MN FL26 AGA Lesson Sample

BY ACCELERATE LEARNING

FL

MATH Lesson Sampler

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Program Overview Algebra 1, Geometr y, and Algebra 2

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Table of Contents Program Overview

................................................................................................................................................ 2

STUDENT WORKBOOKS ......................................................................................................................13 Algebra 1 Solving Quadratics by Factoring ......................................................................................................... 15 Geometry Algebraic Descriptions for Reflections ................................................................................................ 23 Algebra 2 Determining Zeros Within a Real-World Context ................................................................................ 33

TEACHER EDITIONS

......................................................................................................................43

Algebra 1 Solving Quadratics by Factoring ......................................................................................................... 45 Geometry Algebraic Descriptions for Reflections ................................................................................................ 51 Algebra 2 Determining Zeros Within a Real-World Context ................................................................................ 59

INDEPENDENT SKILLS PRACTICE...............................................67 Algebra 1 Quadratic Equations.................................................. 69 Geometry Rigid Motions............................................................. 71 Algebra 2 Quadratic Relationships – Part 2............................... 73

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Program Overview Algebra 1, Geometr y, and Algebra 2

Program Architecture

Program Structure The program is structured by course, unit, and lesson. Each unit includes formative checkpoints throughout and an end-of-unit assessment. Lessons are approximately 45–50 minutes long. They are intentionally designed to fit within the calendar year, so teachers have pacing flexibility. Here is a breakdown by course of the number of units and lessons.

COURSE

UNITS

LESSONS

Algebra 1

14

119

Geometry

13

131

Algebra 2

15

124

HONORS COURSE*

UNITS

LESSONS

Algebra 1 Honors

14

126

Geometry Honors

13

147

Algebra 2 Honors

15

140

13–15 Units per Course

~125 Lessons (~137 for Honors)

45–50 Min.

**

~8–10 LESSONS PER UNIT

* Scopes and sequences are intentionally designed for Florida’s accelerated and honors courses. ***The time estimates in these materials refer to instructional time. Some lessons contain optional activities that provide additional practice for teachers to use at their discretion. Additional Centers and Activities are also available to differentiate instruction.

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Program Overview Algebra 1, Geometr y, and Algebra 2

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Instructional Design

Math Nation Florida is built exclusively for Florida’s B.E.S.T. Standards, providing teachers with a highquality, standards-aligned curriculum that ensures all students can succeed in mathematics.

The design emphasizes: • Student-centered, inquiry-based learning that builds from what students know.

• Scaffolded instruction within and across lessons, ensuring accessibility for all learners.

• Balanced development: conceptual understanding, procedural fluency, and realworld application.

• Multiple representations to deepen understanding and connect math to authentic contexts.

• Collaboration: students learn together and view mistakes as learning opportunities.

• Transfer learning: spiraling and connecting content across courses to highlight the beauty and interconnectedness of mathematics.

• Mathematical Thinking & Reasoning Standards (MTRs): embedded throughout for deeper engagement.

Instruction draws on John Hattie’s Visible Learning research and incorporates: • Practices with the highest impact on student growth.

• Classroom discourse and rich discussion opportunities.

• Clear strategies and content knowledge support in the Teacher Edition.

Teachers also have access to: • Progress monitoring tools within a MultiTiered System of Support (MTSS): embedded assessments, reports, and analytics aligned to Florida’s B.E.S.T. Standards.

• Differentiation supports: remediation, intervention, and enrichment at both unit and lesson levels.

• Professional learning: point-of-use digital resources and a teacher community for continuous support.

• Communication skills: consistent opportunities for oral and written mathematical reasoning.

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Program Overview Algebra 1, Geometr y, and Algebra 2

Math Nation Florida’s Unit and Lesson Guides serve as a companion to our dynamic curriculum. The Unit and Lesson Guides accompany the curriculum for Grades 6–12, acting as an extension and planning partner for further clarification of the purpose, intent, coherence, and language of the lessons.

Unit Guides Each unit includes a Unit Guide that offers additional planning support for the unit as a whole.

Florida’s B.E.S.T. Standards are designed to ensure vertical alignment across grade levels and horizontal alignment within each course. In the Unit Guide, the Unit Narrative section describes the content of the unit, including horizontal connections to other units. The Vertical Alignment section of the Unit Guide provides the vertical alignment for the benchmarks taught in the unit.

Each Unit Guide also incorporates instructional strategies for differentiating, enriching, and implementing lessons throughout the unit.

Unit Narrative

Differentiation

The narrative captures the story of the unit, its progression, and connections to other units and courses. This section also includes a Unit-ata-Glance that gives an overview of the lessons included in the unit.

The differentiation section includes recommendations for incorporating Universal Design for Learning (UDL)™ and information for English Language Learners (ELLs). UDL from CAST© 2008 prioritizes strategies for differentiating instruction that seek to eliminate barriers of entry so that all students can learn. Each Unit Guide offers techniques specific to that unit content through the lens of UDL. Section Overview This section offers visual coherence with benchmarks aligned to lessons and clarifications to inform planning and instruction.

Vertical Alignment This section provides the vertical alignment of the benchmarks taught in the unit by noting the benchmarks from prior courses that are being built upon, and the benchmarks in future courses that students are working toward.

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Program Overview Algebra 1, Geometr y, and Algebra 2

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Lesson Guides Every lesson includes a Lesson Guide that provides insight into the math thinking and reasoning for effective implementation.

Lesson Introduction

Component Guidance

Each Lesson Guide begins with an introduction that captures a snapshot of the lesson content for purposeful preparation. The Lesson Introduction includes the benchmarks and learning targets found in each student lesson. Teacher planning supports include vertical benchmark coherence, essential questions, a brief lesson narrative, a component summary for pacing and adaptation, and resources, including materials and vocabulary.

In the Lesson Guide, specific guidance incorporating instructional strategies is provided for each component of a lesson. This support is shown in line with the answer key of the student lesson for point-of-use implementation in the classroom. These recommendations include teacher moves, additional questions to consider asking, differentiation ideas related to the component, and possible enrichment ideas.

Each Lesson Guide includes additional planning support, such as:

LESSON 3.2 DETERMINING ZEROS WITHIN A REAL-WORLD CONTEXT

• Mathematical Thinking and Reasoning (MTR) Standards: one is highlighted for each lesson for intentional focus when planning, though a lesson may integrate several MTRs.

LESSON INTRODUCTION MA.912.AR.3.2 Given a mathematical or real-world context, write and solve one-variable quadratic equations over the real- and complex number systems.

LEARNING TARGETS ● I can solve a quadratic by factoring when given a real-world context. ● I can interpret solutions as viable or non-viable options given a real-world context.

BENCHMARK COHERENCE

• Common Misconceptions

BUILDING ON MA.912.AR.3.1 Given a mathematical or real-world context, write and solve one-variable quadratic equations over the real-number system.

WORKING TOWARD

• Things to Consider

N/A

ESSENTIAL QUESTIONS

ALGEBRA 2 Teacher Guide

• Literacy and Language Routines

GLOSSARY Key Terms: ● N/A Additional Terms:

• Differentiation Suggestions: found here and also incorporated throughout the lesson components.

● How can the validity of solutions be identified when solving quadratic equations by factoring within a real-world context? ● How can a quadratic equation be solved by factoring within a real-world context?

RESOURCES

● Roots ● Zeros ● Zero-product property

TEACHER TIPS COMMON MISCONCEPTIONS

MATERIALS

ADDITIONAL PREPARATION

NARRATIVE● Consider creating a review of ● (optional)LESSON Graphing utility factoring techniques, including In this lesson, students extend their knowledge of solving quadratic equations by factoring from Lesson 1 to real-world contexts. factoring out the greatest Given a real-world context, students will apply their knowledge of factoring and the zero-product property to solve quadratic common factor, to aid struggling equations. Students will interpret the solutions to quadratic equations within the given context and will further analyze the validity of students. solutions.

COMPONENT SUMMARY 3.2.1 WARM-UP (~5 MIN)

3.2.2 COLLABORATION (20-15 MIN)

“Would You Rather?” activity on sale prices

Solving real-world quadratic equations by factoring and interpreting solutions

3.2.3 GUIDED INSTRUCTION (10-15 MIN)

3.2.4 YOUR TURN! (5-10 MIN)

Solving real-world quadratic equations by factoring and interpreting solutions

Analyzing the flight path of a toy rocket modeled by a quadratic equation

● Students may believe that quadratic equations set equal to zero must always have the zero on the right side of the equation, 3.2.5 WRAP-UP (~5 MIN) causing confusion with function notation.

THINGS TO CONSIDER

Solving a quadratic equation to model fireworks

● Although factoring is a review skill, students still struggle. Consider reviewing all forms of factoring to aid struggling Accelerate Learning – All Rights students. It may also benefit students to utilize multiple©methods, such as Inc. algebra tiles, Reserved area model, grouping, etc.

UNIT 3 – Lesson 2

1

LITERACY AND LANGUAGE ROUTINES Three Reads Consider a Three Reads strategy for 3.2.3 Guided Instruction (struggling students may benefit from utilizing this strategy throughout the entire lesson). Students often struggle with word problems, so the use of this strategy will ensure that students know what they are being asked to do. This strategy supports reading comprehension of questions and negotiating information in a text with a partner in a mathematical conversation.

MATHEMATICAL THINKING AND REASONING (MTR) STANDARD MA.K12.MTR.7.1 Real-World Contexts This lesson extends student understanding of solving quadratic functions by factoring to modeling and solving real-world problems.

In this way, students are connecting the idea of solving quadratic equations by factoring to real-world experiences. Given a realworld context, students are applying the zero-product property to solve quadratic equations and interpreting solutions as viable or nonviable in terms of the given context.

DIFFERENTIATE INSTRUCTION Continue to expose students to the vocabulary by connecting the terms “solutions,” “x-intercepts,” “roots,” and “zeros” of what is being found when solving quadratics. It may benefit struggling students to utilize a graphing utility to draw the connection between zeros and x-intercepts. Students who struggle to understand the zeros of a quadratic equation may benefit from interacting with the corresponding graph.

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UNIT 3 – Lesson 2

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Program Overview Algebra 1, Geometr y, and Algebra 2

Student Lesson Design Student lessons across all courses feature the same structure. The lessons include several components, each with a specific icon associated with it.

Benchmark(s) Focus benchmarks are indicated on the first page of each student lesson.

Learning Target(s) Learning Targets are clearly defined at the beginning of each lesson so student and families have clear expectations.

8th Grade Mathematics

Florida

Unit 8, Lesson 2: Solving Real-World, Multi-Step Equations Benchmark MA.8.AR.2.1 Solve multi-step linear equations in one variable, with rational number coefficients. Include equations with variables on both sides.

Warm-Up Every lesson begins with a Warm-Up. The Warm-Ups are designed to vary so that all students have access and a positive start to each lesson. WarmUps include bell work, STEM career profiles, critical thinking, and more.

Wrap-Up Every lesson ends with a short Wrap-Up. Wrap-Ups include student reflection, cool downs, error analysis, and tickets out the door that can be used for formative assessment.

Learning Target 

I can solve multi-step linear equations from a real-world context.

Green lesson components identify student-led components that are intentionally designed for students to explore, discover, and practice concepts either independently or collaboratively with peers. These activities promote active engagement and deepen understanding of the content.

©Accelerate Learning Inc. – All Rights Reserved

Unit 8 – Lesson 2 | 13

Explorations Explorations are student-led discovery, often involving student collaboration and discussion.

The topics explored in these components may not yet be formalized.

Activity Activities include gallery walks, data collection, and games.

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Collaboration Students collaborate in pairs or small groups.

Try It Students may not have reached mastery when completing Try It. These problems are an opportunity to apply previously learned skills to more formal practice problems.

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Program Overview Algebra 1, Geometr y, and Algebra 2

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Blue lesson components indicate that lesson components are intended to be teacher-led. These provide opportunities for direct instruction and teacher facilitation of learning and are intended to be rich in student collaboration.

Guided Instruction Instruction should be studentcentered during Guided Instruction, but may require more support from the teacher.

Bring it Together Bring it Together components formalize concepts explored in the lesson.

Refresher Vocabulary and concepts learned in previous units or courses are reviewed in Refresher components.

Guided Practice Guided Practice presents an opportunity for students to learn new concepts through practice problems.

