this also apply to equations of perpendicular lines? Perpendicular slopes are opposite reciprocals. This holds true for slopes of perpendicular lines. • DOK-2 How many perpendicular equations can be made from one perpendicular equation? How many can be made passing through one point? There are an inﬁnite number of perpendicular equations that can be made from one equation. There is only one perpendicular equation that can be made passing through a single point. •

Part II 1.

2. 3. 4. 5. 6.

Read the following scenario to the class: Saanvi S decides to program some individual sides outside of the stair program to make individual cuts. Brandon is struggling with how to program an individual line and with not following the staircase program pattern. Use your knowledge of perpendicular lines to decide the best strategy, and then write equations for needed sides. Students should still have their Student Journals. Explain to students that they will work with their groups to decide on a strategy to ﬁnd equations of perpendicular lines. Point out to the class that there is also an option where none of the strategies listed would be the best choice for that scenario. Students will then work together to ﬁnd perpendicular equations and record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 If you are given two points, what is the ﬁrst step to writing a perpendicular equation? Use the slope formula to determine the slope using the points given

DOK-1 Which is the orientation of the y-axis? y It is vertical.

DOK-1 Does y = 3 have the same orientation as the yy-axis? -axis? No, it is a horizontal line. It has the same orientation as the x-axis.

DOK-1 When you are ﬁnding a perpendicular equation to a horizontal or vertical line, is slope-intercept form or point-slope form needed? Why or why not? No. It is the opposite orientation for a line. If it is a horizontal line, then the perpendicular line will be a vertical line through the given point and vice versa.

FACILITATION TIP Post this Math Chat question for students to consider while they collaborate on Part I. Ask them to be prepared to discuss with the whole class.

FACILITATION TIP Consider distributing Part I and II of the Student Journal one at a time as needed.

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Parallel and Perpendicular Lines Explore 3 – Perpendicular Lines e. DOK-1 When an equation is in standard form, what’s the ﬁrst step for identifying the slope? Solve the equation for y to rewrite the equation in slope-intercept form. f.

FACILITATION TIP Consider that some students will still need to create a visual representation of an equation to be able to make accurate observations about the slope and coordinates. 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.

7. 8.

DOK-1 How is slope rewritten in order to be perpendicular? It is rewritten as the opposite reciprocal in order for the product of the two slopes to be −1.

Allow students enough time to complete Part II 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-1 How did you determine what strategy was best for each scenario? I looked to see what I was given. If I had coordinates, then I needed to calculate slope. If I had an equation, I needed to change the slope to the opposite reciprocal. • DOK-1 Why is the slope always a starting point in determining perpendicular equations? The slope changes in perpendicular equations. In order to write the new perpendicular equation, I need to know the slope of the original line to determine the slope of the new line. • DOK-3 Is there only one strategy for completing each scenario? Explain. No, there are multiple ways to determine a perpendicular equation. Another strategy would be graphing an equation to determine the slope and coordinates of the perpendicular equation. I could also graph the original line and use a compass to draw a perpendicular line. Or I could rotate it around the shared coordinate if it is known. •

Post-Explore FACILITATION TIP For this Exit Ticket, provide graph paper as needed for struggling students.

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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Parallel and Perpendicular Lines 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

Parallel and Perpendicular Slopes 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

Parallel 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

Perpendicular Lines 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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Intervention

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ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

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

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

Interactive Practice

Function Junction

Droid Quest

A game to practice the skills established by the standards in the scope

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Intervention and Assessment SMALL-GROUP PLANNER 1

Students who 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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Students

Notes & Comments

Engage

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Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

What does mastery look like?

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I can create and compare equations of two lines that are perpendicular to each other.

I can create and compare equations of two lines that are neither parallel nor perpendicular to each other.

I can compare parallel and perpendicular lines and see the connection as a system of equations.

I can explain if systems are consistent or inconsistent.

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SCOPE 1

Solving Pairs of Linear Equations Scope Introduction SCOPE SUMMARY

Student Expectations

Students will be introduced to the process of solving systems of equations graphically and algebraically. When solving systems of equations, students will be identifying the x value and y value of a point (solution) that would satisfy both equations in the system. Just like with solving linear equations, solving systems of equation solutions can come in various forms. Systems of equations can have one solution, no solution, or inﬁnitely many solutions. Students will be using graphs and solving algebraic equations in order to ﬁnd the solution(s) that will make the two equations equal.

8.FGR.7.1 Interpret and solve relevantmathematical problems leading to two linear equations in two variables. 8.FGR.7.2 Show and explain that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because the points of intersection satisfy both equations simultaneously. 8.FGR.7.3 Approximate solutions of two linear equations in two variables by graphing the equations and solving simple cases by inspection. 8.FGR.7.4 Analyze and solve systems of two linear equations in two variables algebraically to ﬁnd exact solutions.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In Grade 6, students began to determine if given variable values would satisfy a provided equation. Students learned to substitute the given value into the equation or inequality in order to see if the mathematical statement developed by the equation was true. They made the connection that a variable value that satisﬁed or made the equation true was a solution to the equation. In Grade 7, students learned how to solve given equations and inequalities in order to ﬁnd solutions that would satisfy the equation. Students used inverse operations to work an equation or inequality backward to determine the variable value that would make the equation or inequality true. The variable value that is found is the solution to the equation or inequality.

In high school, students will be exploring not only the relationships between systems of linear equations, but they will be looking for solutions between linear equations and functions and quadratic equations and functions. Students will be using systems of equations in all forms (graphically and algebraically) to solve real-world problems involving linear relationships. They will also encounter ﬁnding solutions between systems of inequalities graphically and algebraically.

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

use variables to represent quantities in problems.

•

construct simple equations and inequalities.

•

solve problems by reasoning about the 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: •

solve equations algebraically.

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

Graph Pairs of Linear Equations In this exploration, groups of students will be tasked with helping solve a real-world scenario about representing the number of social media followers for a group of friends over time. Students will: •

graph pairs of linear equations.

•

determine whether there is a single solution, an inﬁnite number of solutions, or no solution.

Explore 2

Explore 1

EXPLORE ACTIVITIES

•

Explore 4

Explore 3

In this exploration, groups of students will be tasked with analyzing equations about the amount of adults and amount of children on each amusement park ride to determine which ride will have the least amount of adults on it. Students will:

In this exploration, students will be tasked with analyzing each friend’s mini-race car equation. Students will: •

analyze pairs of linear equations.

•

determine if there is a single solution, inﬁnite number of solutions, or no solution.

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. Solving with Substitution

Analyze Systems of Linear Equations

Solving with Elimination In this ﬁnal exploration, groups of students will determine the cost of each snack item that two friends ordered. Students will: •

use substitution to solve systems of linear equations.

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use the elimination method to solve systems of linear equations.

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.

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Solving Pairs of Linear Equations Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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Intervention

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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. 8.PAR.3.3 Create and solve linear equations and inequalities in one variable within a relevant application.

Materials

Preparation

Printed •

1 Set 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 using a digital projector.

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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 true.

FACILITATION TIP Alternatively, you could have students write “fact” or “ﬁction” or “true” or “false” on dry erase boards. Have students hold up their boards for the answer.

FACILITATION TIP Have students correct the equation to make it true. It should be 80 = 5x + 50.

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

Identifying Misconceptions • •

Students may not know which value corresponds with the x value, or the rate. Students may not understand what the term deposit means.

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Solving Pairs of Linear Equations Hook – Order Up! ACTIVITY PREPARATION Students will ﬁnd the answer to a real-world problem, determining the costs of different items by solving the equations algebraically.

Materials

Preparation

Printed •

• • •

1 Order Up! (per class)

Reusable •

1 Phenomena Video (per class)

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

PROCEDURE AND FACILITATION Part I: Pre-Explore 1.

FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) Do you go to the movies on a regular basis?; 2) Who do you usually go to the movies with?; 3) What do you like to buy at the movie concession stand?

FACILITATION TIP Have the students make a table of points that ﬁt each equation and/or graph each equation.

4. 5.

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Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: Mr. Lott and Mr. Marquette are neighbors who take their children to the movies the ﬁrst Saturday of every month. It’s a tradition. Each dad usually buys something from the refreshment stand for their kids. One day, they went to a new movie theater and forgot to look at the prices on the menu. However, the dads thought they could ﬁgure out the cost of each item based on what each dad ordered and the total bill each dad paid. 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. Sample student answers: I notice that Mr. Lott and Mr. Marquette are solving for at least one variable. I wonder what snacks the dads bought at the refreshment counter. What did the dads buy that was different, and what did they buy that was the same? I can use math to determine the price of each snack item the dads bought by solving for the variable. Project Order Up!. Explain to students that Mr. Lott and Mr. Marquette each shared what he ordered and what the total cost of his order was. Discuss the following questions: a.

DOK-1 What did Mr. Lott order and what was his total cost? He ordered three large popcorns and three large sodas for a total of $48.00.

DOK-1 What did Mr. Marquette order and what was his total cost? He ordered two large popcorns and three large sodas for a total of $39.50.

DOK-1 What is similar about their orders? They both ordered only two different items — large popcorns and large sodas. They both ordered the same number of large sodas.

DOK-1 What is different about their orders? They ordered different quantities of popcorn.

Engage

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Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II: Post-Explore 1. 2.

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

DOK-1 What is the equation for Mr. Lott’s purchased refreshments? 3p + 3s = $48.00

DOK-1 What is the equation for Mr. Marquette’s purchased refreshments? 2p + 3s = $39.50

DOK-1 When solving algebraically, should you add or subtract these linear equations? Why? We should subtract these linear equations. When we subtract them, the sodas leave zero, but the popcorns show that 1 popcorn is equivalent to $8.50.

DOK-1 Once you know the value of each popcorn, how do you solve for the price of the soda? We can substitute p = $8.50 and solve for s. S is equivalent to $7.50 each.

e. DOK-1 How do you check your answers to make sure they are correct? Plug the values in for each variable and see if they are correct. Mr. Lott: 3p + 3s = $48.00→ 3(8.50) + 3(7.50) = 48.00→ 25.50 + 22.50 = 48.00 Mr. Marquette: 2p + 3s = $39.50→ 2(8.50) + 3(7.50) = 39.50→17.00 + 22.50 = 39.50 f.

FACILITATION TIP For early finishers, have students roll a die once to find a new number of popcorns ordered and again to find a new number of sodas ordered. Have the students calculate the new total cost. FACILITATION TIP Watch out for students who are confused by the variables s and p. Explain that equations do not have to use only x and y, but that any letter can be used.

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DOK-1 How much were the two items purchased at the refreshment stand? One large popcorn was $8.50, and one large soda was $7.50.

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Solving Pairs of Linear Equations Explore 1 – Graph Pairs of Linear Equations ACTIVITY PREPARATION Students will graph pairs of linear equations to determine whether there is a single solution, an inﬁnite number of solutions, or no solution.

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 Exit Ticket (per 2 students)

Plan to separate the class into groups of 2 students. Print a Student Journal for each student. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student will have one.

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) Why do people follow people who post information on social media?; 2) What beneﬁts are there in having a lot of people follow your postings? After reading the scenario, ask the class 3) What equation can you write that models Saanvi’s followers? FACILITATION TIP If students are stuck on how to proceed with drawing the graph, remind them that they can make a table of values. Encourage them to make a table using the same t values for Saanvi’s equation.

2. 3. 4.

FACILITATION TIP Watch out for students who swap the position of the x and y coordinates. Since these equations use the variables t and f, you could have the students cross off the x and replace it with t, and have them cross off the y and replace it with f. FACILITATION TIP This is a good time to review independent versus dependent variables. The variable t corresponds to the x-coordinate and is independent. The variable f corresponds to the y-coordinate and is dependent. 290

Read the following scenario to the class: Saanvi posted a cool video of herself breaking a world record on social media, and now the number of people following her has exploded! She originally had 25 followers, but now her number of followers is increasing at a rate of 10 followers per minute! Saanvi’s best friend, Gabriella, also posted a video on social media and modeled her number of followers over time with an equation as well. Saanvi is determined to get more followers than Gabriella. Help Saanvi determine whether she will ever have more followers, and, if so, when. Give a Student Journal to each student. Explain to students that they will work with their partners to graph Gabriella’s equation of followers over time and answer the questions that follow. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How would you graph the equation f = 50 + 5t? I could plug in values for t to ﬁnd different coordinates on the graph.

