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Georgia Math Teacher Guide

Algebra 1 Teacher Guide

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

ISBN: 979-8-88826-668-7

ALGEBRA 1

GEORGIA MATH A1

GEORGIA

Teacher Guide: Algebra 1 ISBN: 979-8-88826-668-7 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

Properties of Functions....................................................................................... 18

SCOPE 2

Linear Functions ................................................................................................. 44

SCOPE 3

Geometry on the Coordinate Plane ...................................................................... 68

SCOPE 4

Linear Inequalities .............................................................................................. 94

SCOPE 5

Systems of Inequalities..................................................................................... 116

SCOPE 6

Simplify Radicals .............................................................................................. 134

SCOPE 7

Polynomial Operations ...................................................................................... 156

SCOPE 8

Graphs of Quadratic Functions .......................................................................... 184

SCOPE 9

Factors of Polynomials ..................................................................................... 214

TABLE OF CONTENTS

Table of Contents

SCOPE 10 Solve Quadratics .............................................................................................. 244 SCOPE 11 Transform Quadratic Functions ......................................................................... 274 SCOPE 12 Exponential Functions ....................................................................................... 296 SCOPE 13 Exponential Extensions ..................................................................................... 322 SCOPE 14 Compare Function Types................................................................................... 348 SCOPE 15 Statistics.......................................................................................................... 370 SCOPE 16 Model Data ....................................................................................................... 392

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

Properties of Functions Scope Introduction SCOPE SUMMARY

Student Expectations

In this grade level, students will expand their understanding of functions as input and output to sets that represent the domain and range. Students will be able to create examples of what is and what is not a function using a variety of representations, such as a table, a graph, symbols, or a verbal description. They will be able to determine the domain and range given a graph, table, or algebraic representation. Given the context, students will interpret the meaning of statements in function notation. Students will also examine various parent functions and determine features of linear vs. nonlinear functions.

A.FGR.2.4 Use function notation to build and evaluate linear functions for inputs in their domains and interpret statements that use function notation in terms of a mathematical framework. A.FGR.2.5 Analyze the difference between linear functions and nonlinear functions by informally analyzing the graphs of various parent functions (linear, quadratic, exponential, absolute value, square root, and cube root parent curves).

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grade levels, students used functional relationships to create linear models and determine rate of change. Students have recognized and explained a function as each input value gives back exactly one output value, with ordered pairs representing the inputs and outputs.

Students will continue exploring domain and range with quadratic and exponential functions later in Algebra I, as well as building those function types using function notation. Later in this course and in Algebra II, students will learn more about the features of and transform the parent functions that are introduced in this scope.

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

understand a function as a rule that assigns to each input and output.

•

understand a graph of a function is the set of ordered pairs consisting of an input and output.

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: •

see math in scenarios.

•

relate function notation to a realworld 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.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Relations and Functions In this exploration, groups of students will solve a real-world problem as an employee of a widget factory and are asked to analyze inputs and outputs of each widget machine to determine which two are functioning machines and which machine is not functioning properly. Students will: •

analyze properties of functions and non-functions through various types of algebraic representations.

•

develop the deﬁnition of a function.

•

decipher between continuous and discrete functions.

Explore 2

Explore 1

EXPLORE ACTIVITIES

•

determine the domains and ranges of functions.

•

write the domain and range using appropriate formats.

•

determine domain and range as they relate to a problem.

Explore 4

Explore 3

In this exploration, groups of students will solve a scenario about track ticket sales for 7 days prior to a student event to create a report about the anticipated attendance at the event, provide enough food and prizes, and track attendance throughout the 4-hour event. Students will:

In this exploration, students will be tasked with analyzing data about tracking the distance traveled by a nest of leatherback sea turtles over time; and, interpret the data in a graph to complete a report on leatherback sea turtle migration. Students will: •

evaluate linear and quadratic functions in function notation and interpret function notation within various context.

•

interpret data on a graph.

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. Domain and Range

Evaluating Functions

PROPERTIES OF FUNCTIONS

Home

Linear vs. Nonlinear Functions In this exploration, students will compare linear and nonlinear functions using visual analysis of characteristics such as end behavior, increasing and decreasing, domain and range, intercepts, and general curvature. Students will: •

analyze graphs pertaining to various operations at a zoo.

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.

PROPERTIES OF FUNCTIONS

Properties of Functions Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Materials

Preparation

Printed •

• •

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

PROPERTIES OF FUNCTIONS

Home

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

Procedure and Facilitation Points 1. 2.

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

Card 1 matches with card B.

Card 2 matches with card A.

Card 3 matches with card D.

Card 4 matches with card C.

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

Identifying Misconceptions •

Students may struggle to remember how to identify functions. It is beneﬁcial to allow students to investigate the patterns they see when looking at tables, graphs, and equations together.

FACILITATION TIP As students walk around the room, ask the following questions: How do you know whether or not a graph or set of coordinates represents a function? Does Card C represent x-coordinates, y-coordinates, or both? What about Card D?

FACILITATION TIP As they progress through the scope, ask students if a horizontal line can pass through the graph of a function at two points. When students agree on the correct answer, ask them if it is therefore possible to have more than one x-coordinate in a function with the same y-coordinate.

PROPERTIES OF FUNCTIONS

Properties of Functions Hook – Phone Charging ACTIVITY PREPARATION Students will relate function notation to a real-world situation.

Materials

Preparation

Printed •

Reusable • •

Plan to show the video. Prepare to project Phone Charging 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.

• • •

1 Phone Charging (per class)

1 Phenomena Video (per class) 1 Projector (per class)

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

Before showing the video or reading the scenario, ask the class 1) Has anyone had their cell phone run out of power?; 2) How long do you think you used it before it died?; 3) How often do you charge your cell phone?

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: you Mark is charging his phone after its battery died. He is thinking about how to model the amount of charge the phone has as time passes. 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 think the amount of charge increases at a constant rate. I think the charge will plateau as it nears 100%. I think it will take two hours to fully charge. I wonder how long before the phone has 80% charge. I wonder how long it will take for the phone to turn on. Project Properties of Functions. Notes

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

Explain to students that Mark tracked the charge of his phone and wrote a couple of statements to describe what was occurring. Discuss the following questions: a.

DOK-1 Why would the ﬁrst math sentence have an inequality symbol? Allow students to share all ideas. Answers will vary. The amount of charge during one interval is less than another amount of charge.

DOK-1 What does the ff(100) (100) – ff(90) (90) part of this math sentence represent? It represents the charge after 100 minutes minus the charge after 90 minutes.

Share with students that the following guiding question will be used throughout the topic: How do we develop a mathematical shorthand? Complete the Explore activities.

Part II: Post-Explore 1. 2.

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

DOK-1 Do these statements with function notation make more sense after the Explore activities? activities? Yes, I can interpret this notation in the context of the charging phone.

DOK-2 What strategies would you use to explain why these statements make sense? I would write each mathematical statement as a full sentence and ﬁgure out how it makes sense given what I know about phone charging.

DOK-2 What does it mean for ff(100) (100) – ff(90) (90) < ff(30) (30) – f(20)? f The phone increases its amount of charge by less between minutes 90 and 100 than it does between minutes 20 and 30.

DOK-2 What does it mean for ff(150) (150) – ff(115) = 0? The phone probably already reached full charge after 115 minutes.

Intervention

Acceleration

FACILITATION TIP When showing the slide, be sure that students keep track of signs in Statement 1. For the statement, be sure they understand that neither of the values on the left side are less than either of the values on the right side. Rather, the difference of the values on the left side is less than the difference of the values on the right side. FACILITATION TIP

PROPERTIES OF FUNCTIONS

Home

After students answer, ask them which interval in the inequality has a greater rate of change. Then, ask them if the intervals have to be the same length to ﬁnd out which one has a greater rate of change. Have them justify their answers, and encourage them to consider graphs as they do so.

FACILITATION TIP Ask students what the graph of the charge between t = 115 and t = 150 would look like. Answers may vary. Have students assume the phone stays plugged into the charger and the charger is working after t = 150. Ask them if the graph of the function would have the same slope from t = 170 to t = 180 as it does from t = 130 to t = 140.

PROPERTIES OF FUNCTIONS

Properties of Functions Explore 1 – Relations and Functions ACTIVITY PREPARATION Students will analyze the properties of functions and non-functions through various types of algebraic representations in order to develop the deﬁnition of a function. Students will also be able to decipher between continuous and discrete functions.

Standards for Mathematical Practice • • •

MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation • • • •

Printed • • •

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

Reusable • •

•

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

Separate the class into groups of 2 or 3 students. Print a Student Journal for each student. Print a set of Widget Cards for each student. Print an Exit Ticket document for every 2 students. Cut apart the half-page Exit Tickets so each student has one. Gather scissors and glue sticks for each student.

PROCEDURE AND FACILITATION POINTS Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone worked at a paying job?; 2) If so, what were your responsibilities?; 3) What hours did you work?

FACILITATION TIP Give the class six examples of ordered pairs. For each ordered pair, ask the class either which number is the input or which number is the output. Alternatively, take a number from the ordered pair and ask if it is the input or output. Have students answer for each example one at a time.

2. 3.

6. 24

Read the following scenario to the class: Congratulations! You have just been hired part-time at the new widget factory in your hometown. It’s your ﬁrst day on the job. The night-shift employee left you a note that one of the three widget machines you are in charge of is malfunctioning and not producing widgets according to factory standards, but they didn’t tell you which one! Analyze the inputs and outputs of each machine to identify the malfunctioning widget machine so you can shut it down for repairs. Give a Student Journal to each student. Explain to students that they will work with their groups to analyze the inputs and outputs of each widget machine to determine which two are functioning machines and which machine is not functioning properly. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What is meant by the term input? It means “the set of values supplied to a function.” It also represents the domain of a function.

DOK-1 What is meant by the term output? It means “the resulting set of values based on the input of the function.” It also represents the range of a function.

DOK-2 What patterns can you identify in the machines that are working correctly? Every input is matched with only one output.

Avoid conﬁrming groups’ choices for functioning and nonfunctioning machines. Allow groups to create their own decisions and justiﬁcations based on their observations. It is OK for students to misdiagnose the machines at this point. The misconception will be cleared up during the Math Chat. Allow students enough time to complete Part I and answer the questions that follow. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 How did your group determine the machine that was not functioning properly? roperly? We noticed that machine B was the only machine that had the same input (3 blobs) resulting in different outputs (1 widget and 5 widgets). It didn’t make sense that you would put the same input into the machine but the results would be different. • DOK-2 What conditions must be met for a machine to be considered functioning? The machine is functioning if each input value gives back exactly one output value. •

4. 5.

Read the following scenario to the class: Now that machine B has been shut down, your supervisor suspects other widget machines in the factory might also be malfunctioning. She has asked you to analyze the rest of the widget machines to determine if any more need to be shut down for not working properly. Make sure to justify your reasoning for why each machine is or is not functioning to report back to your supervisor. Give a set of Widget Cards, pair of scissors, and glue stick to each student. Explain to students that they will work with their groups to analyze the information given about each machine to decide whether the machines are functioning properly or not functioning properly. They will also justify their choice on each card. Encourage students to analyze, sort, and collaborate with their groups prior to pasting the cards into their corresponding columns. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 On the card labeled “Machine H,” what do the two arrows coming off of the 1 in the Domain column represent? The two arrows represent that the value 1 is listed twice.

DOK-1 On the card labeled “Machine D,” how many output values are displayed when x = 2? There are two output values when x = 2. The values are 1 and −1.

DOK-1 On the cards labeled “Machine F” and “Machine G,” where do you see the input and output values? Input values are represented by the x-coordinate, and output values are represented by the y-coordinate.

d. 6.

FACILITATION TIP After the class answers the question correctly, ask them if Machine A or Machine C is functioning in a linear fashion. Ask them to justify their answer for each machine. If no one mentions constant rate of change in their answer, review rate of change and how it is constant for linear functions with the class. FACILITATION TIP

Part II

2. 3.

Acceleration

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

Math Chat

Intervention

PROPERTIES OF FUNCTIONS

Home

DOK-1 What is the rule we created for identifying functioning machines? Each input must be matched to exactly one output.

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

Math Chat DOK-1 How were you able to determine the inputs and outputs for machines D and E? I looked at the ordered pairs of the coordinates on each graph. The x-coordinate represents the input, and the y-coordinate represents the output. • DOK-1 Find the Widget Card for machine D. What are two points on the graph that tell you this is not a functioning widget machine? Collect all answers. (−1, 0) and (−1, −1) (0, 1) and (0, −1) (1, 1) and (1, −1) (2, 1) and (2, −1) •

After the class answers the question correctly, refresh their memory by asking them if Machine B would be functioning if each input value switched places with its corresponding output value. Ask the same for Machine A and Machine C. FACILITATION TIP Before reading the scenario, ask the class 1) If you had a paying job, who did you report to?; 2) What problems did you encounter when working?; 3) Who did you talk to if you had a problem or question?

FACILITATION TIP After students answer the question, ask, “On the card labeled ‘Machine H,’ what do the two arrows going toward the 5 in the Range column represent?” Then, ask the class “Which column can have a value associated with two arrows and still represent coordinates of a function?” FACILITATION TIP When students establish that Machine D is not functioning, ask them if any one of the line segments would represent a function by themself. Then, ask them if any two segments would together. Finally, ask them if any three segments would together. FACILITATION TIP After they answer the questions that follow Part II, ask students how they would verbally describe the domain and range columns for Machine I. Then, ask them to give ordered pairs that would ﬁt the verbal description of Machine J. Finally, ask them how they would graph the verbal description of Machine K.

PROPERTIES OF FUNCTIONS

Properties of Functions Explore 1 – Relations and Functions DOK-2 What do you notice about the location of each pair of coordinates? What does this tell you about the graphs of functions? Each pair of coordinates is located on the same vertical line. If a graph contains two points on the same vertical line, the data cannot be a function. • DOK-1 What strategies did you use to determine which machines were functioning? For each machine, I looked at the inputs. If none of the inputs were repeating, then the machine was functioning properly. If the inputs repeated, then I checked the outputs. Each input had to be paired with exactly one output. •

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: When a set of data has each input matched with exactly one output, mathematicians call it a function. Post-Explore 1. 2. 3.

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

Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

PROPERTIES OF FUNCTIONS

Home

__________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________

PROPERTIES OF FUNCTIONS

Properties of Functions Explore 2 – Evaluating Functions ACTIVITY PREPARATION Students will evaluate linear and quadratic functions in function notation and interpret function notation within various contexts.

Standards for Mathematical Practice • • •

MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • •

• • •

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

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

Before reading the scenario, ask the class 1) What is your favorite zoo animal?; 2) What information do you know about your favorite animal?; 3) How can you ﬁnd out more information about your favorite animal’s habits? 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.

2. 3.

Read the following scenario to the class: You have recently obtained a position as an assistant zoologist at your local zoo. The head zoologist, Dr. Angola, has assigned you to your ﬁrst research project, in which you are to ﬁnd out more information about the migration behavior of leatherback sea turtles. Dr. Angola’s team has collected data to track the distance traveled by one nest of leatherback sea turtles over time. The data is displayed on a graph, but you need to interpret the graph and use the information to complete the report on leatherback sea turtle migration. Give a Student Journal to each student. Explain to students that they will work with their groups to analyze the information regarding turtle migration on the graph. Students will devise a strategy for completing the missing pieces of data in the report on leatherback sea turtle migration and answer the questions that follow in Part I 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, ask them what they can conclude about the slope of the graph from f(5) to f(10). If students do not mention constant slope in their answer, review rate of change and tie it to slope. Then, review how slope is constant for linear functions with the class.

5. 6.

DOK-1 How can you use ff(5) (5) = 50 to help you ﬁll in the blank for ff(___) = 80? In row A, the 5 represents the days it took to travel 50 miles. So in row B, 8 must go in the blank because it took 8 days to travel 80 miles. The number in the parentheses must represent the days.

DOK-1 What does the coordinate (8, 80) represent on the graph? The coordinate (8, 80) means the turtles traveled 80 miles in 8 days.

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 Normally, a number next to parentheses represents multiplication. Is that the case here? Explain. Here, the number inside the parentheses is telling us the input, or x-coordinate. © Accelerate Learning Inc. - All Rights Reserved

•

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 What is the difference between ff(20) and f( f(x (x) = 20? f(20) is asking us to ﬁnd the distance traveled after 20 days, while f(x) = 20 is asking us to ﬁnd the number of days when the turtles traveled 20 miles. DOK-1 When considering the function in question 1, ff(((xx) = 8x, 8 , what is the dependent variable (output), and what is the independent variable (input)? The dependent variable is f(x), which stands for the distance traveled. This is the value of the output. The independent variable is x, which stands for the number of days. This represents the value of the input. DOK-2 How is f( f(x (x) = 8x different from y = 8x?? How are they similar? They both represent the same pattern and can be used interchangeably.

Explain the following to the class: Function notation was created by mathematicians so they had a shorter way to discuss long or complex equations. Function notation is similar to using LOL or TL;DR—they are all shorthand ways of saying something. If we were analyzing a situation that used multiple equations, we could use function notation. We might use f(x), g(x), or h(x) to name and discuss the functions. The notation g(x) represents the output of a function named g when it is evaluated at a value of x.

Intervention

Acceleration

FACILITATION TIP After students answer the question, remind them what an input is and what an output is. As you remind them, tie input to independent variable and output to dependent variable. Then, ask the class which instance of “20” is an input and which one is an output. FACILITATION TIP

PROPERTIES OF FUNCTIONS

Home

Have the class look at the graph in Part I. Ask them if there is any section before Point D on the graph where f(x)=8x, and have them justify their answer. Use the graph to introduce or further discuss a steeper slope versus a shallower slope.

Part II 1.

4. 5.

Read the following scenario to the class: Dr. Angola was so pleased with your report on leatherbacks that she has asked you to complete the report on multispecies migration. The team has developed functions to represent the distances traveled over any amount of time for each species studied. Explain to students that they will work with their groups to evaluate functions for the given value or ﬁnd the input for a given output. Students will use these calculations to complete the report on multispecies migration in Part II of their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What does ff(14) (14) mean? This means to evaluate the given function with an input of 14.

DOK-1 Once you calculate that ff(14) (14) = 576, how do you determine which number represents days and which number represents miles traveled? I know 14 is the input or independent variable and 576 is the output or dependent variable. In this scenario, the number of days is the independent variable, and the number of miles traveled is the dependent variable.

DOK-1 How do you evaluate a function that has more than one x—for example, g(x ( ) = ((xx + 4)( 4)(xx – 2)? I substitute the given input for each appearance of x and then use order of operations to simplify.

DOK-2 If ff(14) (14) represents the distance traveled by ﬂatbacks in 14 days and g(14) (14) represents the distance traveled by greens in 14 days, what would ff(14) (14) + g(14) represent? This would represent the combined distance traveled by ﬂatbacks and greens in 14 days.

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

FACILITATION TIP Before reading the scenario, ask the class 1) What other animals besides sea turtles do you know of that migrate?; 2) Where do they go when they migrate?; 3) What time of the year do they migrate? FACILITATION TIP Part II, Report table: In the row for ﬂatbacks, watch out for one or more students leaving out the exponent in the equation for f(x). Also, let students know that they can plug the value for x into the equation immediately as long as they account for the exponent in the equation and perform order of operations correctly.

FACILITATION TIP Student Journal Part II, Reﬂect: For Question 1, watch out for students leaving out negative signs or using them incorrectly when solving for the function value. They may forget that squaring a negative value results in a positive value. They may also leave out the negative sign when simplifying the linear value.

PROPERTIES OF FUNCTIONS

Properties of Functions Explore 2 – Evaluating Functions Math Chat DOK-2 For flatback turtles, ff(20) > ff(15). (15). Explain what this means in context and what this information tells you about the graph of f( f(x (x). Since f(20) > f(15), the flatback turtles have traveled farther after 20 days than after 15 days. On a graph, the output, or y value, when x = 20 would be higher than the y value when x = 15. • DOK-2 Is it possible to evaluate a function using a negative number? Does it make sense in this scenario? Explain. Yes, it is possible. You would substitute and simplify just as you would with a positive number. However, it does not make sense in this scenario because our inputs represent days, and the turtles cannot travel for a negative amount of days. • DOK-2 What are some advantages to using function notation? Function notation is more flexible than using y = . It is easier to keep track of different functions in the same scenario using function notation: f(x), g(x), h(x), etc. •

FACILITATION TIP After students answer the question, ask them for other examples of scenarios where it would not make sense to evaluate a function using a negative number. If they are stumped, present length and volume and ask for other examples. Then, ask them for examples of scenarios where it could make sense to evaluate a function using a negative number. If they are stumped, present position and temperature and ask for other examples.

Post-Explore 1. 2. 3.

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

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

PROPERTIES OF FUNCTIONS

Home

PROPERTIES OF FUNCTIONS

Properties of Functions Explore 3 – Domain and Range ACTIVITY PREPARATION Students will determine the domains and ranges of functions and will write those values using appropriate formats. Students will also determine domain and range as they relate to the context of a problem and be able to differentiate between discrete and continuous situations.

Standards for Mathematical Practice • • •

MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Set of Event Cards (per student) 1 Set of Set Notation Notes (per class) 1 Exit Ticket (per 2 students)

• •

Reusable • • •

Separate the class into groups of 2 or 3 students. Print a Student Journal and a set of Event Cards for each student. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student has one. Gather a pair of scissors and a glue stick for each student. Prepare to project the Set Notation Notes.

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

PROCEDURE AND FACILITATION

FACILITATION TIP Before reading the scenario, ask the class 1) What events sell tickets?; 2) What information can events get from ticket sales?

FACILITATION TIP As students work on Page 2 of the Student Journal, ask them what kind of function the graph depicts. Then, ask them if they can think of other real-life scenarios that a quadratic function could graph. FACILITATION TIP Student Journal, Page 1: As students establish that the number of ticket sales is a dependent variable, ask them if the number of sales could ever represent an independent variable. Present a scenario such as the number of ticket sales determining how much money was made from the end-of-year bash. 32

Part I 1.

2. 3.

Read the following scenario to the class: The student event planning committee has one more event to plan for the year, the end-of-year bash. You are in charge of tracking ticket sales for the 7 days leading up to the event so you can report the anticipated attendance at the event. This number is needed so the committee can be sure to have enough food and prizes. You are also in charge of tracking student attendance throughout the 4-hour event so you can help the committee determine if next year’s bash should be extended. The data you have gathered is represented in tables and graphs. Use the data to make your recommendations to the student event planning committee. Give a Student Journal to each student. Explain to students that they will work with their groups to analyze the tables and graphs and then answer the questions about inputs and outputs. They will use their analyses to support their recommendations for ticket sales and attendance on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 Which variable is the independent variable? The variable x is the independent variable.

DOK-1 Which variable is the dependent variable? The variable y is the dependent variable.

DOK-3 Would you want to connect the points on the graph for ticket sales? Why or why not? No, I don’t want to connect the points on this graph because I only collected data for speciﬁc days, and there is nothing that indicates the sales were continuous. © Accelerate Learning Inc. - All Rights Reserved

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

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.

DOK-2 What word did we use to describe the possible x values? The x values are also called inputs.

Explain the following to the class: In algebra, we deﬁne the set of inputs as the domain. When you are asked for the domain of a function, you are looking for the possible x values, or inputs. •

Acceleration

DOK-3 Why would it be okay to connect the points on the graph for the number of students in attendance at the event? This graph shows the number of people at the event over a speciﬁc time interval. Since time is constantly in motion, connecting the points is okay.

Math Chat •

Intervention

FACILITATION TIP Before they start the Math Chat, discuss with the class whether they think a table or a graph tracks ticket sales better. Also, ask students for examples of scenarios they believe are better suited for tables and graphs, respectively. Alternatively or in addition, ask them the same questions for tracking attendance.

PROPERTIES OF FUNCTIONS

Home

DOK-2 What word did we use to describe the possible y values? The y values are also called outputs.

Explain the following to the class: In algebra, we define the set of outputs as the range. When you are asked for the range of a function, you are looking for the possible y values, or outputs. DOK-2 We would say that the domain of the ticket sales is discrete and the domain of the event attendance is continuous. What could these words mean? Think about the differences of the graphs of these two data sets. Continuous would mean that the domain could be any values in between data points (for example, 1.5 hours). The data points on the graph were connected with a line. Discrete must mean that the domain is restricted to specific data points. The data points on the graph were not connected. • DOK-3 What does it mean to say that the domain of a function may be restricted if the function represents a real-world situation? Give an example. It is a reminder that I have to consider what is reasonable for a situation. When a function just represents a relationship and is not in context, the only restrictions are the mathematical restrictions. An example of a restricted domain could be a student walking home from school at a constant rate. They live 1 mile from school. If we represent the distance as a function of time, the time would be limited. The minimum is 0 because negative time doesn’t make sense, and the maximum would be limited because their home is a fixed distance and they are traveling at a constant rate. If we considered the algebraic representation without a context, the domain would not be limited. • DOK-3 What can be helpful in identifying whether a function is discrete or continuous? If there is a graph, look to see whether or not the points are connected. If the domain and range are written with no graph, look to see whether the values are listed or written using inequalities. •

Part II 1.

2. 3. 4.

Read the following scenario to the class: Because of your superb work and recommendations regarding the end-of-year bash, the student event planning committee has selected you as chairperson to make all arrangements for next year’s events. At the last meeting of the year, the committee members were passing around data cards from this year’s events. Clumsily, the Event Cards got all mixed up as members were passing them around the table. Use your knowledge to sort out the information and get it organized for the committee. Give a set of Event Cards, pair of scissors, and glue stick to each student. Have students cut out the cards. Explain to students that they will work with their groups to sort through the cards to match each graph with a corresponding domain and range. Students will glue or tape their own cards on their Student Journals.

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 Before reading the scenario, ask the class 1) Would you consider yourself good at organization?; 2) Who do you know that is good at organization?; 3) What attributes does someone who is good at organization have?

PROPERTIES OF FUNCTIONS

Properties of Functions Explore 3 – Domain and Range 5.

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

DOK-1 What does it mean when you see an open circle at a speciﬁc point on the function? An open circle means that particular value is not included in your domain or range.

DOK-1 What does it mean when you see a closed circle at a speciﬁc point on the function? A closed circle means that particular value is included in your domain or range. It also signiﬁes that the value is the last value in the set.

DOK-2 Which graph(s) show a discrete function? The graph on card 6.

DOK-2 How might the domain and range look for such a graph? There may be a list of speciﬁc numbers.

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

e. DOK-1 Which card has such a list? Card 1 f.

DOK-3 Which domain and range Event Card is likely to correspond to card 9? Explain. Card 10 because it’s the only card that has a 5 in the range and the range for the graph starts at 5.

DOK-1 What does the symbol ∞ mean? Inﬁnity

h. DOK-2 How might we know when to apply the inﬁnity symbol to a graph? There will be arrows. i. DOK-3 What do you think the round brackets, ( ), mean? Explain. They mean “not included” since they correspond to whatever point on the graph that is an open circle. j. DOK-4 What do you think the square brackets, [ ], mean? Explain. They mean “included” since they correspond to whatever point on the graph that is a closed circle. FACILITATION TIP Student Journal Part II, Reﬂect: After students answer Question 2, ask if anyone included time as a continuous situation, and establish that it is one. Then, have them look at the graph on Page 1 of the Student Journal. Note to them that, while days are a measure of time, one cannot sell tickets between consecutive days. Note to the class that, in contrast, the number of people (refer to Page 2 of the Student Journal) could change between consecutive hours.

6. 7.

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

Math Chat Explain the following to the class: Typically, we may use an inequality to describe a span of numbers for the domain and range. However, different notation was used for Part II. Let’s think about how this notation is similar to or different from an inequality. DOK-2 How can you tell if the value of an endpoint is going to be included in a solution set when looking at inequalities? If the inequality is less than (<) or greater than (>), the value will not be included. If the inequality is less than or equal to (≤) or greater than or equal to (≥), the value will be included. • DOK-2 How can you tell if the value of an endpoint is going to be included in a solution set when using brackets? If the value is excluded, round brackets or parentheses, ( ), are used. If the value is included, square brackets, [ ], are used. •

Explain the following to the class: The use of square and round brackets to describe the domain and range is called interval notation. • DOK-3 How would you write the inequality 1 ≤ x < 5, using interval notation? [1, 5) • DOK-2 How would we show that the domain or range includes all real numbers using interval notation? D: (–∞, ∞) or R: (–∞, ∞) • DOK-1 When the graph consisted of discrete points, how was the domain or range written? It included a list of numbers, for example: Domain: D: {x| x = 55, 56, 57, 58, 59, 60} Range: R: {y| y = 137.5, 140, 142.5, 145, 147.5, 150} Explain the following to the class: Notice how this format is different from both the interval notation or inequality alone. This particular way of writing the domain and range is called set notation. (Project the Set Notation Notes, and point out the different features.) 34

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Engage

Explore

Explain

Elaborate

Evaluate

2. 3.

Acceleration

DOK-3 How would I write the domain (–∞, ∞) in set notation? Domain: D: {x| x ∈ ℝ} DOK-3 How would I write the range [ –3, 4] in set notation? Range: R: {y| –3 ≤ y ≤ 4}

Post-Explore 1.

Intervention

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

FACILITATION TIP Ask students when the rock is at its highest position. Then, ask students if the graph indicates how far the rock traveled horizontally. Ask them to justify their answer. If no one mentions that the only variables are time and height, guide them in interpreting what the axes represent and understanding that height is vertical distance rather than horizontal distance.

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PROPERTIES OF FUNCTIONS

Properties of Functions Explore 4 – Linear vs. Nonlinear Functions ACTIVITY PREPARATION Students will explore the difference between linear and nonlinear functions by informally analyzing the graphs of various parent functions. Students will compare linear and nonlinear functions using visual analysis of characteristics such as end behavior, increasing and decreasing, domain and range, intercepts, and general curvature.

Standards for Mathematical Practice • • •

MP.3 Construct viable arguments and critique the reasoning of others. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Set of Graph Cards (per group) 1 Set of Parent Functions (per class) 1 Exit Ticket (per 2 students)

•

Reusable • •

•

1 Resealable bag (per group) 6 sheet protectors (per class)

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 a set of Graph Cards on card stock for durability for each group of students. Cut the cards apart, and place them in a resealable bag. Print the Parent Functions, and place each slide in a sheet protector for durability.

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) Does anyone know what an intern is?; 2) Is there someone you would like to intern for?; 3) What would you expect to learn in your internship?

FACILITATION TIP Ask students what “domain of all real numbers” means in the context of a number line. Then, ask them what it means in the context of a graph. For the context of a graph, they should speciﬁcally mention “values on the x-axis” in their response. FACILITATION TIP Check student understanding of the origin. After they answer the question, ask the class if any lines exist where the x- and y-intercepts have the same coordinate pair. Then, ask them what all of these lines have in common if they have not already mentioned that they all pass through the origin. 36

2. 3. 4.

Read the following scenario to the class: During the summers, you continue to work for Dr. Angola, the head zoologist. She was thoroughly impressed with your previous work as her intern. She decides to entrust you with another task— analyzing graphs pertaining to various operations at the zoo. However, before she gives you the next task, she wants to see your graph analysis skills, so she asks you to categorize several graphs. Give a Student Journal to each student. Explain to students that they will work with their groups to categorize several graphs according to criteria they create. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 Pick two graphs and ask the following questions: How is graph x similar to graph y? How are they different? Answers will vary.

b. DOK-2 Pick three graphs and ask the following questions: Given graphs A, B, and C, which would you say does not belong? Why? Answers will vary. c.

DOK-1 What does “domain of all real numbers” mean? It means every number on the number line from inf to −inf is included.

DOK-1 What is an x-intercept? The x-intercept refers to where the graph crosses the x-axis. It’s the value of x when y equals zero.

e. DOK-2 If a function is increasing from left to right, what do we expect to happen to its y values as you move from left to right (in the positive x direction)? The y values would become larger as you move from left to right (in the positive x direction). © Accelerate Learning Inc. - All Rights Reserved

f. 5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How would you describe end behavior in your own words? Answers will vary.

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 Have two or three groups take turns to come to the board. Let them choose one of their categories and arrange the graphs that are in sheet protectors on the board according to their chosen category. Have the class attempt to guess by what criteria the graphs are sorted. Afterward, begin the Math Chat. •

•

• •

•

DOK-1 How are the absolute value and quadratic Parent Functions similar to each other and different from the rest of the graphs? Both the absolute value and quadratic functions have increasing and decreasing parts, whereas the other graphs are always increasing. DOK-1 Do all the graphs have x-intercepts? -intercepts? Explain. All the graphs except for the exponential function have an x-intercept. The tail end of the exponential function approaches but never crosses the x-axis. As the x values move away from zero in the negative direction, the graph approaches but never reaches zero. DOK-1 Which graph has a restricted domain? Square root DOK-2 What’s unique about the square root function’s end behavior as x values move away from zero in the negative direction? The square root function does not have end behavior in the negative direction. It stops abruptly. DOK-2 Which functions are linear and which are nonlinear? How do you know? The linear function is linear because it is a straight line, whereas all the other functions are nonlinear because they either have curves or a vertex.

Explain the following to the class: In summary, there are many ways to visually analyze the graph of a function—we can observe the end behavior, domain and range, general curvature, intercepts, and whether it’s increasing or decreasing. In Part II, let’s see how we can use these features to explore the difference between linear and nonlinear functions. Part II 1.

2. 3. 4.

Read the following scenario to the class: Great job! Because of how well you sorted the graphs, Dr. Angola would like for you to analyze and compare graphs pertaining to various operations at the zoo. Dr. Angola is waiting for the graphs and is trying to assess whether particular graphs show a linear versus nonlinear trend based on the bits of information her colleague gave her. Work with Dr. Angola to suggest what each graph could be based on the information provided. Explain to students that they will work with their groups to determine whether a graph could be linear or nonlinear based on the information provided. Make sure students know that they can sketch a graph that fulﬁlls the criteria to help them visualize the features. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What does “domain of all real numbers” mean? It means every number on the number line from inf to −inf is included.

DOK-1 What is an x-intercept? The x-intercept refers to where the graph crosses the x-axis. It’s the value of x when y equals zero.

DOK-2 If a function is decreasing from left to right, what do we expect to happen to its y values as you move from left to right (in the positive x direction)? The y values would become more negative as you move left to right (in the positive x direction).

Intervention

Acceleration

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.

PROPERTIES OF FUNCTIONS

Home

FACILITATION TIP After students answer the question, ask them if the range of the graph is restricted as well and how they know. Make sure students understand that the square root of a negative number will not produce a real number value. FACILITATION TIP After students answer the question, briefly review or introduce the concept of an inflection point. Then, ask the class which functions have a vertex and which ones have an inflection point. FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone analyzed graphs outside of school before?; 2) What kind of graphs were they?; 3) What information did they show? 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.

PROPERTIES OF FUNCTIONS

Properties of Functions Explore 4 – Linear vs. Nonlinear Functions d.

DOK-1 What does it mean for a function to have a restricted domain or range? It means that the possible values for x or y are limited to a very speciﬁc range of numbers as opposed to including all real numbers (for example, the square root function).

e. DOK-1 What does it mean for rate of change to be constant? It means the change in x value in relation to the change in y value is a constant ratio. FACILITATION TIP After students answer the question, check their understanding by asking if a constant rate of change would indicate a linear or nonlinear function over an interval. If some answer nonlinear, ask them if they can explain their reasoning, and guide them to the correct answer. FACILITATION TIP Check student understanding of asymptotes. After they answer Question 3 in the Reﬂect section, ask the class which key feature reﬂects a graph’s output getting closer and closer to 3. Then, ask them which parent function(s) have an asymptote.

FACILITATION TIP It may be hard for some students to visualize that the cube root parent function has a range of all real numbers. After they complete the Exit Ticket, present the class with larger numbers and their cube roots so students can see that the graph will eventually reach larger y values.

5. 6.

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

Math Chat DOK-2 How can you tell if a graph is linear or nonlinear by visual analysis alone? A linear function will have a graph that is a straight diagonal or horizontal line, without any curves, whereas a nonlinear graph would have some form of curvature or a vertex. • DOK-2 Is it possible for a linear function to have a graph that is a vertical line? Why or why not? It is not possible because if its graph were a vertical line, it would cease to be a function since there would be multiple outputs for one input. • DOK-2 Is it possible for a linear function to have two x-intercepts? -intercepts? Why or why not? It is not possible for a linear function to have two x-intercepts because a linear function has a constant rate of change. To have two x-intercepts would require the function to change the magnitude or direction of its slope to cross the x-axis twice. • DOK-3 Could a linear graph have end behavior in the positive direction where the output values get closer and closer to 3? Why or why not? No, it could not because a linear graph has end behavior in the inf or −inf direction as x moves away from zero in the positive direction and therefore would not be limited to approaching 3. •

Post-Explore 1. 2. 3. 4.

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

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

PROPERTIES OF FUNCTIONS

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PROPERTIES OF FUNCTIONS

Properties of 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

Relations and Functions 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

Evaluating Functions 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

Domain and Range

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

Linear vs. Nonlinear Functions

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

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

PROPERTIES OF FUNCTIONS

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Intervention and Assessment SMALL-GROUP PLANNER 1

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

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

PROPERTIES OF FUNCTIONS

Properties of Functions

3 42

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

What prompts will be used?

What does mastery look like?

PROPERTIES OF FUNCTIONS

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I can evaluate expressions that use function notation.

I can create and understand linear equations.

I can solve problems involving linear equations.

I can compare the characteristics of a linear graph with those of a nonlinear graph.

I can use graphing calculators and technologies to compare linear graphs with nonlinear graphs.

SCOPE 1

Linear Functions Scope Introduction SCOPE SUMMARY In this grade level, students will expand their understanding of slope to recognize constant rate or constant percent rate per unit interval. Students will also recognize and prove which situations can be modeled and represented by linear functions compared to arithmetic sequences. They will construct linear functions given a graph, verbal or written description of a scenario, as well as two ordered pairs. Student Expectations

A.FGR.2.1 Use mathematically applicable situations algebraically and graphically to build and interpret arithmetic sequences as functions whose domain is a subset of the integers. A.FGR.2.2 Construct and interpret the graph of a linear function that models reallife phenomena and represent key characteristics of the graph using formal notation. A.FGR.2.3 Relate the domain and range of a linear function to its graph and, where applicable, to the quantitative relationship it describes. Use formal interval and set notation to describe the domain and range of linear functions.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grade levels, students learned how to write an equation in slope-intercept, point-slope, and standard form to model a linear relationship. Students also determined the slope and the yy-intercept -intercept from multiple representations. They have constructed a linear function given a description, two ordered pairs, a slope and a point, a table, a graph, or a scatterplot. Students have interpreted the rate of change and initial value of a linear function in terms of the situation, graph, or table.

Later in this course, students will tackle geometric sequences as well and make connections to exponential functions in the same manner that arithmetic sequences and linear functions are compared here. Students will compare linear functions to quadratic and exponential functions later in the course. In future courses, students will analyze arithmetic and geometric series, which are sums of sequences.

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

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

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: •

relate sequences 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. 44

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Identify and Construct Arithmetic Sequences In this exploration, groups of students will evaluate cup stacking sequences for a middle school cup stacking tournament. In addition, groups will need to use the identiﬁed patterns in each competitor’s cup stacking setup to create an explicit equation that can be used to ﬁnd any nth term. Students will: •

analyze data to distinguish between arithmetic and geometric sequences.

•

write explicit equations to ﬁnd the nth terms in arithmetic and geometric sequences.

Explore 2

Explore 1

EXPLORE ACTIVITIES Create Recursive and Explicit Equations

LINEAR FUNCTIONS

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In this exploration, students will determine if enough speciﬁc vegetables and fruits are grown in 7 days to make food for a family gathering. Students will: •

construct recursive formulas for arithmetic and geometric sequences.

•

convert recursive to explicit formulas.

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

Arithmetic Sequences and Linear Functions In this exploration, students will make connections between the explicit formula of an arithmetic sequence and the slope-intercept form of a linear equation, y = mx + b. 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, students will graph linear functions from a slope and point, a table of values, and a verbal description. Students will: •

compare and explain if an arithmetic sequence is better to represent the cost of bikes or a linear equation.

graph the linear function that describes each route in terms of the distance to the destination versus the time traveled.

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

Graph Linear Functions

Graph from an Equation In this exploration, students will graph linear functions from equations in slope-intercept form and standard form. Students will also write equations in slope-intercept form from a graph. Students will: •

graph linear equations given in both slope-intercept and standard form.

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

LINEAR FUNCTIONS

Linear Functions Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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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.

Materials

Preparation

Printed •

•

1 Set of Fact or Fiction Prompts (per class)

•

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. Instruct students to move to one side of the room or the other 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 the remaining prompts. a.

Prompt 1 is ﬁction.

Prompt 2 is fact.

Prompt 3 is ﬁction.

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

Identifying Misconceptions •

Students may struggle identifying the slope from a graph without designated points. Remind students that they should select coordinates that the graphed line passes through.

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.” FACILITATION TIP 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.

LINEAR FUNCTIONS

Linear Functions Hook – Tile Pattern ACTIVITY PREPARATION Students will relate sequences to a real-world situation.

Materials

Preparation

Printed •

Reusable • •

Plan to show the video. Prepare to project Tile Pattern 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.

• • •

1 Tile Pattern (per class)

1 Phenomena Video (per class) 1 Projector (per class)

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

Before showing the video or reading the scenario, ask the class 1) Has anyone seen a room before and after a remodel?; 2) What about the room changed?; 3) Is there anything you would have done differently in the remodel?

FACILITATION TIP After the video, ask students what they think of the pattern. Then, ask them what pattern they would design for ﬂoor tiles.

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: A local community center is laying down a new tile pattern to cover their floors. Each row contains five tiles, and the pattern continues to grow as they fill out more and more of the floor. 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 tiles repeat. I wonder how long the tiles will continue. I could calculate how many tiles it will take to fill a certain area. I wonder if the tiles get put down at a constant rate. Project Tile Pattern.

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

Acceleration

Explain to students that the community center project team created two potential numbers of tiles that are placed on the ﬂoor after each day. Discuss the following questions: a.

DOK-1 What is similar and what is different about the two equations? Allow students to share all ideas. Answers will vary. Both equations have 5s. One equation involves adding ﬁve tiles each time, and the other involves adding three tiles.

DOK-2 Which equation represents faster tile growth? Option 1 represents faster tile growth even though it begins with fewer tiles.

Share with students that the following guiding question will be used throughout the topic: How do we generate rules from patterns? Complete the Explore activities.

Part II: Post-Explore 1. 2.

Intervention

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

DOK-1 Do these equations make more sense after the Explore activities? Yes, both equations are arithmetic sequences. Option 2 is a more familiar linear form.

DOK-1 How are arithmetic sequences different from linear functions? Arithmetic sequences are types of linear functions where the domain is only natural numbers. A common difference of an arithmetic sequence mirrors the slope of a linear function.

DOK-2 Which type of equation makes more sense for laying down tiles in a community center? Answers may vary. The tiles placed on the community center ﬂoor will likely grow at a constant rate. Both options are logical, but the ﬁrst option is faster so will likely be the preferred choice. Budget and labor constraints may force the second option to be used.

DOK-2 What do the numbers in each of the equations represent? In the equation An = 5 + 5(n – 1), the initial 5 is the number of tiles placed after one day, and the next 5 is the number of tiles added each day. In the equation An = 3n + 5, the 5 is the number of tiles initially in place, and then 3 additional tiles are added the next day.

LINEAR FUNCTIONS

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FACILITATION TIP After they answer the question, have the class explain their reasoning. Keep track of this and have them answer the question and explain their reasoning again during the Post-Explore. Discuss with them how their reasoning stayed the same and/or how it changed between the Pre-Explore and PostExplore. FACILITATION TIP After students answer the question, ask them to identify each equation as recursive, explicit, or neither. Have them justify their reasoning.

FACILITATION TIP If they don’t mention it in their answer, note to the class how budget and labor constraints could factor into the equation choice. Ask them if anyone has had to consider budget and labor constraints, either separately or together, when making a decision.

e. DOK-2 How would you connect the arithmetic sequence formula to equations in point-slope form? The sequence modeled by An = 5 + 5(n – 1) would have fallen on the line y – 5 = 5(x – 1) if n mapped to x and An mapped to y. f.

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

LINEAR FUNCTIONS

Linear Functions Explore 1 – Identify and Construct Arithmetic Sequences ACTIVITY PREPARATION Students will analyze data of arithmetic sequences. Students will write explicit equations to ﬁnd the nth th terms in arithmetic sequences.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Set of Cup Stacking Cards (per group) 1 Set of Cup Stacking Charts (per group) 1 Exit Ticket (per 2 students)

• •

Reusable •

Separate the class into groups of 2 or 3 students. Print a Student Journal for each student. Print a set of Cup Stacking Cards for each group of students. Cut the cards apart, and place them in a resealable bag labeled “Part I.” Print a set of Cup Stacking Charts for each group of students. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student has one.

1 Resealable bag (per group)

PROCEDURE AND FACILITATION POINTS Part I FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever helped set up a tournament?; 2) What kind of tournament was it?; 3) What did you do to set it up?

FACILITATION TIP Student Journal, Part I, Question 4: Each student should have time to create a sketch. Afterward, be sure group members collaborate by checking each other’s sketches and comparing them with their respective answers for 5a and 5b If there is a disagreement on a sketch or answer within a group, present it to the class and have a discussion on what is correct.

2. 3. 4.

FACILITATION TIP Some students may think that setups 2 and 3 represent setups that increase and decrease by a common number, respectively. Remind the class that, in this case, “increase” means adding by a number every time and “decrease” means subtracting by a number every time. 50

6. 7.

Read the following scenario to the class: Your team has volunteered to set up and judge the middle school cup stacking tournament. The head judge claims that not all of the sequences of stacking setups follow the criteria for each round of the competition. Help the head judge determine which setups meet the criteria for round 1 and round 2. Give a Student Journal to each student. Give a bag of Cup Stacking Cards to each group. Explain to students that they will work with their groups to analyze the four cup stacking setups on the Cup Stacking Cards to determine which setups meet the criteria for round 1 and which setups meet the criteria for round 2 of the competition. 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 remains constant in each of the cup setups in setups 1 and 4? The number of cups being added or subtracted remains constant and never changes within each cup setup.

DOK-2 What changes do you see in each of the four cup setups? Some cup setups are increasing, while others are decreasing. Some of the cup setups increase or decrease by adding or subtracting the same number between each stack.

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. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Math Chat DOK-1 What are the two ways the cup stacking setups show a change? They change either by adding the same number of cups or by subtracting the same number of cups between each stack of cups. • DOK-2 Mathematicians call these changes common difference. Do you think this term applies to addition/subtraction? Why? The common difference applies to the adding/subtracting change because the word difference is the word for the answer when you subtract. • DOK-1 Mathematicians call sequences with a common difference arithmetic sequences. Which of the cup stack setups are arithmetic, and which are not arithmetic? Setups 1 and 4 are arithmetic sequences. Setups 2 and 3 are not arithmetic sequences. • DOK-1 If predicting the number of cups that would be needed for the 100th stack is too diﬃcult, what do you think could be used to determine the answer easily? Creating an equation where you substitute the value of the nth term would be an easy way to determine how many cups are needed for building the 100th stack. Mathematicians call these equations explicit. Explicit equations allow you to calculate any term in an arithmetic sequence without knowing the value of the previous terms. •

Intervention

Acceleration

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.

LINEAR FUNCTIONS

Home

Part II 1.

2. 3.

Read the following scenario to the class: Now, the head judge trusts your team to start judging some cup stacking competitions. In round 2, competitors are required to stack in sequences, and the judge needs to record the number of cups stacked by each competitor. Your team judged two competitors before you realized that you stopped recording after 4 terms. Luckily, the head judge tells you not to worry; you can calculate any term for each competitor’s sequence. Can you figure out how to save your team members’ reputations as good judges? Give a set of Cup Stacking Charts to each group of students. Explain to students that they will work with their groups to analyze the Cup Stacking Charts to determine the value of the missing fifth term in each competitor’s cup stacking setup. Students will work together to use the identified patterns in each competitor’s cup stacking setup to create an explicit equation that can be used to find any nth term and 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 are the two important parts of an arithmetic sequence? The ﬁrst term and the common difference

DOK-1 What happens to an arithmetic sequence if it has a negative common difference? The sequence will decrease with each term.

DOK-2 If you know what the common difference is, how can you determine the value of the fifth term (or how many cups should be in the fifth stack)? To determine how many cups should be in the fifth stack, you take the number of cups in the fourth stack and either add or subtract by the common difference.

DOK-1 What is a more eﬃcient way to write 3 added to itself four times? Use multiplication, so 3 · 4.

e. DOK-1 What is a more eﬃcient way to write 5 added to itself seven times? Use multiplication, so 5 · 7. f.

FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever served as a judge?; 2) What were you judging?; 3) Would you serve as a judge again? Why or why not?

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.

FACILITATION TIP Some students may overthink this question. It may be clearer to ask, “What is a more eﬃcient way to write 3 + 3 + 3 + 3?”

DOK-2 How can you create an explicit equation to determine any nth term for each competitor’s cup stacking setup? To create an explicit equation that works for any nth term for each competitor, you need to include the d value or common difference as well as the ﬁrst term.

LINEAR FUNCTIONS

Linear Functions Explore 1 – Identify and Construct Arithmetic Sequences 6. 7.

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

Math Chat FACILITATION TIP Some students may answer with a word like “number” or “ﬁnal.” Be sure students clearly articulate what they believe the variable represents. Ask questions like “Number of what?” or “Final what?” as appropriate. FACILITATION TIP Test student understanding of the ﬁrst term. After explaining the explicit formula for an arithmetic sequence, ask the following: If the common difference is cut in half, will the nth term be half of what it was? How do you know?

•

Explain the following to the class: When we write an equation for a sequence, we use the formula that has an output of An. The explicit formula for an arithmetic sequence is An = A1 + d(n – 1), where A1 is the ﬁrst term and d is the common difference. •

•

FACILITATION TIP Check student understanding of the common difference. After students answer the questions, ask them if the common difference can have a decimal value. They should understand that a term can increase by a decimal value just as it can increase by a whole number or fraction value.

DOK-1 The formula for an arithmetic sequence has an output of An. What variable did you use when you wrote the equations in questions 6 and 12? I wrote the words number of cups.

•

DOK-2 What are the values of A1 and d from your Student Journal question 6 in Part II? How would this equation have changed if the common difference had been 7? The value of A1 was 6, and the value of d was 3. If the common difference had been 7, I would have written an equation with a coefficient of 7, not 3. DOK-2 What are the values of A1 and d from your Student Journal question 12 in Part II? How would this equation have changed if the common difference had been 7? The value of A1 was 7, and the value of d was 5. If the common difference had been 7, I would have written an equation with a coefficient of 7, not 5. DOK-2 Describe arithmetic sequences. Arithmetic sequences have a constant difference, where the value of the nth term increases or decreases by adding or subtracting the same number each time. DOK-2 Why is it useful to be able to write equations for arithmetic sequences? Writing equations is useful when finding the value of an nth term that would require an excessive amount of adding or subtracting. Equations allow you to find the value by simply plugging in the nth term to find the correct value. DOK-3 If given an equation for an arithmetic sequence, how could you tell if it is increasing or decreasing? You would be able to tell if an arithmetic sequence is increasing or decreasing with each term by looking at whether the d value is positive or negative within the equation.

FACILITATION TIP

Post-Explore

Later in the scope, students will work with both explicit and recursive equations. After they complete the Exit Ticket, remind them of explicit equations, and note that the Exit Ticket’s answer is an example of one.

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

LINEAR FUNCTIONS

Home

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LINEAR FUNCTIONS

Linear Functions Explore 2 – Create Recursive and Explicit Equations ACTIVITY PREPARATION Students will construct recursive formulas for arithmetic sequences. Students will convert recursive to explicit formulas.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Vegetable Graphs (per group) 1 Exit Ticket (per 2 students)

•

Separate the class into groups of 2 or 3 students. Print a Student Journal for each student. Print a set of Vegetable Graphs for each group of students. If desired, print it on card stock, and laminate it 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) Has anyone ever planted vegetable plants?; 2) If so, what vegetable plants did you plant?; 3) Why did you choose those vegetables?

Part I 1.

2. 3. 4.

FACILITATION TIP After students answer the question, note that the graph for green beans has a steeper slope than that for tomatoes, meaning green beans grow faster. Then, ask the class why they think green beans grow faster than tomatoes.

Read the following scenario to the class: This summer, you’ve decided to build a garden and plant an assortment of fruits and vegetables. One of your goals is to grow enough tomatoes and green beans for your grandma to make her famous tomato and green bean casserole. Today, you got a call from your grandma saying she needs the tomatoes and green beans in 7 days to make the casserole. Will you grow enough tomatoes and green beans in time for your grandma to make her casserole? Give a Student Journal to each student. Give a set of Vegetable Graphs to each group. Explain to students that they will work with their groups to analyze the graphs and answer the questions to determine whether they will grow enough tomatoes and green beans for their grandma’s casserole. 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 How do you know the green beans have a common difference? The number of green beans increases by constant amounts, and the graph forms a straight line.

DOK-1 What does a common difference tell you about a set of data? How does this help you when writing an equation? The common difference shows you the pattern a set of data follows. This means it must be included when writing the equation so the equation will work for determining the value of any term in the given set of data.

DOK-2 What math symbols can be used to represent these changes, and how can you use them to make predictions for unknown data? Addition and subtraction symbols can be used to show these changes. You can identify these mathematical operations within a given set of data (sequences, graphs, and tables) and use them in equations to ﬁnd the value of any unknown term.

FACILITATION TIP If students are thrown off, let them know the question is asking for math operators speciﬁcally, as opposed to other math symbols. Make sure students know that the phrase “these changes” refers to common difference. 54

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

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 can you determine the missing number of tomatoes and green beans for days 5 and 6? You can identify the common difference in the graphs and continue to either add or subtract to determine the following term value until you ﬁnd the value for days 5 and 6. • DOK-2 What three elements must be included when creating an equation that builds on the previous term’s value? You must include the previous term (represented by An – 1), the mathematical operation that’s being done, and the common difference. • DOK-3 Mathematicians call equations that require the value from the previous term to determine the value of the next term recursive equations. Compare and contrast recursive equations to what you already know about explicit equations There are common differences in both recursive and explicit equations. Recursive equations require the value from the previous term to determine the following term’s value. Explicit equations don’t require you to know the value of the previous term. •

Part II 1.

2. 3.

4. 5.

6. 7.

Read the following scenario to the class: Your aunt heard how delicious your grandma’s famous tomato and green bean casserole was and now is requesting your help in growing strawberries and blueberries for her to make her special fruit jam. You would love to help your aunt but are worried that you won’t be able to grow enough strawberries and blueberries for her. Thankfully, you have been keeping track of your fruit growth in a data table and are conﬁdent you can calculate just how many strawberries and blueberries you’ll have for your aunt when she needs them. Will you have enough? Students should still have their Student Journals. Explain to students that they will work with their groups to analyze the given recursive equations for strawberries and blueberries to complete the missing sections in the data tables. Students will work together to create an explicit equation to determine the amount of each fruit on day 10 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 Why is the recursive equation on its own not enough information to ﬁll out the table? Since a recursive formula is based on the previous term, you have to be given a starting number to get started.

DOK-1 How can you use the given equation to determine the missing sections in the data tables? Substitute the previous input into the recursive equation to determine the value of the following term.

DOK-2 How can you use the completed data tables to create an explicit equation? Identify whether it is an arithmetic sequence or not, and then use the d value and ﬁrst term within the formation of your explicit equation.

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

Intervention

Acceleration

FACILITATION TIP As a group answers Question 8 of Part I, ask them what each part of the equation represents without giving away the answers. After you have done this for every group, tell the class as a whole what each part of the equation represents.

LINEAR FUNCTIONS

Home

FACILITATION TIP If no one includes it in their answer, ask students to compare and contrast the general form of arithmetic explicit and arithmetic recursive equations. If someone did compare and/or contrast these equations mathematically, have them share with the class, and discuss with the class if they are accurate. FACILITATION TIP Before reading the scenario, ask the class 1) What are your favorite fruits to eat?; 2) Do you eat the fruit as is, or do you like to use the fruit in a recipe?; 3) What products are made out of your favorite fruit?

FACILITATION TIP Student Journal, Part II, Question 5: Make sure students provide clear reasoning for their answers, as answers will vary. For instance, some students may use “a higher term” instead of “the 100th term” or “all of the previous terms” instead of “the ﬁrst 99 terms.” Give them space to use their own words.

LINEAR FUNCTIONS

Linear Functions Explore 2 – Evaluating Functions Math Chat DOK-1 What are two types of equations that can be used for arithmetic sequences? Recursive and explicit equations • DOK-3 How can you use a recursive equation to create an explicit equation? You can analyze the data in a table to identify the elements needed when creating an explicit equation: the ﬁrst term, the common difference, and whether it is arithmetic or not. •

Post-Explore FACILITATION TIP

Some students may be thrown off seeing a sequence of numbers by itself without context. Remind them to ﬁgure out the type of change so they know which kind of recursive and explicit equations to model.

2. 3.

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

Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

LINEAR FUNCTIONS

Home

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LINEAR FUNCTIONS

Linear Functions Explore 3 – Arithmetic Sequences and Linear Functions ACTIVITY PREPARATION Students will make connections between the explicit formula of an arithmetic sequence, An = A1 + (n ( – 1)d,, and the slope-intercept form of a linear equation, y = mx + b.

Standards for Mathematical Practice • • •

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

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of 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 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.

1 Resealable bag (per group)

PROCEDURE AND FACILITATION

FACILITATION TIP Before reading the scenario, ask the class 1) Where did you go on your last family vacation?; 2) What activities did you do when you were there? FACILITATION TIP Afterward, have a short discussion about what students remember about arithmetic sequences. Have a student write the general form of an explicit arithmetic equation; have another write the general form of a recursive arithmetic equation. Then, write the equation y = mx + b, and ask the class how it compares and contrasts to the arithmetic equations.

FACILITATION TIP After students answer the question, show them the general form of an explicit geometric equation. Then, ask the class how the geometric sequence compares and contrasts to a linear function.

Part I 1.

2. 3.

Read the following scenario to the class: While on vacation, your family decides to rent bikes! There are a lot of shop options for your family to rent from, and you want to get the best deal! You need to ﬁgure out which shop is the best cost for a full day’s ride (8 hours). Your mom says an arithmetic sequence is better to represent the cost of bikes, and your dad says a linear equation is more reasonable to represent the cost of bikes. Who is correct, and why? Give a Student Journal to each student. Explain to students that they will work with their groups to write arithmetic sequences and linear functions from the table given 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 Would you call this domain discrete or continuous? Why? Answers may vary. I would call this a discrete domain because it is only deﬁned for whole numbers.

DOK-2 How are arithmetic sequences and linear functions similar? Both use the slope/common difference and starting point/y-intercept.

DOK-2 How are arithmetic sequences and linear functions different? Can you connect the values in each representation to the scenario? The setup for the equations is different. Linear functions start at x = 0, whereas arithmetic sequences start at n = 1. Both include the $10-perhour rate.

DOK-2 How is graphing a function with a discrete domain different from graphing a function with a continuous domain? Functions with a discrete domain will be points on the grid, whereas functions with a continuous domain will be a line or curve. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

e. DOK-3 Would it make sense to write the equation for the bike shop in recursive notation? Why or why not? Answers may vary. Recursive notation requires using the previous term to determine the next term, so it is more labor intensive to use compared to writing an equation in explicit notation. 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.

Math Chat DOK-2 How is the process for writing these equations similar? Both forms require you to ﬁnd the slope/common difference and starting point. • DOK-2 How is the process different for these equations? The arithmetic sequence starts with n = 1, whereas slope-intercept starts with x = 0. • DOK-3 Which type of function would be better if you are looking for a rental for just part of an hour? Why? Linear would be better because it has a continuous domain that includes parts of the hour, not just the whole hour. • DOK-3 Which function (arithmetic or linear) would be better if you could only rent for whole hours? Why? Arithmetic is reasonable for representing situations with a discrete domain. However, a linear function could be used if we restrict the domain. •

Part II 1.

2. 3. 4.

Read the following scenario to the class: Your parents called the rest of the bike shops and gathered information. They asked you to analyze the data and ﬁnd the cheapest option. Give a set of Bike Shop Cards to each group of students. Students should still have their Student Journals. Explain to students that they will work with their groups to identify whether to write arithmetic sequences or linear functions for the other 4 bike shops and record their answers on their Student Journals. They will also ﬁnd the costs for 8 and 4.5 hours. Once completed, the students will decide which bike shop would be best for 8 and 4.5 hours of rentals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

6. 7.

DOK-1 When the bike shop says they will round up to the nearest hour, how does that affect the domain? Rounding up to the nearest hour means they won’t calculate charges for partial hours or by the minute. That policy means the function that represents the cost will have a discrete domain.

DOK-1 How are you determining whether arithmetic or linear is appropriate? The domain is used to determine if a function is discrete or continuous. If the domain is discrete, then use arithmetic, and if it is continuous, use linear.

DOK-1 Will all of the bike shops calculate charges for 4.5 hours? Explain. Bikes n’ More and Wheels Are Us will not calculate charges for 4.5 hours because they always round up to the nearest whole hour.

LINEAR FUNCTIONS

Home

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.

FACILITATION TIP Before reading the scenario, ask the class 1) If you were going to rent bikes on vacation, how would you determine which bike rental shop had the best rates?; 2) Why might bike rental companies have different rates? FACILITATION TIP Groups will have multiple cards and numerous values to account for. Encourage students to take their time and pay attention to detail as they complete Part II of their Student Journal. FACILITATION TIP After they answer the question, note to the class that this scenario can exist as a step function. Then, graph the points for the bike shop Wheels Are Us for the class to see, and complete the step function. Explain to the class that, though bike shops with this setup won’t calculate charges by the minute, the graph still accurately shows the price for each time you’ve plotted for.

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

Math Chat •

DOK-2 Was there a bike shop where you could not ﬁnd the cost for 4.5 hours? If so, why not? Bikes n’ More and Wheels Are Us had discrete domains and were arithmetic, so you couldn’t ﬁnd an in-between value.

LINEAR FUNCTIONS

Linear Functions Explore 3 – Arithmetic Sequences and Linear Functions DOK-2 When would you use arithmetic sequences? Any time the domain is discrete. • DOK-2 When would you use linear functions? Any time the domain is continuous. • DOK-3 Create some examples of situations where you would use arithmetic sequences instead of linear functions Children/people, speciﬁc times (only days, full hours, etc.), luggage, or tickets • DOK-3 Create some examples of situations where you would use linear functions instead of arithmetic sequences. Anything involving continuous growth or passing of time, money, or distances •

FACILITATION TIP After students answer Question 1a, ask them if it would make sense to graph the scenario as a step function. Have them provide reasoning for their answer.

Post-Explore 1. 2. 3.

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

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

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

LINEAR FUNCTIONS

Home

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LINEAR FUNCTIONS

Linear Functions Explore 4 – Graph Linear Functions ACTIVITY PREPARATION Students will graph linear functions from a slope and point, a table of values, and a verbal description.

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.

Materials

Preparation

Printed • • •

• • •

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

•

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever been on a road trip to the beach?; 2) Which beach was it?; 3) What activities did you enjoy while at the beach? FACILITATION TIP Some students can be overwhelmed with verbal descriptions in math. Encourage students who ﬁnd the cards challenging to understand to underline key phrases for each description to better digest the information. Give them the freedom to underline whatever helps them understand each scenario better.

2. 3. 4.

Read the following scenario to the class: Lashawn and Isaac are planning a road trip during their summer break. They want to visit the beach. Help them analyze the travel information they have gathered to determine the best route. Give a Student Journal to each student. Give a bag of Driving Routes Cards to each group. Explain to students that they will work with their groups to determine what key information is given on the Driving Routes Cards and then use that information to graph the linear function that describes each route in terms of the distance to the destination versus the time traveled. 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.

FACILITATION TIP

DOK-2 To plot a line, what information do you need? I need two points on the line or one point and the slope.

b. DOK-2 If you are given a point on the line and the slope, does it matter where the point is? Does it need to be the y-intercept? No, the point can be anywhere on the line.

For Question 2a of Part I of the Student Journal, it may help if students draw and label lines with different colored pencils. Alternatively, encourage students to alternate between solid and dashed lines by erasing parts of every other line. For solid and dashed lines, they should make sure labels are clearly placed.

c. DOK-2 If you are given two (or more) points on the line, does it matter which points you use to plot the line? No, the two points can be anywhere on the line. d. DOK-2 What does the zero (the x-coordinate of the x-intercept) represent for each line? It is the time it takes to reach the destination. e. DOK-2 What does a steeper slope mean? It means a faster speed (rate of change). 6.

Engage

Explore

Explain

Elaborate

Evaluate

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

Math Chat DOK-2 Given the coordinates for a point on a line, what is the minimum amount of additional information you need to plot that line? I need the slope of that line or the coordinates of another point on the line. • DOK-3 Did plotting these routes on a graph make it easier to visualize or understand the description given? If so, how? It did make it easier to see the behavior of the line over the entire time domain. It was easier to ﬁnd the zeros, which give the duration of the different routes. •

Intervention

Acceleration

FACILITATION TIP After students answer the question, ask them if a change of rate with a fraction value will always have a more shallow slope than a change of rate with a whole number value. They should understand that an improper fraction is larger than one or more whole numbers and therefore can have a steeper slope.

LINEAR FUNCTIONS

Home

Part II 1.

2. 3.

Read the following scenario to the class: Lashawn and Isaac have a budget for their trip, including a ﬁxed amount they can spend on fuel. Help them determine which vehicle they should use for their trip based on gas mileage estimates for each. Students should still have their Student Journals. Explain to students that they will work with their groups to graph linear functions given a table of values, a point and slope, and a verbal description, and then they will analyze the plots to compare the functions and the real-life situations they are modeling. 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 What information can you use to plot Isaac’s vehicle? I can use any two points from the table.

DOK-2 What information are you given for Lashawn’s vehicle that you can use to plot? I can use the initial value ( y-intercept) and the rate of change.

DOK-2 What do the x-intercepts represent? They represent the distance the vehicle can travel before running out of gas.

DOK-2 How do you decide whether the vehicle can make the trip? The line must not intersect the x-axis before 600 miles.

e. DOK-2 Does a steeper downward slope mean better or worse gas mileage? It means worse gas mileage. 5. 6.

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

FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever budgeted for a trip?; 2) What were your expenditures?; 3) If you took the same trip again, would you budget differently? FACILITATION TIP After reading the scenario, ask the class 1) Does anyone know what gas mileage their mom or dad gets with their vehicle?; 2) Did that play a role in which vehicle they chose?; 3) Do they wish they had better gas mileage with their vehicle?

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.

Math Chat DOK-3 What are the advantages of plotting a function? It helps you visualize the behavior of the function (such as the rate of change) over the entire domain and identify speciﬁc points, such as initial values ( y-intercept) or zeros of the function (x-intercepts). • DOK-3 What are the advantages of plotting functions together on the same coordinate grid? It helps us compare their characteristics over the same domain, such as their slopes, initial values, and zeros. •

Post-Explore 1. 2. 3.

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

FACILITATION TIP After they complete the Exit Ticket, have the class use the same coordinate grid to graph the line if Chloe has ridden 12.5 miles after two days and 25 miles after 4 days. Then, ask them if the ﬁrst line they drew or the second line they drew has a steeper slope. 63

LINEAR FUNCTIONS

Linear 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

Identify and Construct Arithmetic Sequences 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

Create Recursive and Explicit 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

Arithmetic Sequences and Linear Functions

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

Graph Linear Functions

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

Show What You Know, Part 5 Graph from an Equation 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

Linear Functions

Can be done independently

LINEAR FUNCTIONS

Home

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

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

LINEAR FUNCTIONS

Linear 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 66

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

LINEAR FUNCTIONS

Home

What does mastery look like?

I can identify situations that can be modeled with an arithmetic sequence.

I can connect arithmetic sequences with linear functions.

I graph a linear function given an equation or scenario.

I can identify key features of the graph of a linear function and connect the features to a context.

I can identify the domain and range of linear functions.

I can determine the domain and range within a given context.

I can change the form of an arithmetic sequence from recursive to explicit and vice versa.

SCOPE 1

Geometry on the Coordinate Plane Scope Introduction SCOPE SUMMARY

Student Expectations

A.GSR.3.1 Solve real-life problems involving slope, parallel lines, perpendicular lines, area, and perimeter.

In this grade level, students will expand their understanding of distance on the coordinate plane as well as their understanding of parallel and perpendicular lines. Students will discover the distance and midpoint formulas through repeated reasoning on the coordinate plane. Next, students will explore shapes on the coordinate plane and use their knowledge of slope and distance to classify them. Finally, students will create parallel and perpendicular lines to fulﬁll requirements set out in different scenarios. Ultimately, students will connect their understanding of geometric shapes and constructions with the algebra work they complete in this course.

