2000 and beyond

Mathematics InstituteMathematics Institutehttp://www.utdanacenter.org/ssi/projects/texteams

Algebra 1:2000 and BeyondAlgebra 1:2000 and Beyond

Dwight D. Eisenhower Professional Development Program, Title II, Part BTexas Education AgencyTexas Statewide Systemic Initiative in Mathematics, Science, and Technology EducationCharles A. Dana Center, The University of Texas at Austin

Permission is given to any person, group, or organization to copy and distribute TexasTeachers Empowered for Achievement in Mathematics and Science (TEXTEAMS)materials for noncommercial educational purposes only, so long as the appropriate credit isgiven. This permission is granted by The Charles A. Dana Center, a unit of the College ofNatural Sciences at The University of Texas at Austin.

Acknowledgements

The TEXTEAMS Algebra I: 2000 and Beyond institute was developed under the directionand assistance of the following:

Academic Advisors/Reviewers

Paul Kennedy Texas Christian UniversityAnne Papakonstantinou Rice University

Writer

Pam Harris Consultant

Advisory Committee

Linda Antinone Fort Worth ISDKathy Birdwell New Braunfels ISDKathi Cook Dana Center, University of Texas at AustinEva Gates ConsultantJuan Manuel Gonzalez Laredo ISDSusan Hull Dana Center, University of Texas at AustinPaul Kennedy Southwest Texas State UniversityLaurie Mathis Dana Center, University of Texas at AustinDiane McGowan Dana Center, University of Texas at AustinBonnie McNemar ConsultantBarbara Montalto Texas Education AgencyAnne Papakonstantinou Rice UniversityCindy Schimek Katy ISDJane Silvey ESC VIIJoAnn Wheeler ESC IVSusan Williams University of Houston

TEXTEAMS Algebra I: 2000 and Beyond Institute

TEXTEAMS Algebra I: 2000 and Beyond iii

Table of Contents

About TEXTEAMS Institutes......................................................................................viiInstitute Introduction.....................................................................................................viiiInstitute Overview.............................................................................................................ixSection Overviews............................................................................................................xMaterials List .......................................................................................................................x v

I. Foundations for Functions

1 Developing Mathematical Models1.1 Variables and Functions .....................................................................................1

Activity 1: Examples of Dependent Relationships .....................................11Activity 2: Independent and Dependent Variables.....................................12Reflect and Apply ............................................................................................13

1.2 Valentine’s Day Idea ........................................................................................14Activity 1: Valentine’s Day Idea .....................................................................24Activity 2: Using Tables to Find the More Economical Offer.......................25Activity 3: Using Graphs to Find the Better Offer.........................................27Activity 4: New Rose Offers ..........................................................................29Activity 5: Using Tables for New Rose Offers.............................................30Activity 6: Using Graphs for New Rose Offers............................................31Reflect and Apply ............................................................................................33Student Activity: Investigate Recursively ...................................................36

2 Using Patterns to Identify Relationships2.1 Identifying Patterns ...........................................................................................41

Activity 1: Painting Towers..............................................................................51Activity 2: Building Chimneys ........................................................................54Activity 3: Constructing Trucks........................................................................57Activity 4: Generating Patterns.......................................................................60Reflect and Apply ............................................................................................62Student Activity: Perimeter of Rectangles ....................................................63

2.2 Identifying More Patterns.................................................................................67Activity 1: Building Blocks...............................................................................72Activity 2: Starting Staircases .........................................................................75Activity 3: Too Many Triangles.......................................................................78Reflect and Apply ............................................................................................81

3 Interpreting Graphs3.1 Interpreting Distance versus Time Graphs.....................................................82

Activity 1: Walking Graphs .............................................................................87Activity 2: Walking More Graphs...................................................................88Reflect and Apply ............................................................................................90Student Activity: Walk This Way ...................................................................91

3.2 Interpreting Velocity versus Time Graphs .....................................................97Activity 1: Matching Velocity Graphs.......................................................... 103Activity 2: Connecting Distance and Velocity Graphs .............................. 105Reflect and Apply ......................................................................................... 107

Table of Contents

TEXTEAMS Algebra I: 2000 and Beyond iv

II. Linear Functions

1 Linear Functions1.1 The Linear Parent Function............................................................................ 108

Activity 1: ACT Scores................................................................................ 115Activity 2: Temperatures.............................................................................. 116Activity 3: Symbolic ..................................................................................... 117Reflect and Apply ......................................................................................... 118Student Activity 1: Age Estimates.............................................................. 119Student Activity 2: Sales Goals.................................................................. 128

1.2 The Y- Intercept............................................................................................. 132Activity 1: The Birthday Gift......................................................................... 139Activity 2: Spending Money....................................................................... 142Activity 3: Money, Money, Money............................................................ 145Reflect and Apply ......................................................................................... 146Student Activity: Show Me the Money! .................................................... 148

1.3 Exploring Rates of Change.......................................................................... 153Activity 1: Wandering Around ..................................................................... 158Activity 2: Describe the Walk ...................................................................... 160Reflect and Apply ......................................................................................... 162Student Activity: What’s My Trend? .......................................................... 163

1.4 Finite Differences............................................................................................ 170Activity 1: Rent Me!...................................................................................... 176Activity 2: Guess My Function .................................................................... 177Activity 3: Finite Differences......................................................................... 179Reflect and Apply ......................................................................................... 180Student Activity: Graphs and Tables ......................................................... 181

2 Interpreting Relationships Between Data Sets2.1 Out for a Stretch ............................................................................................. 183

Activity 1: Stretch It....................................................................................... 190Activity 2: Comparing Graphs .................................................................... 194Reflect and Apply ......................................................................................... 195Student Activity 1: Have You Lost Your Marbles?.................................. 196Student Activity 2: Unidentified Circular Objects (UCO’s)....................... 206Student Activity 3: Going to Great Depths................................................ 215Student Activity 4: Height versus Arm Span ............................................ 224

2.2 Linear Regression.......................................................................................... 232Activity 1: Sum of Squares ......................................................................... 245Activity 2: Lines of Best Fit.......................................................................... 246Activity 3: The Correlation Coefficient ........................................................ 248Reflect and Apply ......................................................................................... 250

3 Linear Equations and Inequalities3.1 Solving Linear Equations .............................................................................. 251

Activity 1: Concrete Models........................................................................ 259Activity 2: Using Concrete Models............................................................. 262Reflect and Apply ......................................................................................... 264

3.2 Stays the Same............................................................................................. 265Activity: Stays the Same............................................................................. 274Reflect and Apply ......................................................................................... 278

3.3 Solving Linear Inequalities ............................................................................ 279Activity 1: Linear Inequalities in One Variable............................................ 288Activity 2: Linear Inequalities in Two Variables.......................................... 290Reflect and Apply ......................................................................................... 294

Table of Contents

TEXTEAMS Algebra I: 2000 and Beyond v

3.4 Systems of Linear Equations and Inequalities............................................ 295Activity 1: Using a Table.............................................................................. 302Activity 2: Solve the System Graphically.................................................. 303Activity 3: Solve the System Symbolically............................................... 305Reflect and Apply ......................................................................................... 306Student Activity: Concrete Models and Systems of Linear Equations .. 307

III. Nonlinear Functions

1 Quadratic Functions1.1 Quadratic Relationships................................................................................. 319

Activity 1: Building a Sandbox.................................................................... 329Activity 2: Projectile Motion.......................................................................... 332Reflect and Apply ......................................................................................... 335

1.2 Transformations.............................................................................................. 336Activity 1: Investigating the Role of a ......................................................... 344Activity 2: Investigating the Role of k.......................................................... 345Activity 3: Investigating the Role of h ......................................................... 346Activity 4: Transformations .......................................................................... 347Reflect and Apply ......................................................................................... 350

1.3 Lines Do It Too .............................................................................................. 351Activity 1: Exploring Slope ......................................................................... 361Activity 2: Exploring Vertical Shifts ............................................................. 362Activity 3: Exploring Horizontal Shifts......................................................... 363Activity 4: A Different Perspective.............................................................. 365Reflect and Apply ......................................................................................... 366

2 Quadratic Equations2.1 Connections ................................................................................................... 367

Activity 1: Roots, Factors, x-intercepts, Solutions..................................... 374Activity 2: Which Form?............................................................................... 377Activity 3: Jump!........................................................................................... 378Reflect and Apply ......................................................................................... 379

2.2 The Quadratic Formula................................................................................... 380Activity 1: Programming the Quadratic Formula......................................... 384Activity 2: Hang Time................................................................................... 385Reflect and Apply ......................................................................................... 387Student Activity: Investigate Completing the Square.............................. 388

3 Exponential Functions and Equations3.1 Exponential Relationships............................................................................ 392

Activity 1: Paper Folding.............................................................................. 405Activity 2: Measure with Paper ................................................................... 407Activity 3: Regions ...................................................................................... 409Activity 4: How Big is a Region?................................................................ 411Reflect and Apply ......................................................................................... 413Student Activity: Recursion Again............................................................... 414

3.2 Exponential Growth and Decay................................................................... 420Activity 1: Exponential Growth.................................................................... 427Activity 2: Exponential Decay..................................................................... 428Reflect and Apply ......................................................................................... 429Student Activity: On the Wall ...................................................................... 430

3.3 Exponential Models...................................................................................... 434Activity 1: Population Growth...................................................................... 441Activity 2: Cooling Down............................................................................. 443Reflect and Apply ......................................................................................... 444

TEXTEAMS Algebra I: 2000 and Beyond vi

4 Interpreting Relationships Between Data Sets4.1 Bounce It!........................................................................................................ 445

Activity 1: Collect the Data........................................................................... 452Activity 2: A Bounce..................................................................................... 453Activity 3: Bounce Height versus Bounce Number.................................. 454Activity 4: Bounce Height versus Drop Height ......................................... 455Reflect and Apply ......................................................................................... 456Student Activity 1: Pattern Blocks............................................................... 457Student Activity 2: Throw Up!..................................................................... 463Student Activity 3: Radioactive Decay....................................................... 467Student Activity 4: Pendulum Decay.......................................................... 474

Calculator Programs...................................................................................................477

TEXTEAMS Algebra I: 2000 and Beyond vii

About TEXTEAMS Institutes

TEXTEAMS Philosophy

• Teachers at all levels benefit from extending their own mathematical knowledgeand understanding to include new content and new ways of conceptualizing thecontent they already possess.

• Professional development experiences, much like the school mathematicscurriculum itself, should focus on few activities in great depth.

• Professional development experiences must provide opportunities for teachersto connect and apply what they have learned to their day-to-day teaching.

Features of TEXTEAMS Institute Materials

Multiple representations (verbal, concrete, pictorial, tabular, symbolic, graphical)Mathematical ideas will be represented in many different formats. This helps both teachersand students understand mathematical relationships in different ways.

Integration of manipulative materials and graphing technologyThe emphasis of TEXTEAMS Institutes is on mathematics, not on learning about particularmanipulative materials or calculator keystrokes. However, such tools are used in variousways throughout the institutes.

Rich Connections within and outside mathematicsInstitutes focus on using important mathematical ideas to connect various mathematicaltopics and on making connections to content areas and applications outside of mathematics.

Questioning strategiesA variety of questions are developed within each activity that help elicit deep levels ofmathematical understanding and proficiency.

Hands-on approach with “get-up-and-move” activitiesInstitutes are designed to balance intense thinking with hands-on experiences.

Math Notes and Reflect and ApplyA feature called Math Notes includes short discussions of mathematical conceptsaccompanying the learning activities. Similarly, the Reflect and Apply feature is designed toextend and apply participants’ understanding of the mathematical concepts.The Charles A. Dana Center is approved by the State Board for Educator Certification as a registered ContinuingProfessional Education (CPE) provider. Hours received in TEXTEAMS institutes may be applied toward the required trainingfor gifted and talented in the area of curriculum and instruction. Individual district/ campus acceptance of these hours forgifted and talented certification is a local decision.


TEXTEAMS Algebra I: 2000 and Beyond viii

Introduction

The Algebra I: 2000 and Beyond Institute is based on the groundbreaking work ofthe 1996 TEXTEAMS Algebra I Institute. Both institutes assert that “Algebra for All”is a realistic and attainable goal. To teach “Algebra for All” will require educators topossess a deep understanding of mathematical content, pedagogy, and strategiesto meet the needs of diverse student populations.

This institute is not meant as a scope and sequence for the Algebra I course, nor isit a set of student activities for use in a classroom without careful thought andmodification on the part of a knowledgeable teacher. The Algebra I: 2000 andBeyond Institute is a rich, carefully designed professional development tool which isintended to deepen teacher content knowledge. It is also intended to model theimportance of content depth by focusing on carefully selected activities that are fewin number and grounded in the mathematics necessary to support teacher andstudent learning. The concepts and ideas explored within this institute areconnected to classroom instruction and key assessments.

In this institute, not all Algebra I topics can be addressed. It is important forteachers to develop a deep and powerful understanding of the concepts and ideasof algebra. This requires educators to understand the mathematics in a differentcapacity from that of the student. Therefore, content within the institute isapproached from a more complex perspective and some topics are addressed ata level that is deeper than would be typical for Algebra I students. Much of theinstitute is built upon learning experiences that develop and promote the power ofusing concrete experiences to introduce and build mathematical concepts.Throughout the institute, multiple representations are utilized as a powerful strategyto assist educators and students in making connections, understandingmathematical concepts, and reasoning in meaningful and complex ways.

The institute assumes some prior participant knowledge. If teachers are unfamiliarwith the following, presenters may need to supplement with introductory materials.• Representing, adding and subtracting polynomials with algebra tiles (area

model)• Using algebra tiles to model monomial and binomial multiplication• Modeling factoring trinomials with algebra tiles• Graphing calculators

The Algebra I; 2000 and Beyond Institute draws on the work of the 1996TEXTEAMS Algebra I Institute; Principles and Standards for School Mathematics,NCTM, 2000; Discovering Algebra, Key Curriculum Press, 2000; and Dr. PaulKennedy, Southwest Texas State University.


TEXTEAMS Algebra I: 2000 and Beyond ix

Institute Overview

I. Foundations for Functions

1 Developing Mathematical Models2 Using Patterns to Identify Relationships3 Interpreting Graphs

II. Linear Functions

1 Linear Functions2 Interpreting Relationships Between Data Sets3 Linear Equations and Inequalities


1 Quadratic Functions2 Quadratic Equations3 Exponential Functions and Equations4 Interpreting Relationships Between Data Sets

I. Foundations of Functions Section Overviews

TEXTEAMS Algebra I: 2000 and Beyond x

I. Foundations of Functions Overview

Activity Overview Materials

Developing Mathematical Models1.1 Variables andFunctions

Participants discuss the concept of function inthe context of a school fund-raising venture.

butcher paper,masking tape

1.2 Valentine’s DayIdea

Participants investigate functionalrelationships for a given problem situationusing tables, graphs, and algebraicrepresentations.

graphingcalculators

Using Patterns to Identify Relationships2.1 IdentifyingPatterns

Participants represent linear relationshipsamong quantities using concrete models,tables, diagrams, written descriptions, andalgebraic forms.

building blocks,color tiles,graphingcalculators

2.2 Identifying MorePatterns

Participants represent non-linearrelationships among quantities usingconcrete models, tables, diagrams, writtendescriptions, and algebraic forms.

building blocks,graphingcalculators

Interpreting Graphs3.1 InterpretingDistance versusTime Graphs

Participants use motion detectors toinvestigate distance over time graphs. Thislays the groundwork for graph reading andfor work with rates of change.

motion detectorconnected to anoverheadcalculator, motiondetectors, datacollection devices,graphingcalculators, 2 or 3transparencies cut-to-fit on theoverheadcalculator screen

3.2 InterpretingVelocity versusTime Graphs

Participants use motion detectors toinvestigate velocity over time graphs. Thiscontinues to build graph-reading skills and tobuild understanding for rates of change.

motion detectorconnected to aoverheadcalculator, motiondetectors, datacollection devices,graphingcalculators

II. Linear Functions Section Overviews

TEXTEAMS Algebra I: 2000 and Beyond xi

II. Linear Functions Overview


Linear Functions1.1 The LinearParent Function

Using contextual situations, participantsinvestigate the linear parent function, theline y x= .

colored pencils orpens, pieces of flatspaghetti, graphingcalculatorsStudent Activity:markers, 1” gridpaper

1.2 The Y-Intercept

Participants use “canned” real lifeexperiences to build the concepts of y-intercept as the starting point and slope asa rate of change, both with contextualsignificance.

graphing calculators

1.3 ExploringRates of Change

Participants use real data from a motiondetector to model motion at a constant rateover time. Participants translate amongalgebraic, tabular, graphical, and verbaldescriptions of linear functions.

motion detectorconnected to anoverhead calculator,motion detectors,graphing calculators,data collectiondevices

1.4 FiniteDifferences

Participants use their cumulative concreteexperiences with the linear model to buildto the abstract symbolic representations ofslope. Finite differences are use to findlinear models and to discover what makesdata linear.

graphing calculatorsStudent Activity:calculator programs

Interpreting Relationships Between Data Sets2.1 Out for aStretch

Participants investigate the relationshipbetween the “stretch” of a rubber bandattached to a container and the number ofmarbles in the container.

Styrofoam cups, 3”long thin rubberbands, marbles ofthe same size, largepaper clips, tape,meter sticks, graphingcalculatorsStudent Activities:blocks, PVC pipe,marbles, tapemeasureflashlights, rulers,cylinders, uniformobjects, water

II. Linear Functions Section Overviews

TEXTEAMS Algebra I: 2000 and Beyond xii

2.2 LinearRegression

Participants write a program to find a leastsquares linear function to model data. Theyuse the program and calculator regressionto find linear models for data and theydiscuss the correlation coefficient, r.

graphing calculator,dynamic geometryprogram with aprepareddemonstration of alinear least squares fitfor data, computerwith a projectiondevice.

Linear Equations and Inequalities3.1 Solving LinearEquations

Participants solve linear equations withconcrete models and make connectionsbetween the concrete model, abstract, andsymbolic representations.

algebra tiles,overhead algebratiles

3.2 Stays theSame

Participants solve linear equations in onevariable, making connections betweenalgebraic solution steps, algebra tilesolution steps, and graphical solutionsteps.

algebra tiles,overhead algebratiles, graphingcalculators, 1” gridpaper, markers

3.3 Solving LinearInequalities

Participants use problem situations andtechnology to explore linear inequalities.

transparencies of theStudent Activity:Age Estimates from2.1.1 The LinearParent Function,graphing calculators

3.4 Systems ofLinear Equationsand Inequalities

Participants use a table to develop asystem of linear inequalities. They solvethe system using various methods andmake connections between a system ofinequalities and a system of equations.

graphing calculatorsStudent Activity:algebra tiles

III. Nonlinear Functions Section Overviews

TEXTEAMS Algebra I: 2000 and Beyond xiii

III. Nonlinear Functions Overview


QuadraticFunctions

1.1 QuadraticRelationships

Participants use lists to develop a quadraticfunction representing the volume of asandbox with a fixed depth. Using thequadratic function, participants solvequadratic equations numerically andgraphically.

graphing calculators,pieces of lumber orcardboard to simulatelumber

1.2Transformations

Participants investigate the effects ofchanging the parameters of quadraticfunction of the form y ax c= +2 . Theyapply this understanding by fitting aquadratic to real data. Participants extendtheir understanding and investigate theeffects of changing the parameter h inquadratic functions of the form y x h= −( )2.

graphing calculators,patty paper or blanktransparencies

1.3 Lines Do It Too Participants connect their knowledge oftransformations with quadratic functions withthe equations of lines. The point-slopeform of a line is looked at from atransformational perspective.

graphing calculators,patty paper or blanktransparencies

Quadratic Equations2.1 Connections Participants make connections between the

roots of quadratic functions and thesolutions to quadratic equations and thefactors of quadratic polynomials and the x-intercepts of a parabola. They connect thisunderstanding to the vertex, polynomial,and factored form of the equation of aparabola. Using this understanding,participants model a vertical jump, findingthe height of the jump.

graphing calculators,data collectiondevices, lightsensors, laserpointers or flashlights

2.2 The QuadraticFormula

Participants program the quadratic formulainto the graphing calculator and use theprogram to solve quadratic equations atappropriate times.

graphing calculators,1” graph paper,markers, meter sticksStudent Activity:algebra tiles

III. Nonlinear Functions Section Overviews

TEXTEAMS Algebra I: 2000 and Beyond xiv

Exponential Functions and Equations3.1 ExponentialRelationships

Participants explore exponential growthand decay situations. Using “canned”situations, participants develop the ideasof the common multiplier or ratio as thebase of an exponential function and thestarting point as the y-intercept of anexponential function.

graphing calculators,sheets of blankpaper

3.2 ExponentialGrowth and Decay

Participants find models for exponentialgrowth and decay situations.

graphing calculatorsStudent Activity:sticky notes, posterboards, large blankpaper, markers, tape

3.3 ExponentialModels

Participants find exponential models forgiven data sets.

graphing calculators

Interpreting Relationships Between Data Sets4.1 Bounce It! Collecting three sets of data from a

bouncing ball experiment, participants findappropriate models and justify theirchoices.

balls, data collectiondevices, motiondetectors, graphingcalculatorsStudent Activity:pattern blocks, balls,stop watches,graphing calculators,soda cans, string,meter sticks

Materials List

TEXTEAMS Algebra I: 2000 and Beyond xv

Materials List

Consumables Non-Consumablesbutcher paper building blocksmasking tape meter sticksblank transparencies (some cut to fitthe overhead graphing calculator)

color tilestape measure

colored pencils or pens PVC pipepieces of flat spaghetti algebra tilesmarkers overhead algebra tiles1” grid paper flashlightsStyrofoam cups cylinders3” thin rubber bandsstring

uniform objects (to put incylinders)

marbles of the same sizelarge paper clipswaterpieces of lumber or cardboardpatty paper or blank transparencies Technologyblank paper graphing calculatorssticky notes overhead graphing calculatorposter boards (large constructionpaper)

data collection devicesmotion detectors

balls (racket, basketball, golf) light sensorssoda cans computer with projection device

laser pointersstop watches

Softwaredynamic geometry program (w/least squares demo)

Calculator Programs:LINEGRPHLINETBLACTCMOVEJUMPITPENDULUM

I. Foundations for Functions 1.1 Variables and Functions: Leaders’ Notes

TEXTEAMS Algebra I: 2000 and Beyond Spring 2001 1

1.1 Variables and FunctionsOverview: Participants discuss the concept of function in the context of a school fund-

raising venture.

Objective: Algebra I TEKS(b.1) The student understands that a function represents a dependence of onequantity on another and can be described in a variety of ways.

Terms: variable, function, independent variable, dependent variable, mathematicalmodeling

Materials: butcher paper, masking tape

Procedures: Participants should be seated at tables in groups of 3 – 4.Traditionally algebra has been taught as an abstract set of algorithms based ondefinitions, properties, and theorems that lead students to “an answer.”Finding x , sketching a graph, factoring an expression, and solving a wordproblem were the orders for the day. Devoid of context, these problems askedstudents to make a jump from their understanding of numbers and operationsto the abstract notions of variables, equations, functions, etc. With context-rich problem situations, and using manipulatives and technologyappropriately, we can bridge the gap between students’ concrete thinking andthe abstract world of algebra.

Transparency #1: Valentine’s Day IdeaIntroduce the school’s drill team money making project.

• What factors might effect the success of the project? [Record at leastone idea from each group.]

Transparency #2: Variables and FunctionsDiscuss list of factors that was acquired from field testing.Introduce the terms variables and functions.

Transparency #3: Non-Mathematical Definition of Function

Activity 1: Examples of Dependent RelationshipsRead the definition of function and have participant groups list at least 7examples of dependent relationships on Activity 1. Have each group writeexamples on butcher paper and post on the walls of the room. Encourageparticipants to use varied language, not only “is a function of” and “dependson,” but also examples like, “The colder it gets, the more I shiver.”

Have participants do a gallery tour and record two or three favorite examples.



Transparency #4: Examples of Dependent RelationshipsDiscuss examples on butcher paper and on Transparency #4.

• Which examples show a clear dependent relationship?• Which examples, if any, do not show a clear dependent relationship?

Activity 2: Independent and Dependent VariablesHave the groups write some of their examples from Activity 1 and some of theexamples from Transparency 4 in the chart as cause-and-effect statements.Some participants may word these in an if-then format. If no one does, helpparticipants word a few in an if-then format. As you are discussing theirexamples, write the word “IF” under “Cause” and “THEN” under “Effect” inthe table.Ask:

• Which of these columns depends on the other? [The Effect columndepends on the Cause column]

Write “Dependent” under “THEN”• If “dependent” describes the second column, what word can we use to

describe the first column? [Independent]Write “Independent” under “IF”

So the table will look as follows:Cause

IF

Independent

EffectTHEN

Dependent

If I get more sleep at night, then I wake up faster.

Math Note:Not all dependent relationships can be written in an if-then format. Considerthe statement: “How I feel depends on how I eat.” You could infer meaningand rewrite it as “If I eat chocolate, then I will be happy.” Or “If I eat quickly,I will feel indigestion.”Also, not all relational statements should imply causation. Just because thingsare related does not necessarily mean that one causes the other. “Height isrelated to shoe size.” “People who spend the most time on the Internet havehigh rates of depression” Does that mean that time on the Internet causesdepression or that depressed people gravitate toward the Internet? Be carefulin the training to not infer causation just because there is relation.Sometimes determining which is the cause and which is the effect in a relationcan be open to interpretation. The time and distance you fly in a plane arerelated. Does the distance you fly in an airplane depend on the time it takes tofly somewhere? Does the time it takes to fly somewhere depend on how faraway the place is?The idea is to not get caught up being too picky. The focus of the activities isto connect participant’s previous understanding of cause and effect to themathematical definition of function, with independent and dependentvariables. Choose some good cause and effect relationships that illustrate



independent and dependent variables and don’t get hung up on those that donot.