DOK-1 For the point (5, 75), which is the x-coordinate? -coordinate? Which is the y-coordinate? 5 is the x-coordinate, and 75 is the y-coordinate.

DOK-1 According to the graph, what does the x-coordinate represent? What does the yy-coordinate represent? The x-coordinate represents time in minutes, and the y-coordinate represents the number of followers.

DOK-1 What does (5, 75) mean in terms of the scenario? At 5 minutes, both Saanvi and Gabriella have 75 followers.

e. DOK-1 What do we call the point where the lines cross each other at (5, 75)? The point of intersection 5.

Have students use Desmos to type in the equation t = 5. Ask the following questions: a.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

b. DOK-2 What does this represent? This represents the point of intersection. c.

6. 7.

DOK-2 What happens to f when you change the value of t? If you raise the value of t, the value of f goes up as well. If you lower the value of t, the value of f goes down as well.

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

Math Chat •

DOK-1 How many equations did we plot on the same graph? Two

Explain the following to the class: Mathematicians call a pair or collection of equations a system of equations. We plotted two simultaneous equations, which is a system of equations, on the graph. •

DOK-1 What do we call the point on the graph where two lines cross each other? The point where the two lines cross is called the point of intersection.

Explain the following to the class: Mathematicians call the point of intersection the solution to a system of equations. •

•

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DOK-2 Why do you think the point of intersection is called the solution to the system of equations? It’s called the solution because it satisﬁes both equations simultaneously. DOK-2 What do you think a graph with no solution would look like? The lines would not cross (or there would be no intersection); the lines would be parallel.

Part II 1.

Read the following scenario to the class: Saanvi’s other three friends, Daisy, Debra, and Devon, want to be a part of the competition too! They also post their videos on social media. Help graph each friend’s equation to determine which friend is the winner of the competition. Explain to students that they will work with their partners to graph each of the equations for Saanvi’s other three friends. Then, they will collaborate with their partners to answer the questions that follow. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What is different about the lines for the equations for Saanvi and Daisy? Daisy’s equation for followers is parallel to the line for Saanvi’s followers.

DOK-1 Where on a graph does the solution to a system of equations occur? The solution to a system of equations occurs at the point of intersection.

DOK-1 Looking at the graph, does the system of equations for Saanvi and Daisy have a solution? There is no solution for a system of equations with parallel lines.

DOK-3 In the context of the problem with Saanvi and Daisy, what does no solution mean? It means that at no point in time will they have the same number of followers.

e. DOK-3 Can you graph the line for Devon as it is given to you? No, we need to change the equation to slope-intercept form. f.

DOK-2 What variable do you need to solve for in order to easily be able to graph? Solve for f to convert the equation from standard to slopeintercept form.

FACILITATION TIP After reading the scenario, ask the class 1) What data would need to change to have Daisy, Debra, and Devon’s followers exceed Saanvi’s followers?

FACILITATION TIP If needed, review the terms parallel and perpendicular from Scope 13. Remind students that the slopes of parallel lines are the same, and these lines will never intersect. Remind students that the slopes of perpendicular lines are the negative inverse of each other, and they form a right angle at their intersection.

FACILITATION TIP Watch out for students who struggle with converting the equation. You may need to do a brief tutorial.

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Solving Pairs of Linear Equations Explore 1 – Graph Pairs of Linear Equations 4. 5.

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

Math Chat •

•

DOK-1 By looking at a graph for a system of equations, how can you determine what the solution is? When looking at a graph for a system of equations, the solution will be the point of intersection. DOK-1 By looking just at the graph, how can you tell if a system of linear equations has one solution, no solution, or an inﬁnite number of solutions? A pair of linear equations has one solution if the lines intersect. A pair of linear equations has no solution if the lines are parallel. A pair of linear equations has an inﬁnite number of solutions if they make a single line. DOK-1 How could you tell if a point was the solution to a system of linear equations without using a graph? You could substitute the point into each equation. If the point makes both equations true, then it’s the solution to the system of equations.

Post-Explore FACILITATION TIP

For this Exit Ticket, some students may need additional graph paper to plot the second line.

2. 3.

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

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Solving Pairs of Linear Equations Explore 2 – Analyze Systems of Linear Equations ACTIVITY PREPARATION Students will analyze pairs of linear equations to determine whether there is a single solution, an inﬁnite number of solutions, or no solution.

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 Exit Ticket (per 2 students)

PROCEDURE AND FACILITATION FACILITATION TIP After reading the scenario, ask the class 1) What information varies throughout the rounds?; 2) Is it possible that Saanvi might never pass her friends? Explain how that could happen. FACILITATION TIP Consider distributing the Student Journal in parts as needed. Part 1 and then later Part II. FACILITATION TIP Watch out for students who are confused about using variables other than x and y. Help students make the connection that t corresponds to x and follows the horizontal axis because it is the independent variable. Help students make the connection that d corresponds to y and follows the vertical axis because it is the dependent variable.

Part I 1.

2. 3.

Read the following scenario to the class: Saanvi and her friends Gabriella and Devon are going to the amusement park and are excited to drive mini-race cars! Saanvi is determined to pass each of her friends. They compete in three rounds of mini-race cars, where their starting points and speeds vary throughout the rounds. Was Saanvi successful in her attempts to pass her friends? Give a Student Journal to each student. Explain to students that they will work with their partners to graph each friend’s equation from the mini-race cars, and then they will collaborate to answer the questions that follow. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 For round 1, is there an intersection point? No

DOK-1 For round 1, what do we call the lines on the graph? The lines for the equations in round 1 are called parallel lines.

DOK-1 When a system of equations results in a graph of parallel lines, how many solutions does that system have? None

DOK-1 For round 2, can you graph the equations as is? No, they need to be converted to slope-intercept form.

e. DOK-1 For round 2, what did you notice about the yy-intercepts and slopes of the equations? The slopes and y-intercepts are the same. f.

FACILITATION TIP Remind students that lines with the same slope are parallel. The only difference between the lines given in round 1 is the y-intercept. 294

DOK-1 For round 3, did Saanvi pass anyone? If so, how do you know? Yes, Saanvi passed Gabriella since the lines of their graphs intersect at (1, 5). She passed Devon shortly after taking off since the lines of their graphs intersect at (0, 0) and her line has a greater slope than Devon’s.

Have students input all 3 equations from round 3 into Desmos. Ask the following questions: a.

b. c. 6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 What do these values represent? They represent how far each person has driven after 5 seconds. DOK-2 Why has Saanvi driven so much farther than the others? Saanvi has a higher slope, which means she’s driving at a faster speed.

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

Math Chat •

DOK-2 How will we know from the graph if anyone passed someone? A point of intersection tells us that one object passed another object.

Explain the following to the class: Mathematicians say that when there is one point of intersection, there is one solution to the system of equations. DOK-1 For round 1, what did you notice about the slopes and yy-intercepts -intercepts of the equations? The slopes were the same, and the y-intercepts were different. • DOK-2 How might your observation about the slopes and yy-intercepts -intercepts explain your graph in round 1? If the slopes are the same and the y-intercepts are different, we will have parallel lines. • DOK-2 What is the relationship between the slopes and yy-intercepts and the type of solution we get for the system of equations? If the slopes are the same and the y-intercepts are different, we get no solution. •

Explain the following to the class: Mathematicians say that when the slope is the same and the y-intercepts are different, then the lines of the graphs are parallel with each other. When the lines are parallel, there are no solutions to the system of equations. DOK-3 What’s a possible scenario with the mini-race cars that could result in a system with no solution? Saanvi and her friends start at different locations or at different times and drive at the same speed as one another, creating parallel lines on a graph. • DOK-2 How might your observation about the slopes and yy-intercepts explain your graph for round 2? Since the slopes and y-intercepts are the same, the equations are the same, and, therefore, the lines for the equations are the same. •

Intervention

Acceleration

FACILITATION TIP Allow early ﬁnishers to go back and come up with additional equations that would produce a line with the same slope and y-intercept as these lines. Make sure they understand that lines can be represented by more than one equation. FACILITATION TIP Allow students to make the connection that at this point of intersection, the equations for the lines are equal. Ask them if they can think of another way to ﬁnd the solution, knowing this information. Solving systems of equations algebraically is covered in Explore activities 3 and 4.

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Explain the following to the class: Mathematicians say the lines of the graphs coincide with each other when they are the same equation. When the graphs coincide, there are an inﬁnite number of solutions. DOK-3 What’s a possible scenario with the mini-race cars that could result in a system with an inﬁnite number of solutions? Saanvi and her friends all start the same distance from the starting point and continue to drive side by side at the same speed. • DOK-1 For round 3, does Saanvi pass any of her friends? If so, when and where? Yes, Saanvi passed Gabriella since the lines of their graphs intersect at (1, 5). She passed Devon shortly after taking off since the lines of their graphs intersect at (0, 0), and her line has a greater slope than Devon’s. • DOK-2 In order for a system of linear equations to have one solution, what must be true about their slopes? Their slopes must be different. •

Part II 1.

Read the following scenario to the class: Feeling conﬁdent, Saanvi has now challenged Bart, the mini-race car owner, to a 4-round mini-race car battle. The equations for both participants are given for each round. Use your knowledge of solutions of systems of equations to determine in which rounds Saanvi wins and passes Bart and in which rounds Bart goes the same speed as Saanvi.

FACILITATION TIP Before beginning Part II, do a quick review of parallel lines from Scope 13. Have students describe how they look on a graph, and explain their slopes. They have the same slope, but different y-intercepts. FACILITATION TIP After reading the scenario, ask the class 1) What are “systems of equations”?; 2) What information in the equations do you need to look at to determine whether Saanvi passes Bart?

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Solving Pairs of Linear Equations Explore 2 – Analyze Systems of Linear Equations 2. 3.

Explain to students that they will work with their partners to determine whether Saanvi passes Bart by looking at the yy-intercepts -intercepts and slopes of each equation. As students collaborate, monitor their work and use the following guiding questions to assess student understanding:

FACILITATION TIP You could distribute blank graph paper and have students graph each system of equations if they need a more visual representation of the problem. Have them attempt to solve this without drawing a graph ﬁrst so students can identify the correlation between equal slopes and parallel lines. 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.

4. 5.

DOK-1 In what form must all the equations be in order to see clearly their slopes and y-intercepts? y All equations must be in slope-intercept form, y = mx + b.

DOK-2 In which round(s) did Saanvi pass Bart? Saanvi passed Bart in round 1 and in round 4.

DOK-2 In which round(s) were there parallel lines, indicating no one passed anyone because they were going the same speed? There were parallel lines in round 3.

DOK-2 In which round(s) were there inﬁnitely many solutions? There were inﬁnitely many solutions in round 2.

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

Math Chat DOK-2 Just by looking at the equations, how can you tell if a system of linear equations has no solution? The slopes are the same, and the y-intercepts are different. • DOK-2 Just by looking at the equations, how can you tell if a system of linear equations has an inﬁnite number of solutions? The slopes are the same, and the y-intercepts are the same. • DOK-2 Just by looking at the equations, how can you tell if a system of linear equations has a single solution? The slopes are different, and the y-intercepts may or may not be different. •

Post-Explore FACILITATION TIP For this Exit Ticket, offer graph paper as needed of students who still need a visual representation to succeed.

1. 2. 3. 4.

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Solving Pairs of Linear Equations Explore 3 – Solving with Substitution ACTIVITY PREPARATION Students will use substitution to solve systems of linear equations.

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 Exit Ticket (per 2 students)

Plan to separate the class into groups of 3 or 4 students. Print a Student Journal for each student. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student will have one.