VERTICAL ALIGNMENT

A.GSR.3.2 Apply the distance formula, midpoint formula, and slope of line segments to solve real-world problems.

Background Knowledge

Future Expectations

In previous grade levels, students explored, proved, and applied the Pythagorean Theorem. Students created and compared parallel and perpendicular lines on the coordinate plane. They also have multiple years of experience ﬁnding the area and perimeter of various polygons.

In Geometry, students will construct midpoints of segments and create perpendicular bisectors with geometric tools like compasses and straightedges. Students will continue their work classifying shapes on the coordinate plane in Geometry.

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

•

create and compare the equations of two lines that are either parallel to each other, perpendicular to each other, or neither parallel or perpendicular. move to one side of the room or the other based on whether they think the prompt is fact or ﬁction.

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

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: •

relate calculating distance and midpoints to a real-world situation.

•

determine the distance between bases on a baseball ﬁeld.

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.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Discover the Distance Formula In this exploration, students will analyze and use the Pythagorean theorem to help them discover the distance formula. Students will: •

Explore 2

Explore 1

EXPLORE ACTIVITIES

analyze the distances between locations in STEM City.

Shapes on the Coordinate Plane In this exploration, students will analyze shapes on the coordinate plane to explore slopes of parallel lines and slopes of perpendicular lines. Students will:

In this exploration, students will analyze and explore the midpoint formula to ﬁnd the midpoint between two points on a coordinate plane. Students will: •

analyze and ﬁnd the middle point between gems on a treasure map.

•

determine the distances and halfway points or midpoints between the gems.

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.

Discover the Midpoint Formula

Parallel and Perpendicular Lines In this exploration, students will analyze lines on the coordinate plane to create parallel lines and perpendicular lines. Students will:

•

describe slopes of parallel lines as equivalent or the same.

•

write the equation of a line that is parallel or perpendicular to another line.

•

describe slopes of perpendicular lines as opposite signs and reciprocals.

•

calculate the distance between landmarks using the distance formula.

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

GEOMETRY ON THE COORDINATE PLANE

Home

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

GEOMETRY ON THE COORDINATE PLANE

Geometry on the Coordinate Plane 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 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.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.

Materials

Preparation

Printed •

1 Set of Fact or Fiction Prompts (per class)

• •

Print one set of Fact or Fiction Prompts to read aloud to students. Another option is to project the prompts using a digital projector.

GEOMETRY ON THE COORDINATE PLANE

Home

Procedure and Facilitation Points 1.

2. 3. 4. 5.

Prompt 1 is ﬁction.

Prompt 2 is ﬁction.

Prompt 3 is ﬁction.

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

Identifying Misconceptions •

•

Students may struggle to identify the slope from a graph without designated points. Remind students that they should select coordinates that the graphed line passes through. Students may be confused about how perpendicular lines look when they intersect. Remind students that the x- and y-axes meet to form a right angle and are perpendicular to each other. Students may believe that lines they do not see intersect in the window of the graph must be parallel, but they need to check the slope of each line to be sure.

FACILITATION TIP Consider using the same prompts for different graphs and presenting them to the class to see which side of the room students go to. For instance, use Prompt 1 but make the lines perpendicular. FACILITATION TIP State whether a prompt is fact or ﬁction before going to the next prompt. If you use the prompts for additional graphs, avoid making all scenarios fact or ﬁction so the students stay engaged.

GEOMETRY ON THE COORDINATE PLANE

Geometry on the Coordinate Plane Hook – Baseball Field ACTIVITY PREPARATION Students will relate calculating distance and midpoints to a real-world situation.

Materials

Preparation

Printed •

• • •

1 Baseball Field (per class)

Reusable • •

1 Phenomena Video (per class) 1 Projector (per class)

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

PROCEDURE AND FACILITATION POINTS Part I: Pre-Explore 1. FACILITATION TIP Before showing the video or reading the scenario, ask the class 1) Has anyone ever helped set up a court or ﬁeld for a sport?; 2) What kind of court or ﬁeld was it?; 3) What did you do to help set it up?

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: 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. He wonders if the mound, which is 60.5 feet from home plate, is closer to home plate or second base. 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 groundskeeper has to create multiple lines. I wonder what shape the bases make. I wonder what tools the groundskeeper needs to use to complete the job and make sure measurements and lines are perfect. Project Baseball Field. Notes

Explore

Explain

Elaborate

Evaluate

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.

Engage

DOK-1 How will the lines connecting the base paths be similar and how will they be different? Allow students to share all ideas. Answers will vary. The two foul lines are perpendicular. The lines between home and third as well as between first and second should be parallel. The lines between second and third and between first and second should intersect on second base. DOK-2 How could you show that the distance between second and first is the same as from first to home? Find the coordinate point of second base, and construct a right triangle where the hypotenuse is the line connecting first and second. See if a right triangle where the hypotenuse connects home plate to first base is the same.

Acceleration

FACILITATION TIP As you explain, trace over the ﬁrst base line to help ensure students make a distinction between a base line and a base. Make sure to point to and identify the equation “y = x – 20” so students do not misread it as “y = x = 20." FACILITATION TIP As you present this question, trace the base paths. It may help to reword the question as “Each base path is a line. How do the base paths relate to each other?”

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Intervention

Show the Phenomena Video again, and restate the problem. Refer to Baseball Field, 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 find the distance? I would find the point for second base by finding the y-intercept of the line that passes through first base and is perpendicular to the foul line that passes through first. I could then use the distance and midpoint formulas.

DOK-2 Is the pitcher’s mound closer to home plate or second base? The midpoint is more than 60.5 feet from home plate, so the mound is closer to the plate.

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

GEOMETRY ON THE COORDINATE PLANE

Home

FACILITATION TIP Leave the Phenomena Video up during the Post-Explore. At the end of the Post-Explore, ask students to identify examples of objects or parts of objects that seem to be parallel or perpendicular to each other.

GEOMETRY ON THE COORDINATE PLANE

Geometry on the Coordinate Plane Explore 1 – Discover the Distance Formula ACTIVITY PREPARATION Students will analyze and use the Pythagorean theorem to help them discover the distance formula.

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.

Materials Printed • • •

1 Student Journal (per student) 1 STEM City Map (per group) 1 Exit Ticket (per 2 students)

Preparation • • • • • •

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 the STEM City Map for each group of students. For students who need additional organizational support, please see our Four Quadrant Coordinate Grid Supplemental Aids elements in the Intervention section. Go Digital! Have students prepare for the work of calculating distance between any points by exploring and revisiting ﬁnding distance in the coordinate plane along vertical or horizontal lines.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP Before reading the scenario, ask the class 1) What are some key landmarks in your city?; 2) Where are they located?; 3) Do you know who built them?

Part I 1.

FACILITATION TIP Check student understanding by asking them for a word to describe a linear distance that is not vertical or horizontal. They should answer “diagonal."

4. 5. 74

Read the following scenario to the class: Some key landmarks in STEM City have been placed on a coordinate grid model of the city. Use the map to answer the questions about the city that follow. Explain to students that they will work with their groups to analyze the distances between locations in STEM City. 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 can you use the grid to ﬁnd vertical or horizontal distances? I can count the number of grid marks between the two points or locations.

DOK-1 Why is it harder to ﬁnd distances that aren’t vertical or horizontal? Counting grid marks no longer gives an accurate answer.

DOK-1 What shape can we create after drawing the line segment of the distance we are trying to calculate? A right triangle

DOK-2 How can you ﬁnd the shortest distance between two points? A straight line between the points is the shortest distance between the two locations.

Engage

Explore

Explain

Elaborate

Evaluate

Math Chat •

•

DOK-2 How did you ﬁnd the distance between two places that weren’t located on the same vertical or horizontal line? I used the Pythagorean theorem by creating a right triangle with endpoints of the hypotenuse as my starting and ending location. DOK-3 Why is the distance found using the Pythagorean theorem considered the shortest distance between the two locations? The distance found using the Pythagorean theorem is considered the shortest distance as it is a straight line connecting the two locations. Otherwise, the distance between the two places would pass through two other legs of a triangle, which must be larger than the third side by the triangle inequality.

Part II 1.

2. 3. 4. 5.

6. 7.

Read the following scenario to the class: You will now use your knowledge of distance to develop a formula that helps to determine where we could put new landmarks on the map. Students should still have their Student Journals. Give a STEM City Map to each group. Explain to students that they will work with their groups to determine the distance the parents have to drive. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 How do you determine the total length of each parent’s drive? Find the distance of each leg of their journey, and add them together.

DOK-2 How do you determine the two components needed to find side lengths a and b in order to use the Pythagorean theorem to find distances on the coordinate plane? We need to find the horizontal distance and vertical distance separately between the two coordinate points.

DOK-2 How do you determine the horizontal distance between 10 and 3, between 20 and 3, between –20 and 3, and between x1 and 3? You can subtract 3 from the ﬁrst value, 10 – 3 = 7, 20 – 3 = 17, –20 – 3 = –23, x1 – 3.

DOK-2 When we write the distance between two points ((x1, y1) and ( 2, y2), explain why this is a formula to calculate distance between any (x two points. Since each coordinate in both points is a variable, we can substitute any value into each coordinate to ﬁnd the distance no matter what the two endpoints are.

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

Math Chat DOK-3 How are the Pythagorean theorem and distance formula related? The distance formula is derived from the Pythagorean theorem such that the hypotenuse corresponds with the distance calculated. a2 + b2 corresponds with _______ 2 2 2 2 (x2 –____________________ x1) + ( y2 – y1) given the formulas c = √ a + b for Pythagorean theorem and d = √ (x2 − x1)2 + ( y2 − y1)2 for the distance formula. Given their relationship, both methods can be used to calculate the length/distance between two points. • DOK-3 When would it be ideal to use the Pythagorean theorem? When would it be ideal to use the distance formula? When provided with a right triangle and the lengths of two sides, the Pythagorean theorem would be ideal. However, if given coordinates, the distance formula would be best. •

Intervention

Acceleration

FACILITATION TIP Students may be confused initially by the idea that they can use the Pythagorean theorem to ﬁnd the distance between any two points not located on a vertical or horizontal line. To help guide them toward understanding, remind them that the shortest distance between two points is a straight line. Build on this by explaining that a line must be vertical, horizontal, or diagonal. FACILITATION TIP Before reading the scenario, ask the class 1) What do you think deﬁnes a key landmark?; 2) What factors do you think determine where a key landmark is placed?; 3) Do you know anyone personally who helped build a key landmark in your city or elsewhere?

GEOMETRY ON THE COORDINATE PLANE

Home

FACILITATION TIP After students answer questions 2 and 3 in Part II of the Student Journal, ask them if they can think of other instances where businesses and/or activities are grouped together strategically. Examples may include boardwalk shops, ﬁlming studios, or university buildings.

FACILITATION TIP After the Math Chat, ask students if they would have arranged the locations on the STEM City Map differently. What would they change or keep the same? Why?

GEOMETRY ON THE COORDINATE PLANE

Geometry on the Coordinate Plane Explore 1 – Discover the Distance Formula • FACILITATION TIP After students complete the Exit Ticket, check their understanding with this prompt: Given that Point A remains in the same location and the line has the same exact direction, what would be the location of Endpoint B if the line’s distance is doubled? How did you ﬁnd out?

•

FACILITATION TIP Afterward, have groups look at their STEM City maps. Use the origin as your city’s town hall. Write a list of the other locations graphed on the map. With the class, use a mapping system to locate those places in your city in relation to your town hall. Then, have groups plot the locations on their STEM City maps and compare the plot sets. Note: Plots should be reasonable rather than exact.

DOK-2 Can we use the distance formula for horizontal and vertical distances? Yes, the distance formula can be used to ﬁnd the distance between any two points, whether they are vertical, horizontal, or diagonal. DOK-2 Will the distance be the same if we use the formula ____________________ ____________________ d = √ (xx2 − x1)2 + ( y2 − y1)2 instead of d = √ (x1− x2)2 ____________________ + ( y1 − y2)2 ? Explain. The distance will remain the same if we use d = √ (x2 − x1)2 + ( y2− y1)2 , as the negative difference will become positive after taking the square. DOK-2 If a restaurant is located at (5, 14), how will you ﬁnd the distance between the restaurant and bowling alley? I will substitute the ordered pairs in the distance formula using the following steps: ____________________ d = √ (x2 − x1)2 + ( y2 − y1)2 ____________________

d = √ (3 − 5)2 + (–4 − 14)2 ______________

________

____

d = √ (−2)2 + (−18)2 = √ 4 + 324 = √ 328

The distance would be about 18.11 km. Post-Explore 1. 2. 3.

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

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

GEOMETRY ON THE COORDINATE PLANE

Home

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GEOMETRY ON THE COORDINATE PLANE

Geometry on the Coordinate Plane Explore 2 – Discover the Midpoint Formula ACTIVITY PREPARATION Students will analyze and explore the midpoint formula to ﬁnd the midpoint between two points on a coordinate plane.

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.

Materials

Preparation

Printed • • •

• • •

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

• •

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 the Map Cards for each group of students. For students who need additional organizational support, please see our Four Quadrant Coordinate Plane Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) Have you ever been on a scavenger hunt?; 2) What were you looking for?; 3) Did you or someone else ﬁnd what you were looking for? FACILITATION TIP Some students may interpret gems' shapes and colors differently than the Student Journal (e.g. the green triangle and gold heart). Establish the shapes and colors of each gem with the class when you ﬁrst hand out the Map Cards.

2. 3. 4.

FACILITATION TIP

e. DOK-1 Which formula helps to ﬁnd the distance between two points on ____________________ a coordinate plane? d = √ (x2 − x1)2 + ( y2 − y1)2

Watch out for students mixing up the x- and y-coordinates of points throughout the Explore activity. Also, watch out for students using an incorrect operation when ﬁnding the coordinates of the midpoint-speciﬁcally, subtracting where they should add.

DOK-2 How will you ﬁnd the distance between two different gems on the map? I will substitute the coordinates of each gem in the distance formula and solve it to ﬁnd the distance between them.

DOK-2 How do you know if a point is the halfway point or midpoint between two gems? I check the distance between the midpoint and each of the gems to see if it is the same.

h. DOK-2 How can you use the graph and slope to ﬁnd the points that are halfway in between two gems? The slope tells us the rise over run from one gem to the next, and the midpoint is half of that rise and half of that run in the same direction as the original segment.

FACILITATION TIP For Question 10f, watch out for students trying to combine variables and constants. Remind them that variables and constants are not like terms.

Read the following scenario to the class: The crew of a pirate ship has discovered a treasure map. There seems to be an overwhelming amount of riches on the map, but the crew is convinced there are more treasures to be found in between the locations on the map. The crew wants to try digging in between the current map locations along the route where the people who buried the treasure would have walked. Give a Student Journal to each student. Give a set of Map Cards (Treasure Map and Route Map) to each group. Explain to students that they will work with their groups to analyze and ﬁnd the middle point between gems on the treasure map. They will use the Treasure Map and Route Map to determine the distances and halfway points or midpoints between the gems and record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding:

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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

Math Chat DOK-2 Why is it easier to find the halfway point or midpoint between two points that lie on the same vertical or horizontal line? The distance between these points is easy to calculate, and then you can divide by 2 to figure out half and how far you need to travel from one endpoint toward the middle. • DOK-2 How can you quickly find the midpoint between the points (0, 0) and (10, 20)? The midpoint would be (5, 10) because you would travel half of the horizontal and half of the vertical distance starting at (0, 0) when headed toward (10, 20). • DOK-2 How do the coordinates of a midpoint connect to the coordinates of the original two endpoints? The x-coordinates of the midpoint are the average or halfway between the x-coordinates of the endpoints, and the same is true for the y-coordinates. •

Part II 1.

2. 3. 4.

Read the following scenario to the class: Captain Crow of the pirate ship knows that ships A and B are worth plundering and that they have to meet to pass a message to them. His plan is to meet them exactly in the middle of their journey to each other so he can save time and attack both at once. The problem at the moment is that he is unsure of their exact location on the high seas and is waiting to get word from spies on the ships’ exact location. He wants to be ready to calculate the middle point as soon as their whereabouts are known. Students should still have their Student Journals and the Treasure Map and Route Map. Explain to students that they will work with their groups to discover the midpoint formula to ﬁnd the midpoint between ship A and ship B. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 How do you determine the coordinates of ship B? Ship B is x1 units to the right and y1 units up. The coordinates of ship B will be (x1, y1).

DOK-2 How do you determine the coordinates of ship A? Ship A is x2 units to the right and y2 units up. The coordinates of ship A will be (x2, y2)

DOK-2 How do you determine where to draw X? X is the point that is equidistant from both ship A and ship B. I will draw a straight line between the two ships and then mark X in the middle.

DOK-2 Explain two ways to find the middle or halfway point between two points on a horizontal line. You can find the total distance between them and then add half of that distance to the smaller x value or subtract that distance from the larger x value. You could also find the average between the two x values.

e. Explain to students that this halfway or middle point is called the midpoint and that the two starting points are called endpoints. f.

DOK-2 Why is the expression for the midpoint from the two ships located at ((x1, y1) and ((x2, y2) a general formula that can help us ﬁnd the midpoint between any two points? Since x1, x2, y1, and y2 are all variables, we can substitute any values in for them to determine the midpoint of a segment.

DOK-3 How is question 10 different from the previous question posed? Most questions so far have provided two endpoints and asked for the midpoint. Question 10 gives us the midpoint and one endpoint and asks us to determine the other endpoint.

FACILITATION TIP After students answer the question, review linear equations with them. Have them ﬁnd the equation for the line with these points in their groups, and have each group state what method they used to get the equation.

GEOMETRY ON THE COORDINATE PLANE

Home

FACILITATION TIP Before reading the scenario, ask the class 1) What stories have you heard about pirates?; 2) Where and when did the stories take place?; 3) Are the stories true or ﬁctional?

FACILITATION TIP As students answer the question, notice if anyone uses the word equidistant in their answer. After they answer the question, discuss the word and its meaning in context. This may be an introduction or review of vocabulary, depending on the student. FACILITATION TIP Afterward, ask for real-life examples of midpoints. Examples may include the center of a globe, the center of a football ﬁeld, or the center of a bridge. Then, ask them how the midpoints can be used to better understand and utilize the respective spaces around them.

GEOMETRY ON THE COORDINATE PLANE

Geometry on the Coordinate Plane Explore 2 – Discover the Midpoint Formula 5. 6.

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 What do you understand by the term midpoint midpoint? The term midpoint means “the exact middle point along a line segment that is the same distance from each of the endpoints.” y1 + y2 x1 + x2 _____ DOK-1 What is the midpoint formula? (______ , 2 ) 2

•

DOK-2 Why does the average of two numbers tell us the value exactly halfway in

• 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.

a+b

between them? With two numbers a and b, if a < b, then the average is ____ , the 2 a+b

b−a

a+b

b−a

– a = ____ , and the distance from b is b – ____ = ____ distance from a is ____ a . So 2 2 2

the distances between a and the average value and b and the average value are identical. DOK-2 How did we come up with a formula to calculate the midpoint between any two points? We followed the pattern from numerical examples to write an expression for the midpoint using two generic points, (x1, y1) and (x2, y2), that could represent any point on the coordinate plane. • DOK-3 How is the midpoint formula similar to and different from the midpoint formula? Both formulas are based on formulas we have previously learned, the Pythagorean theorem and average or mean, respectively. Both use the points (x1, y1) and (x2, y2). The distance formula subtracts x- and y-coordinates to ﬁnd their distance, and the midpoint formula takes their sum before dividing by two. •

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

GEOMETRY ON THE COORDINATE PLANE

Home

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GEOMETRY ON THE COORDINATE PLANE

Geometry on the Coordinate Plane Explore 3 – Shapes on the Coordinate Plane ACTIVITY PREPARATION Students will analyze shapes on the coordinate plane to explore slopes of parallel lines and slopes of perpendicular lines. Students will describe slopes of parallel lines as equivalent or the same and slopes of perpendicular lines as opposite signs and reciprocals.

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.

Materials

Preparation

Printed • • •

• • •

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

Reusable • • •

•

1 Dry-erase marker (per group) 5 Sheet protectors (per group) 1 Scientiﬁc or graphing calculator (per group)

Separate the class into groups of 2 or 3 students. Print a Student Journal and an Exit Ticket for each student. Print a set of Fence Catalog Cards, on card stock for durability, for each group of students. Place each one inside a sheet protector to create an erasable surface. For students who need additional organizational support, please see our Four Quadrant Coordinate Plane Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) Do you know of anyone who has had a fence built?; 2) What does the fence surround?; 3) What material is the fence made out of? FACILITATION TIP After reading the scenario, ask the class how a graph shows whether or not a shape with diagonal lines is a rectangle and whether or not it is a square. Track their responses. With the class, compare the responses with what they ﬁnd as they complete the Explore activity. FACILITATION TIP After handing out the Fence Catalog Cards, ask the class 1) Would you ever consider running a farm?; 2) What would you have on your farm?; 3) What do you think would need fencing?

Part I 1.

2. 3. 4. 5.

Read the following scenario to the class: Farmer John wants to hire a company to build a fence on his farm. He wants to fence in 400 square feet in area, using about 80 linear feet of fencing, and he prefers a square fenced-in area. Farmer John was given 5 options in a fence catalog. The builder told him that all the fence regions are rectangular, but only 1 is a square. Use mathematics to help farmer John see if each region is rectangular like the builder promised. Give a Student Journal to each student. Give a set of Fence Catalog Cards to each group. Explain to students that they will work with their groups to analyze the fences in the Fence Catalog Cards 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 is slope calculated? Using the slope formula or the change in y divided by the change in x.

DOK-1 How can you determine the slope of a line on a graph without calculating slope? Count the change in the y values and the x values between two points, and simplify the fraction.

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.

DOK-1 What makes slopes perpendicular? The slopes are opposite reciprocals and have a product of −1.

e. DOK-2 How does the slope of the bottom side and that of the top side compare? They are equivalent. f. 82

Engage

Explore

Explain

Elaborate

Evaluate

i. DOK-1 How is the slope of the left side different from the slope of the bottom side for fence 2? How is the slope of the right side different from the slope of the top side? One is a positive number and the other is negative. They are also different numbers but can be expressed using the reciprocal of the other. 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 If two lines 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. • DOK-1 If two lines are perpendicular what must be true about their slopes? The slopes must be opposite reciprocals of each other. •

•

2. 3. 4. 5.

6. 7.

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.

DOK-2 Explain how to determine the slope of a line that is perpendicular to a line 1 with a slope of __5 . The slope is −5. First, determine the reciprocal, which is 5, and then change the sign to have a number opposite the original.

Part II 1.

Acceleration

DOK-1 What is it called when two numbers are multiplicative inverses of 1 one another—for example, __2 is the multiplicative inverse of 2? They are reciprocals.

h. DOK-1 How is the slope of the left side similar to the slope of the bottom side for fence 2? How is the slope of the right side similar to the slope of the top side? They both use the number 4. It is the numerator for one slope and the denominator for the other slope.

6. 7.

Intervention

GEOMETRY ON THE COORDINATE PLANE

Home

Read the following scenario to the class: Now that you have helped farmer John verify which fence regions in the catalog are rectangles, we need to find the fence that meets the criteria of being a square and fits the area and perimeter specifications. Farmer John wants the fenced-in region to be 400 square feet in area, using about 80 linear feet of fencing. Help farmer John decide which fence in the catalog he should choose. Give a dry-erase marker and a calculator to each group of students. Students should still have their Student Journals. Explain to students that they will work with their groups to analyze the length, width, area, and perimeter 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 is the formula for the area of a rectangle? The area formula is A = lw.

DOK-1 What is the formula for the perimeter of a rectangle? The perimeter formula is P = 2l + 2w.

DOK-1 What does a number look like if it is rounded to the nearest tenth? To the nearest whole number? When rounded to the nearest tenth, it has one number after the decimal. When rounded to the nearest whole number, it doesn’t have any numbers after the decimal.

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.

FACILITATION TIP Before reading the scenario, ask the class 1) Have you ever built something before?; 2) What did you build?; 3) What factors determined how you built it?

FACILITATION TIP Connect rounding to farming. After students answer the question, ask them for examples of situations on a farm that may call for rounding. FACILITATION TIP For Question 3 of the Reﬂect section, some students may state that slopes of perpendicular lines are reciprocals rather than negative reciprocals or reciprocals with opposite signs. In such a case, provide an example of two diagonal, perpendicular lines on a coordinate grid for the class to see. Then, work with them to ﬁnd and compare the slopes of the two lines.

GEOMETRY ON THE COORDINATE PLANE

Geometry on the Coordinate Plane Explore 3 – Shapes on the Coordinate Plane Math Chat •

FACILITATION TIP Review quadrilaterals and midpoints with the class. After they complete the Exit Ticket, ask them what shape they would have if all of the points were still connected but Point C was located at the current midpoint of Side CD.

DOK-2 Describe the algebraic process to classify a shape on the coordinate plane. Many shapes can be classified by their angles, presence or lack of parallel sides, and length of sides. To determine a right angle, find the slope of both sides. If the slopes multiply to −1, they are perpendicular, and, therefore, they form a right angle. To determine parallel sides, find the slope of both sides. If the slopes are the same, the sides are parallel. To determine the length of sides, use the distance formula.

Post-Explore 1. 2. 3.

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

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

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

GEOMETRY ON THE COORDINATE PLANE

Home

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GEOMETRY ON THE COORDINATE PLANE

Geometry on the Coordinate Plane Explore 4 – Parallel and Perpendicular Lines ACTIVITY PREPARATION Students will analyze lines on the coordinate plane to create parallel lines and perpendicular lines. Students will be able to write the equation of a line that is parallel or perpendicular to another line given the original line and an ordered pair on the new line. They will also calculate the distance between landmarks using the distance formula.

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.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Town Map Blueprint (per group) 1 Town Map (per group) 1 Exit Ticket (per student)

•

Reusable • • •

•

2 Dry-erase markers of different colors (per group) 1 Sheet protector (per group) 1 Scientiﬁc or graphing calculator (per group)

Separate the class into groups of 2 or 3 students. Print a Student Journal and an Exit Ticket for each student. Print the Town Map Blueprint, on card stock for durability, for each group of students. Print the Town Map, on card stock for durability, for each group of students. Place it inside a sheet protector to create an erasable surface. For students who need additional organizational support, please see our Four Quadrant Coordinate Plane Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) Have you ever met a governor or other government official?; 2) What tasks do you think come with being a governor?; 3) What about being a governor do you think would play to your strengths? FACILITATION TIP After reading the scenario, ask the class 1) Do you know of anyone who has filed a complaint to a government official?; 2) What was their complaint about?; 3) What was the outcome? FACILITATION TIP Some students may mix up the numerator and denominator of a fractional slope, such as those for the answers of part 5h. Review how rise and run apply to the slope in an equation and the slope of a graphed line. 86

Part I 1.

2. 3. 4. 5.

Read the following scenario to the class: The governor wants to improve the infrastructure of Joyville City. The people have been complaining about the streets that have deteriorated due to heavy traﬃc. As mayor of this city, you have been assigned to oversee the design of the city master plan to provide the best infrastructure for the people of the city. Review and analyze the proposed new roads from the city planning committee. Give a Student Journal to each student. Give a Town Map Blueprint to each group. Explain to students that they will work with their groups to analyze the criteria for the proposed new roads 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 is always true about lines with the same slope? They are parallel.

DOK-2 If you were to pick a yy-intercept -intercept different from the yy-intercept -intercept of Main Street, what would you pick? How can you make a line parallel to Main Street and through the yy-intercept -intercept you picked? Answers will vary. I would pick (0, 3). I would start at 3 on the y-axis and use the same slope as Main Street, up 3 and right 4. Then, I could connect the points with a ruler. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 Did everyone draw the three possible streets in the same locations? Why or why not? We did not because the location or y-intercept was not speciﬁed. We were allowed to place the street anywhere as long as it was parallel to Main Street.

DOK-2 What is the ordered pair for the location 4 units east of town center? Explain how you know. East of town center would be to the right on the coordinate grid. I would go 4 units right from the origin. That ordered pair would be (4, 0).

e. DOK-2 What is the ordered pair for the location 9 units south of town center? Explain how you know. South of town center would be down on the coordinate grid. I would go 9 units down from the origin. That ordered pair would be (0, –9). f.

DOK-2 Once the ordered pair has been identiﬁed how can a parallel line be drawn? Start at the ordered pair and use the same slope as Main Street. For example, I would start at (4, 0) and move 3 up and 4 to the right to plot another point at (8, 3). Then, I would use a straightedge to connect them and complete the line.

DOK-1 What is the relationship between slopes of perpendicular lines? They are opposite signs and reciprocals.

h. DOK-2 What is the slope of Main Street? What slope would be 3 perpendicular to Main Street? The slope of Main street is __4, and the 4 perpendicular slope would be −__3.

i. DOK-2 How would we know where to start graphing street B? Is there an ordered pair given? The planning committee doesn’t give an ordered pair, but they do tell us it goes through a point 5 units north of town center. That should be 5 units above the origin at (0, 5). j. DOK-2 How would we know where to start graphing street C? Is there an ordered pair given? The planning committee doesn’t give an ordered pair, but they do tell us it goes through a point 6 units west and 1 unit north of town center. That should be 6 units left of the origin and 1 unit up at (−6, 1).

6. 7.

Intervention

Acceleration

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.

GEOMETRY ON THE COORDINATE PLANE

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FACILITATION TIP After students answer the question, explain to them that many cities have a Main Street. Then, ask them why they think a planning committee names a street "Main Street." Examples may include that it's the longest street, that it's the widest street, or that it goes past the most prominent buildings in town.

DOK-1 In slope intercept form, what variable represents slope and what variable represents the yy-intercept? -intercept? Are both the slope and y-intercept y given for street B? The m represents slope, and the b represents the y-intercept. These are both easily determined because we know the 4 slope is −__3. Since we used that to graph the line and (0, 5) was the given ordered pair, it is the y-intercept.

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 If two lines 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. • DOK-1 If two lines are perpendicular, what must be true about their slopes? The slopes must be opposite signs and reciprocals of each other. •

FACILITATION TIP After the Math Chat, present the following scenario to the class: You get to decide where a new street will go in your city. Where would you put it? What locations would it connect? Why would you choose this route? FACILITATION TIP Some students may state that slopes of perpendicular lines are reciprocals rather than negative reciprocals or reciprocals with opposite signs. In such a case, provide an example of two diagonal, perpendicular lines on a coordinate grid for the class to see. Then, work with them to ﬁnd and compare the slopes of the two lines. 87

GEOMETRY ON THE COORDINATE PLANE

Geometry on the Coordinate Plane Explore 4 – Parallel and Perpendicular Lines Part II 1. FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever been in a government building?; 2) Which building was it?; 3) What were you there for? FACILITATION TIP After reading the scenario, ask the class 1) What do you know about city infrastructure?; 2) Do you know of anyone who works with city infrastructure?; 3) What factors do you think are most important in determining how and when infrastructure is built or ﬁxed?

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.

2. 3. 4.

5. 6.

Read the following scenario to the class: The city council has asked the planning committee if the new roads will increase the distance between some of the most used town buildings. Help the planning committee determine whether the new roads will increase the distance, whether the distance will be about the same, or whether the distance will be shorter. Students should still have their Student Journals. Distribute a Town Map, 2 dry-erase markers of different colors, and a calculator to each group. Explain to students that they will work with their groups to analyze the distance between different locations and compare existing streets to proposed new streets using the information provided. Students will then work together to analyze distances using the distance formula 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 determine the distance between two points? I can use the distance formula.

DOK-2 Use one color dry-erase marker to mark the ordered pairs on street A and connect them. Using visual inspection, estimate the distance and explain your process. Answers will vary. I can see it makes a triangle with legs 7 and 7, so the hypotenuse should be greater than 7. I would estimate it is about 9 units.

DOK-2 Use another color dry-erase marker to mark the ordered pairs on street B and connect them. Using visual inspection, estimate the distance and explain your process. Answers will vary. I can see it makes a triangle with legs 4.5 and 6, so the hypotenuse should be greater than 6. I would estimate it is about 7 units.

DOK-2 Use one color dry-erase marker to mark the ordered pairs on the existing roads route from the public library to the town hall. Then, connect the points to show the whole route. Using another color dry-erase marker, mark the ordered pairs on the proposed new road route. Then, connect the points to show the whole route. Using visual inspection, which do you think is shorter and why? Answers will vary. The route using existing roads has one short segment and a really long segment. The route using proposed new roads has two moderate segments. I think the route using proposed new roads will be shorter.

e. DOK-1 Will the distance formula need to be used once or twice to ﬁnd the distance from the public library to the town hall? Explain. It will be used twice because I’ll need to ﬁnd the distance from the library to the intersection and then from the intersection to the town hall. f.

DOK-2 Use one color dry-erase marker to mark the ordered pairs on Main Street and connect them. Then, mark the ordered pair on street A and connect them. Using visual inspection, estimate the distance and explain your process. Answers will vary. The distance from the library to the town center and then to the hospital looks about equivalent. I would estimate about 10 units each since the legs of each triangle are 6 or 8 and the hypotenuse must be greater. The total distance is about 20.

DOK-2 Use one color dry-erase marker to mark the ordered pairs on street 2 and connect them. Then, mark the ordered pair on street B and connect them. Using visual inspection, estimate the distance and explain your process. Answers will vary. The distance traveled on street B looks about twice as long as the distance traveled on street 2. The legs on the smaller triangle are about 5 and 3, so it would be about 6 units. The total distance is about 18 units.

Engage

Explore

Explain

Elaborate

Evaluate

h. DOK-2 When would it be acceptable to estimate the coordinate where two streets intersect? When would the exact value be needed? Answers will vary. If the distance using the existing streets is close to the same as using the proposed new roads, we would want exact values. However, if the distances are not similar, the estimate would be acceptable because it wouldn’t really matter if they were 5 units more or 5.2 units more. i. DOK-1 When might you need to use the distance formula three times? I might need to use the distance formula three times if I make two turns to travel on three different streets. j. DOK-2 What new streets would you take to get from the park to the library? What are the coordinates you would use for the distance formula? Answers will vary. I would take street A to Main St. The coordinates for street A would be (9, −9) and (0, 0). The coordinates for Main St. would be (0, 0) and (8, 6). k.

7. 8.

DOK-2 What existing streets would you take to get from the park to the library? What are the coordinates you would use for the distance formula? Answers will vary. I would take street 1 to street B to street 2. The coordinates for street 1 would be (4, −6) and (6.7, −4). The coordinates for street B would be (6.2, −3.3) and (3.8, −0.1). The coordinates for street 2 would be (3.8, −0.1) and (8, 3).

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 can the distance on a coordinate grid be calculated? The distance can be calculated using the ordered pairs and the distance formula. • DOK-1 If two lines are parallel, what must be true about their slopes? The slopes must be equal. • DOK-1 If two lines are perpendicular, what must be true about their slopes? The slopes must be opposite signs and reciprocals of each other. •

Post-Explore 1. 2. 3. 4.

Intervention

Acceleration

FACILITATION TIP Changes in ground elevation can affect the distance between two points and/or the route chosen between two points. After students answer the question, ask them for examples of physical factors that could affect infrastructure planning. Examples would include mountains, valleys, and hills. Ask them how each example would affect their design of infrastructure.

GEOMETRY ON THE COORDINATE PLANE

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FACILITATION TIP Some students may choose a different route than others. If so, have different students explain to the class why they chose different routes. Discuss any mathematically incorrect part of their explanations with the class to help guide their reasoning. FACILITATION TIP For Question 2 in the Reﬂect section, some students may provide a response using slope-intercept form or point-slope form. In any case, be sure each student has a clear understanding of the process they are describing and that they do not mix up forms of a line. FACILITATION TIP After students answer the question, open a discussion by asking them why it would be helpful to use the distance formula to measure a line's distance rather than just a ruler. Responses may include that the measurement is more exact and that the measurement is not bound by a speciﬁc unit. FACILITATION TIP Students may mix up signs throughout the Exit Ticket. In addition, students may misinterpret the directions for Question 4 given the length of the sentence. Encourage students to read each question carefully, pay close attention to signs, and ask questions if they are unclear with directions.

GEOMETRY ON THE COORDINATE PLANE

Geometry on the Coordinate Plane 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

Discover the Distance Formula 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

Discover the Midpoint Formula 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

Shapes on the Coordinate Plane

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

Parallel and Perpendicular Lines

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

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

Notes

GEOMETRY ON THE COORDINATE PLANE

Home

Intervention and Assessment SMALL-GROUP PLANNER 1

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

GEOMETRY ON THE COORDINATE PLANE

Geometry on the Coordinate Plane

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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?

GEOMETRY ON THE COORDINATE PLANE

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I can calculate the perimeter and area of certain polygons.

I can ﬁnd distances on the coordinate plane.

I can use slope, the distance formula, and the midpoint formula to solve real-life problems.

SCOPE 1

Linear Inequalities Scope Introduction SCOPE SUMMARY Students should be able to identify viable and nonviable solutions algebraically, graphically, and given a context. They have experience working with contexts in which their solutions are both viable and nonviable. Students also have experience explaining their reasoning verbally and in writing. They are able to determine the linear inequality from a graph. Students are also able to graph linear inequalities given a symbolic or contextual representation. Student Expectations

A.PAR.4.1 Create and solve linear inequalities in two variables to represent relationships between quantities including mathematically applicable situations; graph inequalities on coordinate axes with labels and scales. A.PAR.4.2 Represent constraints of linear inequalities and interpret data points as possible or not possible.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grade levels, students solve word problems that lead to one-variable equations and inequalities. They solve equations and inequalities that can be solved using multiple steps. Students interpret solutions in context and graph solutions on a number line. Previously in Algebra I, students graph linear equations on the coordinate plane.

Later in Algebra I, students will graph the solution set to systems of inequalities.

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

use variables to represent quantities.

•

construct simple equations and inequalities.

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: •

relate linear inequalities to a realworld 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.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Solutions of Linear Inequalities In this exploration, groups of students will determine the amount of jewelry that can be made with an initial investment of $500 for a student saving money to purchase a car. Students will: •

use substitution to determine if an ordered pair is a solution.

•

model ordered pairs on a coordinate grid.

•

formalize meaning of a shaded half-plane.

Explore 2

Explore 1

EXPLORE ACTIVITIES

In this exploration, groups of students will determine the amount of time to make jewelry. Students will: •

use a linear inequality to graph.

•

compare inequalities written in different ways.

•

interpret solutions.

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

Graphing Linear Inequalities

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Writing Linear Inequalities In this exploration, students will determine the amount of each type of jewelry to be made to reach the goal of $1200 with an argument supporting their recommendation. Groups will then write an equation to automate a digging machine. Students will: •

use a graph of a linear inequality to determine the related equation and inequality.

•

interpret solutions.

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

LINEAR INEQUALITIES

Linear Inequalities 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 talk 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. 8.PAR.3.3 Create and solve linear equations and inequalities in one variable within a relevant application.

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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 Printed •

1 Four Corners Slides (per class)

Preparation • •

Print one Four Corners Slides. Hang the slides in four separate areas of the classroom, easily visible to all.

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

Ask the students to look at the four corners and think about which corner image best explains linear inequalities. Allow 2 minutes of thinking time. Ask students to move to the corner image they chose. Ask students within each group to discuss why they chose the image. Allow 2–5 minutes of discussion at the corner images. After students have discussed why they chose their answers, talk about the answers with the class and allow students to explain their representations of the problem. 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.

FACILITATION TIP Before this, ask students what they remember about one-variable inequalities. Ask them what they think will be similar or different with linear inequalities. FACILITATION TIP After they explain, ask students if they can think of a different scenario that explains linear inequalities. Encourage them to come up with a scenario that applies to their current life.

Identifying Misconceptions • •

Slide 4 represents a correct inequality/solution. Students may struggle to remember that when using the terms less than or equal to or greater than or equal to, to, the solution must be included in the set.

LINEAR INEQUALITIES

Linear Inequalities Hook – Concert Tickets ACTIVITY PREPARATION Students will relate linear inequalities to a real-world situation.

Materials

Preparation

Printed •

• • •

1 Concert Tickets (per class)

Reusable • •

1 Phenomena Video (per class) 1 Projector (per class)

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

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

Before showing the video and reading the scenario, ask the class 1) Would you charge different prices for tickets to a concert?; 2) What factor(s) would play a role in your decision?

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: you A concert venue sells two types of general admission tickets. There are $25 floor tickets and $15 balcony tickets. The venue aims to make at least $18,000 for each concert they hold. 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 floor seats cost more money. I wonder how many people the venue can hold. I wonder if there are additional VIP or merchandise packages that the venue uses to make more money. I wonder if there are more floor or balcony seats. Project Concert Tickets. Notes

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Explain to students that the concert venue wants to know how many of each type of ticket they could sell in order to reach their revenue goal. Discuss the following questions: a.

DOK-1 Why would this graph have shading on it? Where else have you seen shading in math? Allow students to share all ideas. Student answers will vary. We shaded number lines for single-variable inequalities. The venue wants to make at least $18,000, so it could earn exactly $18,000 or more and still meet its revenue goal.

DOK-1 Can you tell what the values on each axis represent? Allow students to share all ideas. Student answers will vary. The x-axis is the number of ﬂoor seat tickets sold, and the y-axis is the number of balcony tickets sold.

Explain to students that throughout the scope they will make connections between previous topics of linear equations and solving inequalities with one variable and the current topic of linear inequalities. Complete the Explore activities.

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FACILITATION TIP After students answer the question, ask them how they know what the values on each axis represent given the information they have so far. Avoid telling them if their answers are correct so you can present this question again in the Post-Explore.

Part II: Post-Explore 1. 2.

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

DOK-1 Does this graph make more sense after the Explore activities? Yes, the graph is of a linear inequality, and the solid line and everywhere on the graph that is shaded are the solutions.

DOK-1 What strategies would you use to create an inequality to match this graph? I would determine the equation of the boundary line and then test the point (0, 0) to make sure the inequality symbol was positioned to make (0, 0) a nonsolution.

DOK-1 Is it easier to create a linear inequality in slope-intercept or standard form? Standard form is easier because we have a given quantity that we are trying to reach in $18,000.

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

FACILITATION TIP Take the opportunity to further connect the scope with real life. After they answer the question, ask students if they can use this kind of graph in their current life outside of the classroom. If so, how?

LINEAR INEQUALITIES

Linear Inequalities Explore 1 – Solutions of Linear Inequalities ACTIVITY PREPARATION Students will use substitution to determine if an ordered pair is a solution. Students will model these ordered pairs on a coordinate grid and formalize the meaning of the shaded half-plane.

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.

Materials

Preparation

Printed • • •

• • • •

1 Student Journal (per student) 1 Class Data Grid (per class) 1 Exit Ticket (per student)

Reusable • •

1 Marker (per group) 1 Projector or document camera (per class)

•

Consumable •

Separate the class into groups of 3 or 4 students. Print a Student Journal and an Exit Ticket for each student. Print a Class Data Grid for the class. Prepare to project the Class Data Grid on a surface that students can write on. If projecting the grid is not an option, you can create a grid on a sheet of gridded chart paper. For students who need more support in recalling information, please see our Linear Inequalities Supplemental Aid in the intervention section.

1 Sheet of gridded chart paper (optional, per class)

PROCEDURE AND FACILITATION POINTS FACILITATION TIP

Before reading the scenario, ask the class 1) Have you ever run a business before?; 2) What strategies do you think would be useful? After reading the scenario, ask the class 3) What are other real-world scenarios where a businessperson would have to consider more than one start-up cost?

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.

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2. 3.

Read the following scenario to the class: Jahzara will be starting driver’s education class soon, and this has gotten her thinking about saving money for a car. She started making jewelry a few years ago when she couldn’t find the right color pattern to match her favorite outfit. She has decided to turn this hobby into a business and make jewelry to sell at the local market every week this summer. Of course, there are start-up costs, and Jahzara has $500 to put toward starting this business. Materials for each bracelet cost $2, and materials for each necklace cost $3. Jahzara plans to sell each bracelet for $8 and each necklace for $15. She is confident she will sell all of her creations because there is a need in the community for unique jewelry. How many of each type of jewelry should she make? What recommendations would you give to Jahzara, and what math-based argument would you use to convince her your recommendation is reasonable? Give a Student Journal to each student. Explain to students that they will work with their groups to generate a math-based argument to make a recommendation for Jahzara and will record their work on their Student Journals. Continue explaining that, as students are working, you will ask them to put some data on the Class Data Grid posted on the board. Refrain from giving rules such as “shade below the line for less than.” Encourage students to use test points. Allow students to develop this rule on their own if they can explain why it works and in what situations it won’t work (when the coefficient of y is negative). As students collaborate, monitor their work and use the following activity and guiding questions to assess student understanding: a.

As you move around the room, look for students who have ﬁnished question 4 on their Student Journals. Invite students to place at least one ✗ and one ● on the Class Data Grid. Hand the marker to the student when you invite them to place their data on the Class Data Grid. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How do you know if an ordered pair represents a scenario that is over budget, under budget, or meets budget? You have to use the x value to make the calculations for bracelet materials and the y value to make the calculations for necklace materials. Then, you add those together to see if it is greater than, less than, or equal to $500.

DOK-1 How did you decide which inequality symbol to use to represent Jahzara’s start-up costs? When there is a budget, we have to stay under or equal to that value, so I knew the inequality should open toward the number that represents the budget.

DOK-2 If we plot a point on the coordinate plane, how can we tell if it is or is not a solution? It appears they are clustered in groups. If I picked a point at random and it was near an ✗, it is probably also a point that represents a scenario where it would be too much for Jahzara’s startup costs. Likewise, if I pick a point at random and it was near a ●, it is probably also a point that represents a scenario where it would be just right or not too much for her budget.

e. DOK-2 What do the two sets of points represent, ✗s vs. ●s? The ✗s represent non-solutions, and the ●s represent solutions. f.

DOK-2 How can we graph an equation in standard form? Find the x- and y-intercepts, and connect them. The boundary line should go through all points that equal exactly $500. DOK-2 How can we tell where the solution region begins and the non-solution region ends? There is a boundary, like when we solved one-variable inequalities. In one-variable inequalities, it is the number represented by the related equation. In linear inequalities, it is the line represented by the related equation.

Intervention

Acceleration

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.

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FACILITATION TIP Test students' understanding of x- and y-intercepts in the context of equations. Before asking this question, ask them how they would ﬁnd the x- and y-intercepts for 2x + 3y = 500 without a graph and without isolating y.

h. DOK-3 In question 7, it says to shade the half-plane. What is a halfplane? The half-plane is when a line separates the coordinate grid into two regions. Each region is a half-plane. i. DOK-3 How is shading a half-plane like shading a number line when solving one-variable inequalities? When solving inequalities, the solutions are shaded. The numbers and ordered pairs that are shaded are solutions to the inequality. 6. 7.

Allow students enough time to complete questions 1–11 on their Student Journals. The graph below shows the solution set for the number of bracelets and necklaces Jahzara can create based on her start-up costs.

FACILITATION TIP For Question 9, watch out for students saying x and y are greater than or equal to their respective coeﬃcients in the equation 2x + 3y = 500. Also, watch out for students saying x and y are greater than or equal to 1. Students should understand that it is possible, though unlikely, for Jahzara to buy no materials for bracelets and/or necklaces.

STEMscopes Tip

After the Explore activity activity,, invite the class to a Math Chat to share their observations and learning. Students should not complete the reﬂection questions until after the Math Chat.

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. 101

LINEAR INEQUALITIES

Linear Inequalities Explore 1 – Solutions of Linear Inequalities Math Chat •

•

FACILITATION TIP Test students' understanding of how to ﬁnd equations of lines. After students answer the question, ask them how they would ﬁnd the equation of a line with x and y on the same side when given only the x- and y-intercepts.

•

• •

FACILITATION TIP

•

Students may have never referred to the y- and x-intercepts as the vertical and horizontal intercepts in the past. Clarify that the vertical and horizontal intercepts are the same as the y- and x-intercepts, respectively.

•

FACILITATION TIP Answers will vary. Students should provide clear reasoning for their answers. It may help to ask the class, "For instance, would the option at the origin be good and desirable? Why or why not?" after they have looked at this question.

FACILITATION TIP

10.

Different groups can have different recommendations, so long as the math is accurate and reasonable. Students within a group may differ in opinion just as different groups might. Use your discretion as to whether or not students within a group should have the same recommendation.

11. 12.

DOK-3 When solving one-variable inequalities, we used a related equation to find the boundary point. What do you think the related equation of a two-variable inequality represents? The related equation is used to find the boundary. For linear inequalities, the boundary is a line, while for one-variable inequalities, the boundary is a point/number. DOK-2 Once you knew where the boundary line was, how did you decide which side of the line represented the solution region? This situation was about Jahzara’s budget, so it had to be the values that met budget or were under budget. DOK-2 In this situation, what does a point on the line mean? A combination of bracelets and necklaces that Jahzara could have materials for if she spent her entire budget DOK-2 What does the vertical intercept of the graph mean? The number of necklaces she can buy material for if she doesn’t buy any material for bracelets DOK-2 What does the horizontal intercept of the graph tell us? The number of bracelets she can buy material for if she doesn’t buy any material for necklaces DOK-3 Both (10, 160) and (235, 10) are in the solution region. Both points mean a total of $500 spent on materials. Does it make a difference which option is chosen? It does not make a difference in terms of the total spent on start-up costs. It could make a difference in terms of how much she makes when she sells them or how long it takes her to make them. DOK-3 All of the points in the shaded region represent numbers of bracelets and necklaces that are within Jhazara’s budget. Are all of these options equally good and desirable? Probably not; for example, pairs such as (7, 4) and (5, 2) are probably not desirable because they mean less product to sell and, therefore, less money to be made. The point (0, 0) is also in the solution region, but buying no materials is probably not desirable because it would not help Jahzara reach her monetary goals. Explain to students that they will work with their groups to complete the reflection questions and make a math-based recommendation for Jahzara. Point out to the class that when making a math-based recommendation, they must use math to back up their suggestion. Students will then work together to complete the reflection 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.

Students have several values to account for in the table for Question 1. Encourage students to take their time so they can keep x- and y-coordinates straight.

DOK-2 How did you know your ordered pairs were solutions? Solutions to this situation are the values that are just right or not too much. We used ●s to represent those points. These are ordered pairs from the shaded half-plane.

DOK-2 How did you know your ordered pairs were not solutions? Ordered pairs that were not solutions to this situation were the values that were too much. We used ✗s to represent those points. These were ordered pairs from the unshaded half-plane.

FACILITATION TIP

DOK-3 How did you know what math to use to support your recommendation? Answers will vary based on recommendation and math used.

FACILITATION TIP

After students answer Question 3 of the Reﬂect section, ask them if they can think of an instance where decimal values would be reasonable solutions. If they are stumped, present examples such as money and volume.

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13. 14.

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

Engage

Explore

Explain

Elaborate

Evaluate

Math Chat •

•

• •

DOK-2 When solving one-variable inequalities, the number line is shaded to show a graphical representation of the solution set. How is that similar to shading the half-plane when solving linear inequalities? When solving inequalities, the solutions are shaded. The numbers and ordered pairs that are shaded are solutions to the inequality. This is true for both linear inequalities and onevariable inequalities. DOK-2 Is it easier to determine if an ordered pair is a solution to a linear inequality when you are given a graph or given a situation described in words? Why? It is easier to determine solutions given a graph because it is quicker to find a coordinate pair than to work out the multiplication or other operations. DOK-2 When solving one-variable inequalities, we used a related equation to find the boundary point. What do you think the related equation of a two-variable inequality represents? It represents the boundary. In this case, the boundary is a line not a point. DOK-1 How do we know where the boundary line would be for any graph of a linear inequality? It is the graph of a related equation. DOK-1 In general, for any linear inequality, how could we find out which side of the boundary line contains the solutions? We could test one or more pairs of values on each side and see if—when substituted for the variables—they make the inequality true.

Intervention

Acceleration

FACILITATION TIP Students may also note that, conversely, unshaded values for both cases are not solutions to their respective inequality. After students answer the question, ask them how shading is different between the two types of inequalities.

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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.

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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Linear Inequalities Explore 2 – Graphing Linear Inequalities ACTIVITY PREPARATION Students will use a linear inequality to graph and interpret solutions in context. Students will compare inequalities written in different ways.

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.

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

Before reading the scenario, ask the class 1) When did you last have to deal with time constraints outside of school?; 2) Did you use any tools to help you assess or manage your time?; 3) How effective were the tools? Why? FACILITATION TIP Before they begin, show the class the graph of a solid line and the graph of a dashed line. Explain to them that, for inequalities, the solid line is inclusive, while the dashed line is exclusive. Then, ask the class which inequality signs could apply to each line, depending on which half-plane is shaded.

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.

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2. 3.

Read the following scenario to the class: Jahzara has been considering the startup costs for her jewelry business, but she realizes she also has a limited amount of time to make the jewelry. Jahzara uses an app on her phone to help manage her schedule. She inputs data about how long it takes to make one bracelet and one necklace. Her app gives her an inequality to represent all of the available time she has over the next 9 weeks to make jewelry. Help Jahzara ﬁgure out what this inequality means, and determine reasonable amounts of time she can spend making each type of jewelry. Give a Student Journal to each student. Explain to students that they will work with their groups to help Jahzara interpret the inequality from the app, help her interpret the solutions in context, and record their work on their Student Journals. Refrain from giving rules such as “shade below the line for less than.” Encourage students to use test points. Allow students to develop this rule on their own if they can explain why it works and in what situations it won’t work (when the coeﬃcient of y is negative). As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How could we convert the units from hours to minutes? The number of hours would need to be multiplied by the number of minutes per hour, which is 60.

DOK-2 How do you know which coeﬃcient represents the unit rate for making bracelets? The variable x represents the number of bracelets; therefore, the coeﬃcient of x would represent the unit rate for making bracelets.

DOK-2 How do you know which coeﬃcient represents the unit rate for making necklaces? The variable y represents the number of necklaces; therefore, the coeﬃcient of y would represent the unit rate for making necklaces.

DOK-2 Do you think it would be easier to graph Jahzara’s inequality or her aunt’s inequality? Why? Answers will vary. I think it would be easier to graph her aunt’s inequality because it only takes one step to isolate the y variable, and then it is ready to graph. When equations are in slopeintercept form, they are easier for me to graph. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

e. DOK-3 How do you know if negative numbers and/or decimals are reasonable solutions? This has to be analyzed in terms of the situation. Some situations will make sense with negative numbers, while others won’t. It is the same with decimals. For example, decimals make sense when talking about time or distance but do not make sense when talking about ticket sales or jewelry made. 6. 7.

Allow students enough time to complete the Explore activity and answer the questions that follow. After the Explore activity activity,, invite the class to a Math Chat to share their observations and learning. Students should not have completed the reﬂection questions yet.

Math Chat DOK-1 How do we know where the boundary line would be for each graph? It is the graph of a related equation. • DOK-1 For any inequality, how do we ﬁnd out which side of the line contains the solutions? We test one or more pairs of values on each side and see if—when substituted for the variables—they make the inequality true. • DOK-2 For any situation, how do we know which solutions are reasonable? One thing to always consider is if decimals make sense or if the solution should be limited to only whole numbers or only integers. The situation also has to be understood. For example, in Jahzara’s situation, making a small number of jewelry pieces ﬁts the solution set, but it wouldn’t make sense because she is trying to make money, and she wouldn’t make much money just making a few pieces of jewelry. •

8. 9.

10. 11.

Students will then work together to complete the reﬂection questions, make a recommendation for Jahzara, 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 you decide which options were good and desirable? Answers will vary. I decided which options were good and desirable by considering the money she would make from selling those options since the whole reason she’s making and selling jewelry is to make money.

DOK-2 Do all the points in the shaded region meet your criteria for good and desirable? Why or why not? Answers will vary. No, they are not all equally good and desirable. Jahzara is trying to make money, so an ordered pair like (4, 9) would not be as good as (55, 80) because making only 13 pieces of jewelry would not yield much money after she sells them.

DOK-2 What factors should Jahzara consider when deciding how many bracelets and necklaces to make? The start-up cost for each piece of jewelry, the amount for which each type of jewelry is sold, the time it takes to make the jewelry

DOK-2 Given the factors you suggested for consideration, what math could Jahzara do using these factors to help her make a decision? Answers will vary. For a given point in the shaded region, she could multiply the number of possible jewelry pieces by the amount it takes to make them and then add these values together for the total startup cost. Then, she could multiply each type of jewelry by the amount she sells them for and determine the total amount in sales and then subtract the total start-up cost from total sales to see if she will make a substantial proﬁt.

Intervention

Acceleration

FACILITATION TIP After students answer the question, discuss with them instances where negative values would be reasonable solutions. If they are stumped, present examples such as position and temperature.

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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.

FACILITATION TIP Different groups can have different recommendations, so long as the math is accurate and reasonable. Students within a group may differ in opinion just as different groups might. Use your discretion as to whether or not students within a group should have the same recommendation.

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.

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

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Linear Inequalities Explore 2 – Graphing Linear Inequalities Math Chat DOK-2 How is solving a linear inequality different from solving a linear equation? To solve a linear inequality, we need to use a test point to determine which halfplane contains the solutions and then shade the solutions. The beginning, where we graph the related equation, is exactly the same as solving a linear equation. • DOK-2 Explain why a point on the boundary line should not be chosen as a test point. To determine which half-plane contains solutions, we need to test a point in the half-plane not on the boundary line. • DOK-2 How can we create the graph of a linear inequality when it is given in an unfamiliar form (2 (2yy < 5x + 8 for example)? We would need to put the related equation in standard form, use the intercepts to graph or put the related equation in slope-intercept form, and use the slope and y-intercept to graph. To determine where to shade, we could substitute points into the original inequality or our equivalent form. •

FACILITATION TIP Take this opportunity to further illustrate dashed lines in the context of inequalities. After students answer the question, put the equation in standard form or slope-intercept form with their help. Then, show students a coordinate plane and have them gather the key features they need to tell you how to graph the inequality. FACILITATION TIP Continue to build on students' understanding of accounting for multiple factors to graph inequalities. After they complete the Exit Ticket, ask if it is possible to account for the square footage of the yard on the graph for Question 2 and, if so, how.

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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Linear Inequalities Explore 3 – Writing Linear Inequalities ACTIVITY PREPARATION Students will use the graph of a linear inequality to determine the related equation and inequality. Students will use the representations to interpret the solutions in context.

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.

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 Before reading the scenario, ask the class 1) Has anyone ever used tools to track sales before?; 2) What tools did you use?; 3) How effective were they? Why?

Part I 1.

2. 3. 4.

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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Read the following scenario to the class: Jahzara has decided to use a computer program to track her sales. She has entered her product information into the program, and it has produced a graph. She wants to use the graph to help her best decide how many necklaces and bracelets to make. Her goal is to earn an income of more than $1,200. How many of each type of jewelry do you think she should make? Help her interpret the graph, and give a math-based argument to convince her that your recommendation is reasonable. Give a Student Journal to each student. Explain to students that they will work with their groups to interpret the graph and record their work on their Student Journals. Refrain from giving rules such as “shade below the line for less than.” Encourage students to use test points. Allow students to develop this rule on their own if they can explain why it works and in what situations it won’t work (when the coeﬃcient of y is negative). As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How do you determine whether to use an open or closed circle when graphing inequalities on a number line? Inclusive inequalities (greater than or equal to or less than or equal to) tell me to use a closed circle because the value is included in the solution set. Exclusive inequalities (greater than or less than) tell me to use an open circle because the value is excluded from the solution set.

DOK-2 How could this translate into graphing inequalities on a coordinate plane? When graphing on a coordinate plane, the value is replaced by a line, so the solid line would represent inclusive inequalities. For exclusive inequalities, the line would need to have breaks in it.

DOK-2 What do you notice about the line? How is it different from previous graphs? What do you think this means? The line is dashed on this graph, and on other graphs it was solid. I think this means we will use a different inequality. The line is open like an open circle, so we should use > or < inequality symbols. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

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-2 How would you decide whether the boundary line should be solid or dashed? If the situation includes the points on the boundary line, I would use a solid line. If the situation does not include the points on the boundary line, I would use a dashed line. • DOK-2 In this situation, what does a point on the line mean? It means a combination of bracelets and necklaces sold to earn an income of exactly $1,200. This is almost enough to meet her goal. • DOK-2 In this situation, what does a point in the shaded region mean? It means a combination of bracelets and necklaces sold to earn more than $1,200. These are ordered pairs that represent possible combinations that would meet her goal. • DOK-2 In this situation, what does a point in the unshaded region mean? It means a combination of bracelets and necklaces sold to earn less than $1,200. These are ordered pairs that represent possible combinations that would not meet her goal. •

Part II 1.

Acceleration

DOK-2 Why are the points on this line called “almost enough,” but in other situations they were called “just enough”? The points on this line are not solutions. Her goal is to make more than $1,200. The points on the line represent combinations that would earn an income of exactly $1,200, which is almost enough to meet her goal.

e. DOK-2 How could you use the intercepts to determine coeﬃcients of the equations? Substitute in the x- and y-intercepts to the equation Ax + By = 1,200 to determine the coeﬃcients. 6.

Intervention

Read the following scenario to the class: In order to make the most well-informed income goal, Jahzara looks up the income graphs of other jewelry sellers. You will evaluate two new graphs before making a recommendation to Jahzara on the number of bracelets and necklaces she should make to meet her income goal. Be sure to support your recommendation with mathematical evidence from your Student Journal. Explain to students that they will work with their groups to write an inequality, determine reasonable solutions, reﬂect, and make a recommendation to Jahzara using their Student Journals. Refrain from giving rules such as “shade below the line for less than.” Encourage students to use test points. Allow students to develop this rule on their own if they can explain why it works and in what situations where it won’t work (when the coeﬃcient of y is negative). As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-3 Since seller A’s graph has a larger x-intercept than seller B’s, does that mean that seller A charges more or less for bracelets? Seller A charges less for a bracelet because a larger x-intercept means that this seller needs to sell more bracelets to reach $1,200.

DOK-1 What differences do you notice about each graph? The graphs have different intercepts, and the boundary line of one is solid and the other is dashed.

DOK-2 What does the dashed line tell you about seller B? Seller B wants to make more than $1,200, just like Jahzara.

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FACILITATION TIP After they complete Part I, present to the class that Jahzara wants to graph possible combinations if her income goal changes to more than $1,400. Ask them how that would change the original graph and whether the new graph would have more, fewer, or the same amount of solutions.

FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever encountered others in their business, either up close or from afar?; 2) Did it affect your sales?; 3) Did it affect how you ran your business? FACILITATION TIP Different groups can have different recommendations as long as the math is accurate and reasonable. Students within a group may differ in opinion just as different groups might. Use your discretion as to whether or not students within a group should have the same recommendation. FACILITATION TIP After students answer Question 1. ask them what the slope for each graph indicates in terms of number of necklaces versus number of bracelets. Students should be able to answer by stating that for every decrease of _ necklaces, there is an increase of _ bracelets. FACILITATION TIP If no one mentions it in their answer, note to the class that the slopes are different, as well. It may be tricky to spot the difference at ﬁrst glance. For both graphs, students −48 ___ −2 can apply rise over run to ﬁnd m = ____ = 5 120 −60 −3 and m = ____ = ___ for Seller A and Seller B, 5 100 respectively. 109

LINEAR INEQUALITIES

Linear Inequalities Explore 3 – Writing Linear Inequalities 5. 6. FACILITATION TIP After students answer Question 1. ask them how the graph for each seller would change if they started charging the same price for bracelets as they currently do for necklaces. Then, ask them how the graph for each seller would change if they started charging the same price for necklaces as they currently do for bracelets.

FACILITATION TIP Open a brief discussion on proﬁt. After students answer the question, present the scenario that each student has quadrupled an investment in their business. Ask students how they would spend their proﬁt and what percentage would go toward their business.

Math Chat DOK-1 When looking at a graph of a linear inequality, how do you know if the points on the line are solutions? The graph will show a solid boundary line if they are solutions and a dashed line if they are not solutions. • DOK-1 When looking at an algebraic linear inequality, how do you know if the graph should have a solid or dashed boundary line? The graph will have a solid boundary line if the inequality has a less than or equal to (≤) or a greater than or equal to (≥) inequality symbol. The graph will have a dashed boundary line if the inequality has a less than (<) or a greater than (>) inequality symbol. • DOK-2 When you have the graph showing the solutions of a linear inequality, how do you determine the equation of the boundary? To determine the equation of the boundary line, we could use slope-intercept form to write the related equation. Determine the y-intercept and the slope, and substitute them into y = mx + b. If we already know the constraint or target value, we can use the intercepts and standard form Ax + By = C. • DOK-3 Do you think it is possible for Jahzara to quadruple her investment? Why or why not? I do think it is possible, but it will be limited to a very few ordered pairs to reach her goal based on the other limitations. I do think she can quadruple her investment because the shading goes to inﬁnity, so her income is limitless. •

•

FACILITATION TIP Ask the class if the equation y = 2x + 5 has more, fewer, or an equal amount of solutions. The answer may be counterintuitive for students. Remind them that a line can travel endlessly just like a half-plane. They can picture it as taking a line and folding it over a half-plane. Any size of half-plane can be covered by some length of the related equation's line.

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.

DOK-3 Which inequality, y > 2x + 5 or y < 2x + 5, has more solutions? Explain. Both inequalities have an inﬁnite number of solutions. Neither has more solutions than the other. The solution sets are opposite, but one set does not have more elements than the other.

Post-Explore 1. 2. 3. 4.

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Linear 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

Solutions of Linear 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

Graphing Linear Inequalities 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

Writing Linear Inequalities 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

Spiraled Review

Fluency Builder

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

Linear Inequalities

Can be done independently

LINEAR INEQUALITIES

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

PhET Interactive Simulation Student activities using the PhET Interactive Simulations from the University of Colorado Boulder

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

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Linear 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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Explain

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Intervention

Acceleration Acceleration

What prompts will be used?

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What does mastery look like?

I can determine whether an ordered pair is a solution to an inequality.

I can graph linear inequalities.

I can write linear inequalities when given a graph or verbal description.

I can determine the constraints and real-life scenarios with linear inequalities.

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SCOPE 1

Systems of Inequalities Scope Introduction SCOPE SUMMARY In this grade level, students should be able to identify viable and nonviable solutions within the context given and explain their reasoning verbally and in writing. Students should be able to use a test point to determine which half-plane to shade for each inequality and then determine what region on the graph is the solution set for the system.

Student Expectations

A.PAR.4.3 Solve systems of linear inequalities by graphing, including systems representing a mathematically applicable situation.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grade levels, students solved systems of equations and solved and graphed one-variable inequalities on number lines. Earlier in the school year, students graphed two-variable linear inequalities on the coordinate plane. In this grade level, students will expand their understanding of these topics in order to solve systems of linear inequalities by graphing on a coordinate plane and identifying which part of the plane represents the solution set.

In Algebra II, students will expand on their skills by determining possible solutions in the solution set of systems of two or more linear inequalities in two variables. Students will formulate and solve systems of two or more linear inequalities. They will also solve quadratic inequalities.

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

solve word problems leading to inequalities of the form px + q > r or px+ q < rr, where p, q,, and r are speciﬁc rational numbers.

•

graph a solution set of an inequality and interpret it in the context of the problem.

Hook

Accessing Prior Knowledge

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

relate systems of inequalities 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.

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

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Systems of Inequalities In this exploration, groups of students will solve real-world scenarios that are about analyzing data determining some possible combinations of bracelets and necklaces that could be made. Students will: •

use graphs of inequalities.

•

determine solutions in context and model systems of inequalities with and without solutions.

•

compare graphs.

Explore 2

Explore 1

EXPLORE ACTIVITIES Systems of Inequalities Continued In this exploration, students will graph the solution set of systems of two linear inequalities in slope-intercept form for two variables on the coordinate plane. Students will: •

graph inequalities.

•

ﬁnd solutions to systems of inequalities.

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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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Systems of Inequalities Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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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. 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 •

SYSTEMS OF INEQUALITIES

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

• •

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

Procedure and Facilitation Points 1. 2.

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

Card 1 matches with Card C.

Card 2 matches with Card B.

Card 3 matches with Card A.

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

Identifying Misconceptions • • •

Students may struggle to remember that to solve inequalities they still must use the order of operations and combine like terms. Students may not remember that the correct number line represents all solutions to an inequality. Students may struggle to remember that when using “less than or equal to” or “greater than or equal to,” the solution must be included in the set.

FACILITATION TIP Depending on your classes, consider displaying the cards one at a time and guiding students to make observations. Additionally, you could display the numbered cards only and have students record notes, talk with their seat partners, and prepare to ﬁnd the matches on the lettered cards. FACILITATION TIP Card 2 and Card A are text heavy, take time to read them aloud with your classes if needed. Coach students to record essential phrases and values before they go looking for a match. FACILITATION TIP Consider your students' recent experiences with inequalities before this APK. Students may need a quick review of inequality notation before completing this task. Picture vocabulary includes several essential terms.

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Systems of Inequalities Hook – Systems of Inequalities ACTIVITY PREPARATION Students will relate systems of inequalities to a real-world situation.