Transparency #5: Mathematical Definition of FunctionTraditionally many math teachers have introduced functions with themathematical definitions. Students understand the concept of cause andeffect. We use that understanding in Activities 1 – 2 to bridge to thismathematical definition.Introduce the mathematical definition of independent and dependent variablesand discuss cause and effect relationships with variables.

Transparency 6: Stages of Mathematical Modeling Process

As with the Valentine’s Day idea, every business begins with a product orservice that the owners believe will produce a profit. Creating the businessidea is only the beginning stage. Soon decisions have to be made about howto best deliver the product or service so that the most money can be made.Several issues involve numbers, but simple arithmetic is seldom sufficient tohandle all the varying possibilities.

Algebraic methods will allow you (just as they allow businesses) tomathematically define some of the varying situations. You can put togethervariables and relationships into a mathematical model that describe abusiness situation and study questions (i.e., What would happen if one factoris changed?) so that optimum solutions can be identified.Use the following statements in discussing the stages of modeling.Businesses:⇓ begin with a product or service that they believe will produce $$$$$,

⇓ identify issues that involve numbers and the use of algebraic methods tomathematically define some of the varying situations,

⇓ put together variables and relationships into a mathematical model thatdescribes the business situation,

⇓ analyze the model by studying questions (i.e., What would happen if onefactor is changed?) so that optimum solutions can be identified,

⇓ interpret the result within the context of the business situation,

⇓ use the information to formulate conclusions and to make informeddecisions that will profit the business, and

⇓ implement plans based on the information gained through the use of themathematical modeling process.

Within real world situations, you must learn to:



(1) identify problems and discern the important factors (variables/parameters) that affect the problem,

(2) determine the relationships among those factors (variables) anddescribe them mathematically,

(3) analyze the model by applying appropriate mathematical techniquesand draw mathematical conclusions,

(4) interpret the results in context,(5) formulate conclusions and predictions so that optimum choices can be

made, and(6) apply decisions to the real world situation.

Answers to Reflect and ApplyOften we ask students to match situations with graphs with little priorexperience in making, reading, and interpreting such graphs. Ask participantsto reflect on these Exercises as they progress through the institute. Askparticipants to consider how students may be able to better match these graphsafter the experiences suggested in the institute.

1. e2. d.3. c4. a5. b

Summary: Building on the non-mathematical definitions of function and variables,participants are introduced to mathematical functions and mathematicalmodeling.

I. Foundations for Functions 1.1 Variables and Functions: Transparency 1


Transparency 1: Valentine’s Day Idea

The school’s drill team hasdecided on a money-makingproject for February. They planto take orders (and money) forroses in advance and deliverthem to the designated studentson Valentine’s Day.

What factors might affect the success of this money-making project?



Transparency 2: Variables and Functions

The success of the money-making project depends onmany factors such as:

• publicity,

• appeal of project to the students,

• selling price of roses,

• cost of roses,

• willingness of the flower distributors to workwith the student group, and

• cooperation of the administration.

These factors are called variables because they canchange regularly. It is appropriate to say that the successof the money-making project is a function of thosevariables.



Transparency 3: Non-Mathematical Definition of Function

The American Heritage Dictionary of the EnglishLanguage defines function as something closely related toanother thing and dependent upon it for its existence,value, or significance.

Examples of Function1. How fast I wake up in the morning depends on (is a

function of) how much sleep I get.

2.

3.

4.

5.

6.

7.

8.



Transparency 4: Examples of Dependent Relationships

1. How fast I wake up in the morning depends on (is afunction of) how much sleep I get.

2. Our height is a function of our age.

3. The amount of change in my pocket depends on (is afunction of) the type of coins found there.

4. The amount of studying determines the grade we make.

5. As my car gets older, it is worth less.

6. The grass gets greener as I put more fertilizer on it.

7. The amount of money I make at my job depends on (isa function of) the number of hours I work.



Transparency 5: Mathematical Definition of Function

A mathematical function expresses a dependencyrelationship: one quantity depends in a systematic way onanother quantity.

For Example:y x= +2 1 is a function and expresses a dependencyrelationship.

The value of y depends on the value of x.

The variable x is called the input or independent variable.

The variable y is called the output or dependent variable.

In an applied mathematics setting, we must decide whichdecisions (input) influence or produce which results(output). In many cases this is a cause and effectrelationship where the cause is the independent variableand the effect is the dependent variable.



Transparency 6: Modeling with Mathematics

Identify

Create

Analyze/Summarize

Interpret

Formulate

Apply

Real world Situation

Problem, Variables,

Constraints

Mathematical Model

Mathematical Conclusions

Problem Situation

Conclusions, Predictions, Decisions

Entry

Stages of Mathematical Modeling Process

I. Foundations for Functions 1.1 Variables and Functions: Activity 1


Activity 1: Examples of Dependent Relationships

1. How fast I wake up in the morning depends on (is a functionof) how much sleep I get.

2.

3.

4.

5.

6.

7.

8.

I. Foundations for Functions 1.1 Variables and Functions: Activity 2


Activity 2: Independent and Dependent Variables

Identify the independent and dependent variables from some ofthe dependent relationship examples.

Cause Effect

I. Foundations for Functions 1.1 Variables and Functions: Reflect and Apply


Reflect and Apply

Match the following descriptions with a graph:___ 1. The volume of

popcorn popping overtime.

a.

___ 2. Phone deal: $0.50 forthe first 20 minutes and$0.07 per minute after.

b.

___ 3. The worth of my carover time.

c.

___ 4. He walked up and thendown a hill, speedversus time.

d.

___ 5. She walked away andthen walked back,distance versus time.

e.

I. Foundations for Functions 1.2 Valentine’s Day Idea: Leaders’ Notes


1.2 Valentine’s Day IdeaOverview: Participants investigate functional relationships for a given problem situation

using tables, graphs, and algebraic representations.

Objective: Algebra I TEKS(b.1) The student understands that a function represents a dependence of onequantity on another and can be described in a variety of ways.(b.3) The student understands algebra as the mathematics of generalization andrecognizes the power of symbols to represent situations.

Terms: function, independent variable, dependent variable, pattern

Materials: graphing calculators

Procedures: Participants should be seated in groups of 3 – 4.This entire activity is an overview problem to give participants a feel for afunction based approach to algebra and for the power of multiplerepresentations. Thus participants do not need to gain mastery at this point,but work to give them exposure to many of the goals and concepts of theinstitute.

Activity 1: Valentine’s Day IdeaIntroduce the problem situation to the whole group.

• Which seems better:75 cents per rose or 50 cents per rose?$20 fixed cost or $60 fixed cost?75 cents per rose plus $20 fixed cost or 50 cents per rose plus $60

fixed cost?Because students have a tendency to focus on certain parts of a problemsituation while not attending to other parts, this line of questioning assiststhem in considering multiple variables at one time.

• What kind of customer would order from Roses-R-Red? [A customerwho needs a relatively small amount of flowers.]

• What kind of customer would order from The Flower Power? [Acustomer who needs a relatively large amount of flowers.]

Student Activity: Investigate RecursivelyDo the student activity with your participants as appropriate. Helpparticipants generate the recursive routines, first without braces and then withbraces.Note that both the original and the new rose offer are explored.Have participants point out questions that ask for an output value given aninput value, and questions that ask for an input value given an output value.



Doing activities like this helps students gain confidence in reading a mathword problem.

Math Note: Recursion is a process where each successive term is based on theprevious term.

Activity 2: Using Tables to Find the More Economical OfferHave participants complete the chart numerically first, looking for patterns.Develop the patterns live on the transparency of Activity 2 or useTransparency 1. Using the patterns, develop the function rules representingthe cost from each flower shop.

Ask participants to demonstrate the validity of their function rules for eachflower shop.Have half of the group use the statistics capability of graphing calculators toenter the number of flowers sold into a list. Then enter the formula20 0 75 1+ ( ). List into the second list. Enter the formula 60 0 5 1+ ( ). List into athird list. Check that the values generated match the values on the activitysheet.

Have the other half of each group use the table building capability of graphingcalculators to enter y x= +20 0 75. and y x= +60 0 5. .

• What would you expect 240 roses to cost from Roses-R-Red? [Ifparticipants respond $220, then they do not understand that the“doubling effect” does not work here because this is not a proportionalrelation. A proportional relation is in the form y mx= .]

Work through the rest of the Activity, using the following to discuss.

1. The cost of roses from Roses-R-Red is $20 and add $0.75 for every rose.Cost is 20 0 75+ . ( )Number of Roses , C r= +20 0 75. .

2. The cost of roses from Flower Power is $60 and add $0.50 for every rose.Cost is 60 0 50+ . ( )Number of Roses , C r= +60 0 50. .



3. Answers will vary, including increasing, increasing at different rates,increasing by constant amounts.

4. The cost increases as the number of roses purchased increases. Thegraph is linear with a positive slope.

5. Write the equation 65 20 0 75= + . r , then look for the answer in the table.Write r = 60. Do not solve the equation algebraically at this time.

6. Write the equation 65 60 0 5= + . r , then look for the answer in the table.Write r = 10 . As above, do not solve the equation algebraically at thistime. Rather, suggest that this activity could be done before students havelearned to solve one and two step equations. If students see equations likethis in context, learn how to solve them using a table (and later in theactivity, using a graph), then when students learn the algebraic steps tosolving similar equations, they will already have context and purpose.Students will see algebraic methods as another way to find answers.Maybe we can eradicate, “Why do we have to find x anyway?”

7. Roses-R-Red offers the better deal for n < 160, there is not difference forn = 160 , and Flower Power offers the better deal for n > 160.

8. The point of intersection is (160, 140). This point signifies that at n=160roses, the cost ($140) is the same with either flower dealer.

9. 20 0 75 60 0 50+ = +. .r r .

Activity 3: Using Graphs to Find the Better OfferNote that the ways in which two variables are related is not always shownclearly by tables of input-output values. Patterns in the data may be lost amidall the specific numbers. However, when data are displayed in a graph, it isoften much easier to see trends and therefore to make predictions and/orinformed decisions.

First, have participants make a scatter plot of the data that was entered intolists (number of roses sold, cost from Roses-R-Red) and (number of rosessold, cost from Flower Power).

Use the following questions to help participants graph the scatter plots andfunctions on the graphing calculator.

• What do you think the graph will look like? Predict.• What viewing window makes sense for the problem situation?• What does the variable x represent? [Number of roses.]• What values make sense for the number of roses sold? [See Sample

Answers below.]• What does the variable y represent? [Cost of roses]• What values make sense for the cost of roses? [See Sample Answers

below.]



After they have graphed the scatter plots, graph the functions together with theplots.

Make the connection between patterns seen in the tables of values and thegraphs of the same ordered pairs. As the number of roses increases, the totalcost increases.

Make connections between the patterns seen in the graphs and theinterpretation in the context. Note that to get from one point to another one,we must move to the right (increase in quantity) and then up (increase in cost)or we can move to the left (decrease in quantity) and then down (decrease incost).

Sample Answers:xmin=0: The variable x stands for the number of roses. It makes sense to

look at the cost if no one bought a rose.xmax=270: We think that we might sell at most 270 roses.ymin= 0: The variable y stands for the cost of the roses. The cost will not

be negative.ymax=200: This is a reasonable maximum cost for our maximum rose count

of 270.

• What does the ordered pair (210, 165) mean on the graph? [You canbuy 210 roses for $165.]

3. The 75 cent per stem cost is the rate of change or slope of the line. The$20 is the y-intercept and in this case, it makes sense to call it the startingpoint.

4. The 50 cent per stem cost is the rate of change or slope of the line. The$60 is the y-intercept and in this case, it makes sense to call it the startingpoint.

5. The point of intersection is (160, 140), which means that it does not matterfrom whom you purchase 160 stems because it will cost $140 at bothplaces. Show participants the following screen.



6. Roses-R-Red offers the better deal when you purchase less than 160 roses.Flower Power offers the better deal when you purchase more than 160roses. Use the trace cursor to demonstrate.• What are the meanings of the coordinates on the screens?

At Roses-R-Red, 110 stems cost $102.50.

At Flower Power, 110 stems cost $115.

At Roses-R-Red, 197 stems cost $167.75

At Flower Power, 197 stems cost $158.50.

Discuss the meaning of slope and y-intercept within the context of thisproblem. For example, the algebraic rule y x= +20 0 75. yields (a) y-interceptof 20 which indicates the fixed cost that has to be paid initially and (b) slopeof 0.75 which indicates the constant rate of change or the constant increase incost for the purchase of each rose.

Activity 4: New Rose OfferIntroduce the new offer made by the distributors.

Activity 5: Using Tables for New Rose OffersEncourage participants to complete the table so that comparisons between thetable in Activity 2 and this one can be made. Discuss changes from old dealto new deal. Ask participants to share ways to assist students in making theconnection between the table values and the corresponding graphs.



Note: Depending on the experience of your audience, you may want tocontinue to have them write sentences before they write function rules foreach deal.

Transparency 2: Comparing Tables• Looking at the table values for Roses-R-Red, what has happened to all

of the costs, the y-values? [The y-values have all been decreased by20.]

• How has that affected the function representing the cost from Roses-R-Red? [The starting point from the original offer has been decreased by20, so the new function is 20 less than the original.]

• Looking at the table values for Flower Power, what has happened toall of the costs, the y-values? [The y-values have also all beendecreased by 20.]

• How has that affected the function representing the cost from FlowerPower? [The starting point from the original offer has been decreasedby 20, so the new function is 20 less than the original.]

Note that the slight modification made to the old deal is reflected in thefunction rules and their representations. For example,

C n= +20 0 75. old offer by Roses-R-RedC n nn = + − =20 0 75 20 0 75. . new offer by Roses-R-RedC n= +60 0 5. old offer by Flower PowerC n nn = + − = +60 0 5 20 40 0 5. . new offer by Flower Power

Making the connections above is an important precursor for the next section,graphing. Return to the table when you discuss the vertical translation (shift)in Activity 6.

Activity 6: Using Graphs for New Rose Offer1. Encourage participants to always predict what a graph will look like

before looking at the graph on the calculator.

2.

3. The graph of the new rule is the graph of the original rule translated(shifted) down 20. Connect this with your comparisons in the table, whereall of the costs, the y-values decreased by 20.



5.

6. The graph of the new rule is the graph of the original rule translated(shifted) down 20. Connect this with your comparisons in the table,where all of the costs, the y-values decreased by 20.

7. Show the following screens:

The x-coordinates are the same! In other words, the “deal” is the same inthat Roses-R-Red are still better for less than 160 stems and Flower Powerare still better for over 160 stems. The costs are different by $20, but theanswer to the question, “Who has the better deal?” remains the same. Youcan see this in the screens below, where a vertical line shows that the x-coordinates of the intersection points are the same.

8. A graph can be used in comparing the two dependent relations. We cansee from the graphs that the two flower distributors charge the sameamount when 160 roses are purchased. Use the trace cursor on bothgraphs to see that Roses-R-Red is less expensive when n < 160and FlowerPower is less expensive when n > 160.

Note the relationship between the graphical representation of the:original offers 20 0 75 60 0 50+ = +. .x x .new offers 0 75 40 0 50. .x x= + .Although both functions shift 20 units down the y-axis, the intersection of thetransformed graphs has the same input value as the original graphs.

• Why does this make sense algebraically? [Subtract 20 from both sidesof the equation and the result is an equivalent equation, thus having thesame solution. In both equations, x = 160.]



Extension: In the new rose deal, we looked at the effect of subtracting $20from the cost of the roses from both flower dealers. Essentially, the dealremained the same in that Roses-R-Red were better for less than 160 stemsand Flower Power were better for over 160 stems. Now, try looking at theeffect of leaving the fixed costs the same and changing the amount of moneyper stem. Will the deal remain the same?

Answers to Reflect and Apply:1. Flowers-R-Us sells roses for $1.00 a stem. y x=2. All Occasion Roses sells roses for $10.00 plus $0.50 per stem. y x= +10 5.3. The point of intersection (20, 20) tells us that we can buy 20 roses for $20

from either place. Buying less than 20 stems is a better deal fromFlowers-R-Us and buying more than 20 stems is a better deal from AllOccasion Roses.

4. You can purchase roses from R for $15.00 plus $1.25 per stem.5. You can purchase roses from S for $35 plus $1.00 per stem.6. The point of intersection (80, 115) tells us that we can buy 80 roses for

$115 from either place. Buying less than 80 stems is a better deal fromFlowers-R-Us and buying more than 80 stems is a better deal from AllOccasion Roses.

7. With the coupon, roses are $2.00 plus $0.50 per stem. y x= +2 5.8. Lilies cost $5.00 per stem.9. Daisies cost $5.00 plus $0.25 per stem.10.

Summary: Fund-raising by selling roses for Valentines Day is the context for a richproblem where participants make connections between verbal, numerical,graphical, and algebraic representations of a linear situation. Participantsbuild intuition for constant rates of change and y-intercepts of lines.

I. Foundations for Functions 1.2 Valentine’s Day Idea: Transparency 1


Transparency 1: Using Tables to Find the More EconomicalOffer

From the description of the two offers, completethe chart to find an algebraic rule that willdetermine the cost of n roses.

Numberof Roses

ProcessColumn

Cost atRoses-R-Red

ProcessColumn

Cost atFlowerPower

10 20 0 75 10+ ( ). $27.50 60 0 50 10+ ( ). $65.0020 20 0 75 20+ ( ). $35.00 60 0 50 20+ ( ). $70.0030 20 0 75 30+ ( ). $42.50 60 0 50 30+ ( ). $75.0060 20 0 75 60+ ( ). $65.00 60 0 50 60+ ( ). $90.0090 20 0 75 90+ ( ). $87.50 60 0 50 90+ ( ). $105.00

120 20 0 75 120+ ( ). $110.00 60 0 50 120+ ( ). $120.00150 20 0 75 150+ ( ). $132.50 60 0 50 150+ ( ). $135.00180 20 0 75 180+ ( ). $155.00 60 0 50 180+ ( ). $150.00210 20 0 75 210+ ( ). $177.50 60 0 50 210+ ( ). $165.00240 20 0 75 240+ ( ). $200.00 60 0 50 240+ ( ). $180.001000 20 0 75 1000+ ( ). $770.00 60 0 50 1000+ ( ). $560.00

n 20 0 75++ . n 60 0 50++ . n

I. Foundations for Functions 1.2 Valentine’s Day Idea: Transparency 2


Transparency 2: Comparing Tables

Original OfferNumberof Roses

ProcessColumn

Cost atRoses-R-Red

ProcessColumn

Cost atFlowerPower

30 20 0 75 30+ ( ). $42.50 60 0 50 30+ ( ). $75.0060 20 0 75 60+ ( ). $65.00 60 0 50 60+ ( ). $90.0090 20 0 75 90+ ( ). $87.50 60 0 50 90+ ( ). $105.00

120 20 0 75 120+ ( ). $110.00 60 0 50 120+ ( ). $120.00150 20 0 75 150+ ( ). $132.50 60 0 50 150+ ( ). $135.00180 20 0 75 180+ ( ). $155.00 60 0 50 180+ ( ). $150.00210 20 0 75 210+ ( ). $177.50 60 0 50 210+ ( ). $165.00240 20 0 75 240+ ( ). $200.00 60 0 50 240+ ( ). $180.00

n 20 0 75+ . n 60 50+ . n

New OfferNumberof Roses

(new)

ProcessColumn

Cost atRoses-R-Red

ProcessColumn

Cost atFlowerPower

30 0 75 30. ( ) $22.50 40 50 30+ ( ). $55.0060 0 75 60. ( ) $45.00 40 50 60+ ( ). $70.0090 0 75 90. ( ) $67.50 40 50 90+ ( ). $85.00

120 0 75 120. ( ) $90.00 40 50 120+ ( ). $100.00150 0 75 150. ( ) $112.50 40 50 150+ ( ). $115.00180 0 75 180. ( ) $135.00 40 50 180+ ( ). $130.00210 0 75 210. ( ) $157.50 40 50 210+ ( ). $145.00240 0 75 240. ( ) $180.00 40 50 240+ ( ). $160.00n 0 75. n 40 50+ . n

I. Foundations for Functions 1.2 Valentine’s Day Idea: Activity 1


Activity 1: Valentine’s Day Idea

The school’s drill team has contacted several flower distributorsand has narrowed the choice to two companies.

Option 1: Roses-R-Red has offered to sell its roses for a fixeddown payment of $20 and an additional charge of 75 cents perstem.

Option 2: The Flower Power has offered to sell its roses for afixed down payment of $60 and an additional charge of 50 centsper stem.

Which is the more economical offer?



Activity 2: Using Tables to Find the More Economical Offer

From the description of the two offers, completethe chart to find an algebraic rule that willdetermine the cost of n roses.

Numberof Roses

ProcessColumn

(Roses-R-Red)

Cost atRoses-R-

Red

ProcessColumn

(Flower Power)

Cost atFlowerPower

1020306090

1201501802102401000

n

1. Write a sentence and a function rule for the cost of rosesfrom Roses-are-Red.

2. Write a sentence and a function rule for the cost of rosesfrom Flower Power.



3. What patterns do you observe from the table of values?

4. What happens to the cost of the roses as the number of rosespurchased increases? What would a graph of thisrelationship look like?

5. How many roses can you buy from Roses-are-Red for$65.00?

6. How many roses can you buy from Flower Power for$65.00?

7. Which company offers the better deal?

8. Is there a point where the two flower dealers charge the sametotal amount? If so, what is the charge? If not, why do thecosts never equal?

9. Write an equation that represents the point where the twoflower shops charge the same amount.



Activity 3: Using Graphs to Find the Better Offer

1. Find an appropriate viewing window for the graphs of bothfunctions. Sketch both functions here and label.

2. Justify your viewing window choice:xmin:xmax:ymin:ymax:



3. What effect does the 75 cents per stem cost have on the graphof the Roses-R-Red function? What effect does the $20 haveon the graph?

4. What effect does the 50 cents per stem cost have on the graphof the Roses-R-Red function? What effect does the $60 haveon the graph?

5. What are the coordinates of the point of intersection of thetwo functions? What is the significance of this point?

6. Which flower dealer offers the better deal? Justify youranswer.



Activity 4: New Rose Offers

To entice these potential new customers, Roses-R-Red decidesto eliminate its fixed charge of $20. According to its new offer,the drill team pays only for the roses they buy. When theFlower Power learns about the new offer by its competitor, itimmediately enters the price war by reducing its fixed chargealso by $20.

Which new deal is the better offer?

How does the new offer compare to the original offer?



Activity 5: Using Tables for New Rose Offers

From the description of each of the two new deals, complete thechart and write new algebraic rules that will determine the costof n roses.Numberof Roses

ProcessColumn

(Roses-R-Red)

Cost atRoses-R-

Red

ProcessColumn

(Flower Power)

Cost atFlowerPower

1020306090

120150180210240270300n

1. What patterns do you observe in the new table of values?2. Compare the costs on this chart to the costs on the first chart.

What changes do you observe? Predict what the graphs willlook like.

3. Which company offers the better deal?4. Is there a point where the two flower dealers charge the same

amount? If so, what is the charge?5. Write an equation that represents the point where the two

flower shops charge the same amount.



Activity 6: Using Graphs for New Rose Offers

1. On your graphing calculator graph only the original Roses-R-Red function, y x= +20 0 75. . Now, predict what you thinkthe graph of the new offer, y x= 0 75. , will look like.

2. Graph both the original offer, y x= +20 0 75. , and the newoffer, y x= 0 75. , together.

3. What effect does subtracting $20 from the old rule have onthe new graph of the Roses-R-Red function?

4. Turn off the above two graphs. Graph only the originalFlower Power function, y x= +60 0 5. . Now, predict whatyou think the graph of the new offer will look like,y x= +40 0 5. .

5. Graph both the original offer, y x= +60 0 5. , and the newoffer, y x= +40 0 5. , together.

6. What effect does subtracting $20 from the old rule have onthe new graph of the Flower Power function?

7. Graph all four functions at the same time. What are thecoordinates of the point where the two new functionsintersect? What is the significance of this point?

8. Which flower dealer now offers the better deal?



9. Sketch the graphs of all four functions and label the relevant pointsof intersection.

I. Foundations for Functions 1.2 Valentine’s Day Idea: Reflect and Apply


Reflect and Apply

1. Describe the rose deal represented bythe graph with a sentence and afunction.

2. Describe the rose deal represented bythe graph with a sentence and afunction.

3. Using your graphing calculator, graph the above two rosedeals together in the same window, find the point ofintersection and discuss what the point means in the fund-raising context. Sketch the window with the two graphs.

Cos

tC

ost

Number of Roses

Flowers-R-Us

All Occasion Roses

Number of Roses5

$10

$10

5



4. Describe the rose deal represented by the function:R x x( ) = +15 1 25.

5. Describe the rose deal represented by the function:S x x( ) = +35

6. Sketch the graphs of the above two rose deals below, find thepoint of intersection and discuss what the point means in thefund-raising context. Use your graphing calculator toconfirm.

7. The drill team has a coupon for All Occasion Roses for $8off the purchase of roses. Find a function to represent thecost of buying roses from All Occasion Roses if there was nominimum purchase required. Write the function and sketchthe graph.