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What kinds of amusement rides would adults typically not ride?; 2) Would you prefer to ride amusement rides that do not have a lot of adult riders? Why or why not? FACILITATION TIP Before they begin working, make sure they understand what substitution means. Let them know that they will be substituting, or replacing, one variable in one of the equations with the other equation.

2. 3. 4.

Read the following scenario to the class: Saanvi, Gabriella, and Devon decided to go on some of the rides at the amusement park. They wanted to go on the rides that would have the least amount of adults on them. Each of the friends found equations to determine the amount of adults and amount of children on each ride. Help the friends determine which ride will have the least amount of adults on it. Give a Student Journal to each student. Explain to students that they will collaborate with their groups to determine the number of adults and number of children on each ride using the given equations. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 How did you determine how many adults will be on the roller coaster? I know that 30 – 18 = 12, so there must be 12 adults on the roller coaster.

DOK-1 What is substitution? This is where you put one value in for a variable to test if the value is a solution to the equation.

DOK-2 How can you use substitution with two equations? We can ﬁnd the equation that has one variable isolated. Then, we can substitute what this variable is equal to into the other equation for that variable.

DOK-2 What is different about using substitution for Dungeon Drop? Instead of substituting just a number, I have to substitute an expression.

FACILITATION TIP Watch out for students who struggle on the last two systems of equations. These are replacing the variable c with an expression instead of just a constant. FACILITATION TIP While students collaborate, post questions 4b and 4c. Encourage students to be prepared to answer them in their own words during the Math Chat.

e. DOK-2 In Dungeon Drop, what value did you put in equation 1 for c? I put the whole equation for c into equation 1.

FACILITATION TIP Have students who ﬁnish early sort the rides in order from least adults to most adults. 298

DOK-2 In Dungeon Drop, what is the ﬁrst step to solving for a after you substitute a + 2 into the ﬁrst equation? I need to combine like terms. 2a + a + 2 combines to be 3a + 2.

DOK-2 In Dungeon Drop and Bumper Boats, what is the ﬁnal step to solve for a? I need to divide by the coeﬃcient to get a by itself.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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

Math Chat •

• •

•

DOK-2 What do you notice about all of these equations? What does that tell you about equations that can be solved with substitution? One of the equations in each pair must have one of the variables isolated. When one variable is isolated, we can substitute the value of that variable into the other equation. DOK-1 Is graphing the best option here? Why not? No, you would have to solve both equations for a variable, and that is not quickly done here. DOK-2 How would the steps to this problem change if there wasn’t already a variable by itself? You would ﬁrst need to solve for a variable in one of the equations and then complete the steps. DOK-3 How can you write the answer to a system of equations as an ordered pair? Use Bumper Boats as an example. We know that adults equal 5 and children equal 20. We can make an ordered pair of (adults, children), which would be (5, 20).

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Post-Explore 1. 2. 3.

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

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Solving Pairs of Linear Equations Explore 4 – Solving with Elimination ACTIVITY PREPARATION Students will use the elimination method to solve systems of linear equations.

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.

Preparation

Materials Printed • • •

• • •

1 Student Journal (per student) 1 Set of Order Cards (per group) 1 Exit Ticket (per 2 students)

•

Reusable •

1 Resealable bag (per group)

Plan to separate the class into groups of 3 or 4. Print a Student Journal for each student. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student has one. Print a set of Order Cards for each group. Cut out and place each set of Order Cards in a resealable bag. If desired, print them on card stock, and laminate them for future use.

PROCEDURE AND FACILITATION Part I FACILITATION TIP After reading the scenario, ask the class 1) What steps would you take to ﬁnd out how much each item the girls ordered costs?; 2) If the girls ordered more than one item, how could they determine how much one item costs?

2. 3.

FACILITATION TIP If students are struggling with elimination, have them highlight or circle the term in each equation that will be eliminated. This will help them more easily be able to identify systems of equations that are good candidates for the elimination method. FACILITATION TIP Make sure the students understand how the equations are derived. They will need to create their own equations in the next part.

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Read the following scenario to the class: Saavni and Gabriella were hungry and decided to go to the snack station to order food. After the girls ordered, they were curious how much each item cost. Help Saavni and Gabriella determine the cost of each item they ordered. Give a Student Journal to each student. Direct students’ attention to the 2 boxes on their Student Journals with Saanvi’s and Gabriella’s orders. Explain to students that they should look at the order each girl made and the equation for each order. Students will work with their groups to determine how much one order of tots costs and how much one grilled cheese sandwich costs. Then, students will analyze the chart at the bottom of the page and answer the questions that follow. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 Why is the equation for Saanvi’s order 1s + 1t 1t = 3.50? This equation is correct for Saanvi’s order because she ordered one grilled cheese sandwich (1s) and one order of tots (1t) and the total was $3.50.

DOK-1 What is different about Gabriella’s equation and order? Gabriella ordered 2 orders of tots, so she has 2t in her equation. The total for her order was $5.00.

DOK-1 What do you think one order of tots costs? Why? One order of tots costs $1.50 because this is the difference between the two total costs.

DOK-1 How much would that make one grilled cheese sandwich cost? Why? One grilled cheese sandwich will cost $2.00 because if you subtract $1.50, the cost of one order of tots, from Saanvi’s total, you will get $2.00. © Accelerate Learning Inc. - All Rights Reserved

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

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

Math Chat DOK-1 Why would it be useful to subtract the equations? It is useful because it will show you the difference between them. Here, it shows the difference between the orders and the difference between the totals. • DOK-1 How did you arrive at your solution? The difference between the two orders is 1 order of tots, so I subtracted the totals to ﬁgure out how much an order of tots would be. Then, I used that information to substitute back in to ﬁnd out how much a grilled cheese sandwich would cost. • DOK-1 What happened to the s terms when you subtracted the equations? The s terms were eliminated or canceled each other out. •

Intervention

Acceleration

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Explain the following to the class: Mathematicians call this method of two terms canceling each other out the elimination method. •

DOK-2 Where would these lines intersect if we graphed them? How do you know? They would intersect at (1.5, 2). This is based on the solution we found: 1 tots order cost $1.50, and 1 grilled cheese sandwich cost $2.00.

Part II 1.

2. 3. 4.

Read the following scenario to the class: Saavni and Gabriella want to know the cost of other items at the snack station. The girls ﬁnd the last 8 orders and decide to determine the cost of each item from the orders. Help Saavni and Gabriella determine the cost of each item ordered. Give a set of Order Cards to each group. Explain to students that they will collaborate with their groups to use the Order Cards to determine the price for other items on the snack station menu. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What equation did you write for order 3? For order 3, I wrote the equation 1c + 1d = 6.25.

DOK-2 What is different about the coefficients in order 6? Both coefficients in order 6 are 2. Neither of the coefficients is 1 like the other orders.

DOK-2 What is the last step in determining the price for the fries in order 6? The last step is to divide both sides of the equation by 2. This shows that fries cost $2.00.

5. 6.

DOK-2 Why do you think we are subtracting instead of adding? We are subtracting because we want to ﬁnd the difference between the two equations.

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

Math Chat DOK-1 Is graphing the best option here? Why not? No, you would have to solve both equations for a variable, and that is not quickly done here. • DOK-2 When is using the elimination method the best way to solve? Using the elimination method is best when one of the variables in both equations makes a zero pair. For example, one equation has 5x, and the other equation has −5x. •

FACILITATION TIP After reading the scenario, ask the class 1) What equation can you write to determine the cost of each item from the orders? FACILITATION TIP If students are struggling with creating the systems of equations, suggest that they make a table to easily line up which food and quantity each friend ordered. The last row or column of each card could be the total price they paid. They should now more easily be able to pull the information to write their equations.

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Solving Pairs of Linear Equations Explore 4 – Solving with Elimination Post-Explore 1. 2. 3. 4.

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Intervention

Acceleration

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Solving Pairs of Linear Equations 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

Graph Pairs of Linear 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

Analyze Systems of Linear 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

Solving with Substitution

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

Solving with Elimination

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

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

Notes

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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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Solving Pairs of Linear Equations

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Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

What does mastery look like?

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I can solve mathematical problems leading to two linear equations in two variables.

I can use technology tools to graph and interpret systems of two linear equations.

I can approximate solutions of two linear equations and use algebraic strategies to validate the graphical approximations.

I can analyze and interpret solutions to the systems of linear equations.

I can use substitution and elimination to solve systems of linear equations.

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SCOPE 1

Pythagorean Theorem Scope Introduction SCOPE SUMMARY

Student Expectations

In this scope, students will be able to explain the Pythagorean theorem and its converse. They will define the Pythagorean theorem as given a right triangle with sides a and b and hypotenuse c, then a2 + b2 = c2. Students will understand its converse to be if a triangle with sides a and b and hypotenuse c satisfies the equation a2 + b2 = c2, then the triangle is a right triangle. They will use these theorems to calculate the unknown sides of a right triangle as well as find the distance between two points on a coordinate grid. Students will be able to express the sides of triangles in radical form when necessary.

8.GSR.8.1 Explain a proof of the Pythagorean Theorem and its converse using visual models. 8.GSR.8.2 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles within authentic, mathematical problems in two and three dimensions. 8.GSR.8.3 Apply the Pythagorean Theorem to ﬁnd the distance between two points in a coordinate system in practical, mathematical problems.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In 5th grade, students discovered unique triangles, exploring how the angles and sides of triangles represent their classiﬁcation. In 6th grade, students learned how to calculate the distance between two points on the coordinate plane. They found the distance of horizontal and vertical lines through counting. Students have an understanding of irrational and rational numbers and have been introduced to square roots and their meaning. These concepts will be important as they discover the Pythagorean theorem and its converse.

Students will use these concepts in upcoming 8thgrade scopes as they determine the volume of threedimensional ﬁgures. The Pythagorean theorem also establishes the basis for triangle congruence that will be expanded on in high school. Students will use this entry into square roots as they continue to work with radicals and square roots as they extend into simplifying radicals in Algebra 1.

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

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES

explain why the slope, m,, is the same between any two distinct points on a non-vertical line in the coordinate plane.

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

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.

determine missing sides of right triangles using the Pythagorean theorem.

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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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Modeling the Pythagorean Theorem and the Converse of the Pythagorean Theorem In this exploration, groups of students will be tasked with working collaboratively to analyze various aspects of a new mini-golf course and their different holes. Students will: •

ﬁnd a relationship between the three sides of a right triangle.

•

demonstrate the Pythagorean theorem using models.

•

ﬁnd the area of connected squares.

Explore 2

Explore 1

EXPLORE ACTIVITIES Finding an Unknown Side Length in a Right Triangle In this exploration, students will work collaboratively to analyze tent drawings by evaluating different triangles on the tents and tent shadows. Students will: •

PYTHAGOREAN THEOREM

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use the Pythagorean theorem to determine missing side lengths of right triangles when given leg(s) and the hypotenuse.

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

The Pythagorean Theorem in Rectangular Prisms In this exploration, groups of students will be tasked with determining the maximum length a putter can be placed away from the box if a putter is placed diagonally from it. 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.

apply the Pythagorean theorem to threedimensional rectangular prisms.

The Pythagorean Theorem on a Coordinate Grid In this exploration, groups of students will be tasked with designing more holes for an existing golf course using coordinate grids to determine the layout for each hole. 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.

determine the distance between two points on a coordinate grid by using the Pythagorean theorem.

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

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PYTHAGOREAN THEOREM

Pythagorean Theorem Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will examine a series of tables based on the prior standard 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. 7.PAR.4.7 Use similar triangles to explain why the slope, m, is the same between any two distinct points on a non-vertical line in the coordinate plane.

Materials

Preparation

Printed •

PYTHAGOREAN THEOREM

Home

• •

1 Does Not Belong (per student or per group)

Print one Does Not Belong for each student or group. You may place students in groups of two or three.

Procedure and Facilitation Points 1. 2. 3.