Materials

Preparation

Printed •

• • •

1 Systems of Inequalities (per class)

Reusable • •

1 Phenomena Video (per class) 1 Projector (per class)

Plan to show the video. Prepare to project Systems of 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 POINTS Part I: Pre-Explore 1. FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) Does anyone knit, sew, or crochet?; 2) If so, do you sell any of your creations?; 3) Would it be more challenging to manage the time to make an item or the production cost? Why? FACILITATION TIP

Project the text of this complex situation and read it aloud with students. Encourage students to read through it more than once. Have them record important values and phrases. FACILITATION TIP

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: you: A knitting company sells hats and scarves that they produce. It costs the company $4 and takes 3 hours to make each hat. It costs the company $6 and takes 2 hours to produce each scarf. The company has an operating budget of $180 and 70 hours this week to produce hats and scarves. Each hat sale earns a net profit of $10, and each scarf sale earns a net profit of $12. How many of each item should they make this week to maximize their 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. Answers will vary. I notice that it costs more to make a scarf but does not take as long. I notice that the company makes more money for each scarf sale than for each hat sale. I wonder if we have to guess and check different options. I wonder if they will sell as many hats and scarves as the company wants. I wonder if the constraint of time or money is more limiting. Project Systems of Inequalities.

Consider Previewing unique vocabulary (maximize, proﬁt, operating budge, produce) before presenting this scenario. Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________

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Explain to students that the knitting company wants to know how many of each item to make that stays within their constraints but will help maximize their proﬁt. Discuss the following questions: a.

Engage

DOK-1 Why would there need to be multiple inequalities? Allow students to share all ideas. Answers will vary. Because time and money are limiting factors, there is an inequality for each one. DOK-1 What does the 10x + 12y part of this math sentence represent? It represents the amount of proﬁt earned for each hat sold plus the amount of proﬁt earned from each scarf sold.

Explain that a guiding idea throughout this scope will be this question: How do we ensure the best outcome in the face of given constraints? Complete the Explore activities.

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.

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Part II: Post-Explore 1. 2.

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

DOK-1 Does this system of inequalities make more sense after the Explore activities? Yes, I can graph each of the inequalities on a coordinate grid.

DOK-1 What strategies would you use to solve this compound inequality? I would graph each of the inequalities to determine the region of solutions that represents all of the possible combinations of hats and scarves the company can make given the constraints. Then, I would substitute the vertices of each region into the proﬁt equation and see which point yielded the highest value.

DOK-2 How many hats and scarves should the company make this week to maximize profits? Do you think this is a good idea? Making 6 hats and 26 scarves would maximize profits with the given constraints, but this plan may not cater to the needs of the company’s customers. DOK-1 Do you feel that you have a strong understanding of systems of inequalities? Answers will vary based on students’ success during the activity and their confidence level.

FACILITATION TIP Students may struggle to articulate their strategies succinctly given the numerous steps involved. Within reason, give space for lengthier answers so long as they are mathematically correct, complete, and relevant to the question. FACILITATION TIP Rather than ask, "Do you think this is a good idea?", consider asking about how making more hats to maximize proﬁts might affect the company. Conduct a brief discussion regarding supply and demand.

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Systems of Inequalities Explore 1 – Systems of Inequalities ACTIVITY PREPARATION Students will use graphs of inequalities to determine solutions in context. Students will model systems of inequalities with solutions and with no solutions.

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.

Materials

Preparation

Printed • • •

• • •

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

Reusable • •

•

1 Dry-erase marker (per group) 2 Sheet protectors (per group)

Separate the class into groups of 2 or 3 students. Print a Student Journal and an Exit Ticket for each student. Print a set of Graph Cards on card stock for durability. Place each page inside a clear sheet protector to create an erasable surface. Gather a dry-erase marker for each group.

PROCEDURE AND FACILITATION POINTS Part I FACILITATION TIP

Before reading the scenario, ask the class 1) What do you know about business models?; 2) What aspects do you think make a good business model?; 3) If you were to start a jewelry business, what would it be like, and what kind of jewelry would you sell? FACILITATION TIP Consider previewing unique vocabulary for this scenario before projecting it and reading it aloud with students (maximize, proﬁt, start-up costs). FACILITATION TIP

2. 3. 4.

Be sure to allow students time to note down the details in the scenario before they begin to collaborate.

FACILITATION TIP Encourage students to look at each graph carefully. The lines in the Graph Cards look similar, but they are not identical.

Read the following scenario to the class: Jahzara is trying to figure out the best combination of bracelets and necklaces to make and sell in order to maximize her profit. She knows there will be start-up costs to get her business going, and she has to charge enough for each item to make a profit. She has a maximum of $500 to spend on start-up costs. The materials for each bracelet cost $2, and the materials for each necklace cost $3. Jahzara initially projects that she should sell the bracelets for $8 each and the necklaces for $15 each. Her goal is to earn at least $1,500. Jahzara’s home business computer program printed out two graphs to represent the constraints for the start-up costs and the income goal. Help Jahzara determine some possible combinations of bracelets and necklaces she could make to satisfy all of her constraints. Give a Student Journal to each student. Give a set of Graph Cards and a dry-erase marker to each group. Allow students one minute to make observations about the Graph Cards. Avoid telling students what the graphs show. The students should discover this throughout Part I. Explain to students that they will work with their groups to analyze the constraints of the start-up costs and Jahzara’s income goal and determine possible combinations of necklaces and bracelets she could make. 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.

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DOK-1 How did you determine the inequality to represent the constraint? Answers will vary. I used the information in the scenario. Jahzara can spend a maximum of $500, so I knew the inequality would be less than or equal to $500. I used the information on the graph to write the inequality. Since values were shaded less than the y-intercept and the x-intercept, I knew I should use the less than or equal to symbol. © Accelerate Learning Inc. - All Rights Reserved

8. 9.

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Explain

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DOK-2 Is there more than one way to determine if the ordered pair satisfies the constraint? Yes, the ordered pair can be found on the graph that was given to us, or it can be substituted into the inequality. If it is in the shaded region, it satisfies the constraint. If it makes a true statement when substituted, it satisfies the constraint.

DOK-2 What does the ordered pair in question 5 mean in the context of the situation? It means 80 bracelets and 50 necklaces are made and sold. This is possible in terms of the start-up costs, but this doesn’t meet the income constraint, so it does not meet the income goal.

As students are completing Part I, collect a sample from one group that graphed one inequality on top of the other to be shared during the Math Chat. If no students chose this method, demonstrate it during the Math Chat. 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-2 Why do you think only quadrant I is shaded in both graphs? A negative number of jewelry wouldn’t make sense, so the other quadrants don’t contain reasonable solutions. Shading only the first quadrant is a more accurate representation of reasonable solutions. DOK-2 Here is the work of a classmate. What do you notice about the shaded region of the graph they made with the dry-erase marker? Note that if no students have a sample to share, you should demonstrate this method here. Their graph made with the dry-erase marker represents the solutions of the other inequality. It overlaps the shaded region on the printed graph. DOK-2 Students used a variety of methods. Some students tested the ordered pairs by substituting them into the inequality to check for a true statement. Was it more efficient to test ordered pairs in both inequalities or graph both constraints on the same coordinate grid? Explain. Graphing both inequalities on the same coordinate grid is the most efficient method because there are fewer steps than substituting coordinates into the inequality over and over. DOK-2 Students used a variety of methods. Some students found each coordinate on each graph and then decided if it was a solution. Was it more efficient to identify ordered pairs in the shaded region on both graphs or graph both constraints on the same coordinate grid? Graphing both inequalities on the same coordinate grid is the most efficient method because there are fewer steps than checking the coordinates on both graphs over and over. DOK-3 How is a system of linear equations similar to a system of linear inequalities? How are they different? They are both used when a situation has more than one constraint. The graphs of both systems are created using straight lines. The solutions to the system of equations are the point or points where the lines intersect. The solutions to the system of inequalities are the points in the region bounded by the lines. DOK-3 Why do we need to use a system of linear inequalities to represent the solutions to this situation instead of only to individual inequalities? If both constraints in the situation must be met, then we need to find values that satisfy both inequalities.

Explain the following to the class: Two inequalities representing the constraints in the same situation form a system of linear inequalities. The solutions to a system of inequalities are all of the points in the region where the graphs overlap because those points represent all of the pairs of values that make both inequalities true.

Intervention

Acceleration

FACILITATION TIP Consider projecting question 6b before students begin collaborating. Rather than asking, "Is there more than one way...?", consider asking, "What are some different ways....?"

FACILITATION TIP

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Student Journal, Part I, Question 4: Just as (80, 50) does not satisfy the constraint for income, (120, 70) and (60, 68) may not satisfy the constraint for start-up costs. After students complete Question 4, have them verify graphically or algebraically whether or not (120, 70) and (60, 68) satisfy the constraint for start-up costs. 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 Project this question about comparing linear equations and linear inequalities. Have students conduct a think, pair, share. Record appropriate student responses on a chart or list while students take notes. FACILITATION TIP Print and project the explanation about inequalities, solutions, and graphs. Have students read it aloud, take notes, and ask questions before moving on to Part II. FACILITATION TIP After the explanation, compare systems of inequalities with systems of equations. Ask the class if they think the methods of substitution and elimination would work to solve a system of inequalities. Have them explain their reasoning. 123

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Systems of Inequalities Explore 1 – Systems of Inequalities Part II 1.

FACILITATION TIP By the time students get to Question 5 of Part II, some may assume there will be an overlapping region within Quadrant I of the graph. Encourage students to focus on graphing the correct equation and using test points to ﬁnd which half-plane to shade.

3. 4.

Read the following scenario to the class: After thinking it over, Jahzara wants her income to be higher, so she is considering some new income goals. Now she wants her income to be at least $2,000. She is still thinking she should sell the bracelets for $8 each and the necklaces for $15 each. Help her determine combinations of bracelets and necklaces to reach her new income goal. Explain to students that they will work with their groups to analyze Jahzara’s new income goal and determine reasonable combinations of necklaces and bracelets she could make to reach that goal. Students will work together to explore the scenarios and record their work 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, ask them if they have ever had two desires or goals that they could not meet at the same time. Examples may involve a time conflict, insufficient funds, etc. Which desire or goal did they choose, if either? How did they feel about their decision right afterward? How do they feel about it now? FACILITATION TIP After students complete Part II, ask them if Jahzara can do anything to try to reach her new income goal of $2,600. One possible solution is raising the price of at least one of her products. Another possible solution is finding less expensive materials and spending more time making more jewelry. FACILITATION TIP After the Exit Ticket, ask students if they plan to participate in a fundraiser in the next year. Have they decided on one specifically? If so, who will receive the funds, and what will the funds go toward?

5. 6.

DOK-1 When graphing the constraints for Jahzara’s new income goal, which quadrant or quadrants should you shade? Why? All of the solutions will be whole numbers, so I should only shade the ﬁrst quadrant.

DOK-2 Would (250, 0) be a solution when Jahzara’s income goal is $2,000? Why or why not? Yes, it is a solution because it is on the line for both inequalities and both inequalities are inclusive, so the points on the line are solutions.

DOK-2 In which quadrant(s) do the graphs overlap when Jahzara’s income goal is $2,600? What does this mean in terms of the situation? They only overlap in quadrant II. In this situation, quadrant II does not contain reasonable solutions, so the overlapping region does contain solutions to the algebraic representation that are not reasonable in context.

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

Math Chat DOK-2 What is the most efficient method to represent all of the solutions to a system of linear inequalities? Graphing both inequalities on an xy-coordinate grid is the most efficient method. • DOK-3 Why must you use both inequalities to determ If both constraints in the situation must be met, then we need to find values that satisfy both inequalities. •

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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Systems of Inequalities Explore 2 – Systems of Inequalities Continued ACTIVITY PREPARATION Students will graph the solution set of systems of two linear inequalities in slope-intercept form for two variables on the coordinate plane.

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.

Materials

Preparation

Printed • • •

• • •

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

Separate the class into groups of 2 or 3 students. Print a Student Journal and Exit Ticket for each student. Print a set of Graph Cards on card stock for durability. Place each graph inside a sheet protector to create an erasable surface.

Reusable • •

1 Dry-erase marker (per group) 2 Sheet protectors (per group)

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Project the text of this scenario for students to read along with you. Allow students to take notes. FACILITATION TIP Before distributing the Student Journal, consider asking students to write out the inequalities. Coach them to use x and y for the mystery numbers.

2. 3. 4.

FACILITATION TIP Distribute the Student Journal in two parts as needed. Part I is page 1. Part II is pages 2–4.

FACILITATION TIP

Consider projecting the Graph Cards one at a time to the whole class and record some observations without revealing what they show.

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Read the following scenario to the class: Mr. Khan loves to give his students puzzles. Makena always figures them out quickly, but she is having trouble with the most recent puzzle. Help Makena solve the puzzle! The following is Mr. Khan’s puzzle: I’m thinking of two numbers. Twice the first number plus seven is less than the second number. Five times the first number plus three is more than the second number. What are the two numbers? Give a Student Journal to each student. Give a set of Graph Cards and a dry-erase marker to each group. Allow students one minute to make observations about the Graph Cards. Avoid telling students what the graphs show. The students should discover this throughout Part I. Explain to students that they will work with their groups to analyze the two parts of Mr. Khan’s puzzle and determine possible combinations of numbers that solve the puzzle. 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 How did you determine the inequality symbol to represent the ﬁrst part of the puzzle? Answers will vary. I used the information in the scenario. Double the ﬁrst number plus seven is less than the second number; I knew that would mean the inequality needed a less than symbol. I used the information on the graph to write the inequality. I tested the point (0, 0), and since it was not shaded, I used the less than symbol to make sure it was not true. © Accelerate Learning Inc. - All Rights Reserved

8. 9.

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DOK-1 How did you determine the inequality to represent the second part of the puzzle? Answers will vary. I used the information in the scenario. Five times the ﬁrst number plus 3 is more than the second number, so I knew I had to insert a greater than symbol. I used the information on the graph to write the inequality. I tested the point (0, 0), and since it was shaded, I used a greater than symbol to make the inequality true.

DOK-2 Is there more than one way to determine if the ordered pair satisfies the puzzle? Yes, the ordered pair can be found on the graph that was given to us or it can be substituted into the inequality. If it is in the shaded region of both graphs, it satisfies the puzzle. If it makes a true statement when substituted into both inequalities, it satisfies the puzzle.

DOK-2 What does the ordered pair in question 5 mean in the context of the situation? It means that the guess at a solution to the puzzle was that the ﬁrst number is 20 and the second number is 30. This pair meets the second set of criteria but not the ﬁrst set, so it does not solve the overall puzzle.

Math Chat •

•

DOK-2 Why do you think the lines are dashed in both graph cards? If the inequality symbol is inclusive, ≥ or ≤, a solid line is used. If the inequality symbol is exclusive, > or <, a dashed line is used. DOK-2 Here is the work of a classmate. What do you notice about the shaded region of the graph they made with the dry-erase marker? Note that if no students have a sample to share, you should demonstrate this method here. Their graph made with the dry-erase marker represents the solutions of the other inequality. It overlaps the shaded region on the printed graph. DOK-2 Students used a variety of methods. Some students tested the ordered pairs by substituting them into the inequality to check for a true statement. Was it more efficient to test ordered pairs in both inequalities or graph both parts of the puzzle on the same coordinate grid? Explain. Graphing both inequalities on the same coordinate grid is the most efficient method because there are fewer steps than substituting coordinates into the inequality over and over. DOK-2 Students used a variety of methods. Some students found each coordinate on each graph and then decided if it was a solution. Was it more efficient to identify ordered pairs in the shaded region on both graphs or graph both parts of the puzzle on the same coordinate grid? Graphing both inequalities on the same coordinate grid is the most efficient method because there are fewer steps than checking the coordinates on both graphs over and over. DOK-3 How is a system of linear equations similar to a system of linear inequalities? How are they different? They are both used when a situation has more than one constraint. The graphs of both systems are created using straight lines. The solutions to the system of equations are the point or points where the lines intersect. The solutions to the system of inequalities are the points in the double-shaded region bounded by the lines. DOK-3 Why do we need to use a system of linear inequalities to represent the solutions to this situation instead of only one individual inequality? If both constraints in the situation must be met, then we need to find values that satisfy both inequalities.

Intervention

Acceleration

FACILITATION TIP As an alternative to the yes/no question for 6c, consider asking, "What way can we determine the ordered pair that satisﬁes the puzzle? What's another way?"

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FACILITATION TIP For step 7, consider preparing your own sample using this method. Alternatively you could request a speciﬁc student to create one for the class while you monitor collaboration.

FACILITATION TIP Project this question about dashed/solid lines. Have students make notes and sketches for the rule. 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.

FACILITATION TIP If you didn't have time during Explore 1, project this question about comparing linear equations and linear inequalities. Have students conduct a think, pair, share. Record appropriate student responses on a chart or list while students take notes.

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Systems of Inequalities Explore 2 – Systems of Inequalities Continued Explain the following to the class: Two inequalities representing the constraints in the same situation form a system of linear inequalities. The solutions to a system of inequalities are all the points in the region where the graphs overlap because those points represent all pairs of values that make both inequalities true. FACILITATION TIP Project the text of this scenario for students to read along with you. Allow students to take notes. FACILITATION TIP Before distributing Part II of the Student Journal, consider asking students to write out the inequalities. Coach them to use x and y for the mystery numbers. FACILITATION TIP

Part II 1.

2. 3.

4. 5.

Project guiding questions 5a–5c. Preview with students before they begin collaborating. Encourage students to focus their discussions on these prompts.

Read the following scenario to the class: Makena has successfully solved the puzzle. Mr. Khan is impressed. He has decided to challenge Makena by giving her a new puzzle. The following is Mr. Khan’s new puzzle: I’m thinking of two numbers. Half of the first number minus two is less than the second number. Four times the first number plus five is more than the second number. What are the possibilities for my two numbers now? Students should still have their Student Journals. Explain to students that they will work with their groups to analyze Mr. Khan’s new puzzle and determine the possible combinations of numbers that solve the puzzle. Students will then work together to explore the scenarios and record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding:

FACILITATION TIP For question 5e, take time to have students practice reading/saying inequalities in different ways aloud. If needed, display some additional graphs for students to make observations and create statements for.

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.

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DOK-2 How did you graph inequalities in slope-intercept form? Graphing inequalities in slope-intercept form is almost identical to graphing linear equations in slope-intercept form. The y-intercept is then plotted on the graph, and the slope is used to create another point. A straight dotted or dashed line is drawn through the two points.

DOK-2 When graphing the second part of the new puzzle, how would you determine the solution from the graph? When graphing two inequalities, the solution set is the portion of the graph that is double shaded.

DOK-2 Would (40, 20) be a solution to Mr. Khan’s puzzle? Why or why not? Yes, it is a solution because it is in the portion of the graph that is double shaded.

DOK-2 What does the white portion in the graph of question 5 mean in the context of the situation? The white portion of the graph represents all the points that do not satisfy both inequalities of the system. In this situation, it means the white portion has combinations of numbers that do not solve both parts of Mr. Khan’s puzzle.

e. Note for students: We could read each puzzle a different way so that the shading above or below the boundary line seems to match the inequality we are saying. For example, the graph you created in question 5 could have been articulated by saying, “The second number is less than ﬁve times the ﬁrst number minus 50.” 6. 7.

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

Math Chat DOK-2 What is the most efficient method to represent all the solutions to a system of linear inequalities? Graphing both inequalities on an xy-coordinate grid is the most efficient method. • DOK-3 Why must you use both inequalities to determine the combinations of numbers that solve Mr. Khan’s puzzles? If both constraints in the situation must be met, then we need to find values that satisfy both inequalities. • DOK-3 How can the solutions to a system of inequalities be verified algebraically? The solutions to a system of inequalities can be verified algebraically by plugging the points into the inequalities to see if they make both statements true. •

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DOK-3 How will you determine if an ordered pair is in the solution set of a system of two inequalities? I will look at the graph to see if the ordered pair is in the portion of the graph that is double-shaded or by plugging the points into the inequalities to see if they make both statements true.

Post-Explore 1. 2. 3. 4.

FACILITATION TIP This Exit Ticket could also be used as a pre-assessment before this Explore activity. It might provide good data for measuring student growth, or merely inform your pace of instruction.

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Systems of 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

Systems of 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

Systems of Inequalities Continued Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Interactive Notebook

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

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

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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.

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

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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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Students

Notes & Comments

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Acceleration Acceleration

What prompts will be used?

What does mastery look like?

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I can solve systems of linear inequalities by graphing.

I can determine and understand the solution set or feasible region to a scenario that produces a system of inequalities.

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SCOPE 1

Simplify Radicals Scope Introduction SCOPE SUMMARY In this scope, students will simplify numerical and algebraic radical expressions involving square roots and cube roots. Upon simplifying their answers, students will investigate properties of rational and irrational numbers under different arithmetic operations.

Student Expectations

VERTICAL ALIGNMENT

A.NR.5.1 Rewrite algebraic and numerical expressions involving radicals. A.NR.5.2 Using numerical reasoning, show and explain that the sum or product of a rational number and an irrational number is irrational, and the product of a nonzero rational number and an irrational number is irrational.

Background Knowledge

Future Expectations

In previous grade levels, students worked with integer exponents. Students have already been introduced to square roots. They were introduced to positive integers to explain place value in ﬁfth grade and have used properties of integer exponents to generate equivalent expressions.

In Algebra II, students will continue to rewrite radical expressions, but expand to fractional exponents using any rational number. Students will also solve radical equations in one variable as well as graph radical equations in two variables in a future algebra course.

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

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES

explains how to simplify radicals.

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.

how to simplify expressions with square roots.

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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Simplify Square Roots

Explore 2

Explore 1

EXPLORE ACTIVITIES

In this exploration, groups of students will solve problems based on designing a vegetable garden for school. Students will: •

explore perfect squares.

•

multiply any coeﬃcients by any additional numbers that are moved in front of the radical symbol during the simpliﬁcation process.

•

simplify radical expressions involving numbers that may not be perfect squares.

In this exploration, students will be introduced to an extension scenario from the previous Explore activity. Here, students are tasked with helping determine how much fertilizer and topsoil is needed by the Oak Woods High School Garden Club. Students will: •

simplify radical expressions involving multiplication and division.

•

develop methods for simplifying.

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

Simplify Square Root Expressions

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Simplify Cube Root Expressions In this exploration, students will simplify expressions involving cube roots using the properties of exponents. Students will: •

apply and modify their skills to work with cube roots.

•

look for factors that appear three times under the radical.

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

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Simplify Radicals Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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

Materials Printed •

1 Set of Four Corners Slides (per class)

Preparation • •

Print one set of Four Corners Slides. Hang the slides in four separate areas of the classroom, easily visible to all.

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

Ask the students to look at the four corners and think about which corner image best explains simplifying radicals. Allow 2 minutes of thinking time. Ask students to move to the corner image they chose. Ask each group to discuss why they chose the image. Allow 2–5 minutes of discussion at the corner images. After students have discussed why they chose their answers, talk about the answers with the class, and allow students to explain their representation of the problem. If students are struggling to complete this task, do the Foundation Builder to ﬁll the gap in prior knowledge before moving on to other parts of the scope.

Identifying Misconceptions • • •

Slide 3 shows a correct solution for prime factorization. Students may struggle to remember that prime factorization means that all of the factors written in the ﬁnal product must be prime numbers. Students may struggle to remember that prime factorization is written as a product and not a sum of the numbers.

FACILITATION TIP Before asking students to select the best image, project the Four Corners Slides one at time for students to make notes and observations about. Consider facilitating a think, pair, share for each of the four slides. FACILITATION TIP Print the slides on different colored papers so students can quickly locate and move the one they have chosen as the image that best explains simplifying radicals. FACILITATION TIP This Foundation Builder includes six clear slides that could be easily used to review exponents and rational bases. Alternatively, use them as a challenging warm-up.

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Simplify Radicals Hook – Simplify Radicals ACTIVITY PREPARATION Students will begin brainstorming how to simplify expressions with square roots.

Materials

Preparation

Printed •

• •

1 Simplify Radicals (per class)

Reusable •

1 Projector (per class)

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

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

To clarify the task, project the question in step 2 to students before you show Simplify Radicals.

FACILITATION TIP This might be a good time to quickly review divisibility rules with students if needed. Some students may benefit from being reminded how to efficiently determine if 9 fits evenly into a large number.

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 Simplify Radicals behind you: you How can we rewrite and simplify each of these expressions? 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 9 divides 252 evenly. I notice that there is a 10 in the numerator and denominator of the second expression. I wonder how the square roots can be combined in the second expression. I wonder if the expanded version of the ﬁrst expression is easier or more complicated to work with. Explain to students that the goal of the scope will be to rewrite expressions with square roots in ways that make them easier to interpret and work with. Discuss the following questions: a.

DOK-1 How could we write an expanded version of 252 if it did not have the square root? We could write the prime factorization for 252, so it would be written as a longer product of its factors.

DOK-1 Why might condensing the second expression make it easier to work with and understand? Allow students to share all ideas. Answers will vary. We often simplify solutions so that they are represented in a concise way that is easier to utilize and evaluate.

Complete the Explore activities. Notes

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Part II: Post-Explore 1. 2.

Show Simplify Radicals again, and restate the problem. Discuss the following questions: a.

DOK-1 Do these expressions make more sense after the Explore activities? Yes, the simpliﬁed version of the square root of a number often has an integer outside of the radical and a smaller number than the original radicand inside the square root.

DOK-1 What strategies would you use to condense the second expression? I would ﬁnd common factors of the radicands in the numerators and denominators and combine them as well as I could to create integers that are outside of the radicals.

DOK-1 What are the results of the expanded version of expression 1 and the condensed version of expression 2? Expression 1 would become ____ ___________ __ 1 √ 252 = √ 2 · 2 · 3 · 3 · 7 = 6√ 7 . Expression 2 condenses to __ . 2

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

FACILITATION TIP In addition to question 2a, ask students if they can explain how to simplify a square root to another student or younger sibling. Encourage them to use the words radical and radicand in their explanations.

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FACILITATION TIP As a follow up to question 2d, consider having students rate their understanding on a scale of 1–10. Ask them to consider how they would teach a student who was absent how to simplify a radical.

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Simplify Radicals Explore 1 – Simplify Square Roots ACTIVITY PREPARATION A number that has a whole number square root is called a perfect square. Students will simplify radical expressions involving numbers that may not be perfect squares. Students will simplify these expressions by looking for factor pairs within numbers. Students will multiply any coeﬃcients by any additional numbers that are moved in front of the radical symbol during the simpliﬁcation process.

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.7 Look for and make use of structure.

Materials

Preparation

Printed • • •

• • •

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

•

Reusable •

1 Resealable bag (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 a set of Garden Signs 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. For students who need additional organizational support, please see our Factor Tree Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION POINTS FACILITATION TIP

Part I 1.

Before reading the scenario, ask the class 1) Has anyone ever planted a garden?; 2) What did you plant?; 3) What was the most enjoyable part? The most challenging? FACILITATION TIP Print and project the scenario so students can take notes on the values and what they need to discover (side length). FACILITATION TIP Note to the class that Charlotte and Joshua need much more room for tomatoes than for carrots or radishes. After reading the scenario, ask them why they think that is so. Answers may include that tomatoes need room for their vines or that Charlotte and Joshua plan to grow more tomatoes than carrots or radishes.

2. 3.

Read the following scenario to the class: Charlotte and Joshua are planting square gardens at their school for part of an ecology class. They are going to start with 3 crops: carrots, tomatoes, and radishes. After doing some research on how much room each of these vegetables needs to grow, they have decided they need 4 square feet for carrots, 25 square feet for tomatoes, and 8 square feet for radishes. Determine the side length for each vegetable garden. Give a Student Journal to each student. Explain to students that they will work with their groups to simplify the expression for the length of each side of the garden 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 do we know the length of the side of each garden?__ The gardens are square. The area of a square is A = L2, or L = √ A .

DOK-2 Why do we want to pair up factors? We want to pair up factors to create perfect squares. Perfect squares will simplify to an integer and can be rewritten as the coeﬃcient of the radical.

DOK-1 When there is a pair, what do you do with it? Simplify it and rewrite it as the coeﬃcient of the radical.

DOK-1 If there are factors that cannot be removed, what happens to them? They remain as the radicand.

FACILITATION TIP Depending on your students, distribute the Student Journal in two parts as needed. (Part I is page 1 and Part II is pages 2–4). 140

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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 do you know if a number is a prime factor? A number is a prime factor if the only integers it can be evenly divided by are itself and 1. • DOK-2 Will all radicals simplify to a whole number? Explain why or why not. No, if it is a perfect square, then it will be a whole number. If it is not a perfect square, then there will be factors remaining as the radicand. • DOK-1 What is a good way to determine all of the factors of a number? A good way to keep track of factoring is to create a factor tree, working your way down until you have all prime numbers. •

Part II 1.

2. 3. 4. 5.

6. 7.

Read the following scenario to the class: Now that the ﬁrst 2 veggies are growing nicely, Charlotte and Joshua are wanting to add some more produce items. They have created garden signs to help label their produce plots. Each garden sign has the type of produce and the area of the garden plot. Give a set of Garden Signs Cards to each group of students. Students should still have their Student Journals. Explain to students that they will work with their groups to write expressions for the lengths, simplify the lengths, 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 Do you think the order of your factors matters? Why or why not? No, the order of the factors doesn’t matter because they are multiplied, and you can multiply numbers in any order (commutative property of multiplication).

DOK-1 What operation is occurring between the factors taken out of the radical and the radical itself? Multiplication

DOK-1 If there are any numbers outside of the radical before you simplify it, what must you do with the additional coefficients that resulted when simplifying the radical? Multiply them with the original coefficients to simplify them to a single coefficient.

DOK-2 Using your analysis of Charlotte and Joshua’s squash debate, why would Joshua be correct? Explain. The number outside of the radical is multiplying. When we bring factors out, we multiply them. It makes sense that if a number already exists outside of the radical, we would multiply the factors we remove with it for a new product.

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 Are there any scenarios you can think of where it might be better to have a number in the non-simplified form? If you were wanting to compare the size of two expressions, it is easier to do before the radicals are simplified. • DOK-2 What do you think might be an advantage to having a radical simplified? One advantage is that it makes it possible to add radicals since they may have like radicands. • DOK-2 When taking the square root of a number, we pair up factors to create perfect squares. What do you think we would do if we were simplifying a cube root? If we were simplifying a cube root, we would create groups of 3 like factors instead of the 2 we used in a square root. •

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FACILITATION TIP For Part II, Question 1 of the Student Journal, it may be especially helpful for some students to use factor trees as they ﬁll out the table. If students are initially stumped by a radical, have scratch paper handy if they wish to work out factor trees. FACILITATION TIP Before reading the scenario, ask the class 1) What is your favorite vegetable?; 2) Do you know when or where it is grown?; 3) How long does it take to grow? FACILITATION TIP Project some of the guiding questions 5a–5c. Consider reviewing them before students begin collaborating.

FACILITATION TIP Ask students for other real-world examples of squares for which they could ﬁnd the length given the area. Examples may include a patch of grass, a ﬂoor tile, or a framed painting.

FACILITATION TIP Project this question about cube roots and challenge the class to come up with some examples to try simplifying. 141

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Simplify Radicals Explore 1 – Simplify Square Roots •

FACILITATION TIP On this Exit Ticket, support struggling students by reminding them that when taking the root of a large number, they can break the radicand into factors.

DOK-2 Apply your knowledge to reﬂection question 2. What differences, if any, do you believe variables will present? I do not believe there will be any differences. Instead of numbers, variables will be used, but the same process of ﬁnding factors will be applied.

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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Simplify Radicals Explore 2 – Simplify Square Root Expressions ACTIVITY PREPARATION Students will simplify radical expressions involving multiplication and __ ___ Students will look for patterns in the numbers to __ division. develop methods for simplifying. Students will utilize the fact that √ a · √ b = √ ab . Radicands may contain variables.

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.7 Look for and make use of structure.

Materials

Preparation

Printed • •

• • •

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

•

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. For students who need additional organizational support, please see our Factor Tree Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever been part of a school club?; 2) What did you get to do in the club? After reading the scenario, ask the class 3) Why do you think radishes need to stay in their own garden? 2. 3. FACILITATION TIP For Part I, Question 6, some students may answer "They need 3.2 bags" given an area approximation of 320 sq. ft. in Question 5. Remind them that, while their answer is mathematically correct, it is unlikely that they can buy part of a bag of fertilizer.

Read the following scenario to the class: As preparation continues for the Oak Woods High School Garden Club, it was decided it made more sense to combine all of the smaller gardens except the radishes into 2 larger ones. The ﬁrst garden __ __ __ __ will be 5√ 6 · 10√ 2 sq. ft., and the second one will be 10√ 2 · 4√ 6 sq. ft. The next step is to determine how much fertilizer and topsoil they will need. The instructions indicate they will need to use one bag of fertilizer for every 100 sq. ft. of garden. How many bags will they need? Give a Student Journal to each student. Explain to students that they will work with their groups to determine how many bags of fertilizer will be needed 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 do we determine the total combined area of the 3 gardens? We calculate the area of each garden (length times width) and then add them together.

DOK-1 __How could we expand a radical like 5√ 7 ? We could write it as 5 · √ 7 .

√ √ DOK-1 How could __ you simplify 2 · 2 ? Multiply the radicands, and that would give you √ 4 , which equals 2.

√ √ DOK-1 How could __ you simplify 2 · 3 ? Multiply the radicands, and that √ would give you 6 .

e. DOK-2 If you were to add 5 and x, could these terms be combined? If not, how would you give the answer? No, because they don’t have the x in common. You would just leave it as 5 + x. f.

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DOK-2 If we want to know the number of bags we need, can we round like we normally do, with a decimal below 0.5 being rounded down? Why or why not? We need to round up because otherwise we won’t have enough fertilizer. © Accelerate Learning Inc. - All Rights Reserved

Engage

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Explain

Elaborate

Evaluate

__ √x

√

combined into an expression with a single radical? Since a square root is the same as __ a power of a half, we can apply the laws to exponents to x rewrite it as √ _y .

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-2 How is multiplying 2 radicals similar to multiplying 2 variables with coefficients, such as 3xx and 2x2? The process is similar because in both, you group the like items together. With the variables, you first multiply the integers and then combine the variables: (3 · 2)(x · x2). When multiplying radicals, you first multiply the integer and then combine the radicals. • DOK-2 If a radicand can be written as a perfect square, will it simplify to an expression with or without a radical? Explain. It will simplify to an expression without a radical because the perfect square will simplify to a variable or integer. •

Part II 1.

2. 3.

Read the following scenario to the class: Agustin, a student in the garden club, noticed that in the ﬁrst expression with variables, both the expression for area and the expression for depth included a radical, but the product did not. He asked other members of the garden club to explore this and help him summarize what is happening. Agustin suggested examining the sum and product of several numbers. He wants to use some rational numbers and some irrational numbers. Help Agustin and the garden club experiment and ﬁnd some patterns. Students should still have their Student Journals. Explain to students that they will work with their groups to experiment with the sums and products of rational and irrational numbers 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.

Acceleration

DOK-1 Is it possible to combine __2 such that it only has one exponent? y x 2 Yes, the rules of exponents tell us that it would be equal to (_y ) .

__ so it could be h. DOK-1 How could we apply the laws of exponents to ___ y

5. 6.

Intervention

DOK-1 For statement a, which rational numbers did you add? How did you decide which numbers to pick? Was the sum rational or irrational? Answers may vary. I picked 0 and 1 to add. I picked the easiest-looking numbers. The sum was rational.

DOK-1 For statement b, which rational number and irrational number did you add? How did you decide which numbers to pick? Was the sum rational or irrational? Answers may vary. I picked 0 and π to add. I picked the easiest looking numbers. The sum was irrational.

DOK-1 For statement c, which irrational numbers did you add? How did you decide which numbers to__pick? Was the sum rational or irrational? __ Answers may vary. I picked √ 3 and −√ 3 to add. These were opposites and would create a zero pair, so they looked easy. The sum was rational.

DOK-1 For statement d, which rational numbers did you multiply? How did you decide which numbers to pick? Was the product rational or irrational? Answers may vary. I picked 0 and 1 to multiply. I picked the easiest-looking numbers. The product was rational.

FACILITATION TIP Print and project question 4h. Encourage students to think, pair, share and come up with some examples. Have student volunteers demonstrate.

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FACILITATION TIP Test students’ understanding of both percentages and products with identical radicals. After they answer Question 3. ask the class by what percentage is Garden 1’s area greater than Garden 2’s area. They should know that they can disregard the radicals to ﬁnd the percent difference. Use this fact as you guide the class to ﬁnd the percent difference.

FACILITATION TIP Before reading the scenario, ask the class 1) When was a time you asked for help from a member of a group you were in?; 2) Did you know the fellow group member before they joined the group?; 3) What did you need help with from the member? FACILITATION TIP Project the text of this scenario and have students read independently and make notes/observations. Allow students to partner share and then read it all together as a class.

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.

e. DOK-1 For statement e, which rational number and irrational number did you multiply? How did you decide which numbers to pick? Was the product rational or irrational? Answers may vary. I picked 0 and to multiply. Zero times a number is always zero, so these looked really easy. The product was rational.

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Simplify Radicals Explore 2 – Simplify Square Root Expressions f.

DOK-1 For statement f, which irrational numbers did you multiply? How did you decide which numbers to pick? Was the product rational __ 1 __ to multiply. These or irrational? Answers may vary. I picked √ 3 and ___ √3

looked easy because they are reciprocals. The product was rational. FACILITATION TIP Some students may not realize initially that certain scenarios involving adding or multiplying with irrational numbers can result in rational sums and products, respectively. As students begin Part II, remind them that anything times 0 is 0, a rational number. Then, remind them that any nonzero value divided by itself is 1, a rational number. FACILITATION TIP After students answer the question, tell them they are given two radicals, each with an index of 2. Ask them if the radicands have to be the same for the product of the radicals to be__rational. Then, show them the __ expression "√ 1 · √ 4 ", and have them simplify the expression. FACILITATION TIP

5. 6.

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 Describe in your own words how you go about multiplying expressions that contain radicals. When multiplying expressions that contain radicals, you multiply like parts. You multiply the coeﬃcients ﬁrst and then multiply the radicands. Once you do that, you simplify the radical and the whole expression. • DOK-2 Consider the product of two irrational numbers. Will the product always be irrational? Give an example of why or why not. Sometimes this will be true, but __ __ there are times the product will be a whole number. √ 2 · √ 2 = 2 •

Post-Explore 1. 2. 3.

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

On this Exit Ticket, provide struggling students support on Question 2. Have them refer back to their work for Part I, Question 8 on their Student Journal. Speciﬁcally, note that they can focus on simplifying one radical at a time as long as they account for the entire product.

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Simplify Radicals Explore 3 – Simplify Cube Root Expressions ACTIVITY PREPARATION Students will simplify expressions involving cube roots using the properties of exponents. Students will take the skills learned from working with square roots and apply and modify them to work with cube roots. When simplifying cube roots, students will look for factors that appear three times under the radical. It is important to allow students to discover that cube root radicands can be negative, unlike square root radicands. Radicands may contain variables.

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.7 Look for and make use of structure.

Materials

Preparation

Printed • • •

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

Consumable •

1 Resealable bag (per group)

• • • •

•

Separate the class into groups of 3 or 4 students. Print a Student Journal for each student. Print an Exit Ticket document for every 2 students. Cut apart the half-page Exit Tickets so each student has one. Print a set of Storage Cube Cards, on card stock for durability, for each group of students. Cut the cards apart, and place them inside a resealable bag. Label the bag “Part I.” For students who need additional organizational support, please see our Factor Tree Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I 1.

FACILITATION TIP If time is limited, distribute the Storage Cube Cards as a full page rather than cutting apart. FACILITATION TIP Project these guiding questions 5a–5e. Consider previewing them with students before they begin collaboration and/or encourage them to be prepared to clearly answer them after they complete Part I.

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Read the following scenario to the class: Now that the gardens have been completed, the last thing the school would like the students to do is build some cube-shaped storage containers for the extra seeds, topsoil, fertilizer, and gardening equipment. What are the lengths of the sides of the 4 containers? Give a Student Journal to each student. Give a bag of Storage Cube Cards to each group. Explain to students that they will work with their groups to examine the desired volume for the container on each of the Storage Cube Cards, calculate the length of the side for each box, simplify the cube root as much as possible, 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 When simplifying square roots, how many terms do you group together? When we simplify square roots, we ﬁnd pairs of like terms.

DOK-2 When simplifying a cube root, how many terms do you think we want to group together? When we simplify a cube root, we need to ﬁnd groups of 3 like terms.

DOK-1 Can we__expand √ 2 · 23 into 2 different cube roots? Yes, √ 2 · 2 is 3 __ 3 3 √ equal to 2 · √ 2 .

DOK-1 What do you do with the groups of 3 factors you collected in the radicand? They will simplify with the cube root, and integer factors will be written as a coeﬃcient.

3 __

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Intervention

Acceleration

e. DOK-2 Do all factors have to be removed from the radical? No, if there isn’t a full group of 3, the factors remain as part of the radicand. 6. 7.

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

Math Chat DOK-2 If you were to completely factor the radicand and did not have any groups of 3 like terms, what would this mean? If you didn’t have any groups of 3 like factors after writing the prime factorization of your radicand, then you cannot simplify your cube root. • DOK-2 Explain how you can be confident that the radical is completely simplified. If the radicand has no exponents 3 or larger when written as the prime factorization, I know it is completely simplified. • DOK-1 Explain how you approached simplifying the equipment radical. Does a variable have any different approach? I found the prime factorization of the 24 and the cubed x. Then, I looked for groups the same as before. I did not need to change my approach because there was a variable. • DOK-2 How does simplifying cube roots compare to simplifying square roots? The process is similar for both; we now look for factors that are perfect cubes instead of factors that are perfect squares in order to simplify. •

Part II 1.

2. 3.

5. 6.

Read the following scenario to the class: At the end of the summer, the students were able to harvest the crops their garden had grown. They are going to donate the food to the local food bank and need containers for the transport. Students should still have their Student Journals. Explain to students that they will work with their groups to determine the length of the side for each produce container, simplify the cube root as much as possible, 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 After you have factored your number, what is a good way of notating the groups of 3? If you circle the groups of 3, you can easily see what groups you have and what is left over.

DOK-1 What operation do you complete with all removed factors or with all factors remaining as the radicand? Multiplication

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 When we simpliﬁed square roots, we looked for pairs of terms to create perfect squares. What do you think the groups of 3 like terms will create? Three like terms multiplied together create a perfect cube. • DOK-2 Why can cube roots contain a negative and square roots cannot when simplifying? When multiplying three negatives, the factors, as all negatives, will have a negative product. In a square root, there’s an even number of factors, causing a positive product. • DOK-2 Why did we look at volume scenarios for simplifying cube roots but situations involving area for simplifying square roots? The units for volume are cubed since volume measures three dimensions, while area involves units squared since it measures two dimensions. •

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.

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FACILITATION TIP Project this question about cube roots vs. square roots and record appropriate answers regarding similarities and differences.

FACILITATION TIP On Part II, the solution for the prime factorization for some of the vegetables may take up quite a bit of space. Encourage students to record them outside of the charts if desired. FACILITATION TIP Project the question about negatives and have students think, pair, share their responses.

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Simplify Radicals Explore 3 – Simplify Cube Root Expressions Post-Explore 1. FACILITATION TIP This Exit Ticket could be used as a preassessment as well as a post-assessment. It may be an effective tool to measure student growth or merely inform your instruction for this Explore activity.

2. 3. 4.

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Simplify Radicals 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

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

Anchor Chart

Show What You Know, Part 2

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

Simplify Square Root Expressions Independent practice assignment that gives students an opportunity to demonstrate their learning

Interactive Vocabulary

Show What You Know, Part 3

Students form deﬁnitions of mathematical vocabulary words used throughout the scope

Simplify Cube Root Expressions 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

Simplify Radicals

Can be done independently

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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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Simplify Radicals Intervention and Assessment SMALL-GROUP PLANNER 1

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

Students who are still acquiring the concept and need remediation

How to Use the Review

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Students

Notes & Comments

Engage

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Acceleration Acceleration

What prompts will be used?

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What does mastery look like?

I can show that the set of rational numbers is closed under addition and multiplication.

I can classify the result of the sum or product of an irrational and a rational number.

I can simplify radical expressions involving algebraic and numerical expressions.

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Polynomial Operations Scope Introduction SCOPE SUMMARY

Student Expectations

A.PAR.6.1 Interpret quadratic expressions and parts of a quadratic expression that represent a quantity in terms of its context.

In this scope, students will extend their understanding of the order of operations and distributive property to polynomial operations. Students will also rewrite polynomial expressions of degree one and degree two in equivalent forms, using the distributive property. They will be simplifying polynomial expressions of degree one and degree two in this scope by adding, subtracting, and multiplying. Students should be able to explain and identify factors, coeﬃcients, different terms, and like terms. Finally, students will use these algebraic tools to start to explore quadratic expressions and determine when quadratic expressions are equivalent by putting them in standard form.

VERTICAL ALIGNMENT

A.PAR.6.2 Fluently choose and produce an equivalent form of a quadratic expression to reveal and explain properties of the quantity represented by the expression. A.FGR.7.1 Use function notation to build and evaluate quadratic functions for inputs in their domains and interpret statements that use function notation in terms of a given framework.

Background Knowledge

Future Expectations

In previous grade levels, students generated equivalent expressions using the properties of operations. Students also simpliﬁed expressions using the order of operations. They have experience using the distributive property with numeric and algebraic expressions in prior Algebra I scopes.

These skills will be further developed throughout Algebra I when working with quadratics as students will identify key features of functions in standard and other quadratic forms. In Algebra II, students will continue multiplying binomials to create and study polynomials of higher degree.

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

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES

apply the properties of operations to generate equivalent expressions.

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: •

relate polynomial operations 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.

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Intervention

Acceleration

Adding and Subtracting Polynomials In this exploration, groups of students will solve a realworld scenario about an art gallery needing to interpret an artist’s instructions to correctly assemble art mosaics. Students will: •

Explore 2

Explore 1

EXPLORE ACTIVITIES

model adding and subtracting polynomials using algebra tiles.

•

multiply polynomials.

Special Products In this exploration, groups of students will be tasked with helping solve the codes to open four gates leading to a classroom to retrieve a phone. 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 5

apply the distributive property to simplify expressions involving the multiplication of expressions.

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

Explore 4

Explore 3

In this exploration, students will be assisting an architectural landscaping company to help design a garden and pool for a family home. Students will:

In this exploration, students will be tasked with organizing different art collections using the distributive property to simplify expressions involving the multiplication of expressions. Students will also be tasked with organizing the different art collections with tiles that represent various aspects to create an image. 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.

Multiplying Polynomials

Multiplying Polynomials with Models

POLYNOMIAL OPERATIONS

Home

identify patterns in binomials that produce special products.

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

Standard Form of a Quadratic Function In this exploration, students will rewrite quadratic functions in vertex form, factored form, or other forms to create equivalent quadratic functions in standard form. Students will: •

use the function in standard form to determine information about a given scenario.

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

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Polynomial Operations 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 play Always, Sometimes, Never to determine whether statements or claims based on the prior standard are sometimes true, always true, or never true. This element is designed to uncover student misconceptions; it should not be taken for a grade. 6.PAR.6.5 Apply the properties of operations to identify and generate equivalent expressions.

Materials Printed •

Preparation •

1 Always, Sometimes, Never (per student or per group) • •

POLYNOMIAL OPERATIONS

Home

Print one copy of Always, Sometimes, Never for each student or group. Please note that the print document includes a color version and a black-and-white version. Select the version that works best for your classroom. Cut out one set of cards for each student or group. Consider laminating the cards for repeated use.

Procedure and Facilitation Points 1. 2.

3. 4.

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

The order in which two different terms are added does not matter. A When two terms are multiplied, the solution has a higher degree. S A number and its reciprocal, when multiplied, will equal 1. A The order in which two different terms are divided does not matter. N When two terms are divided, the solution has a lower degree. S The order in which two different terms are subtracted does not matter. N

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

Identifying Misconceptions •

• •

Students may think that a term always has to have a variable attached to it. In the second example, they may feel the correct answer is always true. Have them think about situations like (9x (9x)(2) = 18x 18x. The degree does not change in this case, so the correct answer for the second example is, in fact, sometimes true. The same concept can be applied when dividing terms. Offer an example like (10x) ÷ 5 = 2xx to help students understand. Students may also confuse the commutative and associative properties that apply to addition and multiplication but do not apply to subtraction or division.

FACILITATION TIP Before distributing the cards, ask students what they think they know about mathematical operations. If necessary, coach them to focus on addition, subtraction, multiplication, and division. FACILITATION TIP Allow students to jot down notes and ideas on paper as they think. FACILITATION TIP A change of one word can alter if a statement is always, sometimes, or never true. Encourage students to read each statement carefully during their thinking time.

FACILITATION TIP This Foundation Builder includes 18 expressions that can be projected. If desired, challenge students to generate equivalent expressions rather than match them up. It would provide a good review activity for all students.

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Polynomial Operations Hook – Solar Panels ACTIVITY PREPARATION Students will relate polynomial operations to a real-world situation.

Materials

Preparation

Printed •

• • •

1 Solar Panels (per class)

Reusable • •

1 Phenomena Video (per class) 1 Projector (per class)

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

PROCEDURE AND FACILITATION Part I: Pre-Explore 1.

FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) Has anyone had experience with solar panels?; 2) Do you have solar panels on your house?; 3) Can you think of other situations where solar panels would be useful? FACILITATION TIP Project the text of this scenario so students can write down values, variables, and expressions. Read it aloud together and coach students to locate the essential information.

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Explain the situation while showing the video behind you: A local farm sets up solar panels to generate more energy. There are panels that are 150 feet long that need to be fenced in so they are not disturbed by animals. The panels need to be spaced apart so they do not block light from each other and must also be far enough from the fence so that routine maintenance can be performed. If the distance between the panels and fencing is x, write an expression for the area and perimeter of the rectangular fencing for the solar panels. 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 length of a solar panel is 150 feet and that value is in the width expression. I notice that there are 40 panels and that the length is 41x. I notice the length in the video seems longer than the width. I wonder why the length is not 40x because there are 40 panels. I wonder how to ﬁnd the area and perimeter without numerical values. Project Solar Panels. Notes

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Explore

Explain

Elaborate

Evaluate

Explain to students that the goal for this unit is to connect operations with integers to operations with polynomial expressions. Discuss the following questions: a.

Engage

DOK-1 How do you ﬁnd the perimeter and area of a rectangle? Add up the length of the four sides for the perimeter or double the sum of the length and the width. Multiply the length by the width to determine the area.

DOK-2 What kinds of numbers would make sense to substitute in for x? Positive values that would be easy for people on the farm to measure

DOK-1 What does the 150 in the width equation represent, and why is 2x added to it? The 150 represents the length of the solar panel, and the additional term represents the space on either side of the panel before the fence.

Acceleration

FACILITATION TIP As a ﬁnal question for the Pre-Explore, ask students why they think farmers would choose solar energy and what they think farmers could best use it for. FACILITATION TIP Provide a quick review of perimeter and area for other shapes. After students answer the question, ask them how to ﬁnd the perimeter and area for a square and triangle. STEMscopes Tip

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Intervention

POLYNOMIAL OPERATIONS

Home

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

DOK-1 What strategies would you use for ﬁnding the area and perimeter? I would combine like terms and add for the perimeter, and I would use the distributive property to multiply the terms for the area.

DOK-2 How does the work required to ﬁnd the perimeter and area with polynomial expressions connect to the work with integers? The operations of addition and subtraction work similarly with the coeﬃcients of like terms. Multiplication is similar because we can use an area model to help distribute.

DOK-1 What are the perimeter and area of the fence? The perimeter is 86x + 300 feet, and the area is 82x2 + 6,150x square feet.

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

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. FACILITATION TIP Students may overthink or otherwise struggle to understand the question. If so, ask them how ﬁnding the perimeter and area with polynomial expressions is similar to working with integers.

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Polynomial Operations Explore 1 – Adding and Subtracting Polynomials ACTIVITY PREPARATION Students will interpret and deﬁne algebra tiles. Students will model adding and subtracting polynomials using algebra tiles.

Standards for Mathematical Practice • • •

MP.5 Model with mathematics. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

1 Student Journal (per student) 1 Set of Before and After Collection Cards (per group) 1 Exit Ticket (per 2 students)

• • • •

Reusable • • •

1 Set of algebra tiles (per student) 1 Set of colored pencils (per group) 16 Crates (per class)

•

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 has one. Print a set of Before and After Collection Cards, on card stock for durability, for each group of students. Cut the cards apart, and put them in “crates” (crates can be boxes, envelopes, folders, etc.) around your room. Label each crate with the corresponding letter, A–P. 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. For students who need additional organizational support, please see our GEMDAS and Algebra Tiles Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever been to an art gallery before?; 2) Whose work was showcased there?; 3) What was your favorite piece?

FACILITATION TIP Communication is particularly important in this Explore activity, speciﬁcally in Part II. In Part II, groups may use a strategy to sort crates that is more understandable to some members than to others. As much as possible, choose groups before Part I so that each group is led by students who can communicate clearly and patiently with their peers. 162

2. 3. 4.

Read the following scenario to the class: You have been hired by an art gallery to help curate art mosaics by the great artist Algebra Mathematica. The artist has left instructions to help you assemble each piece of art, but they didn’t include a legend or pictures. The art gallery wants you to interpret the artist’s instructions to correctly assemble each piece. Give a Student Journal to each student. Explain to students that they will work with their groups to match each tile piece to a term and then justify their choice and record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding question to assess student understanding: a.

5. 6. 7.

DOK-1 How do you ﬁnd the area of a square or rectangle? To ﬁnd the area, you multiply the length times the width.

Allow students enough time to complete Part I and answer the questions that follow. As students ﬁnish Part I, give a set of algebra tiles to each student. After Part I, invite the class to a Math Chat to share their observations and learning.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Math Chat • •

• •

•

DOK-2 What if the blue tile had dimensions of x units by x units? What term would it represent? The tile would have an area of x2 units. It would represent x2. DOK-2 What if the green tile had dimensions of x units by 1 unit? What term would it represent? The tile would have an area of x units squared. It would represent the variable x. DOK-2 What would it take to represent 4? Model this with your algebra tiles. To represent 4, we would use 4 yellow tiles. DOK-3 What would it take to represent the opposite of 4? Model this with your algebra tiles. We would use the opposite side of the yellow unit tiles. The opposite of 4 would be 4 small red square tiles. DOK-2 What would it take to represent 3A? 3 ? Model this with your algebra tiles. To represent 3A, we would use 3 green rectangles. DOK-3 What would it take to represent the opposite of 3A 3A? Model this with your algebra tiles. We would use red tiles of the same dimension as 3A. This would be a way to represent −3A because the opposite of 3 is negative 3. DOK-2 What would it take to represent 2 2A2? Model this with your algebra tiles. To represent 2A2, we would use two large blue squares. DOK-3 What would it take to represent the opposite of 2 2A2? Model this with your algebra tiles. We would use the opposite side of the A2 tile. The opposite of 2A2 would be two large red squares. DOK-3 If the blue square is A units on all sides and the green rectangle has a dimension of A units, how do those sides compare? Which one is longer? Since they both have a measure of A units, they are the same length. Neither is longer because they have the same measure. DOK-2 What happens when a number and its opposite are combined? They are called zero pairs, and they equal zero. It could also be said that they are zero pairs and they cancel.

FACILITATION TIP Students may overthink this question. They may incorrectly compare it to the ﬁrst Math Chat question and try to take the square root of 4. Emphasize to the class at the beginning of the Math Chat that they are to focus on the types of algebra tiles they are given.

POLYNOMIAL OPERATIONS

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FACILITATION TIP After students answer this question, ask them if the last several questions remind them of a previous scope. Some may remember Systems of Equations, where zero pairs were formed when using elimination to solve a system.

Part II 1.

2. 3.

4. 5.

Read the following scenario to the class: The collection of pieces you are working on ﬁrst is called the Before and After Collection. You have been told the before piece is rather unorganized, and the after piece is a minimalist, organized, and neatly arranged piece. Each piece of art has a before title, before mosaic piece, after title, and after mosaic piece. The art was carefully packed and shipped in crates. The crates were delivered and stored around the room. Visit each crate to retrieve a piece of art, and begin to curate the collection. Students should still have the algebra tiles and their Student Journals. Explain to students that they will work with their groups to curate the collection. They should retrieve one piece of art from each crate and organize the art so each named set has a before title (algebraic expression), before mosaic, after title, and after mosaic. They will record their work on their Student Journals. Point out to the class that modeling the before mosaics with the algebra tiles might help them ﬁgure out the after mosaics and after titles. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How do you know if you should add or subtract the polynomials? When there is a plus symbol between the polynomials, we will add. When there is a subtraction symbol between the polynomials, we will subtract.

FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone seen a mosaic before?; 2) What was it a picture of?; 3) What material was used?

FACILITATION TIP Some set names offer a more obvious clue as to which crates are associated with them than others. Encourage students to focus on matching the correct algebraic representations with the correct models. They can then match each group of crates with their respective set name. FACILITATION TIP Preview question 5a with students before they begin to collaborate.

DOK-2 Why are all of the tiles red if the polynomial is A2 + 2 2A A + 3? All the coeﬃcients are positive, so why would red tiles be used? The tiles are red because each term of the polynomial is being subtracted. To show subtraction, we use the opposite side of the tile.

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Polynomial Operations Explore 1 – Adding and Subtracting Polynomials

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.

DOK-2 This title says 2 2A2 plus 2 2A. Do we combine them? Why or why not? The variables are the same letter, but they have a different exponent. Since they have a different exponent, they are not like terms and we do not combine them.

DOK-1 How is the distributive property used when subtracting polynomials? The subtraction symbol is distributed like a coeﬃcient of negative one.

e. DOK-2 If a before title was evaluated when a = 3 and the corresponding after title was evaluated when a = 3, would they have the same result? Why or why not? They would have the same result because they are equivalent expressions. When equivalent expressions are evaluated using the same number, they will always produce the same result.

FACILITATION TIP Some students may overthink this question. Encourage them to do what the question describes and compare the results to then answer the question.

7. 8.

As you notice that students are finishing organizing their collection, ask them to see if they can find meaning behind the collection name. Thrice Duce has an after title with three twos. Missing Middle has an after title with no middle term, just A2 and a constant. Opposites 7 to 1 has 7 A2 tiles in the beginning, but using opposite pairs, we end up with only 1 A2 tile. So Odd has an after title with all odd coefficients. 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 What did you start with? Did you pick a mosaic and then find the title, or did you pick a title and then find the mosaic? Which way was easier? Answers will vary. • DOK-3 How were the tiles used to represent subtraction? To represent subtracting a polynomial, the tiles were flipped. If it was representing subtracting a positive, the tile was shown with the red side facing up. If it was representing subtracting a negative, the tile was shown with the positive side face up. •

Part III FACILITATION TIP Before reading the scenario, ask the class 1) Think of an art piece you created. What kind of art was it?; 2) How old were you when you made it?; 3) Who did you show it to, if anyone?

2. 3. 4.

FACILITATION TIP Use your discretion as to whether or not students within a group should each have a unique piece. In either case, they should check each other’s work. FACILITATION TIP Ask this question by a show of hands. For students who raise their hands to “creating the title,” ask them if they typically prefer learning with words and numbers. For students who raise their hands to “drawing a model mosaic,” ask them if they typically prefer learning with pictures.

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5. 6. 7.

Read the following scenario to the class: Algebra Mathematica is so impressed by your work that they have offered to mentor you to help you complete your own piece of mosaic art to be included in the Before and After Collection. Use the tiles provided to create your piece for the Before and After Collection. Then, sketch your art in the space provided. Distribute colored pencils to each group. Students should still have their Student Journals, Before and After Collection Cards, and algebra tiles. Explain to students that they will work with their groups to create their own pieces for the Before and After Collection. Point out to the class that their pieces should be unique, but they can draw inspiration from the artwork in the original collection. Students will then work together to create their own pieces for the Before and After Collection and record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding question to assess student understanding: a.

DOK-1 Do you think it will be easier to start by drawing a model mosaic or by creating the title? Why? Answers will vary. Some students will ﬁnd the model an easier place to start, while others will prefer the algebraic representation. If a student is struggling to get started, ask them to get out some tiles and start making a model. Remind them that this is their creation, so their mosaic can’t be wrong.

8. 9.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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-2 How does adding and subtracting polynomials compare to addition and subtraction with integers? Just like integers, when polynomials are added and subtracted, the result is always a polynomial. The coeﬃcients with like terms combine exactly as integers do. DOK-2 Is it possible to add or subtract polynomials and have the result not be a polynomial? As we combine like terms in the polynomials, the result still has variables raised to natural numbers with coeﬃcients, thus remaining a polynomial. DOK-2 Compare and contrast combining like terms with adding and subtracting polynomials. Adding polynomials is the same as combining like terms. Subtracting polynomials has one extra step to distribute the negative sign. DOK-2 Are x2 and x like terms? How do you know? No, x2 and x are not like terms because they have different exponents. Like terms are the same letter or letters with exactly the same exponents.

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.

POLYNOMIAL OPERATIONS

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Post-Explore 1. 2. 3.

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

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Polynomial Operations Explore 2 – Multiplying Polynomials with Models ACTIVITY PREPARATION Students will apply the distributive property to simplify expressions involving the multiplication of expressions of degree two. An area model can be used to represent the multiplication of each term.

Standards for Mathematical Practice • • •

MP.5 Model with mathematics. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • •

• • •

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

Reusable • •

•

1 Set of algebra tiles (per student) 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. 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. For students who need additional organizational support, please see our GEMDAS and Algebra Tiles Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Do you have a favorite visual artist?; 2) What materials (clay, wood, paint, etc.) and creation techniques (sculpture, printmaking, watercolor, etc.) do they use?; 3) Where can one ﬁnd their work?

FACILITATION TIP Some students may have forgotten about or not heard of the term multiplicand. Explain that the multiplicand is the ﬁrst factor and the multiplier is the second factor.

2. 3. 4.

FACILITATION TIP Students may be confused by the wording of Question 3. If so, note that the original equation means the art gallery received Set A, and “lost” is the opposite of “received.” Then, ask them what the opposite of the original equation is. 166

6. 7.

Read the following scenario to the class: Due to the popularity of certain pieces in the Before and After Collection, the art gallery has created a new collection called the Times Up Collection. This new collection uses certain sets from the Before and After Collection. The art gallery has ordered all of the necessary materials to create the new collection, and they have asked you to organize the tiles based on the descriptions from the artist. Each set in the collection has 3 pieces: the multiplicand (factor), the multiplier (factor), and the product. The three pieces can be used to create an image for the series. Give a Student Journal and a set of algebra tiles to each student. Give a set of colored pencils to each group. Explain to students that they will work with their groups and the information provided to identify the multiplicand, multiplier, and product for each art set 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 happens when we multiply a positive value by a negative number? The product is negative.

DOK-1 What happens when we multiply a negative value by a negative number? The product is positive.

DOK-1 How do we write the product of a monomial and polynomial? In descending order by degree

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Math Chat DOK-1 What property is being used during this process of multiplying polynomials? The distributive property is being used. • DOK-2 Do you think the process would change if we had another row in our array? No, the process would remain the same. We would still use the distributive property and just multiply one row at a time. • DOK-1 How do you know how to set up your area model? You look at the two factors you are multiplying. If the ﬁrst factor is a monomial, you only need one row. If the second factor is a binomial, you would need two columns, but if the second factor is a trinomial, you would need three columns. •

POLYNOMIAL OPERATIONS

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Part II 1.

2. 3.

Read the following scenario to the class: There are two final sets in the Times Up Collection. Use the algebraic expressions from the artist to determine the product, image, and associated array for each set. Students should still have their algebra tiles and their Student Journals. Explain to students that they will work with their groups and the information provided to identify the multiplicand, multiplier, and product for each art set and record their work on their Student Journals. Students will then work together to finish piecing together the two final sets in the Times Up Collection and record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding:

FACILITATION TIP Before reading the scenario, ask the class 1) Who has artwork in your house?; 2) What kind of artwork is it?; 3) In what room is it located? FACILITATION TIP

If students struggle to create an image or array, direct them to the image for Set C. Show them how to follow the linear tiles to establish the new quadratic term. Note to a. DOK-1 How can you tell right away what the highest term of your them that, for each set, the array represents product will be? Look at the two highest terms from each factor, and the same factors and product as the use the properties of exponents to add the degree of each term. For image. Therefore, they can follow rows and example, x2 multiplied by x2 would result in x4. columns in the same manner for the array 6. Allow students enough time to complete Part II and answer the reflection questions. as they did for the image. 7. After the Explore activity, invite the class to a Math Chat to share their FACILITATION TIP observations and learning. While students have presumably worked with vertical multiplication, they may Math Chat have forgotten or be unfamiliar with the • DOK-2 Compare and contrast the algebra tile method and the array area model terminology. Before students start the method. If you place the multiplicand and multiplier in the same order when Reflect section, check that everyone multiplying, the algebra tiles and area models will end up looking very similar. understands the term, as they will see it in 2 Generally speaking, the x term is in the upper left corner, constants are in the Question 2. lower right corner, and any x terms are in the alternate corners. Algebra tiles are FACILITATION TIP the visual representation, while the area model is the numerical representation of the multiplication. When you discuss comparing methods for Another difference between the two is that algebra tiles are more limited in what multiplying polynomials, closely check for they can represent. For example, they can only show polynomials with one variable. understanding and fluency with the array area model. Reinforce to students that • DOK-1 Describe the process you used to figure out what the product would look they will need to use the area model on the like for the art pieces. I looked at the highest degree term for each factor I was upcoming Exit Ticket. multiplying. I knew based on the properties of exponents that if I multiplied an x by an x, I would get x2. I had to look at components like that to determine what each piece of art would look like in the end. 5.

Post-Explore 1. 2. 3.

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

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Polynomial Operations Explore 3 – Multiplying Polynomials ACTIVITY PREPARATION Students will multiply polynomials with several terms. The polynomials may contain multiple variables, real-number coeﬃcients, and whole number exponents. Area models will still be used to multiply until students develop more eﬃcient strategies.

Standards for Mathematical Practice • • •

MP.5 Model with mathematics. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning.

Materials Printed • • • •

1 Student Journal (per student) 1 Set of Family Backyard Design Cards (per group) 1 Set of Summer Project Cards (per group) 1 Exit Ticket (per 2 students)

Reusable •

2 Resealable bags (per group)

Preparation • • • •

•

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 a set of Family Backyard Design Cards for each group. Cut out each set of cards, 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 Summer Project Cards for each group. Cut out each set of cards, and place them in a resealable bag labeled “Part II.” If desired, print them on card stock, and laminate them for future use. For students who need additional organizational support, please see our GEMDAS and Algebra Tiles Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever done landscaping work before?; 2) What kind of work did you do?; 3) What season was it?

FACILITATION TIP Remind students that many math assessments include images that are not drawn to scale. The Family Backyard Design Cards are a good example of that common practice. Reassure students that you are aware that they are not drawn to scale, and instruct them to focus on the algebra tiles to complete Part I.

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2. 3. 4. 5.

Read the following scenario to the class: You are interning over the summer with a local architectural landscaping company. They have been super busy recently with projects, and you have been asked to assist the architects with a few of those projects. To start, you will review the plans for the Challa family’s backyard. The ﬁrm has had to make some recent adjustments to their original plans. Review the plans for the pool and garden before taking on your own projects. Give a Student Journal to each student. Distribute the Family Backyard Design Cards to each group. Explain to students that they will work with their groups to analyze the Challa family’s backyard project 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 What is the resulting degree when you multiply a term of degree one by a term of degree one, like x times x? The resulting product is a term with a degree of two.

DOK-2 What is the importance of circling the like terms in your area model? It is important so you don’t forget to combine them in your simpliﬁed product.

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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 Do you see any similarities or differences between the area model for multiplying binomials and the array for multiplying two-digit numbers? Yes, the method is the same. For multiplying two-digit numbers, we combine all of the numbers inside the array. For the binomial area model, we are only combining like terms.

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Part II 1.

2. 3. 4. 5.

Read the following scenario to the class: Now that you have successfully ﬁnished the Challa family’s project, the architecture ﬁrm has tasked you with two more projects to complete. Distribute the Summer Project Cards to each group of students. Students should still have their Student Journals. Explain to students that they will work with their groups to determine the necessary information for each project 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 was the extra step you needed to complete in the vegetable garden project? We had to take the product from multiplying the base 1 1 and height and then multiply that by __2 . Or I multiplied the height by __2 , and then I multiplied the result by the base.

DOK-1 Were there any like terms to combine in the area model you 1 created in the vegetable garden project that was __2 multiplied by a polynomial? Why do you think that is? No, there were not. We did not have more than one row of terms, so there would be no like terms that needed combining.