8. Describe the flower deal represented by the table.

Number of Lilies Cost0 $0.001 $5.002 $10.003 $15.004 $20.00

9. Describe the flower deal represented by the table.

Number of Daisies Cost0 $5.001 $5.252 $5.503 $5.754 $6.00

10. Sketch the graphs of the functions in Exercises 8 and 9 onthe grid below.

I. Foundations for Functions 1.2 Valentine’s Day Idea: Student Activity


Student Activity: Investigate RecursivelyOverview: Using the power of the graphing calculator, students explore problems

recursively.

Objective: Algebra I TEKS(b.1) The student understands that a function represents a dependence of onequantity on another and can be described in a variety of ways.(b.3) The student understands algebra as the mathematics of generalization andrecognizes the power of symbols to represent situations.

Terms: Recursion

Materials: Graphing calculators

Procedures: Student Activity: Investigate Recursively

Do exercises 1 – 6 with students, using an overhead calculator to demonstrate.Then have students complete the New Rose Offers exercises.

1. Help students generate the recursive routines as follows, first withoutbraces and then with braces.

2.

3. $44.00 4. 23 roses

5. $76.00 6. 43 roses



New Rose Offers

1. 2.

3. 4.

5. 6.

7. A fixed charge of $50 with $1.00 per rose.8. A fixed charge of $10 with $1.25 per rose.9. No fixed charge with $0.90 per rose.10. No fixed charge with $0.50 per rose.11. A fixed charge of $30 with $0.35 per rose.

Note: A recursively defined sequence is defined by a starting value and arule. You generate the sequence by using the rule on the starting value andthen again on the resulting value and so on. Traditionally we have focused onfunctions written in closed form in a first year algebra class. In the institute,we use recursion to lead students to writing equations to model situations.

Summary: Students naturally operate recursively. We capitalize on this and use thepower of the graphing calculator to explore some linear situations, looking forboth input and output answers. Students build confidence in reading andunderstanding word problems. Students build intuition for constant rates ofchange and y-intercepts in linear situations.



Student Activity: Investigate Recursively

Roses-R-Red has offered tosell its roses for a fixed downpayment of $20 and anadditional charge of 75 centsper stem.

The Flower Power has offeredto sell its roses for a fixeddown payment of $60 and anadditional charge of 50 centsper stem.

1. Generate a recursive routine to investigate the Roses-R-Redoffer.

2. Generate a recursive routine to investigate the Flower Poweroffer.

Using your recursive routine, answer the following:3. How much would it cost to order 32 roses from Roses-R-

Red?

4. How many roses can you order from Roses-R-Red for$37.25?

5. How much would it cost to order 32 roses from FlowerPower?

6. How many roses can you order from Flower Power for$81.50?



New Rose OffersTo entice these potential new customers, Roses-R-Red decidesto eliminate its fixed charge of $20. According to its new offer,the drill team pays only for the roses they buy. When theFlower Power learns about the new offer by its competitor, itimmediately enters the price war by reducing its fixed chargealso by $20.

1. Generate a recursive routine to investigate the new Roses-R-Red offer.

2. Generate a recursive routine to investigate the new FlowerPower offer.

Using your recursive routine, answer the following:3. Now how much would it cost to order 43 roses from Roses-

R-Red?

4. Now how many roses can you order from Roses-R-Red for$27.00?

5. How much would it cost to order 43 roses from FlowerPower?

6. How many roses can you order from Flower Power for$58.00?



Students decided to check around at other shops. Based on thescreen shots below, what kind of deals did they find?

7.

8.

9.

10.

11.

I. Foundations for Functions 2.1 Identifying Patterns: Leaders' Notes


2.1 Identifying PatternsOverview: Participants represent linear relationships among quantities using concrete

models, tables, diagrams, written descriptions, and algebraic forms.

Objective: Algebra I TEKS(b.1) The student understands that a function represents a dependence of onequantity on another and can be described a variety of ways.(b.2.B) For a variety of situations, the student identifies the mathematicaldomains and ranges and determines reasonable domain and range values forgiven situations.(b.2.C) The student interprets situations in terms of given graphs or createssituations that fit given graphs.(b.3) The student understands algebra as the mathematics of generalizationand recognizes the power of symbols to represent situations.(c.1.B) The student determines the domain and range values for which linearfunctions make sense for given situations.

Terms: function, independent variable, dependent variable

Materials: building blocks, color tiles, graphing calculators

Procedures: Participants should be seated at tables in groups of 3 – 4.Do the Student Activity depending on the level of participants.

Activity 1: Painting TowersDo the Activity together as a whole group, bringing out the following pointsand asking the indicated questions.

1. Encourage participants to write how they found the number of faces topaint in the process column. This can often be done in several ways,which will lead to different, yet equivalent algebraic expressions. This is adesired outcome. Possible equivalent expressions include:

4 1+ 2 2 1+ + 52 4 1( ) + 2 2 2 2 1( ) + ( ) + 5 4+3 4 1( ) + 3 2 3 2 1( ) + ( ) + 5 2 4+ ( )n n4 1 4 1( ) + = + n n2 2 1( ) + ( ) + 5 1 4+ −( )( )n

Ask participants to use the cubes to physically demonstrate the algebraic rulesthey found in the table. In this example, they will mainly be pointing to faceson the cubes and relating them to the heights of the towers.Note: encourage participants to obtain the equivalent expressions from themodel, not by simplifying the algebraic expressions.



2.

Discuss with participants that choosing appropriate windows and thenjustifying window choices is a precursor to students learning about domainand range.Graph the function over the scatter plot to verify as shown above.

3.

4.

4 1 254 24

6

xx

x

+ =

=

=

• What does the ordered pair (8, 33) mean? [For term number 8(figure 8 or a tower 8 cubes high), the numerical term value(number of faces to paint) is 33.]

• Does the ordered pair (12,50) belong to this graph? Why or whynot? [A simple answer might be that there will always be thatextra top face to paint, and therefore the number of faces to paintwill be odd. Thus it is not possible for there to be 50 faces to paint.Also, the symbolic rule suggests odd numbers.]

5. Examples:6 2+ 2 3 1+( )2 6 2( ) + 2 2 3 1( ) +( )3 6 2( ) + 2 3 3 1( ) +( )n n6 2 6 2( ) + = + 2 3 1

2 3 1

n

n

( ) +( ) =+( )



6. Sample answer: For 35 cubes in each column: you need to paint 6 facestimes 35 plus the top 2 faces.

7. F n= +6 2

9.

• What changed in the rule? The figure? The graph? The domain?The range? Compare.

Use the following questions to compare the single tower situation and thedouble tower situation.

• What represents the “changing quantity” and what represents the“fixed quantity” in the pattern? [The addition of 4 faces with eachadditional cube represents the changing quantity in the singletower. The addition of 6 faces with each additional levelrepresents the changing quantity in the double tower. The faces ontops of the towers represents the unchanging one and two faces.]

• What number in the rule affects the slope, steepness, of the line?[The coefficient of x . In this case, it is the number 4 in the singletower and the number 6 in the double tower.]

• What number in the rule affects the starting point for the scatterplot (y-intercept for the line)? [The constant. In the first case, it isthe number 1, and in the second case, the number 2.]

Underline the constants in both of the functions, y x= +1 4 andy x= +2 8 .• What do the constants represent in the functions, y x= +1 4 and

y x= +2 8 ? [faces to paint]Circle the coefficient of x in both of the functions, y x= +1 4 andy x= +2 8 .• What do the coefficients of x represent? [faces to paint per tower

height or faces to paint per figure number.]

Repeat the above questions for Activities 2 – 3.

Activity 2: Building ChimneysHave participants do Activity 2 together in groups. Discuss as a whole group,asking several participants to share their different methods of arranging thecubes to find appropriate expressions.

1. Encourage participants to write how they found the number of blocks inthe process column.



For example:6 + 2(1) 2 + 2(2 + 1)6 + 2(2) 2 + 2(2 + 2)6 + 2(3) 2 + 2(2 + 3)6 + 2n 2 + 2(2 + n)

2. Graph the function over the scatter plot to verify as shown.

A reasonable domain for the situation is 0 to 10 blocks and a reasonablerange is 0 to 26 blocks.

3. You need 6 blocks for the base and then 23 rows of 2 blocks for thechimney. The ordered pair is (23, 52).

4. You need 28 blocks to build a house with a chimney 11 blocks high. Theordered pair is (11, 28).

5. No, the ordered pair (13, 34) does not belong to the graph because if youwere building figure 13, you would need 13 2 6 32( ) + = blocks, not 34.

6. Examples:9 + 1 3 + 3 + (3 + 1)9 + 2 3 + 3 + (3 + 2)9 + 3 3 + 3 + (3 + 3)9 + n 3 + 3 + (3 + n)

7. Sample answer: The total number of blocks equals 9 plus the number ofblocks in the chimney.



9.

• What changed in the rule? The figure? The graph? Compare.

Activity 3: Constructing Trucks

1. Encourage participants to write how they found the number of blocks inthe process column.

For example:3 + 2 2 + 2 + 13(2) + 2 3 + 3 + 23(3) + 2 4 + 4 + 33n + 2 2(n+1) + n

2. Graph the function over the scatter plot to verify as shown.

3. You need 152 blocks for the 50th figure.

4. If you use 242 blocks, you are on the 80th figure (term number).

2 + 3n = 2423n = 240n = 80



5. Examples:3 + 1 2 + 23(2) + 1 2(2) + 33(3) + 1 2(3) + 43n + 1 2n + (n + 1)

6. Sample answer: The total number of blocks equals 1 plus 3 times thefigure number.

8. The second scatter plot, with one block on top of each truck starts (the y-intercept is) lower than the previous plot, with two blocks on the top ofeach truck.

10. The graph starts higher than the original because now you have a constant4 blocks on top of each truck. The graph is steeper than the originalbecause you are now adding 6 blocks each time, instead of 3.

Compare Activities 1 – 3Use the following questions to compare the previous activities.

• What changed in the rule? The figure? The graph? Compare.• What represents the “changing quantity” and what represents the

“fixed quantity” in each of the patterns?• What number in the rule changes the slope of the line?• What number in the rule affects the starting point for the line?

On graphs of the lines generated in the Activities, draw triangles to show theidea that all of these rules have a constant rate of change, each time the samething was changing. See below for an example.



Activity 4: Generating PatternsHave half of the groups generate patterns that model surface area, similar toActivity 1: Painting Towers. Have the other half of the groups generatepatterns that model volume (number of cubes), similar to Activity 2: BuildingChimneys and Activity 3: Constructing Trucks.

Sample answers:1. How many faces to paint:

TermNumber

Visual(Figure)

WrittenDescription

ProcessColumn

Total Facesto Paint

1 Paint 4 lateralfaces and the top

and bottom.

4 2+ 6

2 Paint 4 lateralfaces twice and the

top and bottom.

2 4 2( ) + 10

3 Paint 4 lateralfaces three timesand the top and

bottom.

3 4 2( ) + 14

n 4 2n +

How many blocks to build:

TermNumber

Visual(Figure)

WrittenDescription

ProcessColumn

Number ofBlocks

1 Base of 4 plus 2. 4 2+ 6

2 Base grows by 4plus the 2 on top.

2 4 2( ) + 10

3 Base grows by 4more plus the 2 on

top.

3 4 2( ) + 14

n 4 2n +



2. How many faces to paint:

TermNumber

Visual(Figure)

WrittenDescription

ProcessColumn

Faces toPaint

1 Paint the two faceson the end plus the

8 front and backfaces.

2 8+ 10

2 Paint the 2 newfront and back

faces plus the twoon the end plus theoriginal 8 lateral

faces.

2 2 8( ) + 12

3 Add the 2 newfaces to theprevious.

3 2 8( ) + 14

n 2 8n +

How many blocks to build:

TermNumber

Visual(Figure)

WrittenDescription

ProcessColumn

Number ofBlocks

1 Base of 8 plus 2. 8 2+ 10

2 Base of 8 plus 2rows of 2 on top.

8 2 2+ ( ) 12

3 Base of 8 plus 3rows of 2 on top.

8 3 2+ ( ) 14

n 8 2+ n



Term Number

5

10

15

20

1 2 3 4

Note that both exercises in Activity 4 are examples of doing and undoing – animportant habit of mind for algebraic thinking.

Math Note: The manipulative model (using blocks to build figures torepresent patterns) has inherent domain and range restrictions. The sequencesgenerated in the table are for whole number input (domain) values. Forexample, you would not build figures with 0.5 or 0.3 of a block. Thealgebraic equations developed in this Activity are linear. The domain andrange of a line are all real numbers

Answers to Reflect and Apply:1. The first set of figures show adding 3 blocks every time (3x) to a constant 6

blocks, 3 6x + . The second set of figures show 3 groups of adding a blockevery time, x, to a constant 2 blocks, 3 2x +( ).



2. b3. d4. a5. cAs an extension, ask participants to label the axes with units and explain theirreasoning.

Summary: By using concrete models and the process column, participants model linearpatterns and explore constant rates of change. Participants model both inputand output questions with equations and solve them using tables and graphs.

I. Foundations for Functions 2.1 Identifying Patterns: Activity 1


Activity 1: Painting Towers

Suppose you are painting a tower built from cubes, basedon the pattern below. Use the table to find therelationship between the number of faces to paint and thenumber of blocks in the tower.(Paint only the sides and the top.)

TermNumber(Numberof blocks)

Visual(Figure)

WrittenDescription

ProcessColumn

NumericalValue of Term(Faces to Paint)

1 A 1 cube-hightower has 5

faces to paint.

5

2 9

3

4

n



1. Use the process column to write a function that expresses therelationship between the number of faces to be painted andthe number of cubes.

2. Graph the data from your table on 1” graph paper and/orcreate a scatter plot on a graphing calculator. What is areasonable domain for this situation? A reasonable range?

3. How many faces need to be painted for a 25 cube tower?Explain two ways of getting an answer.

4. If the tower you paint has 25 faces, how many cubes are inthe tower? Explain two ways of getting an answer.



5. Suppose you have two adjacent columns of cubes instead ofthe one column as before. Use your cubes to build the firstfour figures and determine the number of faces that need tobe painted.

TermNumber

Visual(Figure)

WrittenDescription

ProcessColumn

NumericalValue of Term(Faces to Paint)

1

2

3

4

n

6. Write a rule in words to describe how to find the totalnumber of faces that need to be painted for two columns ofcubes with 35 cubes in each column.

7. Write a rule in symbols that expresses the relationshipbetween the number of cubes in each column and the totalnumber of faces to be painted.

8. Predict how the graph of this data differs from the graph ofthe original data. Explain.

9. Graph the above data on 1” graph paper and/or create ascatter plot on a graphing calculator and compare to theprevious graph.



Activity 2: Building Chimneys

Suppose you are building a house with a chimney, based on thepattern below. Use the table to find the relationship between thenumber of blocks you need and the term number.

TermNumber

Visual(Figure)

WrittenDescription

ProcessColumn

NumericalValue of Term

(number ofblocks)

1 A house with achimney 1block high

takes 8 blocksto build.

8

2 10

3

4

n



1. Use the process column to write a function that expresses therelationship between the total number of blocks needed tobuild the house and the term number.

2. Graph the data from your table on 1” graph paper and/orcreate a scatter plot on a graphing calculator. What is areasonable domain for this situation? A reasonable range?Explain.

3. Use words to describe how to use blocks to build a housewith a total of 52 blocks. What is this ordered pair on thegraph?

4. If a house has a chimney that is 11 blocks high, how manyblocks will you need to build the house? What is this orderedpair on the graph?

5. Does the ordered pair (13, 34) belong to this graph? How doyou know?



6. Suppose the chimney is made of 1 block instead of two andthe house is built of three rows of 3 blocks instead of tworows of 3 blocks. Use your cubes to build the first threefigures and record the data below.

TermNumber

Visual(Figure)

WrittenDescription

ProcessColumn

NumericalValue of

Term1

2

3

4

n

7. Write a rule for this new data that expresses the relationshipbetween the total number of blocks and the number of blocksin the chimney for a house.





Activity 3: Constructing Trucks

Suppose you are building a truck, based on the pattern below.Use the table to find the relationship between the number ofblocks you need and the figure number.

TermNumber

Visual(Figure)

WrittenDescription

ProcessColumn

NumericalValue of

Term1 The truck has a

base of 3blocks with 2blocks on top.

5

2 8

3

4

n



1. Use the process column to write a function that expresses therelationship between the total number of blocks needed tobuild the truck and the term number.


3. Find the total number of blocks needed for the 50th figure.

4. If there are a total of 242 blocks, what term number is this?



5. Suppose there is only one block on the top of each truck.Use your cubes to build the first three figures and record thedata below.

TermNumber

Visual(Figure)

WrittenDescription

ProcessColumn

NumericalValue of

Term1

2

3

4

n

6. Write a rule for this new data that expresses the relationshipbetween the total number of blocks and the figure/termnumber.


8. Graph the above data on 1” graph paper and/or create ascatter plot on a graphing calculator and compare to theprevious graph. What effect did changing the number ofblocks on top of the truck have on the graph?

9. Suppose the original trucks (2 blocks on top) were built“double-wide.” Predict how the graph differs from theoriginal.

10. Build the first three “double-wide” trucks and graph the dataon your graphing calculator. How does this graph compareto the graph of the original? Why?



Activity 4: Generating Patterns

1. Given the following graph, use blocks to generate a sequenceof figures that fits the data. Fill in the table and sketch thefigures.

Term Number

5

10

15

20

1 2 3 4

TermNumber

Visual(Figure)

WrittenDescription

ProcessColumn

NumericalValue of

Term1

2

3

n



2. Given the function y x= +2 8, use blocks to generate asequence of figures that fits the function. Fill in the table,sketch the figures, and plot the graph. Label the graph.

TermNumber

Visual(Figure)

WrittenDescription

ProcessColumn

NumericalValue of

Term1

2

3

n

I. Foundations for Functions 2.1 Identifying Patterns: Reflect and Apply


Reflect and Apply

1. Create a physical model to demonstrate 3 6 3 2x x+ = +( ).

Match:2. ____ y x= +3

3. ____ y x=

4. ____ y x= +3 2

5. ____ y x= 2

c

d

a

b

6. Reflect on the activities. How might you adapt the activitiesto use with your students?

I. Foundations for Functions 2.1 Identifying Patterns: Student Activity


Student Activity: Perimeter of RectanglesOverview: Students investigate linear relationships using concrete models, tables,

diagrams, written descriptions, and algebraic forms.

Objective: Algebra I TEKS(b.1) The student understands that a function represents a dependence of onequantity on another and can be described a variety of ways.(b.3) The student understands algebra as the mathematics of generalizationand recognizes the power of symbols to represent situations.

Terms: function, independent variable, dependent variable, pattern

Materials: color tiles, graphing calculator

Procedures: Students should be seated in groups of 3 – 4.

Activity : Perimeter of RectanglesDo the activity together as a whole group, bringing out the following pointsand asking the indicated questions.

1. Encourage students to write how they found the number of perimeter inthe process column. This can often be done in several ways, which willlead to different, yet equivalent algebraic expressions. This is a desiredoutcome. Possible equivalent expression include:

Sample Process Sample Process Sample Process1 1 1 1+ + + 4 1( )2 2 2 2+ + + 4 2( )3 3 3 3+ + + 4 3( )n n n n+ + + 4n

2. Justify: The variable x stands for the figure number and xmin=0 toxmax=5 shows the figures 1 – 4 nicely.The variable y stands for the perimeter and ymin= −2 to ymax=20 showsthe perimeters of 4 to 18 nicely.

3. The perimeter of figure 11 is 44. 4 11 44( ) =4. Figure 12 has a perimeter of 48. 4 48n =



Ask students to use the tiles to physically demonstrate the algebraic rules theyfound in the table. In this example, they will mainly be pointing to sides onthe tiles and relating them to the numbers in the process column.

5.Sample Process Sample Process Sample Process2 1 2 1+ + + 2 2 2 1( ) + ( ) 4 1 2( ) +3 2 3 2+ + + 2 3 2 2( ) + ( ) 4 2 2( ) +4 3 4 3+ + + 2 4 2 3( ) + ( ) 4 3 2( ) +n n n n+( ) + + +( ) +1 1 2 1 2n n+( ) + ( ) 4 2n( ) +

6.

7. Figure 11 has a perimeter of 46. 4 11 2 46( ) + = .8. Figure 13 has a perimeter of 54. 4 2 54n + = .

Ask students to use the tiles to physically demonstrate the algebraic rules theyfound in the table. For example, the rule in the first column above is simplyadding each side in order. The rule in the second column above is noting thatthere are two sides of length n+1 and two sides of length n. The rule in thethird column above is based on the idea of adding two additional sides to asquare of side n.

Ask students to compare the rules, P n= 4 and P n= +4 2, and theirrespective graphs. Note that the lines have the same slope but that the lineP n= +4 2 is the line P n= 4 shifted up two. The perimeters grow by thesame amount each time you change figure numbers by one, but P n= +4 2starts 2 higher than P n= 4 .

Summary Using multiple representations, students gain added understanding for thelinear relationship of a rectangle’s perimeter and the length of a side.



Student Activity: Perimeter of Rectangles

Build these squares and the next three squares in the sequence, usingcolor tiles.Figure number 1, 2, 3Figure

1. Complete the table, using the process column to write a function forfigure n, and graph the relation.

FigureNumber

(length of side)

Process Perimeter

12345

n Length of Side

2. On your graphing calculator, make a scatter plot. Graph the functionover the scatter plot to confirm. Justify your window choice.

Answer the questions and write the equation that represents the question:3. What is the perimeter of figure number 11?

4. What figure number has a perimeter of 48?



Build these rectangles and the next three rectangles in the sequence,using color tiles.Figure number 1, 2, 3

Figure

5. Complete the table, using the process column to write a function forperimeter of the nth figure, and graph the relation.

FigureNumber

(length of side)

Process Perimeter

12345

n Length of Side

6. On your graphing calculator, make a scatter plot. Graph the functionover the scatter plot to confirm.Justify your window choice:

Answer the questions and write the equation that represents the question:7. What is the perimeter of figure number 11?

8. What figure number has a perimeter of 54?

I. Foundations for Functions 2.2 Identifying More Patterns: Leaders' Notes


2.2 Identifying More Patterns

Overview: Participants represent non-linear relationships among quantities usingconcrete models, tables, diagrams, written descriptions, and algebraic forms.

Objective: Algebra I TEKS(b.1.C) The student describes functional relationships for given problemsituations and writes equations or inequalities to answer questions arisingfrom the situations.(d.2.A) The student solves quadratic equations using concrete models, tables,graphs, and algebraic methods.(d.3.C) The student analyzes data and represents situations involvingexponential growth and decay using concrete models, tables, graphs, oralgebraic methods.

Terms: function, independent variable, dependent variable

Materials: building blocks, graphing calculators

Procedures: Participants should be seated at tables in groups of 3 – 4.Have participants do Activities 1 and 2 in their groups. Again encourageparticipants to do the process column in several ways, leading to different, yetequivalent algebraic expressions. Give groups the transparencies of theactivities and have them present their results, using overhead tiles todemonstrate different ways of physically modeling the algebraic rules. Afterthe assigned group has presented, ask if the other participants saw any otherways to physically model the rules and have them demonstrate also.

Ask the following with each activity• What does the ordered pair ( , ) mean?• Does the ordered pair ( , ) belong to this graph? Why or why not?• Trace to any ordered pair on the function. What meaning, if any,

do the coordinates have for the problem situation?

Activity 1: Building BlocksThis sequence of numbers is known as the square numbers (1, 4, 9, 16, 25,...)and provides a beginning point for identifying dependent quadraticrelationships.

1. Sample Process Sample Process Sample Process

1 12

2 2⋅ 22

3 3⋅ 32

n n⋅ n2



2.

3.

4.

6. The fifth figure has 5 rows by 6 rows, so there are 5 6 30⋅ = cubes.

7.Sample Process Sample Process Sample Process1 2⋅ 1 12 +2 3⋅ 2 22 +3 4⋅ 3 32 +n n +( )1 n n2 +

9.

Activity 2: Starting Staircases1.

Sample Process Sample Process Sample Process1 2

2⋅ 1 1

2 32⋅ 1 2+ 1

22 1

222( ) − ( )

3 42⋅ 1 2 3+ + 1

23 1

232( ) − ( )

n n +( )12

? 12

12

2n n−



2.

3. A base of 15 means the 15th triangular number so it has15 15 1

2120+( )

= blocks.

4. For a similar triangle with 36 blocks, the figure number is 8.6.

Sample Process Sample Process Sample Process2 2⋅ 2 2 1+ ( )3 3⋅ 3 2 3+ ( )4 4⋅ 4 2 6+ ( )n +( )1 2

n n n+( ) + +( )1 2 1

2

Flip the blocksin the back upon top to makea square of n+1dimensions.

The centercolumn plus 2triangularnumbers.

8.

Activity 3: Too Many TrianglesNote that TI has a Sierpinski triangle program in the manual.

Sample Process Sample Process11 3⋅1 3 3⋅ ⋅1 3⋅ n



2.

3.

4. We do not have the expectation that participants should solve thisalgebraically. We want participants to understand that the exponentialmodel is accessible to algebra students when approached numerically andgraphically and later symbolically. We spend more time on theexponential model at the end of the institute.

6.Sample Process Sample Process1

1 14

1 14

14

1 14

n

8.

Wrap it up:• Refer to the linear patterns in the previous activities. Do we have

quantities here that are the “changing quantity” and the “fixedquantity” in these patterns? [No, not like the linear patterns. Thepatterns here are a little different. We do have a “starting” point(y-intercept, the y-value when x = 0.]



• What about slope? [The main idea we want to get across here isthat the slope is not constant. The rate of change is changing.Constant rate of change, constant slope means linear relationship.Changing rate of change means non-linear.]