4. 5.

First Slide: Answer Choice D does not belong. Answers A, B, and C all 3 1 simplify to a slope of __5, and D simpliﬁes to a slope of __3.

Second Slide: Answer Choice A does not belong. Answers B, C, and D all 4 simplify to a slope of –__3, and A represents a slope that is undeﬁned.

Third Slide: Answer Choice B does not belong. Answers A, C, and D all 2 1 represent a slope of __3, and B represents a slope of __3.

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

Identifying Misconceptions • •

FACILITATION TIP To coordinate a focused discussion, consider distributing or projecting each slide one at a time. Allow students some independent thinking time with note taking, and then provide some shoulder partner time. FACILITATION TIP If needed, coach students to focus on slopes.

FACILITATION TIP This Foundation Builder includes six very clear graphs that could be used to provide students with extra practice making observations and assessing true/false statements.

Students may struggle to understand that to accurately compare slopes, they should be in the simplest form. Students may also forget that to calculate slope the ratio of the change in y over the change in x must be used.

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Pythagorean Theorem Hook – Pythagorean Theorem ACTIVITY PREPARATION Students will determine missing sides of right triangles using the Pythagorean theorem.

Materials

Preparation

Printed •

• •

1 Pythagorean Theorem (per class)

Reusable •

•

1 Phenomena Video (per class)

Plan to show the video. Prepare to project the Pythagorean Theorem 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) Have you ever played mini golf?; 2) What is the goal of the game?: 3) What strategies do you use when playing a round of mini golf?

FACILITATION TIP The students will probably notice the exponents in the Pythagorean theorem equation. You may need to do a quick refresh on what exponents means.

4. 5.

6. 312

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: Oliver is playing mini golf with his family. As he continues along the course he sees that he will be playing a dogleg left in golf and is trying to determine the best strategy that he can use to play as few strokes per round as possible. He is trying to choose between two options. He could play it safe and hit it short onto the fairway. He can also try to hit it over the trees onto the green. 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. Sample student answers: I noticed that the distance from the tee area to the hole forms a diagonal. I wonder what math can be used in golf to get the fewest number of strokes in a hole. Project the Pythagorean Theorem slide. Explain to students that Oliver’s dad suggested that he try to hit the ball over the trees to the hole. Discuss the following questions: a.

DOK-1 Where do you see math in golf? Allow students to share all ideas. Student answers will vary. Sample student answer: You can measure the distance from the tee area to the hole to determine the fewest strokes to the hole.

DOK-1 What does this sentence represent? Allow students to share all ideas. Student answers will vary. This sentence includes variables, so we can identify the information that we already know and substitute the variables in order to solve the problem.

Engage

Explore

Explain

Elaborate

Evaluate

Show the Phenomena video again and restate the problem. Refer to the Pythagorean Theorem slide and discuss the following questions: a.

DOK-1 Does this math sentence make more sense after the Explore activities? Yes, this is called the Pythagorean theorem.

DOK-1 What strategies would you use to solve for the distance from the tee area to the hole? We can look at the dogleg left golf course as a right triangle from the tee area to the hole. We can measure the distance of each side of the right triangle. Each side would represent the distance. The distance from the tee area to the hole would represent the hypotenuse in a right triangle. We can use the distance of the legs of a right triangle to ﬁnd the hypotenuse.

Acceleration

FACILITATION TIP

Part II: Post-Explore 1. 2.

Intervention

Make sure it is mentioned that this is a right triangle, but we are missing the length of the longest side. This side is called the hypotenuse. It is also the side opposite from the right angle. FACILITATION TIP The hypotenuse of this triangle is approximately 255 yards. Ask the students if that answer makes sense. They should recognize that the hypotenuse should be the longest side length.

PYTHAGOREAN THEOREM

Home

DOK-1 Do you feel that you have a strong understanding of using the Pythagorean theorem? Answers will vary based on students’ success during the activity and conﬁdence level.

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PYTHAGOREAN THEOREM

Pythagorean Theorem Explore 1 – Modeling the Pythagorean Theorem and the Converse of the Pythagorean Theorem ACTIVITY PREPARATION Students will demonstrate the Pythagorean theorem by using models and will ﬁnd the area of three connected squares. Students will also identify right triangles with the converse of the Pythagorean theorem.

Standards for Mathematical Practice • • • •

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

Materials

Preparation

Printed • • • • •

1 Student Journal (per student) 1 Set of Converse of the Pythagorean Theorem Cards (per pair) 1 Modeling the Pythagorean Theorem Work Mat (per pair) 1 Demonstrating the Pythagorean Theorem Triangles (per pair) 1 Exit Ticket (per student)

• • •

•

Plan to separate the class into groups of two to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a Modeling the Pythagorean Theorem Work Mat, a Demonstrating the Pythagorean Theorem Triangles, and a set of the Converse of the Pythagorean Theorem Cards for each pair of students. Gather enough pairs of scissors, glue sticks, and calculators for each pair of students to use one of each for Part II.

Reusable • • •

1 Pair of scissors (per pair) 1 Glue stick (per pair) 1 Calculator (per pair)

PROCEDURE AND FACILITATION Part I: Modeling the Pythagorean Theorem FACILITATION TIP

Before reading the scenario, ask the class 1) Who has golfed on a mini-golf course? Did you ﬁnd it easy or diﬃcult?; 2) What makes mini-golf courses fun to play?; 3) Describe how mini-golf course holes are designed. 2. 3. 4.

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Read the following scenario to the class: Mason plans on opening a mini-golf course in a few months. He begins planning his new course by researching several designs online. Mason notices that a majority of each course’s 18 holes are shaped like right triangles. He notices a small border is placed along the perimeter of each of these right triangle-shaped holes. After looking at the border along each right triangle-shaped hole, Mason is curious whether there is a relationship between the three sides of a right triangle. Let’s help Mason determine the relationship between the 3 sides of a right triangle as he begins planning his mini-golf course. Give a Student Journal to each student. Give a Modeling the Pythagorean Theorem Work Mat, a Demonstrating the Pythagorean Theorem Triangles, and a pair of scissors to each pair of students. Have students use the Modeling the Pythagorean Theorem Work Mat and Demonstrating the Pythagorean Theorem Triangles to represent the Pythagorean theorem. Students should cut out the triangles from the Demonstrating the Pythagorean Theorem Triangles handout and use them on the Modeling the Pythagorean Theorem Work Mat. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Instruct students to use the Demonstrating the Pythagorean Theorem Triangles to demonstrate the Pythagorean theorem. They will place these triangles on the Modeling the Pythagorean Theorem Work Mat. After placing the triangles on the Modeling the Pythagorean Theorem Work Mat, students can draw a square for each side length, using the side lengths of each triangle squared, and cut out each square to represent the side². Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 What is the Pythagorean theorem? Responses may vary. The Pythagorean theorem describes a special relationship between the sides of a right triangle. When applying the Pythagorean theorem, each side of the right triangles is also a side of a square that’s attached to the triangle.

DOK-1 How can you determine the area of a square? Responses may vary. The area of a square is any side multiplied by itself. (For example, a × a = a2.)

DOK-1 How can you use triangles abc, def, and mno to demonstrate the Pythagorean theorem? Responses may vary. Make three squares with sides that are equal to each side of the triangle. Cut out each square to place on each side of the triangle.

PYTHAGOREAN THEOREM

Home

FACILITATION TIP Remind students what exponent means. Showing them in terms of the square will help them understand if they have forgotten the concept.

STEMscopes Tip

DOK-1 Given the area of square a, what operation allows you to determine the side length of square a? Responses may vary. The square root allows us to ﬁnd the side length of a square given the area of the square.

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.

e. DOK-2 Is a2 + b2 = c2 the same as b2 + a2 = c2? Why or why not? Responses may vary. Yes, because of the commutative property of addition, 2 + 4 yields the same answer as 4 + 2. 6. 7.

Allow students enough time to record all of 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-1 Using the area of each square, describe the Pythagorean theorem. The area of squares a and b together equal the area of square c. • DOK-1 How can you determine the side length of a square given its area? We need to find the square root (√) of the area in order to determine the length of one of the square’s sides. • DOK-2 What did you notice about the sum of square a and square b when compared to square c? The sum of the number of tiles it takes to fill square a and square b is the same amount of tiles it takes to fill square c. • DOK-2 Do you think the Pythagorean theorem only applies to right triangles? What about non-right triangles? Answers will vary. I think the Pythagorean theorem applies to all triangles, including non-right triangles. •

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PYTHAGOREAN THEOREM

Pythagorean Theorem Explore 1 – Modeling the Pythagorean Theorem and the Converse of the Pythagorean Theorem Part II: The Converse of the Pythagorean Theorem 1. FACILITATION TIP Before reading the scenario, ask the class 1) In your opinion, which type of mini-golf course holes would be easiest to play: holes with acute, right, or obtuse angles? After reading the scenario, ask the class 2) What is the Pythagorean theorem? FACILITATION TIP Watch out for students who assume hole 12 is a right triangle just because it appears to be one. Make sure they perform the calculation.

FACILITATION TIP Students who ﬁnish early can work on creating a list of triangle side lengths that are right triangles. This list may include triangles that are not in this Explore activity.

Read the following scenario to the class: While researching online, Mason notices that many of the golf courses contain right-triangle-shaped holes, but he’s not 100% conﬁdent that each of these holes is a right triangle. Without a protractor, he’s unsure of how to prove which holes are right triangles and which are not. Let’s help Mason decide whether a triangle is a right triangle by applying the Pythagorean theorem! Give the Converse of the Pythagorean Theorem Cards, a pair of scissors, and a glue stick to each pair of students. A calculator can also be provided to students when evaluating the square of larger numbers. Instruct students to begin by identifying whether hole 7 and hole 12 are right triangles by applying the Pythagorean theorem. Have students ﬁll in the blanks with the legs and assumed hypotenuse of each triangle. Have students solve the equation by simplifying the squares. Instruct students to circle Yes if the equation is true and to circle No If the equation is not true. Have students divide the Converse of the Pythagorean Theorem Cards page into two so each student receives one set of cards. Instruct the class that each student will cut out their own set of 6 cards. Then, have students work together to apply the Pythagorean theorem to each triangle and determine whether the triangle is an example of a right triangle. When students reach a consensus on each triangle, they will glue each card in the appropriate column. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 How do you know which measurements to use when applying the Pythagorean theorem? Responses may vary. We need to know the side lengths of both legs in order to substitute the values for variables a and b. We then substitute the length of the hypotenuse for variable c.

DOK-2 What happens when the Pythagorean theorem creates a true statement? Responses may vary. When the equation created using the Pythagorean theorem generates a true statement, the three side lengths form a right triangle.

DOK-2 What happens when the Pythagorean theorem creates a false statement? Responses may vary. When the equation created using the Pythagorean theorem does not generate a true statement, the three side lengths do not form a right triangle. The three side lengths could, however, create another type of triangle.

6. 7.

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 How can you determine whether a triangle is a right triangle, given only the lengths of each side? We can apply the Pythagorean theorem using the triangle’s side lengths. If the Pythagorean theorem forms a true statement, the triangle is a right triangle. • DOK-2 In your own words, describe the converse of the Pythagorean theorem. A true statement is generated when using the Pythagorean theorem and the lengths of a right triangle’s three sides. If a false statement is created, the triangle is not a right angle. • DOK-2 A triangle has side lengths of 7, 9, and 10. Is this a right triangle? a2 + b2 = c2 72 + 92 = 102 49 + 81 = 100 130 ≠ 100 These side lengths do not form a right triangle. •

316

Engage

Explore

Explain

Elaborate

Evaluate

Post-Explore 1. 2. 3.

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

Intervention

Acceleration

FACILITATION TIP This Exit Ticket could be used as both a pre-assessment and post-assessment to help you modify instruction or show student growth.