FACILITATION TIP Before reading the scenario, ask the class 1) How would you landscape your dream house?; 2) Would you grow any plants?; 3) Would you pay to have it done or do it with friends? FACILITATION TIP Clarify the pool’s concrete border sections, so that students don’t mistake them for algebra tiles. Point out that, if they were algebra tiles, students should expect to see rectangles to represent linear terms.

FACILITATION TIP Allow students enough time to complete Part II and answer the reflection questions. Students may be used to circling the upper After the Explore activity, invite the class to a Math Chat to share their observations right and lower left terms for like terms in and learning. the area model. However, this only works when the terms of the factors are in a certain Math Chat order. For the vegetable garden, remind • DOK-2 What might you have to consider in the garden project in terms of soil? students to circle terms with the same When planting, you have to have layers of soil, so you might have to consider the degree of x. If helpful, they can switch the height of the garden as well. order of a factor’s terms before solving. • DOK-2 If you did consider the height in the garden project, what would you FACILITATION TIP be finding and how might you think it would be represented as a polynomial? It may not come naturally to some students We would need a third expression for volume, and we might expect to see a to think of height. Some may think of other polynomial of degree 3 to represent the volume. aspects, like cost or type of soil. If students • DOK-2 When multiplying multi-digit integers, is using an array the only way, or struggle to understand this question, ask are there multiple methods? If there are other methods, could they also work to them if there is another dimension they may multiply polynomials? Explain. No, using an array is not the only method. There is have to consider with soil. also vertical multiplication. This could also work for multiplying polynomials. • DOK-2 How does multiplying polynomials compare to multiplication with integers? Just like integers, when polynomials are multiplied, the result is always a polynomial. The terms are distributed and combined in the same manner as the integers. • DOK-2 Is it possible to multiply two polynomials together and have the result not be a polynomial? As we multiply polynomials, the coefficients and exponents on variables will likely change, but the results will still be a polynomial since the exponents will be natural numbers and the coefficients will still be real. 6. 7.

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Polynomial Operations Explore 3 – Multiplying Polynomials Post-Explore 1. FACILITATION TIP On this Exit Ticket, be prepared for some students to want to use other methods to multiply the polynomials. When you preview the Exit Ticket with students, clarify your criteria for success related to showing strategies.

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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Polynomial Operations Explore 4 – Special Products ACTIVITY PREPARATION Students will identify patterns in binomials that produce special products. They will recognize that these binomials have predictable terms, and, therefore, there are more eﬃcient ways to multiply them than using an area model.

Standards for Mathematical Practice • • •

MP.5 Model with mathematics. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning.

Materials Printed • • •

1 Student Journal (per student) 1 Set of Gate Code Cards (per student) 1 Exit Ticket (per 2 students)

Reusable • •

Preparation • • • • •

Separate the class into groups of 2 or 3 students. Print a Student Journal and a set of Gate Code Cards for each student. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student has one. Gather a pair of scissors for each student and a glue stick for each group. For students who need additional organizational support, please see our GEMDAS and Algebra Tiles Supplemental Aids elements in the Intervention section.

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

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever left something at school before?; 2) Did you remember where you put it?; 3) Were you able to recover it?

FACILITATION TIP Students may change their answers multiple times. Encourage groups to conclude where all cards go before taping or gluing them in place. It may help to attach cards with sticky notes, which are easier to remove if a student wants to change an answer later on.

2. 3.

4. 5.

FACILITATION TIP Students may put a Code card in its respective Gate space and, granted, it represents an equivalent expression. It may help to verify that groups have the right Gate and Code cards for the ﬁrst gate. Then, you can tell them to pay attention to patterns as they ﬁnd cards for the other gates. 172

Read the following scenario to the class: Yikes! You left your phone on the desk in your algebra class and realized where it was when you got home. You have returned to school with two friends, but everyone is gone except one person from the front oﬃce staff. They have informed you that maintenance has locked all of the gates around the inside of the school, making it impossible for you to get back to your class unless you and your friends can crack the gate codes to unlock each of the four gates leading to your classroom. To retrieve your phone, you need to unlock four separate gates by simplifying each gate and its corresponding code. Give a Student Journal, a set of Gate Code Cards, and a pair of scissors to each student. Give a glue stick to each group. Explain to students that they will work with their groups to identify the patterns they see when multiplying these types of binomials. Students will apply their knowledge of the distributive property and simplifying expressions to solve each piece of the code breaker in Part I. Point out to the class that they will need to cut their set of Gate Code Cards before they begin their work. Note: Not all Gate Code Cards will be used. Students will work together to simplify each gate in the column on the left and the corresponding code in the column on the right. They will share their ideas with their groups 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 types of polynomials are we multiplying? Binomials

DOK-1 What types of polynomials are we seeing as the product of the binomial multiplication? Trinomials

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

DOK-1 What do you notice about the two binomials that are being multiplied together in all of the gate boxes? They are the same binomial.

d. DOK-1 What is a shorter way to write 3 · 3 · 3 · 3? A shorter way to write it is to use exponents, so the expression 3 · 3 · 3 · 3 would be written as 34. e. DOK-1 What is a shorter way to write A · A · A? A shorter way to write it is to use exponents, so the expression A · A · A would be written as A3. f.

DOK-2 When squaring a term that has a variable and a coefficient, why is it important to write it using parentheses? What would happen if we did not write this situation with parentheses? It is important to use parentheses in this type of situation because you want to make sure you know to square both the variable and the coefficient. If we forgot to put the parentheses around the entire term, it only means to square the variable and not the coefficient.

DOK-2 When comparing the two gates on the first page of Part I, what do you notice between the gate and code? Answers will vary. I notice that the first gate uses addition, and the second gate uses subtraction. I notice that the operation signs for the first gate and code are all addition. The second gate uses subtraction only, but the code has addition and subtraction.

STEMscopes Tip The Skills Quiz, located in the Evaluate section, is a short standardsbased assessment where students demonstrate their computational ﬂuency. These assessments include a variety of question types and can be used to formatively evaluate students’ knowledge about topics covered in the scope or to review the content.

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h. DOK-2 Why do you think we see addition in the second code expression on the ﬁrst page of Part I? When you multiply two negative terms together, it creates a positive term. We can see that each of the circles for the corresponding gate has a negative value. Thus, if we multiply a negative circle by a negative circle, our result would be a positive circle squared. 7.

Allow students enough time to complete Part I and answer the questions that FACILITATION TIP follow. After students complete Question 4. 8. After Part I, invite the class to a Math Chat to share their observations and learning. ask them if it matters which shape is designated for which term, and have them Math Chat explain why or why not. Do the same after • DOK-1 Using numbers, what is an example of a perfect square? 3 · 3 = 9, they answer Question 6. 5 · 5 = 25, 7 · 7 = 49 • DOK-2 How does the concept of perfect squares apply to the binomials we saw in Part I? The concept is the same. A perfect square signiﬁes a particular value being multiplied by itself. With the binomials in Part I, we can observe that we have two identical binomials being multiplied together. • DOK-1 In Part I, we saw two identical binomials being multiplied together, and we connected them to perfect squares. If we see that the simpliﬁed product is a trinomial, what can we name it? We can name it a perfect square trinomial. Part II 1.

2. 3. 4.

Read the following scenario to the class: Success! You have retrieved your phone and now need to make your way back out of the school, except there is one problem—you can’t go out the way you came in because the custodians are now mopping the hallway you just came from. You look around and find an alternate route to exit, but you will now have to unlock a few more gates. These gates are a bit different from the first set, so grab your friends and go break those codes! Explain to students that they will work with their groups to establish what patterns are present in the multiplication of this new set of binomials. Point out to the class that they are only using variables, coefficients, and constants. They will no longer be using the shapes to represent each problem. Students will then work together to solve each gate code and create a list of steps that would apply to any situation where these types of binomials are present and record their work on their Student Journals.

FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever been in an escape room before?; 2) What was the theme?; 3) Did you get out in time?

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Polynomial Operations Explore 4 – Special Products 5.

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

DOK-2 What is different about the binomials we see in Part II versus the binomials we saw in Part I? The binomials are almost identical. In Part I, we saw two identical binomials that used the same operation sign to separate the two terms. In Part II, we see that the operation signs separating the terms are different. One set of parentheses uses addition, while the other uses subtraction.

DOK-1 What patterns are present in all of the codes in Part II? The product consists of two perfect square terms separated by a subtraction sign.

DOK-1 What do you notice about the middle terms when completing the multiplication in Part II? The middle terms are a zero pair and cancel each other out when we simplify because one is positive and the other is negative.

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.

FACILITATION TIP Check students’ understanding of operations. After they complete Part II, ask them if (a + b)(b − a) will still yield a2 − b2. Have them answer without doing any work, and have them justify their reasoning.

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-3 The resulting polynomials in Part II were all binomials. What do you notice about each of the terms? What can we name this type of situation? Each term is a perfect square, and they are separated by a subtraction symbol. We can name this situation a difference of two squares. • DOK-2 In the examples in this part of the Explore activity, two binomials were multiplied, and the result was still a binomial. Is it possible to multiply two binomials together and have the result not be some type of polynomial? No, because we have seen that polynomials are closed under addition, subtraction, and multiplication. So even if these special products differ slightly from other results we have seen, the result will always be a polynomial. •

FACILITATION TIP

Post-Explore

After students complete the Exit Ticket, ask them which answer choice in Question 1 does not represent a special product. Then, ask them what they need to change in the given expression to get answer choice D.

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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Polynomial Operations Explore 5 – Standard Form of a Quadratic Function ACTIVITY PREPARATION Students will rewrite quadratic functions in vertex form, factored form, or other forms to create equivalent quadratic functions in standard form. Students will use the function in standard form to determine information about a given scenario.

Standards for Mathematical Practice • • •

MP.5 Model with mathematics. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Blueprint (per group) 1 Set of Student Equations (per group) 1 Exit Ticket (per 2 students)

•

Reusable • • •

•

1 Dry-erase marker (per group) 1 Sheet protector (per group) 2 Resealable bags (per group)

Separate 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 has one. Print a Blueprint, on card stock for durability, for each group of students. Place it inside a sheet protector to create an erasable surface. Print a set of Student Equations Cards, on card stock for durability, for each group of students. Cut the cards apart for Part I, and place them in a resealable bag. Label the bag “Part I.” Cut the cards apart for Part II, and place them in a resealable bag. Label the bag “Part II.”

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario engage students by asking about their experience with home improvement projects, family members who are contractors, or observations about the tile ﬂoors in your school. FACILITATION TIP Project the text of the scenario and read it aloud along with students. Consider showing the Student Equation Part I Cards and Blueprint to students before distributing the Student Journal. Allow students to make observations and ask questions before they begin collaborating.

2. 3. 4. 5.

Read the following scenario to the class: A group of students is doing jobs for their neighbors to earn money over the summer. Sai, Maria, and James are making plans and gathering materials to tile the ﬂoor of their neighbor’s home. They have created equations to represent the area of the home that needs to be retiled. However, they all have different equations. Help them decide who’s correct and how much ﬂooring to purchase. Give a Student Journal to each student. Give a Blueprint, a bag of Student Equations Part I cards, and a dry-erase marker to each group. Explain to students that they will work with their groups to analyze the equations, determine who’s correct, 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 the Blueprint be separated into 4 rectangles? Did any of the students use this method to write their equation? The blueprint can be separated into 4 rectangles by separating the shaded region into 3 rectangles. Sai used this method.

DOK-1 How can the total area of the house be written? What expression would need to be subtracted to represent the shaded area only? Did any of the students use this method to write their equation? The total area can be written using the binomial 2x + 4 for the width and x + 3 for the length. Use the formula for area, A = lw. Then, subtract the expression for the unshaded area, 2x2. Maria used this method.

FACILITATION TIP The scenario asks students to decide which student is correct. Consider whether you want to alert students to the fact that more than one students’ equation might be correct.

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DOK-2 Can the shaded region be separated into two rectangles? How could the area of each rectangle be used to find the total area of the shaded region? Did any of the students use this method? The shaded region could be separated into two rectangles. One way is to use the rectangle with a width of 4 and length of x + 3. Then, use the area of the two rectangles to find the total area by adding them. James used this method. DOK-1 When simplifying Sai’s equation, what operation should be done first, multiplication or addition? Explain. The first operation should be to multiply. After multiplication, the like terms can be combined. The order of operations says we should multiply before adding.

e. DOK-1 When simplifying James’s equation, what operation should be done ﬁrst, multiplication or addition? Explain. The ﬁrst operation should be to multiply, including distribution. After multiplication, the like terms can be combined. The order of operations says we should multiply before adding. f.

DOK-1 When simplifying Maria’s equation, should the binomials ((xx + 3) and (2xx + 4) be multiplied or added? How do you know? The binomials should be multiplied. I know this because when parentheses are used, it means to multiply. If they should be added, there would be an additional symbol between the parentheses.

DOK-2 Given the area of the shaded region is 212 square feet, will that be the input or output for the equation? The 212 square feet is the output, so it can be substituted for A(x).

Intervention

Acceleration

STEMscopes Tip The STEMscopes Teacher Toolbox, located under the Scopes tab on the menu bar, features a variety of resources and tools to help teachers get the get most out of their STEMscopes experience, including essentials like lesson-planning documents, intervention strategies, monitoring tools, mathematical discourse strategies, and data resources.

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Allow students enough time to complete Part I and answer the questions FACILITATION TIP that follow. Consider distributing the Student Journal 7. After Part I, invite the class to a Math Chat to share their observations and learning. one part at a time as needed. Part I is page 1. Part II is pages 2–4. Math Chat • •

DOK-1 When rewriting a quadratic function to standard form, what is typically the first step? Typically, the first step is multiplication to include distribution. DOK-2 How do you know when a quadratic equation is in standard form? What if it only has two terms? The quadratic equation is in standard form when there are at most 3 terms and they are in order of x2 term first, the x term second, and the constant last. If the quadratic equation only has 2 terms, they should still be in order of highest degree first.

Part II 1.

2. 3. 4. 5.

Read the following scenario to the class: The next project they are working on is recording data for a neighbor who is launching a hobby rocket. Each student used their data and generated a different equation. Who, if anyone, is correct? Give a set of Student Equations Part II cards to each group of students. Each student should still have their Student Journal. Explain to students that they will work with their groups to analyze the equations and compare them to the data collected to determine who is correct. Students will then work together to rewrite quadratic functions in vertex form and factored form to create equivalent quadratic functions in standard form. They will use the function in standard form to determine information about the scenario 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 What numbers do you see when you compare the data and the equation? The equation for Sai has a 5 and 90, just like his data. The equation for Maria has an 11, just like her data. James’s equation has a 27.5, just like his data.

FACILITATION TIP This scenario also asks whose equation is correct. Determine ahead of time if you want to alert students to the fact that more than one equation may be accurate.

FACILITATION TIP Print and project guiding questions 6a–6h and determine which ones to preview before students begin collaborating. Encourage students to be prepared with answers when they are ﬁnished with Part II. 177

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Polynomial Operations Explore 5 – Standard Form of a Quadratic Function b.

DOK-2 When you compare the three students’ equations, what number do all the students have in their equation? They all have a coeﬃcient of −2.5.

c. DOK-1 When simplifying Sai’s equation, what operation should be done ﬁrst, exponent, multiplication, or addition? Explain. The ﬁrst operation should be to multiply. After multiplication, the like terms can be combined. The order of operations says we should multiply before adding. d.

DOK-1 When simplifying James’s equation, what operation should be done ﬁrst, multiplication or addition? Explain. The equation is already in standard form. No operations are needed.

e. DOK-1 When simplifying Maria’s equation, should the binomials ((xx + 1) and ((xx – 11) be multiplied or added? How do you know? The binomials should be multiplied. I know this because when parentheses are used, it means to multiply. If they should be added, there would be an additional symbol between the parentheses.

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.

DOK-1 When using substitution to determine ff(0), (0), is zero the input or output? The zero is the input.

DOK-2 Which variable is used to represent seconds? The variable x is used to represent seconds.

h. DOK-2 When a rocket has landed, what is the height above the ground? The height above the ground will be 0 meters. 7. 8.

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 Describe the process to determine if two equations are equivalent algebraically. Rewrite both equations in the same form. Usually, this will be standard form. If they are exactly the same when written in standard form, then the equations are equivalent. • DOK-2 How do you know when an equation is in standard form? The quadratic equation is in standard form when there are at most 3 terms and they are in order of the x2 term ﬁrst, the x term second, and the constant last. •

FACILITATION TIP This Exit Ticket could be used as a preassessment as well as a formative one. Consider using it to measure student growth or inform your instruction. FACILITATION TIP If students still have large conceptual gaps, take time to review the vocabulary included in Picture Vocabulary before moving ahead with further scopes.

Post-Explore 1. 2. 3. 4.

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Polynomial Operations Additional Scope Resources EXPLAIN ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Picture Vocabulary

Can be done independently

Show What You Know, Part 1

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

Adding and Subtracting Polynomials 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 Polynomials with Models 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

Multiplying Polynomials

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

Special Products

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

Show What You Know, Part 5 Standard Form of a Quadratic Function 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

Polynomial Operations

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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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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

POLYNOMIAL OPERATIONS

Polynomial Operations

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Students

Notes & Comments

Engage

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Explain

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Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

What does mastery look like?

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I can multiply, add, and subtract polynomials of degree one and degree two.

I can use the distributive property to develop and recognize patterns when squaring binomials and multiplying by conjugates.

I can create quadratic expressions in standard form and use the expressions to determine information about a given scenario.

I can rewrite a quadratic expression in different equivalent forms.

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Graphs of Quadratic Functions Scope Introduction SCOPE SUMMARY

Student Expectations

A.PAR.6.4 Represent constraints by quadratic equations and interpret data points as possible or not possible in a modeling framework.

In this scope, students will create graphs of quadratic functions for the first time. They will first explore the symmetry and other key features of parabolas on the coordinate plane. Students will see quadratic equations in multiple forms and determine what key information they can extract from these forms in order to create graphs. They should be able to represent solutions to a two-variable equation using a graph. Students will also interpret a graph based on the context of the situation and determine solution sets from graphs and tables created by hand and with technology. They should be able to interpret the terms in an equation as well as the features of the graph in the context of a situation. Students should be able to use multiple representations of quadratic functions flexibly and identify their key features.

A.FGR.7.3 Graph and analyze the key characteristics of quadratic functions. A.FGR.7.4 Relate the domain and range of a quadratic function to its graph and, where applicable, to the quantitative relationship it describes. A.FGR.7.5 Rewrite a quadratic function representing a mathematically applicable situation to reveal the maximum or minimum value of the function it deﬁnes. Explain what the value describes in context.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grade levels, students developed an understanding that rewriting an expression in different forms in a context can shed light on the problem. They previously have written linear functions in different forms to reveal different key features. Students developed an understanding that a function assigns to each input exactly one output. Earlier in the course, students have identiﬁed the standard form of a quadratic, seen the graph of the parent quadratic function, and determined the domain and range of linear functions. Each of these experiences will be leveraged as they explore graphs of quadratic functions in this scope.

In Algebra II, students will continue to graph and analyze quadratic functions in contextual situations. Students will extend their knowledge of factored form and of leading terms and coeﬃcients to help them create and understand the graphs of polynomials with a larger degree.

A.FGR.7.6 Create quadratic functions in two variables to represent relationships between quantities; graph quadratic functions on the coordinate axes with labels and scales.

A.FGR.7.9 Compare characteristics of two functions each represented in a different way.

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

explain a function and graph of a function is the set of ordered pairs consisting of an input and corresponding output.

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

A.FGR.7.8 Write a function deﬁned by a quadratic expression in different but equivalent forms to reveal and explain different properties of the function.

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

understand quadratic equations.

•

graph quadratic 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. 184

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Key Features and Attributes

Explore 2

Explore 1

EXPLORE ACTIVITIES

In this exploration, groups of students will be asked to be consultants to analyze the eﬃciency of a morning school bus route; decide whether it makes sense to make some changes to the route; and, analyze untitled graphs and descriptions to determine the best recommendation. Students will: •

recognize the shape of a quadratic function is a parabola.

•

identify key features of quadratic functions.

Explore 4

Explore 3

In this exploration, students will be helping an architecture ﬁrm to create equations to model the support wires for several suspension bridges. Students will: •

rewrite functions in standard form.

Solving for a in Quadratic Functions In this exploration, students will be asked to help an architecture ﬁrm by developing an equation to represent the opening of the tunnel to produce accurate drawings and blueprints. 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 5

analyze graphs of quadratic functions and quadratic equations in factored form to identify zeros, intercepts, solutions, and roots of a quadratic function.

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

Equations in Vertex Form

write quadratic functions in vertex form.

In this exploration, students will be tasked with analyzing reading data to determine how fast that a student will need to read a 100 page book, as well as write functions about future reading assignments. 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.

•

Roots, Zeros, x-intercepts, and Solutions

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write quadratic functions in standard form or factored form when given real solutions or graphs of the functions.

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

Identifying Key Features and Equivalent Quadratic Functions In this exploration, groups of students will be evaluating equations from the motion detector to determine which student traveled the farthest. Students will: •

identify key features of quadratic functions from equations in vertex form or factored form.

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Graphs of Quadratic 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

Materials

Preparation

Printed •

• •

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

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

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Procedure and Facilitation Points 1. 2.

Card 1 matches with Card C.

Card 3 matches with Card A.

Card 2 matches with Card B.

• •

Before this, ask students what they remember about the graphs of quadratic functions. How are they similar to the graphs of linear functions? How are they different? FACILITATION TIP

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

Identifying Misconceptions •

FACILITATION TIP

Students may struggle to ﬁnd the x- and yy-intercepts. -intercepts. Help students identify those points using effective questioning. Picture Vocabulary slides may help students remember the vocabulary words by providing visual connections. Students may not be used to seeing nonlinear relationships, so you may need to ground their thinking in the connections between tables and graphs for all functions with a linear example.

Alternatively, display Cards 1–3 one at a time. Allow students to make observations and notes. Next, distribute copies of Cards A, B, and C to partner pairs and have students make observations and notes on them. Finally, have students match number cards to letter cards either at their seats or walking around the room. FACILITATION TIP This Foundation Builder includes six different triads of equations, tables, and graphs for students to group together. Consider using a combination of projection and distributed copies to facilitate an effective whole class lesson.

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Graphs of Quadratic Functions Hook – Model Rockets ACTIVITY PREPARATION Students will relate quadratic functions to a real-world situation.

Materials

Preparation

Printed •

• • •

1 Model Rockets (per class)

Reusable • •

1 Phenomena Video (per class) 1 Projector (per class)

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

PROCEDURE AND FACILITATION Part I: Pre-Explore 1.

FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) What do you know about rockets?; 2) How are rockets useful?; 3) Have you launched model rockets before?

2. 3.

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: Students are launching model rockets in the ﬁeld behind their school for physics class. 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 the rocket increases by less and less as time passes. I wonder what the maximum height is. I wonder where and when the rocket will land. I notice the curved trajectory of the rocket because it does not rise at a constant rate. Project Model Rockets. Notes

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Engage

Explore

Explain

Elaborate

Evaluate

Explain to students that the goal throughout the scope will be to explore a new graph and extract important information from equations. Discuss the following questions: a.

DOK-1 How is this graph different from a linear graph? Allow students to share all ideas. Answers will vary. It does not have a constant rate. It has a maximum point. It has a line of symmetry. It has two x-intercepts. It is both increasing and decreasing.

DOK-1 Why might this graph do a good job of modeling the rocket’s ﬂight path? The rocket launches at a high speed before reaching a maximum height and eventually falling back down to the ground.

DOK-1 What features from the equation stand out to you? Allow students to share all ideas. Answers will vary. The negative sign starts the equation. The plus 2 on the end seems to match the y-intercept.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

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

DOK-1 Do this graph and equation make more sense after the Explore activities? Yes, this graph is a parabola, and equations where the variable has an exponent of 2 create these graphs.

DOK-1 What key features might you want to ﬁnd from this graph? I could identify the vertex, axis of symmetry, x-intercepts, and y-intercepts.

DOK-1 What feature does this equation quickly reveal? It is easy to determine the y-intercept from this equation because it is in standard form.

DOK-2 If you know the x-intercepts -intercepts of the graph are at (–0.005, 0) and (24.005, 0), how could you ﬁnd the vertex? The vertex must have an x value in the middle of these two points. So the x value of the vertex is 12, and I could substitute x = 12 into the equation to ﬁnd the y-coordinate of the vertex, or the maximum height the rocket reaches.

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

Intervention

Acceleration

FACILITATION TIP As a ﬁnal discussion for the Pre-Explore, ask students to compare and contrast real rockets with model rockets given what they know. Are there similarities in the ﬂight patterns? What additional factors do they think scientists have to consider when designing real rockets?

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.

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FACILITATION TIP After students answer the question 2e, ask the class how they think attributes of quadratic equations affect the design of rockets, real and model. Students may suggest that the attributes affect materials used, weight, and shape, among other characteristics.

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Graphs of Quadratic Functions Explore 1 – Key Features and Attributes ACTIVITY PREPARATION Students will study motion in real-life scenarios that is described by a quadratic function. By studying graphs of this motion, they will recognize that the shape of a quadratic function is a parabola and will be able to identify key features such as the x-intercept, yy-intercept, -intercept, zeros, maximum value, minimum value, vertex, and axis of symmetry.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.6 Attend to precision.

Materials

Preparation

Printed • •

• •

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

Separate the class into groups of 3 or 4 students. 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) Does anyone play action sports?; 2) If so, which sports?; 3) When did you start playing? 2. 3. FACILITATION TIP Students may mix up the x- and y-coordinates for one or more points as they ﬁll out contestant cards. Ask the class early on which coordinate represents time and which coordinate represents height.

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4. 5.

Read the following scenario to the class: Crush Games, the world’s premier action sports event, will be hosted in your home state this year! One of the events at Crush Games this summer is the skateboard jump. You have been selected from thousands of applicants to serve as a volunteer. You will help with record keeping and identifying the competition winner. To fulﬁll your duties as a volunteer record keeper, you must use the data from the judges’ notes to complete the contestant card and help determine the winner. Give a Student Journal to each student. Explain to students that they will work with their groups to analyze graphs of quadratic functions that model the contestants’ skateboard jumps and record their work on their Student Journals. Point out to the class that each graph will plot the height of the jump versus time. As students collaborate, facilitate their discussion and use the following guiding questions to assess student understanding: a.

DOK-2 What letter represents Amari’s landing position? How many seconds after his jump did he land? Point E represents his landing position. He landed 8 seconds after his jump.

FACILITATION TIP

In case some students don’t mention it in their answer, ask the class what the peak, starting point, and ending point for Amari’s jump represent on a graph. They should answer the vertex and roots (zeros), respectively.

DOK-2 What letter represents the peak of Amari’s jump? How can the x-coordinate -coordinate be determined? Point C represents the peak of Amari’s jump. Since the whole jump took 8 seconds and the judges’ notes state the peak was exactly halfway, this would mean the peak was at 4 seconds, halfway between the start and end (the vertex is exactly between the roots).

DOK-2 What letter represents Amari’s height and time 1 second before his peak? What is the x-coordinate -coordinate of this point? How do you know? Point B represents Amari’s height and time 1 second before his peak. The x-coordinate of this point is 3 because the peak occurred at 4 seconds, and point B is 1 second before the peak. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

d. DOK-2 Since the graph is symmetrical about the maximum, which point is the mirror coordinate for point D? How do you know? Point B is the mirror coordinate because if we folded the graph in half, they would line up. e. DOK-2 Since the curve is symmetrical, what do you know about the horizontal distance between points B and C compared to the horizontal distance between points C and D? What do you know about the vertical height of B vs. D? The horizontal distance, or difference in time, is the same. The vertical height is the same. f.

DOK-2 What letter represents the peak of Harry’s jump? How can the ordered pair be determined? Point C represents the peak of Harry’s jump. The ordered pair can be determined using the judges’ notes. Since the horizontal axis is time and the judges’ notes say 3 seconds, the x-coordinate is 3. Since the vertical axis is height and the judges’ notes say 9 feet, the y-coordinate is 9. The ordered pair for point C is (3, 9).

DOK-2 How can the maximum point be used to determine points A and E? Could this be done if the graph wasn’t symmetrical? Since the graph is symmetrical about the maximum point, we know points A and E are an equal distance from the maximum. The maximum occurred at 3 seconds. We know the jump started at 0 seconds. Since point A occurred 3 seconds before the maximum, we know point E occurred 3 seconds after the maximum. Point E occurred at 6 seconds. This could not be done if the graph wasn’t symmetrical.

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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h. DOK-2 Since the graph is symmetrical about the maximum, which point is the mirror coordinate for point D? How do you know? Point B is the mirror coordinate because if we folded the graph in half, they would line up. i. DOK-2 How can point D be used to determine the coordinate of point B? Since they are mirror coordinates, we know they occurred at the same height or have the same y-coordinate. We can use the distance that point D is from the center of the parabola to determine B. Since D occurred 2 seconds after the maximum, we know point B occurred 2 seconds before the maximum, which gives us an x-coordinate of 1. 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-2 The shape of the quadratic function graph describing the skateboard motion is called a parabola. Is a parabola symmetrical, and how can you tell? Yes, because the graph can fold onto itself, and the output of any x value a certain distance to the right of the vertex has the same output as an x value the same distance to the left of the vertex. DOK-2 The axis of symmetry is a vertical line that divides the parabola into two equal pieces that are the mirror image of one another. Which point (A, B, C, D, or E) on the graph would it pass through? C DOK-2 The vertex of a parabola is the point where the axis of symmetry intersects the parabola. What point is the vertex (A, B, C, D, or E)? C DOK-2 Where is the maximum? The maximum is at the vertex. DOK-2 If the parabola was ﬂipped and opened upward, where would the minimum be? It would be at the vertex. DOK-2 If the parabola was plotted on an xy xy-coordinate -coordinate system, where would the x- and yy- intercepts be (A, B, C, D, or E)? The y-intercept would be at A. The x-intercepts would be at A and E. DOK-2 The parabola is a plot of height as a function of time. At what point is the height zero; that is, what are the roots of the height function? The times at points A and E are the roots.

FACILITATION TIP Some students may notice mirror coordinates and write the same x- and y-coordinates for both points. Note to them that that would mean the points are in exactly the same spot, and instruct them to think about what symmetry means in the context of a parabola.

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.

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Graphs of Quadratic Functions Explore 1 – Key Features and Attributes Part II 1.

FACILITATION TIP After completing Part I, students may ignore the word “Reasonable” in “Reasonable root(s)” and list both roots for the dirt bike contestant cards. If so, ask them if a negative time is reasonable. FACILITATION TIP For the dirt bike contestants, some students may be confused to see starting points that are not at the origin. As they begin working on Part II, instruct students to look at each picture as a whole and note the ramps.

2. 3. 4. 5. 6.

Read the following scenario to the class: Dirt bike jumping and cliff diving are also competitions in the Crush Games. In your volunteer position, you will help with record keeping and identifying the competition winners for dirt bike jumping and cliff diving. To fulﬁll your duties as a volunteer record keeper, you must use the data from the judges’ notes to complete the contestant card and help determine the winner. Students should still have their Student Journals. Explain to students that they will work with their groups to analyze graphs of quadratic functions that model the contestants’ dirt bike jumps and cliff dives. Point out to the class that each graph will plot the height of the jump versus time or the dive depth versus time. Students will work together to complete the contestant cards 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 What does the ordered pair (−2, 0) represent on the graph of Aalia’s jump? Is this reasonable? Explain. It represents that Aalia was 0 feet above the ground at −2 seconds. This is not reasonable because we don’t measure time using negatives. Furthermore, we don’t know that she was going up the hill at that time. She could have been waiting at the top of the hill or some other place. Her approach to the jump is not represented by the same function as the jump.

DOK-2 How can the given roots be used to determine the x value of the vertex? Since the vertex is exactly between the roots, we can ﬁnd the halfway point by counting on a number line or by averaging the x values of the roots.

DOK-2 The axis of symmetry is an imaginary vertical line that goes through the vertex. The vertex for Manuela is located at (3, 16). How can the x value be used to write the equation of the vertical line that represents the axis of symmetry? What information is needed to write the equation of the vertical line that represents the axis of symmetry on Aalia’s graph? If there were a vertical line through the point (3, 16), it must have an equation of x = 3 because the x value of the ordered pair is 3. We must know the x value of the ordered pair of the vertex.

DOK-2 Since the graph is symmetrical about the maximum, which point is the mirror coordinate for the point where Manuela is at 5 seconds? How do you know? Point (1, 12) is the mirror coordinate because if we folded the graph in half, they would line up. The x value of (1, 12) is 2 seconds before the maximum, and the x value of 5 seconds is 2 seconds after the maximum.

FACILITATION TIP Select some of the essential guiding questions; print and project them before students begin to collaborate. Encourage students to be prepared to answer them as they work or after they analyze the graphs.

FACILITATION TIP Students may be thrown off without a visual reference, as they are used to such references from Part I and there is no point marked after (3,16). Ask them how far away 5 seconds is from the x-coordinate of the vertex, and ask again what symmetry means in the context of a parabola.

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e. DOK-2 Does it take longer to reach the maximum height of the jump or to come back down? It takes longer to travel back down from the maximum height to the ground. f.

DOK-2 How can the minimum and the point when Yume emerges from the water be used to determine the point when she enters the water? Since the minimum is exactly between the roots, we can determine that she emerges from the water 3.5 seconds after her minimum, so she entered the water 3.5 seconds before her minimum. Since 6.5 – 3.5 is 3, she entered the water 3 seconds after her dive began.

DOK-2 How can the roots of Aditi’s dive be used to determine the x value of the vertex? Since the vertex is exactly between the roots, the x value of the roots can be averaged to determine the x value of the vertex.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

h. DOK-2 The axis of symmetry is an imaginary vertical line that goes through the vertex. Where is the vertex located for Aditi’s dive, and how can it be used to write the equation of the vertical line that represents the axis of symmetry? The vertex is located at the point (8.5, −42.25). It must have an equation of x = 8.5 because the x value of the ordered pair is 8.5. 7. 8.

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 Describe how the minimum or maximum could be identified or approximated for a quadratic function represented by a table of values. The minimum or maximum is referring to the y value of the ordered pairs. To identify this from a table, we would identify the point that was the maximum or minimum value. To know if the extreme value in the table represents the vertex of the quadratic, we would also want to verify that there are smaller or larger values for the input values on either side of the maximum or minimum, respectively. • DOK-2 For a quadratic function represented by a table of values, how could the roots be identified or approximated? The roots occur when the y value is zero. If the table didn’t show y values of zero, the roots would occur where the y value changes from positive to negative or vice versa. • DOK-3 If a quadratic function were described by a table of ordered pairs, describe what that table might look like. As the x values increased, the corresponding y values would increase and then at some point decrease, or the y values would first decrease and then increase. • DOK-2 For a quadratic equation, the graph is a parabola. If it were plotted in an xy-coordinate coordinate system, from what you know about the shape of a parabola, what is the maximum number of roots the quadratic function can have (that is, how many times could the parabola cross the x-axis)? The maximum number of roots for a quadratic equation is two. •

2. 3.

Check students’ understanding of intercepts for quadratic functions further. After they answer Question 5 in the Reﬂect section, ask the class if a parabola can have 0, 1, or 2 y-intercepts. Have them justify their reasoning.

FACILITATION TIP After the class answers the question, direct them back to the dirt bike contestant cards. Note that a graph or table may or may not include unreasonable points.

FACILITATION TIP

Post-Explore 1.

FACILITATION TIP

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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.

After the Exit Ticket, ask students for other examples in sports of people and/or objects traveling in a parabolic path. Examples may include a basketball during a jump shot or a baseball when it is hit by a bat.

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Graphs of Quadratic Functions Explore 2 – Roots, Zeros, x-intercepts, and Solutions ACTIVITY PREPARATION Students will analyze graphs of quadratic functions and quadratic equations in factored form to identify zeros, x-intercepts, solutions, and roots of a quadratic function and interpret the meaning in context.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

• • •

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

Separate the class into groups of 2 or 3 students. Print a Student Journal and an Exit Ticket for each student. Print a set of Results Cards, on card stock for durability, for each group of students. Place each page inside a sheet protector.

Reusable •

2 Sheet protectors (per group)

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Project this scenario and have student volunteers read it aloud with you and the class. Ask students, “What do we know?” and “What do we need to ﬁnd out?” FACILITATION TIP To support cohesive collaboration, consider distributing the Student Journal in sections for Part I (page 1) and Part II (pages 2 and 3).

2. 3. 4.

5. FACILITATION TIP Before students begin collaborating, clarify the deﬁnition and show some examples of different kinds of factors, x-intercept, y-intercept, maximum, minimum and any other essential vocabulary.

Read the following scenario to the class: Aarav is a recorder of different sports events for Crush Games and specializes in recording data for jumps—the skateboard jump, monster truck jump, and high jump. He recorded the data for each contestant at each event, but somehow, the machine used to record data for the jumps overheated and was malfunctioning. Help Aarav analyze the ﬁnalists’ data that was correctly recorded so later you can help him make sense of that data for other contestants. Give a Student Journal to each student. Give a Part I Results Cards to each group of students. Explain to students that they will work with their groups to analyze the Results Cards to examine the relationship between the zeros of a quadratic function and its linear factors 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 In the equation 5 · 4 = 20, what are the 5 and 4 called? 5 and 4 are factors of 20.

DOK-1 What is the ((xx + 1) in the equation for ﬁnalist C? It is a factor.

DOK-1 Would the initial values be on the left side of the graph or the right? Explain. They would be on the left side. The x-axis represents time, and when we are looking for the starting time, it is when time is 0.

DOK-2 How do you determine each person’s start and end times? On the graph, you look for the value of x when y equals zero (the x-intercept).

e. DOK-2 The ending time for ﬁnalist A was 8 seconds. Is this ending time represented in the equation? If yes, where? Yes, it is the number in the second factor. 194

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How would you determine the highest height from the graph? It would be the coordinate of the maximum, which would occur halfway between the two zeros or x-intercepts.

DOK-2 The maximum height for ﬁnalist A was 16. Is this maximum height represented in the equation? If yes, where? No, it is not obviously represented in this equation.

h. DOK-2 The ending time for finalist B was 6 seconds. Is this ending time represented in the equation? If yes, where? Yes, it is the number in the second factor. i. DOK-2 The ending time for finalist C was 5 seconds. Is this ending time represented in the equation? If yes, where? Yes, it is the number in the second factor. j. DOK-2 What key feature do all of the ending times represent? Do the initial times correspond to the same key feature? The ending times are all x-intercepts. The initial times are x-intercepts, except for finalist C; that is a y-intercept. k.

DOK-2 For ﬁnalist C, what does the coordinate (0, 10) tell you? At 0 seconds, ﬁnalist C starts at a height of 10 feet.

Intervention

Acceleration

STEMscopes Tip Communicate Math – Making Connections is located under the Communicate Math tab of the Teacher Toolbox. Students learn mathematical concepts by linking them to their prior knowledge and experiences. Teachers can emphasize the connections from this page to help students bridge their knowledge from concept to concept. Examples of possible connection types are provided.

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l. DOK-2 If the x and y values of a function’s x-intercept are substituted into the equation for that function, what do you expect to happen? The equation will be true because the ordered pair is a point on the curve. 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-2 What generalization can be made about the relationship between factors and x-intercepts? The factors and x-intercepts will have opposite signs. • DOK-2 What must be true about the factors of an equation if both x-intercepts are positive? Both factors will include subtracting a positive number. • DOK-2 What must be true about the factors of an equation if both x-intercepts are negative? Both factors will include subtracting a negative number or adding a positive number. • DOK-2 What must be true about the factors of an equation if one x-intercept is positive and one is negative? One factor will include subtracting a positive number, and the other will include subtracting a negative number. •

Part II 1.

2. 3. 4.

Read the following scenario to the class: The machine used to record data for the jumps overheated and malfunctioned. Aarav hires you as an intern to help analyze some Results Cards and determine whether there is or is not an error. Analyze the data provided to determine which Result Cards have erroneous equations or graphs and make a decision about the correct start and end time for each contestant. Give a Part II Results Cards to each group of students. Students should still have their Student Journals. Explain to students that they will work with their groups to analyze the equations and graphs on each Results Card to answer the questions that follow 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 would you know whether there is an error on the card? If the graph and equation do not match, then there is an error.

FACILITATION TIP Before students begin Part II, determine if you want to let them know how many errors (if any) there are out of the three cards (A and B have errors, C does not).

FACILITATION TIP Before students begin collaboration, project and ask question 5a to the whole class.

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Graphs of Quadratic Functions Explore 2 – Roots, Zeros, x-intercepts, and Solutions b.

DOK-2 Given the equation for contestant A, what would you expect the x-intercepts to be? Since ordered pairs of (0, 0) or (4, 0) make it true, the x-intercepts would be at x = 0 and x = 4.

DOK-1 Does the graph for contestant A have x-intercepts -intercepts that match the equation? If not, what are the x-intercepts? -intercepts? No, the x-intercepts are x = 0 and x = 2.

DOK-3 What would need to change in the equation for contestant A for the graph and equation to correspond? You could change the equation so the ordered pairs of (0, 0) or (2, 0) make it true: y = −(x – 0)(x – 2). Then, it would match the graph.

e. DOK-3 What would need to change on the graph for contestant A for the graph and equation to correspond? You could change the graph so the x-intercepts are x = 0 and x = 4. Then, the graph would match the current equation.

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.

DOK-2 Given the equation for contestant B, what would you expect the x-intercepts to be? Since ordered pairs of (0, 0) or (−5, 0) make it true, the x-intercepts would be at x = 0 and x = −5.

DOK-1 Does the graph for contestant B have x-intercepts -intercepts that match the equation? If not, what are the x-intercepts? No, the x-intercepts are x = 0 and x = 5.

h. DOK-3 What would need to change in the equation for contestant B for the graph and equation to correspond? You could change the equation so the ordered pairs of (0, 0) or (–5, 0) make it true: y = −(x – 0)(x – 2). Then, it would match the graph. i. DOK-3 What would need to change on the graph for contestant B for the graph and equation to correspond? You could change the graph so the x-intercepts are (0, 0) and (−5, 0). Then, the graph would match the current equation. j. DOK-2 Given the equation for contestant C, what would you expect the x-intercepts to be? Since ordered pairs of (0, 0) or (7, 0) make it true, the x-intercepts would be at x = 0 and x = 7. k.

DOK-1 Does the graph for contestant C have x-intercepts -intercepts that match the equation? If not, what are the x-intercepts? Yes, the x-intercepts are x = 0 and x = 7.

l. DOK-3 What, if anything, would need to change in the equation for contestant C for the graph and equation to correspond? Nothing needs to change. m.

FACILITATION TIP Alternatively, considering the number of guiding questions, you may want to coach students through Part II as a whole group if needed.

6. 7.

DOK-3 What, if anything, would need to change on the graph for Contestant C for the graph and equation to correspond? Nothing needs to change.

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-3 What is always true about the zeros on the graph and the factors of the polynomial? The zeros are the coordinates that, when substituted into the equation, would make it true, and this would also make one of the binomials equal to 0. • DOK-1 How are the terms zeros zeros, x-intercept, solutions, and roots related? Zero is the value of x where an expression is equal to zero; this is the x-coordinate of the x-intercept of the graph. The solution is the value that satisﬁes the equation, called roots, x-intercepts, or zeros. •

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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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Graphs of Quadratic Functions Explore 3 – Equations in Vertex Form ACTIVITY PREPARATION Students will write quadratic functions in vertex form, identify extrema from an equation in vertex form, and interpret key features in context.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.6 Attend to precision.

Materials Printed • • •

1 Student Journal (per student) 1 Bridges A, B, and C (per group) 1 Exit Ticket (per 2 students)

Reusable •

1 20 in. piece of string (per group)

Preparation • • • • •

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 has one. Hang the Bridges A, B, and C posters around the room in different locations for each group. Each location should have posters for bridges A, B, and C. Cut a piece of string for each group. Melt the ends of the string for durability.

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) What do you know about bridges?; 2) When was the last time you crossed a bridge?; 3) What kind of bridge was it? FACILITATION TIP Before you read this scenario, collect some images or video of bridges to show students some cables, support towers, and heights. FACILITATION TIP Rather than have students make observations at the posters around the room, provide each table group with a set of Bridges A, B, and C so they can examine them more closely. FACILITATION TIP Print these guiding questions to use as students collaborate. If students are struggling, consider coaching students through the Student Journal using 5a–5h as a game plan.

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2. 3. 4.

Read the following scenario to the class: You have made it to round 2 of the interview process to work as an intern for an architecture ﬁrm. They give you 3 models, a piece of string, and 3 equations. You are asked to analyze the models of the bridges and three equations. They explain that the cables should be at the same height as the road surface somewhere between the towers. Show that you have the expertise to work for the architecture ﬁrm by explaining the relationship between the equations and the bridges’ dimensions. Give a Student Journal to each student. Send groups of students to the posters around the room. Explain to students that they will work with their groups to analyze the information from the scenario for the second round of the interview 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 many roots does this parabola have? There is only one root since the graph touches the x-axis only once.

DOK-2 Where on the curve is the vertex of the parabola? The vertex will be the minimum (lowest point) of the graph.

DOK-2 What does the midway point between the support towers tell you about the distance from each support tower? If it is midway, it is exactly the middle, so it is an equal distance from both support towers.

DOK-2 If the midway point is 65 m from a support tower, how could that be used to ﬁnd the total distance? If 65 m from one support tower is midway, then it must be 65 m from the other support tower too. If we multiply 65 by 2, we get the total distance. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

e. DOK-2 Since the vertex is exactly between the two towers, how far is it from the yy-axis? -axis? Is this the x value of the vertex or the y value of the vertex? The vertex is 30 feet from the left tower or the y-axis. This value of 30 is the x value of the vertex. f.

DOK-2 Since the cable touches the roadway, what is the y value of the coordinate of the vertex? The y value would be 0 since it is 0 feet above the roadway.

DOK-2 What happened to the value inside the parentheses when the input was 40? How did this impact the output? When 40 was the input, the value inside the parentheses was zero for one equation. This made the output zero because anything multiplied by zero is zero.

h. DOK-2 When the support wire touches the surface, what is a possible y value of the ordered pair? Touching the surface would be 0 m above the roadway, so that would be a y value of 0.

Intervention

Acceleration

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i. DOK-2 For the other bridges, what would happen to the value inside the parentheses when the input is 30? How would this impact the output? When 30 is the input, the value inside the parentheses would be zero. This would make the output zero because anything multiplied by zero is zero. 6.

Allow students enough time to complete Part I and answer the questions that follow. FACILITATION TIP 7. After Part I, invite the class to a Math Chat to share their observations and learning. To facilitate cohesive collaboration and discussions, distribute the two parts of the Math Chat Student Journal only as needed. Part I is • DOK-2 How is the equation related to the x value of the vertex? The number pages 1 and 2. Part II is pages 3–5. subtracted inside the parentheses is the same as the x value of the vertex. The x value will make the expression inside the parentheses equal 0. • DOK-2 If I have the equation f( f(x (x) = ((xx + 3)2, what must the x value of the vertex be? It must be −3 since an x value of −3 will make the expression inside the parentheses equal 0. • DOK-2 Why is the input that makes the quantity inside the parentheses zero the lowest point on the parabola? The output will be the lowest when the quantity inside the parentheses is the lowest. Since squaring a number will always yield a nonnegative result, the smallest number output possible from a quantity squared is zero. Part II 1.

2. 3. 4.

Read the following scenario to the class: Congratulations, you made it to round 3! You have successfully demonstrated your practical knowledge with the models in round 2 of the interview. Now, examine three other bridges to demonstrate your conceptual understanding for the ﬁnal and third round of the interview. Students should still have their Student Journals. Explain to students that they will work with their groups to analyze the information provided for each bridge by answering questions on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 The y value of the ordered pair gives the distance above the surface. Does this change the location of the halfway point between the support towers? The y value changes the position vertically and does not impact the horizontal distance or midway point.

DOK-2 How can you be sure your value for the height is as small as possible? If the term that is squared equals 0, then the output value cannot be any smaller.

DOK-1 How does the y value of the vertex connect to the equation? It is the constant on the outside of the equation.

FACILITATION TIP Before reading the scenario, ask the class 1) What do you know about architects?; 2) What are some speciﬁc things architects do?; 3) Who do they work with?

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Graphs of Quadratic Functions Explore 3 – Equations in Vertex Form d.

5. 6.

DOK-2 Once you have determined the vertex, how can you tell the overall distance of the bridge? The bridge length is twice as far as the x value of the vertex is from the origin.

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 FACILITATION TIP Project these two Math Chat questions as students begin to complete Part II. Encourage students to be prepared to respond during the discussion. Record answers and have students make notes in their notebook or Student Journal.

•

DOK-2 How can you ﬁnd the vertex of an equation in the form y = a(x (x – h)2 + k? (x The x value of the vertex is the value h that will make the inside of the parentheses equal 0. The y value of the vertex is k. DOK-1 If the vertex of a quadratic equation is at (−50, 7), what is one possible equation of the parabola? y = (x + 50)2 + 7 is a possible equation.

Post-Explore

FACILITATION TIP

This Exit Ticket could be used as a pre- and post-assessment. Consider using it to measure student growth or just inform your instruction for this Explore activity.

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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Graphs of Quadratic Functions Explore 4 – Solving for a in Quadratic Functions ACTIVITY PREPARATION Students will write quadratic functions in standard form or factored form when given real solutions or graphs of the functions. Students will use a third point to ﬁnd a.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.6 Attend to precision.

Materials

Preparation

Printed • •

• • •

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

Reusable •

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 has one.

1 Graphing calculator or other graphing technology (per student)

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever been through a tunnel?; 2) What was the shape and size of the tunnel?; 3) Did you walk, ride, or bike through the tunnel? FACILITATION TIP Print this scenario so students can read it along with you and record essential values and constraints. FACILITATION TIP Students may misunderstand how to account for a parabola opening downward. In Question 3. watch out for students solving for a with the equation y = –ax2 + 8. FACILITATION TIP After students answer Question 1. ask them for scenarios where one would need an equation for a parabola that opens upward. Examples may include a bathtub or a skate ramp.

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2. 3.

Read the following scenario to the class: Architects are doing research to build a new parabolic tunnel, so they need to examine other tunnels and the quadratic functions used to model the openings of the tunnels. One tunnel has a parabolic opening that is 8 m wide measured from side to side along the ground. The tunnel entrance is 8 m high when measured at the highest point in the center. The architecture ﬁrm needs to develop an equation to represent the opening of the tunnel to produce accurate drawings and blueprints. Help the architect develop the equations to be used for drawings and blueprints. Give a Student Journal to each student. Explain to students that they will work with their groups to analyze the scenario and use the details of the tunnel to sketch the parabola on the grid 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 Describe the direction the intern’s parabola opens. The intern’s parabola opens upward.

DOK-2 What ordered pairs would represent where the left and right sides of the tunnel meet the ground if it is centered at x = 0? The left will be (−4, 0), and the right will be (4, 0).

c. DOK-2 Describe the location of the roots of the intern’s model. The intern’s model doesn’t cross the x-axis; therefore, it doesn’t have real roots. d.

DOK-2 Describe how to determine the output of the associate’s model for a given input. If the input were 2, we would put 2 in place of x. Square 2, take the opposite, and then add 8 to get the output. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

e. DOK-2 What is the ordered pair for the highest point on the graph of the tunnel opening? How is this related to the equation? The ordered pair is (0, 8), and this is related to the equation because zero is subtracted from the quantity of x, and 8 is added at the end. f.

DOK-2 How can the roots/zeros/x-intercepts roots/zeros/ -intercepts be used to write the linear factors? The linear factors are x plus the opposite of each root.

DOK-2 How can the linear factors be used to write an equation in factored form? The quadratic equations would be written as the product of the linear factors.

FACILITATION TIP Some students may overthink questions in the Explore activity. Encourage them to focus on what they know and trust their intuition.

h. DOK-2 Is every point on a line or curve a solution to the equation that represents the line or curve? Why or why not? Every point on the curve is a solution to the equation that represents the curve. By deﬁnition, an ordered pair that satisﬁes the equation is a solution to the equation.

GRAPHS OF QUADRATIC FUNCTIONS

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i. DOK-2 How do you determine whether your ordered pair satisﬁes the equation? To determine whether my ordered pair satisﬁes the equation, I would need to substitute it into the equation and see if it makes a true statement. j. DOK-2 How are roots of 8 and −8 related to the factors in the equation? The roots are opposite signs of the factors, so the root of 8 would have a factor of (x – 8), and the root of −8 would have a factor of (x + 8). k.

DOK-2 When a quadratic equation is written in factored form like y = a(x (x + 8)( (x 8)(xx – 8), how can the value of the coeﬃcient, a, be determined when there are two other variables, x and yy? We would need to use a point from the parabola to substitute for x and y. That would leave an equation with just one variable.

l. DOK-2 What x value is exactly between the roots? How will this help us find the vertex? The x value exactly between the roots is 0. This will help us find the vertex because we can substitute this value into the equation and find the y value for the ordered pair of the vertex. 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.

Math Chat •

•

DOK-2 What does it mean when the ordered pair doesn’t satisfy the equation? It means the equation doesn’t represent a parabola that goes through the exact roots and the additional point. DOK-2 How was substitution used to determine the coefficient of the quadratic equation? A point that wasn’t used to write the equation was plugged in for x and y. That creates an equation with only one variable, a. This equation can be solved, and the value of a is the coefficient of the quadratic equation. DOK-2 When can the vertex be used as the values to substitute and determine the coefficient of the quadratic equation? The vertex, or any point other than a root, can be used to substitute when the equation is in factored form. DOK-2 When can a root be used to substitute and determine the coefficient of the quadratic equation? The root, or any point other than the vertex, can be used to substitute when the equation is in vertex form. DOK-2 All parabolas have a coefficient that can make the parabola stretch or compress. This will allow us to adjust the equation so it goes through the exact roots and the additional point. How can we use y = a(x (x – 3)( (x 3)(xx + 3) and the additional point to determine the value of a? If we substitute the x and y values from the coordinate point, we will have an equation with one unknown, a. Then, we can simplify and solve for a.

STEMscopes Tip The Visual Glossary, located under the Scopes tab, provides a variety of instructional resources. Browse topics alphabetically in English or Spanish. Each topic includes a visual and/or video featuring key vocabulary and concepts. The visuals include text and a speech button with narration. The videos, featuring real-world examples, are 3–15 seconds in length.

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Graphs of Quadratic Functions Explore 4 – Solving for a in Quadratic Functions Part II 1.

FACILITATION TIP

3. 4.

It may help students to ﬁnd the vertex of the graph before sketching it. Recommend that, if needed, students can plot the known points and then complete Questions 2 and 3 before completing Question 1. They should be able to use the axis of symmetry to plot (–3, 8) if needed.

Read the following scenario to the class: The architecture ﬁrm needs to develop an accurate equation to represent the opening of the tunnel to produce accurate drawings and blueprints. Help the architect reﬁne the equation to be used for drawings and blueprints. Explain to students that they will work with their groups to analyze the scenario and use the details of the tunnel to sketch the parabola on the grid and record their work on their Student Journals. Ensure that students have access to digital or physical graphing calculators. As students collaborate, monitor their work and use the following guiding questions to assess student understanding:

DOK-2 How can you use the symmetry of a parabola to make a fourth point and sketch a more accurate graph? We can reﬂect (3, 8) over the y-axis and also use (−3, 8) to sketch the graph.

DOK-2 How are roots of 7 and −7 related to the factors in the equation? The roots are opposite signs of the factor, so the root of 7 would have a factor of (x – 7) and the root of −7 would have a factor of (x + 7).

DOK-2 What are the possible values to substitute for x and y? The possible values are 3 and 8 or −3 and 8.

DOK-2 Once the value of a is determined, how can that be used to write an equation to represent the opening of the tunnel? We can substitute the value we found for a in the equation y = a(x – 7)(x + 7).

e. DOK-2 Is every point on a line or curve a solution to the equation that represents the line or curve? Why or why not? Every point on the curve is a solution to the equation that represents the curve. By deﬁnition, an ordered pair that satisﬁes the equation is a solution to the equation. f.

DOK-2 How do you determine whether your ordered pair satisﬁes the equation? To determine whether my ordered pair satisﬁes the equation, I need to substitute it into the equation and see if it makes a true statement.

DOK-1 How will you verify whether your chosen point satisﬁes the equation? I will substitute the x and y values to solve the equation. If the point satisﬁes the equation, both sides will have the same answer.

h. DOK-2 How can you use your graphing calculator to verify the complete equation represents a parabola that includes all 4 ordered pairs from the scenario? We can input the equation into our calculator and use the table to verify that the points exist on the parabola. FACILITATION TIP After they answer the question about functions that have the same zeros, ask the class how many quadratic functions can go through two particular zeros. They should know that there are infinite possibilities because there are infinite coefficients. FACILITATION TIP After they answer the question about functions that have the same vertex, ask the class if there are infinite equations that can pass through the same vertex. Students should know that there are infinite equations because they can stretch or compress a quadratic function about the vertex by any one of the infinite factors. 204

5. 6.

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 DOK-3 Will any two functions that have the same zeros create the same graphs? Why or why not? Two functions can have the same zeros but will not always create the same graphs since the parabolas can open upward or downward through the same zeros. Zeros may be the same, but the functions will be different depending on the values of the coeﬃcients. • DOK-3 Will any two functions that have the same vertex create the same graphs? Why or why not? Two functions can have the same vertex but will not always create the same graphs since the parabolas can open upward or downward through the same vertex. The functions may have the same vertices, but the graphs of the functions will be different depending on the values of the coeﬃcients. •

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 What components do you need for writing quadratic functions when the x-intercept(s) -intercept(s) are known? For writing any quadratic function, I need the roots and a third point that exists on the parabola. • DOK-2 What components do you need for writing quadratic functions when the vertex is known? I need the vertex and a second point that exists on the parabola. •

Intervention

Acceleration

FACILITATION TIP

Follow up this question about components needed when the x-intercept is known by asking the class what is the requirement for there to be a quadratic function that goes through two particular zeros and Post-Explore a third point. They should know that the 1. Have students complete the Exit Ticket to formatively assess their understanding x-coordinate of the third point must lie between the x-coordinates of the roots. of the concept. 2. 3.

Complete the Anchor Chart as a class. Have each student complete their Interactive Notebook.

FACILITATION TIP Students may still struggle with signs when forming an equation to solve for a. If this happens, ask them for the generic factored form of a quadratic equation.

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Graphs of Quadratic Functions Explore 5 – Identifying Key Features and Equivalent Quadratic Functions ACTIVITY PREPARATION Students will identify key features of quadratic functions from equations in vertex form, factored form, or standard form. Students will use the key features to determine if the equations represent the same function. The equivalent equations will be veriﬁed by rewriting them both in the same form.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.6 Attend to precision.

Materials

Preparation

Printed • • •

1 Student Journal (per student) 1 Set of Eyewitness Accounts (per group) 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. Print one set of Eyewitness Accounts per group on card stock for durability. Cut the cards apart, and place them in a resealable bag. Label the bag “Part I.”

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION Part I FACILITATION TIP

After engaging students by starting with the Eyewitness Accounts cards, project and read this scenario aloud together. FACILITATION TIP To engage students from the beginning, consider starting with projecting/reading the Eyewitness Accounts together before you read the scenario. FACILITATION TIP Select some of these essential guiding questions to print and project. Consider previewing them with the class before students begin to collaborate to help focus their discussions. FACILITATION TIP Use question 5b to quickly measure how close students are to being able to translate a mathematical scenario into a visual graph. Connecting a story to a shape on a graph and vice versa is a key skill that will be used in later math and science courses. Take time to spot check/assess students’ abilities. 206

2. 3. 4.

Read the following scenario to the class: Your neighbor, Mrs. Evelyn, was walking down the street when a water balloon hit the ground by her feet, splattering her new shoes! She inquires around and gathers information from several eyewitnesses. She asks you and your neighbor, Andres, to help her make sense of the information provided to determine the water balloon’s origin. Give a Student Journal to each student. Give a bag of Eyewitness Accounts cards to each group. Explain to students that they will work with their groups to help Mrs. Evelyn in her quest by sketching a graph and analyzing possible equations that describe the water balloon’s height over time. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 If the height as a function of time were sketched on a graph, what would that look like? It would start above the ground, go up, and then come back down.

DOK-2 Given the eyewitness accounts, what do you expect the general shape of the graph to be? Linear? Parabolic? Why? Parabolic because the eyewitness accounts suggest there is a peak height, and some witnesses see the ball both ascend and descend.

DOK-2 Would the parabola open up or down? Why? Down since there is a peak height or maximum

Engage

Explore

Explain

Elaborate

Evaluate

Acceleration

DOK-2 Would the yy-intercept -intercept be greater than, less than, or equal to zero for this scenario? Explain. Based on the eyewitness accounts, the ball was launched from a building across the street at an elevated height; the fact that the 4th ﬂoor resident only sees the ball descending while other residents such as the 8th ﬂoor tenant see the ball ascending suggests the water balloon was not launched from the ground up but from an elevation.

e. DOK-2 Which feature of the graph describes the time the water balloon hits the ground? Explain. The x-intercept because its y-coordinate is zero, which represents 0 ﬂoors or ground level, and the x-coordinate represents the time at which the water balloon is at this level. f.

Intervention

DOK-2 Which feature of the graph describes the peak height or ﬂoor of the water balloon? Explain. The vertex since it has the maximum y value

g. DOK-2 What does the yy-intercept -intercept represent? The height or ﬂoor from which the water balloon was launched, the water balloon’s starting point h. DOK-2 What key feature is Mrs. Evelyn interested in determining? The y-intercept because it would give her the height (ﬂoor) from which the balloon was launched. i. DOK-1 What form is Mrs. Evelyn’s set of equations? Vertex form

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j. DOK-2 What key feature can readily be determined from this form? The vertex k.

DOK-2 What is the vertex for each of Mrs. Evelyn’s equations? (−1, 9), (2, 9), (3, 9)

l. DOK-1 What do each of the vertices have in common? The y-coordinate is 9. m.

DOK-3 Why would the yy-coordinate -coordinate being 9 for each of Mrs. Evelyn’s equations make sense, and why would the variability of the x-coordinate make sense? 9 makes sense as the y-coordinate of the vertex because it represents the 9th ﬂoor, which was the highest ﬂoor to which the water balloon rose, and the fact that the x-coordinate or time varies makes sense since that is unknown in the situation.

n. DOK-1 What form is Andres’s set of equations? Factored form

6. 7.

DOK-2 What key feature can readily be determined from this form? The roots

DOK-2 How do you ﬁnd the roots from each factor? Find the value of x that makes each factor zero.

DOK-1 What are the roots for each of Andres’s equations? 0 or −6, −2 or 4, −1 or 5

DOK-3 What’s the difference between the roots of each equation? Why does this value make sense? The difference is 6 seconds. It makes sense since from the time the ball was at its highest point to the time it hit the ground, it was 3 seconds. The x-coordinate of the highest point marks the axis of symmetry, and each root is equidistant from it.

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-3 What is the relationship between the vertex and the roots of a quadratic function? The vertex is exactly between the roots. DOK-2 How can the roots be used to determine the x value of the vertex? The x value of the vertex is the average of the x value of the roots.

FACILITATION TIP Take time to clarify the deﬁnitions of roots and vertex in this context. Consider providing sentence frames or starters to encourage students to ﬂuently use the correct vocabulary when making observations about the equations, graphs, and charts.

FACILITATION TIP Before this Math Chat, use the Picture Vocabulary to reinforce and review the terminology to support effective discussions about what students are learning. Consider having students record words, images, and deﬁnitions on their Student Journal or in notebooks. 207

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Graphs of Quadratic Functions Explore 5 – Identifying Key Features and Equivalent Quadratic Functions Part II 1.

2. 3. 4. FACILITATION TIP

Print and project questions 5a–5d. Preview before students begin collaborating on Part II. Encourage students to be prepared to thoughtfully respond as they work and after they complete Part II. Gather and record appropriate student responses before the Math Chat. Have several students (some volunteers and some carefully selected) respond in their own words.

6. 7. FACILITATION TIP By the end of this Math Chat, students should be able to identify the roots, axis of symmetry, vertex, and y-intercept from a quadratic function. Consider that some students may still need to create a graph to be conﬁdently successful at this skill.

Read the following scenario to the class: Mrs. Evelyn and Andres suspect the water balloon was launched from the same ﬂoor, yet they notice they have different equations. Help them determine whether their equations are equivalent. Students should still have their Student Journals and Eyewitness Accounts. Explain to students that they will work with their groups to determine if Mrs. Evelyn’s and Andres’s equations are equivalent. Students will then work together to rewrite the equations in the same form 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 is the x value of all yy-intercepts? -intercepts? The x value of all y-intercepts is zero, (0, some number).

DOK-2 How can we determine the yy-intercept -intercept from an equation? Use substitution and evaluate the equation for x = 0.

DOK-1 What steps would you need to take to convert the equation from vertex form to standard form? Square the binomial, distribute the negative, and combine like terms.

DOK-1 What steps would you need to take to convert the equation from factored form to standard form? Multiply the binomials and combine like terms; distribute the negative if applicable.

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-3 Which quadratic form makes it easiest to find the yy-intercept, and why? The y-intercept can be determined easily by plugging zero in for x. However, the standard form of a quadratic equation can be used to determine the y-intercept by visual inspection since c is the y value of the y-intercept. • DOK-2 If you were asked to identify the maximum or minimum of a quadratic equation, which form would be most efficient, and why? I would prefer to have the equation in vertex form because the values of h and k are the x and y values of the ordered pair that describes the vertex. • DOK-2 Describe two ways to determine if quadratic equations are equivalent. The equations can be rewritten in the same form, or the key features can be identified and compared. • DOK-2 Which method is more efficient? Rewriting equations given in factored form or vertex form to standard form is the most efficient method. •

Post-Explore FACILITATION TIP On this Exit Ticket, struggling students may still want to create a graph or table. Consider providing a graph for support.

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1. 2. 3. 4.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Graphs of Quadratic 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

Key Features and Attributes 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

Roots, Zeros, x-intercepts, and Solutions 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

Equations in Vertex Form

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 for a in Quadratic Functions

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

Show What You Know, Part 5 Identifying Key Features and Equivalent Quadratic Functions Independent practice assignment that gives students an opportunity to demonstrate their learning

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Spiraled Review

Fluency Builder

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

Quadratic Functions

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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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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Graphs of Quadratic Functions

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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 model a scenario with a quadratic function.

I can determine key information from different forms of a quadratic function.

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SCOPE 1

Factors of Polynomials Scope Introduction SCOPE SUMMARY Students have already seen the power of writing quadratic equations in factored and vertex form. In this scope, students will use models to convert from standard form to factored or vertex form. Students will rewrite quadratic equations and expressions in factored form to identify zeros, and vertex form to identify the extreme value and axis of symmetry. They will gain experience with the forms of a quadratic and discern which form is most eﬃcient for the task based on the context and what is asked of them. Student Expectations

A.PAR.6.1 Interpret quadratic expressions and parts of a quadratic expression that represent a quantity in terms of its context.

A.FGR.7.8 Write a function deﬁned by a quadratic expression in different but equivalent forms to reveal and explain different properties of the function. A.FGR.7.9 Compare characteristics of two functions each represented in a different way.

Future Expectations

In previous grades, students learned how to generate equivalent numerical expressions using the order of operations. They are familiar with using whole number exponents and prime factorization to create these expressions. Students also used the distributive property to make equivalent expressions. They have discussed and analyzed graphs of quadratics in standard, factored, and vertex form, and are able to convert between all forms when possible.

In Algebra II, students will continue to factor quadratics using the structure of the expression provided. Later this year, students will use factoring and completing the square to solve quadratic equations, which they will also do in Algebra II in addition to solving quadratic inequalities. Students will also determine linear and quadratic factors of polynomial expressions of degrees three and four. To factor such expressions, they will use methods such as factoring the sum and difference of two cubes and factoring by grouping. They will also use factoring to divide rational expressions involving polynomials.

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

Hook

A.FGR.7.5 Rewrite a quadratic function representing a mathematically applicable situation to reveal the maximum or minimum value of the function it deﬁnes. Explain what the value describes in context.

Background Knowledge

Accessing Prior Knowledge

A.PAR.6.2 Fluently choose and produce an equivalent form of a quadratic expression to reveal and explain properties of the quantity represented by the expression.

VERTICAL ALIGNMENT

write and evaluate numerical expressions involving wholenumber exponents.

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

factor polynomials.

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

In this exploration, groups of students will be following an artist’s instructions to construct a mosaic of tiles and determine the size of the frame that is needed. Students will:

Explore 3

•

Explore 2

Factoring Using Models

factoring trinomials using algebra tiles and area models.

In this exploration, students will work collaboratively to investigate and determine the patterns of the a = 1 trinomial house to help a detective ﬁnd patterns of a crime family. Students will: explore patterns found in the factors of polynomials.

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

Completing the Square – Introduction

Completing the Square – Advanced

In this exploration, students will use algebraic expressions to describe each square block pattern for a quilt; and identify the correct equations for different quilt orders. Students will: •

factor polynomials by completing the square using algebra tiles and area models.