Your Turn!

Your Turn! provides students with the opportunity for independent practice. These problems may also be used for homework.

CE

Warm-Up

P R A CT I

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TIVITIES

Wrap-Up

S IE

AC LED ER

NT-LED A C UDE T IV ST IT

TE AC H

DAILY LESSON STRUCTURE

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Program Overview Algebra 1, Geometr y, and Algebra 2

Instruction and Practice to Meet Rigor Lesson Activities

Algebra 2

Florida

1.3.2 Collaboration: Comparing Key Features of Functions

Lesson Activities serve as the core learning experience in every Math Nation Florida lesson. These engaging tasks:

Work with your partner to complete the following.

1.

Two functions are shown on the coordinate grid.

Function A

• Encourage collaborative, student-led discovery through guided investigation, reasoning, and communication of mathematical ideas

Function B

• Replace front-loaded instruction with problem-solving, pattern recognition, and strategy development • Foster a discovery-based approach where students actively construct mathematical understanding • Purposefully integrate horizontal and vertical alignment, helping students connect concepts deeply and meaningfully

a.

Which function has the larger 𝑥𝑥𝑥𝑥-intercept?

b.

Which function has the smaller 𝑦𝑦𝑦𝑦-intercept?

Florida

Algebra 2

c.

Complete the statements. o o o o

Function A is Function B is Both functions are Neither function is

positive over the interval

©Accelerate Learning Inc. – All Rights Reserved

Unit 1 – Lesson 3 | 35

(−∞, −1.25).

• Include built-in supports in the Teacher Edition for effective student questioning, discourse, exploration, and conceptual clarity

o o o o

Function A is Function B is Both functions are Neither function is

negative over the interval

(−1.25, ∞). d.

2.

For which function could the left end behavior be described by the phrase, “As 𝑥𝑥𝑥𝑥 → −∞, 𝑓𝑓𝑓𝑓(𝑥𝑥𝑥𝑥) → −∞.”?

The table shows three different functions.

Function H

Function M

𝑦𝑦𝑦𝑦 = 5(1.35)𝑥𝑥𝑥𝑥

Function K

Conceptual Understanding

Procedural Fluency

𝒙𝒙𝒙𝒙 −2 −1 0 1 2

36 | Unit 1 – Lesson 3

𝒚𝒚𝒚𝒚 −5 −1 3 7 11

©Accelerate Learning Inc. – All Rights Reserved

RIGOR

Real-World Application

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Program Overview Algebra 1, Geometr y, and Algebra 2

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Student Practice Each lesson includes multiple opportunities for student practice. Here are some examples of meaningful practice and targeted supports in Math Nation Florida: Your Turn!

Collaborative Peer-to-Peer Practice

Your Turn! provides students with a chance for true independent practice of the concepts they are learning in a given lesson.

Math Nation Florida is packed with practice that is designed to have students working collaboratively to solve problems, develop solutions, discover new ideas and strategies, engage in error analysis and much more. With peer-to-peer practice embedded throughout the program, students are provided with ongoing, meaningful collaboration opportunities.

Try It Students may not have reached mastery when completing Try It. These problems are an opportunity to apply skills learned in previous components to more formal practice problems.

Check Your Understanding Check Your Understanding are a set of three digital practice problems available for students at the end of each lesson to inform and check for understanding.

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Program Overview Algebra 1, Geometr y, and Algebra 2

Guided Practice

Stepping Stones

The Guided Practice components provide opportunities for students to practice their skills in a structured, teacher-led setting. The intention is to practice together with guidance to maximize retention and understanding.

Each lesson has a set of digital practice questions called Stepping Stones. These give students targeted practice that moves them toward mastery of lesson-level expectations. They can be used flexibly before or after the main lesson to reinforce understanding.

Hands-On Practice Activities include gallery walks, data collection, and games that allow the students to get additional practice in practical, engaging ways to leverage their critical and creative thinking skills.

Additional Lesson Practice Each lesson has a set of additional practice questions to provide another opportunity for students to independently practice what they learned in the lesson. These can be utilized in class or for homework.

Florida

Florida Independent Skills Practice Book

Algebra 2

1.2.4 Guided Practice: Rational Functions The equation and parent function for a rational function are shown.

Function Type

Parent Function

Graph

Algebra 2

3.

Rational

1.

𝑓𝑓𝑓𝑓(𝑥𝑥𝑥𝑥) =

DaiJa Marcus DaiJa wrote the following and asked, “Could Marcus responded, “The only way a fraction ever equal zero?” for a fraction to equal 0 is if the numerator can equal 0.”

Use the parent function to complete the table of values. 𝒙𝒙𝒙𝒙

2.

1 𝑥𝑥𝑥𝑥

Florida

Marcus noticed that there was also an asymptote at 𝑦𝑦𝑦𝑦 = 0. He asked his partner, DaiJa, to help him figure out why 𝑦𝑦𝑦𝑦 ≠ 0.

An additional Florida Independent Skills Practice Book helps students solidify and refine their skills, either in class or for homework. Practice books include embedded tutorial videos in English and Spanish so students and families have 24/7 support.

𝒚𝒚𝒚𝒚

−4� �

−2� �

−1� �

1� �

1.2.4 Práctica guiada: Funciones racionales

2�

Se muestran la ecuación y la función madre de una función racional.

�

Tipo de función

Review DaiJa’s work and Marcus’ conclusion with your partner. Use DaiJa’s work Notice that the graph has an asymptote at 𝑥𝑥𝑥𝑥 = 0. Use the equation of the parent and Marcus’ conclusion to write an explanation as to why there is an asymptote at function to explain why 𝑦𝑦𝑦𝑦 = 0. 𝑥𝑥𝑥𝑥 ≠ 0.

Función padre

3.

Racional

4.

Complete the table to describe the key features of the parent function, 𝑓𝑓𝑓𝑓(𝑥𝑥𝑥𝑥) = Key Feature

1

𝑥𝑥𝑥𝑥

.

1.

𝑥𝑥𝑥𝑥-intercept(s)

2.

𝑦𝑦𝑦𝑦-intercept(s)

MN FL26 AGA Sampler FINAL.indb 10

DaiJa DaiJa escribió lo siguiente y preguntó: “¿Podría una fracción ser igual a cero?”

1 𝑥𝑥

Marco Marcus respondió: “La única forma de que una fracción sea igual 0es si el numerador puede ser igual a 0”.

Utilice la función padre para completar la tabla de valores. 𝒙𝒙

©Accelerate Learning Inc. – All Rights Reserved range

©Accelerate Learning Inc. – All Rights Reserved

𝑓𝑓(𝑥𝑥) =

Marcus notó que también había una asíntota en 𝑦𝑦 = 0. Le pidió a su compañero, DaiJa, que lo ayudara a descubrir por qué 𝑦𝑦 ≠ 0.

Description

domain

24 | Unit 1 – Lesson 2

Gráfico

𝒚𝒚

−4

−2

−1

1

2

Revisa el trabajo de DaiJa y la conclusión de Marcus con tu compañero. Usa el trabajo de DaiJa y la conclusión de Marcus para escribir una explicación de por

Observe que el gráfico tiene una asíntota en 𝑥𝑥 = 0. Utilice la ecuación la una función quédehay asíntota en 𝑦𝑦 = 0. madre para explicar por qué 𝑥𝑥 ≠ 0.

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Program Overview Algebra 1, Geometr y, and Algebra 2

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Home Support Resources Math Nation Florida is designed to extend support into the home for student success. To help families, we provide:

Take-home Letters

Study Expert Videos

Helpful take-home letters explain how to navigate and use the Math Nation Florida platform effectively. They provide families with specific guidance on how to leverage all of the resources available to best support their student in their math journey.

Families also gain unlimited access to a comprehensive library of multilingual Study Expert videos that can be downloaded on our iOS and Android mobile apps, ensuring access anytime, anywhere, to a math tutor. The videos:

Dear Parent or Guardian, TM

Welcome to Florida’s B.E.S.T. Core Math Curriculum by Math Nation! We know that confidence in math class will help your student succeed in middle school, high school, and beyond. Math Nation is dedicated to providing the best resources and making them accessible to every student.

What is Florida’s B.E.S.T. Math Core Curriculum by Math Nation? ● ● ● ● ● ●

MATH NATION is an all-inclusive, comprehensive math resource created for Florida's B.E.S.T. Standards for Mathematics.

MATH NATION IS AVAILABLE ONLINE, AND CAN BE DOWNLOADED ON IPHONES, ANDROID PHONES, AND IPADS. Your student can access and engage on their own time in a familiar, comfortable, pressure-free environment.

Fun, highly-interactive instructional videos help your student review the material they learn in class. And best yet, there are several Study Experts to choose from for each video, including Spanish and Haitian Creole speaking Study Experts.

Digital workbook lessons align with every video and can be translated into multiple languages. Your student can also use the printed, color workbook to follow along with the videos.

Accessibility features empower students and ensure that all students have the resources they need to be successful.

Auto-graded practice is available for your student via the Stepping Stones and Check Your Understanding tools for each lesson. They can also engage in the Test Yourself! auto-graded practice tool at the end of each unit.

• Come in English, Spanish, and Haitian Creole • Allow families to see how a math concept is taught to better support their student at home • Help students who miss school due to illness or vacation keep up with their studies

What can I do to help my student succeed? ● ● ●

●

●

● ●

●

Go to best.MathNation.com and have your student show you how they sign in.

Ask your student to watch MATH NATION videos to help with homework, to review for a test, or to get caught up if they are absent from class.

Your student can also use Stepping Stones and Check Your Understanding at the end of a lesson and the Test Yourself! Practice Tool at the end of each unit to identify which parts of the material they understood and which videos and topics they need to review.

Visit the Family Area to watch the Welcome Video, which should help you navigate the resource-rich environment. You can access the Family Area in the lower left corner of the digital workbook or from the Menu in the top right corner.

Watch the Family Support Videos which are provided for each and every lesson. These will tell you exactly what your student is learning in each unit and how their learning connects to how you may have learned the very same concepts.

Take advantage of after school and weekend tutoring by having your Algebra 1, Geometry, or Algebra 2 student post their questions on the Student Wall.

We know that education is not one-size-fits-all, so if you worry that your student needs some help getting caught up or is not being challenged, we have solutions! While our OnRamp Personalized Learning Tool is available at 5 different course levels to help students fill any gaps they might have, we also provide enrichment opportunities at the end of each unit.

You can track your student’s performance at any time by clicking the Student Progress tab at the top of the page!

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BY ACCELERATE LEARNING

FL

FL

BY ACCELERATE LEARNING

FL

BY ACCELERATE LEARNING

A2 G √ FL

ALGEBRA

BY ACCELERATE LEARNING

1

Volume 1

Volume

Student Workbook

Student Workbook

© 2025 Accelerate Learning

ALGEBRA 2

My name:

GEOMETRY

1

Volume 1

Volume

Student Workbook

Volume 1

∞

Student Workbook

© 2025 Accelerate Learning

GEOMETRY

Student Workbook

ALGEBRA

ALGEBRA 1

A1

BY ACCELERATE LEARNING

FL

FL My name:

FL

FL

BY ACCELERATE LEARNING

BY ACCELERATE LEARNING

BY ACCELERATE LEARNING

√

BY ACCELERATE LEARNING

Student Workbook Volume

1

FL

Student Workbooks

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BY ACCELERATE LEARNING

FL

WRITINGAND ANDSOLVING SOLVINGQUADRATIC QUADRATICEQUATIONS EQUATIONS WRITING LESSON

SOLVING QUADRATICS BY FACTORING 9.7.1 Warm-Up: Satellite Dishes Valerie Thomas is a scientist and inventor. In 1980, she received a patent for inventing the illusion transmitter. This was an integral piece of technology in developing the digital media formats needed for today’s video games, movies, and 3D television.

Thomas worked at NASA from 1964-1995, where she helped develop real-time computer data systems to support satellite operations. She oversaw the creation of the Landsat program there, which supported the first satellite to transmit images from outer space.

Today, satellite dishes are everywhere. Satellite dishes are designed using a parabolic curve.

1.

2.

What key feature of a parabola is the bottom of the dish?

The arrows show how signals bounce off a satellite dish to the receiver. The receiver is always located at a parabola’s focus. What key feature does the focus lie on?