Notes

PYTHAGOREAN THEOREM

Home

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PYTHAGOREAN THEOREM

Pythagorean Theorem Explore 1 – Modeling the Pythagorean Theorem and the Converse of the Pythagorean Theorem ACTIVITY PREPARATION Students will use the Pythagorean theorem to determine the missing side length of a right triangle when given both legs or one leg and the hypotenuse.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Geoboard (per pair) 1 Set of Finding the Missing Leg of a Right Triangle Cards (per pair) 1 Exit Ticket (per student)

Reusable • • • • •

• • • •

• •

1 Clear sheet protector (per pair) 1 Dry-erase marker (per pair) 1 Calculator (per pair) 1 Glue stick (per pair) 1 Pair of scissors (per pair)

Plan to separate the class into groups of two to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a Finding the Missing Leg of a Right Triangle Card Sort per pair. Print a Geoboard for each pair of students for Part I. Place it inside a clear sheet protector for each pair. If desired, print it on card stock, and laminate it for future use. Gather a dry-erase marker, calculator, pair of scissors, and glue stick for each pair. 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. (Geoboard)

PROCEDURE AND FACILITATION Part I: Finding the Hypotenuse of a Right Triangle FACILITATION TIP Before reading the scenario, ask the class 1) How many holes does a typical mini-golf course have?; 2) What obstacles might be placed in mini-golf course holes?; 3) Why are there borders around the perimeters of the mini-golf course holes? FACILITATION TIP These can be laminated or inserted into sheet protectors. Students can then use dry-erase markers on them. FACILITATION TIP You may need to demonstrate how to use the square root button on your classroom calculators. Some calculators require the number to be entered prior to entering the square root symbol, and some calculators are the opposite. 318

2. 3. 4.

5. 6.

Read the following scenario to the class: Gemma is building a new course at Paciﬁc Mini-Golf inspired by her favorite sci-ﬁ movie. Each layout of the 18 holes on the new course is shaped like a right triangle and will feature several obstacles. A thin border will be placed along the perimeter of each hole. Let’s help Gemma create several right-triangle designs using a geoboard. Then, let’s use the Pythagorean theorem and the geoboard to help calculate the length of each side. Give a Student Journal to each student. Give a Geoboard, a dry-erase marker, and a calculator to each pair of students. Have students recreate hole 1 using the Geoboard. Students will measure the vertical and horizontal distance between pegs to determine the lengths of both legs. Students will apply the Pythagorean theorem to solve for the hypotenuse. Students may also use a calculator to evaluate the square root. Explain to students that they will create three additional right triangles and solve for the hypotenuse of each. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

7. 8.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 What do you think is the smallest right triangle you can create on the geoboard? The smallest triangle I can create is a triangle with legs measuring 1 unit each.

DOK-3 How do you know you’ve created a right angle using both legs of a right triangle on the geoboard? The legs of a right triangle must be perpendicular, so one leg of the triangle on the geoboard should include vertical pegs and the other leg should be formed using horizontal pegs.

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-3 How did you use the Pythagorean theorem to solve for the missing hypotenuse? I identiﬁed the values of both legs and substituted them into the Pythagorean theorem as the a and b variables. I then squared both numbers and added them together. I then found the square root of this total to solve for the missing hypotenuse. • DOK-2 What do you notice about the hypotenuse of hole 1 when compared to holes 2–4? Hole 1’s hypotenuse is an integer, whereas holes 2–4 have hypotenuses that are irrational numbers involving decimals. • DOK-2 Are you able to create a right triangle with leg measurements involving fractions or decimals using the geoboard? Why or why not? No, because the distance between geoboards represents one unit, every line drawn on the geoboard has to connect to each peg and not be between the pegs to represent a rational number. • DOK-2 A right triangle has legs measuring 10 feet and 17 feet. What is the length of the hypotenuse? Round your answer to the nearest hundredth, if necessary. 102 + 172 = c2 100 + 289 = c2 2 389 ____= c √ 389 = c c ≈ 19.72 feet •

Intervention

Acceleration

PYTHAGOREAN THEOREM

Home

Part II: Finding the Missing Leg of a Right Triangle 1.

Read the following scenario to the class: Gemma has space for a few more holes on her mini-golf course. She creates three more designs but forgets to include a key measurement on each of the three right-triangle-shaped designs. Let’s use the Pythagorean theorem to help Gemma determine the length of each of the missing legs. Give a Finding the Missing Leg of a Right Triangle Card sort, a pair of scissors and a glue stick to each pair of students. A calculator can also be provided to students when evaluating the square roots of numbers. Have students divide the Finding the Missing Leg of a Right Triangle Card Sort page into two so each student receives one set of cards. Each student will cut out their own set of 12 cards. Students will work together and identify the given leg and hypotenuse of each right-triangle-shaped hole. (Note that only 9 of the 12 cards will be used. Each card represents either a leg or hypotenuse.) Students will then apply the Pythagorean theorem to each triangle to determine the missing leg. When the students reach a consensus on each triangle, students will glue each card in the appropriate space. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

FACILITATION TIP After reading the scenario, ask the class 1) What is a right triangle?; 2)What are the “legs” in a right triangle?; 2) How can the Pythagorean theorem be used to ﬁnd the length of each of the missing righttriangle legs? FACILITATION TIP Suggest to students that they ﬁgure out where all of their cards should go before they glue anything down. This will make their Student Journals neater and reduce gluing errors.

DOK-1 What are some of the differences when using the Pythagorean theorem to solve for a missing leg instead of a missing hypotenuse? The major difference is using subtraction to solve for a missing leg instead of addition to solve for a missing hypotenuse.

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PYTHAGOREAN THEOREM

Pythagorean Theorem Explore 2 – Finding an Unknown Side Length in a Right Triangle b.

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.

DOK-2 Other than the value of the square root, what is the difference between the square root of 16 and the square root of 20? The square root of 16 is 4. 4 is a whole number and integer. The square root of 20 is an irrational number; however, we can round it to 4.47. The difference between the square root of 16 and the square root of 20 is 0.47.

c. DOK-3 Given that a and b are the measures of both legs and c is the measure of the hypotenuse, is c2 – a2 = b2 the same as c2 – b2 = a2? Yes, since the value of a leg can be either a or b as long as the other leg is the other variable, both equations can be used to solve for the missing leg. 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 You’ve calculated the value of a missing leg. This value is greater than the length of the hypotenuse. Is this possible? No, the hypotenuse should be the greatest length of the three sides of a triangle. You should check your work if this happens. • DOK-2 A right triangle has a leg measuring 11 inches and a hypotenuse measuring 20 inches. What is the length of its missing leg? Round your answer to the nearest hundredth, if necessary. c2 – a2 = b2 202 – 112 = b2 400 – 121 = b2 2 279 ____= b √ 279 = b b ≈ 16.70 inches • DOK-3 Why do you think a geoboard was not used to model solving for the missing leg of a right triangle? It’s a lot harder to create and measure the diagonals of lines on a geoboard than the vertical and horizontal measurements for each leg. •

FACILITATION TIP This Math Chat question provides a good quick check of student understanding. Project it and allow for a think, pair, share to assess students’ skills so far.

Post-Explore FACILITATION TIP

This Exit Ticket could be used as both a pre-assessment and post-assessment to help you modify instruction or show student growth.

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.

320

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

PYTHAGOREAN THEOREM

Home

321

PYTHAGOREAN THEOREM

Pythagorean Theorem Explore 3 – The Pythagorean Theorem in Rectangular Prisms ACTIVITY PREPARATION Students will apply the Pythagorean theorem to three-dimensional rectangular prisms.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

• • • •

1 Student Journal (per student) 1 Set of Putter Boxes (per pair) 1 Exit Ticket (per student)

Reusable • •

•

Plan to separate the class into groups of two to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print a set of Putter Boxes per pair of students. Gather enough calculators and rolls of tape for each pair to have one of each. Gather enough pairs of scissors for each student to have one.

1 Calculator (per pair) 1 Pair of scissors (per student)

Consumable •

1 Roll of tape (per pair)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What type of golf club is used at a minigolf course?; 2) Why do golf clubs come in a variety of lengths? After reading the scenario, ask the class 3) What is a real-life example of a rectangular prism? FACILITATION TIP If you have a hollow box to show the students, it may be helpful for them to see a 3-dimensional example of what this looks like. Many students struggle with 3-dimensional visualizations. FACILITATION TIP You might need to explain why they are doing this. They are ﬁrst ﬁnding the hypotenuse of the right triangle on the base of the box. Then, that hypotenuse becomes one of the legs of the right triangle that is standing up inside the box.

2. 3.

4. 5. 6.

Read the following scenario to the class: Kyle sells miniature golf clubs, known as putters, to miniature golf course owners and managers. Putters come in a variety of lengths depending on the height of the customer. Kyle uses cardboard boxes in the shape of rectangular prisms to ship his putters out to customers. Let’s help Kyle determine the maximum length of a putter he can place in the box if the putter is placed in the box diagonally. Give a Student Journal to each student. Have students use the rectangular prism to ﬁll in the blanks for step 1 and step 2 in order to determine the diagonal length from bottom-most left front corner to the top-most right back corner. You may wish to provide your students with calculators to use when applying the Pythagorean theorem. Distribute the Putter Boxes and rolls of tape to student pairs. Give a pair of scissors to each student. Have students cut out Putter Boxes A and B and use the tape to connect the sides of each to form two rectangular prisms. Students will use the measurements on each prism (Putter Box) to answer the questions on page 2 of their Student Journals. Students can use a calculator when applying the Pythagorean theorem. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

322

DOK-1 What measurements are needed to determine the length of the diagonal of a rectangular prism? I need to know the rectangular prism’s length, width, and height to solve for the diagonal length. © Accelerate Learning Inc. - All Rights Reserved

8. 9.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How do you know the Pythagorean theorem can be applied when solving for a rectangular prism’s diagonal length given the type of angle formed between the base of the prism and one of its lateral sides? The two sides are perpendicular to each other and form a 90-degree angle. The Pythagorean theorem can only be applied to a triangle with a 90-degree angle.

DOK-2 When the measurements of a rectangular prism’s length and width are provided, how many times will you need to apply the Pythagorean theorem when solving for the diagonal of the prism? We will need to use the Pythagorean theorem twice: once for the diagonal of the prism’s base and then once to determine the length of the prism’s diagonal.

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 does the Pythagorean theorem apply to three-dimensional ﬁgures? We can apply the Pythagorean theorem to determine the distance from one of the prism’s lower corners to the opposite side’s upper corner. • DOK-2 What is the diagonal length of a rectangular prism with a length of 12 inches, a width of 5 inches, and a height of 7 inches? Round your answer to the nearest hundredth, if necessary. Diagonal of base, d: 122 + 52 = d2 2 169 ____= d √ 169 = d d = 13 inches Diagonal of prism, x: 132 + 72 = x2 2 169 ____+ 49 = x √ 218 = x x ≈ 14.76 inches • DOK-2 Given Putter Box A or Putter Box B, which rectangular prism has the greater diagonal length and by how much? Round your answer to the nearest inch. Putter Box B has a greater diagonal by about 3 inches. (32.20 – 29.07 = 3.13) •

Post-Explore 1. 2. 3.

Intervention

Acceleration

PYTHAGOREAN THEOREM

Home

FACILITATION TIP If you have any empty boxes (cereal boxes, shipping boxes, shoe boxes, etc), save them. Have students measure the length and width of the outside and then calculate to ﬁnd the inside diagonal length. This can be extra practice for early ﬁnishers. FACILITATION TIP Project this Math Chat question and use it as a quick check for understanding. Consider using a think, pair, share. 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.

FACILITATION TIP

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

For this Exit Ticket, clarify your criteria for success. How many steps are required for students to show their thinking?

323

PYTHAGOREAN THEOREM

Pythagorean Theorem Explore 4 – The Pythagorean Theorem on a Coordinate Grid ACTIVITY PREPARATION Students will determine the distance between two points on a coordinate grid by using the Pythagorean theorem.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision.