Draw triangles on the graphs to demonstrate that the rate of change is notconstant (the triangles are changing—for every constant change in x, thechange is y is different.)

Math Note: Participants may note that for the linear and quadratic patterns,the term number started with 1, but for the exponential patterns, the termnumber started with 0. These ideas will be further explored in the Linearsection and the Exponential Activities.

Answers to Reflect and Apply:1a. All of the tables in 2.1 Identifying Patterns can be produced recursively

using repeated addition.b. The graphs of repeated addition are linear.2a. The two tables in Activity 3 in 2.2 Identifying More Patterns can be

produced recursively using repeated multiplication.b. The graphs of repeated multiplication are not linear. They are

exponential.

Summary: By using concrete models and the process column, participants model non-linear patterns.

I. Foundations for Functions 2.2 Identifying More Patterns: Activity 1


Activity 1: Building Blocks

Suppose you are building square arrays out of cubes, as shownbelow. Use the table to find the relationship between thenumber of blocks needed for each figure and the dimension ofthe square array.

TermNumber

(dimensionof square

array)

Visual(figure)

WrittenDescription

ProcessColumn


(number ofblocks)

1A 1 by 1

square has 1cube.

1

2 4

3

4

n



1. Use the process column to write a function that expresses therelationship between the term number and the number ofcubes.


3. What figure number will have 25 cubes? What is thisordered pair on the graph?

4. How many cubes do you need for the 25th term? What is thisordered pair on the graph?



5. Suppose you add an extra column of cubes to the figuresabove. Use your cubes to build the first four figures andrecord the data below.

TermNumber

Visual WrittenDescription

ProcessColumn

NumericalValue of

Term1

2

3

4

n

6. Write a rule in words to describe how to find the totalnumber of cubes for the fifth figure.

7. Write a rule in symbols that expresses the relationshipbetween the total number of cubes and the term number.





Activity 2: Starting Staircases

Suppose you are building the triangular numbers from cubes,based on the pattern below. Use the table to find therelationship between the number of blocks and the term number.

TermNumber Visual Written

DescriptionProcessColumn

NumericalValue of

Term1 The 1st staircase

needs 1 block. 1

2 3

3

4

n



1. Use the process column to write a function that expresses therelationship between the total number of blocks and the termnumber.


3. Use words to describe how to use blocks to build a similarstaircase with a base of 15 blocks. What is this ordered pairon the graph?

4. If a similar staircase has 36 blocks, what is the figurenumber? What is this ordered pair on the graph?



5. Suppose the figures are as shown. Use your cubes to buildthe first three figures and record the data below.

TermNumber

Visual WrittenDescription

ProcessColumn

NumericalValue of

Term(number of

blocks)1

2

3

4

n

6. Write a rule for this new data that expresses the relationshipbetween the total number of blocks and the term number.





Activity 3: Too Many Triangles

Suppose you are drawing the fractal, based on the pattern below.Use the table to find the relationship between the number ofupward triangles and the term number.




(number ofnew upward

triangles)0 One new

upward triangle 1

1Three new

upwardtriangles

3

2

3

4

n



1. Use the process column to write a function that expresses therelationship between the number of new upward triangles andthe term number.

2. Graph the data from your table on 1” graph paper and/orcreate a scatter plot on a graphing calculator. What is areasonable domain for this situation? A reasonable range?Explain.

3. Find the total number of new triangles for term number 8.

4. If there are a total of 2187 new upward triangles, what termnumber is this?



5. Now consider the area of one of the smallest triangles in eachfigure and record the data below.



NumericalValue of

Term (areaof smallesttriangle)

0The area of the

smallesttriangle is 1

unit

1

1

The area of oneof the smallest

triangles is14

of a unit

14

2

3

n

6. Write a rule for this new data that expresses the relationshipbetween the area of one of the smallest triangles and the termnumber.



I. Foundations for Functions 2.2 Identifying More Patterns: Reflect and Apply


Reflect and Apply

1. Refer to the tables in 2.1 Identifying Patterns.a. Look at the column “Numerical Value of Term.” What

operation when applied to a term produces the next term?In other words, what is happening recursively?

b. What kind of graph does this repeated operation produce?

2. Refer to the tables in Too Many Triangles.a. Look at the column “Numerical Value of Term.” What

operation when applied to a term produces the next term?In other words, what is happening recursively?

b. What kind of graph does this repeated operation produce?

3. Reflect on the activity. How might you adapt the activity touse with your students?

I. Foundations for Functions 3.1 Interpreting Distance versus Time Graphs: Leaders’ Notes


3.1 Interpreting Distance versus Time GraphsOverview: Participants use motion detectors to investigate distance over time graphs.

This lays the groundwork for graph reading and for work with rates of change.

Objective: Algebra I TEKS(b.2.C) The student interprets situations in terms of given graphs or createssituations that fit given graphs.

Terms: rate of change, increasing, decreasing, constant

Materials: motion detector connected to an overhead calculator, motion detectors, datacollection devices, graphing calculators, 2 or 3 transparencies cut-to-fit on theoverhead calculator screen

Procedures: This activity is done in an open area with room for participants to move aboutin groups of 2 – 4.

Have participants complete the Student Activity: Walk This Way. As you dothe whole group introduction, sketch two of the walks of the participants. Dothis by placing a transparency, cut to fit, on the overhead calculator screen andthen sketch the walk with a transparency marker. We will use these sketchesof walks at the end of Activity 2.

Math Note: Scientific convention is to write y versus x when referring to asituation of (x, y). In other words, the dependent variable is always listed first,then the independent variable as in: dependent variable versus independentvariable. Thus, this activity refers to Distance versus Time graphs or Distanceover Time graphs, where time is the independent variable and distance is thedependent variable.

Activity 1: Walking GraphsHave participants answer the questions in their groups. Circulate and answerquestions. Ask a member of each group to present an answer for one of theExercises.Answers will vary. Sample answers:1. The y-intercept of the first segment is the distance away from the motion

detector at time = 0. This tells you where to start walking.2. The x-axis represents time and each tick mark represents a second. The

number of tick marks on the x-axis represents the time to walk for eachsegment.

3. If the segment decreases as time increases on the x-axis, then the distancefrom the motion detector is decreasing. This means you should walktoward the motion detector.



4. If the segment increases as time increases on the x-axis, then the distancefrom the motion detector is increasing. This means you should walk awayfrom the motion detector.

5. If the segment decreased or increased slowly (it was shallow), then thatmeans that the distance from the motion detector was changing slowly astime increased. Therefore, you should walk slowly.

6. If the segment decreased or increased rapidly (it was steep), then thatmeans that the distance from the motion detector was changing rapidly astime increased. Therefore, you should walk quickly.

7. If the segment was horizontal, then that means that the distance from themotion detector was not changing as time increased. The distance wasremaining constant. Therefore you should stand still.

Put up the transparency of the student activity: Practice Walking LinearGraphs. Draw triangles on some of the line segments to show that each linesegment has a constant rate of change, that for every increment in the x-direction, the segment increases or decreases by the same amount in the y-direction. Do not spend too much time here. Simply demonstrate that for agiven line segment, triangles drawn as shown with equal bases are congruent.This means that the heights are equal. In the next activity, we will contrastthese constant rates of change with non-constant rates of change of non-lineargraphs.

Activity 2: Walking More GraphsIn this activity, participants walk non-linear graphs. Have participants collectdata for about 4 seconds. (This may necessitate collecting the data not in real-time. If participants prefer to collect data in real-time (seeing the data as theycollect it), simply have participants ignore the tick marks on the x-axis andcollect data for 15 seconds. Discuss the advantages and disadvantages of bothmethods. In real time, participants can quickly adjust their motion in themiddle of data collection. However, that may mean that they do not analyzeand plan carefully first. Out of real time, participants must analyze and plancarefully. Then after obtaining the resulting graph, they adjust the plan andwalk again.

The big idea in the activity is to get a feel for changing rates of change,therefore the exact starting and stopping points are less important then thegeneral shape of the graph.



Have participants do the activity and answer the questions in their groups.Circulate and help. Ask a member of each group to present an answer for oneof the Exercises.

Answers will vary. Sample answers:1. Start 6 feet from the motion detector. Stand still for less than a second.

Start walking toward the motion detector slowly at first and then speedingup over 4 seconds.

2. Start about 2.5 feet from the motion detector. Start walking away quickly,slowing down over 4 seconds to a dead stop at the end.

3. Start about 2 feet from the motion detector. Start walking away slowly,speeding up over 4 seconds.

4. Start about 6 feet from the motion detector. Start walking toward themotion detector quickly, slowing down over 4 seconds to a dead stop atthe end.

5. Start about 6 feet from the motion detector. Start walking toward themotion detector quickly, slowing down over 2 seconds, stop briefly, thenwalk away from the motion detector slowly, speeding up till the end of thefour seconds.

6. Start about 2 feet from the motion detector. Start walking away from themotion detector quickly, slowing down over 2 seconds, stop briefly, thenwalk toward the motion detector slowly, speeding up till the end of thefour seconds.

7. You know to speed up (accelerate) when the curve gets steeper, when thechange in y over an increment in x is greater than it was before.

8. You know to slow down (decelerate) when the curve gets less steep, whenthe change in y over an increment in x is less than it was before.

9. e, f, g. These graphs show walking at a constant rate because the changein the y-direction is constant for a constant change in the x-direction. Therate of change is constant. For the horizontal line, the rate of change is aconstant zero. Draw triangles to illustrate.

For every change in x , the change in y is zero!



10. a,c. These graphs show a walker speeding up because the change in y isincreasing over a constant change in x. The rate of change is increasing.Draw triangles to illustrate.

You can also draw tangent lines to illustrate that the slopes of the tangent lines(the instantaneous rate of change at the point of tangency) are increasing.

11. b,d. These graphs show a walker slowing down because the change in y isdecreasing over a constant change in x. The rate of change is decreasing.Draw triangles to illustrate.

You can also draw tangent lines to illustrate that the slopes of the tangent lines(the instantaneous rate of change at the point of tangency) are decreasing.

12. e. As shown above, the rate of change for a horizontal line is a constantzero. The walker is standing still because for every change in x, thechange is y is a constant zero. The distance from the motion detector isnot changing.



• Can you produce a vertical line by walking in front of a motiondetector? [No, a vertical line suggests that at one instant in time, thewalker is at an infinite number of locations, every point on that line. Awalker can only be one distance from the motion detector at a time.]

Wrap up these activities as a whole group using the sketches of participants’walks on the cut-to-fit transparencies from the beginning. Put a sketch on acut-to-fit transparency on the overhead calculator. (Consider using a walk thathas varied sections, linear and non-linear.) Ask participants to discuss in theirgroups, how a walker should walk to reproduce the walk. Randomly choose agroup’s description and have a walker walk the description to confirm.Repeat if desired.

Sample Answers to Reflect and Apply:1. I started jogging away from my house (when I was 0.5 kilometers from

my house) and I gradually sped up to a sprint. At 15 minutes (tick marksare at 5 minute increments), I fell on the ground, 4 kilometers from myhouse. I sat there for about 20 minutes, catching my breath. I then startedgradually back home, speeding up as I went until I was again sprinting inthe door.

2. I walked from my locker to my class over two minutes. I stood therechatting for a minute until I realized I had left my book in my friend’slocker. I walked quickly to my friend’s locker, which was thankfullyopen, grabbed the book and started quickly back to class. But I ran intothe principal who walked me back to class, slowing down as we went,because the principal was talking to me about an upcoming event.

Summary: The big idea here is that walking at a constant rate produces a linear distanceover time graph. The rate of change of a line is constant. Speeding up orslowing down, non-constant rates of change, produce non-linear distance overtime graphs. The rates of change are changing.

I. Foundations for Functions 3.1 Interpreting Distance versus Time Graphs: Activity 1


Activity 1: Walking Graphs

Answer the following questions based on your experiences inthe Student Activity: Walk This Way.

1. How did you know where to start walking for each graph?

2. How did you know how long to walk for each segment?

3. How did you know when to walk toward the motiondetector? Use the words “time” and “distance” in youranswer.

4. How did you know when to walk away from the motiondetector? Use the words “time” and “distance” in youranswer.

5. How did you know when to walk slowly?

6. How did you know when to walk quickly?

7. How did you know when to stand still?



Activity 2: Walking More Graphs

Practice walking the following graphs using a motion detectorand a graphing calculator. Describe the walk that you used toproduce each graph.

1. 2.

3. 4.

5. 6.

dist

ance

time timedi

stan

ce

dist

ance

dist

ance

dist

ance

dist

ance

time time

time time



7. How did you know when to speed up?

8. How did you know when to slow down?

Which graph(s) below show:9. a constant rate? Why?

10. a walker speeding up? Why?

11. a walker slowing down? Why?

12. a walker standing still? Why?

a. b. c. d.

e. f. g.

I. Foundations for Functions 3.1 Interpreting Distance versus Time Graphs: Reflect and Apply


Reflect and Apply

Make up a story for the following graphs of distance over time.

Distance over Time Distance over Time

1. 2.

I. Foundations for Functions 3.1 Interpreting Distance versus Time Graphs: Student Activity


Student Activity: Walk This WayOverview: This is an introduction to graphs of motion data, specifically linear data.


Terms: rate of change

Materials: motion detector connected to a viewscreen calculator, a motion detector withgraphing calculator for each group of 3-4 students, and data collection devices

Procedures: Often students are asked to interpret distance over time graphs. In thisactivity, students experience seeing many examples quickly of their ownmotion graphed over time. Students gain intuition for interpreting graphs asthey make connections between their own motion and the graphs of theirmotion.

The classroom should be set up with an aisle down the middle. Set up amotion detector pointing down the aisle, connected to a viewscreen calculator,so the class can see both the students walking down the aisle and the dataprojected from the calculator on a screen in front of the room.

Explain that the motion detector sends out an ultrasonic pulse. The pulsebounces off the walker, and the motion detector records the distance at thattime. The calculator displays the data as a graph with the distance measuredin meters and the time measured in seconds.

Run the CBR Ranger program or a similar program. Use the followingscreens to set up the experiment, and then follow the instructions on thescreen.

Ask a few students to walk one at a time in front of the motion detector.Encourage students to walk differently - slowly, quickly, standing still, towardthe motion detector, away from the motion detector, etc.



After each walk, discuss the following:• What does the starting point represent?• What does a fast walk look like?• What does a slow walk look like?• What does a pause look like?• What does it look like when you walk away from the motion detector

or toward the motion detector?• Discuss a “straight” verses a “curved” graph. Point out that for

constant rates of change, we use a linear model to model the data.

Activity: Practice Walking Linear GraphsEach group of 3 – 4 students will practice walking different linear graphsusing a motion detector and a graphing calculator.

First Time: Hold the motion detector and the calculator. Point the motiondetector at the wall and practice walking the graphs.

Second Time: Have the group hold the motion detector and calculator. Pointthe motion detector at one person in the group. As a group, instruct thewalker on how to walk the graph.

Extension: Have each group come to the front and match a graph. Give thegroup a minute to discuss. Then a member of the group walks to match thegraph. Have the rest of the class rate the group using the following rubric.Have the class rate the group quickly using a show of fingers. Choose themode for each rating to get a quick score for the team.Starting Point: (1 – 5) Did the walker start at the correct point?Rate: (1 – 5) Did the walker walk at the correct rate?Direction: (1 – 5) Did the walker walk in the correct direction?Accuracy: (1 – 5) Was the rate correct but the distance was

incorrect?Teamwork: (1 – 5) Did the group work as a team well?

Assessment: Linear Motion.Now your students should be able to complete the Assessment. Note that thisassessment is intended to provide teachers with sample assessment items. It isnot intended as a stand-alone worksheet or quiz. Teachers can use the itemsas a starting point for creating meaningful assessments.

Answers:1. B2. C3. A4 – 7. Answers will vary.



Sample answers:

8. 9.

10. 11.

Note: this activity was adapted from an activity in the MathematicalModeling Institute for Secondary Teachers.

Summary: Through experience, students learn intuitive notions about distance over timegraphs. Starting points (y-intercepts), rates of change, direction, and timeintervals are among the ideas that are built.



Activity: Practice Walking Graphs

Practice walking the following graphs using a motion detector.



Assessment: Linear Motion

Match the description with the graph.

a. b. c. _____ 1. Start one meter away from the motion detector. Walk slowly

away from the motion detector for about 3 seconds, stand stillfor about 4 seconds, and then walk quickly away from themotion detector for about 3 seconds.

_____ 2. Start 3 meters away from the motion detector, and walk awayfrom it at a moderate rate for about 3 seconds. Stand still forabout 4 seconds, and then walk quickly toward the motiondetector for 3 seconds.

_____ 3. Start 2 meters away from the motion detector, and walk towardit at a moderate rate for about 3 seconds. Stand still for about 4seconds, and then walk toward the motion detector at about thesame moderate rate as earlier for about 3 seconds.

Write a description for a walk that would produce each of these graphs.

4. ____________________ 5. __________________________________________________ ______________________________

6. ____________________ 7. __________________________________________________ ______________________________



Sketch a graph that would match the description.8. Start 4.5 meters from the motion detector, and walk quickly toward it

for 2.5 seconds. Walk slowly toward the motion detector for 3seconds. Walk even more slowly away from the motion detector for4.5 seconds.

9. Start 0.5 meters from the motion detector. Walk slowly away fromthe motion detector for 5 seconds. Walk extremely slowly away fromthe motion detector for 2 seconds. Walk quickly away from themotion detector for 3 seconds.

10. Start one meter from the motion detector and stand still for 2seconds. Walk away from the motion detector quickly for 5 seconds.Walk back slowly toward the motion detector for 3 seconds.

11. Start 3 meters from the motion detector, and walk slowly away fromit for 3 seconds. Stand still for 4 seconds. Walk quickly toward themotion detector for 3 seconds.

I. Foundations for Functions 3.2 Interpreting Velocity versus Time Graphs: Leaders’ Notes


3.2 Interpreting Velocity versus Time GraphsOverview: Participants use motion detectors to investigate velocity over time graphs.

This continues to build graph-reading skills and to build understanding forrates of change.


Terms: velocity, speed, direction, rate of change

Materials: motion detector connected to an overhead calculator, motion detectors, datacollection devices, graphing calculators

Procedures: Set up the room similarly to the Student Activity: Walk This Way, with amotion detector connected to an overhead calculator. The room should havean aisle down the middle in front of the motion detector.

Set up the motion detector to display velocity data, collecting data for 4seconds.

• What is velocity? [Speed and direction.]• What are some common units for speed? [miles per hour, feet per

second, meters per sec, kilometers per hour, etc.]• How will a graph of velocity over time differ than a graph of distance

over time? [Do not answer at this point.]

Have a volunteer walk the following instructions. Collect data for about 4seconds. Before you display each graph, have participants sketch a predictionof the graph. After each walk, discuss the resulting graph, using the questionsfollowing each instruction.

1: Start 1.5 feet away from the motion detector. Stand still briefly when thedata collection starts, then walk away slowly, speeding up as you go.Repeat if necessary.• What does x represent? [elapsed time]• What does y represent? [the speed of the walker]• Why does the graph increase? [As time increased, the speed of the

walker increased.]

Trace to a value on the graph as shown.• What is the meaning of the coordinates

shown? [After about 1.8 seconds, I wasmoving at 0.2 meters per second awayfrom the motion detector.]



2: Start behind the motion detector. Walk past the motion detector and startthe data collection when you are about 1.5 feet in front of it. Continue towalk away, slowing down as you go to a complete stop by the end of thetime. Repeat if necessary.

• What does x represent? [elapsed time]• What does y represent? [the speed of the walker]• Why does the graph decrease? [As time increased, the speed of the

walker decreased.]


shown? [After about 1 second, I wasmoving at 0.5 meters per second awayfrom the motion detector.]

3: Start about 10 feet from the motion detector. Stand still briefly when thedata collection starts, then walk toward the motion detector slowly,speeding up as you go. Repeat if necessary.

• What does x represent? [elapsed time]• What does y represent? [the speed of the walker]• Why does the graph increase? [As time increased, the speed of the

walker increased.]• Why is the graph in the fourth quadrant? [Velocity is speed and

direction. The negative y-values indicate that the walker was movingtoward the motion detector. The magnitude of the y-values indicatethe rate, the speed. Magnitude is the same as the absolute value of thenumber.]


shown? [After about 2 seconds, I wasmoving at 0.3 meters per second towardthe motion detector.]

4: Start about 15 feet from the motion detector, walking quickly toward themotion detector, and begin the data collection when you are about 10 feetin front of it. Continue to walk toward the motion detector, slowing downas you go. Repeat if necessary.

• What does x represent? [elapsed time]• What does y represent? [the speed of the walker]• Why does the graph decrease? [As time increased, the speed of the

walker decreased.]



• Why is the graph in the fourth quadrant? [Velocity is speed anddirection. The negative y-values indicate that the walker was movingtoward the motion detector. The magnitude of the y-values indicatethe rate, the speed. Magnitude is the same as the absolute value of thenumber.]


shown? [After about 2.3 seconds, I wasmoving at 0.34 meters per secondtoward the motion detector.]

5: Start right at the motion detector. Walk away from the motion detector ata constant rate. Begin the data collection when you are at least 1.5 feetfrom the motion detector or when you are walking at a constant rate.Continue to walk at a constant rate until time is up. Repeat if necessary.

• What does x represent? [elapsed time]• What does y represent? [the speed of the walker]


shown? [After about 2.3 seconds, I wasmoving at 0.34 meters per secondtoward the motion detector.]

• Why is the graph so wavy? Was the rate not constant? [Trace on thegraph and note that the values are really quite close together, but thewindow is probably very small, which exaggerates small changes inthe y-values. Change the window as shown below and note that thegraph now looks quite constant.]

Math Note: Velocity is speed and direction. The speed is represented by themagnitude of the velocity. The direction is represented by the sign of the



velocity. If the velocity at time t is –4, then − =4 4 is the speed at time t andthe direction is toward the motion detector. If the velocity at time t is 6, then6 6= is the speed and the direction is away from the motion detector.

When discussing direction and velocity, remind participants of the directionimplied in a different measure of motion, acceleration. The force of

acceleration due to gravity is −32 ftsec2 or −9 8. m

sec2 . The value of

acceleration is negative which means that the object is falling toward theearth, just as the negative in velocity means that the walker is walking towardthe motion detector.

Activity 1: Matching Velocity GraphsHave participants do the activity in groups of 3 – 4, by walking the graphs andthen answering the questions. Ask a member of each group to present ananswer for one of the Exercises.

Answers will vary. Sample answers:1. The y-values stand for velocity so when the y-values have small

magnitudes, then the rate is slower. Therefore, you should walk slower.

2. The y-values stand for velocity so when the y-values have largemagnitudes, then the rate is faster. Therefore, you should walk faster.

3. Since the y-values stand for velocity, when the graph decreases in the firstquadrant, then the velocity should decrease, or in other words, you shouldslow down. In the fourth quadrant, when the graph increases, then thevelocity is decreasing (because the magnitude is getting smaller) and youshould slow down.

4. Since the y-values stand for velocity, when the graph increases in the firstquadrant, then the velocity should increase, or in other words, you shouldspeed up. In the fourth quadrant, when the graph decreases, then thevelocity is increasing (because the magnitude is getting larger) and youshould speed up.

5. Points in the first quadrant have positive y-values. In this case, thepositive velocity means that you were walking away from the motiondetector.

6. Points in the fourth quadrant have negative y-values. In this case, thenegative velocity means that you were walking toward the motiondetector.

7. Since the y-values stand for velocity, points along the x-axis have y-valuesof zero and therefore, the velocity is zero. Therefore, you should stop.



8. You don’t know where to start! You need to start where ever you willhave enough room to do what you need to do. If you need to move towardthe motion detector, you had better start a ways in front of it. If you needto move away from the motion detector, you probably ought to start closerto it.

Activity 2: Connecting Distance and Velocity GraphsDo Exercise 1 together as a whole group. Ask a volunteer to follow theinstructions and first display the distance graph. In order to see both thedistance and the velocity graphs at the same time, do the following. Put anoverhead transparency on the overhead calculator screen and sketch thedistance over time graph. Place this transparency on another overheadprojector. Now display the velocity graph on the viewscreen calculator.With both graphs displayed simultaneously, label them as follows, based onthe questions on the activity page.

1.

quicker rateslower rate

standing still

slowing down

speeding up

quicker rate

slower rate

standing still

slowing down

speeding up

2.

standing still

slowing down

slower rate

quicker rate

speeding up

quicker rate

speeding up

standing still

slowing down

slower rate

Note: This activity provides participants with intuitive experience in thecalculus and physics concepts of a function (a position function) and thefunction’s derivative (a velocity function). However, the main idea of this



activity in the Algebra I Institute is to use the graphs of participant’s ownmotion to build the skills of reading, interpreting, and analyzing graphs.If participants request that you help them connect these experiences with theirprevious experiences in calculus and physics, then briefly help them makethose connections if you wish. Do not suggest that the participants are toteach calculus and physics in an algebra I course.

Answers for Reflect and Apply:Sample Answers:Distance over Time: In PE one day, students were running on the basketballcourt. They started at the free throw line and jogged to the end of the court.Then they stood there and took a breather. Then they sprinted back the wholelength of the court.

Velocity over Time: In PE one day, students were running on the basketballcourt. They were already jogging, and when the time keeper said, “Go,” theysped up to a sprint. They sprinted for a bit and then they slowed down (fasterthan they had sped up) until they came to a stop.

Ask participants to recall the Reflect and Apply Exercises from 1.1.1Variables and Functions.

• Do you think students would complete these Exercises differently afterthe graphing experiences in this Activity?