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WRITING AND SOLVING QUADRATIC EQUATIONS WRITING AND SOLVING QUADRATIC EQUATIONS

BY ACCELERATE LEARNING

FL

9.7.2 Activity: Card Sort 1.

Work with your partner to match the cards.

Equation

Graph

𝒙𝒙-intercept

𝒙𝒙-intercept

Expression T

Expression U

Expression V

Expression W

2𝑥𝑥 2 + 5𝑥𝑥 − 42

2𝑥𝑥 2 − 5𝑥𝑥 − 42

2𝑥𝑥 2 − 19𝑥𝑥 + 42

2𝑥𝑥 2 + 19𝑥𝑥 + 42

P Q R S 2.

Factor each expression.

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BY ACCELERATE LEARNING

3.

FL

WRITINGAND ANDSOLVING SOLVINGQUADRATIC QUADRATICEQUATIONS EQUATIONS WRITING

Work with your partner to match the factored expressions to the 𝑥𝑥-intercepts.

Expression

Factored Expression

T

𝒙𝒙-intercepts

U V W 4.

Discuss with your partner how the factored expression reveals the 𝑥𝑥-intercepts.

Make a conjecture about the connection.

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WRITING AND SOLVING QUADRATIC EQUATIONS WRITING AND SOLVING QUADRATIC EQUATIONS

FL

BY ACCELERATE LEARNING

9.7.3 Guided Instruction: Solving Quadratic Equations by Factoring 1.

In the equation 𝑥𝑥 ∙ 𝑦𝑦 ∙ 𝑧𝑧 = 0, which variable has to equal 0 so that the product is equal to 0?

o o o o

2.

𝑥𝑥 𝑦𝑦 𝑧𝑧 At least one of 𝑥𝑥, 𝑦𝑦, or 𝑧𝑧

Zero Product Property

If the product of two numbers is Ͳ, then at least one of the numbers is Ͳ.

For each expression, use the zero product property to determine the values of 𝑥𝑥 for which the expression would equal 0.

Expression

(𝑥𝑥 + 7)(𝑥𝑥 − 2)

(2𝑥𝑥 + 7)(𝑥𝑥 − 6)

(−3𝑥𝑥 + 11)(4𝑥𝑥 + 18)

𝒙𝒙-values

Each of these expressions could have been written as quadratic equations that are set equal to 0 such as 𝑥𝑥 2 + 5𝑥𝑥 − 14 = 0.

3.

When solving a quadratic equation set the equation equal to 0, the solution(s) will be the value(s) of 𝑥𝑥 when 𝑦𝑦 =_____. Explain how this relates to your conjecture from the card sort.

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Another method of solving quadratics is by factoring. When solving by factoring the quadratic must be set equal to 0. Once the quadratic is factored, the zero product property is applied.

4.

Solve ℎ2 − 10ℎ + 25 = 0 by factoring.

5.

Solve 10𝑓𝑓 2 + 13𝑓𝑓 − 3 = 0 by factoring.

To solve by factoring, the quadratic equation should be in standard form and set equal to zero.

6.

Solve 𝑝𝑝2 + 36 = 13𝑝𝑝 for 𝑝𝑝 by factoring.

7.

Solve 40𝑥𝑥 2 − 18𝑥𝑥 = 12𝑥𝑥 + 135 for 𝑥𝑥 by factoring.

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9.7.4 Try It: Factoring Solve each by factoring. Show your work.

1.

𝑥𝑥 2 − 10𝑥𝑥 − 15 = 9

2.

−3𝑥𝑥 2 + 6𝑥𝑥 + 22 = 2 − 5𝑥𝑥

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9.7.5 Wrap-Up: Error Analysis The work of two students who solved 2𝑥𝑥 2 − 12𝑥𝑥 + 14 = 2𝑥𝑥 − 6 is shown.

Ruchelle

Quinyang

2𝑥𝑥 2 − 12𝑥𝑥 + 14 = 2𝑥𝑥 − 6

2𝑥𝑥 2 − 12𝑥𝑥 + 14 = 2𝑥𝑥 − 6

2(𝑥𝑥 2 − 6𝑥𝑥 + 7) = 2(𝑥𝑥 − 3) 2(𝑥𝑥 2 − 6𝑥𝑥 + 7) 2(𝑥𝑥 − 3) = 2 2 2

𝑥𝑥 − 6𝑥𝑥 + 7 = 𝑥𝑥 − 3 𝑥𝑥 2 − 5𝑥𝑥 + 4 = 0

(𝑥𝑥 − 4)(𝑥𝑥 − 1) = 0

2𝑥𝑥 2 − 12𝑥𝑥 + 14 − (2𝑥𝑥 − 6) = 2𝑥𝑥 − 6 − (2𝑥𝑥 − 6) 2𝑥𝑥 2 − 14𝑥𝑥 + 20 = 0

(2𝑥𝑥 − 5)(𝑥𝑥 − 4) = 0 𝑥𝑥 =

5 2

or 𝑥𝑥 = 4

𝑥𝑥 = 4 or 𝑥𝑥 = 1

1.

Both Ruchelle and Quinyang made one mistake. Circle each mistake.

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ALGEBRAIC DESCRIPTIONS FOR REFLECTIONS 4.2.1 Warm-Up: Moving Circles 1.

Move any three circles so that the pyramid points down.

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4.2.2 Exploration: Discovering Algebraic Descriptions for Reflections 1.

Quadrilaterals 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 and 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 are shown on the coordinate grid. Quadrilateral 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 was reflected across the 𝑦𝑦-axis to create quadrilateral 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇.

a.

Draw a line segment between each set of corresponding vertices.

b.

Describe how the line segments between the vertices are related to the 𝑦𝑦-axis.

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c.

RIGIDMOTIONS MOTIONSAND ANDCONGRUENCE CONGRUENCE RIGID

Complete the table. 𝑯𝑯 to 𝑻𝑻

Distance

𝒁𝒁 to 𝑾𝑾

𝑳𝑳 to 𝑲𝑲

𝑹𝑹 to 𝑩𝑩

Ordered Pair of the Midpoint d.

What do you notice about the midpoints?

e.

The table shows the coordinates of each vertex.

Vertex

𝒙𝒙

𝒚𝒚

Vertex

−5

3

𝑊𝑊

−1

𝐵𝐵

𝐻𝐻

−6

𝑅𝑅

−2

𝑍𝑍 𝐿𝐿

−5

2 1

𝑇𝑇

𝐾𝐾

𝒙𝒙

𝒚𝒚

5

3

6 2 5

2 1

−1

Work with your partner to describe how the coordinates of the preimage, quadrilateral 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻, are related to the coordinates of the image, quadrilateral 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇.

f.

Write a conjecture that you and your partner think will work for the algebraic description of a reflection. (𝑥𝑥, 𝑦𝑦) → (__________, _________)

g.

Discuss with your partner if the algebraic description will work for any reflection. Summarize your conclusion.

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Polygons 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 and 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 are shown on the coordinate grid. Polygon 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 was reflected across the 𝑥𝑥-axis to create polygon 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻.

a.

Describe the relationship that the line segments between each set of corresponding vertices have with the 𝑥𝑥-axis.

b.

Wendall and Samah were working together on writing an algebraic description for reflections across the 𝑥𝑥-axis. Each student’s algebraic description and explanation of their thinking is shown in the table.

Wendall

Samah

(𝑥𝑥, 𝑦𝑦) → (−𝑥𝑥, 𝑦𝑦)

(𝑥𝑥, 𝑦𝑦) → (𝑥𝑥, −𝑦𝑦)

Because the polygon flipped over the 𝑥𝑥-axis, the 𝑥𝑥 becomes its opposite.

Because above the 𝑥𝑥-axis all 𝑦𝑦-values are positive and below the 𝑥𝑥-axis all 𝑦𝑦-values are negative, the 𝑦𝑦 becomes its opposite.

Discuss with your partner which, if any, algebraic description is correct.

Use at least two ordered pairs from polygons 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 and 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 to defend your choice.

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Triangles 𝐻𝐻𝐻𝐻𝐻𝐻 and 𝐻𝐻′𝑊𝑊′𝑇𝑇′ are shown on the coordinate grid. Triangle 𝐻𝐻𝐻𝐻𝐻𝐻 was reflected across the origin to create ∆𝐻𝐻′𝑊𝑊′𝑇𝑇′.

a.

Discuss with your partner how to write the algebraic description for a reflection across the origin. (𝑥𝑥, 𝑦𝑦) → (__________, _________)

b.

Re-examine the algebraic descriptions of rotations. Which rotation is similar to a reflection across the origin?

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Triangles 𝐾𝐾𝐾𝐾𝐾𝐾 and 𝐾𝐾′𝑊𝑊′𝐷𝐷′ are shown on the coordinate grid. Triangle 𝐾𝐾𝐾𝐾𝐾𝐾 was reflected across the line 𝑦𝑦 = 𝑥𝑥 to create ∆𝐾𝐾′𝑊𝑊′𝐷𝐷′.

Use the coordinates to determine the algebraic description for a reflection across the line 𝑦𝑦 = 𝑥𝑥. (𝑥𝑥, 𝑦𝑦) → (__________, _________)

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Triangles 𝐻𝐻𝐻𝐻𝐻𝐻 and 𝑃𝑃𝑃𝑃𝑃𝑃 are shown on the coordinate grid. Triangle 𝐻𝐻𝐻𝐻𝐻𝐻 was reflected across the line 𝑦𝑦 = −𝑥𝑥 to create ∆𝑃𝑃𝑃𝑃𝑃𝑃.

Use the coordinates to determine the algebraic description for a reflection across the line 𝑦𝑦 = −𝑥𝑥. (𝑥𝑥, 𝑦𝑦) → (__________, _________)

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4.2.3 Bring It Together: Algebraic Descriptions for Reflections Reflections Across the 𝒙𝒙-axis

Across the 𝒚𝒚-axis

The _____ −coordinate changes to its __________.

The _____ −coordinate changes to its

(𝑥𝑥, 𝑦𝑦) → (_____, _____)

(𝑥𝑥, 𝑦𝑦) → (_____, _____)

Across 𝒚𝒚 = 𝒙𝒙

Across 𝒚𝒚 = −𝒙𝒙

The coordinates _________.

(𝑥𝑥, 𝑦𝑦) → (_____, _____)

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__________.

The coordinates _________ and become ___________.

(𝑥𝑥, 𝑦𝑦) → (_____, _____)

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4.2.4 Wrap-Up: Cool Down A reflection can be thought of as folding or "flipping" an object over the line of reflection. A squish painting is a fun application of a line of reflection.

To make a squish painting, a large sheet of construction paper is folded in half then opened up.

Paint is dotted along one side of the fold. The paper is folded, and the folded paper is rubbed, pounded, and squished to move the paint around. The paper is then opened up to reveal the squish painting.

1.

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How does a squish painting show that in a reflection, each point of the original figure and its image are the same distance away from the line of reflection?

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DETERMINING ZEROS WITHIN A REAL-WORLD CONTEXT 3.2.1 Warm-Up: Would You Rather?

1.

Which deal would you rather choose – buy 1, get 2 free for $120, or buy 1 pair that is already 60% off for $48?

2.

Justify your answer mathematically.

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3.2.2 Collaboration: Using Quadratics in the Music Business 1.

Organizers of a music festival charge $60 for admission. The organizers polled the attendees to determine how increasing the price would affect attendance. The equation 𝑃𝑃(𝑥𝑥) = −20,000𝑥𝑥 2 − 20,000𝑥𝑥 + 600,000 models profit as a function of 𝑥𝑥, where 𝑥𝑥 is each $1 increase in the admission fee. a.

Discuss with your partner which factors would affect your attendance at a music festival. List your top three factors.

b.

What admission fee would you consider to be too high?

c.

If 𝑥𝑥 = 2, what is the admission fee to the festival?

d.

Solve −20,000𝑥𝑥 2 − 20,000𝑥𝑥 + 600,000 = 0 by factoring. Show your work.

e.

Interpret the solutions in terms of the context.

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Ask two other classmates for their interpretations of the solutions to −20,000𝑥𝑥 2 − 20,000𝑥𝑥 + 600,000 = 0. Write a summary of their interpretations of the solutions. You are the only person who should write in the first column of the table. Have each person initial next to your summary stating that your summary was accurate.