Preparation

Materials Printed • •

• • •

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

Plan to separate the class into groups of two to complete this activity. Print a Student Journal and an Exit Ticket for each student. Gather enough calculators and number cubes for each pair to have one.

Reusable • •

1 Calculator (per pair) 1 Number cube (per pair)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What could help mini-golf course designers visualize their golf course designs? After reading the scenario, ask the class 2) What is a coordinate grid?; 3) How can using a coordinate grid help Yvette determine the layout for mini-golf course holes?

2. 3. 4.

FACILITATION TIP Once students have the ordered pairs for all three points of their triangle, have them calculate the slope of the hypotenuse. Have them identify the side with a slope of 0 and the side with an undeﬁned slope. This will be a review of a previously learned concept.

FACILITATION TIP If you want students to calculate with larger or smaller numbers, consider giving students a third grid and changing the scale on the axes. This could be an additional exercise for early ﬁnishers. FACILITATION TIP Specify what place value you would like the students to round their answers to. The answer key rounds to the nearest hundredths place, but you can round to whichever place is standard for your classroom. 324

Read the following scenario to the class: Miniature golf course designer Yvette is adding several right-triangle-shaped holes to the existing golf course. She begins planning by using a coordinate grid to determine the layout for each hole. Let’s help Yvette by mapping out several outlines she can use for her additions. Give a Student Journal to each student. Give a calculator and a number cube to each pair of students. Explain to students that they will create addition 1. Instruct students to begin by plotting point A on the coordinate grid. Then, have students use the number cube to generate the ordered pairs for points B and C. Have students form a triangle using these three points. Have students record the measurements of each side using the table provided. Allow students to use calculators when applying the Pythagorean theorem. Then, have students repeat this process by creating addition 2, a second right triangle. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 How many ordered pairs are needed to create a right triangle on a coordinate grid? Three ordered pairs are needed to create the right triangles.

DOK-1 How are the two perpendicular legs of a right triangle measured on a coordinate grid? We count the number of grid spaces between the two points that form each leg.

DOK-2 What types of numbers did you use when rolling the number cube to generate ordered pairs? Whole numbers and integers. Because the number cube has whole numbers 1–6, there was no way for us to generate an ordered pair with fractions or decimals.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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

Math Chat DOK-3 How is using the coordinate grid helpful when creating right triangles? The coordinate grid is a helpful guide when drawing straight lines, especially for the two legs of a right triangle. The grid lines on the coordinate plane can also help us determine the lengths of both legs. • DOK-3 How can you determine which sides would be the legs? I can determine the legs of a right triangle by ﬁnding the two sides that meet to form a right angle. • DOK-2 Right triangle XYZ is drawn on a coordinate grid. Point X is located at (5, 5), point Y is located at (10, 5), and point Z is located at (5, 6). What is the distance between points Y and Z? Distance from point X to point Y = 5 units Distance from Point X to point Z = 1 unit Distance from point Y to point Z = 52 + 12 = c2 25 + 1 = c2 2 26 ___= c √ 26 = c c ≈ 5.10 units • DOK-2 Which of your group’s two additions had the smaller hypotenuse? By how much smaller? Round your answer to the nearest hundredth. Addition 2 had the smaller hypotenuse by about 0.23. •

Post-Explore 1. 2. 3. 4.

PYTHAGOREAN THEOREM

Home

FACILITATION TIP Project this Math Chat question and use it as a quick check for understanding. Consider using a think, pair, share. Consider having students plot the shape on a coordinate plane. STEMscopes Tip Students take notes, express ideas, and/or process the information presented in class using the Interactive Notebook element, located in the Explain section of each scope. These cut-and-glue activities provide an interactive way for students to showcase the concepts and skills learned in the Explore activities and can be added to a notebook for future reference.

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PYTHAGOREAN THEOREM

Pythagorean Theorem Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Picture Vocabulary

Show What You Know, Part 1

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

Modeling the Pythagorean Theorem and the Converse of the Pythagorean Theorem Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

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

Finding an Unknown Side Length in a Right Triangle 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

The Pythagorean Theorem in Rectangular Prisms

Interactive Notebook

Show What You Know, Part 4

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

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

The Pythagorean Theorem on a Coordinate Grid Independent practice assignment that gives students an opportunity to demonstrate their learning

326

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

Pythagorean Theorem

PYTHAGOREAN THEOREM

Home

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

327

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

PYTHAGOREAN THEOREM

Pythagorean Theorem

3 328

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

PYTHAGOREAN THEOREM

Home

What does mastery look like?

I can explain a proof of the Pythagorean theorem and its converse using visual models.

I can use the Pythagorean theorem to determine unknown side lengths in right triangles within mathematical problems in two and three dimensions.

I can use the Pythagorean theorem to ﬁnd the distance between two points in a coordinate system.

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SCOPE 1

Volume Scope Introduction SCOPE SUMMARY In this scope, students will become familiar with the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. They will find the height and length of figures as needed in order to fill in the missing pieces of volume formulas. Students will understand when to use radicals versus decimals to represent the solution to the equations. Student Expectations

8.GSR.8.4 Apply the formulas for the volume of cones, cylinders, and spheres and use them to solve in relevant problems. 8.PAR.3.6 Use algebraic reasoning to ﬂuently manipulate linear and literal equations expressed in various forms to solve relevant, mathematical problems.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous years, students learned about the volume of three-dimensional figures. In 5th grade, students used unit cubes to find the volume of right rectangular prisms. They learned the relationship between length, height, and width of a prism with its volume. In 7th grade, students extended this knowledge to find the volume of triangular prisms and cylinders. Students also learned about the areas of figures. Knowing the area of a shape will help the students in remembering the formula for the volume of a figure based on its base.

Although students have just learned the formulas for the volumes of cylinders, cones, and spheres, they will not fully understand the explanations for these formulas until later in high school. As they continue work in volumes, they will begin to determine the volume of solids that are not traditional geometric ﬁgures. They will decompose these into several solids that they recognize in order to solve. These will progress as students use other algebraic tactics to determine percentages of these volumes being ﬁlled.

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

ﬁnd the volume of a right rectangular prism with fractional edge lengths.

•

apply the formulas V = lwh and V = bh to ﬁnd volumes of right rectangular prisms with fractional edge lengths.

•

examine a series of rectangular prisms.

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 cylinders, cones, and spheres.

•

use formulasto determine volume.

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

Cylinders In this exploration, groups of students will be tasked with solving a scenario that involves helping a canning company to determine the volume of cans (cylinders) with a variety of bases, sizes, and height. Students will: •

discover the formula for the volume of a cylinder.

•

solve problems for the radius, base, height, and volume of a cylinder.

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.

Cones In this exploration, students will be tasked with solving a real-world scenario about helping farmers by ﬁnding the volume of cones for ice cream to determine the amount of ice cream each cylinder container can serve. Students will: •

discover the formula for the volume of a cone.

•

solve problems to ﬁnd the volume of a cone.

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

Spheres In this exploration, groups of students will be helping the farmer determine the volume formula for different fruits for a juice bar based on what they know about the volume formula for a cone. Students will: •

learn the formula for the volume of a sphere.

•

solve problems to ﬁnd the volume of a sphere.

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

331

VOLUME

Volume Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

332

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

VOLUME

Home

ACCESSING PRIOR KNOWLEDGE Students will examine a series of cylinders and right prisms 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. 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.

Materials

Preparation

Printed •

1 Does Not Belong (per student or per group)

• •

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

Procedure and Facilitation Points 1. 2. 3.

Give one Does Not Belong to each student or group. Explain that each table on the handout contains four options. Three of the options go together, and one does not belong. Instruct students to determine which letter does not belong in each group and to explain their thinking. a. B does not belong. The answer is incorrect. The formula for volume is 1 length times width times height. The volume is the product of 5(__2), which 1 __ 1 __ left out a dimension. Volume should be the product of 5(2)(2) instead.

4. 5.

B does not belong. The formula is V = Bh, where the area of the base should be the base of the triangle times the height of the triangle divided by 2. For B, the base and height of the triangle were multiplied together and then multiplied by 2, instead of dividing by 2.

D does not belong because the diameter was used instead of the radius. The formula should be pi times radius squared times height or π • 82 • 9 = 1,809.56 cubic feet.

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

Identifying Misconceptions • • • •

Students may forget to use cubic units when labeling volume. Students may forget the formulas to use for each three-dimensional ﬁgure. Students may forget to use the radius when ﬁnding the volume of a cylinder and instead use the diameter. Students may think that each rectangular side of each triangular prism is equal in length and width when they are not all the same.

FACILITATION TIP Depending on your students’ recent experience with 3-D shapes, consider providing some hands-on time with geometric solids or display some under the document camera. FACILITATION TIP Review the attributes of rectangular prisms and cubes. Discuss what surface area is and what volume is – describe what the terms mean and how you calculate them. FACILITATION TIP In addition to reviewing the attributes of prisms and cubes, consider providing a quick drawing lesson to model how a few 3-D ﬁgures are represented in 2-D. FACILITATION TIP Have students calculate the correct solutions for the “does not belong” items. For example, have students calculate the correct volume for page 1 where B does not 1 belong (1 __4 cubic units). FACILITATION TIP Post the formulas for students before beginning this scope.

333

VOLUME

Volume Hook – 3-D Figures ACTIVITY PREPARATION Students will determine the volume of cylinders, cones, and spheres by using formulas.

Materials

Preparation

Printed •

• • •

1 3-D Figures (per class)

Reusable •

1 Phenomena Video (per class)

Plan to show the video. Prepare to project 3-D Figures 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 a sculpture?; 2) If you wanted to make a sculpture, what materials would you use?

4. 5.

DOK-1 How do we know that we are ﬁnding the volume? We are looking for how much clay is inside the ﬁgures.

DOK-1 What do all these ﬁgures have in common? They all have circular bases.

Complete the Explore activities. Notes

334

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

VOLUME

Home

Part II: Post-Explore 1. 2.

Show the Phenomena Video again, and restate the problem. Refer to 3-D Figures, and discuss the following questions: a.

DOK-1 How can you determine the volume of the cylinder? The volume of the cylinder can be found by using the formula V = Bh = πr2h = π(72)(3) = π(49)(3) = 147π ≈ 461.58 cm3.

DOK-1 How can you determine the volume of the cone? The volume of the cone can be found by using the formula 275 1 1 1 1 V = __3 Bh = _3_ πr2h = __3 π(52)(11) = _3_ π(25)(11) = ____ π ≈ 287.83 cm3. 3

DOK-1 How can you determine the volume of the sphere? The volume of the sphere can be found by using the formula 2,048 4 4 4 V = __3 πr3 = __3 π(83) = __3 π(512) = _____ π ≈ 2,143.57 cm3. 3

FACILITATION TIP Let half of the class solve the problem by converting to a decimal, but have the other half solve by converting to a fraction. Have them compare their answers. FACILITATION TIP Relate this formula to the formula for the volume of a rectangular prism. They are both found by calculating the area of the base times the height.

335

VOLUME

Volume Explore 1 – Cylinders ACTIVITY PREPARATION Students will discover the formula for the volume of a cylinder and solve mathematical and real-world problems for the radius, base, height, and volume of a cylinder.

Standards for Mathematical Practice • •

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 Can It Cards (per pair) 1 Cylinder Net (per pair) 1 Volume of a Cylinder Work Mat (per pair) 1 Exit Ticket (per student)

• •

Reusable • • • • • • • •

• • •

24 Linking cubes (per teacher) 160 Centimeter cubes (per pair) 1 Gallon-sized resealable bag (per pair) 1 Clear sheet protector (per pair) 1 Resealable bag (per pair) 1 Dry-erase marker (per pair) 1 Pair of scissors (per pair) 1 Glue stick (per pair)

•

Plan to divide the class into groups of 2 to complete the activity. Print a Student Journal and an Exit Ticket for each student. Prepare a model of a rectangular prism using linking cubes to demonstrate to the class with the following measurements: 3 units long, 2 units wide, and 4 units high. Place 160 centimeter cubes in a gallon-sized resealable bag for each pair. Print a Cylinder Net for each pair. Place it into a clear sheet protector or print it on card stock. Print a Volume of a Cylinder Work Mat for each pair. Place it into a clear sheet protector to create an erasable surface. If desired, print it on card stock. Print a set of Can It Cards for each pair. If desired, print the cards on card stock and laminate them for future use. Cut them out, and place each set in a resealable bag. Gather enough dry-erase markers, glue sticks, and pairs of scissors for each pair to have one of each.