Summary: The big idea here is to build intuition for interpreting graphs. With the use oftechnology, participants get instant feedback on distance and velocity graphsand can see many examples rapidly. They can monitor and adjust quickly togain added understanding. By comparing a distance graph for a situation witha velocity graph of the same situation, participants make valuable connectionsbetween distance and velocity and about analyzing graphs.

I. Foundations for Functions 3.2 Interpreting Velocity versus Time Graphs: Activity 1


Activity 1: Matching Velocity Graphs

Practice walking the following graphs using a motion detectorand a graphing calculator. Describe the walk that you used toproduce each graph.a. b.

Time(0, 0) Time(0, 0)

c.

Time(0, 0)

d.

Time(0, 0)

e.

Time(0, 0)



1. How did you know when to walk slowly?

2. How did you know when to walk quickly?

3. How did you know when to slow down?

4. How did you know when to speed up?

5. How did you know when to walk away from the motiondetector?

6. How did you know when to walk toward the motiondetector?

7. How did you know when to stop?

8. How did you know where to start walking for each graph?

9. How did you know what speed to start walking?



Activity 2: Connecting Distance and Velocity Graphs

Read the following directions and sketch your prediction of theresulting graphs. Then collect the data (for about 4 seconds.)

1. Start about 1.5 feet in front of the motion detector and walkaway quickly for about 2 seconds. Stand still for 1 secondand then walk toward the motion detector slowly for theremaining time.Predict :

a. Distance versus Time b. Velocity versus Time

The actual results:c. Distance versus Time d. Velocity versus Time

Label the following sections in the graphs above:A. The walker is standing still.B. The walker is slowing down.C. The walker is speeding up.D. The walker is traveling at a quicker rate.E. The walker is traveling at a slower rate.



2. Start about 5 feet in front of the motion detector and walktoward the motion detector slowly for about 1 second. Standstill for 1 second and then walk away from the motiondetector quickly for the remaining time.

Predict :a. Distance versus Time b. Velocity versus Time

The actual results:c. Distance versus Time d. Velocity versus Time

Label the following sections in the graphs above:A. The walker is standing still.B. The walker is slowing down.C. The walker is speeding up.D. The walker is traveling at a quicker rate.E. The walker is traveling at a slower rate.

I. Foundations for Functions 3.2 Interpreting Velocity versus Time Graphs: Reflect and Apply


Reflect and Apply

Make up two different stories based on the graph below. Onestory should be based on the graph representing distance overtime. The other story should be based on the graph representingvelocity over time.

Story for distance versus time: Story for velocity versus time:

II. Linear Functions 1.1 The Linear Parent Function: Leaders’ Notes


1.1 The Linear Parent FunctionOverview: Using contextual situations, participants investigate the linear parent function,

y x= .

Objective: Algebra I TEKS(b.2.A) The student identifies and sketches the general forms of linear ( y x= )and quadratic ( y x= 2 ) parent functions.(b.2.C) The student interprets situations in terms of given graphs or createssituations that fit given graphs.

Terms: input, output

Materials: colored pencils or pens, pieces of flat spaghetti, graphing calculators, programACTSCRS

Procedures: Because of the role of the line y x= as the linear parent function, we beginour work with linear functions using contextual situations to develop intuitivenotions about the line y x= .

Participants should be seated at tables in groups of 3 – 4.Complete the Student Activity, Age Estimates, with participants. Talkthrough the assessment.Talk through the Student Activity, Sales Goals, with participants.

Activity 1: ACT ScoresHave participants run the program ACTSCRS. The program stores the dataon Transparency 1 into five lists in the calculator.Create the indicated scatter plots and graph the line y x= , finding appropriatewindows to be able to see the comparisons.

Sample General Statements:1. Most states have higher mean Mathematics scores than that state’s mean

English scores.2. Most states have higher mean Reading scores than that state’s mean

Mathematics scores.3. Most states have higher mean Sci Reasoning scores than that state’s mean

Mathematics scores.4. Most states have higher mean Composite scores than that state’s mean

Mathematics scores.



Sample Scatter Plots:

English versus Mathematics Reading versus Mathematics

17.0

19.0

21.0

23.0

17.0 19.0 21.0 23.0

Mathematics

17.0

19.0

21.0

23.0

17.0 19.0 21.0 23.0

Mathematics

Sci Reasoning versus Mathematics Composite versus Mathematics

17.0

19.0

21.0

23.0

17.0 19.0 21.0 23.0

Mathematics

17.0

19.0

21.0

23.0

17.0 19.0 21.0 23.0

Mathematics

There is other comparative information about ACT scores athttp://www.act.org/news/data/99/99data.html

• Did any of the results surprise you?• How did you use the line y x= to make general statements about the

scores?• What are some of the factors that may influence the results? [Some

states have large populations of students who take the ACT, wherethere are other states whose student populations primarily take theSAT. In those SAT states, why might students choose to take the



ACT? Perhaps those students are desiring to go out of state, lookingfor specific scholarships, or going to private schools.]

• Is it true that the higher the mathematics score, the higher the other testscore tend to be? In other words, is there a positive correlationbetween math scores and the other scores? [Yes]

Activity 2: TemperaturesIn the Student Activity and Activity 1, participants explore data in the firstquadrant, comparing the data to the line y x= . In this activity, participantswill explore data in all 4 quadrants, using average temperatures around theworld.

Have participants work in groups to complete the activity. Ask a group topresent their results. Use Transparency 3: World Map to determine thelocation of indicated places. Note that for places north of the equator seasonsare opposite of those south of the equator.Below find the average temperatures used to make the graph.

Place July (C˚) December (C˚)1 Austin, TX 3 5 16.72 Detroit, MI 28.5 1.83 Siberia 1 4 -144 Santiago, Argentina 1 3 26.95 Nome, Alaska 10.2 -13.96 Greenland icecap -11 -477 Antarctica -57 -188 Butlers Gorge, Australia -0.4 5.4

-60 -40 -20 0 20 40 60

-20

-40

-60

20

40

60

Average Temperatures (C˚) in Selected Locations

Average July Temperatures (C˚)

12

35

6

7

8

4

• How many points are in the second quadrant? [One]



• Describe the place that would have a point in the third quadrant. [Theplace would have both a cold average July temperature (belowfreezing) and a cold average December temperature (below freezing).The place with the colder July temperature would need to be in thesouthern hemisphere, our example is Antarctica. The place with thecolder December temperature would need to be in the northernhemisphere, our example is the Greenland icecap.]

• Describe the place that would have a point in the second quadrant.[The place would have a cold average July temperature (belowfreezing) and a warmer average December temperature. Thus, itwould need to be in the southern hemisphere, our example is Butler’sGorge, Australia.]

Activity 3: SymbolicHave participants work in groups to complete the activity. Ask a group topresent their results.

1. A, B, C, K, L2. E, F, G, H, I3. D, J4. A, G5. Any point in the second quadrant, points in the first quadrant above the

line y x= , points in the third quadrant above the line y x= .6. Any point in the fourth quadrant, points in the first quadrant below the line

y x= , points in the third quadrant below the line y x= .7. Points on the line y x= .8. Points on the line y x= − .

• In what ways have the concrete problem situations in the Activitychanged the way you approached these symbolic questions?

• In what ways can experience with similar concrete problem situationsaid students in their understanding of similar symbolic questions?

Answers to Reflect and Apply1. The walker is moving at about 1 foot per second.2. From 0 to 1 seconds, 4 to 5 seconds, 5 to 6 seconds3. From 1 to 2 seconds, 2 to 3 seconds, 7 to 8 seconds

Summary: The line y x= is the parent function for the linear function family. In thisactivity we build intuition for the line y x= as an important starting point forwork with linear functions.

II. Linear Functions 1.1 The Linear Parent Function: Transparency 1


Transparency 1: ACT Scores by State 1999

State Name English Math Reading Sci

Reasoning

Composite

Alabama 20.3 19.5 20.5 19.9 20.2Alaska 20.1 21.0 21.7 21.1 21.1Arizona 20.7 21.4 21.9 21.2 21.4Arkansas 20.6 19.2 20.7 20.1 20.3California 20.5 21.9 21.5 20.8 21.3Colorado 20.9 21.1 21.9 21.7 21.5Connecticut 21.2 21.7 21.8 21.1 21.6Delaware 19.7 20.1 21.1 20.4 20.5Florida 20.0 20.5 21.0 20.5 20.6Georgia 19.3 20.0 20.3 20.0 20.0Hawaii 20.3 22.7 21.5 21.4 21.6Idaho 20.7 20.8 22.1 21.4 21.4Illinois 20.9 21.4 21.6 21.3 21.4Indiana 20.6 20.9 21.7 21.2 21.2Iowa 21.5 21.6 22.2 22.1 22.0Kansas 21.0 21.0 21.9 21.4 21.5Kentucky 19.9 19.3 20.6 20.2 20.1Louisiana 19.7 18.9 19.7 19.5 19.6Maine 21.4 21.8 22.8 21.8 22.1Maryland 20.2 20.9 21.3 20.8 20.9Massachusetts 21.7 22.0 22.5 21.4 22.0Michigan 20.6 21.1 21.5 21.5 21.3Minnesota 21.4 22.0 22.4 22.3 22.1Mississippi 18.8 17.9 18.9 18.6 18.7Missouri 21.3 20.9 22.0 21.5 21.6Montana 20.9 21.2 22.5 21.9 21.8Nebraska 21.3 21.4 21.9 21.7 21.7Nevada 20.7 21.3 22.1 21.4 21.5NewHampshire

21.7 22.0 23.0 21.7 22.2

II. Linear Functions 1.1 The Linear Parent Function: Transparency 2


Transparency 2: ACT Scores by State 1999 (cont.)

New Jersey 20.1 20.8 20.9 20.4 20.7New Mexico 19.5 19.5 20.7 20.3 20.1New York 20.8 22.2 22.2 22.2 22.0North Carolina 18.6 19.4 19.6 19.5 19.4North Dakota 20.6 21.2 21.7 21.6 21.4Ohio 20.8 21.1 21.9 21.4 21.4Oklahoma 20.4 19.8 21.0 20.5 20.6Oregon 21.8 22.1 23.5 22.4 22.6Pennsylvania 20.8 21.2 21.9 21.3 21.4Rhode Island 22.5 22.3 23.4 21.9 22.7South Carolina 18.6 19.0 19.3 19.2 19.1South Dakota 20.5 20.9 21.5 21.5 21.2Tennessee 19.8 19.1 20.4 19.8 19.9Texas 19.7 20.2 20.6 20.4 20.3Utah 21.0 20.7 22.0 21.3 21.4Vermont 21.3 21.5 22.7 21.8 21.9Virginia 20.2 20.4 21.0 20.4 20.6Washington 21.9 22.2 23.4 22.3 22.6Washington, DC 18.0 18.8 18.7 18.5 18.6West Virginia 20.1 19.0 20.9 20.3 20.2Wisconsin 21.6 22.2 22.4 22.4 22.3Wyoming 20.7 20.9 22.0 21.6 21.4

II. Linear Functions 1.1 The Linear P

TEXTEAMS Algebra I: 2000 and Beyond Spring 2001

Transparency 3: World Map

1 Austin, Texas 2 Detroit, Michigan 3

Siberia4 Santiago, Argentina 5 Nome, Alaska 6

Greenland7 Antarctica 8 Butler’s Gorge, Australia

5

1

2

4

7

3

6

II. Linear Functions 1.1 The Linear Parent Function: Activity 1


Activity 1: ACT Scores

Run the program ACTSCRS. The program stores the followingACT test score data for 1999 for each state in the US and theDistrict of Columbia into lists in your calculator: the meanscores for Mathematics, Reading, English, Sci Reasoning andthe mean composite score.

• Set up the scatter plots 1 – 4 listed below.• Graph the line y x= over each scatter plot.• Find windows that will help you compare the scores.• Make a general statement about each plot, comparing the two

scores.• Identify at least four coordinate pairs to justify your summary.

1. English versus Mathematics

2. Reading versus Mathematics

3. Sci Reasoning versus Mathematics

4. Composite versus Mathematics



Activity 2: Temperatures

-60 -40 -20 0 20 40 60

-20

-40

-60

20

40

60

Average Temperatures (C˚) in Selected Locations

Average July Temperatures (C˚)

1. Austin, Texas2. Detroit, Michigan3. Siberia4. Santiago, Argentina5. Nome, Alaska6. Greenland Icecap7. Antarctica8. Australia

Ave

rage

Dec

embe

r Tem

pera

ture

s (C

˚)



Activity 3: Symbolic

For the following points (x, y), choose a point or points thatmatch the description.1. x y<2. x y>3. x y=4. x y= −

AB

CD

EF

GHIJ

K

L

Describe where the point (x, y) appears in general when:5. x y<6. x y>7. y x=8. y x= −

II. Linear Functions 1.1 The Linear Parent Function: Reflect and Apply


Reflect and Apply

The scatter plot represents a walker’s distance from a motiondetector.

Time (sec)2 4 6 8

2

4

6

8

1. Judging by the graph, about how fast do you think the walkerwas moving?

2. Which one-second interval(s) show the walker moving fasterthan that rate?

3. Which one-second interval(s) show the walker movingslower than that rate?

II. Linear Functions 1.1 The Linear Parent Function: Student Activity 1


Student Activity 1: Age EstimatesOverview: Students develop intuitive notions about the line y x= .

Objective: Algebra I TEKSb.2.a The student identifies and sketches the general forms of linear ( y x= )and quadratic ( y x= 2 ) parent functions.b.2.c The student interprets situations in terms of given graphs or createssituations that fit given graphs.

Terms: trend line

Materials: colored pencils or pens, pieces of flat spaghetti or brightly colored string,graphing calculator

Procedures: Students should be seated at tables in groups of 3 – 4.

Activity: How Old? Read 12 names from the list below and ask students to guess the person’scurrent age, filling in the first and second columns of the table. It is importantthat you list a variety of ages, from old to young.Have students fill in the third column as you read the actual ages based on thebirth dates below.

Name BirthdateJulie Andrews 10-01-1935Bill Gates 10-28-1955Ronald Reagan 02-06-1911George W. Bush 07-06-1946Shania Twain 08-28-1965LeAnn Rimes 08-28-1982Sophia Loren 09-20-1934Bill Cosby 07-12-1937Britney Spears 12-02-1981McCaughey septuplets 11-19-1997Jennifer Love Hewitt 02-21-1979Jennifer Aniston 02-11-1969Charlton Heston 10-04-1924Leonardo DiCaprio 11-11-1974Harrison Ford 07-13-1942Tim Allen 06-12-1953Oprah Winfrey 01-29-1954Michelle Pfeiffer 04-29-1958Michael J. Fox 06-09-1961Jodie Foster 11-19-1962



Ben Affleck 08-15-1972Drew Barrymore 02-22-1975Frankie Muniz 12-05-1985Mary Kate and Ashley Olsen 06-13-1986Haley Joel Osment 04-10-1988Jonathan Lipnicki 10-22-1990

• How well did you guess?• How do you think we could judge if you are a good guesser?• How do you think we could tell if you are an “over” guesser?• How do you think we could tell if you are an “under” guesser?

Have students plot the data by hand on the grid and label the axes and a fewordered pairs.Tell students that we want to use a trend line to get a feel for how good theyestimate ages.Instruct students to use a piece of spaghetti or string to sketch a trend line fortheir data. (Use the same color for the trend line as the color of the datapoints.) Do the same on the transparency of the Activity.Instruct groups to compare their graphs within their group.Based on the graphs:

• Who is the better guesser?• Who is more of an “under” guesser? How can you tell?• Who is more of an “over” guesser? How can you tell?

Instruct students to use a different color and sketch a “perfect-guess” line, aline that represents perfect guessing.Ask students to write a sentence to describe the perfect-guess line and thentranslate to symbols. An example:

“All my guesses are the same as the actual ages”Guess = ActualG = A or y x=

Check that the line students sketched as their “good-guess” line is indeed theline y x= .2. Have students create a scatter plot (guess, actual).Sample graph:

3. Sample answer: The variable x stands for the guessed age so x min is 0years and x max is 100 years. The variable y stands for the actual age soymin is 0 years and y max is 100 years.



4.

5. If the student’s points lie mostly above the line, they are an under-guesser.If the student’s points lie mostly below the line, they are an over-guesser.

Over or UnderHave students complete the activity and then discuss as a group.1.

guess10 20 30 40 50

10

20

30

40

50

perfect guess

guess10 20 30 40 50

10

20

30

40

50

perfect guess

2. Underguess because all of the guesses are smaller when the actual ages arelarger.

3. Overguess because all of the guesses are larger when the actual ages aresmaller.

4.

guess guess10 20 30 40 50

10

20

30

40

50

5. Accept answers between 30 and 40 years old.6. Accept answers between 20 and 30 years old.



7.

guess

Sample Assessment:1. You guessed B’s age better.2. You overguessed C’s age.3. y x= .4. [0, 55], [0, 45]5. The variable x stands for my partner’s guesses of people’s ages, so [0, 40]

shows all of the guesses listed. The variable y stands for the actual ages ofthe people listed, so [0, 45] is a reasonable choice for those ages. Thischoice allows one to see the origin, which may be helpful in orienting thereader to whether the student is a good age estimator.

Note: In this activity students draw trend lines and a “perfect guess” line.The trend lines are approximations or estimates of their guessing, showing thegeneral trend of how they guessed. Refrain from calling trend lines “lines ofbest fit.” The “perfect guess” line is the line y x= .

Summary: The line y x= is the parent function for the linear function family. In thisactivity intuition for the line y x= is developed as an important starting pointfor work with linear functions.



Student Activity 1: How Old?

Name Guess (Age) Actual (Age)

1. Sketch a graph of the data.



2. Using your data, create a scatter plot on your graphingcalculator. Sketch it here.

3. Justify your viewing window choice.

4. Using your calculator, graph the “perfect-guess” line over thescatter plot.

5. Are you an “over-guesser” or an “under-guesser”? Explain.



Over or Under?

Graph #1 Graph #2

guess10 20 30 40 50

10

20

30

40

50

guess10 20 30 40 50

10

20

30

40

50

1. For each graph above, sketch in the “perfect-guess” line,actual = guess, and label it.

2. For graph #1, did the person over-guess or under-guess?Explain.

3. For graph #2, did the person over-guess or under-guess?Explain.

4. For each graph above, sketch a trend line for the data andlabel it “my trend”.

5. Use your trend line on graph #1. If I guessed an age of 24,what is the actual age of the person?(24, ______)

6. Use your trend line on graph #1. If the person really was 36years old, what did I probably guess? (______, 36)

7. On the back, create a scatter plot of a person who guessedreally well. Label the axes.



Sample Assessment

1. Consider A and B. Whose age did youguess better? Why?

2. Did you overestimate or underestimateC’s age?

3. How do you tell your graphing calculator to graph the bestguess line, guess = actual ?

Your partner had the following guesses.Name Guess Actual

Mr. Jackson 50 42Ms. Chi 42 37Mr. Beyer 45 40Ms. Harris 40 28

4. Which would be a good viewing window for a scatter plot ofthe above data?

Guess

A

B

C



Your partner had the following guesses.Name Guess Actual

Mr. King 37 42Ms. Alcini 30 37Mr. Golm 35 42Ms. Cline 25 30

5. What would be a good viewing window for a scatter plot?Explain each choice.

Xmin:

Xmax:

Ymin:

Ymax:



Student Activity 2: Sales GoalsOverview: Students use the context of sales goals to further refine their understanding of

the line y x= by contrasting points above and below the line.

Objective: Algebra I TEKSb.2.a The student identifies and sketches the general forms of linear ( y x= )and quadratic ( y x= 2 ) parent functions.b.2.c The student interprets situations in terms of given graphs or createssituations that fit given graphs.

Terms: linear parent function

Materials: markers, 1” grid paper

Procedures: Activity: Sales GoalsBegin by orienting students to the graphs using the following discussionquestions.

The management for a clothing store sets up weekly sales goals for theiremployees. The graphs show the results for a quarter of a year (12 weeks).Each data point represents a week’s (actual sales, sales goals) for anemployee.

• What is represented on the x-axis? [Actual sales.]• What is represented on the y-axis? [Sales goals.]• What is the meaning of an ordered pair in this situation? [An ordered

pair is (actual sales, sales goals).]• Why might some employees have higher goals set for them? [Some

examples may include: more experienced employees, employees withstrong sales in the previous quarter, employees that work at peak salestimes of the day or week.]

• Why do employees have such different levels of goals in the samequarter? [If these graphs represent the second quarter of the year, thegoals for the two weeks before Father’s Day would be higher than theweek following. The goals for the weeks previous to Easter would behigher than the weeks after Easter.]

Pick a specific point on a graph and discuss the meaning. For example,circle the lowest point on Amber’s graph.• What does this point mean for Amber? [Amber set a goal to sell about

$2600 for a particular week. She actually sold about $6800, exceedingher goal.]

• What are the meanings for the points under the line goals = actual?[The employees exceeded their goals.]

• What are the meanings for the points over the line goals = actual?[The employees failed to meet their goals.]



Have students answer the questions on the following page in their groupsand sketching their graphs for Exercises 6 – 8 on 1” grid paper. Havestudents view other groups’ graphs.

1. Enrique and Amber met and exceeded their goals.2. Seth failed to meet his weekly goals.3. Moesha most often met her weekly goals.4. Moesha and Enrique had the higher goals.5. Seth and Amber had the lower goals.

6. Sample answer: 10000

Actual Sales ($)

8000

6000

4000

2000

7. Sample answer: 10000

Actual Sales ($)

8000

6000

4000

2000

8. Sample answers: 10000

Actual Sales ($)

8000

6000

4000

2000

Wardrobe consultant with middlegoals who often matched the goals.

10000

Actual Sales ($)

8000

6000

4000

2000

Wardrobe consultant with middlegoals who often failed to meetthe goals.

Summary: Reading and interpreting scatter plots of points above and below the liney x= helps students further refine their understanding of the linear parentfunction.



Student Activity 2: Sales Goals

A local clothing store sets weekly sales goals for theiremployees. The graphs below show the quarterly results forfour employees, (actual sales, sales goals).

10000

Actual Sales ($)

Moesha

8000

6000

4000

2000

10000

Actual Sales ($)

Seth

8000

6000

4000

2000

10000

Actual Sales ($)

Enrique

8000

6000

4000

2000

10000

Actual Sales ($)

Amber

8000

6000

4000

2000



1. Which consultant(s) most often met and exceeded theirweekly goals?

2. Which consultant(s) most often failed to meet their weeklygoals?

3. Which wardrobe consultant(s) most often matched theirweekly goals?

4. Which consultant(s) had the higher goals?

5. Which consultant(s) had the lower goals?

6. Sketch a graph of a consultant who has low goals andconsistently matched them.

7. Sketch a graph of a consultant who has high goals and didnot meet them.

8. Name a scenario that is not represented by the four originalgraphs or in Exercise 6 and 7 above. Sketch a graph to matchthe scenario.

II. Linear Functions 1.2 The Y-Intercept: Leaders’ Notes


1.2 The Y-InterceptOverview: Participants use real life experiences to build the concepts of y-intercept as the

starting point and slope as a rate of change.

Objective: Algebra I TEKS(c.1.C) The student translates among and uses algebraic, tabular, graphical, orverbal descriptions of linear functions.(c.2.A) The student develops the concept of slope as rate of change anddetermines slopes from graphs, tables, and algebraic representations.(c.2.B) The student interprets the meaning of slope and intercepts insituations using data, symbolic representations, or graphs.

Terms: y-intercept, slope, rate of change, increasing, decreasing, recursion


Procedures: Participants should be seated at tables in groups of 3 – 4.Depending on the participants, briefly talk through or work through theStudent Activity, which connect recursion with graphing.

Activity 1: The Birthday GiftWork through Activity 1 with participants, modeling good pedagogy: askleading questions, use appropriate wait time, have teachers present their work,etc.Introduce the scenario.1. Guide participants in filling in the table, using language similar to the

following:At time zero, Susan started with $25.

Time (weeks) Process Amount Saved0 $25 $25

After the 1st week, Susan had the $25 she started with and $2.50.Time (weeks) Process Amount Saved

0 $25 $251 $ $ .25 2 50+ $27.50

After the 2nd week, Susan had the $25+$2.50 from week 1 and another $2.50.In other words, Susan had the $25 she started with and two $2.50’s.

Time (weeks) Process Amount Saved0 $25 $251 $ $ .25 2 50+ $27.502 $ $ . $ . $ $ .25 2 50 2 50 25 2 2 50+ + = + ( ) $30.00



Note that the above step is not a natural step for many students. They aremore apt to operate recursively on the previous term , adding $2.50 to $27.50.Teachers need to be aware that this is difficult for some students.

Time (weeks) Process Amount Saved0 $25 $251 $ $ .25 2 50+ $27.502 $ $ . $ . $ $ .25 2 50 2 50 25 2 2 50+ + = + ( ) $30.003 25 2 5 2 5 2 5 25 3 2 50+ + + = + ( ). . . . $32.504 25 2 5 2 5 2 5 2 5 25 4 2 50+ + + + = + ( ). . . . . $35.00

2. Write the sentence in words and then abbreviate to variables:After t weeks, Susan will have the $25 she started with and t ($2.50’s).Because of convention, mathematicians write 2.5t , instead of t (2.5).

Time (weeks) Process Amount Savedt 25 2 5 25 2 5+ = +t t( . ) . 25 2 5+ . t

3. Use questions to lead participants to find a suitable viewing window.• What does x represent in this problem? [Elapsed time in weeks]• What values make sense for x in this problem? [Answers will vary.

Sample answer. Zero weeks to 10 weeks.]• What does y represent in this problem? [Total money saved]• What values make sense for y in this problem? [Answers will vary.