My Summary of Their Interpretation of the Solutions in Terms of the Context

g.

2.

Initials

Review your classmates’ interpretations with your partner. If necessary, adjust your interpretation.

The compact disc (CD) was invented in 1979, many years before online music existed. At the time it was the most sophisticated way to store and play music.

In 1982, CDs were made available to the public. CDs quickly became the most efficient way of storing music, which replaced vinyl records and cassette tapes.

The function 𝐶𝐶(𝑥𝑥) = −2𝑥𝑥 2 + 38𝑥𝑥 + 40 models the sales, in hundreds of millions of dollars, of compact discs in the years since 1990.

a.

Solve −2𝑥𝑥 2 + 38𝑥𝑥 + 40 = 0 by factoring.

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Interpret the year that each zero represents.

Zero

Year

c.

Discuss the interpretation of each zero with your partner. Then, determine whether or not the interpretation makes sense. Summarize your discussion.

d.

In 2001, iTunes was launched by Apple. For the first time, people could buy songs online and store them on mp3 players, iPods, or on a computer. Then, in 2008, the music streaming site Spotify was started.

Discuss with your partner why one of the zeros shows that online access to music may have led to CD sales eventually reaching zero.

Summarize your discussion.

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3.2.3 Guided Instruction: Finding Zeros 1.

A landscape designer is adding a garden to a client’s backyard. One side of the garden will be alongside an existing fence. The client has 180 feet of fencing that is left over from another project. The client would like the garden to use all of the fencing.

The equation 𝐴𝐴 = 180𝑤𝑤 − 2𝑤𝑤 2 can be used to find the area of the garden, where 𝑤𝑤 is the garden’s width, in feet. a.

Find the zeros of the equation 𝐴𝐴 = 180𝑤𝑤 − 2𝑤𝑤 2 .

b.

Interpret both of the zeros in terms of the context.

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A statistician gathers data over several years, starting in 1995, on the average number of trips a household makes to run errands. The statistician uses the data to find the equation, 𝐸𝐸(𝑥𝑥) = −2𝑥𝑥 2 + 68𝑥𝑥 + 480, which models the average annual number of trips for errands per household for years since 1995. The graph of the equation is shown.

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a.

Interpret 𝐸𝐸(12). Include units.

b.

Determine the zeros of the function.

Show your work.

c.

Interpret both of the zeros in terms of the context.

d.

Discuss with your partner whether or not both zeros are viable. If a zero is non-viable, explain why.

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3.2.4 Your Turn!

1.

The equation 𝐻𝐻(𝑥𝑥) = −16𝑥𝑥 2 + 64𝑥𝑥 + 192 models the height, 𝐻𝐻, in feet, of a toy rocket after 𝑥𝑥 seconds that it is launched from the top of a 16-story building. a.

Determine the zeros of the function.

Show your work.

b.

Interpret both of the zeros in terms of the context.

c.

Determine whether or not both zeros are viable. If a zero is non-viable, explain why.

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3.2.5 Wrap-Up: Ticket Out the Door A firework is launched from a hill that is 80 feet (ft) above a lake. The firework will descend into the lake after it has exploded at its maximum height. The firework’s height, in ft, above the lake’s surface is given by ℎ(𝑡𝑡) = −16𝑡𝑡 2 + 64𝑡𝑡 + 80, where 𝑡𝑡 is the time in seconds.

1.

Determine the zeros of the function. Show your work.

2.

Determine the viable solution and interpret in terms of the context.

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WRITINGAND ANDSOLVING SOLVINGQUADRATIC QUADRATICEQUATIONS EQUATIONS WRITING

LESSON 9.7 LESSON SOLVING QUADRATICS

BY FACTORING SOLVING QUADRATICS

BY FACTORING LESSON INTRODUCTION MA.912.AR.3.1 Given a mathematical or real-world context, write and solve one-variable quadratic equations over the real number system.

LEARNING TARGETS ● I can solve a quadratic by factoring.

BENCHMARK COHERENCE BUILDING ON MA.8.AR.2.3 Given an equation in the form of x2 = p and x3 = q, where p is a whole number and q is an integer, determine the real solutions.

WORKING TOWARD MA.912.AR.3.2 Given a mathematical or real-world context, write and solve one-variable quadratic equations over the real number system.

ESSENTIAL QUESTIONS ● How can my knowledge of factoring help solve quadratic equations? ● When is factoring an appropriate method for solving quadratic equations?

LESSON NARRATIVE In this lesson, students will apply their knowledge of factoring polynomial expressions to solving quadratic equations. Students will be introduced to the zero-product property and learn how their ability to factor can help solve certain quadratic equations.

COMPONENT SUMMARY 9.7.1 WARM-UP (~5 MIN.)

9.7.2 ACTIVITY (15 MIN.)

Reading a passage about Valerie Thomas and relating satellite dishes to parabolas

Matching equations, graphs, intercepts, and factored expressions Card Sort

9.7.3 GUIDED INSTRUCTION (10 MIN.)

9.7.4 TRY IT (10 MIN.)

Introduction to solving a quadratic equation by factoring

Solving quadratic equations by factoring practice

9.7.5 WRAP-UP (~5 MIN.) Error Analysis activity on solving a quadratic by factoring.

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UNIT 9 – Lesson 7

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RESOURCES GLOSSARY Key Terms:

MATERIALS ● Cards for Card Sort Activity

● Zero product property Additional Terms: ● Factoring

ADDITIONAL PREPARATION ● Work through each of these questions prior to teaching.

After each question, highlight where students may have misconceptions during the factoring process.

TEACHER TIPS COMMON MISCONCEPTIONS ● ● ● ●

Students may choose the wrong factors.

Students may use the wrong sign of the factors.

Students may make calculation errors when factoring.

Students may forget to find both solutions after factoring.

THINGS TO CONSIDER ● Students have already learned how to factor in earlier lessons. Some students may need additional support for factoring, and not as much on using the zero-product property. Consider your student population when designing your lesson.

LITERACY AND LANGUAGE ROUTINES Notice and Wonder The 9.7.1 Warm-Up for this lesson contains a passage about Valerie Thomas and her role in developing satellite dishes. Students can discuss what they notice and wonder about the relationship between the satellite dish and the parabola.

MATHEMATICAL THINKING AND REASONING (MTR) STANDARD MA.K12.MTR.4.1 Discussion and Reflection In the 9.7.5 Wrap-Up, students will be provided with two work samples from two different students solving the same question.

Students will be asked to determine the error in each work sample and provide an explanation of how to correctly solve the question.

DIFFERENTIATE INSTRUCTION 9.7.4 Try It provides students with an opportunity for solving quadratic equations by factoring. Students will be able to show mastery of multiple components of the question, allowing for more easily identifiable differentiation opportunities.

9.7.2 Activity provides students with an opportunity to engage in a card sort. Consider having students do all three questions with real cards for kinesthetic learners to connect to the content.

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UNIT 9 – Lesson 7

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9.7.1 WARM-UP: SATELLITE DISHES 9.7.1 Warm-Up: Satellite Dishes

TEACHER MOVES Consider a Notice and Wonder learning routine for this Warm-Up. Have students share out everything they notice and wonder as a class. Then release the students to work with a partner or group to answer the remaining questions. Position students to analyze the relationship between the key features of a parabola and the satellite dish. Probe student thinking about other objects in the real-world that may have also been designed using a parabolic curve.

Valerie Thomas is a scientist and inventor. In 1980, she received a patent for inventing the illusion transmitter. This was an integral piece of technology in developing the digital media formats needed for today’s video games, movies, and 3D television.

Thomas worked at NASA from 1964-1995, where she helped develop real-time computer data systems to support satellite operations. She oversaw the creation of the Landsat program there, which supported the first satellite to transmit images from outer space.

Today, satellite dishes are everywhere. Satellite dishes are designed using a parabolic curve.

1. What key feature of a parabola is the bottom of the dish? vertex

ENRICHMENT What are other objects in the real-world that are shaped as a parabola?

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2.

The arrows show how signals bounce off a satellite dish to the receiver. The receiver is always located at a parabola’s focus. What key feature does the focus lie on?

Axis of symmetry

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9.7.2 ACTIVITY: CARD SORT 9.7.2 Activity: Card Sort 1.

Work with your partner to match the cards.

Equation P Q R S

2.

Graph J L K M

𝒙𝒙𝒙𝒙-intercept D A B A

𝒙𝒙𝒙𝒙-intercept C B C D

Factor each expression.

Expression Expression Expression Expression T U V W 2𝑥𝑥𝑥𝑥 2 + 5𝑥𝑥𝑥𝑥 − 42� 2𝑥𝑥𝑥𝑥 2 − 5𝑥𝑥𝑥𝑥 − 42� 2𝑥𝑥𝑥𝑥 2 − 19𝑥𝑥𝑥𝑥 + 42� 2𝑥𝑥𝑥𝑥 2 + 19𝑥𝑥𝑥𝑥 + 42� (𝑥𝑥𝑥𝑥 + 6)(2𝑥𝑥𝑥𝑥 − 7) (𝑥𝑥𝑥𝑥 − 6)(2𝑥𝑥𝑥𝑥 + 7) (𝑥𝑥𝑥𝑥 − 6)(2𝑥𝑥𝑥𝑥 − 7) (𝑥𝑥𝑥𝑥 + 6)(2𝑥𝑥𝑥𝑥 + 7) Work with your partner to match the factored expressions to the 𝑥𝑥𝑥𝑥-intercepts.

Factored 𝒙𝒙𝒙𝒙-intercepts Expression Expression T P (𝑥𝑥𝑥𝑥 + 6)(2𝑥𝑥𝑥𝑥 − 7) 𝑥𝑥𝑥𝑥 = −6 and 𝑥𝑥𝑥𝑥 = 3.5 Q (𝑥𝑥𝑥𝑥 − 6)(2𝑥𝑥𝑥𝑥 + 7) 𝑥𝑥𝑥𝑥 = 6 and 𝑥𝑥𝑥𝑥 = −3.5 U R (𝑥𝑥𝑥𝑥 − 6)(2𝑥𝑥𝑥𝑥 − 7) 𝑥𝑥𝑥𝑥 = 6 and 𝑥𝑥𝑥𝑥 = 3.5 V W S (𝑥𝑥𝑥𝑥 + 6)(2𝑥𝑥𝑥𝑥 + 7) 𝑥𝑥𝑥𝑥 = −6 and 𝑥𝑥𝑥𝑥 = −3.5

DIFFERENTIATE INSTRUCTION Consider having students do all three questions with real cards and then posting them on a large visual like chart paper for kinesthetic and visual learners to connect to the content. In the interest of time, it would be beneficial to have the cards cut out and ready prior to the lesson, to allow more time to be spent on the activity.

3.

4.

Discuss with your partner how the factored expression reveals the 𝑥𝑥𝑥𝑥-intercepts. Make a conjecture about the connection.

Answers will vary. The 𝑥𝑥𝑥𝑥-intercept is the opposite sign of each factor. The 𝑥𝑥𝑥𝑥-intercepts are the numbers inside the parenthesis, but with different signs.

TEACHER MOVES Consider making each question an instructional event in the classroom.

• Review the instructions for a question • Set a time to complete • Call time • Select groups to share their response • Poll class for agree/disagree • Move to next activity

DIFFERENTIATE INSTRUCTION Try having students switch groups/partners for each question to have different experiences and conversations.

ENRICHMENT What is the relationship between the factored form and the x-intercepts?

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UNIT 9 – Lesson 7

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9.7.3 GUIDED INSTRUCTION: SOLVING QUADRATIC EQUATIONS BY FACTORING

9.7.3 Guided Instruction: Solving Quadratic Equations by Factoring 1. In the equation 𝑥𝑥𝑥𝑥 ∙ 𝑦𝑦𝑦𝑦 ∙ 𝑧𝑧𝑧𝑧 = 0, which variable has to equal 0 so that the product is equal to 0?

TEACHER MOVES

o o o o

Make a connection to what students learned about factoring in a previous unit. Highlight key differences, including that in this scenario they are working with equations, not expressions.

𝑥𝑥𝑥𝑥 𝑦𝑦𝑦𝑦 𝑧𝑧𝑧𝑧 at least one of 𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦, or 𝑧𝑧𝑧𝑧

Zero Product Property

2. �

If the product of two numbers is 0, then at least one of the numbers is 0.