In this exploration, students will build on the previous exploration to use algebraic notation to describe how many quilt pieces were used and how many are left. 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 5

Factoring Using Patterns

•

Explore 4

Explore 1

EXPLORE ACTIVITIES

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factor polynomials by completing the square using algebra tiles and area models.

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

Quadratic Equations – Key Features In this exploration, students will investigate quadratic equations without the use of technology to help prepare for an exam. Students will: •

determine and use the most eﬃcient method to analyze a situation and identify key features.

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

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Factors of Polynomials Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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Explain

Elaborate

Evaluate

Intervention

Acceleration

ACCESSING PRIOR KNOWLEDGE Students will listen to prompts about the prior standard and communicate whether they feel the prompts are fact or ﬁction by walking to the designated sides of the classroom. This element is designed to uncover student misconceptions; it should not be taken for a grade. 7.PAR.2.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coeﬃcients.

Materials

Preparation

Printed •

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1 Set of Fact or Fiction (per teacher)

• •

Print one set of Fact or Fiction 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.

Prompt 1 is ﬁction.

Prompt 2 is fact.

Prompt 3 is fact.

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

Identifying Misconceptions • •

Students may not understand that they can use the distributive property to check any factoring work. Students may be uncomfortable working with fractions and may need some examples as a reminder.

FACILITATION TIP Before this, ask students to explain the order of operations; review as needed. Then, let them know their skills will be put to the test with a game of Fact or Fiction. FACILITATION TIP Consider creating additional prompts by taking the given prompts and switching exponents and/or the placement of parentheses. In such case, change the value of the answers, as well. If you make additional prompts, avoid making them all fact or all ﬁction so the students stay engaged. FACILITATION TIP Consider using the Foundation Builder with the whole class either as a challenge or review activity. It includes eight sets of expressions for students to analyze for equivalency.

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Factors of Polynomials Hook – Breaking It Down ACTIVITY PREPARATION Students will relate factoring polynomials to a real-world situation.

Materials

Preparation

Printed •

• • •

1 Breaking It Down (per class)

Reusable • •

1 Phenomena Video (per class) 1 Projector (per class)

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

PROCEDURE AND FACILITATION Part I: Pre-Explore 1. FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) Have you ever been in a car that has broken down?; 2) What happened to the car?; 3) How was the issue resolved?

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: Greg’s car breaks down unexpectedly, and he takes it to a local mechanic. Greg’s mind is racing with all of the things that could be wrong with the car before the mechanic takes a look. Consider how the mechanic will approach this challenge. 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 mechanic seems to be doing work under the hood of the car. I wonder how old the car is. I wonder what part of the car is not working. I wonder what the mechanic checked ﬁrst. I wonder how the cost of the repair changes based on what part of the car is not working. I wonder if there are any tests that the mechanic can run. Project Breaking It Down. Notes

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Explore

Explain

Elaborate

Evaluate

Explain to students that the goal of the scope will be to break down large, complicated polynomials into parts that are easier for us to understand and interpret. Discuss the following questions: a.

DOK-1 How is the 120 broken down? The 120 is written with its prime factors. The small prime numbers on the right side all multiply to 120.

DOK-1 Why might it be harder to break down the large polynomial expressions? Allow students to share all ideas. Answers will vary. It is less obvious what multiplies to the quadratic polynomial. There are three terms to consider instead of one number.

DOK-1 How do you think the mechanic should approach Greg’s car? You do not need any car expertise to answer this question. How would you approach the challenge? Allow students to share all ideas. Answers will vary. I think the mechanic should look at smaller parts of the car at a time. I think the mechanic should look for signs of certain car problems. I think the mechanic should start by looking at the engine. I think the mechanic should start by looking for the most common problems that other people have.

Complete the Explore activities.

Intervention

Acceleration

FACILITATION TIP As a ﬁnal discussion for the Pre-Explore, ask if anyone knows a mechanic personally. Has the mechanic shown them or told them about their work? Do they know which car diagnostic is the most challenging for the mechanic to make?

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FACILITATION TIP After students answer the question, ask the class for examples of things that could make Greg’s car break down. Which ones do they think are more problematic and costly? What other things could be wrong with the car that wouldn’t make it break down?

Part II: Post-Explore 1. 2.

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

DOK-1 Do the factors make more sense after the Explore activities? Yes, the binomials multiply to create the initial trinomial.

DOK-1 What strategies would you use to factor the polynomials? I would look for two values that add to the middle term’s coeﬃcient and multiply to the product of the two outer coeﬃcients. I would then create an area model.

DOK-1 Which trinomial represents a special product, and how do you know? Part d is a perfect square trinomial because half of the middle term’s coeﬃcient squared is the constant term.

DOK-1 What are the equivalent factored expressions of each polynomial? x2 + 15x + 56 = (x + 8)(x + 7); x2 – 3x – 70 = (x – 10)(x + 7); x2 – 24x + 144 = (x – 12)2

e. DOK-2 How does factoring help you break down a complex polynomial in the same way the mechanic has to examine a complex machine? The factors of these polynomials are linear terms that we can use to determine roots of a polynomial. A mechanic looks at individual parts of a car and can determine what needs to be ﬁxed when looking at it piece by piece instead of as one daunting project. f.

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

FACILITATION TIP As a ﬁnal discussion for the Post-Explore, ask students if a trinomial could possibly be used to describe business for the mechanic. If students are stumped, suggest an aspect like time of labor or cost, and work with them to articulate a quadratic equation that could represent an aspect of the mechanic’s business.

FACILITATION TIP In addition to question 2f, ask students to try to explain factoring polynomials to a peer partner. Next, ask them to model how they would you explain it to a parent, grandparent or younger student.

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Factors of Polynomials Explore 1 – Factoring Using Models ACTIVITY PREPARATION Students will explore factoring trinomials using algebra tiles and area models.

Standards for Mathematical Practice • • •

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

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Algebra Tiles Factoring Mat (per group) 1 Set of Art Pieces Cards (per group) 1 Exit Ticket (per 2 students)

•

Reusable • •

•

1 Set of algebra tiles (per group) 1 Resealable bag (per group)

• •

•

Separate the class into groups of 2 or 3 students. Print a Student Journal for each student. Print an Algebra Tiles Factoring Mat for each group. If desired, print it on card stock, and laminate it for future use. Print a set of Art Pieces Cards for each group. Cut the cards apart, and place them 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 a set of algebra tiles for each group. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. For students who need additional organizational support, please see our Algebra Tiles Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) What experiences have you had in art class?; 2) What is one fond memory you have of art class?; 3) What type of art and materials do you like best?

2. 3. 4.

Read the following scenario to the class: You are preparing an art exhibit in your school that will feature the work of an artist who uses algebra tiles to form mosaics. Mosaics are pictures made of smaller pieces of tile, glass, or other material such as algebra tiles. The artist shipped each work to you in pieces, and you must assemble the art according to the instructions and determine the dimensions of the art. Distribute the algebra tiles and Algebra Tiles Factoring Mat to each group of students. Explain to students that they will work with their groups to model a rectangle that is 16 square units. Allow students enough time to complete the task, and then invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 What is the area of the rectangle? 16 square units DOK-1 What are the dimensions of the rectangle? Possible answers include 16 × 1, 1 × 16, 2 × 8, 8 × 2, or 4 × 4. • DOK-1 Which dimensions would look like a square? 4 × 4 would look like a square because each dimension is the same and 42 is 16. The other dimensions would create rectangles. • •

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Engage

Explore

Explain

Elaborate

Evaluate

•

DOK-2 Why are there different possible answers? The length and width of the rectangle are the factors of the number. There are different combinations of factors that multiply to equal 16. Also, multiplication is commutative, so the order of the factors does not matter.

Read the following scenario to the class: The artist did not provide a picture of the assembled art. She gave the following three rules for creating the mosaic:

6. 7. 8. 9.

10.

The algebra tiles must be arranged in the shape of a rectangle.

The large square(s) must be in the upper left corner of the rectangle.

The small square(s) must be in the lower right corner of the rectangle. Follow these rules to assemble the artwork, and prepare the collection for display at the art exhibit.

Give a Student Journal to each student. Give a set of Art Pieces Cards to each group of students. Explain to students that they will work with their groups to arrange the tiles to prepare for the art exhibit and record their work on their Student Journals. Point out to students the meaning of the algebra tiles. The large square tile has dimensions of x units by x units and represents x2. When the large square is a shaded square, it represents −x − 2. The rectangle has dimensions of x units by 1 unit and represents x.. When the rectangle is shaded, it represents −x. − The small square has dimensions of 1 unit by 1 unit and represents 1. When the small square is shaded, it represents −1. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 What are the dimensions of each algebra tile? x2 tiles have dimensions of x and x. The x tiles have dimensions of 1 and x. Unit tiles have dimensions of 1 and 1.

DOK-1 What is the area of the tiles for art A? Explain your reasoning. The area is x2 + 5x + 6 because there is one large square tile that represents x2, ﬁve rectangle tiles that represent 5x, and six small square tiles that represent 6.

DOK-1 Which algebra tiles should be placed on the Algebra Tile Factoring Mat ﬁrst? First, place the large square, or x2, tile in the upper left corner. Second, place the 6 small square tiles in the lower right corner. The arrangement of the small square tiles is unclear now. We need to investigate further to determine if the small square tiles form a 1 × 6 or a 2 × 3 rectangle.

DOK-1 After you have arranged your tiles in a rectangle, how can you determine the dimensions of the frame of the artwork? Look at the length and width of the larger rectangle. The longer algebra tile sides represent x, and the shorter algebra tile sides represent 1. Add each tile to determine the length and width.

e. DOK-2 How can the tiles be arranged so they follow all of the rules? Remember, all algebra tiles must be used, and there should not be any holes. Answers may vary. I can move these rectangles to have some on each side. Then, the pieces will make a rectangle, and there will be no holes. f.

DOK-1 How do you calculate the total area of the art piece? Add the area of each algebra tile together by combining like terms.

DOK-2 In this activity, is there only one correct way to form a rectangle? If you follow the rules from the artist, there is only one way to form a rectangle. The difference between my classmate and me is that I assembled my rectangle in portrait orientation, and my classmate assembled the rectangle in landscape orientation.

Intervention

Acceleration

FACILITATION TIP Before reading the scenario, ask the class to describe a time when they gave instructions to someone. Were they speciﬁc in their instructions? How well was the person they instructed able to understand and follow the instructions?

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FACILITATION TIP Project the scenario and the three constraints for students to read along with you and write down the speciﬁcs. FACILITATION TIP Project these explanations about the meanings of the algebra tiles so students can read along with you and refer to as needed. FACILITATION TIP Before students begin Question 2. make sure they understand what each column title refers to. Then, remind them of the commutative property of multiplication while noting that they should write factors in a consistent order so that they can connect algebraic and visual representations.

FACILITATION TIP Entire groups, if not the whole class, may use one orientation for one or more mosaics. When the class has completed the table, have a student draw the ﬁrst mosaic for everyone to see. Ask if anyone’s mosaic looks different. If so, have them draw it. If not, draw the other orientation yourself. Do this for all mosaics. 221

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Factors of Polynomials Explore 1 – Factoring Using Models 11. 12. FACILITATION TIP For Question 4a, watch out for students just answering “Two rectangles”. Have them look back at the table for Question 2, and ask them what types of rectangles are there to work with. What types of rectangles should they represent here in 4a?

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-2 Can every quadratic trinomial expression be factored using algebra tiles? No. For example, it is impossible to make a rectangle with algebra tiles for x2 + x + 1 because the trinomial cannot be factored. You also cannot factor expressions that contain decimals or fractions with algebra tiles. DOK-1 Describe the process you used to ﬁnd the missing pieces to the artwork in Part I. The x terms total 1x. There is already 1x, so there can only be 0 added. I have to add 1x and −1x, which is adding a zero pair. DOK-2 When are algebra tiles a useful model for factoring? Predict why algebra tiles might not be useful. Algebra tiles are useful for visualizing how the area is related to its dimensions. It is useful for rewriting an expression from standard form into factored form. Algebra tiles might not be useful if the expression cannot form a rectangle. For example, an expression could be prime. Algebra tiles are not useful for expressions with very big numbers, expressions with more than one variable, or expressions of degrees larger than 2.

Part II 1. FACILITATION TIP Before reading the scenario, ask the class 1) When was a time you were able to work smarter, not harder?; 2) What activity were you performing?; 3) How much time, energy, money, etc. did you save?

3. FACILITATION TIP After students answer the first part of Question 5, write the dimensions for everyone to see. Then, add a negative sign to the first term in the first factor. Ask the class if they can tell through quick eye inspection how many terms in the new area will have a different sign than in the given area.

Read the following scenario to the class: You did so well displaying the artist’s work that she wants to hire you to be her assistant. The artist will provide the total area of her art, and you will use area models to help you determine the dimensions of the frames. You want to find a more efficient way to determine the dimensions of the frames rather than building each piece of art. Use an area model to help you quickly find the dimensions of each frame. Explain to students that they will work with their groups to determine a more efficient way to represent the area, length, and width of the artwork 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 What is the relationship between the b term, 7x,, and the numbers you wrote on the area model? 3 + 4 = 7, so the two numbers must add up to equal b.

DOK-2 What is the relationship between a, c (2x2 and 6), and the numbers you wrote on the area model? Both sets of numbers are factor pairs for 12. 2 · 6 = 12 and 3 · 4 = 12

DOK-2 Summarize the process of ﬁlling in the boxes of the area model. The upper left square is ax2. The lower right square is c. The two numbers in the remaining squares must add to equal b and multiply to equal the product of a and c.

DOK-2 What is the relationship between each row/column and the dimensions? Each term of the dimension is the greatest common factor (GCF) of each row/column.

e. DOK-1 What method do you use to ﬁnd the two numbers in the upper right/lower left squares? The two numbers must add to equal b and multiply to equal ac. f. FACILITATION TIP Students should remember the commutative property of multiplication. Encourage them to answer Question 1 of the Reﬂect section through quick eye inspection. 222

4. 5.

DOK-1 What method do you use to ﬁnd the dimensions of the rectangle? Each term of the dimension is the GCF of each row/column.

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.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Math Chat •

DOK-2 Compare and contrast the algebra tile method and the area model method. Algebra tiles and area models have similar formats. The x2 term is in the upper left corner, the constant terms are in the lower right corner, and the x terms are in the other corners. The area model is a numerical representation of the algebra tiles. Algebra tiles are more limited in what they can represent. For example, they can only show polynomials with one variable.

Post-Explore 1. 2. 3.

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

FACILITATION TIP Students may be intimidated to see that both coeﬃcients and the constant are all prime numbers. Explain that this means there are fewer factors they have to consider, and encourage them to focus on the strategy they have used to complete the area model.

FACTORS OF POLYNOMIALS

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FACTORS OF POLYNOMIALS

Factors of Polynomials Explore 2 – Factoring Using Patterns ACTIVITY PREPARATION Students will explore patterns found in the factors of polynomials. Students will analyze the relationships between polynomials and their factors to uncover strategies for special products.

Standards for Mathematical Practice • • •

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

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Set of Quadratic Bandits Wanted Posters (per student) 1 Exit Ticket (per 2 students)

• • •

Reusable • •

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

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 a set of Quadratic Bandits Wanted Posters for each student. Gather a glue stick for each group and a pair of scissors for each student. For students who need additional organizational support, please see our Algebra Tiles Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) Who likes to read mystery novels?; 2) Which ones do you read?; 3) Do you ever imagine yourself as one of the characters?

Part I 1.

FACILITATION TIP To help engage students, project the text of this scenario and have students read it aloud with you. FACILITATION TIP Consider making time to preview/review the essential vocabulary terms in Picture Vocabulary before moving through these next four scopes. Have students record notes on their Student Journals or in notebooks. FACILITATION TIP To facilitate cohesion, consider distributing the Student Journal in parts as needed. Part I (pages 1 and 2), Part II (pages 3 and 4), and Part III (pp 5-6). FACILITATION TIP Watch out for students missing or incorrectly using negative signs or mixing up the target product and target sum. 224

2. 3.

Read the following scenario to the class: You have been hired as an algebra super sleuth. The detective firm has assigned you with learning the patterns of the crime family known as the Mysterious Quadratics. It is essential you help them recognize the patterns and predict the factored form before these Mysterious Quadratics get out of control and take over the world! The Mysterious Quadratics family is separated into houses, and some houses have smaller groups called pods. The firm has asked you to first investigate and determine the patterns of the a = 1 trinomial house. The bandits leave diamond calling cards at every crime scene. What is the pattern? What are the missing values? Give a Student Journal to each student. Explain to students that they will work with their groups to analyze the relationships between a trinomial in standard form, ax2 + bx + c where a = 1, and its factors. Students will record their answers and explanations on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a. DOK-2 Can you combine numbers on the calling card using basic arithmetic operations (+, –, ×, ÷) to obtain other numbers on the card? Yes, I can combine two of the numbers using both addition and multiplication to obtain the other two numbers. b.

DOK-2 How can the numbers on the left and right be used to determine the number on the top? Their product is the top number.

DOK-2 How can the numbers on the left and right be used to determine the number on the bottom? Their sum is the bottom number.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

DOK-2 If you didn’t know the numbers on the left and right, how could the top number help you narrow down the options? The only possibilities for the left and right would have to be factors of the top number. Listing the factors would give a short list of possibilities.

e. DOK-2 How can it be determined if both factors are positive? If the target product and target sum are positive, then both factors are positive. f.

DOK-2 How can it be determined if both factors are negative? If the target product is positive and the target sum is negative, then both factors will be negative.

DOK-2 How can it be determined if one factor is positive and one factor is negative? If the target product is negative, then one factor is positive, and the other is negative.

FACTORS OF POLYNOMIALS

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h. DOK-2 When one factor is positive and one factor is negative, how is it determined which number should have which sign? If the target sum is positive, then the larger factor is positive. If the target sum is negative, then the larger factor is negative. i. DOK-1 What is a trinomial? A trinomial is a polynomial with three terms. Those terms can be constants, variables, or variables raised to a nonnegative exponent. j. DOK-2 When transforming standard form to factored form, what is the ﬁrst step? The ﬁrst step is to determine the factors of c. k.

DOK-2 Once the factors of c have been determined, what’s the next step to writing the expression in factored form? The next step is to determine which set of factors add to b.

l. DOK-2 How do we know we have the right numbers to write the expression in factored form? We know they are the right numbers if they meet the target product and target sum criteria. m.

5. 6.

DOK-2 Why is multiplication used to verify that the factors are correct? Multiplication is used because factoring is the opposite of multiplying. To check, we use the opposite process, similar to checking a division fact with multiplication.

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-2 How can multiplication be used to check if the factored form is equivalent to the quadratic given in standard form? Multiplying the binomial factors is the reverse operation of factoring the original quadratic in standard form. If the factoring was done correctly, multiplying those factors should produce the original quadratic. • DOK-2 How is factoring using patterns different when c is not positive? The two constants in the factored form will have opposite signs. • DOK-2 How is factoring using patterns different when b is not positive? One or both of the constants in the factored form will have a negative sign. • DOK-1 How can perfect square numbers be determined? Perfect square numbers are numbers that are the product of two equal integers—for example, 4 · 4 = 16, 5 · 5 = 25, 6 · 6 = 36, 7 · 7 = 49, and so on. •

FACILITATION TIP After students answer the question, explain to the class what types of terms can be in a trinomial in case some students don’t mention it in their answer. Then, ask them why trinomials can’t have a variable raised to a negative exponent. Do they think this applies to all polynomials? Why or why not? This may be a review for some students.

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FACTORS OF POLYNOMIALS

Factors of Polynomials Explore 2 – Factoring Using Patterns Part II 1. FACILITATION TIP Before reading the scenario, ask the class 1) Have you ever watched shows or movies based on mystery novels?; 2) Have you also read the novels?; 3) If so, in what ways were the books and movies the same and different?

Read the following scenario to the class: You were so successful with the a = 1 trinomial house that the ﬁrm has made you the lead detective investigating the remaining 2 crime houses of the Mysterious Quadratics family. Use the information your boss compiled to apprehend the crime families. Explain to students that they will work with their groups to identify patterns for factoring quadratics that are perfect square trinomials or that are the difference of squares. Students will record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding question to assess student understanding: a.

FACILITATION TIP To clarify question 3 on the reﬂection questions, consider discussing it as a class. Write a perfect square trinomial for everyone to see, and discuss each answer option in the context of the trinomial. Once students conﬁrm which answer works, you can present another couple of perfect square trinomials to help verify it is true. FACILITATION TIP Check students’ understanding of the structure of perfect square trinomials. After students answer the question, ask them which term in a perfect square trinomial is affected by the sign of the constant in the factors and how so. FACILITATION TIP On the last Math Chat questions, make sure students know that it refers to one of the “Bandit in Standard Form” entries in Part II. FACILITATION TIP Before reading the scenario, ask the class 1) Who is your favorite mystery hero or villain?; 2) What mystery is the character a part of?; 3) What do you like about them?

FACILITATION TIP Students may change their answers multiple times. Encourage groups to conclude where all posters go before taping or gluing them in place. It may help to attach posters with sticky notes, which are easier to remove if a student wants to change an answer later on.

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4. 5.

DOK-2 In perfect square trinomials and difference of squares, which terms are perfect square numbers? ax2 and c

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 Describe the pattern for factoring quadratics in the perfect square trinomial house. In the perfect square trinomial house, the product is positive and a perfect square number, so the constant term is identical in both factors. That constant can be either positive or negative. • DOK-2 Describe the pattern for factoring quadratics in the difference of squares house. In the difference of squares house, the product is a negative perfect square, so the constant terms of the two factors have the same absolute value (magnitude) but with opposite signs. • DOK-3 One of these quadratics doesn’t belong in the a = 1 trinomial house, but it does belong in the family. Which is it? What is different about this quadratic? Could we use a pattern to factor this quadratic? What is the pattern? The 4x2 – 9 quadratic does not belong in the a = 1 trinomial house because a ≠ 1 in this case. However, this quadratic is the difference of two squares. In this case, (2x)2 = 4x2 and 32 = 9, so there is a pattern. For ax2 – c where a and c are both perfect square numbers, the factored form is (√ax – √c)(√ax + √c). •

Part III 1.

2. 3. 4.

Read the following scenario to the class: You have apprehended the major players of the crime houses. However, some members were not discovered until after the initial investigation was concluded. Examine the wanted posters, and determine which house each quadratic belongs to so they can be apprehended and charged. Give a set of Quadratic Bandits Wanted Posters and a pair of scissors to each student. Give a glue stick to each group. Have students cut out the Quadratic Bandits Wanted Posters. Explain to students that they will work with their groups to classify quadratics as either (1) trinomials in standard form, ax2 + bx + c where a = 1; (2) difference of squares; or (3) perfect square trinomials. Students will glue or tape the wanted posters on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 Which house is made up only of quadratics that are a difference of two terms? The difference of squares house includes quadratics that are the difference of two terms.

DOK-1 Which other house is made up of quadratics that have a perfect square as the constant term but have more than two terms? The perfect square trinomial house includes quadratics that have a perfect square for c but also have a bx term.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

FACILITATION TIP Allow students enough time to complete Part III and answer the reflection questions. Invite each group to factor one quadratic expression from each house. Check students’ understanding of the 7. After the Explore activity, invite the class to a Math Chat to share their structure of perfect square trinomials observations and learning. again. After students answer Question 2 of the Reflect section, ask them if switching Math Chat the signs of the first two terms in the given expression would result in a new perfect • DOK-2 Why is understanding the relationships between a quadratic and its square trinomial. How do they know? factors useful? Understanding the relationships allows me to see patterns. Knowing those patterns reduces the number of possible choices for constants that I have to consider in the binomial factors. This can make factoring faster and easier. 6.

FACTORS OF POLYNOMIALS

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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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FACTORS OF POLYNOMIALS

Factors of Polynomials Explore 3 – Completing the Square – Introduction ACTIVITY PREPARATION Students will factor polynomials by completing the square using algebra tiles and area models.

Standards for Mathematical Practice • • •

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

Materials

Preparation

Printed • • • • •

1 Student Journal (per student) 1 Algebra Tiles Factoring Mat (per student) 1 Set of Quilt Orders Cards (per student) 1 Math Chat – Other Area and Dimension Expressions (per class) 1 Exit Ticket (per student)

Reusable • • • •

1 Set of algebra tiles (per group) 1 Pair of scissors (per student) 1 Glue stick (per group) 1 Projector or document camera (per class)

• • • • • • • •

•

Separate the class into groups of 2 or 3 students. Print a Student Journal, Algebra Tiles Factoring Mat, and Exit Ticket for each student. Print a set of Quilt Order Cards for each student. Print a Math Chat – Other Area and Dimension Expressions for the class. Gather a glue stick and a set of algebra tiles for each group. Gather a pair of scissors for each student. Be prepared to project the Math Chat – Other Area and Dimension Expressions for the class to see. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. For students who need additional organizational support, please see our Algebra Tiles Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Does anyone own a homemade quilt?; 2) How often do you use it?; 3) What colors and patterns does it have? FACILITATION TIP Print and project this detailed scenario so students can take notes as you all read it aloud together. FACILITATION TIP Some students may not be familiar with the term “Abuela.” Ask if anyone knows what it means. If so, have them explain it. If not, explain to the class that the word means “grandmother” in Spanish. 228

2. 3. 4.

Read the following scenario to the class: Miranda is learning how to make a quilt from her Abuela Florencia. Abuela Florencia is a retired engineer, so she often uses algebraic patterns and expressions to design her quilts. Abuela Florencia starts each quilt by planning the individual block patterns. Her designs are based on perfect squares. The squares can be larger or smaller to enable her to make quilts of any size to ﬁt a lap or bed. She calls the length of the side for the standard square x, and she determines the area of the standard square is A(x) = x2. Abuela’s quilts are very popular, and she sells out each month when she sells quilts at the ﬂea market. Abuela tells Miranda she will teach her how to make a quilt. Miranda wants a quilt that is 5 inches larger than the standard square. Abuela uses algebraic expressions to describe each square block pattern. Help Miranda describe her square block pattern with algebraic expressions. Give a Student Journal to each student. Give a set of algebra tiles to each group. Invite the class to a Math Chat to discuss the algebra tiles.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Math Chat • •

•

DOK-1 What are the dimensions of each type of tile? The large square is x units by x units. The rectangle is x units by 1 unit. The small squares are 1 unit by 1 unit. DOK-2 Use 4 unit tiles to make a perfect square. What are the dimensions of the square? What multiplication fact does this show? The dimensions of the square are 2 units by 2 units. This shows 2 times 2, or 2 squared. DOK-2 Use 9 unit tiles to make a perfect square. What are the dimensions of the square? What multiplication fact does this show? The dimensions of the square are 3 units by 3 units. This shows 3 times 3, or 3 squared. DOK-2 Use 16 unit tiles to make a perfect square. What are the dimensions of the square? What multiplication fact does this show? The dimensions of the square are 4 units by 4 units. This shows 4 times 4, or 4 squared. DOK-3 The ﬁrst 3 perfect square numbers are 4, 9, and 16. Why are these perfect square numbers? What is the related math fact to determine these numbers? These are perfect square numbers because they can model a perfect square. They can be determined by raising a number to the second power. DOK-3 How would the next 3 perfect square numbers be determined? What are the next 3 perfect square numbers? The next 3 perfect square numbers would be determined by calculating 5 squared, 6 squared, and 7 squared. The next 3 perfect square numbers are 25, 36, and 49. Explain to students that they will work with their groups to help Miranda with her square block patterns and algebraic expressions. The algebra tiles may be used to model square block patterns and may be helpful when answering questions. Explain to students that 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 do you show ﬁve more than a number algebraically? Since “more than” indicates adding, we would use a variable plus 5.

DOK-1 How do you show the area of a square? Can that be shown using an exponent? Can it be shown using multiplication? The area of a square can be shown using the length of a side squared. It can also be shown as multiplication by multiplying the length of a side times the length of a side.

DOK-2 What are the dimensions of the algebra tiles? The big square is x by x, the rectangles are x by 1 unit, and the small squares are 1 unit by 1 unit.

DOK-2 If you were given 1 large square, 10 rectangles, and 25 small squares, how could you express the area of the square block pattern? The area of the square block pattern would be all the tiles added together. We would use x2 + 10x + 25 because the x2 represents the large square’s area, the 10x represents the rectangles’ area, and the 25 represents the small squares’ area.

e. DOK-2 Describe how you know what algebra tiles to use to represent the expression x2 + 10x 10x + 25. We know x2 is represented by a large square tile, x is represented by a rectangle tile, and the constants are represented by the small square tiles. Based on this equation, we would need one large tile, 10 rectangle tiles, and 25 small square tiles. f.

7. 8.

DOK-2 If you know the area of a square block pattern is described by the expression ((xx + 5)2, how could you ﬁnd an equivalent expression for the area of the square block pattern? We could multiply it by itself to determine an equivalent quadratic in standard form.

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.

FACTORS OF POLYNOMIALS

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FACILITATION TIP Some students may not yet connect the sides of a diagram with algebraic expressions. Remind the class of the dimensions of all three algebra tile shapes. Then, draw the diagram from Question 3 on the Student Journal, and have a student label each section of the top and left of the diagram with its corresponding dimension.

FACILITATION TIP Check students’ understanding of subtraction in the context of algebra tile models. After students answer the question, ask them how the model would change if the given expression said “ –10x “ instead of “ +10x.”

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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Factors of Polynomials Explore 3 – Completing the Square – Introduction Math Chat DOK-2 If you were given 1 large square, 4 rectangles, and 4 small squares, how could you express the area of the square block pattern? The area of the square block pattern would be all the tiles added together. We would use x2 + 4x + 4 because the x2 represents the large square’s area, the 4x represents the rectangles’ area, and the 4 represents the small squares’ area. • DOK-2 Describe how you know what algebra tiles to use to represent the expression x2 + 4 4xx + 4. We know x2 is represented by a large square tile, x is represented by a rectangle tile, and the constants are represented by the small square tiles. Based on this equation, we would need one large tile, 4 rectangle tiles, and 4 small square tiles. • DOK-1 How could ((xx + 1)2 be written as multiplication? We would multiply it by itself: (x + 1)(x + 1). • DOK-2 If you know the area of a square block pattern is described by the expression ((xx + 1)2, how could you ﬁnd an equivalent expression for the area of the square block pattern? We could multiply it by itself to determine an equivalent quadratic in standard form. •

FACILITATION TIP Clarify to the class that the question is asking how to write the given expression as factors. After they answer the question, check their understanding of zeros by asking how many zeros would there be given the equation y = (x + 1)(x + 1). Then, ask what the zero(s) is(are).

Part II

FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever been to a ﬂea market?; 2) What did you ﬁnd there?; 3) What did you purchase, if anything? FACILITATION TIP Consider having the Quilt Orders Cards precut.

2. 3. 4. 5.

FACILITATION TIP Watch out for students matching b, c, and h with the wrong operations in their answers. If needed, work through the question as a class. 230

6. 7.

Read the following scenario to the class: Miranda’s baby sister ripped the orders to be completed for customers at the next flea market. Help Miranda match up Myra and Kofi’s orders with the correct equations and diagrams. Then, help Miranda determine what, if any, patterns occur. Give a set of Quilt Orders Cards and a pair of scissors to each student. Give a glue stick to each group of students. Have students cut out the Quilt Orders Cards. Explain to students that they will work with their groups to help Miranda make sense of the orders and describe patterns that can be used to make the process more efficient. Explain to students that they will glue the cards to the page for questions 1 and 2. Point out to the class that they will have cards left over after question 1. The remaining cards should be used for question 2. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 How are the equations for area and dimensions related? Answers will vary. The binomials used in the dimension equation can be multiplied to get the area equation.

DOK-2 What is the result if the value of b is divided by 2? Is half of the value of b used in an expression? The results will vary. Half the value of b is used in the dimension expression.

DOK-2 In the square block pattern, the large square represents x2, and the small squares represent units. How are these terms represented in the array? In the array, we see the x2 in the upper left corner and the units in the lower right corner.

d. DOK-3 What do you notice about the square block pattern and the array? How are they similar? Explain how the position of the tiles in the square block pattern relates to the position of the terms in the array. I notice that the x squared tile and x squared term are both in the upper left. I also notice that the unit tiles are in the lower right of the square block pattern and the constant from the area equation is in the lower right of the array. Similarly, the x tiles are divided evenly between the upper right and lower left of the square block pattern. In the array, the term that includes x is divided evenly between the upper right and lower left of the array.

Engage

Explore

Explain

Elaborate

Evaluate

e. DOK-3 Explain how you could use the value of b to determine if an area equation was a perfect square. If we use the value of b and ﬁnd half, that gives us the constant in the binomial, h. If we square the value of h, then we have the value of c if the equation is a perfect square.

8. 9.

Intervention

Acceleration

FACILITATION TIP

For reﬂection questions 3–5, address multiple relationships that may still confuse some students when observed together. Allow students enough time to complete Part II and answer the reﬂection questions. Have students refer back to part e of the guiding questions as needed to help explain After the Explore activity, invite the class to a Math Chat to share their relationships. observations and learning.

Math Chat DOK-2 How is the total number of rectangles in the square block pattern related to the constant, h,, in the dimension expression? The value of the constant, h, in the dimension expression can be determined by ﬁnding half of the total number of rectangles. • DOK-2 How is the constant, c, in the area expression related to the number of small squares in the square block pattern? The value of the constant, c, is the same as the number of small squares in the square block pattern. •

Display the Math Chat – Other Area and Dimension Expressions for the class, and then ask the following questions. DOK-3 Miranda also examined other area and dimension expressions she found among previous orders. Does the pattern work for these expressions? Can the value of b be used to determine the value of h?? Why or why not? The pattern still works for these expressions. It will always work because b represents the total number of rectangles, and h represents the number of rectangles on each side when they are distributed evenly. If they are distributed evenly, half go on each side. Based on the samples shown, this works when b is positive and negative. • DOK-3 Does this work for the other expression Miranda found on previous orders? Can the value of h be used to determine the value of c? Why or why not? This pattern does work on these expressions. It works because h is the number of rectangles on each side. The number of rectangles on each side deﬁnes the number of unit tiles that will be needed. The small tiles are h units by h units, so it would make sense that h2 could be used to determine the number of unit tiles. •

STEMscopes Tip

FACTORS OF POLYNOMIALS

Home

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

FACILITATION TIP

When you preview this Exit Ticket with students, consider that some students may be intimidated when factoring using an array given a large number like “441.” Ask students which corners of the array they can already ﬁll in based on quick eye inspection. Then, ask them how the Post-Explore coeﬃcients of the upper right and lower left 1. Have students complete the Exit Ticket to formatively assess their understanding terms relate to b and c of the given perfect of the concept. square trinomial. 2. Complete the Anchor Chart as a class. 3. Have each student complete their Interactive Notebook. Notes __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________ __________________________________________________________________________________________________________________________________________________

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Factors of Polynomials Explore 4 – Completing the Square – Advanced ACTIVITY PREPARATION Students will factor polynomials by completing the square using algebra tiles and area models.

Standards for Mathematical Practice • • •

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

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Algebra Tiles Factoring Mat (per group) 1 Set of Bags of Fabric Cards (per group) 1 Exit Ticket (per student)

•

• •

Reusable • •

1 Resealable bag (per group) 1 Set of algebra tiles (per group)

•

Separate the class into groups of 2 or 3 students. Print a Student Journal and an Exit Ticket for each student. Print an Algebra Tiles Factoring Mat for each group. If desired, print it on card stock, and laminate it for future use. Print a set of Bags of Fabric 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. Gather a set of algebra tiles for each group. Go Digital! Have students explore or present their solutions using virtual manipulatives! The manipulatives used in this lesson can be found in the Explore drop-down menu and can be digitally assigned to students. For students who need additional organizational support, please see our Algebra Tiles Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION FACILITATION TIP

Part I

Before reading the scenario, ask the class 1) What are some patterns quilters include in their quilts?; 2) Why do you think they use different patterns?; 3) If you were going to make a quilt, what type of pattern would you use?

FACILITATION TIP For Question 1 encourage students to look at the entries for Miranda’s bag carefully. As they look, have the class explain the relationship between the values for b, c, and h before they complete the other columns. FACILITATION TIP For Question 1 monitor how students explain diagrams to each other within their groups, and clear misconceptions as necessary. If a whole group is stuck on a diagram, encourage them to observe how b was represented in the diagram for Miranda’s bag. Note to them that they should use the same pattern for Bags 1 and 2. 232

2. 3. 4.

5. 6.

Read the following scenario to the class: Abuela Florencia gave Miranda quilting fabric and orders to make square block patterns. Abuela is very picky and wants all fabric accounted for. Miranda has to use algebraic notation to describe how many pieces she used and how many are left over after she creates the square block pattern. Use the pieces in bag 1 and bag 2 from Abuela Florencia to help Miranda make the square block pattern, and record your information in the table like Miranda did. Give a Student Journal to each student. Give a set of algebra tiles, an Algebra Tiles Factoring Mat, and a set of Bags of Fabric Cards to each group. Explain to students that they will work with their groups to help Miranda with her square block patterns and algebraic expressions. Then, they will record their work on their Student Journals. Have students pull out Miranda’s bag card. Tell students this card has already been solved as an example for them on the table on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 In the Bags of Fabric Cards, what are the values of the large square, the rectangles, and the small squares? The large square represents x2. Each rectangle represents x. Each small square represents the constant 1. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 How many small squares does Miranda have in her bag? How many does she need to complete the block pattern? How many are left after completing it? She starts with 13 and uses 9 to complete the pattern. 4 are extra.

DOK-2 What is the area of the block pattern Miranda creates? The area is (x + 3)2.

DOK-2 How are the extra squares represented in the dimension equation? This number is added to the area of the block pattern to give the complete dimension equation and account for all of the material.

Intervention

Acceleration

FACILITATION TIP Preview the guiding questions and select some essential ones to project and discuss before, during, and after collaboration.

FACTORS OF POLYNOMIALS

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e. DOK-2 When using an array, what goes into the upper left box? The x2 term goes in the upper left. f.

DOK-2 What goes into the upper right and lower left boxes of the array? The linear term that includes x is divided evenly between those two boxes.

DOK-2 In problem 4, the square has equal height and width. What is this side dimension value? x + 12

h. DOK-2 How do you determine the value for the lower right box of the array, and what is it? The value is the square of the side dimension’s constant term, 12, which is 144. i. DOK-2 To account for all of the material in the original area equation, what amount remains after completing the constant term in the lower right box, and where can this extra amount be recorded? The original area equation had a constant term of 146, so 2 remains after taking 144, and this extra remaining amount can be recorded in the circle below the array. j. DOK-2 Where does this extra amount appear in the dimension equation? The dimension equation is the sum of the area of the array and this extra amount. 7. 8.

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-2 How is the area equation related to a quadratic equation in standard form? The area equation is a quadratic equation in standard form, y = ax2 + bx + c. DOK-2 How is the dimension equation related to a quadratic equation in vertex form? The dimension equation is a quadratic equation in vertex form y = (x – h)2 + k DOK-2 How do you know how to set up the linear terms or x tiles in the block pattern? The value of h is half of b, so half of the x tiles are arranged on each side of the x2 tile. DOK-2 How can you determine the value of k from h and c? What would it be in terms of b and c? The value of k is the difference of c – h2, or the original constant term from the area equation, c, minus the constant amount needed to b b complete the square, h2. Since h = __2 , then k = c – ( __2 )2.

Part II 1.

Read the following scenario to the class: Miranda’s cousin has been helping Abuela too. She has constructed partial square block patterns and doesn’t have time to ﬁnish. Help Miranda decide how many unit squares she needs from Abuela’s supply closet to ﬁnish each square block pattern, and record them algebraically for Abuela. Explain to students that they will work with their groups to help Miranda with her square block patterns and algebraic expressions. Then, they will record their work on their Student Journals.

FACILITATION TIP Before reading the scenario, ask the class 1) What is something you started working on but didn’t finish?; 2) Why were you unable to finish it?; 3) Did you come back and finish it, or did someone else finish it for you? 233

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Factors of Polynomials Explore 4 – Completing the Square – Advanced 3.

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

FACILITATION TIP Students have several entries to account for in the table for Question 1. Instruct students to remember to shade tiles, when need be, without indicating when.

DOK-2 Does the initial area equation given represent a complete square pattern block? No, it represents the partially completed block shown in the top diagram.

DOK-1 What does shading represent in the diagram? Shaded pieces have a negative value.

DOK-2 Are the two diagrams given in each column equivalent? Why or why not? Yes, they are equivalent because an equal number of small squares were added (unshaded) and subtracted (shaded).

DOK-2 Is the dimension equation equivalent to the area equation? Why or why not? Yes, they are equivalent because the constant term that needed to be added to complete the square term was also subtracted, which maintains the equivalence.

e. DOK-2 In the array diagrams, what must be true about the term in the lower right-hand box and the term in the circle? They must sum to the original constant term in the area equation. f.

FACILITATION TIP For Question 6 students may forget to account for the factor of 2 when adding the zero pair to complete the square. It may help to work through this question as a class. Have the class discuss each step and explain their reasoning without giving away any steps or the answer.

FACILITATION TIP Project the last question of the Math Chat and encourage students to think, make notes, and then share their responses with a partner. Check for understanding by calling on individual students to share with the whole class. FACILITATION TIP For this Exit Ticket, students should apply their experiences using arrays to complete the square for quadratics with larger numbers. Instruct them to use the same method they used on the Student Journal.

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4. 5.

DOK-3 In problem 4, what kind of square block pattern does A(x ( ) (x represent? How many does it represent? It represents two identical square block patterns that each need an additional 4 small squares.

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 What are two methods that can be used to visualize completing the square? Making a block diagram or using an array are two methods that can be used. • DOK-2 If the equation involves large numbers, which is more appropriate to use, and why? The array method is better because it would be tedious to draw a large number of boxes and rectangles for a block diagram. • DOK-2 When completing the square to represent a quadratic expression, there are three possibilities. Two of them are as follows: (1) the quadratic is a perfect square; (2) the quadratic expression has enough to make a perfect square with an extra constant term left over. What is the third possibility? The quadratic expression does not have enough, so a constant term must be added but also subtracted to maintain equivalence. • DOK-2 How would you describe completing the square to a friend that missed class today? Give steps as part of your description. 1. If there is a leading coefficient, then factor it out. 2. Group the first- and second-degree terms together. 3. Using a zero pair, add and subtract a constant to complete the square formed by the first- and second-degree terms. 4. Write the expression in vertex form: a(x + b)2 + k. •

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

FACTORS OF POLYNOMIALS

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Factors of Polynomials Explore 5 – Quadratic Equations – Key Features ACTIVITY PREPARATION Students will determine and use the most eﬃcient method to analyze a situation and identify key features.

Standards for Mathematical Practice • • •

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

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Set of Note Cards (per group) 1 Set of Supplemental Cards (per group, optional) 1 Exit Ticket (per student)

•

Reusable •

•

1 Resealable bag (per group)

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 Note 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. Optionally, print a set of Supplemental Cards for each group of students to use in Part II. These cards are larger images of graph A and graph B. For students who need additional organizational support, please see our Algebra Tiles Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION FACILITATION TIP

Part I

Before reading the scenario, ask the class 1) Do you use technology to help you complete assignments or study for tests?; 2) How is using the technology useful?; 3) If you were told you couldn’t use technology for assignments or to study, what would you use instead?

FACILITATION TIP After students answer the question, ask if anyone answered that it is in vertex form. Note to the class that the equation is also in vertex form, y = a(x – h)2 + k, where h = 0 and the parentheses are taken away. Write out f(x) = 2(x – 0)2 – 3 to clarify. FACILITATION TIP Check students’ understanding of transformations. After they answer the question, ask the class how the negative sign of the leading coeﬃcient affects the equation’s graph. FACILITATION TIP These equations are found at the top of page 2 of the Student Journal. After they answer the question, ask the class how they would transform h(x) to get m(x). 236

2. 3.

Read the following scenario to the class: Taio and Myra have been working together to understand quadratics for the upcoming test on Wednesday. They have been using technology at home when practicing for their math test. The teacher just told them today that there will be no technology use when taking the test. Help Taio and Myra make sense of the quadratics they are investigating. Give a Student Journal to each student. Explain to students that they will explore and analyze quadratic equations and help Taio and Myra make sense of the quadratics for their test. 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 In what form is the equation f( f(x (x) = 2x2 – 3? The equation is in standard form.

DOK-1 In what form is the equation g(x ( ) = −( (x −(xx – 1)2 + 4? The equation is in vertex form.

DOK-1 What is the leading coeﬃcient of f( f(x (x)? The leading coeﬃcient is 2.

DOK-1 What is the leading coeﬃcient of g(x ( )? The leading coeﬃcient (x is −1.

e. DOK-1 When a quadratic function opens upward, does it have a minimum or maximum? It has a minimum. f.

DOK-1 What do the equations h(x ( ) and m(x (x ( ) have in common? They (x are both written in vertex form, they both have a negative leading coeﬃcient, and they both have the same constant added. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

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 When does a parabola have a minimum? When a parabola opens upward, the graph has a minimum. DOK-1 What is the value of a when a parabola has a minimum? When a parabola has a minimum, the value of a > 0. DOK-1 When does a parabola have a maximum? When a parabola opens downward, the graph has a maximum. DOK-1 What is the value of a when a parabola has a maximum? When a parabola has a maximum, the value of a < 0. DOK-2 How will you use the equation in factored form to find the roots of the equation? The factored form of the equation tells us the roots—in other words, the x-intercepts. The roots have the opposite sign of the linear factors. DOK-2 How will you use the equation in vertex form to find the vertex of the graph? I will compare the given equation in vertex form with y = (x – h)2 + k. The point (h, k) will tell me about the vertex of the parabola. DOK-2 How will you use the equation in standard form to find the y-intercept? y I can find the y-intercept by looking at the coefficient c of the equation in standard form.

2. 3. 4. 5.

FACILITATION TIP

Part II 1.

Acceleration

DOK-1 What is the relationship between the roots and vertex of a parabola? The vertex is exactly between the roots.

h. DOK-1 Which form of a quadratic equation is most efficient for determining the vertex? Vertex form is most efficient because the vertex is easily identified from h and k. 5.

Intervention

FACTORS OF POLYNOMIALS

Home

Read the following scenario to the class: Taio and Myra have a math test on Wednesday. Myra made note cards so they both could study them before the test. However, she accidentally dropped them on her way to study, and the notes all got mixed up! Give a set of Note Cards to each group of students. Optionally, give a set of Supplemental Cards to each group. Explain to students that they will work with their groups to sort the Note Cards to help Taio and Myra get organized to study. Students will then work together to analyze the quadratic functions 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 On which axis is the yy-intercept -intercept located? It is located on the y-axis.

DOK-1 What does the y value of the yy-intercept have in common with the quadratic function in standard form? The y value of the vertex is the constant at the end of the equation when written in standard form.

DOK-1 What pattern can be used to factor a quadratic equation when a is 1? The pattern of determining what numbers multiply to c and add to b—those two numbers will be used in the linear factors.

DOK-1 What do the linear factors and the zeros have in common? They use the same numbers but are opposite signs.

e. DOK-1 What is the relationship between the roots and vertex of a parabola? The vertex is exactly halfway between the roots.

Before reading the scenario, ask the class 1) What strategy or method do you use use to study for a test?; 2) Do you do better on the test when you study alone or with a friend?; 3) Why do you think that is? FACILITATION TIP Provide a quick review of what students can easily ﬁnd from each form of a quadratic. Then, advise them to ﬁrst sort the cards with equations into standard, vertex, and factored form. Students can then take those cards one at a time and match them with the key features they can easily pick out to sort the cards. FACILITATION TIP Some students may overthink some familiar concepts that are simply presented in a different way with different wording. As students start working, explain that some of the guiding questions will review what they have learned. FACILITATION TIP For guiding questions 5a–5n, consider coaching the whole class through them while you work together on the Student Journal for Part II. 237

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Factors of Polynomials Explore 5 – Quadratic Equations – Key Features f.

DOK-1 Does the ﬁrst graph have a maximum or minimum? The graph has a minimum.

DOK-1 What will be the value of the coeﬃcient of a graph with a minimum? The value of a > 0.

h. DOK-1 Does the second graph have a maximum or minimum? Graph B has a maximum.

i. DOK-1 What will be the value of the coeﬃcient of a graph with a maximum? The value of a < 0. j. DOK-1 Where on a parabola is the vertex located? The vertex is the minimum or maximum. It is the turning point where the curve changes from increasing to decreasing or vice versa. k.

DOK-1 How is the axis of symmetry related to the coordinate of the vertex? The axis of symmetry is the vertical line that goes through the vertex. To create the equation, use the x value of the vertex.

l. DOK-1 What happens if a quadratic function in vertex form is expanded by multiplying the linear factor, distributing, and simplifying? The new function would be a quadratic function in standard form. m.

DOK-1 How do you know if a quadratic equation is in standard form? It is written with the terms going from highest degree to lowest, and there are no linear factors, just terms.

n. DOK-1 When completing the square, what is the step after grouping the first- and second-degree terms? Next, we figure out what number will complete the square created by the first- and second-degree terms. We add that number inside the parentheses and subtract that number outside the parentheses, creating a zero pair and keeping the equation balanced. FACILITATION TIP Check students’ understanding of transformations. After they answer Question 13 ask the class how the graph of the resulting equation would change if the value of a was changed to 3.

6. 7.

Math Chat • •

•

FACILITATION TIP Encourage students to pay close attention as they work through the Exit Ticket to avoid algebraic and graphing errors. At this point, students have experience ﬁnding key features of quadratics and graphing their equations. 238

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.

DOK-1 What point does the axis of symmetry pass through? The axis of symmetry passes through the vertex, or the maximum/minimum. DOK-2 How will you use the equation in standard form to find the roots of the equation? I will use the factoring method to convert the equation in standard form to factored form to find the roots of the equation. DOK-2 How will you use the equation in standard form to find the vertex of the graph? I will use the completing the square method to convert the equation in standard form to vertex form to find the vertex of the graph. DOK-2 How will you use the equation in factored form or vertex form to find the y-intercept? I will substitute zero for the independent variable and simplify to y-intercept? determine the value of the dependent variable. DOK-2 What is the relationship between the x-intercepts -intercepts and the vertex? The axis of symmetry lies equidistant from each of the x-intercepts and passes through the vertex. Thus, the x value of the vertex is halfway between the values of the x-intercepts.

Post-Explore 1. 2. 3. 4.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

FACTORS OF POLYNOMIALS

Home

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Factors of Polynomials 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

Factoring Using Models 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

Factoring Using Patterns 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

Completing the Square – Introduction

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

Completing the Square – Advanced

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

Show What You Know, Part 5 Quadratic Equations – Key Features Independent practice assignment that gives students an opportunity to demonstrate their learning

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Spiraled Review

Fluency Builder

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

Rewrite Quadratics

Can be done independently

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

FACTORS OF POLYNOMIALS

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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

FACTORS OF POLYNOMIALS

Factors of Polynomials

3 242

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

What prompts will be used?

What does mastery look like?

FACTORS OF POLYNOMIALS

Home

I can identify all parts of a quadratic function presented algebraically.

I can determine the meaning of individual terms and factors.

I can rewrite a quadratic function in a different form.

I can factor and complete the square using models.

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SCOPE 1

Solve Quadratics Scope Introduction SCOPE SUMMARY

Student Expectations

In this grade level, students will expand their understanding of interpreting the solution as the place where the lines intersect on a graph, or where the same ordered pairs exist in a table. Students should be able to use models, algebra tiles, and algebraic representations to explain and understand equations. They should solve an equation by using strategies such as completing the square, factoring, and the quadratic formula. Students should be able to complete the square to derive the quadratic formula. They should be able to explain why one solution method is more suited to a particular problem context than another. Students will be able to use the discriminant to determine the number of real solutions to a quadratic equation.

A.PAR.6.3 Create and solve quadratic equations in one variable and explain the solution in the framework of applicable phenomena. A.PAR.6.4 Represent constraints by quadratic equations and interpret data points as possible or not possible in a modeling framework.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grade levels, students created and solved linear equations based on real-world and mathematical situations. They will build on those skills as they create quadratic equations to model situations and data. Students have graphed quadratic equations and factored and completed the square already. As a preface to the quadratic formula, students have taken square roots of perfect squares as well as simpliﬁed radicands that were not perfect squares.

In Algebra II, students will continue to solve quadratic equations and will also solve quadratic inequalities. As students solve quadratic equations in future grades, they will extend their knowledge of the number system to include complex numbers as solutions. Students will also solve systems of quadratic and linear equations using many of the techniques from this scope in addition to their knowledge of points of intersection and systems of equations.

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

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES

use agree/disagree strategy to justify the steps of a one-solution equation or inequality.

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.

solve quadratics that relate to realworld situations.

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.

244

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Solve Quadratics by Taking Square Roots In this exploration, students will solve quadratic equations using the method of taking square roots. Students will: •

Explore 2

Explore 1

EXPLORE ACTIVITIES

solve quadratic equations of the form s2 = A where s is the side length and A is the area of a square.

•

solve quadratics by factoring an equation.

•

apply the zero product rule to ﬁnd solutions.

•

determine which solution is a reasonable answer.

use their knowledge of completing the square to solve quadratic equations.

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

Explore 4

Explore 3

In this exploration, groups of students will solve a realworld scenario about working for a pool design and installation company and are tasked with designing a pool and deck to meet a customer’s speciﬁcations. Students will:

In this exploration, students will be tasked with determining the winner of a pumpkin throwing contest using a catapult. Students will: •

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

Solve Quadratics by Factoring

Solve Quadratics by Completing the Square

SOLVE QUADRATICS

Home

Using the Quadratic Formula In this exploration, students will be asked to develop an application for a computer software company to solve quadratic equations. Students will: •

solve quadratic equations using the quadratic formula.

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

After students have had time to explore and solve the scenario, they will discuss their learning and complete an Exit Ticket for assessment. Choosing the Best Method In this exploration, students will be tasked with helping solve different business models that have an quadratic equation representing the different business processes. Students will: •

analyze quadratic equations and determine which method would be the most eﬃcient for solving.

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SOLVE QUADRATICS

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

Tuesday

Wednesday

Thursday

Friday

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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

SOLVE QUADRATICS

Home

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

Materials

Preparation

Printed •

•

Print one Agree or Disagree for each student.

1 Agree or Disagree (per student)

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

Instruct students to complete the Agree or Disagree independently. Once students have completed the activity on their own, have them stand up. Instruct all students to walk around the classroom with their 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. You may then continue as many times as you want with different partners. Discuss the responses as a class. Allow students to explain their reasoning for each problem. a.

Agree with Shalyn

Disagree with Kiy

Disagree with Austin

FACILITATION TIP Consider distributing or displaying each of the three items one at a time to help you facilitate a cohesive discussion. Cutting the sheet into thirds would still give students enough room to show some of their strategies for solving on their own. FACILITATION TIP Before having students begin to complete problems, clarify whether students are to solve for the variables in the equations. FACILITATION TIP

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

If needed, brainstorm a list of words that students might need to create a clear justiﬁcation (inverse operation, order of operations, steps, balanced). Additionally, you could provide sentence starters or frames.

Identifying Misconceptions • • • •

Students may struggle to remember the properties used to justify steps in solving equations. Students may struggle to remember the order of operations so they can properly use inverse operations to solve. If students are struggling with the abstract algebraic representations, provide algebra tiles for students to model solving. Students may need to be reminded that they can substitute a potential solution into the equation to see whether it is correct. Notes

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SOLVE QUADRATICS

Solve Quadratics Hook – Stone Throw ACTIVITY PREPARATION Students will relate solving quadratics to a real-world situation.

Materials

Preparation

Printed •

• • •

1 Stone Throw (per class)

Reusable • •

1 Phenomena Video (per class) 1 Projector (per class)

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

PROCEDURE AND FACILITATION Part I: Pre-Explore 1. FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) If you’ve tried skipping or throwing stones in a pond, how did you do it?; 2) Did you compete for distance, splash, skips, or anything else?; 3) What method might you use to accurately measure who wins?

2. 3.

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: Dan is throwing stones into a pond. How far did Dan throw the ﬁrst stone before it landed in the pond? 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 Dan throws the stone from about shoulder height. I notice that the stone travels in an arc. I wonder if the stone would go farther if Dan threw it overhand. I wonder which of the three throws traveled the farthest. I wonder what equation would model the path of the stone. Project Stone Throw. Notes

248

Engage

Explore

Explain

Elaborate

Evaluate

Acceleration

Explain to students that the equation models the ﬁrst stone throw. The y value is the height of the stone measured in feet, and the x value is the horizontal distance traveled measured in feet. Discuss the following questions: a.

DOK-1 Why would this model have a negative leading coeﬃcient? Because the path of the stone increases in height before falling to the water, the parabola should open down, and it needs a negative leading coeﬃcient.

DOK-1 What does the + 4 represent at the end of the model? Allow students to share all ideas. Answers will vary. It represents the height of the stone when Dan released it.

DOK-1 What forms would be most helpful to determine how far the stone travels before hitting the water? Intercept or vertex form

Complete the Explore activities.

Part II: Post-Explore 1. 2.

Intervention

SOLVE QUADRATICS

Home

FACILITATION TIP At the end of the Post-Explore, ask students how they think the path of water hit by a thrown stone would compare with the path of the stone. Give space for them to justify their reasoning.

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

DOK-1 How would you begin to ﬁgure out how far the stone travels before it hits the water? Set the value of y equal to 0.

DOK-1 What strategies would you use to solve the equation you create? I would like to factor, but that does not seem possible so I would complete the square to solve for x.

DOK-1 What is your new equation after completing the square? Solve your equation. 0 = −0.1(x – 10)2 + 14 x ≈ −1.832 and x ≈ 21.832.

DOK-2 Why do we not always use both solutions to a quadratic equation? When there are two solutions, they may not both make sense in a given scenario. In this case, the negative x value does not make sense when trying to ﬁnd the distance the stone traveled before hitting the water.

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

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Solve Quadratics Explore 1 – Solve Quadratics by Taking Square Roots ACTIVITY PREPARATION Students will solve quadratic equations using the method of taking square roots.

Standards for Mathematical Practice • • •

MP.3 Construct viable arguments and critique the reasoning of others. 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)

•

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. For students who need additional organizational support, please see our Algebra Tiles Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) What kind of art do you like to make?; 2)What classes have you taken?; 3) What skills can art classes teach you? FACILITATION TIP

2. 3.

Consider distributing the Student Journal in sections as needed. Part I is pages 1 and 2; Part II is pages 3 and 4. 4.

DOK-1 In the area expression A = s2, what operation does the 2 indicate? It indicates a square or exponent of 2 (not multiplication by 2).

DOK-2 Add/subtract and multiply/divide are examples of operation pairs that undo each other. What is the operation to undo a square? The operation to undo a square is to take the square root.

DOK-2 How are the expressions 52 and (−5)2 similar? How are they different? They both have the same value when they are evaluated. One is squaring a positive number, and one is squaring a negative number.

FACILITATION TIP

Check students’ understanding of order of operations further. After they answer the question, ask the students how – (5)2 is similar to and different from the given expressions.

DOK-2 Squaring a number always gives a positive result, so when reversing the operation and taking the square root, how do you know if the original number was positive or negative? I don’t know, so I need to consider both the positive and negative square roots.

e. DOK-2 To ﬁnd all possible solutions, how many equations do you have to solve? I have to solve two equations.

FACILITATION TIP After students answer the question, ask them what the expression inside a radical sign is called. Then, ask them what an index is. Remind them that an index of 2 is the understood index of a square root and is unwritten just as an understood coeﬃcient of 1 is typically unwritten.

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Read the following scenario to the class: Mateo is enrolled in an art class he enjoys at school. Right now, his class is learning about patterns. The instructor assigns projects involving patterns. Can you help Mateo with some of the math calculations? Give a Student Journal to each student. Explain to students that they will work with their groups to solve quadratic equations of the form s2 = A, where s is the side length and A is the area of a square. Instruct them to ﬁrst look at the sketch provided for each art assignment to determine the side length and area of the square. Have them set up the quadratic equation based on this information and solve. Have them record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding:

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 Does the answer make sense based on the real-life context? Only nonnegative answers make sense for a real-world length.

DOK-2 How can you check that your answer is correct? I can substitute my answer back into the original equation. If the equation is true, then my solution is correct.

h. DOK-2 In assignment 2, does 900 square inches represent information about the area, A,, or does it represent information about the side length, s? It represents information about the area, A. We would substitute this value for the variable A. i. DOK-2 What information is given for the side length of the poster? How can that information be used to represent the side length of the poster? It says the side length of the photograph, x, and that the photograph should be increased by 22 inches. The side length of the photograph plus 22 would represent the side length of the poster, s = x + 22.

Intervention

Acceleration

FACILITATION TIP

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After students answer the question, ask them how they know that “900 square inches” represents area rather than length even without any context. They should understand that any square unit, such as square inches, indicates area.

j. DOK-2 To solve the equation from question 7, ((xx + 22)2 = 900, what is the ﬁrst operation (outermost operation) to undo? The ﬁrst operation to undo is the square. k.

DOK-2 How many equations do you have to solve from this point? I have to solve two equations—one is equal to 30, and the other is equal to −30.

l. DOK-1 After undoing the square, what is the next operation to undo? The next operation to undo is addition. 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.

Math Chat • • • •

• • •

DOK-1 What type of equation is the area equation A = s2? It is a quadratic equation. DOK-2 In these problems, what is true about the expression containing the unknown variable? It is a perfect square. DOK-1 What is true about the other side of the equation? It is a constant. DOK-2 We are using the perfect square principle, which says that the side of the equation containing the unknown variable is a perfect square and the term on the other side is a constant. What can we do to both sides to solve the equation for the unknown? We can take the square root of both sides. DOK-2 How many solutions do you get when you apply the perfect square principle? I expect two solutions if the quantity squared equals a positive value. DOK-2 How do you decide if they are valid? I can check if the answer physically makes sense. DOK-2 How did you check your ﬁnal answer? I substituted the answer back into the original equation.

FACILITATION TIP After students read Assignment 2, ask the class if they have tried to resize a picture in a word document before. Did it change the picture’s quality? Note that one can typically select the word document’s picture, hover over one of its corner points, and click and drag to enlarge or shrink the picture while keeping its quality. It may help to show an example on a screen.

Part II 1.

Read the following scenario to the class: An exhibition of student artwork is planned for Friday in the school’s auditorium. Mateo and his classmates will each show a collection of their prints. Each is responsible for installing their own display following the guidelines provided by the instructor. Explain to students that they will work with their groups to once again solve quadratic equations utilizing the area formula s2 = A,, where s is the side length and A is the area of a square. However, this time they will be considering multiple areas at the same time. Have them set up the quadratic equation based on the information provided and solve. Have them record their work on their Student Journals.

FACILITATION TIP Before reading the scenario, ask the class 1) Where have you seen artwork (maybe your own) displayed?; 2) What media was it (painting, sculpture, print, etc.)?; 3) What objects were included in the artwork?

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Solve Quadratics Explore 1 – Solve Quadratics by Taking Square Roots 3.

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

DOK-1 In problem 1, besides the square area enclosing the art installation, what other area is included in the overall allocated space? The area of the empty space serving as a buffer between displays is also part of the overall allocated space.

DOK-2 How do you evaluate the area of the buffer of empty space, and is it a constant or a variable expression? The area of the buffer is 20% of the total allocated and is a constant.

DOK-1 How do you evaluate the area of the actual art installation with side length s?? Use the area formula A = s2.

DOK-1 What do the area of the buffer and the area of the installation sum to? They sum to the allocated wall space of 20 square feet.

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.

e. DOK-1 How do you isolate a perfect square here? Subtract 4 on each side to isolate a perfect square on one side and a constant on the other side.

FACILITATION TIP After students answer Problem 2 ask them if it would be practical to measure out a sheet of paper with exactly the maximum dimensions. Students should realize that this would not be practical given that the maximum dimensions are irrational.

DOK-1 In problem 2, how many square areas are there? There are three square areas. Mateo and two other students each have an art installation.

DOK-2 If the students are sharing the paint equally so that their areas are equal, will the side dimensions of their square art installations be the same? Yes, they will be the same.

h. DOK-1 For side dimension s,, what is the expression for the area of each student’s installation? The area of each student’s installation is s2. i. DOK-1 What do their areas sum to? Their areas sum to 54 square feet. j. DOK-2 Can you isolate a perfect square? Yes, you can isolate a perfect square.

FACILITATION TIP After students answer Question 3 in the Reﬂect section, ask them for the quickest way to solve the given equation algebraically. They should conclude that the quickest way to solve the equation algebraically is by factoring.

4. 5.

Math Chat • •

• FACILITATION TIP After they complete the Exit Ticket, ask the class if they could solve the second equation by taking square roots if “7s” was gone. Some may initially think you can’t because the coeﬃcient “2” is not a perfect square. Remind them that they can divide both sides by 2 and how then all of the criteria for the perfect square principle are met.

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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.

•

DOK-3 Can this method of isolating perfect squares be used to solve any quadratic? No, this method cannot be used to solve any quadratic. DOK-2 What type of problem cannot be solved using this method? Equations that involve a linear term cannot be solved using this method. You cannot solve the equation unless you can isolate a constant on one side and a perfect square on the other side of the equal sign. DOK-2 If not, give an example of a quadratic that cannot be solved with this approach. x2 – 5x = 0 DOK-3 Do you think solving by taking square roots is easier than factoring or completing the square in the problems you just completed? A “yes” response will probably be common.

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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Solve Quadratics Explore 2 – Solve Quadratics by Completing the Square ACTIVITY PREPARATION Students will use their knowledge of completing the square to solve quadratic equations.

Standards for Mathematical Practice • • •

MP.4 Model with mathematics. 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 Pumpkin Launch Notebooks (per group) 1 Exit Ticket (per 2 students)

• • • •

Reusable • •

1 Scientiﬁc calculator (per group) 2 Resealable bags (per group)

• •

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 has one. Print a set of Pumpkin Launch Notebooks for each group. Cut out the Part I cards, and place them in a resealable bag labeled “Part I.” Cut out the Part II cards, and place them in a resealable bag labeled “Part II.” If desired, print them on card stock, and laminate them for future use. Gather a scientiﬁc calculator for each group. For students who need additional organizational support, please see our Algebra Tiles Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) What is a catapult?; 2) Have you ever built a catapult?; 3) What types of things can be launched from a catapult? FACILITATION TIP Project this scenario for the class to read along with you. Have student volunteers read sections aloud while students write down important values, constraints, math phrases, and ideas.

2. 3. 4.

FACILITATION TIP Depending on your class, consider guiding them through some of the steps as a whole class. FACILITATION TIP Students may be initially confused by the handout. Explain that students will ﬁnd the solutions to each quadratic equation and compare the solutions with their respective team’s projection. 254

Read the following scenario to the class: Hope Mountain High School’s engineering class is having a catapult competition. The class has been divided into 2 teams. Each team designed and built their own catapults with 2 arms of different lengths. Each team is given a pumpkin to launch. Based on the design of the catapult, the base, and the weight of their pumpkin, each team has calculated a quadratic equation to estimate the distance their pumpkin will travel. The ﬁrst round of the competition will be with the shorter arm. The team whose pumpkin comes closest to their projection will win the round. Give a Student Journal to each student. Give a Part I Pumpkin Launch Notebook and a scientiﬁc calculator to each group. Explain to students that they will work with their groups to solve the quadratic equations to determine how far each team projected their pumpkins will launch 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 many solutions are there to x2 = 9? Why? There are 2 solutions because both 3 and −3 squared equal 9.

DOK-2 How could you go about solving an equation like x – 2 = ± √9? You would simplify the square root as much as possible or ﬁnd the decimal approximation and then add 2 to both sides of the equation. You would then have to compute 2 + 3 and 2 – 3 to ﬁnish solving.

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-1 What happens when you take the square root of a number that is not a perfect square? You factor the number and pull out all of the perfect squares you can form from the factors. The ones that do not pair up remain inside the radical. You can also use a calculator to determine the decimal approximation.

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-2 Although there are two solutions to each of these quadratics, only one can reasonably be the distance the pumpkin will fly. Which of the solutions is the reasonable answer? Why? The positive number is the reasonable solution because we expect the pumpkin to fly forward, not backward. DOK-2 How can you get from the standard form of the quadratic equation to the vertex form? You can convert a quadratic equation from the standard form into vertex form by completing the square. DOK-2 Why did we need to isolate the square term on one side of the equation? Solving equations is the undoing of the operations. If we were doing number calculations, exponents would be first. Conversely, when solving, the exponents are the last operations to be undone.

Intervention

Acceleration

FACILITATION TIP Some students may be initially intimidated by a word problem that asks for sentence order rather than calculations. For Question 1 of Part I, it may help to provide a simple quadratic for students to solve using square roots. As they solve it, they can track what they did to then sort the steps for Question 1.

SOLVE QUADRATICS

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FACILITATION TIP Review students’ understanding of radicals. Before they answer the question, have them take the square root of a few numbers that are not perfect squares as a class. Let one radicand be a double-digit prime number and two radicands be composite numbers.