Sample answer. No money to $60.]

4. Sample answer. The variable x stands for elapsed time in weeks so zero to10 weeks shows a reasonable amount of time. The variable y stands fortotal money saved, so $0.00 to $60.00 will show all the savings and the x-axis.

5. 25 2 5 7 42 50+ ( ) =. . . Susan will have $42.50 after 7 weeks.

6. 25 2 5 139 99+ =. .t . After 46 weeks, Susan will have more than $139.99,enough to buy the ring.You may have to open up your window. We did as follows:



As an extension, note that the question really asks for an inequality:25 2 5 139 99+ ≥. .t .

7. Susan’s starting value is lower, so the line will “start” on the y-axis at 15instead of 25. The y-intercept changed. The slope, or amount of moneyshe saved every week, did not change. The two lines are parallel, with thenew line translated down from the original.

8. Susan’s starting value is higher, so the line will “start” on the y-axis at 40instead of 25. The y-intercept changed. The slope, or amount of moneyshe saved every week, did not change. The two lines are parallel, with thenew line translated up from the original.

9. Susan’s rate of saving has changed so the amount of money will not growas fast, so the line will be less steep. Susan’s rate of saving has changed.Her starting point, or the y-intercept, did not change. The new line is notparallel to the original line because the rate of saving has changed.

10. When the rate of saving changes, the slope of the line changes.11. When the starting value in Susan’s saving’s plan changed, the starting

point, or y-intercept of the line, changed.12. The point (0, y) is where a line intersects the y-axis. This point represents

the starting value of Susan’s savings plan.

Activity 2: Spending Money

Have participants work through Activity 2 in their groups. Encourage them topractice the language they plan to use when teaching their students.1.

Time (weeks) Process Amount ofMoney

0 $1090 $10901 $ $1090 30− $10602 $ $ $ $ $1090 30 30 1090 2 30− − = − ( ) $10303 1090 30 30 30 1090 3 30− − − = − ( ) $10004 1090 30 30 30 30 1090 4 30− − − − = − ( ) $35.00

2. Write the sentence in words and then abbreviate to variables:After t weeks, Manuel will have the $1090 he started with minus t ($30’s).Because of convention, mathematicians write 30 , instead of t (30).

Time (weeks) Process Amount of Moneyt 1090 30 1090 30− = −t t( ) 1090 30− t



3. Use questions to lead participants to find a suitable viewing window.• What does x represent in this problem? [Elapsed time in weeks]• What values make sense for x in this problem? [Answers will vary.

Sample answer. Zero weeks to 10 weeks.]• What does y represent in this problem? [Total amount of money]• What values make sense for y in this problem? [Answers will vary.

Sample answer. $800 to $1090.]

4. Sample answer. The variable x stands for elapsed time in weeks so zero to10 weeks shows a reasonable amount of time. The variable y stands fortotal amount of money, so $800 to $1090 will show all his money.

5. 1090 30 11 760− ( ) = . Manuel will have $760 after 11 weeks.

6. 1090 30 0− =t . After 36 weeks, Manuel will only have $10. He willcannot spend the whole $30 the next week, only $10 and then he will beout of money.You may have to open up your window. An example:

7. Manuel’s starting value is higher, so the line will “start” on the y-axis at1300 instead of 1090. The y-intercept changed. The slope, or amount ofmoney he spent every week, did not change. The two lines are parallel,with the new line translated up from the original.

8. Manuel’s starting value is lower, so the line will “start” on the y-axis at890 instead of 1090. The y-intercept changed. The slope, or amount ofmoney he spent every week, did not change. The two lines are parallel,with the new line translated down from the original.



9. Manuel’s rate of spending has changed so now the amount of money willnot deplete as fast, so the line will be less steep. Manuel’s rate of savinghas changed. His starting point, or the y-intercept, did not change. Thenew line is not parallel to the original line because the rate of saving haschanged.

Activity 3: Money, Money, MoneyWork through Activity 3 with participants.

1. Use questions to find a suitable viewing window.• What does x represent in this problem? [Elapsed time in weeks]• What values make sense for x in this problem? [Answers will vary.

Sample answer. Zero weeks to 38 weeks.]• What does y represent in this problem? [Total amount of money]• What values make sense for y in this problem? [Answers will vary.

Sample answer. No money to $1100.]

Sample answer. The variable x stands for elapsed time in weeks so zero to38 weeks shows the time it takes Manuel to spend all of his money. Thevariable y stands for total amount of money, so $0.00 to $1100 will showboth graphs.

2. 25 2 5 7 42 50+ ( ) =. . . Susan will have $42.50 after 7 weeks.

3. 25 2 5 1090 30+ = −. t t . They never do have the same amount of moneybecause they are saving or spending each week, not in the middle of theweek. This is shown in the table as we choose the increment to be a weeknot a part of a week. After week 33, Susan has $107.50 and Manuel has$100, which is the closest they get to each other.



4. From earlier work, we found that Manuel had only $10 to spend after 36weeks. So we will say that after 36 weeks, Manuel is out of money. Sothe question is now, how much money does Susan have after 36 weeks?Susan has $115 after 36 weeks.To find this answer, we solved 1090 30 0− =t and used the solution tosolve 25 2 5 36 115+ ( ) =. .

Answers to Reflect and Apply:1. a. Yen started with $20.

b. Lira started with $0.00.c. Lira is saving $30 a month.d. Mark is saving $10 a month.

2. a. Frank started with $80.b. Ruble started with $40.c. Peso is spending $30 a month.d. Ruble is spending $10 a month.

3. ii, b4. iii, c5. iv, a6. i, d

Use the following questions to summarize and connect activities:• What changes in the situation resulted in a change in the steepness of the

line? [Changing the rate of spending per week, the amount of moneyspent per week. Encourage participants to use the word “rate”.]

• What changes in the situation resulted in a change in the starting point ofthe line? [Changing the starting amount of money, initial amount ofmoney.]

• Look at your function rules. What does the constant represent in thisproblem? [The initial, or starting, amount of money]

• Look at your function rules. What does the coefficient of x represent inthis problem? [Encourage the words “rate of spending”]

• Look at your function rules. If the coefficient of x is negative, what doesthis represent in this problem? [Spending]



• Look at your function rules. If the coefficient of x is positive, what doesthis represent in this problem? [Saving]

Summary: Using real life situations, participants investigate the effects of changing thestarting point and the rate of change of a line.

II. Linear Functions 1.2 The Y-Intercept: Activity 1


Activity 1: The Birthday Gift

Susan’s grandmother gave her $25 for her birthday. Instead ofspending the money, she decided to start a savings program bydepositing the $25 in the bank. Eachweek, Susan plans to save anadditional $2.50.

1. Make a table of values for the situation.

Time (Weeks) Process Amount Saved$25

2. Write a function rule for the amount of money Susan willhave after t weeks.



3. Find a viewing window for the problem situation.

Sketch your graph: Note your window:

Xmin:Xmax:Xscl:Ymin:Ymax:Yscl:

4. Justify your window choices.

Use your graph and table to find the following:

5. How much money will Susan have after 7 weeks? Write thisequation. Show how you found your solution.

6. Susan wants to buy a school ring. When will she haveenough money to buy the $139.99 ring? Write this equation.Show how you found your solution.



7. How will the line change if Susan deposits only $15 of the$25? Graph the line. What changed? What did notchange?

8. How will the line change if Susan deposits the $25 from hergrandmother plus another $15 she already had? Graph theline. What changed? What did not change?

9. How will the line change if Susan deposits the $25 from hergrandmother, but decides she can only save $2.00 a week?Graph the line. What changed? What did not change?

10. What changes in the situation resulted in a change in thesteepness of the line?

11. What changes in the situation resulted in a change in thestarting point of the line?

12. Write the coordinates of the point where a line intersects they-axis. This point is called the y-intercept. What do thesecoordinates represent in this problem?



Activity 2: Spending Money

Manuel worked all summer and saved $1090.He plans to spend $30 a week.

1. Make a table of values for the situation.

Time (Weeks) Process Amount of Money$1090

2. Write a function rule for the amount of money Manuel willhave after t weeks.








5. How much money will Manuel have after 11 weeks? Writethis equation. Show how you found your solution.

6. When will Manuel be out of money? Write this equation.Show how you found your solution.



7. How will the line change if Manuel had initially earned$1300? Graph the line. What changed? What did notchange?

8. How will the line change if Manuel spent $200 on schoolclothes and started the year with only $890? Graph the line.What changed? What did not change?

9. How will the line change if Manuel starts with the $1090, butdecides he will only spend $25 a week? Graph the line.What changed? What did not change?



Activity 3: Money, Money, Money

1. Manuel has $1090 and he will spend $30 aweek. Susan has $25 and will save $2.50a week. Find a viewing window thatincludes both situations.





3. When will Manuel and Susan have the same amount ofmoney? Write this equation. Show how you found yoursolution.

4. How much money will Susan have when Manuel is out ofmoney? Show how you found your solution.

II. Linear Functions 1.2 The Y-Intercept: Reflect and Apply


Reflect and Apply

1. The graphs represent the savings of three students.

Months1 2 3 4

Lira

Months1 2 3 4

Mark

1 2 3 4Months

Yen

a. Which student started with the most money? Explain.b. Which student started with the least money? Explain.c. Which student is saving the fastest? Explain.d. Which student is saving the slowest? Explain.

2. The graphs represent the spending habits of three students.

Months1 2 3 4

Ruble Frank

Months1 2 3 4

Months1 2 3 4

Peso

a. Which student started with the most money? Explain.b. Which student started with the least money? Explain.c. Which student is spending the fastest? Explain.d. Which student is spending the slowest? Explain.

II. Linear Functions 1.2 The Y-Intercept: Reflect and Apply


Match the recursive routine with an equation and with a graph:

3. _____1000 ENTERAns – 20, ENTER,ENTER, . . .

i. y x= −35 3

4. _____35 ENTER,Ans + 3, ENTER,ENTER, . . .

ii. y x= −1000 20

a

b

5. _____1000 ENTERAns + 20, ENTER,ENTER, . . .

iii. y x= +35 3

6. _____35 ENTER,Ans – 3, ENTER,ENTER, . . .

iv. y x= +1000 20 c

d

II. Linear Functions 1.2 The Y-Intercept: Student Activity


Student Activity: Show Me the Money!Overview: Students connect recursive operations with graphs.

Objective: Algebra I TEKS



6.Time

(weeks)MoneySaved

0 $10901 $10602 $10303 $10004 $9705 $9406 $9107 $880

$880

$940

$ 1,000

$ 1,060

$ 1,120

0 1 2 3 4 5 6 7

Time (weeks)

M

o

n

e

y

7. Subtraction, which can also be thought of as repeated addition of a negativenumber.

8. linear, decreasing. Emphasize that repeated subtraction is the same as repeatedaddition of a negative number.

Assessment Answers:1. c2. f3. d4. b5. g6. e7. a8. h

Summary: Using a recursive routine, students generate points on a graph and makegeneralizations. Repeated addition results in a linear graph. Repeatedaddition of a positive number is an increasing line. Repeated addition of anegative number is an decreasing line.



Student Activity: Show Me the Money!

Susan’s grandmother gave her $25 for her birthday. Instead ofspending the money, she decided to start a savings program bydepositing the $25 in the bank. Each week, Susan plans to savean additional $2.50.

1. Write a recursive routine to model Susan’s savings plan.

2. Fill in the table and sketch a graph to model Susan’s savingsplan:

Time(weeks)

Money

3. What operation did you repeat in your recursive routine?

4. How does repeated addition “look” in a graphicalrepresentation?



Manuel worked all summer and saved $1090. He plans to spend$30 a week.

5. Write a recursive routine to model Manuel’s spending plan.

6. Fill in the table and sketch a graph to model Manuel’sspending plan:

Time(weeks)

Money

7. What operation did you repeat in your recursive routine?

8. How does repeated subtraction “look” in a graphicalrepresentation?



Assessment

Match the recursive routines with the graphs:

1. ___ 300 ENTERAns – 20, ENTER, ENTER, . . .

2. ___ 90 ENTERAns + 20, ENTER, ENTER, . . .






8. ___ 90 ENTERAns - 50, ENTER, ENTER, . . .

a

b

c

d

ef

g

h

II. Linear Functions 1.3 Exploring Rates of Change: Leaders’ Notes


1.3 Exploring Rates of ChangeOverview: Participants use real data from a motion detector to model motion at a

constant rate over time. Participants translate among algebraic, tabular,graphical, and verbal descriptions of linear functions.

Objective: Algebra I TEKS(b.1.B) The student gathers and records data, or uses data sets, to determinefunctional (systematic) relationships between quantities.(c.2.A) The student develops the concept of slope as rate of change anddetermines slopes from graphs, tables, and algebraic representations.(c.2.B) The student interprets the meaning of slope and intercepts insituations using data, symbolic representations, or graphs.(c.2.D) The student graphs and writes equations of lines given characteristicssuch as two points, a point and a slope, or a slope and y-intercept.

Terms: rate of change, constant rate

Materials: motion detector connected to an overhead calculator, a motion detector withgraphing calculator for each group of 3-4, data collection devices, graphingcalculators

Procedures: The room should be set up with an aisle down the middle. Set up a motiondetector pointing down the aisle, connected to an overhead calculator, so thegroup can see both the participants walking down the aisle and the dataprojected from the calculator on a screen in front of the room.

Work through the Student Activity, Rates of Change, with participants. Talkthrough the assessment. Make sure each participant gets a chance to write anequation for their own motion over time.

Activity 1: Wandering AroundWork through Exercise 1 with participants. This activity takes participantsfrom a verbal description of a situation to a graph, table, and rule representingthe situation.

1. You know that Ryan was at 9 feet at 3 seconds. Label the table and fill inthe (3, 9) as shown.

TableTime Distance

3 9



Since you know that Ryan was walking at 2 feet per second, count back tofind his starting point as shown below.

Table GraphTime Distance

0 31 52 73 9

222

A sentence to describe Ryan’s walk: Start at 3 feet and walk away fromthe motion detector at 2 feet per second.Rule: y x= +3 2

Have participants do Exercise 2 and then discuss their results.

2. You know that Madeline was at 9 feet at 2 seconds. Label the table andfill in (2, 9) as shown.

TableTime Distance

2 9

Since you know that Madeline was walking at 3 feet per second, countback to find her starting point as shown below.


0 151 122 9

-3 -3

A sentence to describe Ryan’s walk: Start at 15 feet and walk toward themotion detector (decrease the distance) at 3 feet per second.Rule: y x= −15 3



Do Exercises 3 – 4 with participants.3. You know that Robyn started at 1 foot and she was at 9 feet at 2 seconds.

Label the table and fill in (0, 1) and (2, 9) as shown.Table

Time Distance0 1

2 9

2 8

From the table, you can tell that Robyn covered 8 feet in 2 seconds.• If Robyn walked 8 feet in 2 seconds, how far did she walk in one

second? [4 feet]• So how fast was she walking per second? [4 feet per second]


0 11 52 9

44

A sentence to describe Robyn’s walk: Start at 1 foot and walk away at 4feet per second.Rule: y x= +1 4

4. You know that Chet was at 6 feet at 1 second and he was at 1 foot at 2seconds. Label the table and fill in (1, 6) and (2, 1) as shown.

TableTime Distance

1 62 11 -5

From the table, you can tell that Chet covered 5 feet in 1 second, thus hisrate was 5 feet per second.


0 111 62 1

11

-5-5



A sentence to describe Chet’s walk: Start at 11 feet and walk toward themotion detector at 5 feet per second.Rule: y x= −11 5

Activity 2: Describe the WalkThis activity takes participants from a table representing a situation to a verbaldescription, graph, and rule of the situation.

Have participants work through Exercises 5 – 8 in their groups and thendiscuss their results.

1.Table Graph

Time Distance0 151 212 27

11

66

Rule: y x= +15 6Sample verbal description: Start 15 feet in front of the motion detector andwalk away at 6 feet per second.

2.Table Graph

Time Distance0 241 232 223 214 20

11111 5 19

11111

Rule: y x= −24Sample verbal description: Start 24 feet in front of the motion detector andwalk toward the motion detector (distance decreases) at 1 foot per second.

3.Table Graph

Time Distance0 13 76 13

33

66

Rule: y x= +1 2



Sample verbal description: Start 1 foot in front of the motion detector andwalk away at 2 feet per second.

4.Table Graph

Time Distance0 -11 22 56 17

1144 10 29

33

1212

Rule: y x= − +1 3Sample verbal description: Start 1 foot behind the motion detector and walkpast it. Continue to walk away at 3 feet per second. (Think outside, or thistime, behind the box!)

Answers to Reflect and ApplyThe scatter plot is a finite set of points. It is based on the data collected by themotion detector and is stored in the lists on the calculator.

The function rule is an infinite set of points.

One way to tell the difference is to trace. When tracing in the scatter plot, thecalculator will show each data point from the lists. When tracing on thefunction rule, the calculator can evaluate the function at any x-value in thedomain determined by the window settings (and limited by the calculator’srounding off.) Often there is confusion when a participant traces on thescatter plot and wants to know why the calculator will not let them trace to aparticular value.

Summary: By using real data generated from their own motion to determine linearmodels, participants further develop the concepts of the y-intercept as astarting point and slope as a rate of change. They gain facility in translatingamong representations: algebraic, tabular, graphical, and verbal descriptionsof linear functions.

II. Linear Functions 1.3 Exploring Rates of Change: Activity 1


Activity 1: Wandering Around

Label the table and graph. Fill in the table, sketch the graph, andwrite a symbolic rule for the situation.

1. Ryan was walking away from the motion detector at 2 feetper second. You missed where he started but you know thathe was at the 9 foot mark when the timer called out the 3rd

second.

Table Graph

Rule:

2. Madeline was walking toward the motion detector at 3 feetper second. You missed where she started, but you know thatshe was at the 9 foot mark at the 2nd second.

Table Graph

Rule:



3. Robyn started 1 foot from the motion detector. You lookedup and she was at 9 feet at the 2nd second.

Table Graph

Rule:

4. You looked up and Chet was walking! He was at the 6 footmark at the 1st second and the 1 foot mark at the 2nd second.

Table Graph

Rule:



Activity 2: Describe the Walk

Label the table and graph. Sketch the graph. Write a symbolicrule and a description of the walk.1.


0 15

1 21

2 27

Verbal Description:

Rule:

2.Table Graph

Time Distance

3 21

4 20

5 19

Verbal Description:

Rule:



3.Table Graph

Time Distance

0 1

3 7

6 13

Verbal Description:

Rule:

4.Table Graph

Time Distance

2 5

6 17

10 29

Verbal Description:

Rule:

II. Linear Functions 1.3 Exploring Rates of Change: Reflect and Apply


Reflect and Apply

What is the difference between the scatter plot created by themotion detector and the graph of the function rule created by thefunction grapher?

Scatter plot created by the motion detector:

Function rule:

II. Linear Functions 1.3 Exploring Rates of Change: Student Activity


Student Activity: What’s My Trend?Overview: Students investigate the linear model with motion data. Students also use

numeric techniques to write the equation of a line.

Objective: Algebra I TEKSb.1.B The student gathers and records data, or uses data sets, to determinefunctional (systematic) relationships between quantities.c.2.A The student develops the concept of slope as rate of change anddetermines slopes from graphs, tables, and algebraic representations.c.2.B The student interprets the meaning of slope and intercepts in situationsusing data, symbolic representations, or graphs.

Terms: Linear model, rate of change

Materials: motion detectors, overhead calculator, data collection devices, graphingcalculators

Procedures: The classroom should be set up with an aisle down the middle. Set up amotion detector pointing down the aisle, connected to a viewscreen calculator,so the class can see both the students walking down the aisle and the dataprojected from the calculator on a screen if front of the room.

Activity: What’s My Trend?1. Relate the following situation to your class: Suppose you start 2 feet from

a chair and walk away at 1.5 feet per second. Complete the table to showwhere you are at each second.

0 21 3.52 5

10 ?t ?

• How can you find the distance from the chair at 10 seconds? [Multiplyby 1.5 and add 2]

• Write a sentence to describe how you can find the distance if you knowthe time. [Distance is 2 plus 1.5 times the time.]

• Translate the sentence to an equation. [Distance is 2 + 1.5 * time.d t= +2 1 5. ]

0 21 3.52 510 17t 2 + 1.5 t



Make a scatter plot of the data.

Show that your equation contains the points by graphing it.

2. Run the motion detector Ranger program. Use the following screens to setup the experiment and then follow the instructions on the screen.

Ask a student to walk in front of the motion detector as follows. Start about2 feet from the motion detector and then walk away from it.

When you have a satisfactory graph, press ON, quit, and graph. You shouldsee the graph again.

Trace to the points where time is 0 seconds, 1 second, and 2 seconds. Havestudents fill in the table.

x y0 1.41 2.92 4.4t



Ask:• How can you use the points to find how fast you were going? [Take the

difference over 1 second, about 1.5 feet per second.]• Where did you start? [about 1.4 feet]• How can you use your rate and where you started to figure out where you

will be in 10 seconds? [1.4 plus 10 times 1.5]• Write a sentence to describe how you can find the distance if you know the

time. [Distance is 1.4 plus 1.5 times the time.]• Translate the sentence in words to a sentence in symbols.

[Distance = 1.4 + 1.5 * time. d t= +1 4 1 5. . ]

Type the equation into the y= menu and graph.

Use the table to check your prediction for where you will be in 10 seconds.

Now repeat the above procedure for the rest of the walks. Find an equation tofit the data and check your prediction with a table.

Examples of different walks follow:Ask a student to start 11 feet away and walk toward the motion detector.

(Where were you 2 seconds before?)

Ask a student to stand approximately 4 feet from the motion detector and standstill for the whole 4 seconds.



Assessment: What’s My Trend?

1. b 6. y x= +1 5 2.2. a 7. y x= +0 5 1.3. c 8. y x= +4 0 2.

9. y x= −5 210. y x= 0 4.11. y x= +24.

5.

12. You rode your bike to the parkat a rate of one mile per minutefor 3 minutes. Then you talkedto your friend at the park for 2minutes. You both walked backto your house at a rate of 0.6miles per minute for 5 minutes.

Note: This activity was based on the activity Rates of Change in theTEXTEAMS Mathematical Modeling Institute for Secondary Teachers.

Summary: Using motion detectors to gather data for their own motion, students developthe concept of slope as a rate of change as students write functions to modelthe collected data.



Student Activity: What’s My Trend?

1. Start 2 feet away fromthe motion detector andwalk away at 1.5 ft/sec.

2. Walk slowly awayfrom the motion detectorat a constant rate.

t d t d0 01 12 210 tt

3. Walk quickly awayfrom the motion detector ata constant rate.

4. Walk slowly towardthe motion detector atconstant rate.

t d t d0 01 12 2t t

5. Walk quickly towardthe motion detector at aconstant rate.

6. Stand still about 6 feetin front of the motiondetector.

t d t d0 01 12 2t t



Assessment: What’s My Trend?

Match the following equations with the graphs they represent.

___ 1. y x= −5 13 ___ 2. y x= +1 0 5. ___ 3. y x= +2 1

a. b. c.

Draw a graph of each of the following:4. Start 1 foot from the motion detector and stand still for 3 seconds.

Then walk away from the motion detector at a rate of 2 feet persecond for 2 seconds. Then walk toward it at a rate of 0.2 feet persecond for 5 seconds.

5. Start 5 feet from the motion detector and walk toward it at a rate of 2feet per second for 1 second. Then walk toward the motion detectorat a rate of 0.3 feet per second for 3 seconds. Now walk away fromthe motion detector at a rate of 0.5 feet per second for 6 seconds.



Write an equation for each of the following:6. Start 2 feet from the motion detector and walk away at a rate of 1.5

feet per second.

7. Start 0.5 feet from the motion detector and walk away at a rate of 1foot per second.

Write an equation for each of the following:

8. _______________ 9. _______________

10. _______________ 11. _______________

12. Write a story for the following graph using units of hours for timeand miles for distance.

II. Linear Functions 1.4 Finite Differences: Leaders’ Notes


1.4 Finite DifferencesOverview: Participants use their cumulative concrete experiences with the linear model to

build to the abstract symbolic representations of slope. Finite differences areused to find linear models and to discover what makes data linear.

Objective: Algebra I TEKS(c.2.A) The student develops the concept of slope as rate of change anddetermines slopes from graphs, tables, and algebraic representations.(c.2.B) The student interprets the meaning of slope and intercepts insituations using data, symbolic representations, or graphs.(c.2.D) The student graphs and writes equations of lines given characteristicssuch as two points, a point and a slope, or a slope and y-intercept.

Terms: rate of change, constant rate, constant differences, finite differences


Procedures: Participants should be seated at tables in groups of 3 – 4.

Activity 1: Rent Me!Here we introduce rental problems, another good real world linear situation.Have participants work through the activity and share their results.

TableTime Cost

123 12.25456 14.50

3

7

2.252 25

30 75. .=

Time Cost0 10.001 10.752 11.503 12.254 13.005 13.756 14.507 15.25

Function rule:y x= +10 00 0 75. .



Activity 2: Guess My Function!Play Guess My Function using an overhead graphing calculator as shown:Put the Beach Rental equation y x= +10 0 75. in the graphing calculatorwithout showing the participants, and using a table, ask for y when x = 2and x = 4. Give them a minute to figure out the equation.

• How can you find the equation? [Take differences and work backwards.]

Then enter y x= +14 1 5. and give them 3 and 7 for x.

Discuss their methods for finding the rate and the starting point.

Have participants play Guess My Function with a partner using the tables inActivity 2.