For each expression, use the zero product property to determine the values of 𝑥𝑥𝑥𝑥 for which the expression would equal 0.�

Expression (𝑥𝑥𝑥𝑥 + 7)(𝑥𝑥𝑥𝑥 − 2)� 𝒙𝒙𝒙𝒙-values

𝑥𝑥𝑥𝑥 = −7���� �𝑥𝑥𝑥𝑥 = 2�

(2𝑥𝑥𝑥𝑥 + 7)(𝑥𝑥𝑥𝑥 − 6)�

𝑥𝑥𝑥𝑥 = −3.5���� �𝑥𝑥𝑥𝑥 = 6�

(−3𝑥𝑥𝑥𝑥 + 11)(4𝑥𝑥𝑥𝑥 + 18)� 11

𝑥𝑥𝑥𝑥 = 3 ����𝑥𝑥𝑥𝑥 = −4.5�

� Each of these expressions could have been written as quadratic equations that are set equal to 0�such as 𝑥𝑥𝑥𝑥 2 + 5𝑥𝑥𝑥𝑥 − 14 = 0.

3.

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When solving a quadratic equation set the equation equal to 0, the solution(s) will be the value(s) of 𝑥𝑥𝑥𝑥 when 𝑦𝑦𝑦𝑦 = 0. Explain how this relates to your conjecture from the card sort.

Answers will vary. This would be similar to finding the 𝑥𝑥𝑥𝑥-intercepts during the card sort activity. I set the 𝑦𝑦𝑦𝑦 equal to zero during the card sort activity.

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Another method of solving quadratics is by factoring. When solving by factoring the quadratic must be set equal to 0.

Once the quadratic is factored, the zero product property is applied.

4.

5.

Solve ℎ2 − 10ℎ + 25 = 0 by factoring. (ℎ − 5)(ℎ − 5) = 0 (ℎ − 5) = 0 (ℎ − 5) = 0 ℎ=5 ℎ=5

Solve 10𝑓𝑓𝑓𝑓 2 + 13𝑓𝑓𝑓𝑓 − 3 = 0 by factoring.

(2𝑓𝑓𝑓𝑓 + 3)(5𝑓𝑓𝑓𝑓 − 1) = 0 (2𝑓𝑓𝑓𝑓 + 3) = 0 (5𝑓𝑓𝑓𝑓 − 1) = 0 3 1 𝑓𝑓𝑓𝑓 = − 2 𝑓𝑓𝑓𝑓 = 5

To solve by factoring, the quadratic equation should be in standard form and set equal to zero.

6.

7.

Solve 𝑝𝑝𝑝𝑝2 + 36 = 13𝑝𝑝𝑝𝑝 for 𝑝𝑝𝑝𝑝 by factoring. 𝑝𝑝𝑝𝑝2 − 13𝑝𝑝𝑝𝑝 + 36 = 0 (𝑝𝑝𝑝𝑝 − 9)(𝑝𝑝𝑝𝑝 − 4) = 0 (𝑝𝑝𝑝𝑝 − 9) = 0 (𝑝𝑝𝑝𝑝 − 4) = 0 𝑝𝑝𝑝𝑝 = 9 𝑝𝑝𝑝𝑝 = 4

Solve 40𝑥𝑥𝑥𝑥 2 − 18𝑥𝑥𝑥𝑥 = 12𝑥𝑥𝑥𝑥 + 135 for 𝑥𝑥𝑥𝑥 by factoring.

40𝑥𝑥𝑥𝑥 2 − 30𝑥𝑥𝑥𝑥 − 135 = 0 5(8𝑥𝑥𝑥𝑥 2 − 6𝑥𝑥𝑥𝑥 − 27) = 0 5(4𝑥𝑥𝑥𝑥 − 9)(2𝑥𝑥𝑥𝑥 + 3) = 0 (4𝑥𝑥𝑥𝑥 − 9) = 0 (2𝑥𝑥𝑥𝑥 + 3) = 0 9 3 𝑥𝑥𝑥𝑥 = 4 𝑥𝑥𝑥𝑥 = − 2

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9.7.3 GUIDED INSTRUCTION: SOLVING QUADRATIC EQUATIONS BY FACTORING (CONTINUED) TEACHER MOVES Have students relate what they did in their 9.7.2 Activity to what they are experiencing now. Students will begin making the connection between their knowledge of factoring and their ability to solve quadratic equations. Help students build their understanding of the zero-product property by naming the property when setting each factor equal to 0.

DIFFERENTIATE INSTRUCTION Differentiation may be necessary at this stage if students did not do well with the factoring pieces of the 9.7.2 Activity. If students struggle with factoring, this additional step may add too much complexity for the student.

ENRICHMENT • How do questions 6 and 7 looks different than questions 4 and 5?

• What would we need to do before factoring?

• How does this relate to applying the zero-product property?

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9.7.4 TRY IT: FACTORING 9.7.4 Try It: Factoring

DIFFERENTIATE INSTRUCTION Consider grouping students together who have a deficit in factoring. Evidence from their performance on the factoring unit and their performance on the 9.7.2 Activity and 9.7.3 Guided Instruction in this lesson should be used to determine which students will need support. Work with these students on the factoring component so that they can complete the work of this concept.

Solve each by factoring. Show your work.

1.

2.

𝑥𝑥𝑥𝑥 2 − 10𝑥𝑥𝑥𝑥 − 15 = 9 𝑥𝑥𝑥𝑥 2 − 10𝑥𝑥𝑥𝑥 − 24 = 0 (𝑥𝑥𝑥𝑥 − 12)(𝑥𝑥𝑥𝑥 + 2) = 0 (𝑥𝑥𝑥𝑥 − 12) = 0 (𝑥𝑥𝑥𝑥 + 2) = 0 𝑥𝑥𝑥𝑥 = 12 𝑥𝑥𝑥𝑥 = −2

−3𝑥𝑥𝑥𝑥 2 + 6𝑥𝑥𝑥𝑥 + 22 = 2 − 5𝑥𝑥𝑥𝑥 −3𝑥𝑥𝑥𝑥 2 + 11𝑥𝑥𝑥𝑥 + 20 = 0 −(3𝑥𝑥𝑥𝑥 2 − 11𝑥𝑥𝑥𝑥 − 20) = 0 −(𝑥𝑥𝑥𝑥 − 5)(3𝑥𝑥𝑥𝑥 + 4) = 0 (𝑥𝑥𝑥𝑥 − 5) = 0 (3𝑥𝑥𝑥𝑥 + 4) = 0 4 𝑥𝑥𝑥𝑥 = 5 𝑥𝑥𝑥𝑥 = − 3

ENRICHMENT To factor, what must the equation be set equal to? Ask students their strategies for question 2. Students may need to be reminded that the polynomial on the left side of the equation could be moved to the right side of the equation, resulting in a positive leading coefficient.

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9.7.5 WRAP-UP: ERROR ANALYSIS 9.7.5 Wrap-Up: Error Analysis The work of two students who solved 2𝑥𝑥𝑥𝑥 2 − 12𝑥𝑥𝑥𝑥 + 14 = 2𝑥𝑥𝑥𝑥 − 6 is shown.

Ruchelle

Quinyang

2𝑥𝑥𝑥𝑥 2 − 12𝑥𝑥𝑥𝑥 + 14 = 2𝑥𝑥𝑥𝑥 − 6�

2𝑥𝑥𝑥𝑥 2 − 12𝑥𝑥𝑥𝑥 + 14 = 2𝑥𝑥𝑥𝑥 − 6�

2(𝑥𝑥𝑥𝑥 2 − 6𝑥𝑥𝑥𝑥 + 7) 2(𝑥𝑥𝑥𝑥 − 3) = � 2 2

2𝑥𝑥𝑥𝑥 2 − 14𝑥𝑥𝑥𝑥 + 20 = 0�

2(𝑥𝑥𝑥𝑥 2 − 6𝑥𝑥𝑥𝑥 + 7) = 2(𝑥𝑥𝑥𝑥 − 3)� 2𝑥𝑥𝑥𝑥 2 − 12𝑥𝑥𝑥𝑥 + 14 − (2𝑥𝑥𝑥𝑥 − 6) = 2𝑥𝑥𝑥𝑥 − 6 − (2𝑥𝑥𝑥𝑥 − 6)�

𝑥𝑥𝑥𝑥 2 − 6𝑥𝑥𝑥𝑥 + 7 = 𝑥𝑥𝑥𝑥 − 3�

(2𝑥𝑥𝑥𝑥 − 5)(𝑥𝑥𝑥𝑥 − 4) = 0�

𝑥𝑥𝑥𝑥 2 − 5𝑥𝑥𝑥𝑥 + 4 = 0�

(𝑥𝑥𝑥𝑥 − 4)(𝑥𝑥𝑥𝑥 − 1) = 0�

𝑥𝑥𝑥𝑥 =

5 2

Both Ruchelle and Quinyang made one mistake. Circle each mistake.

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UNIT 9 – Lesson 7

MN FL26 AGA Sampler FINAL.indb 52

This activity will provide differentiation data for which students need additional supports in which component of the process.

The component the students miss and/or get correct in the error analysis could determine what a student understands to provide differentiation and support if necessary.

or 𝑥𝑥𝑥𝑥 = 4

𝑥𝑥𝑥𝑥 = 4 or 𝑥𝑥𝑥𝑥 = 1

1.

DIFFERENTIATE INSTRUCTION

QUESTIONING For an additional challenge, have students explain or correct the mistake in the process.

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RIGIDMOTIONS MOTIONSAND ANDCONGRUENCE CONGRUENCE RIGID

LESSON 4.2 LESSON ALGEBRAIC DESCRIPTIONS FOR REFLECTIONS ALGEBRAIC DESCRIPTIONS FOR REFLECTIONS LESSON INTRODUCTION MA.912.GR.2.1 Given a preimage and image, describe the transformation and represent the transformation algebraically using coordinates.

LEARNING TARGETS ● I can relate a reflection on the coordinate plane to an algebraic notation of the transformation.

BENCHMARK COHERENCE BUILDING ON MA.8.GR.2.1 Given a preimage and image generated by a single transformation, identify the transformation that describes the relationship.

WORKING TOWARD N/A

ESSENTIAL QUESTIONS ● What is a reflection? ● How can reflections be represented algebraically?

LESSON NARRATIVE The main part of the lesson, 4.2.2 Exploration, immerses students in an in-depth exploratory discovery of the effects of reflections across a variety of lines. Students focus on the ordered pairs of the vertices and the midpoint of line segments that connect corresponding vertices across the line of reflection. Students engage in language routines and math thinking and reasoning to draw conclusions leading toward generating algebraic descriptions of reflections.

COMPONENT SUMMARY 4.2.1 WARM-UP (~5 MIN.)

4.2.2 EXPLORATION (25-30 MIN.)

Critical thinking problem

Exploration of reflections across a variety of lines of reflection

4.2.3 BRING IT TOGETHER (5-10 MIN.)

4.2.4 WRAP-UP (~5 MIN.)

Students summarize and synthesize their findings from the Exploration

Students relate elements of reflections to a squish painting

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UNIT 4 – Lesson 2

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GEOMETRY Teacher Guide

RESOURCES GLOSSARY Key Terms: ● N/A

MATERIALS

ADDITIONAL PREPARATION

● (Optional) Integer chips or other chips for 4.2.1 Warm-Up ● (Optional) Squish painting

● N/A

Additional Terms: ● N/A

TEACHER TIPS COMMON MISCONCEPTIONS ● Students may struggle differentiating between axes, values in an ordered pair, and their opposites. ● Students who struggle in spatial reasoning might find it difficult to navigate the visual elements of reflections.

THINGS TO CONSIDER ● Consider using sentence stems to actively convert student verbal descriptions into mathematical language. For example, when students are describing the effects of reflection like, “They are in the same place,” provide scaffolded phrases like, “Vertex __ is the same distance away from __ as Vertex __.”

LITERACY AND LANGUAGE ROUTINES Notice and Wonder 4.2.2 Exploration questions 1c-f provide two opportunities for students to make sense of their observations in mathematically meaningful ways. Encourage students to utilize Notice and Wonder techniques for each element to help each other discover and produce algebraic descriptions based on the Exploration.