Part II 1.

2. 3. 4.

6. 7.

Read the following scenario to the class: After the first round of pumpkin launching, the students were allowed to make some adjustments to their catapults to try to get better accuracy. The second round of the competition will be using the longer arms of their catapults. Give a Part II Pumpkin Launch Notebook to each group. Students should still have their Student Journals and scientific calculators. Explain to students that they will work with their groups to solve the quadratics by first completing the squares to determine how far the pumpkins should fly. Then, they will calculate which launch was the closest to the predicted value. The students will work together 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 is the requirement to be able to complete the square that has to do with the coeﬃcient in front of x2? The coeﬃcient in front of x2 has to be 1 in order for you to be able to complete the square.

DOK-1 What can you do if this requirement is not met? If the coefficient in front of x2 is not 1, you can factor the leading coefficient out of the first- and second-degree terms.

FACILITATION TIP Before reading the scenario, ask the class 1) How accurate do you think most objects launched from a catapult are in hitting their target?; 2) How might a catapult be changed or adjusted to increase its accuracy? FACILITATION TIP Note to students that they will need their calculators for Question 3 of Part II. Have scratch paper available so students have ample room to solve the equations.

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 When is it a good idea to use completing the square? It is a good idea to use completing the square when it is the only available method or the most eﬃcient method. • DOK-2 Can all quadratic equations be solved using completing the square? Yes, all quadratic equations can be solved using completing the square. • DOK-2 Are there always solutions to a quadratic equation? No, if the quadratic equation does not cross the x-axis, there are no real solutions to the equation. •

FACILITATION TIP Check students’ understanding of graphs and transformations. After students answer the question, show the class a parabola with two solutions. Then, ask them for examples of transformations that would produce a graph with no solutions. 255

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Solve Quadratics Explore 2 – Solve Quadratics by Completing the Square DOK-1 Is there always only one way to solve a quadratic equation? No, there are usually multiple ways of solving a quadratic equation. It is kind of like having multiple tools in your toolbox. Many tools can be used for the same task; you just need to decide which tool is most eﬃcient for the task. • DOK-1 What are some of the ways we know to solve equations? We can solve them by factoring, completing the square, graphing, or using the quadratic formula. • DOK-2 When completing the square, if b is an even number, we get an integer when taking b/2. What would happen if b is odd? Could we still complete the square? How would the vertex form of the equation look different? Yes, you could still complete the square if b is an odd number. The vertex form of the equation would simply have a fraction rather than an integer inside the squared term. •

FACILITATION TIP Project the essential questions about ways to solve equations. List appropriate student answers and encourage students to take notes, volunteer answers, and ask follow-up questions. Call on some select students to check for understanding.

Post-Explore FACILITATION TIP When you preview this Exit Ticket with students, clarify your criteria for success. Consider how many steps students are to show and any other constraints.

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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Explain

Elaborate

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Intervention

Acceleration

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Solve Quadratics Explore 3 – Solve Quadratics by Factoring ACTIVITY PREPARATION Students will solve quadratics by factoring the equation and applying the zero product rule to ﬁnd the solutions. They will then determine which solution is a reasonable answer.

Standards for Mathematical Practice • • •

MP.4 Model with mathematics. 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 Exit Ticket (per 2 students)

•

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 has one. For students who need additional organizational support, please see our Algebra Tiles Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) Where have you been swimming?; 2) When you swim indoors, what pool design is best for swimming laps?; 3) What pool design do you think is best for a play pool? FACILITATION TIP Consider distributing this Student Journal in two sections as needed. Part I is pages 1 and 2; Part II is pages 3–5. FACILITATION TIP Project questions 4a and 4b before students begin to collaborate. Clarify and post the answers if needed for support.

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2. 3.

Read the following scenario to the class: Maria just got a job working for a pool design and installation company. She has been tasked with designing a pool and deck to meet a customer’s speciﬁcations. The customer would like the pool itself to have an area of 24 square yards and for the length of the pool to be 2 yards longer than the width. Help Maria determine the dimensions of the pool. Give a Student Journal to each student. Explain to students that they will work with their groups to use factoring to determine the length and width of the pool 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 is the formula for the area of a rectangle? The formula for the area of a rectangle is A = lw.

DOK-1 What is the standard form of the quadratic equation? The standard form of the quadratic equation is ax2 + bx + c = 0.

DOK-2 How did rewriting the equation in standard form help us write it in factored form? To rewrite an equation in factored form, we must either model it or have it written in standard form ﬁrst. Rewriting it in standard form was the most eﬃcient method to rewrite the equation in factored form.

DOK-2 Why do you need to put the quadratic equation into standard form? To be able to solve using the zero product rule, one side of the equation needs to be equal to zero. You also need to be able to factor the other side, and standard form is the best starting spot for factoring a quadratic. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

e. DOK-2 Could you determine the value of b in question 5c? Why or why not? I could determine the value of b because I knew that b + 5 = 0, so I could solve the simple equation and see that the value of b is −5. 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.

Math Chat

STEMscopes Tip

SOLVE QUADRATICS

Home

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.

DOK-2 How can we use the zero product rule to solve a quadratic equation? You can factor the equation and then set the individual factors to zero and solve those equations for the solutions to your original equation. • DOK-2 Could you have simply applied the zero product rule to the equation w(w + 2) = 24? No, one side of the equation needs to be equal to zero to use the zero product rule. • DOK-2 Will all solutions to the equation always be reasonable? No, sometimes one or more of the solutions will not work in the given scenario. Like with dimensions, the number needs to be positive. • DOK-2 How can we check to see if our answer is correct? We can multiply the length times the width to see if we get the correct area. Then, we can look to see if our dimensions meet the criteria set out in the problem. • DOK-3 In the equation a · b = 0, what must be true about a or b? If ab = 0, then either a or b or both a and b must be equal to zero. FACILITATION TIP • DOK-2 Could we use the zero product rule on an equation such as a · b = 5? No, one side of the equation has to be equal to zero for the zero product rule to apply. Check students’ understanding of inverse operations in the context of the zero Part II product rule. After students answer the 1. Read the following scenario to the class: Maria is now working on the design and question, ask them if they can use the zero dimensions for the pool deck to go around the pool. It will be the same size on 3 product rule if they bring “5” to the left side sides, but one side will be 4 times wider so they can set up a patio table. The total of the given equation. Why or why not? area of the pool and deck will be 66 square yards. What will be the dimensions of FACILITATION TIP the pool, including the new deck? Before reading the scenario, ask the class 2. Students should still have their Student Journals. 1) Typically, there is a concrete or stone 3. Students will work together to use factoring to determine the length and width of deck around a pool. What deck size do the pool deck and record their work on their Student Journals. you think would be best to go around a 4. As students collaborate, monitor their work and use the following guiding pool?; 2) When might a bigger sized deck questions to assess student understanding: be desired?; 3) When might a smaller sized deck be necessary? a. DOK-1 When you have a quadratic equation in standard form and you FACILITATION TIP notice a greatest common factor, what can you do? You can factor out the greatest common factor. Project this scenario for the class to read along with you. Have student volunteers b. DOK-2 Why might you want to factor out the greatest common factor read sections aloud while students write before doing any other factoring? Removing the greatest common down important values, constraints, math factor usually simpliﬁes the equation, making it easier to factor further. phrases, and ideas. It will either lower the order of the equation or reduce the size of the coeﬃcients. •

5. 6.

DOK-2 When applying the zero product rule, why don’t we set the greatest common factor (GCF) equal to zero? Could there be an equation where the GCF would be set equal to zero using the zero product rule? We do not set it equal to zero in this case because it is an integer GCF. If the GCF included a variable, we would set it equal to zero and solve.

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.

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Solve Quadratics Explore 3 – Solve Quadratics by Factoring Math Chat •

• FACILITATION TIP Check student retention of polynomial operations. After they answer the question, have them write down which special product the given equation describes without saying it aloud. Then, reveal the correct answer. FACILITATION TIP Some students may notice a familiar structure on the left side of a given equation and begin solving it without ﬁrst moving “2” to the left side. Encourage students to look at the initial equation attentively before solving it.

•

• •

DOK-2 Does having a constant come into a factored equation affect the zero product rule? Use 2ab 2ab = 0 as an example. No, a constant does not affect the solutions. Both sides can be divided by that constant and not change the outcome. DOK-2 Why did we not use the zero product rule on (6 + 5x)(4 5 )(4 + 2x) 2 = 66? To use the zero product rule, you have to have one side of the equation equal to zero. DOK-2 How can we know if our solution is correct? We have the dimensions of the deck, and we know what the area should be. To verify our answer is correct, we simply need to multiply our dimensions to see if we get the desired area. DOK-3 What would be the solutions to ((xx – a)(x )(x – b) = 0? The solutions to (x – a) (x – b) = 0 would be a and b. DOK-3 What would be the solutions to x2 – a2 = 0? x2 – a2 = 0 can factor to (x – a) (x + a) = 0, so the solutions would be a and −a.

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 Quadratics Explore 4 – Using the Quadratic Formula ACTIVITY PREPARATION Students will solve quadratic equations using the quadratic formula.

Standards for Mathematical Practice • • •

MP.3 Construct viable arguments and critique the reasoning of others. MP.6 Attend to precision. MP.7 Look for and make use of structure.

Materials

Preparation

Printed • •

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

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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. For students who need additional organizational support, please see our Algebra Tiles Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I: Analysis and Development Phases FACILITATION TIP Before reading the scenario, ask the class 1) What apps are you familiar with?; 2) How do the apps help you?; 3) If you could develop an app, what would it be? FACILITATION TIP Consider distributing the Student Journal in two parts as needed; Part I (pages 1 and 2), then Part II (pages 3–5). FACILITATION TIP

2. 3.

Read the following scenario to the class: Compute-It Software Corp. wants to develop an application to solve quadratic equations. Since quadratic functions describe many real-world behaviors, they are sure the application would be marketable. Help the development team decide on the right approach to power the app. Give a Student Journal to each student. Explain to students that they will work with their groups to evaluate the best approach for developing a general formula to be used to solve quadratic equations and that they will record their work in their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding:

For Question 1, encourage students to use the graph on the page rather than a calculator and to count the grid lines carefully to locate the x-coordinates of the x-intercepts. Remind them that they can check each solution by plugging it into the given equation 0 = x2 – 4x + 3.

DOK-1 In problem 1, where along the curve does y = 0? y = 0 when the curve crosses the x-axis.

DOK-1 In problem 1, what are the x-coordinates -coordinates where the y-coordinate y is 0? At x = 1 and x = 3, y = 0.

DOK-2 For team A’s approach, what values are you trying to ﬁnd? I need two numbers that multiply to 8 and add up to 6.

FACILITATION TIP

DOK-2 Once you have factored the quadratic expression, what do you need to do to ﬁnd the solution? You need to use the zero product property and set each factor equal to 0.

Check students’ understanding of completing the square. Ask them how they are able to find the constant term “9.” They should know that they can find the constant by dividing the linear coefficient by 2 and squaring the result.

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e. DOK-1 What is the constant term of the perfect square trinomial that starts with x2 + 6xx? The constant term would be 9 because x2 + 6x + 9 = (x + 3)2. f.

DOK-2 How will team B’s approach lead us to getting two solutions? We need to take the positive and negative square root using a ± sign to get two solutions. © Accelerate Learning Inc. - All Rights Reserved

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DOK-2 How can you check that these values of x are solutions? I can substitute these values of x into the quadratic equation and verify that y = 0.

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-2 What were the pros and cons of the strategy used by team A? Team A’s strategy of factoring required less work than team B’s strategy. Once a quadratic expression is factored, it is really easy to see the solutions. The biggest downside of team A’s strategy is that we cannot solve all quadratics by factoring. • DOK-2 What were the pros and cons of the strategy used by team B? Team B’s strategy of completing the square made it possible to solve quadratic equations when factoring was not an option. While we can solve more equations with this method, it is a longer and more labor-intensive process. • DOK-3 Have we reached our goal of being able to solve any quadratic equation without needing to graph? Why or why not? We could use completing the square to solve any quadratic, but it will not be an eﬃcient method for solving. •

STEMscopes Tip

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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.

Part II: Test Phase 1.

4. 5.

Read the following scenario to the class: Now that the Compute-It teams have found a formula to use in their application, help them test the app by using it to solve quadratic equations, and then check that the new formula matches your earlier work. Explain to students that they will work with their groups to solve quadratic equations using the quadratic formula. They will record all of 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 What is meant by the term inputs inputs? Inputs are the values a, b, and c that I will put into the solver (substitute into the equation) to evaluate the solution, x.

DOK-2 Where do you ﬁnd the inputs? These are the coeﬃcients of the variable terms. The constant multiplying the x2 term is a. The constant multiplying the x term is b. The constant by itself (multiplying the x0 term) is c.

DOK-2 How many outputs can you have? I can have a maximum of two solutions because of the ± in front of the radical.

DOK-2 How can you verify your answer? I can substitute the x value(s) back into the equation or graph the quadratic and identify the zeros.

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.

FACILITATION TIP Before reading the scenario, ask the class 1) Once an app is created, what process do you think it goes through before it is released to the public?; 2) What types of things does an app have or do that makes it successful? FACILITATION TIP Project these guiding questions and review them before, during, and after collaboration. Check for students’ understanding, especially for inputs/outputs. 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.

Math Chat DOK-2 In what form must you ﬁrst put the quadratic equation in order to use the quadratic formula? I need to write it in standard form ﬁrst. • DOK-2 The equation the app is using is known as the quadratic formula. When would you use this formula instead of factoring or completing the square? I would use this formula when the equation I am using is not easily solved by taking the square root and is not factorable. It would be especially useful when a ≠ 1. •

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Solve Quadratics Explore 4 – Using the Quadratic Formula DOK-3 Have we reached our goal, and can we use the quadratic formula to solve any quadratic equation? Yes, it is completely general, and I can use it to solve any quadratic equation. Once we have the equation in standard form, we just take the coeﬃcients a, b, and c. • DOK-3 Could you use the quadratic formula to solve ((xx – 3)2 = 9, and would you? Yes, I could use it to solve this equation if I rewrote it in standard form, but it would be quicker and easier to simply take the square root of both sides and solve. • DOK-3 Would this quadratic formula give solutions to every quadratic equation? The formula will always give solutions, but it would not give values on the real number line if the result inside the square root is negative. These types of solutions are known as complex numbers. •

FACILITATION TIP Check students’ understanding of the commutative property of multiplication. After they answer the question, ask the class if it makes a difference whether the factor “2” is before or after the factor “x – 3”. Why or why not? FACILITATION TIP Some students__may try to simplify their answer to 2 ± √ 5 . Remind them that the radical sign acts as parentheses and that order of operations still applies here.

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 Quadratics Explore 5 – Choosing the Best Method ACTIVITY PREPARATION Students will analyze quadratic equations and determine which method would be the most eﬃcient for solving.

Standards for Mathematical Practice • • •

MP.3 Construct viable arguments and critique the reasoning of others. 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)

•

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 has one. For students who need additional organizational support, please see our Algebra Tiles Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever had a paid job?; 2) If so, what were your responsibilities? FACILITATION TIP Before projecting and reading this scenario as a class, preview speciﬁc vocabulary (market research, price setting, supply chain, analyst) as needed.

2. 3.

FACILITATION TIP

Project and preview guiding questions 4a–4e before students begin to collaborate. Record appropriate student responses and encourage students to take notes as needed. Check for understanding by asking for volunteers/carefully selecting students to respond in their own words.

Read the following scenario to the class: You are starting a new job as a business analyst. The company that hired you has many projects, such as market research, price setting, supply chain review, etc. These processes are often modeled by quadratic equations. You will be solving these equations frequently, so you want to do so as eﬃciently and accurately as possible to impress your new boss. You decide to investigate the best approach to use. Give a Student Journal to each student. Explain to students that they will work with their groups to compare quadratic equations in order to determine what type of quadratic equation is best solved by each of the solution methods they have learned and that 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 What do you look for in a quadratic equation if you are considering solving it by taking the square root? The term involving the unknown is a perfect square, and the other side of the equation is a constant. b.

DOK-2 Can you always use this technique on any quadratic equation? No, I cannot always use it. I might have to complete the square ﬁrst if I want to take a square root.

c. DOK-2 What do you look for in a quadratic equation that would make you consider completing the square? Completing the square is a technique I might use if the leading coeﬃcient is a = 1 and the linear term coeﬃcient b is even because then you are not dealing with fractions. 266

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Acceleration

DOK-2 What do you look for in a quadratic equation that would make you consider trying to factor it? I look for integer coeﬃcients that are not particularly large. Then, I try to see if they factor in such a way that the product gives ac and the sum gives b, where a, b, and c are the coeﬃcients from the quadratic equation in standard form.

e. DOK-2 What methods can you fall back to if you can’t easily factor, take a square root, or complete the square? I can use the quadratic formula or graph. f.

DOK-2 Can you use the quadratic formula or complete the square on any quadratic equation? Yes, those two methods always work.

DOK-2 When you have large or decimal coeﬃcients, what method might you decide to use for convenience? Graphing could be used for an approximate answer. The quadratic formula could be used for an exact answer.

h. DOK-2 For a given quadratic equation, the quadratic formula gives the result 1 ± √2. Graphing gives the solution ±2.41414. Which is more accurate? The solution given by the quadratic equation is exact and not limited by the number of signiﬁcant digits read from the plot. i. DOK-2 If the quadratic equation has complex solutions, will graphing provide those like the quadratic formula will? No, it will not. 5.

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FACILITATION TIP After students answer question 4h, ask them if it would ever be possible to use an exact irrational value in an equation. Students should know that it is impossible because an irrational number has an inﬁnite number of digits after the decimal point.

Allow students enough time to complete Part I and answer the questions FACILITATION TIP that follow. 6. After Part I, invite the class to a Math Chat to share their observations and learning. Check students’ understanding of graphs and equations further. Ask them to explain, Math Chat without looking at a graph, why they can • DOK-3 Are some problems more easily solved by one method than another? Yes. have a real graph without real solutions. For example, if an equation factors easily, I don’t need to plug it into the quadratic They should understand that the term “solution” in this instance depends on a formula. specific condition–namely, a quadratic • DOK-2 Do you expect different methods to give different results? No, I expect crossing the x-axis. all of the methods to give me the same result. There may be a difference in FACILITATION TIP accuracy, such as in graphing, where I may only get an answer to a certain number of significant digits. After students answer the question about • DOK-2 Which methods seem as though they would be easier if you could use expectations, ask them why different them? It would be easier if I could take the square root, find factors, or graph methods don’t render different results. using technology. They should know that only one result or set of results can be plugged back into the • DOK-2 Which methods will always work for any quadratic equation but may not always be the most efficient? I can always use the quadratic formula or complete original equation to make it true. the square. FACILITATION TIP • DOK-2 How can you check your solutions? I can substitute them back into the Discuss the quadratic formula and original equation to make sure the equality holds. completing the square further. After students answer the question, ask them Part II which method they think is more efficient 1. Read the following scenario to the class: Another new business analyst is hired, and which one they prefer to use. and you are tasked with helping him get up to speed. You will review with him the FACILITATION TIP best way to decide what method to use in solving the quadratic equations that frequently arise in your projects. Before reading the scenario, ask the class 2. Students should still have their Student Journals. 1) Has anyone ever been responsible for training someone to do a job?; 2) If so, 3. Explain to students that they will work with their groups to identify the most what job was it?; 3) How did you train the efficient method for solving a given quadratic equation. person? 4. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 What methods would you consider ﬁrst because they would be most eﬃcient if you could use them? I would try to see if I could factor or take the square root.

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Solve Quadratics Explore 5 – Choosing the Best Method

FACILITATION TIP In the Reﬂect section, some students may answer Question 2 differently than expected. Give them space to stick with their answer as long as they provide reasoning.

5. 6.

DOK-2 If you see non-integer coeﬃcients, what method would you consider? I would probably try to graph for an approximate answer or use the quadratic formula for an exact answer.

DOK-2 If the leading coefficient is 1, the coefficient of the linear term is even, and all three coefficients are not too large, what technique would be fairly straightforward assuming the equation does not easily factor? In that case, I could complete the square.

DOK-2 When other techniques are not easily applied, what is an allpurpose method you could use that gives an exact answer? I could use the quadratic formula.

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 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.

• •

DOK-1 What are the different methods you can use to solve a quadratic equation? • Graphing • Factoring • Taking the square root • Completing the square • Quadratic formula DOK-3 Are certain methods more appropriate for solving a particular quadratic equation? Yes, sometimes a particular method may be quicker and easier to use. DOK-2 What are some things to look for when solving a quadratic equation to help you decide which method to try? What method does each “look for” suggest? I would look for the following things: • The term involving the unknown is a perfect square (taking the square). • There are factors that give a product of ac and a sum of b (factoring). • There are non-integer coeﬃcients (graphing or quadratic formula). • The leading coeﬃcient is a = 1, and b is even (completing the square). • I only need an approximate answer (graphing).

FACILITATION TIP

Post-Explore

After students complete the Exit Ticket, ask the class by a show of hands who likes graphing the best to solve a quadratic equation. Then, ask those who raised a hand if they typically enjoy learning better with pictures or with words and symbols. Repeat this for each method to solve a quadratic equation.

1. 2. 3. 4.

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Solve Quadratics 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

Solve Quadratics by Taking Square Roots Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

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

Solve Quadratics by Completing the Square 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 Quadratics by Factoring

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

Using the Quadratic Formula

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

Show What You Know, Part 5 Choosing the Best Method 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

Find Quadratic Intercepts

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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 Quadratics 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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Acceleration Acceleration

What prompts will be used?

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What does mastery look like?

I can solve quadratic equations using different methods, both with and without technology.

I can rewrite a quadratic function in a form featuring a new term and different operations.

I can determine what the zeros of a quadratic function describe in a given context.

I can understand the beneﬁt of rewriting a quadratic expression in a different form in order to solve an equation.

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Transform Quadratic Functions Scope Introduction SCOPE SUMMARY In this scope, students should be able to graph and identify key features by hand given a simple quadratic function and use parameter changes from the parent functions to graph a transformed quadratic function. Students should be able to identify the effect on the graph when applying transformations to any quadratic function (including the parent function). They should be able to use multiple representations of quadratic functions and make connections between the representations. Student Expectations

A.FGR.7.1 Use function notation to build and evaluate quadratic functions for inputs in their domains and interpret statements that use function notation in terms of a given framework. A.FGR.7.2 Identify the effect on the graph generated by a quadratic function when replacing f(x) with f(x) + k, kf(x), f(kx), and f(x + k) for speciﬁc values of k (both positive and negative); ﬁnd the value of k given the graphs.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

Students worked with linear functions in previous grades, so they should have an understanding of some of the key features of function graphs. They have prior experience using tables, graphs, and equations when working with linear and quadratic relationships. Students have compared properties of two linear functions that are represented in different ways. They have graphed quadratic equations in vertex form and will continue to utilize this form to best understand transformations.

Students will compare and transform exponential functions in later scopes of Algebra I. In future courses, students will use transformations to analyze many function families as well as polygons and other shapes. When working with trigonometric functions, transformations are particularly important to analyze amplitude and period.

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

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES

verify experimentally the properties of rotations, reﬂections, and translations.

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: •

relate transforming quadratic functions to a real-world situation.

•

create parabolas and transform quadratic 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.

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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Translations of Quadratics In this exploration, groups of students will help solve a real-world scenario about a video game where ground squirrels burrow to get acorns in a quadratic pathway. Students will: •

analyze quadratic functions given as equations, graphs, or tables.

•

identify the transformations of quadratic functions.

Explore 2

Explore 1

EXPLORE ACTIVITIES

identify the combinations of dilations and/or reﬂections •

analyze quadratic functions given as equations, graphs, or tables.

•

identify the transformations of quadratic functions.

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

Dilations of Quadratics

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Transformations of Quadratics analyze quadratic functions given as equations, graphs, or tables •

analyze quadratic functions given as equations, graphs, or tables.

•

identify the transformations of quadratic functions.

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

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Transform Quadratic Functions Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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ACCESSING PRIOR KNOWLEDGE Students will read different student responses to a posed question on the prior standard, decide whether they agree or disagree with the student, and explain their reasoning. 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.

Materials

Preparation

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Print one Agree or Disagree for each student.

1 Agree or Disagree (per student)

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

Disagree with Trevor

Agree with Lana

Agree with Howard

Disagree with Alex

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

Identifying Misconceptions • •

FACILITATION TIP Depending on your students, consider distributing the graphs one at a time. After students complete each scenario, conduct a stand up/hand up collaboration. FACILITATION TIP Before students stand up, check their answers and reasonings. Don’t give away the answers, but make sure the reasonings are clear so they can discuss them effectively. FACILITATION TIP After they explain their reasoning for Question 1 ask the class if Trevor can reﬂect the image about the line y = x to get it in Quadrant IV. Have them explain their reasoning, and clear up any misconceptions. FACILITATION TIP This Foundation Builder includes ﬁve clear graphs to project. Consider using them to facilitate an effective review.

Students may want tracing paper to help them visualize shifting the lines on the graphs up or down. Students may not remember the word translation and can be reminded of its meaning. Notes

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Transform Quadratic Functions Hook – Transform Quadratic Functions ACTIVITY PREPARATION Students will relate transforming quadratic functions to a real-world situation.

Materials

Preparation

Printed •

1 Transform Quadratic Functions (per class)

Reusable • •

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1 Phenomena Video (per class) 1 Projector (per class)

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

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

Before showing the video and reading the scenario, ask the class 1) Have you ever snowboarded?; 2) How did you learn to snowboard?; 3) What skills do you need to have to snowboard?

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: Junior is snowboarding a slalom course and has to make sure he passes through each gate to not receive a score deduction from the judges. 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 Junior just has to touch the gate instead of going through the exact center. I notice that the slalom requires a lot of aim and directional precision. I wonder how much a deduction is. I wonder how snowboarders decide to approach each gate. I wonder how hard it is for snowboarders to control their aim if they are traveling fast. Project Transform Quadratic Functions. Notes

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Explain to students that they will examine how to move quadratic functions around the coordinate plane in this scope. Discuss the following questions: a.

DOK-1 How does the task on the coordinate grid connect to the snowboarder? Allow students to share all ideas. Answers will vary. Both tasks are trying to create a path that will go through multiple gates to succeed.

DOK-1 Can you picture two different parabolas that could both pass through the gates? Allow students to share all ideas. Answers will vary. A parabola that opens up with its vertex in quadrant four could pass through both as well as a parabola that opens down with its vertex in the ﬁrst quadrant.

Intervention

Acceleration

FACILITATION TIP As a ﬁnal discussion for the Pre-Explore, ask if anyone has watched a professional snowboarder. What event(s) did they see them perform in? What tricks would they like to learn that they saw them perform?

Complete the Explore activities.

Part II: Post-Explore 1. 2.

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

DOK-1 What quadratic form would be easiest to complete this task? Vertex form of a quadratic equation

DOK-2 What strategies would you use to create your parabolas? I would sketch a parabola that passes through both gates and then use the vertex to start creating my equation in the form y = a(x – h)2 + k. I would use another point on my parabola to ﬁnd a and make sure that the parabola opens in the correct direction.

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

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FACILITATION TIP As a ﬁnal discussion for the Post-Explore, ask students if there are more than two quadratic functions that would pass through both gates in the graph of the given problem. Be sure they realize that there are more than two and that some functions would yield more of their graphs within the given coordinate plane than others.

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Transform Quadratic Functions Explore 1 – Translations of Quadratics ACTIVITY PREPARATION Students will analyze quadratic functions given as equations, graphs, or tables and identify the transformations.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • •

• • •

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

Reusable •

•

1 Graphing calculator (per student)

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) What video game is your favorite to play?; 2) What characters are in the video game?; 3) What is the goal of the game?

Part I: Ground Squirrels 1.

FACILITATION TIP Project the scenario in print so students can read it along with you and note important phrases and values.

2. 3.

FACILITATION TIP

The acorn graphics are bigger than the plotted points. Explain to the class that they should look for a coordinate pair of integers toward the middle of an acorn to establish its location on the graph. FACILITATION TIP Consider saying, “All of the function selections in Question 1 feature x2 minus a constant.” Use this sentence structure for parts 4c–4e.

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Read the following scenario to the class: You are playing a game with squirrels, and they are hungry! The squirrels for levels 1 and 2 are ground squirrels. You can aim them at the ground, and they burrow in a quadratic path. Your goal is to help write functions for the squirrels to reach the acorns underground before they emerge back above the ground. The red dot indicates the speciﬁc part of the path where the squirrel needs to be to get the acorn. Give a Student Journal to each student. Explain to students that they will work with their groups to graph the equations shown and determine which equation will transform the function so the red dot touches the acorn. 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 Is there a quick way to use the calculator to see if the equation contains the desired point? Once you enter the equation into Y = on the calculator, you could look at the table (second graph) to see if the equation contains the point desired.

DOK-2 All of the functions in question 1 have minus some numbers. What do the vertices of each of those functions have in common? The vertices of all of these functions are below the x-axis.

DOK-2 All of the functions in question 3 have plus some numbers. What do the vertices of each of those functions have in common? The vertices of all of these functions are above the x-axis.

DOK-2 All of the functions in question 5 have minus some numbers. What do the vertices of each of those functions have in common? The vertices of all of these functions are to the right of the y-axis. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

e. DOK-2 All of the functions in question 6 have plus some numbers. What do the vertices of each of those functions have in common? The vertices of all of these functions are to the left of the y-axis. 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.

Math Chat •

•

DOK-1 Describe how k changes the function when f( f(x (x) = x2 is transformed to 2 g(x ( ) = x + k.. Include an explanation for positive k and negative k values. If k is (x positive, it will shift the graph up k units. If k is negative, it will shift the graph down k units. DOK-1 Describe how h changes the function when f( f(x (x) = x2 is transformed to g(x ( ) = ((xx – h)2. Include an explanation for positive h and negative h values. If h is (x positive, it will shift the graph right h units. If h is negative, it will shift the graph left h units.

Part II: Level Up 1.

2. 3.

5. 6.

Read the following scenario to the class: For level two, you are asked to adjust the path to gather the acorn, but the squirrel has to get the acorn at the minimum point in his dive. If you do not, you will have to try again before you can move on! Students should still have their Student Journals. Explain to students that they will work with their groups to identify two shifts needed to move the vertex (red dot) to the acorn. The shift must move the vertex to the acorn. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 When performing translations, how do you get from the original to the transformed graph? The easiest way is to use one speciﬁc point (vertex to vertex, red point to acorn, etc.) and ﬁgure out how to get from the original point to the new point.

DOK-2 How could we tell if our new, transformed equation is correct? Plug in the point the acorn lies on, and see if it makes your new equation true.

DOK-2 When performing a horizontal and vertical translation, how does the equation that represents the function change? The value of h and k will change.

DOK-3 When translating a function horizontally, does x plus a positive number move the function left or right? Why does this happen? It will move the function some units to the left. This occurs because typically functions are written to express y in terms of x, so when we are considering the translation of x in terms of y, it is the opposite of what we would expect.

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.

Intervention

Acceleration

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FACILITATION TIP Before reading the scenario, ask the class 1) What is the highest level you ever got to when playing a video game?; 2) Each time you level up, does the game get more diﬃcult, easier, or stay the same? FACILITATION TIP Project these guiding questions 4a–4d. Have students complete some think, pair, share time and take notes. Record appropriate student answers. Check for understanding before students complete this Exit Ticket.

FACILITATION TIP After students answer Question 10. ask them how they know that the vertex for squirrel A’s path is (5, –4) without knowing the zeros of the function. They should know that, for a parabola, the vertex is equidistant from any two points with the same y-coordinate.

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Transform Quadratic Functions Explore 1 – Translations of Quadratics Math Chat DOK-1 When performing horizontal and vertical shifts, does the order matter? No, as long as you perform both shifts, the order doesn’t matter. However, typically we perform the operations inside parentheses ﬁrst. • DOK-1 When writing a function g(x ( ) as a function of f( (x f(x (x), what steps do you recommend completing? 1. Identify the shifts visually from the graph that would change from f(x) to g(x). 2. Determine if that means the h and k values will increase or decrease from f(x) to perform the shifts and create the g(x) function. •

FACILITATION TIP If students struggle with the Exit Ticket, ask them to describe the transformation from f(x) to f(x + 4). Then, ask them if the x values, y values, or both will change for coordinate pairs, and by how much. Next, ask them how to change the coordinate pair of the y-intercept of f(x) to ﬁnd a point on the new graph.

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. vNotes

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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TRANSFORM QUADRATIC FUNCTIONS

Transform Quadratic Functions Explore 2 – Dilations of Quadratics ACTIVITY PREPARATION Students will analyze quadratic functions given as equations, graphs, or tables and identify the transformations.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation • • •

Printed • •

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

•

Reusable •

1 Graphing calculator (per student)

PROCEDURE AND FACILITATION Part I: Ground Squirrels FACILITATION TIP

Before reading the scenario, ask the class 1) When playing your favorite video game, are new characters introduced at each level?; 2) Who are they?; 3) What is their role in the game?

2. 3.

FACILITATION TIP

Students may ﬁnd the workings of horizontal stretches and compressions to be counterintuitive compared to their vertical counterparts. For Question 4 on the Student Journal, remind them they can plug in the x value for the acorn into each function selection and see which yields the y value of the acorn.

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Read the following scenario to the class: Today, we level up with our squirrel game. In these next levels, you have an option for a new squirrel—a ﬂying squirrel! If the acorn is underground, you will still use the ground squirrel. However, if the acorn is on a tree branch, you will need to use the ﬂying squirrel. Give a Student Journal to each student. Explain to students that they will work with their groups to graph the equations shown and determine which equation will transform the function so the red dot touches the acorn. 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-2 All of the functions in question 1 have a coeﬃcient greater than 1. What does the width of each of those functions have in common? The width of each of these functions is smaller or more narrow than the initial function.

DOK-2 In question 1, what is the y value of the ordered pair where the red dot is located? What is the y value of the ordered pair where the acorn is located? What would you have to multiply the y value of the red dot by to get to the y value of the acorn? The y value for the red dot is 1, and for the acorn, it is 5. I would need to multiply 1 times 5 to get 5.

DOK-2 All of the functions in question 2 have a coeﬃcient smaller than 1. What does the width of each of those functions have in common? The width of each of these functions is greater or wider than the initial function. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

e. DOK-2 In question 2, what is the y value of the ordered pair where the red dot is located? What is the y value of the ordered pair where the acorn is located? What would you have to multiply the y value of the red dot by to get to the y value of the acorn? The y value for the red dot is 9, 1 and for the acorn, it is 3. I would need to multiply 9 times __3 to get 3. f.

DOK-2 What input in the parent function would you have to plug in to get an output of 9? What would you do to the acorn’s input of 1 to get the same output? The input of x = 3 gives the parent function an output of 9. To get the same output for the acorn’s x value of 1, you would need to multiply the input by 3.

DOK-2 All of the functions in question 4 have a coeﬃcient inside the parentheses. What does the width of each of those functions have in common? The width of each of these functions is wider or more narrow than the initial function.

h. DOK-2 The function in question 6 has a negative coefficient. How does the direction of the opening change when the coefficient changes signs? The direction of the opening changes from opening up to opening down when the sign of the coefficient changes. i. DOK-2 In the graph for question 6, are the orange dot and the acorn on the same side of the x-axis? -axis? What type of transformation would change the ordered pair from one side of the axis to the other? No, they are on different sides. A reflection would change the ordered pair from one side to the other. 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.

Math Chat •

DOK-1 How does the value of the coeﬃcient a in front of the x2 affect the graph? If a is greater than 1, it dilates the graph by stretching it vertically. If a is less than 1 but greater than 0, it dilates the graph by compressing it vertically. If a is negative, it reﬂects the whole graph over the x-axis.

Part II: Level Up 1.

2. 3.

Read the following scenario to the class: In level four, you are asked to adjust the path to gather the acorn, but the squirrel’s path may involve more than one transformation. Students should still have their Student Journals. Explain to students that they will work with their groups to identify the combinations of dilations and/or reﬂections that would allow the squirrel to get to the acorn. The shift must move the orange dot to the acorn. Explain to students that they need to make the decision to use ground squirrels or ﬂying squirrels. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How do you decide whether to use a ﬂying squirrel or a ground squirrel? If the path goes down and then back up, I would need a ground squirrel. If the path goes up and then down, I would need a ﬂying squirrel.

DOK-2 When performing horizontal and vertical dilations or reﬂections, how will the equation that represents the function change? The value of a and b will change.

FACILITATION TIP Check students’ understanding of horizontal stretches and compressions further. After students answer the question, ask them which functions are wider and which functions are more narrow than the initial function.

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FACILITATION TIP Check students’ understanding of dilations further. After students answer Question 8 ask what additional dilation is needed given squirrel B’s current path for it to reach the acorn. FACILITATION TIP Project this Math Chat question. Encourage students to take notes, respond in their own words, and quiz a partner.

FACILITATION TIP Before reading the scenario, ask the class 1) Do you have to change your strategy at different levels when playing video games?; 2) If so, how does your strategy change?; 3) What things in the game are the same and different in each level?

FACILITATION TIP Check students’ understanding of translations. After they answer Question 1., ask the class what action they would take if they could only use translations. Then, ask them what actions they would take if they had to use a reﬂection and one translation.

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Transform Quadratic Functions Explore 2 – Dilations of Quadratics c.

DOK-2 In question 1, what is the y value of the ordered pair where the orange dot is located after reﬂecting it over the x-axis? What is the y value of the ordered pair where the acorn is located? What would you have to multiply the y value of the orange dot by to get to the y value of the acorn? The y value after the reﬂection is −1, and the y value of the acorn is −4. I would need to multiply −1 times 4 to get −4.

DOK-2 In question 4, what is the y value of the ordered pair where the orange dot is located after reﬂecting it over the x-axis? -axis? What is the y value of the ordered pair where the acorn is located? What would you have to multiply the y value of the orange dot by to get to the y value of the acorn? The y value after the reﬂection is −8, and the y value of the acorn is −4. I would need to multiply −8 times 1/2 to get −4.

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.

e. DOK-2 Since the basic path was modeled by the function f( f(x (x) = 2x2, how 1 __ does compression by a factor of 2 change the function? The coeﬃcient 1

of the new function won’t be __2 because we started with a coefficient other than 1. Since we started with a coefficient of 2, it would make 1 sense to multiply 2 times __2 to get the new coefficient.

FACILITATION TIP To clarify, consider asking, “In Question 7 what is the y value of the new ordered pair when the red dot is reﬂected over the x axis?” 5. 6.

DOK-2 In question 7, what is the y value of the ordered pair where the orange dot is located after reﬂecting it over the x-axis? What is the y value of the ordered pair where the acorn is located? What would you have to multiply the y value of the orange dot by to get to the y value of the acorn? The y value after the reﬂection is 5, and the y value of the 4 acorn is 4. I would need to multiply 5 times __5 to get 4.

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

Math Chat FACILITATION TIP

•

Project this Math Chat question. Encourage students to take notes, respond in their own words, and quiz a partner. FACILITATION TIP If students struggle with the Exit Ticket, ask them how reﬂecting a graph about the x-axis affects the value of k. Then, ask them to compare the y values of the two points that are separated vertically.

DOK-1 Describe what the coeﬃcients a and b do for the function f( f(x (x) = a(bx)2. Explain positives and negatives as well as numbers larger and smaller than 1. If a is negative, it reﬂects the function over the x-axis. If it has a magnitude larger than 1, it stretches the function vertically. If it has a magnitude between 0 and 1, it compresses the function vertically.

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. vNotes

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Transform Quadratic Functions Explore 3 – Transformations of Quadratics ACTIVITY PREPARATION Students will analyze quadratic functions given as equations, graphs, or tables and identify the transformations.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation • • •

Printed • •

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

•

Reusable •

Separate the class into groups of 3 or 4 students. Print the 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. Gather a graphing calculator for each student.

1 Graphing calculator (per student)

PROCEDURE AND FACILITATION Part I: Choose a Squirrel FACILITATION TIP Before reading the scenario, ask the class 1) When playing video games, how do you choose which character to be?; 2) What power or special ability does the character you choose have?; 3) Do you always choose the same character each time you play the game? Why or why not? FACILITATION TIP For Question 1 students may try to use translations that will result in the graph for h(x)=(x – 2)2 + 1. While this path is correct, the option is not available for Question 1. Encourage students to follow the order of operations when performing transformations.

2. 3.

Read the following scenario to the class: For level ﬁve, there are multiple transformations you must perform to get to the acorns. The orange dot must pass through one acorn, but you must gather both acorns with your path. You can use a ground squirrel or a ﬂying squirrel. Give a Student Journal to each student. Explain to students that they will work with their groups to graph the equations shown and determine which equation will transform the function so the orange dot touches the acorn. 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 When identifying transformations, does it matter in which order you perform the transformations? Follow the order of operations and perform transformation inside the parentheses ﬁrst, multiplication and division next, and ﬁnally addition and subtraction.

DOK-2 All of the functions in question 1 have a constant added at the end. What does the vertex of each of those functions have in common? The vertex of all of these functions is above the x-axis.

DOK-2 This is a flying squirrel. Does the parabola for the path of a flying squirrel open up or down? What transformation from the basic path will produce that type of parabola? The parabola for the path of a flying squirrel will open down. The transformation to produce that would be a reflection over the x-axis, which would mean we need a negative coefficient in our equation.

FACILITATION TIP After they answer the question 4b, have the class graph y = x2 + 3. Then, instruct them to reflect it about the x-axis. Watch out for students reflecting the graph about the vertex rather than the x-axis. Be sure they know that, given a reflection about the x-axis, the x values stay the same and the y values are opposite of the original graph.

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Explain

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Evaluate

Intervention

Acceleration

e. DOK-2 In question 1, what is the y value of the ordered pair after the function has been reﬂected where the orange dot is located? What is the y value of the ordered pair where the highest acorn is located? What would you have to add to the y value of the orange dot to get to the y value of the acorn? The y value for the orange dot is 0, and for the acorn, it is 5. I would need to add 5 to 0 to get 5.

5. 6.

DOK-2 In question 4, what is the y value of the ordered pair after the function has been reﬂected where the orange dot is located? What is the y value of the ordered pair where the highest acorn is located? What would you have to multiply the y value of the orange dot by to get to the y value of the acorn? The y value for the orange dot is 0, and for the acorn, it is 4. I would need to add 4 to 0 to get 4.

DOK-2 Can the squirrel get to both acorns using only a reﬂection and a shift, or does it need a third transformation? After performing a reﬂection and a shift up 4 units, the parabola does not go through both acorns, so I need a third transformation.

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-1 When performing transformations, how should you use the vertex to your advantage? Try putting the vertex on one of the acorns, and see if any of the points work on the other point. If not, switch the vertex to the second point, and see if any points work. If not, try performing a vertical stretch or shrink.

Part II: Level Up 1.

2. 3.

Read the following scenario to the class: For level six, you are asked to adjust the path to gather the acorn, but be careful to look at the original path! There are already shifts being performed, so you need to adjust the path to go through both acorns. The acorns should be hit by two points right next to each other, and one of those points should be the orange dot. Students should still have their Student Journals. Explain to students that they will work with their groups to identify the transformations that would allow the squirrel to get to the acorn. The shift must move the orange dot to the acorn. 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 When rewriting g(x ( ) as a function of f( (x f(x (x), what is the best course of action? First, identify the changes from f(x) to g(x). Then, apply those changes to the f(x) function to ﬁnd the new function g(x).

DOK-2 In question 1, what is the y value of the ordered pair where the orange dot is located? What is the y value of the ordered pair where the lowest acorn is located? What would you have to add to the y value of the orange dot to get to the y value of the acorn? The y value for the orange dot is −4, and for the acorn, it is −1. I would need to add 3 to −4 to get −1.

DOK-2 In question 2, what is the y value of the ordered pair where the orange dot is located? What is the y value of the ordered pair where the highest acorn is located? What would you have to add to the y value of the orange dot to get to the y value of the acorn? The y value for the orange dot is 3, and for the acorn, it is 0. I would need to add −3 to 3 to get 0.

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FACILITATION TIP Before the Math Chat, explain to the class that reflecting a parabola about the y-axis leaves its graph the same when the vertex is at the origin. Then, ask them how such a reflection would affect the graph when the vertex is not at the origin. Be sure they know that, given a reflection about the y-axis, the y values stay the same and the x values are opposite of the original graph. FACILITATION TIP Before reading the scenario, ask the class 1) What is the biggest video game challenge you have had to overcome in order to win the game?; 2) Were you successful mastering the challenge?; 3) If so, what strategies did you use to win? FACILITATION TIP Project the text of this scenario while you read it along with students. Guide them to write down important constraints and values. STEMscopes Tip Each Explore activity includes a Student Journal that students complete collaboratively while participating in group work. Students use the journal to develop metacognitive skills by reflecting on how and what they are learning. Communicating mathematical thinking leads to a deeper conceptual understanding of the skills at hand.

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Transform Quadratic Functions Explore 3 – Transformations of Quadratics d.

DOK-2 In question 3, what is the y value of the ordered pair where the orange dot is located? What is the y value of the ordered pair where the lowest acorn is located? What would you have to add to the y value of the orange dot to get to the y value of the acorn? The y value for the orange dot is 0, and for the acorn, it is −1. I would need to add −1 to 0 to get −1.

e. DOK-2 In question 3, what is the x value of the ordered pair where the orange dot is located? What is the x value of the ordered pair where the lowest acorn is located? What would you have to add to the x value of the orange dot to get to the y value of the acorn? The x value for the orange dot is 0, and for the acorn, it is 1. I would need to add 1 to 0 to get 1. f.

FACILITATION TIP To clarify, consider asking, “In question 3, can a new parabola model the squirrel’s path using only translations of the given graph? Why or why not?” After students answer the questions, ask them if it’s possible to form the new model with only dilations and/or reﬂections. Why or why not?

5. 6.

DOK-2 In question 3, can a parabola model the squirrel’s path using only translations? Why or why not? It cannot because the basic path does not show two consecutive points that are only a difference of a half unit above or below each other. The closest consecutive points are the vertex and the point on either side of the vertex. The basic path shows those are 1 unit above/below each other.

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 •

FACILITATION TIP After students come up with a valid answer, ask them if the graph of their function would be the same if reﬂected about the y-axis. How do they know? They should realize the graph would not be the same because their original function shifted from the parent function, meaning the vertex is not at the origin.

DOK-2 What is the difference between writing a new function g(x ( ) and writing g(x (x ( ) (x as a function of f( f(x (x)? When writing the function g(x), you only have to identify the changes for g(x) from the parent function x2. When writing the function g(x) as a function of f(x), you have to identify the changes from f(x) and write the new function g(x) while only representing the changes made. For example, if both f(x) and g(x) have the same vertical stretch, you don’t have to represent that in g(x).

Post-Explore 1. 2. 3. 4.

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Transform Quadratic 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

Translations of Quadratics 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

Dilations of Quadratics 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

Transformations of Quadratics 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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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

Transform Quadratics

Can be done independently

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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Transform Quadratic Functions

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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 connect function notation used for quadratic functions to graphs and scenarios.

I can use graphing technologies to demonstrate translations, dilations, and reﬂections of quadratic functions.

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SCOPE 1

Exponential Functions Scope Introduction SCOPE SUMMARY

Student Expectations

In this grade level, students will model with exponential functions. Students should be able to model a situation using an exponential function and identify individual parts of an expression to interpret them in context. They should be able to use the rules of exponents to rewrite exponential functions that represent a context. Students should be able to identify the exponential function that matches a graph, describe key features of the function, sketch simple graphs by hand, and use technology to graph more complex exponential functions.

A.PAR.8.1 Interpret exponential expressions and parts of an exponential expression that represent a quantity in terms of its framework.

A.FGR.9.1 Use function notation to build and evaluate exponential functions for inputs in their domains and interpret statements that use function notation in terms of a context. A.FGR.9.2 Graph and analyze the key characteristics of simple exponential functions based on mathematically applicable situations.

Future Expectations

In previous grade levels, students looked at exponents most deeply when exploring scientiﬁc notation and developing properties of integer exponents with a numerical base. Students have utilized patterns to develop these properties and performed operations with scientiﬁc notation with and without technology to develop a strong intuition about exponents and orders of magnitude.

Students will transform and model situations with exponential functions later in Algebra I. They will also compare exponential growth to linear and quadratic growth in later scopes. In Algebra II, students will solve exponential equations using the properties of logarithms.

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

Hook

A.PAR.8.4 Represent constraints by exponential equations and interpret data points as possible or not possible in a modeling environment.

Background Knowledge

Accessing Prior Knowledge

A.PAR.8.3 Create exponential equations in two variables to represent relationships between quantities, including mathematically applicable situations; graph equations on coordinate axes with labels and scales.

VERTICAL ALIGNMENT

use proportional relationships to solve multistep ratio and percent problems with examples.

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: •

relate exponential functions to problems.

•

write exponential 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.

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Attributes of Exponential Functions In this exploration, groups of students will analyze the behavior of different types of microbe colonies inside Petri dishes. Students will:

Explore 2

Explore 1

EXPLORE ACTIVITIES

In this exploration, students will analyze samples of bacteria from various places around the school to determine a reasonable domain and range for each. Students will:

•

examine graphs of exponential functions.

•

identify key features.

•

apply what is applicable to graphs of exponential functions.

identify the domain and range of exponential functions.

•

discuss asymptotes for the ﬁrst time.

•

use knowledge of domain and range with linear functions and apply those skills to exponentials.

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

Initial Value and Constant Multiplier In this exploration, students will assist park rangers to create a report for the rangers regarding the impact the wolves have had on other mammal populations. Students will also recommend which bird species may need urgent protection in the park. The students are tasked with interpreting models and recommend which bird species may need urgent protection in the park. 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

Domain and Range

look for patterns in exponential functions and data in tables.

•

interpret models.

•

notice constant multipliers that correspond to values in the function.

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Write Exponential Functions In this exploration, groups of students will identify whether other mammal populations have seen a rise or decline in numbers since the reintroduction of the wolves in the Yellowstone National Park. Students will also identify initial values and percent increase/decrease (factor of change) within populations. Students will: •

analyze data tables.

•

calculate the initial values and constant multipliers.

•

write exponential functions.

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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Exponential 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

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

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.

4. 5.

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FACILITATION TIP

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.

Set 1: B does not belong. A, C, and D equal 400. B does not.

Set 2: D does not belong. A, B, and C are equivalent. D is not.

Set 3: B does not belong. A, C, and D have an answer that equals 36%. B does not.

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

Identifying Misconceptions

Before distributing Does Not Belong, ask students what they know or think they know about constant rate of change. Then, ask them what is another way a function could increase or decrease in a uniform manner. Without answering, tell the class they will ﬁnd out in this scope. FACILITATION TIP Consider creating a Know, Want to Know, Learned chart for ratios, rate of change, and percent. FACILITATION TIP Depending on your students, consider projecting or distributing each table one at a time to facilitate cohesive discussions. FACILITATION TIP

•

Students may struggle with calculating percent increase and decrease. Encourage them to start by calculating with multiple steps of using percentages they can calculate mentally.

•

On the second page, a misconception may be that 64% is 6.4 as a decimal. Encourage students to use “64 percent” and “64 out of 100.” Students can then 64 read ____ as “64 hundredths,” which is 0.64. 100

Students may look to match keywords rather than ﬁnd equivalent values. Encourage students to look at and compare entire scenarios on a page rather than rely solely on keywords. FACILITATION TIP This Foundation Builder includes six practice problems with matching answers. Consider conducting a good whole class or partner review about percents and decimals (discounts, taxes, tips).

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Exponential Functions Hook – Algae Growth ACTIVITY PREPARATION Students will relate exponential functions to a real-world situation.

Materials

Preparation

Printed •

• • •

1 Algae Growth (per class)

Reusable • •

1 Phenomena Video (per class) 1 Projector (per class)

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

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

Before showing the video and reading the scenario, ask the class 1) What do you know about marine biology?; 2) Has anyone seen scuba diving or snorkeling on TV or movies before?; 3) Has anyone experienced underwater diving in real life?; 4) What did you notice about the plants and animals?

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: Marine biologists study the growth of algae in a local pond. They notice that the algae cells grow much more rapidly in warmer temperatures. They want to model and predict the algae growth during days when there are temperatures above 95°F. 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 cells seem to be multiplying. I notice that the growth is rapid. I wonder how many cells there are. I wonder how quickly the growth is occurring. I wonder what size the cells are. I wonder if we will be able to create an equation to match the algae cell growth. Project Algae Growth. Notes

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Engage

Explore

Explain

Elaborate

Evaluate

Explain to students that the goal of the scope will be to examine how to model rapid multiplicative growth. Discuss the following questions: a.

DOK-1 How will the values in this table differ from the values in a table that displays linear growth? In a linear table, the values would increase by a constant amount, but that will not happen in this table because the number of cells multiplies itself by 5 every hour instead of growing by the same value each hour.

DOK-1 How do you predict the model equation will be different from a linear equation? Allow students to share all ideas. Answers will vary. I think the y-intercept will still be important. I don’t think there will be an addition or subtraction sign. I don’t think the model will contain a slope value.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

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

DOK-1 Does the idea of exponential growth make more sense after the Explore activities? Yes, when tables grow by a common factor, we have exponential growth.

DOK-1 What strategies would you use to create a model for the algae growth? I would ﬁnd the initial value and then use the equation y = a(bx).

DOK-1 What equation would you use to model this growth? 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 exponential functions? Answers will vary based on students’ success during the activity and their conﬁdence level.

Intervention

Acceleration

FACILITATION TIP After the explanation, ask students if they can think of another real-world scenario that could involve rapid multiplicative growth. If they are stumped, present examples such as the number of students enrolled in online schools, the number of laptops used in the workforce, or followers on social media sites.

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FACILITATION TIP Students may ﬁnd it challenging to take the information they’ve been given so far to predict an equation’s structure. Encourage them to trust their intuition, and have them look back at their prediction after the Explore activities to compare what they perceived with what they learned.

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Exponential Functions Explore 1 – Attributes of Exponential Functions ACTIVITY PREPARATION Students will examine graphs of exponential functions and identify key features such as intercepts, increasing and decreasing intervals, and end behavior. Students will draw on knowledge from linear graphs and apply what is applicable to graphs of exponential functions.

Standards for Mathematical Practice • • •

MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

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

• • • •

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever used a Petri dish before?; 2) Did you use it in a class?; 3) If so, which class, and what did you use it for? Take time to locate a Petri dish or provide some visual images or video. FACILITATION TIP Consider distributing the Student Journal in two parts as needed (Part I page 1, Part II pages 2–4). FACILITATION TIP Depending on your class, consider providing every student the page of Microbe Game Cards so they can make observations up close and perhaps make notes/marks on the graphs or tables.

2. 3. 4.

Read the following scenario to the class: Let the games begin! You are conducting an experiment in your science class where you are analyzing the behavior of different types of microbe colonies inside Petri dishes. The experiment has been set up in competition fashion, where the fastest-growing microbe will be crowned the winner of the Microbe Games! The microbes have had 3 days in class to compete for the title. To start, you’ll have to compare the growth of your ﬁrst two contestants, microbe A and microbe B. Give a Student Journal to each student. Give a set of Microbe Games Cards to each group. Explain to students that they will work with their groups to analyze the information for microbe A and microbe B 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 Does each graph have an x-intercept? No, only microbe A shows an x-intercept on the graph.

DOK-2 What do you notice about the shape of each graph? Microbe A is graphed as a straight line, and microbe B is graphed as a curved line.

DOK-3 What is the pattern seen in the y values for microbe A? The y values are increasing by an addition of 3 for each increase of 1 in the x values.

DOK-3 What is the pattern seen in the y values for microbe B? The y values are increasing by a multiple of 3 for each increase of 1 in the x values.

FACILITATION TIP Have students analyze the Microbe Games Cards further. As they work, ask them between which two days Microbe A and Microbe B have the same population. FACILITATION TIP Project these guiding questions for students to view either before, during, or after they collaborate. 302

Engage

Explore

Explain

Elaborate

Evaluate

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

Math Chat DOK-2 What do you notice about the variable in each of the two equations? The x is used as a multiplier in the equation for microbe A. In the equation for microbe B, the x is used as an exponent. • DOK-3 Compare the growth of the linear function to the function with a variable exponent. Which one increases more quickly? Explain why this occurs. The function with the variable exponent increases more quickly. Raising a number to a power greater than 1 will always cause a number to increase (or decrease) more than just multiplying or adding would. •

Intervention

Acceleration

FACILITATION TIP After students complete Part I, discuss Question 4 further. Ask them if they can explain mathematically why the exponential function will never cross 0.

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Explain the following to the class: Functions that change based on a variable exponent are called exponential functions. Part II 1.

2. 3.

4. 5.

6. 7.

Read the following scenario to the class: For round 2, you will be comparing the results of your third and fourth contestants, microbe C and microbe D, to your first two contestants. However, this time, you have only been given the equation for each microbe’s growth over the last 3 days. Use the equations to fill in the tables and graph and figure out who the overall winner will be! Students should still have their Microbe Games Cards from Part I. Explain to students that they will work with their groups to fill in the information on the tables and the graph using only the equations provided. They will compare the information from microbe C and microbe D to the information from microbe A and microbe B in Part I. They will then record all work on their Student Journals Point out to the class that they will need to review the tables and graphs for each microbe to come up with the overall winner. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 The graph of microbe C looks most similar to the graph of which microbe from Part I? Microbe B

DOK-2 How are the graphs of microbe C and microbe D similar? How are they different? They are similar in the sense that to the right of the y-axis, they grow rapidly toward inﬁnity. They are different because microbe D also has a piece that goes to the left of the y-axis toward negative inﬁnity. Microbe D is linear, while microbe C is curved and increasing by more and more instead of a constant amount.

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 Describe what an exponential function is in your own words. An exponential function is a function that has a base raised to an exponent, x. • DOK-3 What is the difference between a constant rate and a constant multiplier? A constant rate means the value is changing by the same amount during equal intervals. For example, each time you increase by 1 on the x-axis, you add 3 on the y-axis. A constant multiplier means that the value is changing by the same multiple each time. For example, each time you increase by 1 on the x-axis, you multiply by 3 on the y-axis. • DOK-3 Create an equation for a microbe that would increase more rapidly than microbe C, y = 4x. y = 5x •

FACILITATION TIP Before reading the scenario, ask the class 1) If you had to be a biologist, what would your favorite animal, plant, or single-celled organism be to study?; 2) Why?; 3) What questions might you have about how it grows and repopulates over time? STEMscopes Tip The Math Chat, embedded in each Explore lesson outline as well as in printable form, provides a forum where students collaboratively discuss their ideas and strategies and develop their number sense, mathematical vocabulary, and math thinking skills. Discussing the concepts taught helps students formulate stronger reasoning and critical thinking skills.

FACILITATION TIP For the tables on Page 2 of their Student Journals, note that the third column for each microbe represents the expressions of the respective second column when simpliﬁed. Encourage students to ﬁll in the tables one column at a time to avoid confusion.

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Exponential Functions Explore 1 – Attributes of Exponential Functions Post-Explore 1. FACILITATION TIP After students complete the Exit Ticket, ask them if there is a way to simplify the expression 2(3x). Some students may say it simpliﬁes to 6x. Remind the class of order of operations, and plug in a value for both expressions to show that 2(3x) and 6x are not equivalent.

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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Exponential Functions Explore 2 – Domain and Range ACTIVITY PREPARATION Students will identify the domain and range of exponential functions. Students will extend their thinking to the entire shape of the graph, not just the part that is visible, and will discuss asymptotes for the ﬁrst time. Students will recall their experiences with domain and range with linear functions and apply those skills to exponentials.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

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

• • •

Separate the class into groups of 2 or 3 students. Print a Student Journal and an Exit Ticket for each student. Print a set of School Bacteria 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 II.”

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) What do you know about bacteria?; 2) Can you name any common bacteria?; 3) What jobs need to be able to work with bacteria?

2. 3.

FACILITATION TIP Be prepared with some visual examples (video) of bacterial growth to engage students. FACILITATION TIP Bacteria can spread to numerous surfaces within a school building. After students look at the cards, ask them for other examples of surfaces in a school that could carry bacteria. FACILITATION TIP Some students may be confused to see the graph for Part I decreasing given a positive initial value and constant multiplier, especially since the initial value is large. Remind them that the constant multiplier is raised to a power x and a decimal times a decimal results in a smaller value. 306

Read the following scenario to the class: We previously looked at microbe growth and determined that the function that grew the fastest had a variable in the exponent and the largest base. Today, we will examine ways to effectively combat bacterial growth! Give a Student Journal to each student. Explain to students that they will work with their groups to review the information on the School Bacteria Cards and record their work on their Student Journals. Remind students that an unrestricted domain looks at all x values from negative infinity to positive infinity, while a restricted domain gives a specific range of values to observe. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How does this graph differ from a linear graph? This graph is curved and decreases by less and less as x increases.

DOK-1 How does this graph differ from the Microbe Games Cards graphs? What do you notice about the range when the domain is unrestricted? This graph is decreasing, and the domain is x ≥ 0.

DOK-2 If 70% is removed each minute, why will there never be a negative amount left? When 70% is removed, there will always be 30% left, even if that 30% grows smaller and smaller.

As students complete question 2, note that when a function approaches a y value, it doesn’t quite reach as the x values approach negative or positive inﬁnity. We call this line an asymptote. This graph has a horizontal asymptote at y = 0. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

e. DOK-2 How does restricting the domain impact the range? The range has two ﬁxed values as endpoints when the domain is restricted between two values. 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-2 Why do large x values never produce a value below 0 for the function f( f(x (x) = 100(0.3)x? When x is a large value like 1,000, the expression 100(0.3)1,000 is computed by multiplying 0.3 by itself 1,000 times, then by 100. Since all of these factors are positive, the result will be a positive number despite being very close to 0. DOK-2 Where is the horizontal asymptote on this graph, and how can you tell? The horizontal asymptote is at y = 0 because the graph gets closer and closer to the x-axis without ever reaching it. DOK-2 When might there be added constraints on the domain and range of exponential functions? There might be added constraints when the example is modeling a real-world situation. For example, there might be added constraints when negative input values are not acceptable. DOK-3 What does it mean to have a reasonable domain and range? It means that you are restricting the input and output values to only include values that make sense for the scenario you are examining.

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Part II 1.

2. 3.

Read the following scenario to the class: You collect samples of bacteria from various places around the school to determine a reasonable domain and range for each. Give a set of School Bacteria Cards to each group of students. Explain to students that they will work with their groups to identify a reasonable domain and range for each scenario on the School Bacteria Cards and record their work on their Student Journals. Encourage students to ﬁll out the table on the cards to help them make their decisions. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 How can you quickly identify if there are restrictions on a domain or range? If there is a set value within the inequality

DOK-1 What would the domain and range be for the restroom sink bacteria if the domain was not restricted? Domain: all real numbers; range: y > 0

DOK-3 Why do the domain and range for the restroom sink bacteria have an upper and lower extreme value, but the library computer keyboard bacteria domain and range do not? In the restroom sink scenario, there is a speciﬁed interval of time. For the library computer keyboard scenario, there is not; it only indicates that the number of bacteria is declining over x hours.

DOK-3 For the lunch table bacteria, why does the domain include 0, but the range does not? The domain includes 0 because that is where we see the y-intercept, or our initial value. It also refers to the time “0 minutes.” The range cannot be 0, though, because there is a horizontal asymptote at y = 0. This means the bacteria will continue to decrease in population but never actually disappear entirely.

FACILITATION TIP Before reading the scenario, ask the class 1) How might someone actually discover an outbreak of bacteria?; 2) What would you use to ﬁght the outbreak?; 3) How would you ask for someone’s help?

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Exponential Functions Explore 2 – Domain and Range 5. 6. FACILITATION TIP Relatively high and low temperatures affect bacterial populations. Tell this to the class, and ask them if an unrestricted domain would make sense for a graph showing bacterial population as a function of temperature. Would they expect the equation for the new relationship to be exponential? Why or why not? Give space for all answers. FACILITATION TIP Before they begin the Exit Ticket, ask students to describe a time they took medication. How was it given to them? Who administered it? Did it work well?

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 What does it mean if you are given a domain or range that has values other than positive to negative infinity or all real numbers? This means the domain and/or range is restricted. DOK-2 When is it beneficial to look at restricted domains? It is beneficial when you are looking at real-world scenarios in which a negative input would not make sense. For example, when looking at growth over a period of several days, you would not look at a negative value for the number of days.

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

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Exponential Functions Explore 3 – Initial Value and Constant Multiplier ACTIVITY PREPARATION Students will identify and interpret the initial value and constant multiplier. Students will look for patterns in exponential functions and their data represented in tables. Students will notice that there is a constant multiplier that corresponds to a value in the function and begin to make observations about growth and decay.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

1 Student Journal (per student) 1 Set of Ranger’s Notebook Cards (per group) 1 Exit Ticket (per 2 students)

Reusable • •

• • • • •

1 Graphing calculator (per student) 1 Resealable bag (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 a set of Ranger’s Notebook Cards on card stock for durability. Cut the cards apart, and place them in a resealable bag. Gather a graphing calculator for each student.

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever been to a national park before?; 2) Which park did you go to?; 3) What did you do and see there?

FACILITATION TIP Students may mix up the terms “constant rate of change” and “constant factor of change” as they work through the Explore activity. If they remember that a constant multiplier applies to an exponential function, it may help to encourage them to think “constant factor-constant multiplier” as they would “factor-multiplication.”

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2. 3. 4.

Read the following scenario to the class: Congratulations! You have just secured your spot as a summer intern for Yellowstone National Park, the region of the United States with the largest concentration of mammals. To get you acquainted with the park, the head park ranger has given you some pages from his notebook with information regarding certain animal populations in the park. You must review this information and report back to the rangers regarding the impact the wolves have had on other mammal populations in the years following their reintroduction to the area. Give a Student Journal and a graphing calculator to each student. Distribute the Ranger’s Notebook Cards. Explain to students that they will work with their groups to review the information on the Ranger’s Notebook Cards and record their work on their Student Journals. They may use the graphing calculator to check any part of the information presented on the Ranger’s Notebook Cards or to help deepen their understanding. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 In the equation that represents the wolf population, where do you see the initial value represented? (0, 20); the value 20

DOK-2 What do you think the value 1.25 represents in the equation? 1.25 represents the percentage the wolf population is increasing each year.

DOK-1 In the equation that represents the elk population, where do you see the initial value represented? (0, 12000); the value of 12,000 is the initial value. © Accelerate Learning Inc. - All Rights Reserved

Engage

Explore

Explain

Elaborate

Evaluate

6. 7.

Acceleration

DOK-2 What do you think the value 0.75 represents in the equation? 0.75 represents the percentage the elk population is decreasing each year.

e. DOK-1 How can you determine the percent increase or percent decrease from the constant multiplier? I know 1 is 100%, so I can look at the value of the constant multiplier and compare it to 1. If the constant multiplier is less than 1, it is a decrease. If it is greater than 1, it is an increase. f.

Intervention

DOK-3 When would you expect to see the wolf population start to settle in terms of its size? We could expect to see the wolf population settle when the area has reached its carrying capacity. This means that nature has balanced itself out in terms of predator/prey ratios.

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 thus far.

Math Chat

FACILITATION TIP Print and project the question about percent increase and decrease. Do a quick check for understanding. Ask several volunteers and carefully select a few students to answer it in their own words.

EXPONENTIAL FUNCTIONS

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FACILITATION TIP Students may misinterpret a constant multiplier when ﬁnding percent increase or decrease. In such case, review how decimal values relate to percentages.

DOK-1 What are the key characteristics of exponential functions? Exponential functions include a common ratio. This means the y values are being multiplied by a common amount at each equal interval. The graphs of exponential functions are always increasing or always decreasing and will always include a horizontal asymptote. • DOK-1 What is the initial value in regard to exponential functions? The initial value is the starting point for the data set you are examining. The initial value can also be called the y-intercept. • DOK-1 What is the constant multiplier in regard to exponential functions? The constant multiplier is the factor at which an exponential function is growing or decaying. •

Part II 1.

2. 3.

Read the following scenario to the class: More than 100 species of birds live in Yellowstone National Park. The park rangers and parks services teams have done their best to model the growth and decline of several different species. Your job is to interpret their models and recommend which species may need urgent protection. Remind students that they should continue to ﬁll in their answers on their Student Journals. As students are collaborating on their work, monitor their understanding by asking the following guiding questions: a.

DOK-1 What part of the equation do you need to look at to identify if the population is growing or decaying? The base of the exponential

DOK-1 What part of the equation do you need to look at to identify the yy-intercept? -intercept? The constant in front of the equation

DOK-2 Why is the number in front of the equation ff(x) (x) = a(bx) the yy-intercept? -intercept? When x = 0, f(0) = a(b0), and b0 = 1, so f(0) = a.

DOK-2 How can you determine the percent increase or decrease from a function? You need to determine how much greater than or less than 1 the constant multiplier is.