Answers:1. y x= +11 52. y x= +1 3 1 2. .3. y x= −4 24. y x= −25 75. y x= +16 46. y x= − +4 9 3.7. y x= −248. y x= −41 59. y x= +11 610. y x= −1511. y x= +4 2

12. y x= −25 12

13. y x= +6 214. y x= −8 05 25. .15. y x= −2

16. y x= +5 12

Transparency: How Did You Do It?



Now that participants have many concrete experiences finding rates of change(slopes), we formalize the slope of a line as an algebraic rule.

You may use either Transparency 1 or 2.Use Transparency 1 to discuss the process of finding the rate of change bothnumerically and generally.Use Transparency 2 to develop the ideas live, using Transparency 1 as aguide.

• What do the values in the table have to do with the points on thegraph? [Use the graph to show that the change in y , y y2 1− , is thevertical distance between the points. Also show that the change in x ,x x2 1− , is the horizontal distance between the points.]

• Where do you find ∆∆

yx

in your rule? [The rate of change is ∆∆

yx

.]

• How can the development of rate of change and slope throughconcrete experiences enhance student understanding?

Extension: In these activities, an algebraic rule for finding the slope betweentwo points on a line was developed. However, up to this point we havealways found the y-intercept (starting point) by counting back in the tableusing the rate of change. Later in the institute, we formalize the point-slopeform of a line, but at this point there is another way to find the equation of aline when you have found the slope between the two points. See the examplebelow.Participants are now comfortable with the form of a line, y b mx= + ,y y intercept rate x= − + ( ). We use that to our advantage.We will use the information given in Activity 1: Rent Me!Find the rate of change between (3, 12.25) and (6, 14.50) which is $0.75 perhour. Substitute this into the equation of the line, y b mx= + :y b x= + 0 75. .Now choose one of the given points, (3, 12.25) and substitute into the above:12 25 0 75 3. .= + ( )b and solve for b, b = 10 .So now we have y x= +10 0 75. .

Activity 3: Finite DifferencesWork through the Activity with participants. Point out that in all the linearactivities up to this point in the institute, we have assumed a constant rate ofchange. Here we want to emphasize that in order for a function to be linear,the first level of differences must be constant and conversely, if the firstdifferences are constant, the function is linear.

∆x Term Number Process Column Value of Term ∆y



0 b1 b a+ b a+12 b a a+ + b a+ 23 b a a a+ + + b a+ 34 b a a a a+ + + + b a+ 4

1111

n b an+

aaaa

1. If first differences are constant, then the data is linear.

2. If the data is linear, then first difference are constant.

Sample Answers for Reflect and Apply:

1.x y0 111 162 21

Function rule: y x= +11 5

Real world scenario: Rent a bicycle built for two at a base charge of $11.00and $5.00 per hour.Another possible real world scenario: Start with a base of 11 block. Add 5blocks to each figure.

Fig 1:

Fig 2:

Fig 3:

Summary: Building on participants’ previous experiences in the institute with rate ofchange, we formalize the concept of slope and finite differences.

II. Linear Functions 1.4 Finite Differences: Transparency 1


Transparency 1: How Did You Do It?The Slope of a Line

With a specific situation:Table Graph

Time Distance2 44 104 2− 10 4− } 10-4

4-2

(2, 4)

(4, 10)

Rule: y starting point x= + −−

10 44 2

In general:Table Graph

x yx1 y1x2 y2

x x2 1− y y2 1−

y2-y1

x2-x1

(x1, y1)

(x2, y2)

Rule: y starting point y yx x

x= + −−

2 1

2 1

, y starting point rate x= + ( )

rate of change = change in change in

= yx

yx

y yx x

= −−

∆∆

2 1

2 1

II. Linear Functions 1.4 Finite Differences: Transparency 2


Transparency 2: How Did You Do It?The Slope of a Line

With a specific situation:Table Graph

Time Distance2 44 10

Rule:

In general:Table Graph

x y

Rule:

rate of change =

II. Linear Functions 1.4 Finite Differences: Activity 1


Activity 1: Rent Me!

1. At the beach in Galveston, you decide to rent an umbrellafrom a beach side vendor. Upon asking for the cost to rentthe umbrella for 8 hours, the vendor pulls out a worn and wettable of prices.

“Oh, no. What will I do?” exclaimed the vendor!Help the vendor reproduce the price list. What linearfunction rule can you use?

Time Cost123 $12.25456 $14.5078

Rule:



Activity 2: Guess My Function!

Write the symbolic rule below each table:1. x y 2. x y

0 11 0 1.31 16 1 2.52 21 2 3.7

3. x y 4. x y0 4 0 251 2 1 182 0 2 11

5. x y 6. x y1 20 5 10.12 24 6 13.13 28 7 16.1

7. x y 8. x y3 21 10 -94 20 11 -145 19 12 -19



9. x y 10. x y0 11 0 152 23 5 104 35 10 5

11. x y 12. x y0 4 0 253 10 10 206 16 20 15

13. x y 14. x y2 10 11 5.35 16 15 4.38 22 19 3.3

15. x y 16. x y-7 14 -10 0-5 10 -4 3-3 6 2 6



Activity 3: Finite Differences

When only one level of differences is necessary to obtain aconstant value, the algebraic rule which generates the terms ofthe sequence is linear and can be written in the form b an+ . Theterms of a linear sequence are in the form b, b a+1 , b a+ 2 ,b a+ 3 , b a+ 4 , . . . , an b+ .

∆x Term # Process Column Value of Term ∆y

0 _____________ b___ ___

1 _____________ b a+1___ ___

2 _____________ b a+ 2___ ___

3 _____________ b a+ 3___ ___

4 _____________ b a+ 4

n a n b( ) + an b+

1. If first differences are constant, then ___________________

2. If the data is linear, then ____________________________

II. Linear Functions 1.4 Finite Differences: Reflect and Apply


Reflect and Apply

Refer to the tables in Activity 2. Choose 4 tables from theactivity. List them below. Graph each scatter plot and thefunction rule. Make up a real world scenario for each. Includeone real world scenario that can be physically built, such as withcentimeter cubes, etc.

II. Linear Functions 1.4 Finite Differences: Student Activity


Student Activity: Graphs and TablesOverview: Students use graphing calculator programs to find the equations of lines given

graphs or tables.

Objective: Algebra I TEKS(b.3.B) Given situations, the student looks for patterns and representsgeneralizations algebraically.(c.1) The student understands that linear functions can be represented indifferent ways and translates among their various representations.

Terms: rate of change, y-intercept

Materials: graphing calculators, LINEGRPH program, LINETBL program

Procedures: Link the two programs to students, LINEGRPH and LINETBL. Brieflydemonstrate how to run each. An example of each is shown below.

The program LINEGRPH:When you run the program, it graphs a randomly generated line in the window[-4.7, 4.7] [-3.1, 3.1].

After examining the graph, enter the equation of the line in Y2 and changethe graph style of Y2 to the point tracer. This allows you to see graphmore easily. Below is an example of a correct equation because thegraphs are the same.

If you had entered an incorrect equation, it would graph accordingly andyou can quickly see that it is incorrect, as shown below.

II. Linear Functions 1.4 Finite Differences: Student Activity


The program LINETBL:When you run the program, it displays a randomly generated table of a line.

After examining the table, enter the equation of the line in Y2 . Below is anexample of a correct equation because the tables return the same values.

If you had entered an incorrect equation, it would display a table where thevalues generated are not the same and you can quickly see that it isincorrect, as shown below.

Summary: Using the power of technology to examine many examples quickly and getinstant feedback, students gain facility in finding the equations of lines givengraphs or tables.

II. Linear Functions 2.1 Out for a Stretch: Leaders’ Notes


2.1 Out for a StretchOverview: Participants investigate the relationship between the “stretch” of a rubber band

attached to a container and the number of marbles in the container.

Objective: Algebra I TEKS(b.1.B) The student gathers and records data, or uses data sets, to determinefunctional (systematic) relationships between quantities.(b.1.E) The student interprets and makes inferences from functionalrelationships.(c.1.A) The student determines whether or not given situations can berepresented by linear functions.(c.1.C) The student translates among and uses algebraic, tabular, graphical, orverbal descriptions of linear functions.(c.2.B) The student interprets the meaning of slope and intercepts insituations using data, symbolic representations, or graphs.

Terms: trend line, linear model, rate of change, slope, y-intercept

Materials: for each group of 3 –4 participants: one 8 oz. Styrofoam or paper cup, 3” longthin rubber bands, one 8 oz. cup of marbles of the same size, large paper clips,tape, meter sticks, graphing calculators

Procedures: Participants should be in groups of 3 – 4.

Complete the teacher activity with all of the participants, modeling goodpedagogical practices for data collection activities. After completing theteacher activity as a whole group, divide participants into 4 groups, each ofwhich will then do one of the student activities in small groups of 3 – 4.When they have completed their respective student activities, have one smallgroup for each student activity present their activity to the large group. Theyshould include a demonstration of the experiment, a scatter plot of their data,their trend line and discussion of its meaning, and any conclusions.In this way, each participant will actively participate in two linear datacollection activities and see three others.

If participants have experienced the Hooke’s Law activity in the TEXTEAMSMathematical Modeling Institute for Secondary Teachers, you may want tochoose one of the student activities to do as the whole group activity. Thenhave any teachers who are not as familiar with the activity do the “Stretch It”teacher activity, while the other teachers complete the remaining studentactivities.



Activity: Stretch ItBriefly demonstrate the experiment, clearly showing each of the distances tobe measured.

1. Stress the importance of predicting results of the experiment before theyperform the experiment.Have participants predict each relationship, one at a time, as youdemonstrate the distance to be measured.• What will the values in List 2 represent? [The distance from the table

to the top of the cup.] Demonstrate this measurement and haveparticipants sketch their prediction of the graph of (list 1, list 2).

• What will the values in List 3 represent? [The distance from the tableto the bottom of the cup.] Demonstrate this measurement and haveparticipants sketch their prediction of the graph of (list 1, list 3).

• What will the values in List 4 represent? [The distance from thebottom of the cup to the floor.] Demonstrate this measurement andhave participants sketch their prediction of the graph of (list 1, list 4).

2. Have participants create their “hanging containers” as shown and collectthe data. Have participants fill in the table accordingly. Make sureparticipants measure the distances from the free hanging cup, before anymarbles are added.• Which measure is the dependent variable and which is the independent

variable? Justify your response. [The number of marbles is theindependent variable because we are controlling this number. Thedistances from the table are dependent because they rely, depend, onhow many marbles are in the cup.]

Sample data:Number of

marbles

List 1

Distance fromtable to top of cup

(cm)List 2

Distance from tableto bottom of cup

(cm)List 3

Distance frombottom of cup to

floor (cm)List 4

0 16 26.1 51.65 16.5 26.4 51.210 16.9 27 50.815 17.2 27.3 50.520 17.5 27.6 50.225 17.8 27.8 49.830 18.2 28.2 49.4

Have the recorder enter the data in a graphing calculator and link with theother members of the group.Have one group write their data on the transparency of Activity 1 to use withthe whole group.



First find a trend line for List 2 (distance from the table to the top of the cup)versus List 1 (number of marbles) together as a group.

• Is the graph linear or non-linear? [Answers will vary. It looks linear.]• How can you decide if the data is linear? [Based on the work in 2.1.4

Finite Differences, if the first differences are relatively constant, itmakes sense to model the data with a linear model.]

4. Have participants use mental math to find first differences on the data inList 2 as you find first differences on the transparency of Activity 1.

Number ofmarblesList 1

Distance from tableto top of cup (cm)

List 2

Distance from table tobottom of cup (cm)

List 3

Distance from bottomof cup to floor (cm)

List 40 16 26.1 51.65 16.5 26.4 51.210 16.9 27 50.815 17.2 27.3 50.520 17.5 27.6 50.225 17.8 27.8 49.8

555555 30 18.2

0.50.40.30.30.30.4 28.2

0.30.60.30.30.20.4 49.4

0.40.40.30.30.40.4

To estimate a rate of change per marble, find change in ychange in x

. Take an average

of the differences in List 2 and divide by 5 (the constant difference in List 1.)

For the sample data above, rate of change ≈ =. .375

0 074 cm/marble.

5. We do not need to estimate because we found the distance for no marblesin the cup. For our sample data, the y-intercept is 16 cm.

6. For our sample data, y x= +16 0 074.

7.

8. The units of slope are centimeters per marble.9. The y-intercept represents the distance from the table to the top of the cup

when the cup is empty.



11. To estimate a rate of change, find change in ychange in x

. Take an average of the

differences in List 3 divided by 5 (the constant difference in List 1.) For

the sample data above, rate of change ≈ =. .355

0 07cm/marble.

12. We do not need to estimate because we found the distance for no marblesin the cup. For our sample data, the y-intercept is 26.1 cm.

13. For our sample data, y x= +26 1 0 07. .

14.

15. The units of slope are centimeters per marble.16. The y-intercept represents the distance from the table to the bottom of the

cup when the cup is empty.

18. To estimate a rate of change, find change in ychange in x

. Take an average of the

differences in List 4 divided by 5 (the constant difference in List 1.) For

the sample data above, rate of change ≈ =. .375

0 074cm/marble.

19. We do not need to estimate because we found the distance for no marblesin the cup. For our sample data, the y-intercept is 51.6 cm.

20. For our sample data, y x= +51 6 0 074. .

21.

22. The units of slope are centimeters per marble.23. The distance from the floor to the bottom of the cup when the cup is

empty.

Activity 2: Comparing Graphs

1.



2. For the sample data:#1: y x= +16 0 074.#2: y x= +26 1 0 07. .#3: y x= −51 6 0 074. .

3. The slope for trend line #3 is negative because the distance from the floorto the bottom of the cup is decreasing. The slopes of trend lines #1 and #2are positive because the distances from the table to the top or bottom ofthe cup are increasing. The magnitudes of the slopes of the three trendlines are very close to one another. This is because the slopes all representthe number of centimeters the cup moved for each marble added.

4. For the sample data the difference between the y-intercept of trend line #1and the y-intercept of trend line #2 is 26 1 16 10 1. .− = cm. This distance isthe length of the cup.

5. Sample data. One way to find when the cup will touch the floor is to findwhen the distance from the bottom of the cup to the floor is zero. Thisrelationship is represented by trend line #3, y x= −51 6 0 074. . . Therefore,find x when 0 51 6 0 074= −. . x .

Another way to find when the cup will touch the floor is to find when thedistance from the table to the bottom of the cup is equal to the height ofthe table. So, the height of the table is the distance from the floor to thetable equals the distance from the table to the bottom of the cup and thedistance from the floor to the bottom of the cup which is:26 1 51 6 77 7. . .+ = cm . So the question is when does trend line #2 equal77.7? Therefore, find x when 26 1 0 07 77 7. . .+ =x .



(The above two methods produce different solutions because they usedifferent trend lines that used slightly different rates of change.)

6. Sample data: 16 0 074 42 19 108+ ( ) =. .

Answers to Reflect and Apply:1. Either Group 1’s marbles are much heavier than Group 2’s marbles or

Group 1’s rubberband is stretchier than Group 2’s.2. Group 1’s table is taller than Group 2’s. Group 1’s cup is taller than

Groups 2’s by 0.5 cm.3. The intersection point is when the distance from the table to the top of the

cup equals the distance from the floor to the bottom of the cup. It is thenumber of marbles that move the cup to a point halfway between the tableand the floor.

Math Note:Hooke’s Law: The force exerted by a spring is proportional to the distance thespring is stretched or compressed from it’s relaxed position, that is that thetension exerted by a stretched string is (within certain limits) proportional tothe extension, or, in other words, that the stress is proportional to the strain.Robert Hooke, 1635- 1703, had many varied interests from physics andastronomy, to chemistry, biology, and geology. He made many importantscientific contributions.Note that Hooke’s law is based on a spring. In this activity, we use a rubberband instead of a spring to achieve similar results. If desired, use a spring forthe most accurate results. Another alternative is to use a slinky to simulate aspring.

Note: This activity was based on the activity “Linear Modeling in Science” inthe TEXTEAMS Mathematical Modeling Institute for Secondary Teachers.

Note on Student Activity Sample Assessments: The sample items areintended to provide teachers with possible ways to assess the data collectionactivities. They are not intended to show the best or only assessmentspossible. Have participants suggest other possible techniques. Haveparticipants compare the Sample Assessments and discuss the differentobjectives of each. For example, one Sample Assessment prompts studentsfor a graphical solution, another for a tabular solution, and another for twosolutions.



Summary: Participants collect and analyze data to find the relationships between thenumber of marbles in the cup and selected distances from the cup. Byconcluding that a linear model is a reasonable model, participants demonstratethat a spring stretches at a constant rate, Hooke’s Law. By using differencesand estimating rates of change with real data, participants further cement theconcept of the linear model.

II. Linear Functions 2.1 Out for a Stretch: Activity 1


Activity 1: Stretch It

What is the relationship between the number of marbles in the cupand the distances shown below?

Use large paper clips and an 8 oz. paper cup to form a “hangingcontainer.” Measure the distances indicated below as marblesare added to the cup.

List 2

List 3

List 4Floor

Table

Rubberband

Paperclips

1. Predict a graph of the relationship between the number ofmarbles in the cup and the distances shown above. Youshould predict 3 different graphs.



2. Measure the indicated distances and record in the tablebelow. Next, add 5 marbles to the container and measureeach distance as before. Record the new measurements.Continue this process: add 5 marbles to the container,measure, and record.

Numberof

marblesList 1

Distance fromtable to top of

cup (cm)List 2

Distance fromtable to bottom

of cup (cm)List 3

Distance frombottom of cupto floor (cm)

List 405

1015202530

Trend line #1: Consider the relationship between the distancefrom the table to the top of the cup (List 2) and the number ofmarbles (List 1.)

3. Create a scatter plot using a graphing calculator.4. Estimate a rate of change by finding first differences in

your data.

5. Estimate the y-intercept (starting point.)

6. Use the estimated rate and y-intercept to find a trend line foryour data.



7. Graph the trend line over the scatter plot. Adjust theparameters y-intercept and rate of change, if necessary, fora better fit.

8. What are the units of slope for the trend line?

9. What is the meaning of the y-intercept in the trend line?

Trend line #2: Consider the relationship between the distancefrom the table to the bottom of the cup (List 3) and the numberof marbles (List 1.)

10. Create a scatter plot using a graphing calculator.

11. Estimate a rate of change by finding first differences in thedata.








Trend line #3: Consider the relationship between the distancefrom the floor to the bottom of the cup (List 4) and the numberof marbles (List 1.)

17. Create a scatter plot using graphing calculator.18. Estimate a rate of change by finding first differences in the

data.








Activity 2: Comparing Graphs

1. Graph all three scatter plots and trend lines on your calculatorin the same window. Sketch:

2. Write the equations of the trend lines.Trend line #1, List 2 vs List 1:Trend line #2, List 3 vs List 1:Trend line #3, List 4 vs List 1:

3. Compare the slopes of the trend lines. What do you find?

4. Find the difference between the y-intercept of trend line #1and the y-intercept of trend line #2. Where is this distance inthe experiment?

5. Use your trend line to determine when the cup will touch thefloor. Describe your strategy.

6. Use your trend lines to determine how far the top of the cupwould be from the table if you added 42 marbles. Describeyour strategy.

II. Linear Functions 2.1 Out for a Stretch: Reflect and Apply


Reflect and Apply

Group 1 Group 2

1. What can you conclude about Group 1’s marbles and rubberband compared to that of Group 2?

2. What can you conclude about Group 1’s table and cupcompared to that of Group 2?

3. What is the meaning of the intersection point shown belowfrom Group 1?

II. Linear Functions 2.1 Out for a Stretch: Student Activity 1


Student Activity 1: Have You Lost Your Marbles?Overview: Students investigate the relationship between the height of an object and the

distance the object rolls.

Objective: Algebra I TEKS(b.1.B) The student gathers and records data, or uses data sets, to determinefunctional (systematic) relationships between quantities.(b.1.E) The student interprets and makes inferences from functionalrelationships.(c.1.A) The student determines whether or not given situations can berepresented by linear functions.(c.1.C) The student translates among and uses algebraic, tabular, graphical, orverbal descriptions of linear functions.(c.2.B) The student interprets the meaning of slope and intercepts insituations using data, symbolic representations, or graphs.(b.2.D) In solving problems, the student collects and organizes data, makes andinterprets scatter plots and models, predicts, and makes decisions and criticaljudgments.

Terms: rate, slope

Materials: each group needs 5 – 6 building blocks, 39 cm of PVC pipe, marble or steelball bearing that will fit inside and roll freely through the PVC pipe, metrictape measure, graphing calculators


Activity: Have You Lost Your Marbles?Briefly describe and/or demonstrate the experiment. Make sure studentsmeasure the distance the marble rolls once it leaves the end of the pipe.

1. Stress how important it is for students to predict the results of theexperiment before they perform the experiment. Encourage students tothink about and anticipate the results of the experiment before they begincollecting data.

2. Sample data:Height

(blocks)Distance

(cm)1 192 53.53 744 1005 137.5



3. Sample data:

4. Sample data:

An estimate for the rate of change is about 30 cm/block.5. Since the rate of change is about 30, then the y-intercept is about

19 30 11− = − .6. Using our sample data, y x= − +11 30

7. Sample data:

8. The units of slope are centimeters per block.

9. The real world meaning of the y-intercept is that for zero blocks themarble does not roll out of the pipe at all, it covers no distance.

10. The equation is y = − + ( )13 30 20 . Some solution methods:

11. For our sample data, solve: − + =13 30 60x .Some solution methods:

Table:

OtherTable:



Trace:

Trace to theIntersection point.

Guess and check.

Solve algebraically, if you are at a place in your curriculum where itmakes sense for students to do so: − + =13 30 60x

− + + = +13 30 13 60 13x1

3030 73 1

30

=

x

x = 2 43.

12. The more blocks placed under the pipe, the farther the marble travels.

• Did the height of the pipe increase the distance the marble rolled?[Yes.]

• Why or why not? [The marble has more potential (stored) energy withgreater height.]

• If the slope keeps increasing, will the marble roll farther each time?Why or why not? [Up to a point, yes. When the slope is completelyvertical, the marble will not roll far because its energy is absorbed bythe ground.]

Extensions. Find the trend lines for the following and compare.• Use different sized marbles (as long as they still travel freely through

the pipe),• Use spheres with different masses (i.e., golf balls, ping pong balls,

steel ball bearings),• Use different surfaces (i.e., rug, cement, dirt, table top, sheets).

Answers to Sample Assessment



1.

2. With 7 blocks, the marble will roll approximately 163 cm.Some solution methods are shown below.

Trace to x = 7

Solve using the trendline and arithmetic.

Use a table.

3. x ≈ 10 . Students’ answers should be close, depending on their trend lines.Some solution methods are shown below:Trace to theintersection ofy x= +2 23and y = 232

Use a table, intwo ways

Trace to y = 232



Summary: By collecting data and finding a trend line, students investigate therelationship between the height of an object and the distance it rolls. Studentsuse real data to further their conceptualization of the linear function.



Student Activity 1: Have You Lost Your Marbles?

What is the relationship between the height of the pipe and thedistance the marble rolls?

Measure

Roll the marble from heights of 1, 2, 3, 4, and 5 blocks. Releasethe marble at the opening of the pipe. Measure the distance themarble rolls from the end of the pipe.

1. Sketch a graph predicting the relationship between the heightof the pipe and the distance the marble rolls.



2. Data CollectionTasks:• one person rolls the marble,• one person holds the blocks and pipe,• one person marks where the marble stops,• one person measures the distance the marble traveled.

Height(blocks)

Distance(cm)

3. Make a scatter plot using a graphing calculator. Sketch below.



4. Use first differences to estimate a rate of change.


6. Find a trend line for the data using the estimated rate and y-intercept.

7. Graph your trend line over the scatter plot. Adjust theparameters y-intercept and rate of change, if necessary, fora better fit.



10. Use your trend line to determine how far the marble wouldroll if you placed 20 blocks under the pipe. Write anequation and solve in at least four ways.



11. Use your trend line to determine how many blocks areneeded for the marble to roll 60 cm. Write an equation andsolve in at least four ways.

12. Make a general statement about the relationship between thenumber of blocks and the distance the marble travels.



Sample Assessment

A group collected the following data for “Have You Lost YourMarbles?”

Height(blocks)

Distance(cm)

1 252 47.53 73.754 925 117

1. Create a scatter plot and find a trend line. Sketch both in anappropriate window.

2. Use the graph to determine how far the marble would rollwith 7 blocks. Solve in two ways. Show your work.

3. Use the graph to determine how many blocks it would takefor the marble to roll 232 cm. Solve in two ways. Showyour work.



Student Activity 2: Unidentified Circular Objects (UCO’s)Overview: Students investigate the relationship between the diameter of a circular light

on a surface produced by a flashlight and the distance of the flashlight fromthe surface.

Objective: Algebra I TEKS(b.1.B) The student gathers and records data, or uses data sets, to determinefunctional (systematic) relationships between quantities.(b.1.E) The student interprets and makes inferences from functionalrelationships.(c.1.A) The student determines whether or not given situations can berepresented by linear functions.(c.1.C) The student translates among and uses algebraic, tabular, graphical, orverbal descriptions of linear functions.(c.2.B) The student interprets the meaning of slope and intercepts insituations using data, symbolic representations, or graphs.

Terms: diameter, trend line, linear model, rate of change, slope, y-intercept

Materials: flashlights (one per group), rulers, yardstick or meter stick, graphingcalculators


Activity 1: Unidentified Circular ObjectsBriefly describe and/or demonstrate the experiment. Make sure students holdthe meter stick perpendicular to the surface on which the light is shining.