PROCEDURE AND FACILITATION Part I: Understanding the Volume of a Cylinder Formula FACILITATION TIP

Use the Picture Vocabulary to show students how to accurately draw a 2-D model of a cylinder with labels. FACILITATION TIP Before reading the scenario, ask the class 1) What happens at a food processing plant?; 2) What shape are food cans?; 3) Why do you think canned goods come in all different sizes?

2. 3.

Read the following scenario to the class: One of your friends and classmates is named Madeline Dupont. Madeline’s family owns a food processing plant. One branch of the plant involves canning foods. Because a large number of employees are absent today, Madeline’s mom has asked if she has any friends who are good at math who could help with the canning process. Your job today is to show that you understand the volume of cylinders and can determine the volume of cans (cylinders) with a variety of bases, sizes, and heights. Give a Cylinder Net, a dry-erase marker, a glue stick, and a pair of scissors to each pair of students. Review students’ prior knowledge of ﬁnding the volume of a rectangular prism by asking the following questions: a.

336

DOK-1 How do you ﬁnd the volume of a rectangular prism? (Please note that you should show students the rectangular prism model that you prepared.) Student responses will vary. We ﬁnd the volume by multiplying the length times width times height. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 When you use the formula V = lwh,, what formula is being used when you multiply the length times the width? Explain why the formula is being used. (Note that you should point to the top base and the bottom base of the rectangular prism.) Student responses will vary. A rectangular prism is a 3-D ﬁgure, and to ﬁnd the volume, you can multiply the length times width times height. The base of a rectangular prism is a rectangle, so the formula length × width is used to determine the area of the base.

DOK-2 What is the formula that is used to ﬁnd the volume of a rectangular prism? V = Bh, where B = area of the base. So V = B × height, or length × width × height.

DOK-2 What is the volume of this rectangular prism? (Note that students should recognize that the dimensions of the rectangular prism are 3 units long, 2 units wide, and 4 units high.) The volume of the rectangular prism is 24 units cubed.

Explain to students that they will work with their partners to understand how to find the volume of a cylinder using what they know about finding the volume of a rectangular prism. Have students work with their partners to cut out the net of the cylinder and create the cylinder. (Note that students will need to leave one base open to fill the base with centimeter cubes.) Instruct students to arrange the centimeter cubes in a single layer at the bottom of the cylinder and fit as many cubes into the layer as possible. They will also need to find how many layers of cubes fit in the cylinder and make a stack of cubes along the inside of the cylinder. Students will use the Volume of a Cylinder Work Mat to calculate the area of the base and the volume of the cylinder. (Note that you should show the students how to use the base to determine the radius and how to determine the height of the cylinder using the formula.) Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 What shapes are in a cylinder? 1 rectangle and 2 circles

DOK-1 What is the shape of the bases on a rectangular prism and cylinder? (Note that you should refer to the rectangular prism that was created using the linking cubes and the cylinder that was created using the Cylinder Net.) The bases on rectangular prisms are rectangles, while the bases on a cylinder are circles.

DOK-2 How might finding the volume of a cylinder be similar to finding the volume of a rectangular prism? I can find the area of the base and then find the product of the base and height.

DOK-2 How might ﬁnding the volume of a cylinder be different from ﬁnding the volume of a rectangular prism? The base of a rectangular prism is a rectangle, so I will use the formula for area of a rectangle (l × w), and the base of a cylinder is a circle, so I will use the formula for area of a circle (πr2).

Intervention

Acceleration

VOLUME

Home

e. DOK-2 How many layers of cubes did it take to ﬁll the cylinder? 12 f.

DOK-2 What is the area of the cylinder’s base? Express your answer in terms of π. 4π

DOK-2 What is the volume of the cylinder? Express your answer in terms of π. 48π

h. DOK-2 How can you determine the approximate number of cubes that will ﬁt in the cylinder? Why is this an approximate number of cubes? You can place cubes in the cylinder until the cubes are ﬁlled to the top. This represents an approximation because there are empty spaces around the cubes when they are inside the cylinder.

337

VOLUME

Volume Explore 1 – Cylinders

FACILITATION TIP Tell students to use 3.14 for pi. Remind them that pi is an irrational number that continues on and on, but we just use 3 digits for a more simple calculation.

i. DOK-2 How can you calculate the volume of the cylinder? You can look at the base of the cylinder to get the radius and then use the area of a circle to ﬁnd the area of the base: 12.56 cm2. Use the area of the base and multiply times the height 12 to get a volume of 150.72 cm3. 7.

Math Chat DOK-2 What formula is used to find the volume of a cylinder? V = Bh B represents the area of the base, or πr2. • DOK-2 What process is used to find the volume of a cylinder? First find the area of a circular base. Then, multiply it by the height. You can also find the area of the base and express the answer in terms of π. Then, multiply the area of the base times the height to get the volume expressed in terms of π, and then multiply by π to get the volume of the cylinder. • DOK-1 How would you figure out the radius of a cylinder if you were given the diameter and height? We would divide the diameter in half because the radius is half of the diameter. •

FACILITATION TIP

Part II: The Cannery

Before reading the scenario, ask the class 1) What information do you need to determine the volume of cans?; 2) What types of foods have you seen in cans that have a large volume?; 3) What types of foods have you seen in cans that have a little volume?

FACILITATION TIP Post the text of this scenario and have student volunteers read it aloud to the class.

2. 3. 4.

FACILITATION TIP Remind students that the radius is half of the diameter. The radius is used to ﬁnd the area of the base, and thus the volume.

FACILITATION TIP Show students how to create a quick sketch of a cylinder if needed.

FACILITATION TIP If students ﬁnish early, have them order the cans from smallest to largest volume. Let the students evaluate whether or not the cans are also in order of smallest to largest height. Have them discuss what other factor contributes to the volume besides the height.

338

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

6. 7. 8.

Read the following scenario to the class: Madeline’s family was very pleased with how well you could determine the volumes of cylinders. At their plant, all types of fruits, vegetables, soups, and sauces are canned. Today, a large number of employees are away at a training session, so Madeline’s family needs help with the canning. Your job today is to ﬁgure out the volume of various cans and report the volumes to the supervisor so she can determine how many cans are needed for each product being canned. Give a Student Journal to each student. Give a set of Can It Cards to each pair of students. Explain to students that they will work with their partners to determine the radius, height, area of the base, or volume of cylinders by using information about the cans that are being used at the cannery. Have students draw a model and label the dimensions for each can. They will also calculate the missing dimension of each cylinder. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 How do you know which base to use to ﬁnd the volume of the cylinder? Student responses will vary. We could use either base because in a cylinder, the bases are congruent.

DOK-2 How did you determine the area of the base of a cylinder? Since the base is a circle, we found the area of the circle by using the formula πr2.

DOK-2 If you already know the volume of a cylinder but are missing either the radius or the height, how would you find the missing information? Since we know the formula for volume is V = Bh, we fill in the information that we know and solve to find the missing information.

Give students enough time to read each Can It Card, record the given information, determine the missing information, and draw a model on their Student Journals. Allow time for students 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.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

VOLUME

Home

Math Chat DOK-2 Why are the units of volume in cubic units? The area of the base is in square units, such as cm2. Then, it is multiplied by height, which is in units. Square units × units = cubic units. cm2 × cm = cm3. • DOK-2 If a cylinder were on its side, how would you know which measurements to use for the dimensions to find the volume of the cylinder? When using the formula V = Bh, we would need the radius for the measurement r to find B (area of the base) first. The height always represents the distance between the bases. • DOK-2 Compare the following two cylinders, and determine which cylinder has the greater volume: Cylinder 1: diameter of 20 cm, height of 5 cm Cylinder 2: radius of 5 cm, height of 10 cm Cylinder 1 volume: 1,570 cm3 Cylinder 2 volume: 785 cm3 Cylinder 1 has the greater volume. • DOK-2 In what other situations in the real world would you need to be able to find the volume of a cylinder? How much oil will fit in an oil barrel? How much rain will a rain barrel hold? How much soup will fit in a slow cooker? How much water fits in a water bottle for hydration purposes? How much paint fits in a paint can to make sure there is enough to paint the walls in a space? •

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.

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 For this Exit Ticket, determine ahead of time if students will be allowed access to volume formulas.

339

VOLUME

Volume Explore 2 – Cones ACTIVITY PREPARATION Students will discover the formula for the volume of a cone and solve mathematical and real-world problems to ﬁnd the volume of a cone.

Standards for Mathematical Practice • •

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 Cone It Cards (per group) 1 Set of Cylinder and Cone Nets (per group) 1 Exit Ticket (per student)

Reusable • • •

• • • •

•

1 Resealable bag (per group) 1 Pair of scissors (per teacher) 1 Glue stick (per group)

Plan to divide the class into groups of 3 or 4 to complete the activity. Print a Student Journal and an Exit Ticket for each student. Print one set of Cylinder and Cone Nets for each group of students. If desired, print them on card stock. Print a set of Cone It Cards for each group of students. If desired, print them on card stock, and laminate them for future use. Cut out the cards, and put them in a resealable bag for each group. Gather enough bags of rice, pairs of scissors, and glue sticks for each group to have one of each.

Consumable •

1 Bag of rice (per group)

PROCEDURE AND FACILITATION FACILITATION TIP

Part I: Understanding the Volume of a Cone Formula

Use Picture Vocabulary to help students sketch a 2-D model of a cone and a cylinder.

FACILITATION TIP Before reading the scenario, ask the class 1) What shape is an ice cream cone?; 2) When you were ﬁlling an ice cream cone with ice cream, how can you prevent the ice cream from dripping? FACILITATION TIP If you don’t have rice, you could also use sand or something else that has small grain sizes.

2. 3.

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Read the following scenario to the class: Madeline’s family added another fun component to their farm. The Dupont family purchased dairy cows and started making ice cream. They built an ice cream parlor on their farm. They package the ice cream in cylinder containers. They serve it only in cones and fill the cones only until they are full to the level of the opening because this prevents dripping and messes. The Duponts would like to know how to find the volume of the cones so they know how many cones can be served from each cylindrical container. Your job today is to discover a formula for the volume of a cone. Give one set of Cylinder and Cone Nets, a pair of scissors, a glue stick, and a bag of rice to each group. Explain to students that they will work in their groups to discover the formula for volume of a cone by using what they know about the formula for the volume of a cylinder. (Note that they will look at models of a cone and a cylinder with congruent bases and identical heights.) Have students cut out the Cylinder and Cone Nets to create the cylinder and cone. (Make sure students understand that the base of the cone and one of the bases of the cylinder should be open so that rice can be poured in both figures.) Once the cylinder and cone have been created, have students fill the cone with rice to the top with a level surface and pour it into the cylinder. They will repeat this process and see how many times it takes until the cylinder is full right to the top with a level surface. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

a. DOK-1 How are a cone and a cylinder similar? Student responses will vary. They are both 3-D ﬁgures and both have a circular base and are curved. b.

DOK-1 How are a cone and a cylinder different? Student responses will vary. A cone has only one base and an apex, while a cylinder has two bases.

DOK-1 What do you notice about the bases of the cylinder and the cone? Student responses will vary. They are congruent, and they are circles.

DOK-1 What do you notice about the heights of the cone and the cylinder? Student responses will vary. They are equal.

e. DOK-1 Did the cylinder and the cone hold equal volumes of rice? If not, which held more? The cylinder and the cone do not hold equal volumes of rice. The cylinder holds more rice than the cone. f. 5.