FACILITATION TIP Before reading the scenario, ask the class 1) What do you know about species in our local natural areas?; 2) What animals or plants might need protection?; 3) What data do we need to make important decisions about different population growths in our natural areas? FACILITATION TIP Guiding questions 3a–3d are essential for student understanding. Print and project them to discuss before or after they collaborate. Check for understanding, encourage students to take notes, and carefully select students to respond in their own words.

FACILITATION TIP Allow students enough time to complete Part II and answer the reﬂection questions. After students answer the Reﬂect After the Explore activity, invite the class to a Math Chat to share their questions, ask them if it is possible to observations and learning. have a negative constant multiplier. Would Math Chat the function be exponential? How do they know? • DOK-2 Why does the base, or constant multiplier, of 1.7 not mean that the 4. 5.

population grows by 7 percent each year? 1.7 as a percent is 170% not 107%. Therefore, the population grows by 70% each year. The constant would need to be 1.07 to show 7% growth.

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Exponential Functions Explore 3 – Initial Value and Constant Multiplier • FACILITATION TIP Before students begin the Exit Ticket, ask them if it would ever make sense to insert a decimal value or fraction for x for an equation of form a(bx). They should know that it depends on the situation, equation, and value for x. They should know it depends because noninteger values for x can result in radical expressions, and the situation may call for whole-number solutions.

DOK-2 Why does the base, or constant multiplier, of 0.7 not mean that the function decreases by 70% each year? The constant multiplier captures the percent remaining versus the percent lost, which means that if the constant multiplier is 0.7, then the population actually decreases by 30% each year since 70% remains.

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

EXPONENTIAL FUNCTIONS

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Exponential Functions Explore 4 – Write Exponential Functions ACTIVITY PREPARATION Students will write exponential functions to model real-world situations. Students will analyze data tables, calculate the initial values and constant multipliers, and write exponential functions. Exponential functions will be written in the form f( f(x (x) = abx. Students will make connections between the b term and whether the function is representing growth or decay.

Standards for Mathematical Practice • • •

MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Set of Ranger Hint Cards (per group) 1 Set of Animal Population Cards (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 an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student has one. Print a set of Ranger Hint 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.” Print a set of Animal Population Cards, on card stock for durability, for each group of students. Cut the cards apart, shuﬄe them, and place them in a resealable bag. Label the bag “Part II.”

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) If you had to conduct a summer science research project, what data would you like to collect?; 2) Who do you know that might get to do this kind of work? ; 3) What local wildlife might be a good subject for research or data collection?

2. 3. 4.

FACILITATION TIP Consider distributing the Student Journal in two parts as needed. Part I is pages 1 and 2, Part II is pages 3 and 4.

FACILITATION TIP Watch out for students switching the order of values when dividing to calculate a constant multiplier. Remind them that an increasing function should have a constant multiplier greater than 1, while a decreasing function should have a constant multiplier between 0 and 1. 6. 314

Read the following scenario to the class: Now that you are settled in, the lead park ranger has assigned you a summer research project. You are tasked with identifying whether other mammal populations have seen a rise or decline in numbers since the reintroduction of the wolves. Give a Student Journal to each student. Give the Ranger Hint Cards to each group. Explain to students that they will work with their groups to identify the key features of exponential functions, and they will use those features to write the corresponding equations. 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 do you determine the constant multiplier, b? Divide each output by the preceding output if the change in input values is 1.

DOK-1 How do you determine the initial value of an exponential function? The initial value is the value of the function when the input is zero.

DOK-1 When do we see exponential growth within a function with respect to the b value? Exponential growth occurs when b > 1.

DOK-1 When do we see exponential decay within a function with respect to the b value? Exponential decay occurs when 0 < b < 1.

Engage

Explore

Explain

Elaborate

Evaluate

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

Math Chat DOK-2 How do you determine whether there is a constant factor of change and not a constant rate of change? Constant rate of change occurs if there are equal factors over equal intervals. If the function grows or decays by a constant percent rate per unit interval, then there is a constant factor of change, called the constant multiplier. • DOK-2 How could you create an inequality that shows when the cougar population reaches at least 60, and what would the solutions mean? The inequality would be 15(1.28)x > 60, and x would tell us what years the population of cougars would be above 60. • DOK-2 If the initial population of cougars was actually 30 and the population grew annually by 20%, what would the new function representing their population be? f(x) = 30(1.2)x •

Intervention

Acceleration

FACILITATION TIP Project these guiding questions 5a–5d. Record appropriate student responses, have students take notes, and conduct a quick check for understanding. Call on volunteers and some carefully selected students to assess progress.

EXPONENTIAL FUNCTIONS

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Part II 1.

2. 3. 4.

5. 6.

Read the following scenario to the class: Oh, no! You accidentally left your ranger log outside overnight, and the wind blew all of your notes away! You have collected all of the sheets of paper, but they are now all mixed up. You will need to sort them out to get all of your data back in order and report back to the lead park ranger to present your summer research. Give a set of Animal Population Cards to each student. Each group should still have the Ranger Hint Cards from Part I if they choose to use those to guide them. Explain to students that they will work with their groups to organize the cards into sets. Each set of cards should include a written description, table, graph, and equation. 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 If we look at the river otters, what are some ways of getting from 100 to 150 mathematically? Add 50 or multiply by 1.5.

DOK-1 Which of those ways could you use consistently to get from 150 to 225? Multiply by 1.5.

DOK-1 What do you notice about the population size of the bighorn sheep? The population is decreasing over time.

DOK-1 What are some ways of getting from 400 to 360? Subtract 40 or multiply by 0.9.

FACILITATION TIP Before reading the scenario, ask the the class 1) When was a time when you misplaced information of some kind?; 2) What information was involved?; 3) Were you able to recover the information? FACILITATION TIP Students have numerous cards to organize. Recommend that they sort the cards into stacks ﬁrst and then ﬁll in the table in Question 1 one row at a time.

e. DOK-1 Which of those ways could you use consistently to get from 360 to 324? Multiply by 0.9. f. 7. 8.

DOK-1 If the constant multiplier is 0.9, what is the percent increase/ decrease? 10% decrease

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 What do all increasing exponential functions have in common? A constant multiplier that is greater than one • DOK-1 What do all decreasing exponential functions have in common? A constant multiplier that is between zero and one •

FACILITATION TIP Watch out for students answering “A large initial value” or “An increasing exponent,” the latter of which applies to all exponential functions when using larger and larger values of x. Direct students back to the table in Part II to answer the ﬁrst two questions of this Math Chat. 315

EXPONENTIAL FUNCTIONS

Exponential Functions Explore 4 – Write Exponential Functions •

DOK-2 How would you explain to someone how to sketch a graph of a function in the form f( f(x (x) = a(bx)? I would have them plot the y-intercept at (0, a) and then use the value of b to determine whether the function was increasing or decreasing. I would plot the point at (1, ab) and then sketch the graph toward its asymptote at y = 0.

FACILITATION TIP

Post-Explore

After the Exit Ticket, explain to the class that they had to account for several cards for each animal in Part II. Ask them why or if they think it was necessary to track data in so many ways. Is there anything in the project they would have changed? Why or why not?

1. 2. 3. 4.

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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

EXPONENTIAL FUNCTIONS

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Exponential 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

Attributes of Exponential Functions 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

Domain and Range 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

Initial Value and Constant Multiplier

Interactive Notebook

Show What You Know, Part 4

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

Write Exponential Functions

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

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Spiraled Review

Fluency Builder

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

Exponential Functions

Can be done independently

EXPONENTIAL FUNCTIONS

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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 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

EXPONENTIAL FUNCTIONS

Exponential Functions

3 320

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

What prompts will be used?

EXPONENTIAL FUNCTIONS

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What does mastery look like?

I can connect the parts of an exponential function to features in a table, on a graph, or in a scenario.

I can determine the meaning of individual terms and factors within an exponential expression.

I can create and graph exponential functions.

I can identify a reasonable domain and range given an exponential function or a situation modeled by an exponential function.

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SCOPE 1

Exponential Extensions Scope Introduction SCOPE SUMMARY

A.PAR.9.3 Identify the effect on the graph generated by an exponential function when replacing f(x) with f(x) + k, and k f(x), for speciﬁc values of k (both positive and negative); ﬁnd the value of k given the graphs. A.PAR.9.4 Use mathematically applicable situations algebraically and graphically to build and interpret geometric sequences as functions whose domain is a subset of the integers. A.FGR.9.5 Compare characteristics of two functions each represented in a different way.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

Earlier in the year, students explored and graphed exponential functions. Students have seen graphs that have yy-intercepts -intercepts at points other than (0, 1) and horizontal asymptotes at lines other than y = 0. Students also worked with arithmetic sequences and deﬁned them explicitly and recursively.

In future courses, students will also look at the inverse of exponential functions, logarithms, and compare their graphs. Students will also solve more complex exponential equations by using logarithms.

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

Hook

A.PAR.8.2 Create exponential equations in one variable and use them to solve problems, including mathematically applicable situations.

Accessing Prior Knowledge

Student Expectations

In this grade level, students should be able to graph simple exponential functions from existing functions by hand, and to use what they know about the parameter changes from the parent function to graph a transformed function. Students should be able to use multiple representations of exponential functions and compare the properties of functions in different representations. They should be able to identify growth and decay functions. Students will solve exponential equations in one variable using tables, graphs, and technology. They will also explore exponential functions through a new lens by exploring geometric sequences.

match cards about applying the properties of integer exponents to generate equivalent numerical expressions.

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: •

connect transforming exponential functions with transformations of shapes.

•

determine how to transform shapes on a coordinate plane.

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

Solve Exponential Equations In this exploration, students will solve exponential equations in one variable using tables and graphs, as well as by inspection. Students will interpret their solutions in context. Students will: •

Explore 2

Explore 1

EXPLORE ACTIVITIES

determine the weights of puppies at different times in their growth.

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

Exponential Asymptotes and Intercepts In this exploration, groups of students will help solve a real-world scenario about helping a detective determine which suspect was at a coffee shop during the time of a crime by analyzing temperatures of coffee. Students will also help a detective by analyzing car engine temperatures to determine which car has been stationary. Students will: •

analyze exponential functions given as an equation or graph.

•

identify transformations.

•

focus on vertical and horizontal dilations.

EXPONENTIAL EXTENSIONS

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Geometric Sequences In this exploration, students will determine if enough speciﬁc vegetables and fruits are grown in 7 days to make food for a family gathering. Students will: •

construct recursive and explicit formulas for geometric sequences.

•

convert recursive to explicit formulas.

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. Geometric Sequences and Exponential Functions In this exploration, students will make connections between the explicit formula of a geometric sequence and exponential form. Students will: •

write geometric sequences and exponential functions from the table given.

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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Exponential Extensions 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

Materials

Preparation

Printed •

• •

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

Print one set of Match Around the Room Cards. Hang the cards in a random order around the room. FACILITATION TIP

Procedure and Facilitation Points 1. 2.

Card 1 matches with Card B.

Alternatively, after students have numbered their papers, display Card 1 and allow quiet seat time for observations/solving. Next, provide think time with a neighbor. Continue on this way and display Card 2 and so on. Finally, allow students to move around the room to look for matching letter cards (or display one at a time).

Card 2 matches with Card C.

FACILITATION TIP

Card 3 matches with Card A.

If students need a challenge, have them create equivalent expressions for the letter cards using the same base but different exponents.

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

Identifying Misconceptions •

EXPONENTIAL EXTENSIONS

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Students may struggle to remember the exponent rules. It is beneﬁcial to allow students to investigate the patterns when working with exponents with the same base. Encourage students to construct rules based on observed patterns.

FACILITATION TIP Depending on your students’ recent experience with exponent rules, consider using this Foundation Builder as a review activity before Match Around the Room. It includes eight good practice samples.

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EXPONENTIAL EXTENSIONS

Exponential Extensions Hook – Transform Exponential Functions ACTIVITY PREPARATION Students will connect transforming exponential functions with transformations of other functions and shapes.

Materials

Preparation

Printed •

1 Transform Exponential Functions (per class)

Reusable • •

• • •

1 Phenomena Video (per class) 1 Projector (per class)

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

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

Before showing the video and reading the scenario, ask the class 1) What games do you play involving shapes?; 2) Do the shapes change in position or size? How so?

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: Andrea has played games where she transformed shapes. She is wondering how she can do the same things to functions on the coordinate plane. 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 tiles in the image can rotate. I notice that the tiles can shift from left to right as they move down. I notice that the shapes cannot reﬂect in one move. I wonder if shapes can rotate in both directions. I wonder how the programming works to make the shapes move in the way the user wants them to. Project Transform Exponential Functions. Notes

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Engage

Explore

Explain

Elaborate

Evaluate

Explain to students that they will see how they can move functions in the coordinate plane throughout this scope. Discuss the following questions: a.

DOK-1 What transformations from the original function in black do you see on this graph? Allow students to share all ideas. Answers will vary. The blue graph is a reﬂection. All the graphs have been translated. The purple function ﬂips over the y-axis. The red function is three units lower than the black function.

DOK-1 What transformations from previous functions like the quadratic parent function f( f(x) = x2 do you remember? Allow students to share all ideas. Answers will vary. A negative sign in front led to a reﬂection. A number added on the end shifted the function up. The function g(x) = |x – 4| would move the parent function four units to the right.

Intervention

Acceleration

FACILITATION TIP As a ﬁnal question for the Pre-Explore, ask students if they can think of other real-world scenarios where shapes are transformed. Examples may include inﬂating a basketball (dilation) or driving a car (translation).

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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 Transform Exponential Functions, and discuss the following questions: a.

DOK-1 Do these graphs make more sense after the Explore activities? Yes, they are all transformations of the parent function f(x) = 2x.

DOK-1 What strategies would you use to create equations for the red, blue, and purple functions? I would identify the transformations of each one and then adjust the function f(x) until I have created a match.

DOK-1 How can you distinguish whether a function in an equation has been reflected over the x-axis or y-axis? y-axis? −f(x) is a reflection over the x-axis, and f(−x) is a reflection over the y-axis.

DOK-1 What are the equations for g(x ( ), h(x (x ( ), and j(x (x ( )? g(x) = 2x – 3, h(x) = (x 2(−(x + 4)) + 1 or h(x) = 2−x – 4 + 1; j(x) = −2x – 1 + 2

FACILITATION TIP After the discussion, ask students if they can think of examples of exponential functions in their everyday life. If they are stumped, present examples such as a skateboard ramp or the volume of a car as it leaves a parking lot. FACILITATION TIP In addition to this yes/no question, ask students to follow up with how they would explain the graphs to a younger student, parent, or the PE teacher.

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

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Exponential Extensions Explore 1 – Solve Exponential Equations ACTIVITY PREPARATION Students will solve exponential equations in one variable using tables and graphs, as well as by inspection. Students will interpret their solutions in context.

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 Exit Ticket (per student)

Separate the class into pairs of students. Print a Student Journal and an Exit Ticket for each student.

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before projecting the scenario, provide students some time to share their experiences with puppies and their growth. Consider showing a visual example of puppies growing.

2. 3.

FACILITATION TIP Consider distributing the Student Journal in two parts as needed. Part I is pages 1 and 2 and Part II is pages 3 and 4. Use two colors of paper to differentiate.

FACILITATION TIP Project these guiding questions. Take time to preview them with students before they begin collaborating on the puppy tables/ graphs on the Student Journal.

FACILITATION TIP Consider addressing the fact that Peaches the teacup dog is described in both pounds and grams. Some students may be curious about converting metric grams into US customary pounds. 328

5. 6.

Read the following scenario to the class: Many puppies grow exponentially in their ﬁrst few months before the growth steadies. Use the information provided to determine the weight of two German shepherd puppies as well as a morkie, which is a small teacup dog. Give a Student Journal to each student. Explain to students that they will work with their groups to determine the weights of puppies at different times in their growth 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 is solving an exponential equation different from substituting in a value? When solving, we are trying to determine the value for x that makes the equation true instead of taking a given value and substituting it into an equation.

DOK-2 How can a table help us solve an exponential equation? We can see when output values of the table match one side of the equation we are trying to solve.

DOK-1 What are the initial values for Gus and Duke? What are the constant multipliers? Gus has an initial value of 1.5 pounds and a constant multiplier of 2, and Duke has an initial value of 1 pound and a constant multiplier of 3.

DOK-2 How does creating a graph help us solve the exponential equation in question 9? Similar to a table, looking at the output values of a graph can help us ﬁnd the input value that makes the equation true.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Math Chat •

•

DOK-1 Do you ﬁnd looking at a graph or table easier for determining solutions to exponential equations? I prefer using a graph because I can quickly ﬁnd the output value I am looking for. DOK-2 What do you think must be true about the exponents in the equation 3x 3x + 5 = 310? Since the base is the same, the exponents must be equal. In this case, x + 5 = 10, so x would equal 5. DOK-2 How does solving an exponential equation using a graph compare to solving another type of equation using a graph? The process is the same regardless of the function type. DOK-2 How does solving an exponential equation using a table compare to solving another type of equation using a table? The process is the same regardless of function type. DOK-2 How can we check that our solution to an exponential equation is correct? Like any other equation, we can substitute our value in for the variable and verify that both sides produce the same value.

Part II 1.

2. 3. 4.

Read the following scenario to the class: Chelsea starts posting pictures on her STEMstagram page of Gus and Duke playing together in matching bandanas. Her number of followers starts to explode! Help Chelsea determine when she will meet certain follower milestones. Students should still have their Student Journals. Explain to students that they will work with their groups to determine the solutions to equations involving Chelsea’s exponential growth in followers. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

5. 6.

DOK-2 Why could the model for Chelsea’s number of followers not continue for a full year or two? The value would exceed the number of people that could reasonably follow one account if the tripling trend continued.

DOK-2 What would change about the model for Chelsea if she were losing followers instead of gaining them? The base in the model would be less than 1 instead of greater than 1.

DOK-1 What is the prime factorization of 27? 64? 33 and 26.

DOK-2 What point on the graph are we looking for to solve our equation? We are looking for the point of intersection.

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-3 Why are x-coordinates -coordinates where the graphs of f( f(x (x) and g(x ( ) intersect the (x solutions to the equation f( f(x (x) = g(x ( )? The points of intersection represent an x (x value where the output of f(x) and g(x) are equal. Thus, this x value solves the equation where f(x) = g(x) since it produces an equal output on each side of the equation. • DOK-2 Why would solving 2x = 7 be harder than solving 2x = 7? To solve 2x = 7, we can divide by 2 on both sides and substitute x = 3.5 into both sides of the equation to check our work. We do not yet have an inverse operation to solve 2x = 7, but we know it is slightly less than 3 since 23 = 8. •

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FACILITATION TIP In addition to these Math Chat questions, ask students to help you list some other real-world examples for exponential growth. Perhaps coach them to consider social media followers to engage them in Part II before you begin. FACILITATION TIP Project the text of this scenario for students to read along with you. FACILITATION TIP Consider distributing the Student Journal in two parts as needed. Part I is pages 1 and 2 and Part II is 3 and 4. Use two colors of paper to differentiate. FACILITATION TIP Project these guiding questions for students to see while they collaborate.

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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Exponential Extensions Explore 1 – Solve Exponential Equations DOK-3 Do you think all exponential equations have solutions? Explain why or why not. They do not because an equation like 2x = −1 does not have a solution. The range of the graph of f(x) = 2x is x > 0, so no x value could produce a negative output and no value would make this equation true. • DOK-2 How can you rewrite 243 so the equation 3x = 243 is easy to solve? 243 = 35, so 3x = 35 and it is clear that x = 5. •

FACILITATION TIP Project the last two Math Chat questions and allow students some think and partner chat time on both of them before a whole class discussion.

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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Exponential Extensions Explore 2 – Exponential Asymptotes and Intercepts ACTIVITY PREPARATION Students will analyze exponential functions given as an equation or graph and identify the transformation; they will focus on vertical and horizontal dilations.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

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

• • •

Separate the class into pairs of students. Print a Student Journal and an Exit Ticket for each student. Print a set of Cooling Curves Cards, on card stock for durability, for each group of students. Place the cards inside sheet protectors to create erasable surfaces.

Reusable • •

1 Dry-erase marker (per group) 2 Sheet protectors (per group)

PROCEDURE AND FACILITATION Part I FACILITATION TIP Before reading the scenario, ask the class 1) When was a time where you had to perform an investigation of some kind?; 2) What was the investigation about?; 3) What questions did you ask in order to solve the investigation?

2. 3. 4.

Read the following scenario to the class: Detective Ace is investigating a crime that occurred 3 hours ago at a coffee shop. All three suspects say they were at the coffee shop only within the last 2 hours. To determine the number of hours each suspect was present, Detective Ace takes the current temperature of each of their beverages and compares it to the cooling curve for that beverage. Are they all telling the truth? Who, if anyone, will be caught in a lie? Give a Student Journal to each student. Give a Part I Cooling Curves Card to each group. Explain to students that they will work with their groups to determine which suspect was at the coffee shop during the time of the crime 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 Where on the graph would you ﬁnd the initial temperature? The initial temperature will be the y value for x = 0.

DOK-2 As the initial values increase (40, 90, 110, etc.), what happens to the curve? The curve stretches upward along the y-axis. The coordinate that represents the initial value has a greater x value. It appears the same is true along the curve.

DOK-1 What is the temperature for a cup of coffee that has been cooling for 1 hour? How do you know? The temperature is about 103 degrees. The graph of the function that represents the temperature of coffee as it cools goes through the point (1, 103).

FACILITATION TIP Students may use the graph or the equation to approximate temperature. After they answer the question, ask them to compare the advantages of the two representations in the context of exponential functions. 332

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• •

•

Acceleration

DOK-2 How would you determine from the graph the amount of time each beverage has been cooling? Identify the current temperature of the beverage from the table in the detective’s notes, and then ﬁnd the corresponding x value for that temperature. The x value will be the number of hours the beverage has been cooling.

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 •

Intervention

DOK-2 As the initial values increase (40, 90, 110, etc.), what happens to the curve? The curve stretches upward along the y-axis. DOK-2 What do you think will happen to the curve as the initial value approaches values between 0 and 1 if the horizontal asymptote is at y = 0? The curve will become more and more compressed, almost to the point of resembling a straight horizontal line. DOK-2 How does the constant k in y = abx + k transform the parent exponential function? The value of k determines the horizontal asymptote of the function. DOK-2 How does the constant a in y = abx + k transform the parent exponential function? The value of a determines how far from the horizontal asymptote the y-intercept is. DOK-2 What part of the equation y = abx + k should we look at to determine whether the function is increasing or decreasing? We would look at the value of b. If b > 1, the function is increasing, and if b is between 0 and 1, the function is decreasing.

FACILITATION TIP After students complete Part I, ask them if there are any factors that, if present, would have demanded a different graph for the three drinks. Examples may include adding a cube of ice or pouring more hot water into a drink after a time.

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Part II 1.

2. 3. 4.

Read the following scenario to the class: With Suspect B successfully caught, Detective Ace is ready for his next case. Three suspects, D, E, and F, were found sitting in their cars, close to a different crime scene. They all claim to have been sitting in their cars for 2–4 hours with the engines off, and the crime was committed about an hour ago. Detective Ace takes the temperature of each of the engines and compares the temperatures to the cooling curve for each type of engine to determine the length of time each car has been stationary. What will he discover? Will anyone be caught in a lie? Give a Part II Cooling Curve Card to each group of students. Students should still have their Student Journals. Explain to students that they will work with their groups to determine which suspect’s alibi is most questionable 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 Compare the three equations for the graphs. What’s similar and what’s different about the three equations? All aspects of the equations are the same except for the base that is raised to the power of t.

DOK-2 Compare the graphs of the three equations. What’s similar and what’s different about the three graphs? The graphs have the same y-intercept and horizontal asymptote but decrease at different rates.

DOK-3 In these scenarios, what does the asymptote represent, and does it make sense in the physical situation that that would be the asymptote? The asymptote is the surrounding temperature. It makes sense that that would be the asymptote because after a long enough time, everything should equilibrate to the same temperature, which is the surrounding temperature.

FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever been caught in a lie?; 2) What was the lie regarding?; 3) Who uncovered the lie?

FACILITATION TIP Show the class a larger rendering of the Part I Cooling Curve Card. Have students compare and contrast the graphs and equations of the two cards.

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Exponential Extensions Explore 2 – Exponential Asymptotes and Intercepts d.

FACILITATION TIP After students ﬁnish Part II, ask them to describe other scenarios where cooling curves would be useful. Examples may include testing three sizes of cooling fans for a room or testing one cooling fan for three rooms of different sizes. FACILITATION TIP The conditions for horizontal dilations can seem counterintuitive compared to those for their vertical counterparts. The Math Chat will provide an opportunity for students to gain a better understanding of horizontal dilations by relating them to vertical dilations.

FACILITATION TIP Some students may see the curve for g(x) as a shift to the left of f(x) and answer “horizontal dilation” for the ﬁrst part of 1a. If this happens, ask them what key feature changed from f(x) to g(x) (They should realize that the y-intercept changed.). Then, ask them if the change was in the vertical direction or the horizontal direction.

6. 7.

DOK-2 How would you determine from the graph the amount of time each car had been cooling? Identify the current temperature of the engine from the detective’s notes, and then ﬁnd the corresponding x value for that temperature. The x value will be the number of hours the car has been cooling.

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-3 What strategies did you use to compare different representations of exponential functions? Convert one representation to another for easy comparison (for example, ﬁnd the equation of a graph and compare it to a given equation, or place all values in a table for comparison purposes). • DOK-2 How can you tell when an exponential function does not have an asymptote at y = 0? There is a constant value added or subtracted at the end of the equation. • DOK-2 Describe how to solve for a in the equation y = a(b bx) + k when you know the growth/decay factor, the horizontal asymptote, and the y-intercept. Substitute in the growth/decay factor for b and then the horizontal asymptote for k. Then, substitute the y-intercept into the equation and solve for a. • DOK-2 If you are looking at a graph of a function in the form y = a(bx) + k and the horizontal asymptote is at y = −3 and the value of a is 9, what is the y-intercept, y and how do you know? The y-intercept would be at (0, 6); the equation would be y = 9(bx) – 3 since the value of a is given and we know the horizontal asymptote. Substituting x = 0 into the equation yields y = 6 regardless of the value of b. Also, the value of a can be added to k to determine the y-coordinate of the y-intercept. •

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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Exponential Extensions Explore 3 – Geometric Sequences ACTIVITY PREPARATION Students will construct recursive and explicit formulas for geometric sequences. Students will convert recursive to explicit formulas.

Standards for Mathematical Practice • • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

• • •

1 Student Journal (per student) 1 Green Beans Graph (per group) 1 Exit Ticket (per 2 students)

•

Separate the class into groups of 2 or 3 students. Print a Student Journal for each student. Print a Green Beans Graph for each group of students. If desired, print it on card stock, and laminate it 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 Part I FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever planted vegetable plants?; 2) If so, what vegetable plants did you plant?; 3) Why did you choose those vegetables?

2. 3. 4.

Read the following scenario to the class: This summer, you’ve decided to build a garden and plant an assortment of fruits and vegetables. One of your goals is to grow enough green beans for your grandma to make her famous green bean casserole. Today, you got a call from your grandma saying she needs the green beans in 7 days to make the casserole. Will you grow enough green beans in time for your grandma to make her casserole? Give a Student Journal to each student. Give a Green Beans Graph to each group. Explain to students that they will work with their groups to analyze the Green Beans Graph and answer the questions to determine if they will grow enough green beans for their grandma’s casserole. 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:

FACILITATION TIP

After students answer the question, note that the graph for green beans forms an exponential curve. Let them know that they will address exponential functions in later scopes.

DOK-1 How do you know the green beans do not have a common difference? The number of green beans increases by greater amounts as time goes on, and the graph does not form a line.

DOK-1 What does a constant ratio tell you about a set of data? How does this help you when writing an equation? The constant ratio shows you the pattern a set of data follows. This means it must be included when writing the equation so the equation will work for determining the value of any term in the given set of data.

DOK-2 What math symbols can be used to represent these changes, and how can you use them to make predictions for unknown data? Multiplication and division symbols can be used to show these changes. You can identify these mathematical operations within a given set of data (sequences, graphs, and tables) and use them in equations to ﬁnd the value of any unknown term.

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Acceleration

DOK-2 How do you determine the constant ratio based on a graph? After every integer value of x, I look to see by how much each y value has been multiplied. This gives me the constant ratio.

e. DOK-2 How were you able to determine the number of green beans on subsequent days not included in the graph? I took the y value from day 4 and then multiplied by 2 (the constant ratio). That gave me day 5. I multiplied again by 2 to get day 6. f.

6. 7.

DOK-2 How can you create an explicit equation to determine any nth term for the green bean output? To create an explicit equation that works for any nth term for each competitor, you need to include the r and the ﬁrst term, as well as consider at what term you begin multiplying or dividing.

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 can you determine the missing number of green beans for days 5 and 6? You can identify the constant ratio in the graph and continue to multiply or divide to determine the following term value until you ﬁnd the value for days 5 and 6. • DOK-2 If predicting the number of green beans on the 100th day is too diﬃcult, what do you think could be used to determine the answer easily? Creating an equation where you substitute the value of the nth term would be an easy way to determine how many green beans there are on the 100th day. Mathematicians call these equations explicit. We used explicit equations to calculate any term in an arithmetic sequence without knowing the value of the previous terms earlier this year. •

Explain the following to the class: When we write an equation for a sequence, we will use the formula that has an output of An. The explicit formula for a geometric sequence is An = A1(r)n – 1, where A1 is the ﬁrst term, r is the common ratio, and An is the nth term in the sequence. •

•

DOK-2 What three elements must be included when creating an equation that builds on the previous term’s value? You must include the previous term (represented by An – 1), the mathematical operation that’s being done, and the constant ratio. DOK-3 Mathematicians call equations that require the value from the previous term to determine the value of the next term recursive equations. Compare and contrast recursive equations to what you already know about explicit equations. There are constant ratios in both recursive and explicit equations. Recursive equations require the value from the previous term to determine the following term’s value. Explicit equations don’t require you to know the value of the previous term.

Part II 1.

2. 3.

Read the following scenario to the class: Your aunt heard how delicious your grandma’s famous green bean casserole was and now is requesting your help in growing blueberries and blackberries for her to make her special jam. You would love to help your aunt but are worried that you won’t be able to grow enough fruit for her. Thankfully, you have been keeping track of your fruit growth in a data table and are conﬁdent you can calculate just how many blueberries and blackberries you’ll have for your aunt when she needs them. Will you have enough? Students should still have their Student Journals. Explain to students that they will work with their groups to analyze the given recursive equations for blueberries and blackberries to complete the missing sections in the data tables.

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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 If no one includes it in their answer, ask students to compare and contrast the general form of arithmetic explicit and arithmetic recursive equations. Do the same for the general form of geometric explicit and geometric recursive equations. If someone did compare and/or contrast these equations mathematically, have them share with the class, and discuss with the class if they are accurate. FACILITATION TIP Before reading the scenario, ask the class 1) What are your favorite fruits to eat?; 2) Do you eat the fruit as is, or do you like to use the fruit in a recipe?; 3) What products are made out of your favorite fruit?

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Exponential Extensions Explore 3 – Geometric Sequences 4. FACILITATION TIP

Students will work together to create an explicit equation to determine the amount of each fruit on day 10 and record their work on their Student Journals. As students collaborate, monitor their work and use the following guiding questions to assess student understanding:

Student Journal, Part II, Question 5. Make sure students provide clear reasoning for their answers, as answers will vary. For instance, some students may use “a higher term” instead of “the 100th term” or “all of the previous terms.” instead of “the ﬁrst 99 terms”. Give them space to use their own words.

FACILITATION TIP Student Journal, Part II: It may not be intuitive for students to understand the advantages and disadvantages of using recursive formulas versus explicit formulas. After the class completes Question 2 have a short discussion to build on the reasoning behind the answer. Do the same for Question 5. FACILITATION TIP Coach students to come up with some real-world applications for arithmetic and geometric sequences (Consider a ball bouncing or stacking items like a pyramid). FACILITATION TIP Some students may be thrown off seeing a sequence of numbers by itself without context. Remind them to ﬁgure out the type of change so they know which kind of recursive and explicit equations to model.

6. 7.

DOK-1 How can you use the given equation to determine the missing sections in the data tables? Substitute the previous input into the recursive equation to determine the value of the following term.

DOK-2 How can you use the completed data tables to create an explicit equation? Identify whether it is a geometric sequence, and then use the r value and ﬁrst term within the formation of your explicit equation.

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 What are two types of equations that can be used for geometric sequences? Recursive and explicit equations DOK-3 How can you use a recursive equation to create an explicit equation? You can analyze the data in a table to identify the elements needed when creating an explicit equation: the first term, the constant ratio, and whether it is geometric. DOK-2 Why is it useful to be able to write equations for arithmetic and geometric sequences? Writing equations is useful when finding the value of an nth term that would require an excessive amount of adding, subtracting, multiplying, or dividing. Equations allow you to find the value by simply substituting in the nth term to find the correct value.

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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Exponential Extensions Explore 4 – Geometric Sequences and Exponential Functions ACTIVITY PREPARATION Students will make connections between the explicit formula of a geometric sequence, An = A1(rrn – 1), and exponential form, y = a(bx).

Standards for Mathematical Practice • • •

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

Materials

Preparation

Printed • • •

1 Student Journal (per student) 1 Set of Theater Scenario 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. 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 Theater Scenario 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.

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Preview with students what refurbish and revenue mean before reading the scenario. FACILITATION TIP Print and project this scenario. Have student volunteers read it aloud. Read through it once without allowing students to take notes. Next, encourage students to write down the important math phrases and values.

2. 3.

FACILITATION TIP

Consider distributing the Student Journal in two parts as needed. Part I is pages 1 and 2 and Part II is pages 3 and 4.

Read the following scenario to the class: Ishaan owns a movie theater that people often visit. Over the years, the seats have become worn out and stained. He decides to refurbish all the seats of the theater. Since doing all the seats at once would be too expensive, he decides to refurbish 25 seats the ﬁrst year and plans to double the number of seats he refurbishes each year thereafter as revenue increases. His two assistants, Sai and Emma, create graphs to represent the number of refurbished seats over time. Give a Student Journal to each student. Explain to students that they will work with their groups to write geometric sequences and exponential functions from the table given 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 Would you call this domain discrete or continuous? Why? Answers may vary. I would call this a discrete domain because it is only deﬁned for whole numbers.

DOK-2 How are geometric sequences and exponential functions similar? Both use a common ratio or factor to move from one term to the next.

DOK-2 How are geometric sequences and exponential functions different? Can you connect the values in each representation to the scenario? The setup for the equations is different. Exponential functions start at x = 0, whereas geometric sequences start at n = 1. Both include a 2 as a base, showing the number of refurbished seats doubling each year.

FACILITATION TIP Select the essential guiding questions to print, project, and preview with students before they begin collaborating.

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e. DOK-3 Would it make sense to write the equation for the refurbished seats in recursive notation? Why or why not? Answers may vary. Recursive notation requires using the previous term to determine the next term, so it is more labor intensive to use compared to writing an equation in explicit notation. Since we know that A1 = 25, a recursive equation would work in this scenario. 5.

Intervention

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-2 How is the process for writing equations of exponential functions and geometric sequences similar? Both forms require you to ﬁnd the common ratio/ constant multiplier. • DOK-2 How is the process different for these equations? The arithmetic sequence starts with n = 1, whereas the exponential form y = a(bx) requires the y-intercept where x = 0. • DOK-3 Which type of function would be better if some of the chairs are constantly being refurbished? Why? Exponential would be better because it has a continuous domain that includes parts of the year, not just a yearly update. • DOK-3 Which function (geometric or exponential) would be better if Ishaan refurbishes his chairs for the year all at once? Why? Geometric is reasonable for representing situations with a discrete domain. However, an exponential function could be used if we restrict the domain. •

STEMscopes Tip

EXPONENTIAL EXTENSIONS

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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.

Part II 1.

2. 3. 4.

Read the following scenario to the class: Ishaan wants to gather information about other aspects of the theater. He wants to make sure he uses the best method for data collection. Help Ishaan determine what equations he should use and why. Give a set of Theater Scenario Cards to each group of students. Students should still have their Student Journals. Explain to students that they will work with their groups to identify whether to write geometric sequences or arithmetic functions for the four theater scenarios and record their answers on their Student Journals. They will create the appropriate function for each scenario. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-1 Which of the scenarios are ongoing as opposed to discrete events? The movie ticket sales and repair costs are ongoing as opposed to one-time events each week or year.

DOK-1 How are you determining whether geometric or exponential is appropriate? The domain is used to determine if a function is geometric because of a discrete domain or exponential because of a continuous domain.

FACILITATION TIP Consider distributing the Student Journal in two parts as needed. Part I is pages 1 and 2 and Part II is pages 3 and 4. FACILITATION TIP Alternatively, you could guide students through some or all of the four scenarios as a whole class.

FACILITATION TIP c. DOK-1 What key pieces of information do you need to create each function? An initial value and the constant multiplier or common difference Post questions 5c and 5d. Emphasize to students that understanding these d. DOK-1 How can you use a table or graph to write a geometric or questions will support their success on the exponential function? First, identify an initial value, and then divide upcoming Exit Ticket. consecutive output values to determine the constant multiplier or common difference. © Accelerate Learning Inc. - All Rights Reserved

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Exponential Extensions Explore 4 – Geometric Sequences and Exponential Functions 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 How are the formulas for geometric sequences and basic exponential functions similar and different? Both have a base that is the same but different values in the equation to show that the sequence starts at an input value of 1, and the exponential function contains the y-intercept, which is when the input value is 0. • DOK-2 When would you use arithmetic sequences? Any time the domain is discrete. • DOK-2 When would you use linear functions? Any time the domain is continuous. • DOK-3 Create some examples of situations where you would use geometric sequences instead of exponential functions. Children/people, speciﬁc times (only days, full hours, etc.), luggage, or tickets • DOK-3 Create some examples of situations where you would use exponential functions instead of geometric sequences. Anything involving continuous growth or passing of time, money, or distances • 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.

Post-Explore 1. 2. 3. 4.

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EXPONENTIAL EXTENSIONS

Exponential Extensions 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

Solve Exponential 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

Exponential Asymptotes and Intercepts 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

Geometric Sequences

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

Geometric Sequences and Exponential Functions

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

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Explain

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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

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

EXPONENTIAL EXTENSIONS

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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

EXPONENTIAL EXTENSIONS

Exponential Extensions

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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?

EXPONENTIAL EXTENSIONS

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I can use exponential equations to solve real-world problems.

I can transform the parent exponential function with and without the use of graphing technologies.

I can construct geometric sequences.

I can convert between recursive and explicit forms of geometric sequences and relate exponential functions with geometric sequences when I am given an applicable situation.

I can compare the behavior and characteristics of two exponential functions.

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SCOPE 1

Compare Function Types Scope Introduction SCOPE SUMMARY

Student Expectations

In this grade level, students will build on their current understanding of linear, quadratic, and exponential functions. Students will examine average rate of change to explore how quadratic functions and exponential functions change over certain intervals relative to linear functions which always change at a constant rate. They will calculate the average rate of change of functions represented symbolically, in tables, and in graphs. Students will then compare linear, exponential, and quadratic functions by looking at tables, graphs, equations, and verbal descriptions. They will pay close attention to key distinguishing features of the functions that they analyze to discuss the growth, end behavior, intercepts, and intervals of increase and decrease.

A.FGR.7.7 Create quadratic functions in two variables to represent relationships between quantities; graph quadratic functions on the coordinate axes with labels and scales. A.FGR.9.5 Compare characteristics of two functions each represented in a different way.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grade levels, students explored linear functions and continued that work in Algebra I. Students previously explored exponential and quadratic functions by using function notation to build functions over their domains. They have explored the different properties of the graphs of parent linear, quadratic, and exponential functions.

Students will continue to explore and work with all linear, quadratic, and exponential functions in future grades. They will speciﬁcally compare exponential functions to their inverses, logarithms. Students will continue to discuss features like average rate of change, intercepts, zeros, domain and range, and asymptotes of various function types in future courses.

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

Hook

Accessing Prior Knowledge

ENGAGE ACTIVITIES

describe qualitatively the functional relationship between two quantities by analyzing a graph.

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: •

connect different function types and their growth to a real-world scenario involving stock prices.

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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Average Rate of Change Introduction

Explore 2

Explore 1

EXPLORE ACTIVITIES

In this exploration, students will solve a real-world scenario about analyzing food sales from the previous year and identifying average changes in sales for speciﬁc time periods for a Student Celebration Day for the school. Students will: •

calculate and interpret average rates of change over speciﬁed intervals.

•

connect the rates of change across representations.

•

make generalizations about behaviors of functions over time.

In this exploration, groups of students will determine the amount of caffeine in the body over time to help determine the amount of time two students can be studying. Students will: •

create linear and exponential functions.

•

examine exponential functions from different representations.

•

compare key features of the functions.

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

Explore 3

Compare Linear and Exponential Representations

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Compare Linear, Exponential, and Quadratic Relationships In this exploration, students will examine and compare linear, exponential, and quadratic relationships. Students will: •

explore the difference of quadratic functions using tables.

•

examine when a function representing exponential growth will overtake a quadratic function.

•

distinguish between these three function types from different representations.

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

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Compare Function Types 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 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.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.

Materials

Preparation

Printed •

COMPARE FUNCTION TYPES

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• •

1 Set of Fact or Fiction (per class)

Print one set of Fact or Fiction 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. Instruct students to move to one side of the room or the other 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 peers. Before reading the next prompt, allow students to move back to their starting point. Repeat with remaining prompts. a.

Prompt 1 is ﬁction. The function has 3 x-intercepts.

Prompt 2 is fact.

Prompt 3 is ﬁction. The distance vs. time graph would increase at an increasing rate.

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

Identifying Misconceptions • •

•

FACILITATION TIP Before beginning the Fact or Fiction activity, ask students what they know about interpreting graphs. Consider starting a KWL chart (Know, Want to Know, Learned) Next, let them know their skills will be put to the test with a game of Fact or Fiction. FACILITATION TIP Students may read a prompt too quickly and/or miss a detail on the graph. As students discuss a prompt, allow them to switch sides before calling them back to their starting point. FACILITATION TIP This Foundation Builder includes ﬁve examples that provide excellent practice and review for interpreting graphs.

Students may be confused about the x- and y-intercepts. Help students make other vocabulary connections with the preﬁxes max- and min- to remind students that the maximum is the highest point on the function, and the minimum is the lowest point. Students may want to consider a speciﬁc situation that could have the same velocity graph to decide what the distance vs. time graph might look like.

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Compare Function Types Hook – Stock Growth ACTIVITY PREPARATION Students will connect different function types and their growth to a real-world scenario involving stock prices.

Materials

Preparation

Printed •

• • •

1 Stock Growth (per class)

Reusable • •

1 Phenomena Video (per class) 1 Projector (per class)

Plan to show the video. Prepare to project Stock Growth 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

Consider your students’ prior experience with stocks and investing. Depending on their interests, ﬁnd a way to connect stock growth to something more familiar to them. You might use the same functions with a different real-world connection.

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: Tasha is looking into three stocks in which to potentially invest. A stockbroker presents her with models for the price of each stock where x is in days. Because stock prices can change quickly and drastically, the models are only good for the next two weeks. Project Stock Growth.

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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 graphs in the video have linear and nonlinear parts. I notice that the three equations are linear, quadratic, and exponential. I notice that Stock 1 has the highest y-intercept. I wonder which stock is worth the most at the end of the two weeks. I wonder what impacts the prices of stocks most often. I wonder what it means to buy a stock? Are there any other purchasing options on the stock market? Explain to students that during this scope they will compare the growth and features of different functions they have studied throughout the year. Discuss the following questions: a.

DOK-1 What are the different types of functions that are represented by the models for the three stocks? Stock 1 is a linear function. Stock 2 is a quadratic function. Stock 3 is an exponential function.

b. DOK-1 What information about each function do you remember from previous scopes? Allow students to share all ideas. Answers will vary. Linear functions create straight lines on a graph and increase or decrease at a constant rate. Quadratic functions form parabolas on graphs and are symmetric about a vertical line that passes through the vertex. All parabolas have increasing and decreasing intervals. An exponential function has a horizontal asymptote and is either always increasing or always decreasing. Exponential functions can grow quickly. 6.

Complete the Explore activities.

Part II: Post-Explore 1. 2.

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

DOK-1 Do these equations make more sense after the Explore activities? Yes, they are all functions that we can graph, analyze, and compare.

DOK-1 What strategies would you use to compare the growth rate of each stock over the next two weeks? I would determine the average rate of change for each function on the interval [0, 14].

DOK-1 How can you distinguish between function types on an equation or graph? A(x) is linear because it is in slope-intercept form and because its graph is an increasing line. B(x) is a quadratic function because it has a degree or highest exponent of 2 and creates a U shape or parabola on a graph. C(x) is an exponential function because it has a variable in its exponent, its graph has a horizontal asymptote, and it is increasing at an increasing rate.

DOK-1 When does each stock have the most value? Stock 1 has the greatest value on days 0 through 8. Stock 2 has the greatest value on days 9 through 13. Stock 3 has the greatest value on day 14.

Intervention

Acceleration

FACILITATION TIP This information is essential for success in this scope. Take time to record vocabulary, images, and deﬁnitions with and for students. Use the Picture Vocabulary under the Explain tab.

COMPARE FUNCTION TYPES

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FACILITATION TIP In addition to this yes/no question, ask students what they learned and record it on the KWL chart if you have one from the APK.

FACILITATION TIP As a follow up to this yes/no question, ask students if they can name some real-world applications for understanding function types. Be prepared with some visual examples for how these skills are used in careers and later math/science courses.

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

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Compare Function Types Explore 1 – Average Rate of Change Introduction ACTIVITY PREPARATION Students will calculate and interpret the average rates of change over speciﬁed intervals. Students will connect the rates of change across representations and make generalizations about the behaviors of functions over time.

Standards for Mathematical Practice • • •

MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

1 Student Journal (per student) 1 Set of Food and Beverage Options Cards (per group) 1 Exit Ticket (per student)

• • •

Separate the class into groups of 2 or 3 students. Print a Student Journal and an Exit Ticket for each student. Print a set of Food and Beverage Options Cards, on card stock for durability, for each group of students. Do not cut the cards apart, as students will compare the two options for each category.

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever been on a committee?; 2) If so, what committee were you on?; 3) What did you do on the committee?

2. 3.

FACILITATION TIP As students complete the hamburger sales table on the Student Journal, have them look at the graph. Ask them if there is any time period where there was no change in sales. When they ﬁnd that time period, ask them what the slope of the graph for that time period is.

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Read the following scenario to the class: You are a part of the student event planning committee at your school. This committee is in charge of planning the Student Celebration Days that will take place periodically throughout the school year. Your team is in charge of ordering food for the events, and you want to ensure that you choose foods that have proven popular with students during the times in which the Student Celebration Days will occur. In order to plan what items to buy, you need to analyze food sales from the previous year and identify average changes in sales for speciﬁc time periods. Give a Student Journal to each student. Explain to students that they will work with their groups to analyze the data provided for hamburger and pizza slice sales. Students will calculate changes in sales over time and complete the information for each food type. 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-2 What would it mean to get a negative number for the Average Change in Sales per Month column? A negative number in that column would mean the sales rate would be decreasing over that time period.

DOK-1 What would the slope of that line look like when graphed? The line would have a negative slope. The slope would go downhill when graphing from left to right.

DOK-2 If you were ﬁnding the average of 4 numbers, you would divide the total by 4. How does this relate to ﬁnding the average change in sales? I divide the total change by the number of months in the interval. This is exactly like calculating average. © Accelerate Learning Inc. - All Rights Reserved

5. 6.

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DOK-2 In the Pizza Slice Sales Chart, what information do you need to know to be able to calculate an average change in sales per month from month 3 to month 5? I would need to look at the change in sales and the change in months. I would divide the change in sales by the change in months to get the average change in sales per month.

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. FACILITATION TIP

Math Chat DOK-3 When figuring out the change in number of sales each month, what things do you need to be aware of/consider? I need to be aware of the time period. If I am not calculating the average change in sales from one month to the next, then I will need to divide by the number of months. I also need to pay attention to the scale of the axes. • DOK-2 How is the average change in sales similar to slope? I calculated the average change in sales exactly how I would calculate slope. I found the change in the outputs and divided it by the change in the inputs. • DOK-2 How is the average change in sales different from slope? A linear function has the same slope no matter what interval I am observing. The pizza slice sales data and the hamburger sales data were not linear because they did not have a constant rate of change. I could find the average change in sales when I was given specific intervals. •

Before they start the Math Chat, explain to the class that hamburger sales were presented in a graph while pizza sales were presented in a chart. Discuss with the class whether they think a graph or a chart compares sales better. Also, ask students for examples of scenarios they believe are better suited for graphs and charts, respectively.

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Explain the following to the class: For nonlinear functions, the rate of change is not constant, so mathematicians refer to average rates of change over an interval. Part II 1.

2. 3.

Read the following scenario to the class: For all of the Student Celebration Days, the committee wants to provide a healthy food option, a dessert, and a beverage. You have been provided with data for these three categories based on last year’s sales. Analyze the data, and perform the necessary calculations to make your recommendation for each of the three categories. Give a set of Food and Beverage Options Cards to each group of students. Explain to students that they will work with their groups to compare the data for the two options in each category. Students will follow the prompts on page 3 of their Student Journals, perform the necessary calculations, and make a recommendation for each category. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-2 How can you determine which healthy food option had the highest average rate of change from month 1 to month 5 just by looking at the graphs? Since the intervals of time are the same and the scale of the graphs are the same, I can just compare the graphs. I can tell that the salad graph had a larger increase in sales from month 1 to month 5.

DOK-2 How is calculating the average rate of change similar to calculating slope of a line? They are similar because I am looking at the change in the y value divided by the change in the x value. With a linear function, it doesn’t matter what part of the graph I observe, the slope is consistent. With nonlinear graphs, I must pay close attention to the intervals.

DOK-1 How do you calculate unit rate when given a rate? I have to divide the quantity associated with the dependent variable by the quantity associated with the independent variable.

FACILITATION TIP Before reading the scenario, ask the class 1) What is your favorite food?; 2) Is your favorite food considered a healthy food source?; 3) If not, what food do you eat that is considered healthy?

FACILITATION TIP As students calculate slopes on Page 3 of the Student Journal, make sure they understand that the y values belong in the numerator and the x values belong in the denominator. Also, make sure they understand that the point whose y value is written ﬁrst in the numerator must have its x value written ﬁrst in the denominator.

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Compare Function Types Explore 1 – Average Rate of Change Introduction d.

DOK-2 How is this similar to calculating the average rate of change? It is the same as calculating the average rate of change. I know the beverage costs will be linear.

e. DOK-1 What months should you analyze for the interval 2 ≤ x ≤ 9? This notation tells me that I should look at the change in sales from month 2 to month 9. FACILITATION TIP Student Journal Part II, Reﬂect: For Question 2, neither axis in the graph counts by ones. Before students answer the question, remind them to note the scale of both axes without giving away the scales or the answer to the question. FACILITATION TIP Exit Ticket: Upon completion, have the class draw a dotted line on the graph in Question 2 from (–6, 9) to (–4, 9). Explain to them that this represents a rate of change of 0. Then, have them draw another dotted line from (–4, 9) to (0, 0). Have them ﬁnd the average rate of change from x = –6 to x = 0 given both graphs. Have them compare the average rates. Guide them to conclude that they are equal because the endpoints and interval are the same for both graphs.

5. 6.

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 Why is it helpful to use average rates of change on nonlinear graphs? Nonlinear graphs do not have constant rates of change, so using the average over an interval can help generalize the behavior of the function. DOK-2 How did you calculate the average rate of change within each set of data? What general rule can you come up with that would help in any scenario? Calculate the change in the outputs over a speciﬁed interval, and divide that number by the change in inputs over the same interval.

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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Compare Function Types Explore 2 – Compare Linear and Exponential Representations ACTIVITY PREPARATION Students will create linear and exponential functions. Students will also examine exponential functions from different representations. These can be graphs, equations, tables, or written descriptions. They will compare key features of the functions and consider how models would differ if a linear function was used compared to an exponential one.

Standards for Mathematical Practice • • •

MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

• • •

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

Separate the class into groups of 2 or 3 students. Print a Student Journal and an Exit Ticket for each student. Print a set of Coffee Shop 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 II.”

Reusable •

1 Resealable bag (per group)

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Does anyone drink caffeinated drinks?; 2) What kinds of caffeinated drinks do you drink?; 3) Why do you drink them?

2. 3. FACILITATION TIP Some students may be overwhelmed initially by Question 5 thinking they have to account for two equations at once or temporarily forgetting how to approach a linear scenario. Instruct them to look at each scenario in Question 5 as the start of a new function and remember what they have learned about linear functions.

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Read the following scenario to the class: Lucy loves to drink coffee. As the caffeine from the coffee leaves her system, she crashes, meaning she experiences increased fatigue and less productivity. The amount of caffeine in Lucy’s body over time is modeled in the table on the Student Journal. Lucy has a big exam tomorrow for which she still needs to study. She guzzles down 2 large mugs of coffee and hopes it will sustain her for the next 10 hours. Her performance is optimal when she has at least 25 mg of caffeine in her system. Will Lucy still be energized and alert to study after 10 hours? Lucy asks her classmate Hannah to study with her. Hannah drinks coffee as well, and the amount of caffeine in Hannah’s system over time is provided in a graph. Hannah is also alert and optimal when she has at least 25 mg of caffeine in her system. Will she be able to stay awake and study with Lucy? If so, for how long? Give a Student Journal to each student. Explain to students that they will work with their groups to determine how long Lucy and Hannah can study together 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 is the initial value for an exponential function? The initial value is the output value when the input value is zero.

DOK-2 How would I ﬁnd the initial value on a graph? Look for the y-intercept.

DOK-2 How would I identify the initial value on a table? Look for the corresponding output value when the input is 0.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

e. DOK-1 Point to the ordered pair on the graph where the amount of caffeine is 25 mg. What is the corresponding time? 10 hours f.

DOK-2 How does Lucy’s decline differ from Hannah’s? Lucy experiences a linear decline, while Hannah’s is exponential.

DOK-2 How can we use the initial point and one other point to write the equation of a line? If we use slope-intercept form, y = mx + b, we can use the initial value as b and calculate the slope between the two points for m.

FACILITATION TIP After students answer the question, ask them for possible reasons for the different types of decline between Lucy and Hannah. Answers may include differences in genetics, diet, sleep quality, etc.

h. DOK-2 What information do we need to write an exponential equation in the form y = a(bx)? We need the initial amount and the growth/decay factor or constant multiplier. 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.

Math Chat • • • •

•

DOK-1 What is the initial value for an exponential function? The initial value is the output value when the input value is zero. DOK-2 How would I find the initial value on a graph? Look for the y-intercept, where the x value of the ordered pair is zero. DOK-2 How would I identify the initial value on a table? Look for the corresponding output value when the input is 0. DOK-3 How would I create an exponential equation based on two points? First, use the y-intercept to know the initial value. Then, substitute the other point into the equation and solve for the constant multiplier. DOK-2 How does creating an exponential equation differ from finding a linear equation? How is it similar? I need to find the slope to create a linear function and need to figure out the constant multiplier to create an exponential equation. Both equations use the initial value or y-intercept. DOK-2 How does a graph decreasing linearly differ from a graph decreasing exponentially? A linear graph will keep decreasing at the same rate forever. An exponential graph will decrease by less and less and approach a horizontal asymptote that it will never reach.

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FACILITATION TIP For Question 5e watch out for students solving for b2 and using that as their constant multiplier rather than b. If this happens, encourage students to pay close attention to the exponential form expressed in the question. 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.

Part II 1.

2. 3.

Read the following scenario to the class: Lucy goes to different coffee shops all over town. A few of them are large chains and a couple are smaller coffee shops that have been steadily growing since a spike in the 2018 coffee market. Answer a series of questions based on the four Coffee Shop Cards to determine a model for the amount of sales at each shop. Give a set of Coffee Shop Cards to each group of students. Explain to students that they will work with their groups to answer a series of questions based on the four Coffee Shop Cards to assess the sales growth of each shop since 2018. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

DOK-3 For coffee shop A, how else could you represent the relationship between number of sales and years if the current written description is diﬃcult to understand? Create a table

DOK-2 When looking at a table, how can you tell whether a function is exponential or linear? If the function is linear, the same constant is added or subtracted to get the next output value. If the function is exponential, you get the same ratio when you divide each successive term by the previous term.

FACILITATION TIP Print and project this scenario for students to read along with you.

FACILITATION TIP Consider setting students up for success by projecting and reviewing the guiding questions 4a–4g before they begin collaborating.

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Compare Function Types Explore 2 – Compare Linear and Exponential Representations c.

DOK-2 How can you tell whether the graph of a function is linear or nonlinear? The graph of a linear function will have a constant rate of change.

DOK-1 What is the y-intercept? y It’s the output value when the input value is zero.

e. DOK-2 How do we determine the average rate of change between two points? Determine the slope (the change in y divided by the change in x) between the two designated points.

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.

5. 6.

DOK-2 To write a linear function in the form of y = mx + b, what information do you need? We need to know the slope and y-intercept.

DOK-2 To write an exponential function in the form, y = a(bx), what information do you need? We need to know the initial value and constant multiplier.

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 Share the following observation with the class: For the last reﬂection question, you were able to create a linear and exponential function based on the same two points. Therefore, it seems that a linear and exponential function can both pass through the same two points. DOK-2 How would the graphs of the linear and exponential functions be the same or different between the two points? The graph of the linear function would have a constant rate of change between the two points, whereas the graph of the exponential function would have a varying rate of change between the two points. • DOK-2 How would the average rate of change between the two common points for each function compare? Explain. The average rate of change would be the same for both functions given the two points since the average rate of change is not dependent on what happens between the two points but the two points themselves. • DOK-3 How can you decide whether a scenario is best represented by linear or exponential growth? As x moves farther away from zero in the positive direction, both functions get larger and larger. However, as x moves farther away from zero in the negative direction, the linear function’s end behavior approaches get more and more negative, while the exponential function approaches a horizontal asymptote. • DOK-3 How can you decide whether a scenario is best represented by linear or exponential growth? Linear functions grow by equal differences of equal intervals, and exponential functions grow by equal factors over equal intervals. Ultimately, exponential functions will exceed linear functions, so situations that grow more rapidly and by constant factors as opposed to constant rates require exponential functions. •

FACILITATION TIP As a follow up to this question, help students list some additional real-world applications that result in linear and exponential growth. Be prepared with some visual career examples. FACILITATION TIP On this Exit Ticket, some students may need read aloud support, a masking tool, or other accommodations. Consider telling some students how many statements are true. FACILITATION TIP Students may select answer choice A or C if they are not careful. Without being speciﬁc, encourage them to pay attention to all signs and relevant properties as they compare and solve for 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.

Engage

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Explain

Elaborate

Evaluate

Intervention

Acceleration

COMPARE FUNCTION TYPES

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Compare Function Types Explore 3 – Compare Linear, Exponential, and Quadratic Relationships ACTIVITY PREPARATION Students will examine and compare linear, exponential, and quadratic relationships. Students will explore the second difference of quadratic functions in tables. Students will examine when a function representing exponential growth will overtake a quadratic function. Students will be able to distinguish between these three function types from different representations.

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.7 Look for and make use of structure.

Materials

Preparation • • •

Printed • •

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

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 with students. Provide some additional examples of careers that require being able to read and analyze data. FACILITATION TIP Consider distributing the Student Journal in two sections as needed; Part I (pages 1–3) and Part II (pages 4–6). FACILITATION TIP Print and project these guiding questions (4a–4f). Consider going over them before students begin to collaborate if needed.

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. 362

2. 3. 4.

Read the following scenario to the class: Isabel was just hired as a data coordinator for the Riverside school district. She wants to review different types of models before she examines data about students’ ﬁrst languages and enrollments in local community college courses. Give a Student Journal to each student. Explain to students that they will work with their groups to represent and model data in different ways 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 can you tell if there is a linear or exponential relationship from a table? There will be a common difference between each output if the function is linear and a common ratio between each output if the function is exponential.

DOK-1 What clue lets you know what type of equation b((xx) = 2x2 + 1 is? The highest degree of the equation is 2, so it is quadratic.

DOK-1 What is a y-intercept? y It’s a point that has an x value of 0 that intersects the y-axis.

DOK-1 How would I determine the output for a given input using a table? Search the table for the given input value, and then ﬁnd the corresponding output value in the next column.

e. DOK-1 How would I determine the output for a given input if I have an equation? Substitute the given input into the equation for x and solve for y. f.

DOK-1 How would I determine the output for a given input on a graph? Find the given input value along the x-axis, and then ﬁnd the y-value on the graph that corresponds with the x-coordinate. © Accelerate Learning Inc. - All Rights Reserved

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

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

Intervention

Acceleration

DOK-2 If we saw the graph of a function and wanted to determine whether it was linear, quadratic, or exponential, what key features would we look for? End behavior: • Does the function have a horizontal asymptote? If yes, then it is exponential. • As x goes farther from zero in the negative direction and positive direction, is the end behavior the same? If yes, then it is quadratic. • As x goes farther from zero in one direction, does the function approach ∞, and as x goes farther from zero in the other direction, does the function approach −∞? If yes, then it is linear. Rate of change: • Does the function have a constant rate of change? If yes, then it is linear. • Does the rate of change become increasingly large and the function is always increasing? If yes, it is likely exponential. FACILITATION TIP • DOK-2 If we saw the equation of a function and wanted to determine whether it was linear, quadratic, or exponential, what key features would we look for? If Create an if/then graphic to record answers there is an exponent, we would look to see if it is a variable or number. If it is a to some of these Math Chat questions. variable, then it is likely an exponential function. Furthermore, if the equation is Allow students to record notes. written in the form y = mx + b, then it is linear; if it is written in a form equivalent to y = ax2 + bx + c, then it is quadratic; if it is written in the form y = abx, then it is exponential. • DOK-3 What strategy would you use to determine whether a relationship was linear, quadratic, or exponential given a table? First, I would check to see whether there is a common difference between the outputs, which would indicate a constant rate of change and a linear relationship. If I find this is not the case, then I would check to see whether there is a common ratio between output values, which would indicate an exponential relationship. If there isn’t a common ratio, then I would determine a first and then second difference to see whether there is a common difference for the first difference, which would indicate a quadratic relationship. • DOK-2 How does the table of a linear function compare to the table of a quadratic function? Both tables require the determination of differences between outputs. However, if the difference between outputs is a common difference, then there is a linear relationship. However, if the difference between the differences of outputs or if the second difference is a common difference, then the table shows a quadratic relationship. •

FACILITATION TIP

Part II 1.

2. 3. 4.

COMPARE FUNCTION TYPES

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Read the following scenario to the class: Now that Isabel has explored the growth of three function types, she wants to dive into data about students and use her knowledge to form future predictions. Students should still have their Student Journals. Explain to students that they will work with their groups to write expressions for the lengths, simplify the lengths, 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 a table represents a linear relationship, what must be true about the outputs? There is a common difference between outputs.

DOK-1 If a table represents an exponential relationship, what must be true about the outputs? There is a common ratio between outputs.

Project this scenario for students and provide some time to explore why a data analyst might want to form future predictions. List real-world reasons and examples with students. FACILITATION TIP Consider distributing the Student Journal in two sections as needed; Part I (pages 1–3) and Part II (pages 4–6). FACILITATION TIP Print and project these guiding questions for students before they begin to collaborate. 363

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Compare Function Types Explore 3 – Compare Linear, Exponential, and Quadratic Relationships DOK-1 If a table represents a quadratic relationship, what must be true about the output’s differences? There is a common difference between the output’s differences, also known as a common second difference.

DOK-2 For question 1, which of the three models displayed the fastest growth? Why do you think that is? The exponential model displayed the fastest growth because it has an ever-increasing and steeper rate of change, even more so than the quadratic model.

e. DOK-2 How can you use the common second difference of quadratic functions to determine outputs without an equation? Use the second difference to create a table of ﬁrst differences that will be linear. Use these ﬁrst differences to create successive outputs.

5. 6.

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-3 When comparing different types of functions that have different representations, what strategies did you use? I changed the functions I was comparing to the same form of representation (such as an equation, table, or graph) and then made the comparison. I found different ways to determine speciﬁc features of each function depending on the representation before making comparisons. • DOK-2 If I have two functions, f( f(x (x) = 10x2 and g(x ( ) = 3x,, as the x values get larger (x and larger, which function will have a greater value, and why? As x values get larger and larger, function g will have a greater value than function f because its rate of change increases faster and will eventually be greater than f(x)’s rate of change and exceed its values. • DOK-2 Explain how to continue a linear, quadratic, and exponential pattern when given only 3 values. The linear pattern will continue by adding or subtracting the same common difference or rate of change each time. The quadratic pattern will have a constant second difference, so a table of ﬁrst differences can be created that will help determine the next output values. The exponential pattern will have a common ratio or multiplier that we can continue using to continue the pattern. •

Post-Explore FACILITATION TIP For this Exit Ticket, consider providing tables/graphs/scratch paper for students to show their thinking.

1. 2. 3. 4.

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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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Compare Function Types 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

Average Rate of Change Introduction Independent practice assignment that gives students an opportunity to demonstrate their learning

Anchor Chart

Show What You Know, Part 2

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

Compare Linear and Exponential Representations 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 Linear, Exponential, and Quadratic Relationships Independent practice assignment that gives students an opportunity to demonstrate their learning

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

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Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

ELABORATE ELEMENTS ELEMENT USE KEY

Can be assigned digitally

Contains printable handouts

Can be done independently

Spiraled Review

Fluency Builder

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

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

COMPARE FUNCTION TYPES

Home

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Intervention and Assessment SMALL-GROUP PLANNER 1

Students who are still acquiring the concept and need remediation

How to Use the Review

Students who have mastered the concept and need extension

Students who are approaching mastery and need review

COMPARE FUNCTION TYPES

Compare Function Types

3 368

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

What prompts will be used?

What does mastery look like?

COMPARE FUNCTION TYPES

Home

I can identify key differences between linear, quadratic, and exponential functions from tables, graphs, and scenarios.

I can calculate and explain the average rate of change over intervals of various functions.

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SCOPE 1

Statistics Scope Introduction SCOPE SUMMARY

Student Expectations

A.DSR.10.1 Use statistics appropriate to the shape of the data distribution to compare and represent center (median and mean) and variability (interquartile range, standard deviation) of two or more distributions by hand and using technology. A.DSR.10.2 Interpret differences in shape, center, and variability of the distributions based on the investigation, accounting for possible effects of extreme data points (outliers).

In this grade level, students will expand their understanding of dot plots, box plots, and histograms. They will use their previous skills to represent data in a variety of ways and explore the different features each data representation possesses. Features include but are not limited to exact measures of center and spread, summary statistics, and skewness. Students will develop an understanding of the similarities, strengths, and weaknesses of each data representation in context. They will expand their understanding of absolute mean deviation to standard deviation and standard deviation of distributions. Students will develop numerical strategies and methods for determining outliers of data sets.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grade levels, students summarized and described distributions. They related the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. Students have drawn informal comparative inferences about two populations. They have gained an understanding that a set of data collected to answer a statistical question has a distribution that can be described by its center, spread, and overall shape. Students have recognized that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

In Algebra II, students will use the mean and standard deviation of a data set to ﬁt it to a normal distribution and to estimate population percentages. Students will apply the empirical rule for data distributions and explore the bell curve and make inferences with data.

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

know the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.

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: •

use statistics to analyze a realworld 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. 370

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

STATISTICS

Home

Shape of Data In this exploration, groups of students will be asked to analyze various data collected in a classroom from cell phone usage, to test results, surveys, and other research. Students will: •

represent data on number lines.

•

use data representations to analyze the shape of data.

•

determine the best measure of center and variability.

Explore 2

Explore 1

EXPLORE ACTIVITIES

In this exploration, students will be asked to analyze data from a class Trashketball tournament. Students will: •

create or use data to calculate and analyze standard deviation.

•

determine the relationship between mean and standard deviation.

•

alter data to determine the effect on mean and standard deviation.

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 Deviation

Outliers In this exploration, students will analyze test data that is on a box and whisker plot to investigate the outliers to see if the class is owed a retake. Students will: •

use the IQR 1.5 rule to calculate outliers.

•

determine the effect of outliers on measures of center and variability.

•

analyze the source of outliers and when to include outliers in data.

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

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STATISTICS

Statistics 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

STATISTICS

Home

ACCESSING PRIOR KNOWLEDGE Students will play Always, Sometimes, Never to determine whether statements or claims based on the prior standard are sometimes true, always true, or never true. This element is designed to uncover student misconceptions; it should not be taken for a grade. 7.PR.6.6 Use appropriate graphical displays and numerical summaries from data distributions with categorical or quantitative (numerical) variables as probability models to draw informal inferences about two samples or populations.

Materials Printed •

Preparation •

1 Always, Sometimes, Never (per student or per group) • •

Procedure and Facilitation Points 1. 2.