MATHEMATICAL THINKING AND REASONING (MTR) STANDARD MA.K12.MTR.4.1 Discussion and Reflection 4.2.2 Exploration question 2b provides an opportunity for students to engage in mathematical discussion by analyzing and critiquing the thinking of others. Use this as an opportunity to provide students with practice analyzing the thinking of others and then articulating their own thinking in response.

DIFFERENTIATE INSTRUCTION For a kinesthetic entry point, consider using physical manipulatives such as patty paper or cut-outs for students to explore the reflections in 4.2.2 Exploration.

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4.2.1 WARM-UP: MOVING CIRCLES 4.2.1 Warm-Up: Moving Circles

TEACHER MOVES

1.

Move any three circles so that the pyramid points down.

The arrows provided here are for answer key purposes. Students will only have the blue dots visible.

ENRICHMENT Consider prompting questions to target specific actions of Growth Mindset.

• Was the puzzle a struggle, or was it easy for you?

• What did you look for first when making pyramid point down?

• What did you try first? What strategy or different way of looking at the pyramid helped you know what to do next?

DIFFERENTIATE INSTRUCTION Do you have integer chips or similar manipulatives? Students could interact with this pyramid prompt for a kinesthetic learning experience. This could also be done with digital manipulatives.

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1.

4.2.2 Exploration: Discovering Algebraic Descriptions for Reflections

Quadrilaterals 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 and 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 are shown on the coordinate grid. Quadrilateral 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 was reflected across the 𝑦𝑦𝑦𝑦-axis to create quadrilateral 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇.

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4.2.2 EXPLORATION: DISCOVERING ALGEBRAIC DESCRIPTIONS FOR REFLECTIONS TEACHER MOVES This Exploration is scaffolded to allow students to engage with it with their partner rather than being guided through by the teacher. There will be an opportunity to summarize and synthesize what is discovered in the next component. Students may revise their thinking throughout the exploration as they engage with each new reflection.

Observe and listen as students work with their partners. Make note of student ideas that you can ask them to share in the whole group Bring it Together.

a.

Draw a line segment between each set of corresponding vertices.

Shown on figure.

b.

Describe how the line segments between the vertices are related to the 𝑦𝑦𝑦𝑦-axis.

Answers will vary. Each vertex is the same distance from the 𝑦𝑦𝑦𝑦-axis as its corresponding vertex. The 𝑦𝑦𝑦𝑦-axis bisects each line segment. The line segments between the vertices are perpendicular to the 𝑦𝑦𝑦𝑦-axis.

Additional scaffolding questions are provided, if needed, but this should be a primarily student-directed exploration.

TEACHER MOVES During the exploration, allow students to describe relationships using their own words first. Students informally experienced reflections in Unit 1, so the exploration helps them recall that information. For students who are struggling with descriptions, refer them to Unit 1. As you circulate, if student descriptions show misunderstanding, then ask probing questions like: • What is the relationship (ZW) ̅has with the y-axis?

• Is that relationship the same for the other line segments?

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4.2.2 EXPLORATION: DISCOVERING ALGEBRAIC DESCRIPTIONS FOR REFLECTIONS (CONTINUED)

c.

Complete the table.

𝑯𝑯𝑯𝑯 to 𝑻𝑻𝑻𝑻 𝒁𝒁𝒁𝒁 to 𝑾𝑾𝑾𝑾 12 10 Distance Ordered (0,2) (0,3) Pair of the Midpoint

d.

TEACHER MOVES Notice and Wonder Routine

e.

1c-1d and 1e-1f provide the space and opportunity for students to discover how the ordered pairs of the vertices after a reflection can be used to generate algebraic descriptions. Students may not be engaging in a formal Notice and Wonder routine, but they can be prompted to notice and wonder about what they see for additional entry points when stuck in parts like question 1e.

f.

Emphasize the word “opposite” in student discussion, as this word is seen in later questions of this lesson.

TEACHER MOVES MA.K12.MTR.5.1: Patterns and Structure The exploration provides students the opportunity to use patterns and structure to help understand and connect previously learned concepts (reflections) to new concepts (representing them with algebraic descriptions).

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g.

𝑹𝑹𝑹𝑹 to 𝑩𝑩𝑩𝑩 10

(0,1)

(0, −1)

What do you notice about the midpoints?

Answers will vary. All 𝑥𝑥𝑥𝑥-values are zero. All of the midpoints are on the 𝑦𝑦𝑦𝑦-axis.

The table shows the coordinates of each vertex. 𝒚𝒚𝒚𝒚 Vertex Vertex 𝒙𝒙𝒙𝒙 𝐻𝐻𝐻𝐻 � −6 2 𝑇𝑇𝑇𝑇 𝐻𝐻𝐻𝐻� −5 3 𝑇𝑇𝑇𝑇 �� −2 1 𝑇𝑇𝑇𝑇 �� −5� −1 𝑇𝑇𝑇𝑇

Notice the intentional scaffolding of language routine elements embedded within the Exploration. Questions

𝑳𝑳𝑳𝑳 to 𝑲𝑲𝑲𝑲 4

𝒙𝒙𝒙𝒙 6 5 2 5

𝒚𝒚𝒚𝒚 2 3 1 −1

Work with your partner to describe how the coordinates of the pre-image, quadrilateral 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻, are related to the coordinates of the image, quadrilateral 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇.

Answers will vary. The 𝑥𝑥𝑥𝑥-values are opposites and the 𝑦𝑦𝑦𝑦-values are the same.

Write a conjecture that you and your partner think will work for the algebraic description of a reflection.

Answers will vary. Since the polygon is reflecting over the 𝑦𝑦𝑦𝑦-axis, the 𝑥𝑥𝑥𝑥-values change to their opposite sign. (𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦) → (−𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦)

Discuss with your partner if the algebraic description will work for any reflection. Summarize your conclusion.

Answers will vary. The point or line of reflection matters. Reflections over the 𝑦𝑦𝑦𝑦-axis will follow that description, but if the line of reflection is not the 𝑦𝑦𝑦𝑦axis, then a new algebraic description would be needed.

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2.

Polygons 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 and 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 are shown on the coordinate grid. Polygon 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 was reflected across the 𝑥𝑥𝑥𝑥-axis to create polygon 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻.

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4.2.2 EXPLORATION: DISCOVERING ALGEBRAIC DESCRIPTIONS FOR REFLECTIONS (CONTINUED) ENRICHMENT

a.

b.

Describe the relationship that the line segments between each set of corresponding vertices have with the 𝑥𝑥𝑥𝑥-axis.

Answers will vary. Each vertex is the same distance from the 𝑥𝑥𝑥𝑥-axis as its corresponding vertex. The line segments between corresponding vertices are perpendicular to the 𝑥𝑥𝑥𝑥-axis.

Wendall and Samah were working together on writing an algebraic description for reflections across the 𝑥𝑥𝑥𝑥-axis. Each student’s algebraic description and explanation of their thinking is shown in the table.

Wendall (𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦) → (−𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦)

Samah (𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦) → (𝑥𝑥𝑥𝑥, −𝑦𝑦𝑦𝑦) Because above the 𝑥𝑥𝑥𝑥-axis all 𝑦𝑦𝑦𝑦-values are positive and Because the polygon flipped over the 𝑥𝑥𝑥𝑥-axis, the 𝑥𝑥𝑥𝑥 below the 𝑥𝑥𝑥𝑥-axis all 𝑦𝑦𝑦𝑦-values becomes its opposite. are negative, the 𝑦𝑦𝑦𝑦 becomes its opposite.

Monitor student discussion for question 2a. If necessary, prompt students to use the scaffolding from question 1 similarly in question 2. Draw line segments between each set of corresponding vertices from polygon MCRTA to polygon HVGNW. Prompting questions could include the following: • What do you notice about the midpoints of the line segments?

• What do you notice about the coordinates of each ordered pair?

• What do you notice about the line of reflection and the line segments?

• How are these relationships similar to and different from the previous reflection?

TEACHER MOVES MA.K12.MTR.4.1: Discussion and Reflection Students have the opportunity to critique the reasoning and thinking of others. Consider having students or their partners read this passage out loud for others to listen and respond to in conversation. The error analysis of question 2b has students look at the common misconception of a reflection across the x-axis resulting in the x-value changing.

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4.2.2 EXPLORATION: DISCOVERING ALGEBRAIC DESCRIPTIONS FOR REFLECTIONS (CONTINUED)

Discuss with your partner which, if any, algebraic description is correct. Use at least two ordered pairs from polygons 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 and 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 to defend your choice. Write a summary of your discussion.

Samah is correct.

Answers will vary. Samah is correct because 𝑀𝑀𝑀𝑀(4, −1) becomes 𝐻𝐻𝐻𝐻(4,1) and 𝑀𝑀𝑀𝑀(2, −4) becomes 𝐻𝐻𝐻𝐻(2,4). You can see that the 𝑥𝑥𝑥𝑥-values remained the same but the 𝑦𝑦𝑦𝑦-values are now the opposite sign.

QUESTIONING Question 3b seeks to build a conceptual connection to the previous lesson. Consider how to best engage students with the two elements of question 3 by beginning with a question like, “Does this look familiar? Why or why not?” and suggest they start with question 3b before finishing on question 3a.

3.

Triangles 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝑀𝑀𝑀𝑀 and 𝐻𝐻𝐻𝐻′𝐻𝐻𝐻𝐻′𝑀𝑀𝑀𝑀′ are shown on the coordinate grid. Triangle 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝑀𝑀𝑀𝑀 was reflected across the origin to create ∆𝐻𝐻𝐻𝐻′𝐻𝐻𝐻𝐻′𝑀𝑀𝑀𝑀′.

a.

b.

QUESTIONING

4.

If time permits, transform your classroom into a coordinate plane and challenge students to create a mathematically accurate reflection using elements like desks, chairs, bags, and more!

5.

Discuss with your partner the algebraic description for a reflection across the origin. (𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦) → (−𝑥𝑥𝑥𝑥, −𝑦𝑦𝑦𝑦)

Re-examine the algebraic descriptions of rotations.

Which rotation is similar to a reflection across the origin?

180° rotation

Triangles 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 and 𝐾𝐾𝐾𝐾′𝐾𝐾𝐾𝐾′𝐾𝐾𝐾𝐾′ are shown on the coordinate grid. Triangle 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 was reflected across the line 𝑦𝑦𝑦𝑦 = 𝑥𝑥𝑥𝑥 to create ∆𝐾𝐾𝐾𝐾′𝐾𝐾𝐾𝐾′𝐾𝐾𝐾𝐾′.

Use the coordinates to determine the algebraic description for a reflection across the line 𝑦𝑦𝑦𝑦 = 𝑥𝑥𝑥𝑥. (𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦) → (𝑦𝑦𝑦𝑦, 𝑥𝑥𝑥𝑥)

Triangles 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 and 𝑃𝑃𝑃𝑃𝐾𝐾𝐾𝐾𝑃𝑃𝑃𝑃 are shown on the coordinate grid. Triangle 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 was reflected across the line 𝑦𝑦𝑦𝑦 = −𝑥𝑥𝑥𝑥 to create ∆𝑃𝑃𝑃𝑃𝐾𝐾𝐾𝐾𝑃𝑃𝑃𝑃.

Use the coordinates to determine the algebraic description for a reflection across the line 𝑦𝑦𝑦𝑦 = −𝑥𝑥𝑥𝑥. (𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦) → (−𝑦𝑦𝑦𝑦, −𝑥𝑥𝑥𝑥)

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4.2.3 Bring It Together: Algebraic Descriptions for Reflections Reflections Across the 𝒙𝒙𝒙𝒙-axis Across the 𝒚𝒚𝒚𝒚-axis The 𝑦𝑦𝑦𝑦 −coordinate changes to The 𝑥𝑥𝑥𝑥 −coordinate changes its opposite. to its opposite. (𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦) → (𝑥𝑥𝑥𝑥, −𝑦𝑦𝑦𝑦) (𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦) → (−𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦) Across 𝒚𝒚𝒚𝒚 = 𝒙𝒙𝒙𝒙 Across 𝒚𝒚𝒚𝒚 = −𝒙𝒙𝒙𝒙 The coordinates switch The coordinates switch places and become places. opposite. (𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦) → (𝑦𝑦𝑦𝑦, 𝑥𝑥𝑥𝑥) (𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦) → (−𝑦𝑦𝑦𝑦, −𝑥𝑥𝑥𝑥)

4.2.3 BRING IT TOGETHER: ALGEBRAIC DESCRIPTIONS FOR REFLECTIONS TEACHER MOVES Use this section to encourage synthesis, not memorization.