DOK-1 How many times were you able to ﬁll the cylinder with rice from the cone? Three times

Intervention

Acceleration

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FACILITATION TIP Create a T-chart to list responses to questions 4a and 4b. 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.

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

Math Chat •

DOK-1 What is the formula for the volume of a cylinder? Volume = Bh or V = πr2h

DOK-2 What is the volume of a cone compared to the volume of a cylinder with a 1 congruent base and an equal height? A cone has __3 the volume of a cylinder with a congruent base and an equal height. 1 • DOK-2 What is the formula for the volume of a cone? Volume = _3_ πr2h

•

Part II: Solving Volume of a Cone Problems 1.

2. 3. 4.

Read the following scenario to the class: The Dupont family is adding a big menu board behind the counter at their ice cream parlor on the farm. Each menu item will be described. It is half advertisement and half information. The menu is almost ready to be completed and hung, but the family is missing one piece of information about the size of each cone. Your job is to use the formula for the volume of a cone to determine the missing information regarding the different sizes of cones the Dupont family sells. The missing information might be the radius or diameter of the opening, the area of the opening, the height of the cone, or the volume of ice cream the cone holds when filled to exactly the level of the opening. This information will allow customers to select the best-size cone for their situations, creating more satisfied customers. Give a Student Journal to each student. Give a bag containing a set of Cone It Cards to each group. Explain to students that they will work in their groups to find the volume of a cone. Instruct students to use the information on the Cone It Cards and find the volume of each cone. Have students draw a model of the cone and label its dimensions and then find the area of the base, express the area in terms of π, and find the volume of the cone. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a.

DOK-1 What happens to the volume of a cone as its height increases? Student responses will vary. Volume increases.

DOK-1 What happens to the volume of a cone when the radius increases? Volume increases.

DOK-1 How do you ﬁnd the volume of a cone? We use the volume 1 formula for a cone (_3_ πr2h) and plug in the radius and height.

DOK-2 How do you ﬁnd the height or the radius of a cone? We use the volume formula but solve for radius or height.

FACILITATION TIP Post these Math Chat questions, have students note the appropriate formulas in their Student Journals or in notebooks as you lead the discussion. FACILITATION TIP Before reading the scenario, ask the class 1) What types of information might a menu board in a ice cream parlor show?; 2) What size ice cream cone would you prefer? Why? After reading the scenario, ask the class 3) What is the formula for ﬁnding the volume of a cone? FACILITATION TIP Project this scenario in print so students can read it along with you.

FACILITATION TIP Students should be using the volume formula, which was determined in Part I. Answers should be left in terms of pi, not calculated with 3.14. 341

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Volume Explore 2 – Cones 6. FACILITATION TIP If students ﬁnish early, give them some practice problems where they are given the volume and either the radius or height. They must ﬁnd the missing dimension.

Math Chat •

• 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.

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

•

DOK-2 Which affects the volume of a cone more—increasing the radius or increasing the height? Why? Increasing the radius affects the volume of the cone more than increasing the height because the radius is squared to ﬁnd the volume. DOK-2 What did you notice about the relationship between the height and the volume of a cone? The greater the height, the greater the volume of the cone. DOK-2 Do you think the relationship between the volume of a cylinder and the volume of a cone would work if the radii and heights were not the same? Explain. 1

•

No, in order for the volume of a cone to represent _3_ of the volume of a cylinder, the radius and the height must be the same. DOK-2 What other real-world situations would require people to know the volume, height, or radius of cones? What should be the height of traﬃc cones so people can see them and not run over them? What should be the radius/diameter of a conifer Christmas tree to know whether it will ﬁt in a living room? What should be the volume of cone-shaped paper water cups to make sure we serve enough water to athletes playing in the heat?

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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Volume Explore 3 – Spheres ACTIVITY PREPARATION Students will learn the formula for the volume of a sphere and solve mathematical and real-world problems to ﬁnd the volume of a sphere.

Standards for Mathematical Practice • •

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 Juice It Cards (per group) 1 Exit Ticket (per student)

•

Reusable • • •

1 Resealable bag (per group) 1 Dry-erase marker (per group) 3 Fillable geometric volume shapes (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. Gather 3 ﬁllable geometric volume shapes (cone, sphere, and cylinder) for each group of students. Print a set of Juice It Cards for each group of students. Cut out the cards, and put them in a resealable bag for each group. If desired, print the cards on card stock, and laminate them for future use. Gather enough bags of rice and dry-erase markers for each group to have one of each.

Cone Sphere Cylinder

Consumable •

1 Bag of rice (per group)

PROCEDURE AND FACILITATION FACILITATION TIP

Part I: Understanding the Volume of a Sphere Formula

Use Picture Vocabulary and show students how to draw a sketch of a sphere.

FACILITATION TIP Before reading the scenario, ask the class 1) What is your favorite type of juice?; 2) What ingredients are used to make juice?; 3) What types of fruit are shaped as spheres? FACILITATION TIP Students may also use sand, beans, or water to ﬁll their 3-D shapes. Anything that is small and granular or liquid would work.

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2. 3.

Read the following scenario to the class: Madeline’s family was so successful when they added their ice cream parlor that her parents decided to trust Madeline’s newest idea and add a juice bar to their list of ventures. Madeline wants to call it the Sphere Juicery and only use fruits grown on the Dupont farm that grow into perfect spheres. To make juice, Madeline wants to use the whole inside of the fruit—the pulp, juice, and meat of the fruit. This way it will be healthier than just straight juice. In order to know how many fruits to grow and use, Madeline wants to learn the average volume of each type of fruit. Your job today is ﬁrst to discover the volume formula for a sphere based on what you know about the volume formula for a cone. Give a dry-erase marker, cone, sphere, cylinder, and bag of rice to each group of students. Explain to students that they will ﬁrst work in their groups to discover the formula for the volume of a sphere by using what they know about the formula for the volume of a cone and the volume of a cylinder. They will look at models of a cone, cylinder, and sphere with equivalent radii/diameters and heights. © Accelerate Learning Inc. - All Rights Reserved

Engage

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Explain

Elaborate

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Intervention

Acceleration

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Have students ﬁll a cone with rice and then pour the rice into the sphere. They will repeat this process and see how many times it takes until the sphere is full to the top. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a. DOK-1 How many times did it take for the cone to ﬁll the sphere? 2 times b. c. d.

DOK-1 What is the formula for the volume of a cone? __3 π r2h

DOK-1 What is the height of a sphere? The height of a sphere is represented by 2r (twice the radius, or the diameter). DOK-1 What aspects of a cone and a sphere are similar? Student responses will vary. Both ﬁgures are 3-D solids. Both have a radius and a height. The height of a sphere is 2r (twice the radius). Both are curved ﬁgures.

e. DOK-1 What aspects of a cone and a sphere are different? Student responses will vary. A sphere has no faces, no vertices/apexes, and no edges, while a cone has one face and one apex. 5.

Have students use dry-erase markers to divide the cylinder into thirds. (Note that students should label the cylinder in thirds.) Fill the sphere with rice, and then pour the rice into the cylinder. Monitor and assess student understanding as each group collaborates by asking the following guiding questions: a. b. c.

DOK-1 How much rice was poured into the cylinder? __3

DOK-1 What is the formula for the volume of a cylinder? Bh or πr2h DOK-1 What aspects of a cylinder and a sphere are similar? Student responses will vary. Both ﬁgures are 3-D solids. Both are curved ﬁgures.

d. DOK-1 What aspects of a cylinder and a sphere are different? Student responses will vary. A sphere has no bases and a cylinder has two bases. 6.

FACILITATION TIP Watch out for students who confuse the formulas for the volume of a sphere and the volume of a cone. They are similar, but 1 the fractions are different: __3 in the cone 4 __ equation and 3 in the sphere equation, and the exponents for the radius are different. This is because a sphere has no height, so the radius is factored in 3 times.

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

Math Chat •

DOK-2 How did you discover the formula for a sphere? We discovered that a sphere with the same radius and height as a cone holds twice the volume of the 1 cone. We used the volume of a cone formula, V = __3 πr2h. Since the height of a sphere is equivalent to its diameter, which is equivalent to two radii, we replaced the h with 2r and simpliﬁed the formula. 2

We also discovered that a sphere ﬁlls __3 of a cylinder. The cylinder’s height is equal to twice the radius of the sphere. •

DOK-1 What is the formula for the volume of a sphere? __3 πr3

•

DOK-1 If the radius of a sphere is given, how do you determine the diameter of the sphere? The radius of the sphere is half of the diameter on a sphere.

•

DOK-2 If a sphere and a cylinder have the same radius, which ﬁgure would have the greater volume? It would depend on the height of the cylinder.

Part II: Solving Volume of a Sphere Problems 1.

2. 3.

Read the following scenario to the class: Madeline’s family has ﬁnalized the menu to show the fruit that will be used for different types of juices made at the juice bar. Your next job is to use the volume formula for a sphere to ﬁnd the volume of given pieces of spherical fruit grown on the Dupont farm that will be used to make juice. Give a Student Journal to each student. Give a bag containing a set of Juice It Cards to each group.

FACILITATION TIP After reading the scenario, ask the class 1) What information is needed to ﬁnd the volume of a sphere?

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Volume Explore 3 – Spheres 4. 5. FACILITATION TIP Students should use 3.14 as the value for pi. They should round their answers to the nearest hundredth.

Explain to students that they will work in their groups to determine the volume for each sphere-shaped piece of fruit by using the volume formula for a sphere. Have students use the Juice It Cards to draw a model and label its dimensions. They will then ﬁnd the volume of each sphere. Monitor and assess student understanding as each group collaborates by asking the following guiding questions:

6. 7.

DOK-1 If the diameter of a sphere is given, how do you determine the radius of the sphere? Divide the diameter by 2.

DOK-2 How can you ﬁnd the volume of a sphere? Substitute the dimensions of each sphere into the formula. Use 3.14 as the value for π. Simplify the formula.

DOK-2 If you already know the volume of a sphere, how can you find 4 the radius? Since we know the formula for Volume is V = __3 πr3, we fill in 4 the volume in place of the V. Then, we can multiply __3 by π to get 4.19. Finally, we divide both sides by 4.19 and find the cube root of each side to find the radius.

Allow time for students to complete Part II on 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 If a sphere and a cone have the same radius and height, which figure has the greater volume? The sphere will have a volume that is twice the volume of the cone. • DOK-2 What is the relationship between the volume of a sphere and the volume of a cylinder? The sphere takes up two-thirds of the volume of the cylinder. • DOK-2 If a sphere and a cylinder have the same radius, which figure would have the greater volume? It would depend on the height of the cylinder. • DOK-2 In what other real-world scenarios would people need to know the volume of a sphere? How much air will fit in a beach ball? How much water do I use in a spherical water balloon? How much clay will an artist need to use to create a sculpture in the shape of a sphere? How much gas is contained in a star or a gasgiant planet? How much liquid can be injected into a hollow chocolate ball by a chocolatier? •

Post-Explore FACILITATION TIP When you preview this Exit Ticket with students, clarify how many steps they need to show in the work space.

1. 2. 3. 4.

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Evaluate

Intervention

Acceleration

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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

Cylinders 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

Cones 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

Spheres 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

Evaluate

Intervention

Acceleration

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ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

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

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

Fluency Builder Volume – Cylinders 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

3 350

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

VOLUME

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What prompts will be used?

What does mastery look like?

I can apply the formulas for volume of cones and use them to solve relevant problems.

I can apply the formulas for volume of cylinders and use them to solve relevant problems.

I can apply the formulas for volume of spheres and use them to solve relevant problems.

I can solve for an unknown dimension of a ﬁgure when given the volume.

I can use a variety of methods and strategies to solve problems eﬃciently and accurately.

I can rearrange formulas and use the same reasoning when solving equations to solve mathematical problems.

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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

8 Georgia Math Teacher Guide

Grade 8 Teacher Guide

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

ISBN: 979-8-88826-667-0

8 GEORGIA

MATH G8