3. 4.

5. • • • • • • 6.

Distribute the Always, Sometimes, Never cards to students. Tell students you are going to read or project a series of statements to them, and they will determine whether each statement is sometimes true, always true, or never true. After having some time to think, students hold up cards with their answers, and they must be able to justify their responses. Ask students to justify their responses to an elbow partner or within their groups. Choose volunteers to explain their reasoning to the whole group. For “sometimes” statements, students should be able to explain when they are true. Ask students how they would rewrite the statements so they are always true or never true. Scenario statements and answers are provided below: The mode of a set of data is the value that occurs most frequently. A The median and mode of a data set are the same. S The mean of a data set is the average. A If the range of a data set is 10, then the median is 5 greater than the minimum. S The mean absolute deviation (MAD) of a data set is the average distance between each data value and the mean. A The range of a data set is a negative number. N If students are struggling to complete this task, do the Foundation Builder to ﬁll the gap in prior knowledge before moving on to other parts of the scope.

Identifying Misconceptions •

•

A misconception may be that the mean, median, and mode cannot ever be the same number. Have students generate their own data sets and try to create examples and counterexamples. Students often confuse the terms and their meanings. Providing contextual examples can help generate student thinking.

FACILITATION TIP Before distributing the cards, ﬁrst ask students what they remember about data distribution or statistics. Consider starting a KWL (Know, Want to know, Learned) chart. Next, let them know their knowledge will be tested through a round of Always, Sometimes, or Never. FACILITATION TIP To create a cohesive class discussion about the statements, display them one at a time on the projector.

FACILITATION TIP After they complete the APK activity (and the Foundation Builder, if needed) ask the class for scenarios where the mode would be useful. Examples may include choosing a restaurant based on reviews, planning inventory based on sales, or buying balls for gym class based on students’ favorite sports. FACILITATION TIP These Always, Sometimes, Never Cards provide a great opportunity to review statistics vocabulary. As you read the statements, create a vocabulary list with students (mode, frequent, median, data set, mean, average, range, minimum, maximum, MAD).

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Statistics Hook – Home Buying ACTIVITY PREPARATION Students will use statistics to analyze a real-world situation.

Materials

Preparation

Printed •

• • •

1 Home Buying (per class)

Reusable • •

1 Phenomena Video (per class) 1 Projector (per class)

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

PROCEDURE AND FACILITATION Part I: Pre-Explore 1. FACILITATION TIP Before showing the video and reading the scenario, ask the class 1) Have you ever bought a high price item?; 2) If so, what things did you consider before making the purchase?; 3) Did you compare prices of other similar items before purchasing the item? FACILITATION TIP Depending on your students, consider that some families may live in rented or subsidized residences. Students may have limited experience with prices of homes, new cars, or big ticket items.

3. 4.

6. 374

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 Enriquez family is considering purchasing a house at the end of Hickory Lane that has a list price of $288K or $288,000. Before buying the house, the family looked up the most recent sale price of each house on the street according to city records. They want to analyze the values of the other houses to make sure that they are getting a good deal. The family wants to know if there are any outliers that they should remove when conducting their analysis. Project Home Buying. 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 one house sold for more than $600,000. I notice that one house had a sale price of $1. I wonder how the houses and properties differ. I wonder how the prices have changed over the past few years. I wonder if determining the average or median home price would help. Explain to students that the Enriquez family wants a way to determine mathematically if any of the sale prices are outliers that should not be included in their calculations. Discuss the following questions: a.

DOK-1 Why might the family want to remove outliers for their analysis? Allow students to share all ideas. Answers will vary. The outliers would drastically change the average sale price. The outliers do not accurately represent the price of a home on Hickory Lane.

DOK-1 What statistical measures might be helpful to determine whether the listed price is reasonable? The mean, median, range, and average deviation would all be helpful values to know.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Part II: Post-Explore 1. 2.

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

DOK-1 Does this question about outliers make more sense after the Explore activities? Yes, I know more about the impact of outliers.

DOK-1 What strategies would you use to ﬁnd outliers? I would arrange the data in order from least to greatest then calculate Q1, median, and Q3. I would ﬁnd the IQR and see what values lie 1.5 times the IQR above Q3 and below Q1.

DOK-1 Where are the outliers on Hickory Lane? Why do you think their prices fell so far outside the normal range? The house with a sale price of $1 and the house with a sale price of $610K are both outliers on this street. The sale for $1 was likely to pass ownership between family members. The house that sold for $610K may have a much larger yard or more bedrooms and bathrooms than the other houses on the street.

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

STATISTICS

Home

FACILITATION TIP As a ﬁnal discussion for the Pre-Explore, have students look at the list prices for the houses. Ask them if, from a quick examination, the prices seem evenly distributed or if more prices seem closer to the highest or lowest price. FACILITATION TIP As a ﬁnal discussion for the Post-Explore, ask students if they know anyone in real estate. Do they know anything about the tasks they perform in their work? In what other ways might they use statistics in their work?

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Statistics Explore 1 – Shape of Data ACTIVITY PREPARATION Students will represent data on number lines. Students will use various data representations to analyze the shape of the data and determine the best measure of center and variability.

Standards for Mathematical Practice • • •

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

Materials

Preparation • • •

Printed • •

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

Separate the class into groups of 3 or 4 students. Print a Student Journal and an Exit Ticket for each student. For students who need additional support, please see our Mean, Median, Mode, Range Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I: Mrs. Ortega’s Class Data FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever participated in a survey?; 2) If so, what was the survey about?; 3) What was the purpose of the survey?

2. 3.

FACILITATION TIP

Some students may mix up “skewed left” and “skewed right,’ thinking that each name coincides with the concentration of data. Be sure to clear this misconception early on in the Explore activity. Encourage students to think of the name as being the opposite of the data concentration.

Read the following scenario to the class: Mrs. Ortega has collected a variety of data for her class–some from student surveys, some from student test results, and some from outside research. She is hoping you can help lead her class to make sense of it all. Give a Student Journal to each student. Explain to students that they will work with their groups to analyze Mrs. Ortega’s class data. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a. DOK-2 How does your scenario compare to a partner’s? Answers will vary.

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5. 6.

DOK-1 How can you tell whether data is skewed left or right? Data is skewed to the side of the “tail.” The majority of the data is on the opposite side of the skew.

DOK-2 Compare how to identify the shape of the data in the box plot and dot plot. In both, identifying where a majority of the data lies can help you identify the shape. In a box plot, only the median can be determined, but in a dot plot, both the mean and median can be identiﬁed.

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 causes a data set to be skewed? When more data is located to the left or right, the data is skewed. When the mean and median are similar, the data tends to be symmetrical; if one is greater than the other, the data tends to be skewed. • DOK-1 How can you identify whether data is symmetrical or skewed? If data is symmetrical, the data is evenly distributed, but if it is skewed, there is more data to the left or right. •

• •

Engage

Explore

Explain

Elaborate

Evaluate

DOK-3 Give an example of a situation that is skewed right. Situations in which younger people are involved much more often is skewed right. DOK-2 Give an example of a situation that is uniform. Situations determined by luck or pure chance are uniform.

Part II: Mrs.Ortega’s Class Research Continued 1.

Read the following scenario to the class: Mrs. Ortega decided to collect data to see how much time students spend on their phones. You will analyze this data to explore the relationships. Each classmate answered the question, “On a typical school day, how many hours do you spend on your phone outside of class?” Help Mrs. Ortega’s class analyze the data in the dot and box plots, and draw conclusions about the relationships. Explain to students that they will work with their groups to identify the shape of the data. Some of the graphs have more unique shapes, so they should describe the shapes to the best of their ability. Point out to the class that there can be multiple scenarios for a single graph, and they need to work together to think of what scenario would make the most sense to create that data shape. Students will then work together to describe the graphs, predict what could cause the shapes, and determine the best measures of center and spread. They are to 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 you calculate the mean? Find the sum of all of the data values, and then divide by how many values there are.

DOK-1 How do you calculate the median? Order the data values, and ﬁnd the middle value. If there are two middle values, ﬁnd the mean of those values.

DOK-1 Does the mean indicate the average or middle of the data? What does the median indicate? The mean is the average, and the median is the middle value.

e. DOK-2 Compare the mean and median of the histogram on the left. The mean is larger than the median. f.

DOK-2 Compare the mean and median of the histogram on the right. The mean and median are similar. There is a difference of one-tenth.

DOK-3 Describe what the shape of the data tells you about the location of the mean and median. When more data is to the right, the median is greater than the mean. When more data is to the left, the median is less than the mean. When the data is evenly distributed, the mean and the median are similar.

Intervention

Acceleration

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FACILITATION TIP After students give an example, explain that depending on the situation and number of possible outcomes, it may take numerous results before uniformity is seen in the data. Provide examples such as spinning a number wheel, rolling dice, or selecting a certain color of marble out of a bag.

FACILITATION TIP Before reading the scenario, ask the class 1) How much time do you think you spend on your phone?; 2) Do you spend more time talking or texting with people?; 3) Besides talking and texting, what else do you use your phone for?

FACILITATION TIP Delve deeper into how students learn through visual representation. Have them write their reasoning as they answer Question 3 on the Student Journal.

h. DOK-3 Describe how the data shape will affect the choice for the best representation of the measure of variability. If the data is skewed, the interquartile range (IQR) will be used because the data would not be evenly distributed. When the data is symmetrical and evenly distributed, the standard deviation would be the best representation of the measure of variability. 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.

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STATISTICS

Statistics Explore 1 – Shape of Data Math Chat • 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. FACILITATION TIP After students answer the question, note that uniform data can be viewed as a special type of symmetrical data. They should see that uniform data can also be split into two halves that more or less mirror each other. Ask the class what is the difference between bell-shaped/ symmetrical data and uniform data. They should know that one has a distinct peak while the other does not.

FACILITATION TIP On this Exit Ticket, consider encouraging students to make observations about the data before looking at the questions. Sometimes it’s good practice to examine the images and absorb the information before tackling the assessment questions.

•

DOK-3 Explain the differences between the box and dot plots for finding the shape of the data. In a box plot, identifying where the data is located in the box and the location of the median can help identify the shape of the data. In a dot plot, identifying both where the data is located and the mean and median can help identify the shape of the data. DOK-2 Describe when you’d rather have a box plot, dot plot, or histogram for your data display. All three allow you to view how the data is spread and make predictions related to the mean and median. A dot plot and histogram have the advantage of displaying all data, so mean and median can be calculated. DOK-2 Mrs. Ortega’s class data had a very similar mean and median. How did this affect your choice of the best measure of center? I saw that the data was not strongly skewed because the mean and median were close. I still chose to use the median because the data was skewed, and that is the usual choice when data is skewed. DOK-2 Explain how the mean and median, which are the measures of center, affect the shape of data. When more data is to the right, the median is greater than the mean. When more data is to the left, the median is less than the mean. When the data is evenly distributed, the mean and the median are similar. DOK-3 Based on your experience, predict what a bimodal distribution would look like when graphed as a dot plot or histogram. What does the prefix of bimodal distribution tell you? Bi- means “two,” like in bicycle, so I think there will be two peaks. DOK-3 Reflecting on your knowledge of data shapes, what do you predict a uniform distribution would look like when graphed as a dot plot or histogram? Think of what the word uniform means to help. In sports, a uniform is the same for the entire team. So I think the shape will be the same amount for each data value.

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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Statistics Explore 2 – Standard Deviation ACTIVITY PREPARATION Students will create or use data to calculate and analyze standard deviation. Students will determine the relationship between mean and standard deviation. Students will alter data to determine the effect on mean and standard deviation.

Standards for Mathematical Practice • • •

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

Materials

Preparation

Printed • • • • •

1 Student Journal (per student) 1 or 2 Sets of Trashketball Tourney Player Data Cards (per group) 1 Set of Trashketball Tryouts Shooting Instructions (per group) 1 Set of Trashketball Tryouts Standard Deviation Calculation Instructions (per group) 1 Exit Ticket (per student)

• • • • • •

Reusable •

1 Small trash can or another receptacle (per group)

•

Consumable • •

•

9 Sheets of scrap paper (per group) 1 Roll of masking tape (per class)

Separate students into groups of 4 to 8 to complete this activity. Print a Student Journal and an Exit Ticket for each student. Print 1 or 2 sets of Trashketball Tourney Player Data Cards, on card stock for durability, for each group, depending on group size. Print a set of Trashketball Tryouts Shooting Instructions, on card stock for durability, for each group of students. Print a set of Trashketball Tryouts Standard Deviation Calculation Instructions, on card stock for durability, for each group of students. Create a trashketball shooting station for each group. Place a trash can or other receptacle at each station, with 9 balled-up pieces of scrap paper. Use masking tape to mark off a shooting location. Place the Trashketball Tryouts Shooting Instructions and the Trashketball Tryouts Standard Deviation Calculation Instructions at each station. Be prepared to write each group’s data somewhere the students can see it. For students who need additional support, please see our Mean, Median, Mode, Range Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever participated in a tournament?; 2) If so, what was your experience?; 3) Did the tournament last one day or did it continue for several days? FACILITATION TIP Make sure students can interpret what the cards are saying. For each card, they should see each dot as one of the nine days and the number line as representing the number of shots made. FACILITATION TIP Depending on your students’ ﬂuency with statistics vocabulary, consider creating a list with deﬁnitions before they begin collaborating. Encourage students to use the accurate mathematical language as they work together. 380

Part I 1.

2. 3. 4. 5.

Read the following scenario to the class: Mr. Williams’s class has decided to have a trashketball tournament. Each player gets 6 shots each day over the course of 9 days. The results for four classmates are shown on your Trashketball Tourney Player Data Cards. Give a Student Journal to each student. Give the Trashketball Tourney Player Data Cards to each group. Explain to students that they will work with their groups to analyze the data plots, mean, and standard deviation 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 measure of center is another name for average? Mean is another term for average.

DOK-1 What statistical measures have you used to look at data previously? Possible answers include the following: mean, median, range, mode, interquartile range (IQR), mean absolute deviation (MAD), and standard deviation (SD). © Accelerate Learning Inc. - All Rights Reserved

6. 7.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 Why would ﬁnding the mean, median, range, and mode not paint a complete picture of each shooter’s results? These measures would not give us a picture of how spread out the data is throughout or how erratic a player is at shooting.

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

Acceleration

FACILITATION TIP

STATISTICS

Home

As students mention the measures of range, mode, IQR, and MAD, ask them what each measure represents. Then, have them explain the value of each measure for statistics as best as they can remember. Intervene in their explanations as necessary.

Math Chat •

•

DOK-2 Would you expect a high or low standard deviation if the data was very close together and close to the center of the distribution? Explain. I would expect a low standard deviation because the closer the data was to the mean, the lower the standard deviation was. DOK-2 Would you expect a high or low standard deviation if the data was spread apart and not very close to the center of the distribution? Explain. I would expect a high standard deviation because Amari had data that was 4 away and not just 2 away from the mean, and his standard deviation was the highest. DOK-2 Would you expect a high or low standard deviation if consistency was being examined? Explain. I would expect a low standard deviation because Reyansh had a standard deviation of 0 when all of his data was the same. That would make him the most consistent, so the closeness to 0 would mean the data is almost identical. DOK-2 What would happen to Amari’s standard deviation if the outlier were removed? Would it increase or decrease? I would expect Amari’s standard deviation to decrease because the data would be closer to the mean. It seemed that the closer the data was to the mean, the smaller the standard deviation. DOK-2 What would happen to Amari’s data if there were two outliers? I would expect the standard deviation to increase because the data is further spread out and not as close to the mean.

Part II 1.

2. 3.

6. 7.

Read the following scenario to the class: Trashketball tryouts are today! Everybody in your group is going to take 9 shots. Record your data, create your own shooting results, and then ﬁll out your group data to complete the calculations. Give each group a name (Group A, for example). Set up a trashketball shooting station around the room for each group. Have students follow the Trashketball Tryouts Shooting Instructions posted to complete Part II. Set a timer, and instruct students to each shoot 9 times with their groups. If a student hasn’t completed their data when the time expires, assign them a number of makes to use with their groups. Students will then work together to gather data, calculate the mean and standard deviation, and record their work on their Student Journals. Groups should reference the Trashketball Tryouts Standard Deviation Calculation Instructions to ﬁnd the standard deviation of their own data. Each group will have one mean and one standard deviation value. Have each group share their mean and standard deviation with the rest of the class to complete the awards on their Student Journals. Write each group’s data somewhere the students can see and compare the data. As students collaborate, monitor their work and use the following guiding questions to assess student understanding: a.

FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever had to try out for a sports team, contest, band, etc?; 2) If so, what was your experience?; 3) Were you successful?

FACILITATION TIP Students may be overwhelmed by the process of calculating the standard deviation. It may help to set up an example problem and work through it with the class before they begin Part II on their Student Journal.

DOK-1 What does it show you about data if the standard deviation is low? What does it show you when it’s high? A low standard deviation indicates data close to the mean, and a high standard deviation indicates data farther from the mean.

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Statistics Explore 2 – Standard Deviation

FACILITATION TIP After students answer the question, ask the class if anyone mentioned that it is possible to get a 0 if all of the data is the same. If so, have them brieﬂy explain. If not, discuss why the statement is true with the class. Then, ask them what kind of shape this kind of data produces. They should know the shape would be uniform. FACILITATION TIP Some students may have left a negative sign incorrectly somewhere in their standard deviation calculations. Make sure group members agree on their standard deviation calculations, and check the work of one student from each group before they tackle this question. If a group’s members disagree, intervene as necessary. FACILITATION TIP When using this Exit Ticket, determine if you will require students to show their thinking or calculations.

9. 10.

DOK-3 Did your standard deviation surprise you based on the results of your group’s shooting? Answers will vary.

DOK-2 Describe what stands out to you about standard deviation that could be award-worthy. Answers will vary. It is possible to get a 0 if all of the data is the same.

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 When calculating standard deviation, why do you think we start by subtracting each data point from the mean? Since standard deviation measures how spread out the data is, we start by ﬁnding how far away it is from the center. • DOK-3 Based on your calculations for standard deviation, why does it always yield a nonnegative value? We square all of the differences between the data and the mean, so the values will never be negative. • DOK-2 What information can we learn from standard deviation that we cannot from the mean? Standard deviation tells us about the spread of the data, while mean only gives us a measure of center. •

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.

382

Engage

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Explain

Elaborate

Evaluate

Intervention

Acceleration

Notes

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Home

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Statistics Explore 3 – Outliers ACTIVITY PREPARATION Students will use the IQR · 1.5 rule to calculate outliers. Students will determine the effect of outliers on measures of center and variability. Students will analyze the source of outliers and when to include outliers in data.

Standards for Mathematical Practice • • •

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

Materials

Preparation

Printed • •

• • •

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

Separate the class into groups of 3 or 4 students. Print a Student Journal and Exit Ticket for each student. For students who need additional support, please see our Mean, Median, Mode, Range Supplemental Aids elements in the Intervention section.

PROCEDURE AND FACILITATION Part I: Understanding Outliers FACILITATION TIP Before reading the scenario, ask the class 1) Where have you seen something that is related to other things yet very different from them in some way?; 2) What was different about it?; 3) What might be the reason it was so different in that way?

2. 3. 4.

FACILITATION TIP After students answer the question, ask for real-life scenarios where it would be likely to see an outlier. Examples may include measurements for certain scientiﬁc experiments, a smaller tip at a restaurant, or a student who is noticeably shorter than everyone in their class.

FACILITATION TIP Check students’ understanding of medians further. After they answer the question, ask the class when they would have two middle values. 384

Read the following scenario to the class: Mr. Thomas was bragging to his class about how well everyone did on their history test. He displayed a box and whisker plot to his class to show them how well they did. A few students even got a perfect 100%! Ishaan is skeptical when he sees the box plot and believes there are several outliers. He thinks the class is probably owed a retake according to the school handbook’s retake policy. Help Ishaan and Mr. Thomas investigate the outliers to see if the class is owed a retake. Give a Student Journal to each student. Explain to students that they will work with their groups to determine what makes an outlier 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 is an outlier? An extreme data value that signiﬁcantly impacts the data

DOK-1 Comparing Mr. Thomas’s and Ishaan’s graphs, what is the disadvantage to using a box plot to identify outliers? Mr. Thomas’s graph doesn’t identify the outlier, but Ishaan’s does. Without the specific data points, it is more difficult to be sure all of the outliers are identified.

DOK-2 Explain why calculating the outlier is more accurate than analyzing a plot alone. Calculating an outlier gives the exact outliers; analyzing a plot results in opinions of what would be an outlier.

DOK-1 Why would outliers affect the shape of the box plot? The outlier is an extreme value, which means the whisker length will be affected.

e. DOK-2 Describe how you ﬁnd the median and why an outlier may not cause that value to change. When data is ordered least to greatest, the middle value or average of two middle values is the median. An outlier may not cause the value to change because the median is found by the position of the numbers and not calculated using the value of the outlier. © Accelerate Learning Inc. - All Rights Reserved

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

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-3 Consider how you calculated the outliers. Justify why the interquartile range (IQR) is multiplied by that value. Multiplying the IQR by 1.5 allows the whiskers to have similar lengths when grouping the data. This will not greatly affect the box’s values but will help identify the extreme values and shorten the whiskers. • DOK-1 How did the different box plots increase your understanding of the effect of outliers? I found it helpful to see that the whiskers changed lengths but the spread and median were not greatly affected. • DOK-2 Construct an argument explaining whether or not all data will have outliers. Not all data will have extreme data. Even if data appears to have an extreme, if the value is within 1.5 times the value of the IQR, it will not be an outlier. • DOK-1 Reﬂect on the changes that occurred in the data and representations. What parts of data analysis will outliers affect the most? Outliers affect the shape and spread. I believe they will affect the mean and standard deviation, but the median and IQR will not be affected. •

Intervention

Acceleration

STATISTICS

Home

Part II: The Impact of Outliers 1.

2. 3.

4. 5.

Read the following scenario to the class: Ishaan’s understanding of the school’s policy on retakes is based on outliers because the extreme data values signiﬁcantly impact the class’s data. Ishaan and Mr. Thomas want to investigate to learn what speciﬁc impact outliers have on the data. Explain to students that they will work with their groups to determine how outliers affect data analysis. Point out to the class that they need to consider how the data analysis is calculated as they work through the effect of the outliers. For example, the median is calculated by putting the data values in order and calculating the middle value. Students will work together to compare graphs determining the effect of outliers 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 Consider how mean and median are calculated. Why is the mean more affected than the median by outliers? Mean uses the data values to calculate the average, meaning it will be affected by extremes. Median uses the position of numbers; this means it is less likely to be affected by extremes.

DOK-2 Consider how IQR and standard deviation are calculated. Why is IQR not as affected as standard deviation? IQR is calculated using positions of the data, whereas standard deviation uses all values when it is calculated. IQR is based on the median, while standard deviation is based on the mean, which is more impacted by outliers.

DOK-2 Reﬂect on how you calculate the standard deviation. Would including an outlier increase or decrease the standard deviation? A single extreme value could increase the standard deviation and misrepresent the data.

DOK-1 Why would removing an outlier decrease the standard deviation? Removing an outlier would decrease the variation and distance from the center, which would decrease the standard deviation.

FACILITATION TIP Before reading the scenario, ask the class 1) Is there a test retake policy at our school?; 2) If so, what does the policy say?; 3) If not, do you think there should be test retake policy? Why or why not? FACILITATION TIP After students answer Question 1. have them look again at Period 2. Ask them how the measures of center for Period 2 would compare with those of Period 1 if three dots were added to the right of “65.” They should know that the median would be the same as in Period 1 and the mean would increase from its previous value for Period 2. FACILITATION TIP After they answer Question 2. ask them, “How is it that the median did not change when values were only added to one side?” They should note that there are three instances of the value “55.” They should then see that the median was represented by the second instance but is now represented by the third instance.

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Statistics Explore 3 – Outliers 6. 7. FACILITATION TIP Test students’ ability to interpret box plots further. After they answer Question 5. ask the class how the IQR and standard deviation would compare to Version A if Mrs. Paola included 95 and 100 and excluded 10.

Math Chat •

•

• 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.

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.

•

DOK-1 Why is the median chosen over the mean if there is an outlier? The outlier affects the mean more than the median because the value is used in the calculation. DOK-1 What specifically happens to the mean if there is a high outlier or a low outlier? If there is a high outlier, the mean will increase; if there is a low outlier, it will decrease. DOK-2 Is it fair to say outliers do not affect the IQR? Explain. It is not fair to say that the IQR is not affected. While IQR may not change significantly as standard deviation does, outliers can cause the IQR to be slightly increased. DOK-2 You are given a graph only and not told specifically about outliers. How would a skewed or approximately symmetric data shape help you decide on the choice of mean or median as the best measure of center? If it was approximately symmetric, the mean would be chosen over the median because it uses all of the data values. If it was skewed, the median would be chosen over the mean because the data to the left or right of the skewed values has a more significant impact on calculating the mean instead of the median. DOK-1 What effect would removing an outlier have on standard deviation and the analysis you would make of the data? Removing an outlier could decrease the standard deviation. This would give me a better idea of what is more typical for the data.

Part III: Analyzing the Source and Inclusion of Outliers FACILITATION TIP Before reading the scenario, ask the class 1) Is anyone on a sports team?; 2) If so, what is your role on the team?; 3) What statistics might the coaches keep about your performance? FACILITATION TIP Students may ﬁnd it challenging to ﬁll in the middle column for Question 2. After students complete what they can in the table, discuss possible unclear scenarios as a class. Provide unclear scenarios if students are stumped. Examples include, “He missed some games due to unforeseen circumstances” and “He joined the team halfway through the season.”

2. 3. 4. 5. 6.

7. 8.

386

Read the following scenario to the class: Word of Ishaan’s knowledge of data and outliers has spread throughout the school. Ishaan’s football coach hears of his expertise and asks him for his help. The coach has the data from the defensive players’ tackles they have made this season. He gives Ishaan the data to have him decide what he should use to analyze his players’ progress. Students should still have their Student Journals. Explain to students that they will work with their groups to determine where outliers come from and if they should be used to analyze the data. Point out to the class that they need to reﬂect not only on what data is affected by outliers but also where outliers come from. Students will then work together to compare graphs determining the effect of outliers 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 do you calculate an outlier? First, calculate the IQR by subtracting Q3 and Q1. Then, multiply the IQR by 1.5. Take the product and add it to the Q3, and take the product and subtract it from the Q1. Any values that are above or below the sum and difference are outliers.

DOK-1 Consider how statistics are collected. Is the data recorded always authentic? Explain. No, there can be errors in the collection or recording.

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.

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Math Chat DOK-1 When should an outlier be excluded? It should be excluded when the outlier is from an error or isn’t authentic to the data. • DOK-1 When should an outlier be included? It should be included when removing the outlier would cause the data not to be authentic any longer. • DOK-2 Explain why the source of an outlier needs to be considered. If an outlier is an error, the data will not be authentic and the analysis will be affected. Removing an authentic outlier causes data to be false, and the analysis will be false. • DOK-3 Create a scenario when you would exclude an outlier. Nahla picks cards from a deck and records what she picked. Her recorded data includes 82. There is not a card that is an 82 in a deck of cards, so that is an error that should be excluded.

STATISTICS

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FACILITATION TIP After students explain, ask them to describe a time when they recorded data and found outliers. Did they take any measures beforehand to make sure outliers were authentic? If so, what measures?

Post-Explore 1. 2. 3. 4.

FACILITATION TIP After students complete the Exit Ticket, ask for real-life scenarios where it would be desirable to note an outlier. Examples may include a record-scoring output in a sport, a larger tip at a restaurant, or a record-low temperature.

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Statistics 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

Shape of 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

Standard Deviation 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

Outliers 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

388

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Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

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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

Statistics

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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Statistics 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

3 390

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 determine and draw conclusions from the mean, median, and interquartile range of a data set with and without technology.

I can explain standard deviation and its use in solving problems.

I can identify outliers and their impact on a data set.

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SCOPE 1

Model Data Scope Introduction SCOPE SUMMARY

Student Expectations

In this grade level, students will expand their understanding of linear models to explain the slope and intercepts in terms of the context being modeled. Students should be able to use technology and tools to represent and model data, make inferences, test conjectures, explore correlation, and create lines of best ﬁt supported by their respective correlation coeﬃcients. They should be able to describe how variables are related in the context of the situation (as the x variable increases, the y variable decreases, or as the x variable increases, the y variable increases, for example). Students should be able to describe associations with reference to direction and strength. They should be able to analyze different residual plots for different predictive models and recognize which points are overestimates and underestimates. Students should be able to use the context of the situation to determine if their model has potential for extrapolation or predicting the future.

A.DSR.10.3 Represent data on two quantitative variables on a scatter plot and describe how the variables are related. A.DSR.10.4 Interpret the slope (predicted rate of change) and the intercept (constant term) of a linear model based on the investigation of the data. A.DSR.10.5 Calculate the line of best fit and interpret the correlation coefficient, r, of a linear fit using technology. Use r to describe the strength of the goodness of fit of the regression. Use the linear function to make predictions and assess how reasonable the prediction is in context.

VERTICAL ALIGNMENT Background Knowledge

Future Expectations

In previous grade levels, students calculate and interpret slope and the yy-intercept. -intercept. Students construct a function to model a linear relationship between two quantities. Students informally ﬁt a straight line and informally assess the model ﬁt by judging the closeness of the data points to the line. Students construct and interpret scatterplots for bivariate measurement data to investigate patterns of association between two quantities, describing patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

In Algebra II, students will predict and make judgements based on regression information calculated from linear, quadratic, and exponential data, as well as model data with new functions they learn. Students will use the skills learned in Algebra I and apply them to explore the concept of statistical signiﬁcance.

A.DSR.10.6 Decide which type of function is most appropriate by observing graphed data.

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

understand straight lines are used to model relationships between two quantitative variables.

•

understand scatter plots that suggest a linear association, informally ﬁt a straight line, and informally assess the model ﬁt by judging the closeness of the data points to the line.

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

Hook

Accessing Prior Knowledge

A.DSR.10.7 Distinguish between correlation and causation.

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 a set of data.

•

calculate linear regression statistics.

•

use information to make predictions about the variables.

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. 392

Engage

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Explain

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Acceleration

Correlation and Causation In this exploration, groups of students will be asked to analyze information in newspapers to determine if each headline is an accurate representation. Students will: •

Explore 2

Explore 1

EXPLORE ACTIVITIES

determine whether correlation and causation are present.

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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Linear Regression In this exploration, students will be asked to evaluate newspaper cartoons that are drawn accurately based on real human body proportions. Students will also measure and collect data to determine linear models that predict body measurements of cartoon drawings.Students will: •

write an equation representing the line of best ﬁt for data following a linear trend represented in a table or scatterplot.

•

use linear functions to estimate solutions and make predictions.

•

determine linear models.

Residuals In this exploration, students will analyze sales report data for different items in the school store to determine the amount of inventory to order. 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.

plot and analyze residuals.

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

Quadratic and Exponential Regression In this ﬁnal exploration, students determine a function type that can model the change in population of species at a national park over time. Students will also be tasked with estimating solutions and making predictions for future populations. Students will: •

write an equation representing the best ﬁt for data following either a quadratic or exponential trend represented in a table or scatterplot.

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

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Model Data Lesson Calendar and Accessing Prior Knowledge PLAN YOUR LESSONS Monday

Tuesday

Wednesday

Thursday

Friday

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ACCESSING PRIOR KNOWLEDGE

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Students will identify two truths and a lie by reading statements about the prior standard. This element is designed to uncover student misconceptions; it should not be taken for a grade. 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.6.3 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.

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

FACILITATION TIP

The second statement is incorrect because based on the scatterplot, there is not a majority of students who slept 5 hours. You must pay attention to the axes’ labels to help make sense of the data.

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.

To prevent students being overwhelmed and jumping to conclusions, consider projecting only the graph and allowing students to make observations using think, pair, share. Next, project the three statements. FACILITATION TIP Students may be overwhelmed initially by numerous points. Encourage them to pay attention to the axes’ labels and remember how coordinate pairs work to interpret the graph. FACILITATION TIP After the activity, ask students if they know about many hours of sleep they get per night. Do they get less sleep when they spend more time on the phone than usual? If so, do they think it affects their studies?

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Model Data Hook – Baseball Scatterplot ACTIVITY PREPARATION Students will analyze a set of data, calculate linear regression statistics, and use that information to make predictions about the variables.

Materials

Preparation

Printed •

Reusable • •

Plan to show the video. Prepare to project Baseball 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.

• • •

1 Baseball Scatterplot (per class)

1 Phenomena Video (per class) 1 Projector (per class)

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

Before showing the video or reading the scenario, ask the class 1) Is anyone on an organized sports team?; 2) If so, what is your role on the team?; 3) What data is kept about your performance? FACILITATION TIP Project the text of this scenario and have student volunteers read it aloud. Consider that some students may be aware of a related book or movie very similar to this scenario.

3. 4.

Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. Read the following scenario while showing the video behind you: A famous baseball manager used data analysis to choose and trade baseball players with the goal of creating a high-performing team out of the baseball organization for which he worked. He analyzed player statistics and chose underrated players strictly based on their data. Sometimes, this meant releasing the high-profile, highsalaried players in exchange for a larger quantity of younger and less-expensive players. Despite his low budget, he managed to create a high-performing team, and his methods of data analysis were copied by many professional teams across a variety of sports. Project Baseball Scatterplot. Explain to students that this is an example of the type of data that the manager would analyze to predict the team’s future ability to score runs. This scatterplot shows the relationship between a player’s on-base percentage and runs scored during his first five seasons. 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 know batting averages are calculated using math, but I wonder what other statistics are useful. I wonder what statistics the manager used to make his predictions. Complete the Explore activities. Notes

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Part II: Post-Explore 1. 2.

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

DOK-1 How can we use this data to make predictions about the future of this player? I can draw a line of best ﬁt, calculate a linear regression equation, and use that equation to predict the number of runs scored for any on-base percentage.

DOK-1 What type of function would best model this relationship? I think a linear function would work best.

DOK-1 How can you ﬁnd the linear regression equation by hand? I can sketch the line of best ﬁt and then identify the slope and y-intercept of that line.

DOK-1 Do you feel that you have a strong understanding of modeling data and using regression equations to make predictions? Answers will vary based on students’ success during the activity and conﬁdence level.

Intervention

Acceleration

FACILITATION TIP

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As a ﬁnal discussion for the Post-Explore, ask students for scenarios where data modeling could be useful for a teacher. Would correlation and/or causation ever come into play? Which kind(s) of regression equations would be helpful, if any?

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Model Data Explore 1 – Correlation and Causation ACTIVITY PREPARATION Students will analyze data to distinguish between correlation and causation.

Standards for Mathematical Practice • • •

MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

• • •

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

Reusable •

•

1 Resealable bag (per group)

Separate the class into groups of 2 or 3 students. Print a Student Journal for each student. Print and cut out a set of Proposal Cards for each group of students. Place each set of cards into 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.

PROCEDURE AND FACILITATION FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever read a newspaper, magazine, or online article?; 2) If so, why did you choose to read that article?; 3) Did the article’s title or headline accurately represent what the article was about? Why or why not? FACILITATION TIP Emphasize that causation means that if the entire ﬁrst part holds true, the entire second part will always hold true. Encourage students struggling with Questions 1 and 2 of Part I to pay attention to the presence or absence of the words “could be” in the headlines. Would those words suggest causation in context?

Part I 1.

2. 3.

FACILITATION TIP After students answer the question, ask them if it is possible to have a speciﬁc scenario where ice cream sales would boost sunglasses business. Then, present the case where a boardwalk vendor sells a pair of sunglasses at half price when someone buys a gallon of ice cream. Explain that, without this explicit description, the second headline is inaccurate. 398

5. 6.

Read the following scenario to the class: You are training to become the new editor of The Daily Times, your school newspaper. As the new editor, you are committed to only printing headlines that accurately represent information. Analyze the information, and determine if each headline is an accurate representation. Give a Student Journal to each student. Explain to students that they will work with their groups to analyze each scatterplot and corresponding headline to determine whether the headline is an accurate representation of the information. Students will record their answers and explanations on their Student Journals. Monitor students as they collaborate on their work. Use the following guiding questions to assess student understanding: a.

DOK-1 What is meant by the term positive correlation? correlation? The term positive correlation means that as one variable increases, the other variable also increases.

DOK-1 What is meant by the term negative correlation? The term negative correlation means that as one variable increases, the other variable decreases.

DOK-2 What are some reasons the second headline does not accurately represent the data? Both variables could be increasing because of warmer temperatures. When it’s warmer outside, more people may buy sunglasses and more people may want ice cream.

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

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

Math Chat DOK-2 What are some reasons for a positive correlation between ice cream sales and sunglasses sales? Warmer temperatures could cause both variables to increase. The ice cream store and sunglasses store could be located close to one another, prompting the same shoppers to visit both stores. • DOK-2 Mathematicians call these reasons lurking variables.. Why do you think that term is used? Lurking means “hidden,” so these are variables that could be hidden in the story or information that affects the relationship between the two main variables. • DOK-1 Complete this statement using the information from question 1: If more caffeine is consumed, then ______. If more caffeine is consumed, then fewer hours are spent sleeping. • DOK-1 Is this if-then statement accurate? Why or why not? Yes, this statement is accurate because consuming more caffeine causes a person to sleep less. •

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FACILITATION TIP Students don’t necessarily use the term “lurking” regularly in conversation. Similarly, they may not immediately understand how the word would ﬁt in this context. Before they try to answer the question, it may help to ask them what lurking means.

Read the following to the class: Mathematicians would say that this relationship represents causation, where one event is affected by the other event. We are going to continue exploring causation in Part II. Part II 1.

2. 3.

Read the following scenario to the class: As the editor, you must approve the data and proposed headlines that writers submit to you. Remember, you are committed to only printing headlines that accurately represent the information. Analyze each Proposal Card to determine whether correlation and causation are present, and use that information to help you determine whether or not to approve the proposed headline. Give a set of Proposal Cards to each group. Explain to students that they will work with their groups to analyze each Proposal Card to determine if there is correlation between the variables. Then, students will use each headline to write if-then statements to help determine causation. Give groups one minute to analyze and discuss proposal A. Ask students the following questions: a.

DOK-1 What are the variables in proposal A? The variables are the number of bus stops and the time it takes to complete the route.

DOK-1 Is there a correlation between these variables? If so, what type? Yes, the two variables represent a positive correlation. As the number of bus stops increases, the time it takes to complete the route also increases.

Instruct students that the if-then statement is created by writing, “If the ﬁrst event happens, then the second event happens.” Read the if-then statement from the Student Journal for proposal A.

DOK-1 Does the “if” variable cause the “then” variable to happen? Yes, an increase in the number of bus stops will directly cause the time to complete the route to increase.

e. Remind students that when one event is affected by the other event, that relationship is causal. 5. 6.

Encourage students to collaborate with their groups to complete the rest of Part II 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.

FACILITATION TIP Before reading the scenario, ask the class 1) When you complete a writing assignment, what type of feedback do you get?; 2) Who gives you the feedback?; 3) What happens if you haven’t accurately represented the facts or data? STEMscopes Tip The Planner, accessed along the menu bar, provides a calendar planning tool for teachers. Download, print, save, or share your plans. Use the Elements tab on the left to access grade-level scopes and virtual-learning options with embedded links to all scope elements. Drag the elements you want to implement into the calendar, and click on each element to enter element details and personal planning notes.

FACILITATION TIP Correlate the “if” and “then” variables with function variables. After students answer the question, ask them which variable would be the independent variable in a function and which would be the dependent variable.

DOK-1 How can you determine whether the variables are positively correlated or negatively correlated? If both variables are increasing, then they are positively correlated. If one variable increases while the other decreases, then they are negatively correlated.

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Model Data Explore 1 – Correlation and Causation b.

7. 8.

DOK-2 What are some lurking variables involved in proposal B? Student responses may vary. Maybe the higher-priced items are located at more popular stadiums. More popular stadiums can charge more money because they have more customers.

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-3 Compare and contrast correlation and causation. They both analyze the relationships between two variables. Correlation means that there is a positive or negative relationship between the two variables. Causation means that one variable directly affects the other variable. • DOK-1 Does correlation imply causation? Why or why not? No. Two variables can be correlated, but not causal. •

FACILITATION TIP For this Exit Ticket, consider having students record their reasoning for their multiple-choice selection.

Post-Explore

FACILITATION TIP

After the Exit Ticket, ask students for examples of correlation in their current life. Do they have enough information to determine if their correlations are causal?

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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Model Data Explore 2 – Linear Regression ACTIVITY PREPARATION Students will write an equation representing the line of best fit for data represented in a table or scatterplot. Students will determine the line of best fit manually and technologically. Students will use these linear functions to estimate solutions and make predictions for real-world problems. Students will calculate the correlation coefficient, rr,, to determine the strength of the linear fit for the data.

Standards for Mathematical Practice • • •

MP.3 Use appropriate tools strategically. MP.4 Model with mathematics. MP.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • • •

1 Student Journal (per student) 1 Set of Data Cards (per group) 1 Set of Data Collection Tables (optional, per group) 1 Exit Ticket (per 2 students)

Reusable • • • • •

1 Quart-size resealable bag (per group) 1 Measuring tape (per group) 1 Ruler (per student) 1 Graphing calculator (per student) 1 Projector or document camera (per class)

• • •

•

• • •

Separate the class into groups of 2 or 3 students. Print a Student Journal for each student. Print and cut out a set of Data Cards for each group of students. Place each set into a quart-size resealable bag. If desired, print the Data Cards on card stock, and laminate them for future use. Be prepared to project the Data Collection Tables on the front board. Make sure the surface you use to project can be written on. Optionally, you can print one for each group. Gather a measuring tape for each group. Gather a graphing calculator and a ruler 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

Before reading the scenario, ask the class 1) Has anyone ever drawn a cartoon character?; 2) If so, what character did you draw?; 3) What did it look like?

2. 3. 4. FACILITATION TIP For Question 3 on the Student Journal, some students may come up with a linear question of y = 5x. Before they start working on the question, encourage the class to pay attention to the axis scale for each axis. 402

Read the following scenario to the class: In the previous Explore activity, you analyzed proposed headlines for the school newspaper, The Daily Times, to determine if each headline was an accurate representation of the provided information. You are now going to analyze the work submitted for the comics or advertisements sections of the newspaper. You want to make sure the cartoons submitted are drawn accurately based on real human body proportions. You will measure and collect data about yourselves and use that data to determine linear models that predict body measurements of cartoon drawings. Give a Student Journal, ruler, and calculator to each student. Distribute measuring tape to each group. Project the Data Collection Tables on the front board. Optionally, you can print one for each group. Explain to students that they will work with their groups to measure their height and wingspan using a tape measure. Model how to measure the wingspan by raising your arms parallel to the ground at shoulder height, making a letter T. Measure from ﬁngertip to ﬁngertip. Students should send one person from each group to record their group’s measurements on the projected class Data Collection Tables. Encourage students to work with their groups to complete Part I of their Student Journals. © Accelerate Learning Inc. - All Rights Reserved

8. 9.

Engage

Explore

Explain

Elaborate

Evaluate

• •

•

DOK-1 Does the data appear to be linear, quadratic, or exponential? The data appears to be linear.

DOK-1 Describe the strategy used to write the equation of the straight line. Answers will vary. I used a ruler to estimate a line that ﬁts most of the data points. Then, I calculated the slope and y-intercept of the line. Note: The line should be approximately y = x since arm span and height are the same on average.

DOK-2 If you look for patterns in the data, what do you notice about the wingspan when the height increases? The wingspan increases as the height increases. It appears that for each height increase of one inch, the wingspan also increases by one inch.

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

DOK-1 How well does this line fit our data? (How close are the data points?) The line should fit the data points very well. There may be outliers in the class. Possible outliers can be caused by differences in growth rates of teenagers, measurement errors, differences in human anatomy, etc. The overall pattern should be very close to y = x. DOK-1 Describe the correlation of the data. There is a strong positive correlation in the data. DOK-2 The line of best fit gives the yy-intercept -intercept of our model. Is this reasonable in the context of height and wingspan? The y-intercept is (0, 0.25), but that doesn’t make sense; if someone has a height of zero, they can’t have a wingspan because they don’t exist. DOK-2 If we know a person’s height, how can we predict their wingspan, or vice versa? Because the line that models this data is y = x, the height should equal the wingspan. DOK-2 How can we use this information to make accurate cartoons? If a data set has a strong linear correlation, this means the line fits the data very closely. We can calculate a line that fits the data and extend the line to humans that are larger or smaller than the sample size. In the case of drawing cartoons, we can substitute the height of the cartoon as the x value of the equation, and the output, or y value, is the wingspan. This tells the artist how long to draw the arms so the cartoon looks accurate based on real human ratios.

Explain the following to the class: Mathematicians would say that this straight line that models a linear trend of data is called the “line of best ﬁt.” We are going to continue exploring the line of best ﬁt and correlation in Part II. 10.

Acceleration

Monitor students as they collaborate on their work. Use the following guiding questions to assess student understanding:

Math Chat •

Intervention

Read the following prompt and step a to the class: You can also use technology to calculate a line of best ﬁt for data. I will demonstrate how to use technology using the ﬁrst 5 data points from our class data. a.

In Desmos, you can add a table using the + button in the top left. You can also copy and paste any data in from a table. To create a linear model for your data, look at the headers of your table, such as x1 and y1. To create a line of best ﬁt, type in a new row: y1 ~ mx1 + b. Desmos will tell you the slope and y-intercept for this line. More detailed instructions can be found at the link in footnote I at the bottom of this page.

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FACILITATION TIP Check students’ understanding of the line of best fit in the context of correlation and causation. Ask them if the line of best fit is causal. Is any line causal, in fact? What about a function of any other degree? How do you know? FACILITATION TIP After reading the prompt and providing the demonstration, ask students for real-life scenarios where it would be especially helpful to use technology to find the line of best fit. Explain to the class that, technology is especially helpful when there is a large amount of data.

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Model Data Explore 2 – Linear Regression Part II 1.

FACILITATION TIP Students may not have heard of the term correlation coefficient before this scope. Be sure to review the definition of the term with the Picture Vocabulary. Make sure to explain how to find both the line of best fit and the correlation coefficient before groups begin Part II.

2. 3.

FACILITATION TIP Some students may not notice that neither column in Data A increases in value all the way down. Encourage students to observe consecutive ordered pairs carefully and not make assumptions about the order of values.

Read the following scenario to the class: There is more to cartoon drawings than just height and wingspan. Work with your groups to analyze the Data Cards and calculate an equation that can be used to determine if the cartoon characters are drawn accurately. Some sets of data may be useful, while others may not be useful. Give a set of Data Cards to each group of students. Explain to students that they will work with their groups to analyze each Data Card to calculate the line of best fit and the correlation coefficient. Students will also graph the data to describe the correlation of each data set. Students should look for patterns to help them define the correlation coefficient. Give groups one minute to analyze and discuss data A. Ask students the following questions: a.

DOK-1 What are the variables in data A? The variables are height and femur bone length.

DOK-1 Is there a correlation between these variables? If so, what type? Yes, the two variables represent a positive correlation. As the height increases, the femur bone length also increases.

Instruct students to type the data into their calculator: STAT > Edit. Students should calculate the line of best ﬁt for the data: STAT > CALC > 4:LinReg(ax+b). It may be helpful to post the calculator steps on an anchor chart.

DOK-1 Look at the graph of your data: Zoom > 9. Describe the correlation of the data. Explain your reasoning. There is a strong positive correlation between height and femur bone length. This is because both variables are increasing. As the height increases, the femur bone length also increases.

FACILITATION TIP Note to students that the femur bone is located in the leg. This will help students visualize the positive correlation between the femur bone and height. FACILITATION TIP Students will learn of the correlation coefficient value r, where –1 ≤ r ≤ 1. Be sure to explain to them early on that the correlation coefficient does not indicate slope and is separate from the coefficient of the line of best fit.

FACILITATION TIP Some students may wonder if there is a curve of best ﬁt for data following a quadratic or exponential pattern. Let the class know that they will learn about quadratic and exponential regression in a later Explore activity.

e. Remind students that they should look for patterns to help them determine the meaning of the correlation coeﬃcient (r (r value). 5. 6.

DOK-1 Do you recall the steps for calculating a line of best fit using your technology? Answers will vary depending on the graphing technology used. If using the TI-84 calculator, enter data into the graphing calculator: STAT > Edit. Use the linear regression feature to calculate the line of best fit and correlation coefficient: STAT > CALC > 4:LinReg(ax+b)

DOK-1 What does a scatterplot look like when it has a strong correlation? What does it look like when it has a weak correlation? If the points closely follow a pattern such as linear, quadratic, exponential, etc., I can predict that there is a strong correlation among the data. If the points are more spread out and do not follow a line, parabola, or other function, the correlation is weak. If there does not seem to be a pattern among the variables, there is likely no correlation.

DOK-1 What does a scatterplot look like when it has a positive correlation? What does it look like when it has a negative correlation? If the points are increasing and linear, the correlation is positive. If the points are decreasing and linear, the correlation is negative.

d. DOK-2 Is a linear model always appropriate for a set of data? For data A, data C, and data D, a linear model appears to be the best model for the data. If the points are graphed, they appear to approximately form a line. For data B, there does not appear to be any correlation between the variables. A linear model is not useful for predicting values in this data set. 404

Engage

Explore

Explain

Elaborate

Evaluate

Intervention

Acceleration

FACILITATION TIP 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 After students answer Question 4 of the observations and learning. Reflect section, ask them for examples of other things that could be associated with Math Chat the same person but have no correlation. If they are stumped, present examples such • DOK-1 Does a correlation coefficient of 0.9 indicate a better linear fit than a as body weight and forearm length, age and correlation coefficient of −0.9? No. Both 0.9 and −0.9 indicate a strong linear correlation. This means the line fits the data very closely. A correlation coefficient type of pet, and length of first name and last name. of 0.9 indicates a positive correlation, while a correlation coefficient of −0.9 indicates a negative correlation. • DOK-2 How is a strong positive correlation different from a weak positive correlation? A strong correlation has a correlation coefficient closer to −1 or 1. The points are closer to the line of best fit. A weaker correlation has a correlation coefficient closer to 0. The points are more spread out from the line of best fit. • DOK-3 Suppose a set of data relating height and hair length has a correlation coefficient of 0.85. Does this mean the line of best fit can be accurately used to make predictions? Analysis of real-world variables is key. Although the correlation coefficient indicates a strong correlation, height cannot be used as a predictor of hair length. 7. 8.

MODEL DATA

Home

Desmos Instructions 1.

TI Instructions a.

Step 1: If using the TI-84 calculator, enter data into the graphing calculator. STAT > Edit

Step 2: Use the linear regression feature to calculate the line of best fit and correlation coefficient. Round to the nearest hundredth. STAT > CALC > 4:LinReg(ax+b) i. Note that students must have their settings correct to see the correlation coefficient. 2nd Catalog > DiagnosticOn > Enter

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Model Data Explore 2 – Linear Regression a.

Ask students the following questions: i. DOK-1 What do you notice? What do you wonder? Accept all answers. Students should see a linear equation that is similar to the one created by hand.

Read the following explanation to the class: In the next part of the Explore activity, you will look for patterns and analyze the meaning of the r value, which is called the correlation coeﬃcient.

Post-Explore FACILITATION TIP

When assigning this Exit Ticket, determine whether you want students to include some evidence for their thinking.

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

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Intervention

Acceleration

Notes

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Model Data Explore 3 – Residuals ACTIVITY PREPARATION Students will plot and analyze residuals, which are the collection of differences between corresponding coordinates on a line of best ﬁt and the actual data value for a variable, to informally assess the ﬁt of a function.

Standards for Mathematical Practice • • • •

MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.6 Attend to precision.

Materials

Preparation

Printed • • • •

• • •

1 Student Journal (per student) 1 Sales Report (per group) 1 Set of Graph Cards (per group) 1 Exit Ticket (per 2 students)

•

Reusable •

•

1 Quart-size resealable bag (per group)

Separate the class into groups of 2 or 3 students. Print a Student Journal for each student. Print a Sales Report for each group. If desired, print it on card stock, and laminate it for future use. Print and cut out a set of Graph Cards for each group of students. Place each set of cards into a quart-size resealable bag. If desired, print the cards on card stock, and laminate them for future use. Print an Exit Ticket for every 2 students. Cut apart the half-page Exit Tickets so each student has one.

PROCEDURE AND FACILITATION Part I FACILITATION TIP

Before reading the scenario, ask the class 1) Has anyone ever worked in a store?; 2) If so, how did the store determine what items to sell?; 3) How did the store make its proﬁt?

FACILITATION TIP Before starting the Math Chat, ask students for their initial observations. Is each graph of actual data increasing or decreasing? Which is changing at a faster rate overall?

2. 3. 4.

Read the following scenario to the class: The school store uses its profits to fund extra events for students. The school store volunteers have not been managing their inventory well, and they are at risk of not having any profits. They are constantly running out of items that are in high demand while filling their shelves with things that no one ever buys. You have been asked to step in and help the school store manager get the inventory under control. The store manager suggests analyzing the sales report data for different items. Use this data to help determine what inventory items to order for next year. Give a Sales Report to each group. Explain to students that they have about 2 minutes to analyze the data with their groups. Encourage students to look for patterns in the data. Invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 How well does a linear model fit each scatterplot? Both lines have a correlation coefficient very close to 1, so it appears the lines fit the data well. However, the data representing pen sales looks more like a line. The fidget spinner data points appear to be curving away from the line at the beginning and end, so the line may not fit perfectly. • DOK-1 In month 1, what is the difference between the actual number of pens sold and the predicted number of pens sold? The actual number of pens sold was 13. The line of best fit predicted that there would be 20 pens sold because 21 · 1 – 1 = 20. The difference is −7 because 13 – 20 is −7. • DOK-1 Why is the difference negative? The difference is negative because the actual number of pens sold was less than the predicted number. •

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•

5. 6.

Engage

Explore

Explain

Elaborate

Evaluate

DOK-2 Infer what it means if the difference is close to zero. This means the actual and predicted values are almost the same. This indicates that a linear model ﬁts the actual data very well. Give a Student Journal to each student. Explain to students that they will work with their groups to calculate the difference between the actual data and the predicted data. They will create the residual plots to notice any patterns and make conclusions about what residuals and residual plots can tell us. Students will record their answers and explanations on their Student Journals. Monitor students as they collaborate on their work. Use the following guiding questions to assess student understanding: a.

DOK-1 What method do you use to calculate the predicted values? I use the equation for the line of best ﬁt. I substitute the month in the x value, and that gives me the predicted value.

b. DOK-1 How are the variables of the difference graphs different from the variables of the sales report scatterplots? The independent variable in both types of graphs is time (months). The dependent variable in the scatterplots is the number of items sold. The dependent variable in the difference graphs is the difference between the actual and predicted values. 8. 9.

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-2 What patterns do you notice in the table and difference graph for the pen sales? I do not see any patterns. Some values are positive and some are negative. This tells me that the difference is random. • DOK-2 What patterns do you notice in the table and difference graph for the fidget spinner sales? The differences decrease and then increase, and the graph appears to be a parabola. This tells me that the data may not be linear. It may be quadratic. • DOK-2 Which data is better represented by a linear model? How do you know? The data for pen sales is better represented by a linear model. The distance from the actual values and the line of best fit is evenly distributed around the line. The data for the fidget spinner sales is curving away from the line of best fit at the ends of the data provided. This shows that the data trend is not a straight line but could be a quadratic curve. The difference graph also shows a pattern of the difference increasing over time. •

Explain the following to the class: Mathematicians would say that the difference between the actual y value (from the scatterplot) and the predicted y value (from the regression equation line) is called the “residual.” The graph of the residuals is called the “residual plot.” Part II 1.

2. 3.

Read the following scenario to the class: Last school year, the inventory for the school store was bulk ordered at the beginning of the school year. This was a problem because there was not enough storage space and some products ran out of inventory. This year, you would like to set up a monthly ordering system to receive inventory every month. Receiving a set amount of inventory every month is a linear model. Analyze the residual plots on the Graph Cards to determine if sales of each product followed a linear model. Give a set of Graph Cards to each group. Explain to students that they will work with their groups to analyze each scatterplot and corresponding residual plot to determine if a linear function is a good ﬁt for the data. Students will record their answers and explanations on their Student Journals.

Intervention

Acceleration

FACILITATION TIP After students answer the question, explain to them that a set of residuals close to zero is different from a correlation coeﬃcient close to zero. Emphasize to them how this applies to the line of best ﬁt.

MODEL DATA

Home

FACILITATION TIP For the tables in Part I, students may mix up the order of entries when subtracting for their respective difference. If this happens, direct them to the equation under “Difference” in the second row. Be sure they keep track of signs, as well. FACILITATION TIP After students complete the tables on Page 1 ask them if all of the predicted values make sense. Why or why not? Then, ask them for scenarios where a negative predicted value would make sense. If they are stumped, present examples such as position and temperature.

FACILITATION TIP Print and project this statement. Students should record the information in their notebooks. If needed, also refer to Picture Vocabulary. FACILITATION TIP Before reading the scenario, ask the class 1) Does anyone buy items on a regular basis?; 2) Where do you go to get the items?; 3) Is there a more eﬃcient way to regularly receive the items than going to the store or ordering them on line when you need them? Explain your ideas.

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MODEL DATA

Model Data Explore 3 – Residuals 4.

Give groups time to analyze and discuss graph A. Ask students the following questions: a.

DOK-1 What are the variables of the two graphs? The independent variable of both graphs is time (months). The dependent variable of the scatterplot is total inventory. The dependent variable of the residual plot is the residual values.

DOK-1 What information does each graph communicate? The scatterplot shows how much inventory remains at the end of each month. The residual plot shows the difference between the actual data and the predicted data.

DOK-1 What does a linear scatterplot look like? A linear scatterplot looks approximately like a line.

DOK-1 What does a nonlinear scatterplot look like? A nonlinear scatterplot looks curved, like a parabola, or random with no correlation.

In case some students don’t mention it in their explanation for Graph E in the table, explain that the scatterplot and residual plot look very similar. Emphasize that the resemblance lies particularly in how the upand-down nature of actual sales is reﬂected by that of the residuals.

FACILITATION TIP

410

7. 8.

DOK-1 What do these scatterplots show? These scatterplots show the actual sales data.

DOK-1 What do these residual plots show? These residual plots show the difference in the actual data and the predicted data from the line of best ﬁt.

DOK-2 What characteristics are you looking for to determine if a linear model would ﬁt the data? I am looking for a scatterplot that appears to be in a straight line, as well as a residual plot that appears random.

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.

After students ﬁnish their explanation for Graph E in the table, ask them in what way are the residuals random. Some may overthink this initially. They should realize that the residuals are random in that they do not follow a pattern resembling a line or curve deﬁned by a function.

Math Chat

FACILITATION TIP

Post-Explore

After students complete the Exit Ticket, ask for other real-life scenarios where residual plots could be useful. If they are stumped, present examples such as assessments of race car performance, forecasts, or annual bluebird populations of an area.

DOK-1 How can a residual plot help us understand if a linear model is a good fit for a data set? In general, if the residual plot shows no pattern, it indicates that a straight line will represent the data the best. The points should be evenly distributed about the horizontal axis on the residual plot. • DOK-2 When should we use scatterplots, and when should we use residual plots? We should use scatterplots when we need to see trends in data. We should use residual plots when we need to know if our chosen model (linear, quadratic, exponential, etc.) is appropriate. • DOK-2 What is the difference between the correlation coefficient and residuals? What does this information tell you? The correlation coefficient measures how strong a relationship is between two variables. For example, in Part I, both scatterplots show a strong relationship between the variables because the correlation coefficient is 0.98. • DOK-1 Why is a residual plot useful? It is a visual way to represent the difference between the actual and predicted values. Residual plots help to visualize patterns. •

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

Notes

MODEL DATA

Home

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Model Data Explore 4 – Quadratic and Exponential Regression ACTIVITY PREPARATION Students will write an equation representing the best ﬁt for data following either a quadratic or exponential trend represented in a table or scatterplot. Students will use these functions to estimate solutions and make predictions for real-world problems

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.8 Look for and express regularity in repeated reasoning.

Materials

Preparation

Printed • • •

• • •

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

Reusable •

•

1 Graphing calculator (per student)

Separate the class into groups of 2 or 3 students. Print a Student Journal for each student. Print and cut out two sets of Population Cards. Tape the Population Cards around the classroom to create 2 sets of stations. 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 Part I FACILITATION TIP Before reading the scenario, ask the class 1) Has anyone ever been to a national park?; 2) How could understanding data help a national park?; 3) What populations might we want to collect data about in a national park?

Read the following scenario to the class: You are interning at a national park, and you are responsible for monitoring the populations of different animal species. You have received population data for the animals from 2015 through 2021. Determine a function type that can model the change in population over time. This can help predict if the population is increasing or in decline. Invite the class to a Math Chat to share their observations and learning.

Math Chat DOK-1 What function types can you recall? Linear, quadratic, and exponential DOK-1 What are the key characteristics of linear functions? Linear functions have a constant rate of change. The graph of a linear function is a straight line that is either always increasing or always decreasing. • DOK-1 What are the key characteristics of quadratic functions? Quadratic functions increase and then decrease or decrease and then increase. Graphs of quadratic functions are called parabolas. Quadratic functions have a minimum or maximum value at their vertex. • DOK-1 What are the key characteristics of exponential functions? Exponential functions have a common ratio, meaning the y values are multiplied by a common number at each step. The graphs of exponential functions are always increasing or always decreasing. The graphs also have an asymptote. • •

FACILITATION TIP Students will have numerous values to observe throughout the Explore activity. Encourage them to read tables carefully as they look for relationships within data. 412

3. 4.

Give a Student Journal and graphing calculator to each student. Explain to students that they will use their graphing calculators and work with their groups to analyze the animal populations. They will determine which function type best models each change in population over time. Students will record their answers and explanations on their Student Journals. © Accelerate Learning Inc. - All Rights Reserved

Explore

Explain

Elaborate

Evaluate

DOK-1 Is the population of each species increasing or decreasing over time? The animal populations are all increasing initially, but the black bear population decreases after year 4.

•

Acceleration

FACILITATION TIP

After students answer the question, ask them for reasons the black bear population may have decreased after year 4. Then, DOK-1 What does the scatterplot of the coyote population look like? (You may need to remind students that Zoom > 9:ZoomStat allows them ask them if they think the decrease in to view their scatterplots.) The scatterplot is curving up, which looks like population would affect populations of other animals. Why or why not? an exponential growth curve.

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 •

Intervention

Monitor students as they collaborate on their work. Use the following guiding questions to assess student understanding: a.

7. 8.

Engage

MODEL DATA

Home

DOK-1 How do you know if a set of data is well represented by a quadratic model? The population increases and then decreases or decreases and then increases. Also, if you look at the scatterplot, it appears to be a parabola. DOK-1 How do you know if a set of data is well represented by an exponential model? The population is always increasing or always decreasing over time. The amount of increase or decrease is not constant. The change is represented by an approximately common ratio. Also, if you look at the scatterplot, the points are curved like an exponential function. DOK-1 What process did you use to calculate the line of best fit for the bobcat population data? I entered STAT > Edit and input the data in L1 and L2, and then I entered STAT > CALC > 4:LinReg(ax+b). DOK-1 How could we use the equation for the line of best fit to predict the bobcat population in the year 2022? Substitute 7 for x because 2022 – 2015 = 7. So 10.1 · 7 – 7.1 = 63.6, but you should round to 64 because the animal population should be represented by a whole number. DOK-1 How could we predict the black bear or coyote population in the year 2022? We could plot the points and draw an approximate curve through the points. We could extend the curve to the year 2022. We could possibly use technology to generate an equation that fits the data points like we did with the linear model. Read the following prompt and step a to the class: You can use technology to calculate quadratic and exponential models. This is called “quadratic regression” or “exponential regression.” Let’s work through the steps to use technology to calculate quadratic and exponential regression. a.

In Desmos, you can add a table using the + button in the top left. You can also copy and paste any data in from a table. To create a quadratic or exponential model for your data, look at the headers of your table, such as x1 and y1. To model the data with a quadratic function, type in a new row: y1 ~ ax12 + bx1 + c. To model the data with an exponential function, type in a new row: y1 ~ a(bx1) or y1 ~ a(bx1) + c. Desmos will tell you the coeﬃcients a, b, and c that best model the data.

More detailed instructions can be found at the link in footnote I at the bottom of this page.

FACILITATION TIP Before reading the scenario, ask the class 1) What technology have you used to help you solve problems?; 2) What problems did you solve?; 3) What other technologies do you think could be used to solve the same problems?

Part II 1.

Explain to students that the Population Cards are posted around the room. Students will work with their groups to analyze the animal populations to determine which function types best model the changes over time. Then, students will use their graphing calculators to calculate the regression functions. They will use the functions to make predictions about the future populations of the species.

FACILITATION TIP Before this, ask the class 1) What is your favorite animal?; 2) Why is it your favorite animal?; 3) Have you seen the animal in its natural habitat?

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Model Data Explore 4 – Quadratic and Exponential Regression 2.

3. 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. 7.

Ask students to look around the walls of the classroom and make note of the locations of the Population Cards. Explain to students that there are two sets of the same 7 Population Cards so that one area does not get too congested. Encourage students to pay close attention so they do not repeat an animal. Assign each group of students one Population Card to begin with. Instruct students to progress to the remaining cards at their own pace. They will have about 15 minutes to complete all of the cards. Encourage students to collaborate with their groups to complete the rest of Part II 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 How should you round your answers? For the equations, the instructions say to round to the nearest tenth. For the predicted animal populations, you should round to the nearest whole number because you cannot have a fraction of an animal.

DOK-2 How can you distinguish between linear and exponential models? Linear models will appear as a straight line in a scatterplot. They will also change by an approximately constant rate, such as +10 each year or −50 each year. Exponential models are curved like an exponential function in a scatterplot. They also change by an approximately constant ratio such as ·1.1 or ·0.5 each year.

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 FACILITATION TIP After students compare and contrast, check their understanding of outliers and trends. Ask them if the best fit line or curve for any table or graph would change if one or two of its y values was changed significantly. They should remember that a best fit line or curve just needs to be relatively close to most of the coordinate pairs.

DOK-2 Compare and contrast linear, exponential, and quadratic models. Exponential and linear models are always increasing or always decreasing. Quadratic models increase and then decrease or decrease and then increase. All three models can be represented by an equation calculated using a graphing device. • DOK-2 Can a model always be used to predict the future? When considering reallife data, models are useful for representing trends and predicting the near future. However, equations should not be used to predict values in the far future. There are too many factors that can impact real-life data. •

Desmos Instructions 1.

TI Instructions a.

Black bear step 1: If using the TI-84 calculator, enter data into the graphing calculator. STAT > Edit

Black bear step 2: Use the quadratic regression feature to calculate the quadratic function that best models this data. Round to the nearest hundredth. STAT > CALC > 5:QuadReg i. Note that students may notice that the correlation coeﬃcient, r, r

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Elaborate

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Intervention

Acceleration

has been replaced by the coefficient of determination R2. Explain to students that correlation coefficients are only used to describe the relationship between two variables in a linear situation. The coefficient of determination shows the percentage of data that is closest to the line of best fit.

MODEL DATA

Home

STEMscopes Tip

Coyote step 1: If using the TI-84 calculator, enter data into the graphing calculator. STAT > Edit

Coyote step 2: Use the exponential regression feature to calculate the exponential function that best models this data. Round to the nearest hundredth. STAT > CALC > 0:ExpReg

FACILITATION TIP Let students know they can solve the problem without plugging numbers into the equations by comparing the behavior of function types with the data. Check everyone’s answer as a class. Then, plug some x values into the correct equation. Ask them why the results don’t match perfectly with the data, and give them space to remember the equation is a best ﬁt curve. Post-Explore 1. 2. 3. 4.

FACILITATION TIP Before returning to the Hook, ask students for other real-life scenarios where a quadratic or exponential regression equation could be appropriate for predicting data. Examples could include the number of ticket sales for a concert over time, the average high temperature for all months of a year, or a country’s population increase.

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Model 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

Correlation and Causation 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

Linear Regression 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

Residuals

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

Quadratic and Exponential Regression

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

Spiraled Review

Fluency Builder

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

Model Data

MODEL 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

PhET Interactive Simulation Student activities using the PhET Interactive Simulations from the University of Colorado Boulder

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Model 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

3 418

Students

Notes & Comments

Engage

Explore

Explain

Elaborate

Evaluate Evaluate

Intervention

Acceleration Acceleration

ASSESSMENT PLANNER

MODEL DATA

Home

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

What prompts will be used?

What does mastery look like?

I can describe the association between two variables.

I can create and interpret linear models of data in context.

I can use graphing technologies to assess the strength of a line of best ﬁt.

I can determine the best function type to model a data set.

I can discern the difference between correlation and causation.

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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

Georgia Math Teacher Guide

Algebra 1 Teacher Guide

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

ISBN: 979-8-88826-668-7

ALGEBRA 1

GEORGIA MATH A1