Call on students to share their ideas following the exploration to highlight student voice and solidify takeaways from the Exploration.

4.2.4 WRAP-UP: COOL DOWN 4.2.4 Wrap-Up: Cool Down A reflection can be thought of as folding or "flipping" an object over the line of reflection. A squish painting is a fun application of a line of reflection.

To make a squish painting, a large sheet of construction paper is folded in half then opened up. Paint is dotted along one side of the fold. The paper is folded, and the folded paper is rubbed, pounded, and squished to move the paint around. The paper is then opened up to reveal the squish painting.

1. How does a squish painting show that in a reflection, each point of the original figure and its image are the same distance away from the line of reflection?

QUESTIONING Consider what this Cool Down might look like as a performance task or interdisciplinary project with an art class. Students could be immersed into an exploration of reflections in art to establish a strong foundation for key elements of reflections, while the 4.2.2 Exploration can serve to quantify and make mathematical sense of their findings of reflections in art.

Answers will vary. In a squish painting, you can see how close each part of the painting is to the crease in the paper.

You can see that the picture is a mirror image. The fold is the line of reflection.

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LESSON 3.2 LESSON DETERMINING ZEROS WITHIN ZEROS A REAL-WORLD CONTEXT DETERMINING WITHIN A REAL-WORLD CONTEXT LESSON INTRODUCTION MA.912.AR.3.2 Given a mathematical or real-world context, write and solve one-variable quadratic equations over the real- and complex number systems.

LEARNING TARGETS ● I can solve a quadratic by factoring when given a real-world context. ● I can interpret solutions as viable or non-viable options given a real-world context.

BENCHMARK COHERENCE BUILDING ON MA.912.AR.3.1 Given a mathematical or real-world context, write and solve one-variable quadratic equations over the real-number system.

WORKING TOWARD N/A

ESSENTIAL QUESTIONS ● How can the validity of solutions be identified when solving quadratic equations by factoring within a real-world context? ● How can a quadratic equation be solved by factoring within a real-world context?

LESSON NARRATIVE In this lesson, students extend their knowledge of solving quadratic equations by factoring from Lesson 1 to real-world contexts.

Given a real-world context, students will apply their knowledge of factoring and the zero-product property to solve quadratic equations. Students will interpret the solutions to quadratic equations within the given context and will further analyze the validity of solutions.

COMPONENT SUMMARY 3.2.1 WARM-UP (~5 MIN)

3.2.2 COLLABORATION (20-15 MIN)

“Would You Rather?” activity on sale prices

Solving real-world quadratic equations by factoring and interpreting solutions

3.2.3 GUIDED INSTRUCTION (10-15 MIN)

3.2.4 YOUR TURN! (5-10 MIN)

Solving real-world quadratic equations by factoring and interpreting solutions

Analyzing the flight path of a toy rocket modeled by a quadratic equation

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RESOURCES GLOSSARY Key Terms:

MATERIALS ● (optional) Graphing utility

● N/A Additional Terms:

ADDITIONAL PREPARATION ● Consider creating a review of factoring techniques, including factoring out the greatest common factor, to aid struggling students.

● Roots ● Zeros ● Zero-product property

TEACHER TIPS COMMON MISCONCEPTIONS ● Students may believe that quadratic equations set equal to zero must always have the zero on the right side of the equation, causing confusion with function notation.

THINGS TO CONSIDER ● Although factoring is a review skill, students still struggle. Consider reviewing all forms of factoring to aid struggling students. It may also benefit students to utilize multiple methods, such as algebra tiles, area model, grouping, etc.

LITERACY AND LANGUAGE ROUTINES Three Reads Consider a Three Reads strategy for 3.2.3 Guided Instruction (struggling students may benefit from utilizing this strategy throughout the entire lesson). Students often struggle with word problems, so the use of this strategy will ensure that students know what they are being asked to do. This strategy supports reading comprehension of questions and negotiating information in a text with a partner in a mathematical conversation.

MATHEMATICAL THINKING AND REASONING (MTR) STANDARD MA.K12.MTR.7.1 Real-World Contexts This lesson extends student understanding of solving quadratic functions by factoring to modeling and solving real-world problems.

In this way, students are connecting the idea of solving quadratic equations by factoring to real-world experiences. Given a realworld context, students are applying the zero-product property to solve quadratic equations and interpreting solutions as viable or nonviable in terms of the given context.

DIFFERENTIATE INSTRUCTION Continue to expose students to the vocabulary by connecting the terms “solutions,” “x-intercepts,” “roots,” and “zeros” of what is being found when solving quadratics. It may benefit struggling students to utilize a graphing utility to draw the connection between zeros and x-intercepts. Students who struggle to understand the zeros of a quadratic equation may benefit from interacting with the corresponding graph.

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3.2.1 WARM-UP: WOULD YOU RATHER? 3.2.1 Warm-Up: Would You Rather?

ENRICHMENT Based on the pricing, what do you think the sale items may be?

TEACHER MOVES

1. Which deal would you rather choose – buy 1, get 2 free for $120, or buy 1 pair that is already 60% off for $48?

Answers will vary.

Turn and Talk Engage students in a Turn and Talk after completing the Warm-Up. Position students to consider the mathematics behind the sale pricing when making their decision. A Turn and Talk routine will allow students to think individually about the questions and then discuss their thoughts with a partner.

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2. Justify your answer mathematically.

Answers will vary. Sample response – I would choose option 1 because if you are buying 3 items for $120, each item would be $40 each, which is cheaper than option 2.

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3.2.2 Collaboration: Using Quadratics in the Music Business 1. Organizers of a music festival charge $60 for admission.

The organizers polled the attendees to determine how increasing the price would affect attendance. The equation 𝑃𝑃𝑃𝑃(𝑥𝑥𝑥𝑥) = −20,000𝑥𝑥𝑥𝑥 2 − 20,000𝑥𝑥𝑥𝑥 + 600,000 models profit as a function of 𝑥𝑥𝑥𝑥, where 𝑥𝑥𝑥𝑥 is each $1 increase in the admission fee. a.

Discuss with your partner which factors would affect your attendance at a music festival. List your top three factors.

Answers will vary. Sample response – Cost of festival, location, and type of music.

b.

TEACHER MOVES Students may see the negative zero in question 1 parts d and e and immediately think that a negative would not work in terms of the given context. Position students to connect negative x-values to a decrease in price.

If x strictly represented the actual price of a ticket, would it still be viable for x to be negative?

TEACHER MOVES

If 𝑥𝑥𝑥𝑥 = 2, what is the admission fee to the festival? $62

d.

3.2.2 COLLABORATION: USING QUADRATICS IN THE MUSIC BUSINESS

What admission fee would you consider to be too high?

Answers will vary.

c.

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Solve −20,000𝑥𝑥𝑥𝑥 2 − 20,000𝑥𝑥𝑥𝑥 + 600,000 = 0 by factoring.

Show your work. −20,000(𝑥𝑥𝑥𝑥 − 5)(𝑥𝑥𝑥𝑥 + 6) = 0

Students who struggle to identify the greatest common factor may not recognize the need to factor out -20,000 in question 1 part d. Consider facilitating a brief class discussion to review the process of identifying the greatest common factor.

𝑥𝑥𝑥𝑥 = −6, 𝑥𝑥𝑥𝑥 = 5

e.

Interpret the solutions in terms of the context.

If the price increases $5 or decreases $6, then the profit will be $0.

f.

Ask two other classmates for their interpretations of the solutions to −20,000𝑥𝑥𝑥𝑥 2 − 20,000𝑥𝑥𝑥𝑥 + 600,000 = 0.

Write a summary of their interpretations of the solutions. You are the only person who should write in the first column of the table. Have each person initial next to your summary stating that your summary was accurate.

My Summary of Their Interpretation of the Solutions in Terms of the Context Answers will vary.

g.

Initials

Review your classmates’ interpretations with your partner. If necessary, adjust your interpretation.

ENRICHMENT Position students to see the relationship between the validity of solutions and the given context.

• What key factors make a negative root okay or not okay?

• Can you think of an example where a negative root would not be okay?

TEACHER MOVES

Answers will vary.

2. The compact disc (CD) was invented in 1979, many years before online music existed. At the time it was the most sophisticated way to store and play music. In 1982, CDs were made available to the public.

CDs quickly became the most efficient way of storing music, which replaced vinyl records and cassette tapes.

The function 𝐶𝐶𝐶𝐶(𝑥𝑥𝑥𝑥) = −2𝑥𝑥𝑥𝑥 2 + 38𝑥𝑥𝑥𝑥 + 40 models the sales, in hundreds of millions of dollars, of compact discs in the years since 1990. a.

Solve −2𝑥𝑥𝑥𝑥 2 + 38𝑥𝑥𝑥𝑥 + 40 = 0 by factoring. −2(𝑥𝑥𝑥𝑥 − 20)(𝑥𝑥𝑥𝑥 + 1) = 0 𝑥𝑥𝑥𝑥 = 20, 𝑥𝑥𝑥𝑥 = −1

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Students may struggle with negative leading coefficients when factoring. Position students to factor out a negative when the leading coefficient is negative.

DIFFERENTIATE INSTRUCTION Continue to expose students to the vocabulary by connecting the terms “solutions,” “x-intercepts,” “roots,” and “zeros” of what is being found when solving quadratics. It may benefit struggling students to utilize a graphing utility to draw the connection between zeros and x-intercepts.

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b.

c.

Do you know anyone who still uses CDs? How does this impact your answer for question 2 part d? d.

Interpret the year that each zero represents.

Zero 𝑥𝑥𝑥𝑥 = 20 𝑥𝑥𝑥𝑥 = −1

Year 2010 1989

Discuss the interpretation of each zero with your partner. Then, determine whether or not the interpretation makes sense. Summarize your discussion.

Answers will vary. Sample response - 𝑥𝑥𝑥𝑥 = 20 means that in 2010 the sales of compact discs was 0 and 𝑥𝑥𝑥𝑥 = −1 means that in 1989, the sales was also 0.

In 2001, iTunes was launched by Apple. For the first time, people could buy songs online and store them on mp3 players, iPods, or on a computer.

Then, in 2008, the music streaming site Spotify was started.

Discuss with your partner why one of the zeros shows that online access to music may have led to CD sales eventually reaching zero. Summarize your discussion.

Answers will vary. Sample Response – when 𝑥𝑥𝑥𝑥 = 20, the year was 2010, which was 9 years after Apple launched iTunes so it would make sense that the sales of CDs would be 0.

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3.2.3 GUIDED INSTRUCTION: FINDING ZEROS 3.2.3 Guided Instruction: Finding Zeros 1. A landscape designer is adding a garden to a client’s backyard. One side of the garden will be alongside an existing fence. The client has 180 feet of fencing that is left over from another project. The client would like the garden to use all of the fencing.

The equation 𝐴𝐴𝐴𝐴 = 180𝑤𝑤𝑤𝑤 − 2𝑤𝑤𝑤𝑤 2 can be used to find the area of the garden, where 𝑤𝑤𝑤𝑤 is the garden’s width. a.

Find the zeros of the equation 𝐴𝐴𝐴𝐴 = 180𝑤𝑤𝑤𝑤 − 2𝑤𝑤𝑤𝑤 2 . −2𝑤𝑤𝑤𝑤(𝑤𝑤𝑤𝑤 − 90) = 0

𝑤𝑤𝑤𝑤 = 0 and 𝑤𝑤𝑤𝑤 = 90

b.

Interpret both of the zeros in terms of the context.

A width of 0 feet is not realistic in the context. If the landscaper creates the garden with a width of 90 feet, there will not be enough left over fencing material for any length for the garden.

2. A statistician gathers data over several years, starting in 1995, on the average number of trips a household makes to run errands. The statistician uses the data to find the equation, 𝐸𝐸𝐸𝐸(𝑥𝑥𝑥𝑥) = −2𝑥𝑥𝑥𝑥 2 + 68𝑥𝑥𝑥𝑥 + 480, which models the average annual number of trips for errands per household for years since 1995. The graph of the equation is shown. a.

b.

Interpret 𝐸𝐸𝐸𝐸(12). Include units.

In 2007, 12 years after 1995, the average number of trips to run errands a household makes per year is 1008.

c.

Interpret both of the zeros in terms of the context.

Determine the zeros of the function. Show your work. 𝑥𝑥𝑥𝑥 = 40, −6

TEACHER MOVES Three Reads Consider a Three Reads strategy for the questions in 3.2.3 Guided Instruction. Students who struggle with interpreting word problems and making sense of the given context may benefit from using this strategy throughout the entire lesson. Students should first read through the word problems to make sense of the context. Then they should reread the problem to make sense of the mathematics involved. Students will then read the problem a third time to design a strategy for solving the problem.

ENRICHMENT Which zero makes more sense in terms of the given context?

TEACHER MOVES Students who struggle with function notation may struggle with question 2 part a. Position students to connect the notation E(12) to evaluating the function for x=12.

TEACHER MOVES Though this lesson largely utilizes the standards in MA.K12.

MTR.7.1 (real-world contexts), students are also utilizing the standards in MA.K12.MTR.6.1 (reasonableness of solutions). For instance, in question 2 part d, students are asked to interpret the reasonableness of the solutions.

The zero of 𝑥𝑥𝑥𝑥 = 40 means that in 2035, 40 years after 1995, and the zero 𝑥𝑥𝑥𝑥 = −6 means that in 1989, 6 years before 1995; households are expected to make an annual average of 0 trips to run errands. d.

Discuss with your partner whether or not both zeros are viable. If a zero is non-viable, explain why.

Both zeros are not viable. The zero 𝑥𝑥𝑥𝑥 = −6 is not viable because people were running errands in 1989 before this data was collected. Also, 𝑥𝑥𝑥𝑥 = 40 is unlikely to be viable as it is unrealistic that the average number of trips to run errands for a household is zero.

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b.

c.

Do you know anyone who still uses CDs? How does this impact your answer for question 2 part d? d.

Interpret the year that each zero represents.

Zero 𝑥𝑥𝑥𝑥 = 20 𝑥𝑥𝑥𝑥 = −1

Year 2010 1989

Discuss the interpretation of each zero with your partner. Then, determine whether or not the interpretation makes sense. Summarize your discussion.

Answers will vary. Sample response - 𝑥𝑥𝑥𝑥 = 20 means that in 2010 the sales of compact discs was 0 and 𝑥𝑥𝑥𝑥 = −1 means that in 1989, the sales was also 0.

In 2001, iTunes was launched by Apple. For the first time, people could buy songs online and store them on mp3 players, iPods, or on a computer.

Then, in 2008, the music streaming site Spotify was started.

Discuss with your partner why one of the zeros shows that online access to music may have led to CD sales eventually reaching zero. Summarize your discussion.

Answers will vary. Sample Response – when 𝑥𝑥𝑥𝑥 = 20, the year was 2010, which was 9 years after Apple launched iTunes so it would make sense that the sales of CDs would be 0.

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3.2.5 WRAP-UP: TICKET OUT THE DOOR 3.2.5 Wrap-Up: Ticket Out the Door A firework is launched from a hill that is 80 feet (ft) above a lake. The firework will descend into the lake after it has exploded at its maximum height. The firework’s height, in ft, above the lake’s surface is given by ℎ(𝑡𝑡𝑡𝑡) = −16𝑡𝑡𝑡𝑡 2 + 64𝑡𝑡𝑡𝑡 + 80, where 𝑡𝑡𝑡𝑡 is the time in seconds.

1. Determine the zeros of the function. Show your work.

QUESTIONING Consider using the Ticket Out the Door as a formative assessment to gather data on student progress and understanding from this lesson.

𝑡𝑡𝑡𝑡 = −1, 𝑡𝑡𝑡𝑡 = 5

2. Determine the viable solution and interpret in terms of the context.

The viable solution is 𝑡𝑡𝑡𝑡 = 5 because it means that it takes 5 seconds for the rocket to touch the lake after take off.

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INDEPENDENT SKILLS PRACTICE

ISBN: 979-8-89192-615-8

ISBN: 979-8-89192-616-5

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MATH INDEPENDENT SKILLS PRACTICE

ALGEBRA 2

ISBN: 979-8-89192-614-1

LEARNING

THE LEADER in STEM EDUCATION

ALGEBRA 2

MATH

GEOMETRY

ALGEBRA 1

ALGEBRA 1

INDEPENDENT SKILLS PRACTICE

LEARNING

THE LEADER in STEM EDUCATION

GEOMETRY

MATH

MATH INDEPENDENT SKILLS PRACTICE

MATH INDEPENDENT SKILLS PRACTICE

MATH INDEPENDENT SKILLS PRACTICE

LEARNING

THE LEADER in STEM EDUCATION

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Independent Skills Practice

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L-7

Quadratic Equations

SKILL

I can solve quadratics by factoring.

GUIDED PRACTICE Use the guiding tips to solve the problem. Scan the QR code to watch a video tutorial.

1

Solve each quadratic equation by factoring. Show your work. a. x 2 − 3x − 28 = 0

L-7

Name: ___________________________________________________

GUIDING TIPS Use these if you need help.

• To solve a quadratic equation by factoring, the quadratic equation must first be written in standard form, a x 2 + bx + c = 0.

• A quadratic equation can be solved by factoring using the zero product property.

• The zero product property states that if the product of two numbers is 0, then one of the numbers is 0.

b. x 2 − 12x = − 20

c. 2 x 2 − 3x = − 10x + 15

VIDEO TUTORIAL A video about how to solve

ALGEBRA 1

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Independent Skills Practice, Algebra 1

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Quadratic Equations

L-7

BY ACCELERATE LEARNING

FL

I can solve quadratics by factoring.

INDEPENDENT PRACTICE Solve the following questions.

Use the following information to answer questions 2 and 3.

4

Whitman solved the quadratic equation x 2 + 11x + 28 = 54 by factoring, but he made an error.

His work is shown.

Step 1

Work x 2 + 11x + 28 = 54

2

(x + 4)(x + 7) = 54

3

x+4 = 0

4 2

Serenity started solving the quadratic equation 7 x 2 − 12 = x 2 − x by factoring, but she got stuck.

Her partial work is shown.

Step

Work

1

7x − 12 = x 2 − x

2

6x 2 + x − 12 = 0

3 4

x+7 = 0

5

x = − 4, − 7

Help Serenity solve the equation by completing the steps to solve by factoring.

Explain where Whitman made his first error.

5

Select all of the solutions to the equation 2x 2 + 6x − 80 = 0.      

3

2

Find the correct solution(s) of the quadratic equation by factoring. Show your work.

Solve the quadratic equation − 12x 2 − 81x = − 21 by factoring. Show your work.

ALGEBRA 1

6

−8 −5 0 2 5 8

390 Independent Skills Practice, Algebra 1

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E-3

Rigid Motions

SKILL

Date: ___________________

E-3

Name: ___________________________________________________

I can relate a reflection on the coordinate plane to an algebraic notation of the transformation.

GUIDED PRACTICE Use the guiding tips to solve the problem. Scan the QR code to watch a video tutorial.

1

Polygons PFCG, SZ S WV V, LTD T H, and KBRM are shown on the coordinate TD plane.

7

y = –x

F

P

y

Z

y=x

5

• The algebraic description for reflecting a point across the y-axis is (x, y) → ( − x, y).

4 3

G

2

C

R

M

W

V

1

-8 -7 -6 -5 -4 -3 -2 -1 0 -1 -2 -3

• The algebraic description for reflecting a point across the line y = x is (x, y) → (y, x).

x 1

2

3

4

5

6

7

8

H

D

• The algebraic description for reflecting a point across the line y = − x is (x, y) → ( − y, − x).

-4 -5

K B

-6 -7

T

Use these if you need help.

• The algebraic description for reflecting a point across the x-axis is (x, y) → (x, − y).

S

6

GUIDING TIPS

L

a. What is the line of reflection that maps polygon PFCG to polygon KBRM?

b. Write an algebraic description for this reflection.

c. Which polygon is the reflection of polygon LTD T H across y = x?

TD

d. Write an algebraic description for this reflection.

e. What is the line of reflection that maps polygon KBRM to polygon S WV SZ V?

Write an algebraic description for this reflection.

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A video about how to solve

GEOMETRY

f.

VIDEO TUTORIAL

Independent Skills Practice, Geometry

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I can relate a reflection on the coordinate plane to an algebraic notation of the transformation.

INDEPENDENT PRACTICE Solve the following questions.

Use the following information for questions 2–4.

Parallelogram KP KPS PSR, with vertices K(− 9, 0), P(− 2, 0), S(− 2, − 10), and R( − 9, − 10), undergoes the transformation (x, y) → (x, − y).

2

Use the following information to answer questions 5 and 6.

7 6

Write the coordinates of the vertices of the image that results from the transformation.

5

y N

B

L

4 3 2

R

X

1 -4 -3 -2 -1 0 -1

x 1

2

3

4

5

6

Pentagon LXR X BN is shown on the coordinate plane.

XR 3

Describe the transformation that occurred on parallelogram KP KPS PSR.

4

Triangles HRB and TL T X are shown on the coordinate plane. Write the algebraic description for the reflection that maps ∆ HRB to ∆ TL T X. 7

B

y

6 5 4

X

5

What are the coordinates of Bʹ and Lʹ after pentagon LXR X BN is reflected across the y-axis?

XR

6

Write the algebraic description for the transformation of pentagon LXR X BN. XR

T

3 2 1

-7 -6 -5 -4 -3 -2 -1 0 -1

x 1

2

-2

H

-5

4

5

6

7

L

-3 -4

3

R

GEOMETRY

-6

104 Independent Skills Practice, Geometry

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G-9

I can solve a quadratic equation by factoring when given a real-world context, interpreting solutions as viable or nonviable.

GUIDED PRACTICE Use the guiding tips to solve the problem. Scan the QR code to watch a video tutorial.

1

A student club is selling raffle tickets for a fundraiser. The profit function for ticket sales is represented by P(x) = x 2 + 3x − 10, where x is the number of tickets sold. a. Determine the zero(s) of the function. Show your work.

GUIDING TIPS Use these if you need help.

• When solving a quadratic equation by factoring, look for common factors, and apply the zero-product property.

• Always consider the real-world context when interpreting solutions; some solutions may not make sense in context.

Quadratic Relationships – Part 2

SKILL

Date: ___________________

G-9

Name: ___________________________________________________

b. Determine whether the zero(s) is viable in this context. If a zero is nonviable, explain why.

c. Interpret the viable zero(s) in context.

VIDEO TUTORIAL A video about how to solve

ALGEBRA 2

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I can solve a quadratic equation by factoring when given a real-world context, interpreting solutions as viable or nonviable.

G-9

Quadratic Relationships – Part 2

FL

INDEPENDENT PRACTICE Solve the following questions.

2

A splash pad shoots a stream of water into the air such that the height of the water stream in meters (m) can be modeled by the quadratic equation h(t) = − 3t² + 9tt, where t is time in seconds.

4

a. Determine the zero(s) of the function.

Show your work.

a. Determine the zero(s) of the function. Show your work.

b. Determine whether the zero(s) is viable. If a zero is nonviable, explain why.

b. Interpret the zero(s) in terms of the context.

c. Determine whether the zero(s) is (are) viable.

If a zero is nonviable, explain why.

3

A landscape architect is designing a rectangular garden. The length of the garden is represented by x, and the width is represented by 10 − x.

A manufacturer’s cost function for producing a certain item is C(x) = 2x² − 9x + 9, where x is the number of items produced.

Use the following information to answer questions 5 and 6.

A local movie theater charges $12 for a regular movie ticket. The theater management is considering changing the ticket prices. They want to know how potential price changes could affect their total profit. The equation P(x) = − 1,500x 2 + 1,500x + 9,000 models the total weekly profit as a function of x, where x is the potential change in the ticket price in dollars.

5

Determine the zero(s) of the function. Show your work.

6

Interpret the solution(s) in terms of the context.

a. Determine the zero(s) of the function that represents the area of the garden. Show your work.

ALGEBRA 2

b. Determine whether the zero(s) is (are) viable.

If a zero is nonviable, explain why.

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