2. Sample data:Distance (cm) Diameter

1 6.52 8.53 10.64 12.45 14.56 16.39 22.310 24.6



3. Sample data:

4. Rate of change is approximately 2 cm/cm5. Estimated starting point is 4.5 cm.6. y x= +4 5 2.

7. Sample data:

8. The units of slope are centimeters per centimeters.9. The real world meaning of the y-intercept is that if the flashlight was no

centimeters from the surface, the circular light pattern would have the y-intercept as its diameter.

10. The equation is y = + ( ) =4 5 2 15 34 5. . . Some solution methods:

11. For our sample data, solve: 4 5 2 18. + =x

Table:

OtherTable:

Trace:




Guess and check.

Solve algebraically, if you are at a place in your curriculum where it makessense for students to do so:

4 5 2 18. + =x4 5 2 4 5 18 4 5. . .+ − = −x

12

2 13 5 12

=

x .

x = 6 75.

12. The further from the surface the flashlight is, the larger the circular lightpattern produced.

Answers to Sample Assessment:

1.

2. Trace to x = 3 5.

3. Two graphical methods:

Trace to y = 20



Trace to theintersection ofy x= +3 6 1 5. .and y = 20

Summary: By collecting data and finding a trend line, students investigate therelationship between the diameter of the circular light pattern produced by aflashlight at varying distances from the surface. Students use real data tofurther their conceptualization of the linear function.



Activity 2: UCO’s

What is the relationship between the diameter of the circularlight pattern cast by a flashlight and the flashlight’s distancefrom the circular light pattern?

Distance

Diameter

Vary the distance of the flashlight from the surface and measurethe diameter of the circular light pattern cast by the flashlight.

1. Sketch a graph predicting the relationship between thediameter of the circular light pattern cast by a flashlight andthe flashlight’s distance from the circular light pattern.



2. Data CollectionHold a yardstick perpendicular to a flat surface, such as a table,with the end starting at 0 on the flat surface. Hold a flashlightnext to the meter stick so that it will cast light on the flat surface.Place the rim of the flashlight (light source end) at 1 cm andmeasure the diameter of the distinct circular pattern formed onthe flat surface. Record in the table below. Continue to vary thedistance of the flashlight from the table and record the diameterof the circle formed.

Distance (cm) Diameter1234569

10







7. Graph your trend line over the scatter plot and adjust theparameters y-intercept and rate of change, if necessary, fora better fit.



10. Use the trend line to determine the diameter of the circlewhen the flashlight is 15 cm from the flat surface. Write theequation and solve in at least three ways.



11. Use the trend line to determine how far the rim of theflashlight is from the flat surface if the diameter of the circleis 18 cm. Write the equation and solve in at least four ways.

12. Make a general statement about the relationship between thedistance of the flashlight from the surfaces and the diameterof the circular light pattern produced on the surface.



Sample Assessment

A group collected the following data for Unidentified CircularObjects.

Distance (cm) Diameter (cm)1 52 6.43 84 9.65 11


2. Use the graph to determine what the diameter of the circularlight pattern is when the flashlight is 3.5 cm from the surface.Show on the graph how you found the answer.

3. Use the graph to determine how far the flashlight is from thesurface when the diameter of the circular light pattern is 20cm. Show on the graph how you found the answer.



Student Activity 3: Going to Great DepthsOverview: Students investigate the relationship between the height of water in a cylinder

and the number of bolts that have been added to the cylinder.

Objective: Algebra I TEKS(b.1.B) The student gathers and records data, or uses data sets, to determinefunctional (systematic) relationships between quantities.(b.1.E) The student interprets and makes inferences from functionalrelationships.(c.1.A) The student determines whether or not given situations can berepresented by linear functions.(c.1.C) The student translates among and uses algebraic, tabular, graphical, orverbal descriptions of linear functions.(c.2.B) The student interprets the meaning of slope and intercepts in situationsusing data, symbolic representations, or graphs.(b.2.D) In solving problems, the student collects and organizes data, makes andinterprets scatter plots and models, predicts, and makes decisions and criticaljudgments.

Terms: rate, slope

Materials: each group needs a cylinder (obtain from a science class, use a pharmacymedication bottle that is cylindrical, or use a cylindrical flat-bottomeddrinking glass), uniform objects that will fit in the cylinder and sink (golfballs, marbles, centimeter cubes), water, metric ruler, graphing calculators


Note: for the sample data below, we used a cylinder and 4 golf balls. Youcan also use marbles but instead of adding one at a time, add 5 marbles at atime. You want the displacement to be enough to be able to measure easily.If you use 5 marbles each time, adjust the questions accordingly.

Activity: Going to Great DepthsBriefly describe and/or demonstrate the experiment. Make sure studentsmeasure the water level before adding any objects.


2. Sample data:



Number of Objects Height(cm)

0 91 10.22 11.53 12.74 13.9

3. Sample data:

4. For our sample data, an estimated rate of change is 1.2 centimeters perobject.

5. For our sample data, the y-intercept is the original water level, 9 cm.6. For our sample data, a trend line is y x= +9 1 2. .

7.

8. The units of slope are centimeters per object.9. The real-world meaning of the y-intercept is that for zero objects the water

level is the original level before there were any objects added.10. The equation is y = + ( )9 1 2 9. . Some solution methods:

11. For our sample data, the highest water level recorded was 13.9centimeters so we need to find how many objects would cause the water torise to 13 9 6 19 9. .+ = cm. Solve: 9 1 2 19 9+ =. .x .Some solution methods:

Table:



Trace:


Guess and check.

Solve algebraically, if you are at a place in your curriculum where itmakes sense for students to do so:

9 1 2 19 9+ =. .x9 1 2 9 19 9 9+ − = −. .x

11 2

1 2 10 9 11 2.

. ..

=

x

x = 9 083.

12. The more objects placed in the cylinder, the higher the level of the water.13. The rate of change would be higher because larger objects would displace

more water. Therefore the line would be steeper, have a higher slope.14. If you added the same amount of water, the original water level would be

higher. Therefore the line would shift up. Also, while the objects wouldstill displace the same amount of water, this amount of water displaced ina smaller container would make the rate of change increase. Therefore theslope of the line would be steeper.

Answers to Sample Assessment

1.



2.

Using a table.

3. x ≈ 31. Students answers should be close, depending on their trend lines.

Using a table, in2 ways

Summary: By collecting data and finding a trend line, students investigate therelationship between the height of water in a cylinder and the number ofuniform objects added to the cylinder. Students use real data to further theirconceptualization of the linear function.



Student Activity 3: Going to Great Depths

What is the relationship between thenumber of uniform objects added to thecylinder and the height of the water inthe cylinder?

Add uniform objects to the cylinder.Measure the height of the water witheach additional object.

1. Sketch a graph predicting therelationship between the height of thewater and the number of objects added:



2. Data Collection

Number ofObjects

Height(cm)

3. Make a scatter plot using a graphing calculator. Sketchbelow.









10. Use your trend line to determine how high the water wouldrise with 9 objects in the cylinder. Write the equation andsolve in at least three ways.



11. Use your trend line to determine how many objects areneeded to make the water rise 6 cm higher than the highestwater level recorded (assuming the container could hold thatmuch water.) Write the equation and solve in at least fourways.

12. Make a general statement about the relationship between thenumber of uniform objects and the height of the water.

13. Suppose you used larger objects. Predict how the graphwould change.

14. Suppose you used a cylinder whose diameter is half that ofthe original cylinder. Predict how the graph would change.



Sample Assessment

A group collected the following data for Going to Great Depths.

Number ofObjects

Height (cm)

0 151 16.12 173 18.24 19


2. Use a table on a graphing calculator to determine how highthe water level would be with 10 added objects. Show howyou found your answer.

3. Use the table on a graphing calculator to determine howmany objects were added if the water level is 45 cm,(assuming the container is tall enough.) Show how youfound your answer.



Student Activity 4: Height versus Arm SpanOverview: Students investigate the relationship between the height of person and the

person’s arm span.

Objective: Algebra I TEKS(b.1.B) The student gathers and records data, or uses data sets, to determinefunctional (systematic) relationships between quantities.(b.1.E) The student interprets and makes inferences from functionalrelationships.(c.1.A) The student determines whether or not given situations can berepresented by linear functions.(c.1.C) The student translates among and uses algebraic, tabular, graphical, orverbal descriptions of linear functions.(c.2.B) The student interprets the meaning of slope and intercepts insituations using data, symbolic representations, or graphs.(b.2.D) In solving problems, the student collects and organizes data, makes andinterprets scatter plots and models, predicts, and makes decisions and criticaljudgments.(c.2.G) The student relates direct variation to linear functions and solve problemsinvolving proportional change.

Terms: rate, slope, arm span, proportional relationship

Materials: metric measuring tape or meter stick(s), graphing calculators


Activity: Height versus Arm SpanBriefly describe and/or demonstrate the experiment.


2. Sample data:Arm Span (cm) Height (cm)

171.5 170169 166.5169 169170 175159 166179 184



3. Sample data:

4. For our sample data, an estimated rate of change is 1 centimeters per 1centimeter.

5. For our sample data, we could reason that the y-intercept is the theoreticalheight of a person with no arm span, therefore the y-intercept is zero.

6. For our sample data, a trend line is y x= +0 1 .

7.

8. The units of slope are centimeters of height per centimeters of arm span.Since the units (centimeters) are the same,cmcm

= 1 and therefore, the slope is this case is dimensionless. To illustrate,

consider if the measurements would have been made in inches, feet,cubits, or pencil lengths. The units of slope would be the same,unit of measureunit of measure

= 1.

9. The real world meaning of the y-intercept is that for a theoretical personwith no arm span, the person would have no height.

10. y = ( ) =1 137 137. Students could solve using a table, tracing on the graph,and evaluating on the home screen. See the other student activities forexamples of solution methods.

11. 1 214x = . Students could solve using a table where y x= , using a tablewhere y x= and y = 214, tracing on the graph, finding the intersection ofy x= and y = 214, and using guess and check on the home screen. Seethe other student activities for examples of solution methods.

12. The longer your arm span is, the taller you are.13. Neither variable is the independent or dependent variable. There is not a

dependent relationship inherent in this situation. The two relationships(arm span, height) and (height, arm span) are inverse relations.

14. The ratios should be somewhat constant and approximately equal to 1.

15. The average of the ratio yx

should be very close to the slope of the trend

line, approximately 1.

16. If yx

k= , then k is the constant of proportionality.



Answers to Sample Assessment:

1 – 2. The points in the scatter plot for Group A all lie below the line y x= .This means the people measured by Group A have longer arm spans thantheir heights. However, the points in the scatter plot for Group B all lieabove the line y x= . This means the people measured by Group B aretaller than their arm spans.Also, noting the different windows, the points in the scatter plot for GroupA must have generally higher coordinates than those for Group B. Thefolks measured by A are taller than those measured by Group B.

3. The points should have relatively high coordinates and lie below the liney x= .

Summary: By collecting data and finding a trend line, students investigate therelationship between the height of a person and the person’s arm span.Students use real data to further their conceptualization of the linear function,specifically of the form y mx= , a proportional relation.



Student Activity 4: Height versus Arm Span

What is the relationship between the height of a person and the lengthof the person’s arm span?

Measure the height and arm span of students.

1. Sketch a graph predicting the relationship between the heightof the person and the person’s arm span:



2. Data Collection

Arm Span(cm)

Height(cm)










10. Use the trend line to determine how tall a person is with anarm span of 137 cm. Write an equation and solve in at leastthree ways.



11 . Use your trend line to determine what arm span a 214 cmtall person would have? Write an equation and solve in atleast four ways.

12. Make a general statement about the relationship between theheight of a person and the person’s arm span.

13. Which is the independent variable and which is thedependent variable in this problem situation?

A linear relationship that contains the origin is called aproportional relationship and is in the form y mx= .

14. In the table, find the average of the ratios, yx

.

15. Compare the average ratio yx

above to the slope in your

trend line. What do you find?

16. If yx

k= , what is k called?



Sample Assessment

Two groups collected data for height versus arm span. Theirscatter plots in relation to the line y x= are shown below.

Group A Group B

1. Name one difference between the people measured by GroupA and the people measured by Group B.

2. Name another difference between the people measured byGroup A and the people measured by Group B.

3. What would a graph look like of mostly tall people whosearm spans tended to be greater than their heights? Sketch thegraph and include the line y x= .

II. Linear Functions 2.2 Linear Regression: Leaders’ Notes


2.2 Linear RegressionOverview: Participants write a program to find a least squares linear function to model

data. They use the program and calculator regression to find linear models fordata and they discuss the correlation coefficient, r.

Objective: Algebra I TEKS(c.2.C) The student investigates, describes, and predicts the effects of changesin m and b on the graph of y mx b= + .

Terms: trend line, line of best fit, linear regression, residual, r, correlation coefficient

Materials: graphing calculator, dynamic geometry program with a prepareddemonstration of a linear least squares fit for data, computer with a projectiondevice.


Math Note:With the advent of graphing calculators, many have begun to use calculatorregression to find models for data. This activity is designed to helpparticipants understand how a linear regression model is calculated, to discusswhen and how to use calculator linear regression, to think about thepedagogical issues associated with calculator linear regression, and tocorrectly understand how to use (or not to use) the correlation coefficient, r.

The least squares method of finding a line of best fit is accessible for teachers,especially if looked at geometrically, using a dynamic geometry program andalso if looked at numerically, using a graphing calculator.

Begin by showing a geometric demonstration of the least squares method forfinding a line of best fit and discuss as follows.Help orient participants by pointing out the data set (points) and the trend line.Make sure the trend line is not close to the line of best fit so that the “squares”can be seen.When finding a line of best fit, we desire to minimize the distance between they-values of the data and the function values of the line. (A statistician mightsay that we want to minimize the difference between an observed value of theresponse variable and the value predicted by the regression line.) The closerthe line is to the data, the smaller the differences will be. Because thedifferences may be positive (if the data point is above the line) or negative(the data point is below the line), we then look at the squares of thedifferences. Hence we are really looking to minimize the squares of thedifferences. This can be shown on the dynamic geometry program as shownbelow.



The following graphs come from a demonstration sketch from the Geometer’sSketchpad. The original sketch:

- 2 2 4

2

- 2

P 1

P 2

P 3

P4

P 5

P6

y i n t

Slope

Points P1 through P6 represent data points. A line is drawn through the points and from each data point to the line a square is constructed.Drag the y-intercept and slope of the line so that the sum of the areas of the squares is minimized. That line is the least squares regression line for the data.Bill Finzer, 3/95

Total Area = 0.82 inches2

Show participants the data set and the trend line. Point out the differencesbetween the data and the trend line, and the visuals representing the square ofthe differences (shaded squares) as shown below. Use the Transparency toillustrate.

2

- 2

- 2 2 4

P 6

P5

P4

P3

P2

P1

y int

Slope

}} }

}

The differences betweenthe y-values of the points and the

function values of the line.

This square representsy f x1 1

2− ( )( ) for

point P1 (x1, y1)



Use the dynamic geometry program to raise the y-intercept of the trend lineand watch the differences get larger, and hence, the squares get bigger. Notethe total area gets bigger as the squares get bigger.

- 2 2 4

2

- 2

P1

P2

P3

P4

P5

P6

y int

Slope


Use the dynamic geometry program to lower the y-intercept of the trend lineand watch the differences get smaller, and hence, the squares get smaller.Note the total area gets smaller as the squares get smaller.

- 2 2 4

2

- 2

P1

P 2

P 3

P 4

P 5

P6

y int

Slope




Use the dynamic geometry program to change the slope of the trend line tomake it steeper and watch the squares change. Note the total area.

- 2 2 4

2

- 2

P 1

P 2

P 3

P 4

P 5

P 6

y int

Slope


Use the dynamic geometry program to change the slope of the trend line tomake it less steep and watch the squares change. Note the total area.

- 2 2 4

2

- 2

P1

P2

P3

P4

P5

P6

y intSlope




Change both the y-intercept and the slope to minimize the total area. Whenthe area is as small as you can get it, you have a line of best fit using a leastsquares method. If you desire, you can display the equation of your line ofbest fit as shown.

2

- 2

- 2 2 4

P 6

P 5

P 4

P3

P 2

P1

y int

Slope


y = 0.39x + 0.33

• Using the method of least squares, why do we square the differencesand then sum them? Why do we not just sum the differences? [Someof the differences might be positive, while others might be negative.Adding these together, they may cancel each other out. You would geta small sum but you might not have a very good fit. Also the pointsthat are furthest from the line adds much more weight than points closeto the line, when their differences are squared.]

• How has this geometrical approach added to your understanding oflinear regression?

Math Note: The difference between the y-value of the data point and thefunction value of the trend line is called the residual. Therefore, we want tominimize the sum of the squares of each of the residuals to achieve a betterfit. A method of discussing the appropriateness of a model is to look at aresidual plot, (x-value, residual). This leads more into statistics and modelingand will not be discussed more in the institute.



Activity 1: Sums of SquaresNow participants use a numeric approach to finding a model using leastsquares. To do this, write a program that does the following with participants.

The program should:Find the differences between the y-values of the data points and the associatedy-values on the line.Square the differences.Sum the squares.Display the sum.

To do this, the participant should first put the data in List 1 (x-values) and List2 (y-values) and the first guess of a trend line in the function grapher (y=).Help participants write the program, using questions like the following:

• How do you denote the difference between a point’s y-value and thefunction’s y-value? [Given the point (L1, L2) and the function y1, thedifference is denoted L y L2 1 1− ( ).]

• Where should we put those differences? [Store the difference in List3.]

• What do we need to do with the differences? [Square them, (List 3)2]• Where should we put those squares? [Store the squares in List 4.]• What do we need to do with the squares? [Sum them, sum(List 4)]• What do we want to do with the sum of the squares once it is

calculated? [See it on the calculator screen. Display the sum.]

Using the following data, here is a sample of how you can demonstrate theprogram: Enter the data. Graph the data.

Propose a trend line and find the sum of squares.

Adjust the trend line and try for a smaller sum.



Adjust the trend line again and try for a smaller sum.

Adjust the trend line again and try for a smaller sum.

When you are satisfied with your accuracy, when you have minimized thesum of squares, then you have a reasonable fit for your data. A line of best fitis the line that minimizes the sum of squares. Try to minimize the sum ofsquares to get the best fit you can.

Activity 2: Line of Best FitHave participants work through the activity in groups, comparing their trendlines and sums of squares with each other in their group.

1. Enter the data

2. Enter a trendline.

3. Use the program to findthe sum of squares.



Repeat to find better and better trend lines as the sum of squaresdecreases.

Equation Sum of Squaresy x= +111 0 9. 3881y x= +100 1 6. 88

y x= +99 86 1 63. . 76.0252

Note that the person whose equation most closely matches the linearregression line of best fit shown below will have the least sum of squares.We will return back to this after the next activity.

5. Answers will vary.Discuss with participants the different processes used to find a trend line.In this activity, we looked for the smallest sum of squares. In the datacollection activities originally we took first differences, estimated a rate ofchange and a starting point, and adjusted the function based on the graph.• How do these methods compare?• What different things do you learn from each method?• Which method do you think you will use in the rest of the institute?

Why?

Math Note: The least sum of squares that you find with the above programwill vary dramatically from data set to data set. The more linear the data, thesmaller the least sum of squares can be. The less linear the data is, the biggerthe residuals will be, the larger the least sum of squares will be. Also, thenature of the data can affect the magnitude of the least sum of squares. Forinstance, if the data deals with relatively large numbers such as distancesbetween planets, the least sum of squares will probably be a similarly large



number (unless, of course, the data is completely linear, in which case the sumof squares is zero.) Likewise, if the data deals with relatively small numberssuch as the size of insect eyes, the least sum of squares will probably be asimilarly small number. This is all about scale. The least square sums are notscale insensitive. So if you are measuring in centimeters versus meters, thesums of squares may be very different in magnitude.

Activity 3: The Correlation Coefficient

A big part of thinking algebraically is doing and undoing. Here we start withfirst differences and ask participants to make data sets based upon the givenfirst differences. Here we again return to the idea that if a data set hasconstant first differences, it is linear.

1. Sample data sets. Note: for each different first value, there will be adifferent data set.

a. b. c. d.x y x y x y x y0 100 0 0.2 0 -10 0 20001 129 1 -14.8 1 10 1 20042 89 2 6.2 2 18 2 20083 124 3 21.2 3 23 3 20124 137 4 17.2 4 40 4 20165 119

29-403513-18 5 25.2

-152115-48 5 72

2085

1732 5 2020

44444

2.

4.

Not linear data, low rvalue

Slightly more lineardata, slightly higherbut still low r value

More linear, higher rvalue

Exactly linear, r = 1

5. As the data is more linear, the r gets closer to 1. See the following noteabout data that correlates negatively.



Note that all of the r values shown above are positive. This is because thedata correlated positively. If the data correlates negatively, the r values arenegative.You can quickly demonstrate this by looking at data that correlates negatively.

Note that r is close to –1. Data that is linear with a negative slope has an rclose to –1. The same pattern holds true. That is that the more linear the data,the better a line will model the data. This is indicated by r close to 1, thistime by r being close to –1.. In other words, if a linear function models thedata well, you will have an r value close to 1. The next part of the activityexamines the converse of this statement.

6.

7. Note that the second differences below are constant. This means that thedata is quadratic, not linear.

x y FirstDifferences

SecondDifferences

0 0 1 21 1 3 22 4 5 23 9 74 16

8. Just because you have r close to 1, you may not have found a good (anappropriate) model. The data above is quadratic, yet a linear model yieldsr close to 1.

If you have a good model, the r will be close to 1. If you have a modelthat yields r close to 1, then you may or you may not have a good model.The correlation coefficient, r, measures the strength and direction of thelinear associate between two variables. An underlying key to thediscussion is that one must look at the data to see if a linear model isappropriate, and then interpret the r value in that context.



Math note: Statisticians often find it useful to square the correlationcoefficient, r. This statistic, called the coefficient of determination, is ameasure of the proportion of total variation in the observed values of y (thedependent variable) that is explained by the observed values of x (theindependent variable). The value of the coefficient of determination may varyfrom zero to one. A coefficient of determination of zero indicates that none ofvariation in the dependent variable is explained by the independent variable.On the other hand, a coefficient of determination of one indicates that 100%of the variation in y has been explained by the regression equation. Thus, if aresearcher finds that there is a correlation coefficient of +0.5 between IQ andreading speed, then the r-squared value of 0.25 tells us that 25% of thevariation in reading speed of the subjects is related to the individual IQ's.Note that this also means that 75% is related to other factors, so that much ofreading speed is not accounted for by IQ.If r2 is large (98%, 89%, etc.), the model is providing a good fit to the dataand we can have confidence in its ability to predict. If r2 is small (10%, 18%,25%, etc.), the model is not providing a good fit. If the data fall perfectly allalong a straight line, then the model is a perfect fit, and r2 is 1.0. In general,the extent to which the data points are lined up along the line or scatteredaway from it determines the strength of the correlation r2 . Keep in mind thatwhile r2 indicates the strength of correlation, one still requires r to indicatethe direction of the correlation (+ or -). So, one needs both statistics to tell thewhole story.

Math Note: The value of r can vary, depending on the size of the data set.Larger data sets yield more confidence in the trend and therefore the model.A linear model can often fit reasonably well over a small set of data, but doesnot represent the trend over the long run behavior of the data. Teachersshould use caution when using models to make predictions far from the dataset, especially when the data set is small.

Caution: It is suggested that you not become involved in a discussion ofstatistics. The intention in this activity is not to teach a statistics course. It isto caution teachers from making inferences about r that are not true and aboutlines of best fit when the model may not be appropriate.

Discuss with participants the difference between trend lines and lines of bestfit.

• When in the institute did we find trend lines?• When in the institute did we find lines of best fit?• How do trend lines and lines of best fit compare?

Note: Throughout the institute, we have found trend lines. We have used reallife problem situations to develop the concepts of slope as a rate of change andy-intercept as a starting point. We have also used first differences to find ratesof change and y-intercepts. Also, in the data collection activities, we used first



differences to estimate rates of change and y-intercepts for trend lines. Wethen adjusted the parameters to visually obtain a better fit.Our objectives in all of these activities finding and using trend lines was tobuild understanding of the linear function. Our objective was not to obtain aline of best fit in the easiest way. Had we entered table values into lists andfound regression lines as “end-all” answers, we would have missedopportunities to learn about the attributes of linear functions.When and how to use regression models is a topic for discussion. Certainlycalculator regression can be used effectively in the midst of a larger problem,where the objective is to use the regression model to learn about a concept.Teachers may have to wrestle with the fact that some students may know howto find regression models with technology before the teacher wants students tohave that knowledge. Effective assessment is essential in provoking studentsto really think, using technology as a tool for understanding and not as acrutch that hinders further progress.

Sample Answers to Reflect and Apply:1. A trend line is an estimate for a linear function to model a situation. It

may or may not be a line of best fit. A line of best fit is the best linearfunction to model a situation, usually found by linear regression. In thecase of perfectly linear data, it is simply the line that contains those points.

4. Enter the two points into lists as shown below and find the line thatcontains them by using linear regression. Note that the correlationcoefficient is 1 or –1 because through any two points there is exactly oneline.

For example: find the equation of the line that contains (0, 0) and (1, 10).

Find the equation of the line that contains (4.3, 20.4) and (5.1, -10.5)

Summary: By looking at a geometric and numeric approach to finding lines of best fit bythe method of least squares, participants gain added understanding ofcalculator linear regression models. Calculator linear regression should beused to further understanding of the linear model or as an intermediate step ina bigger problem.

II. Linear Functions 2.2 Linear Regression: Transparency


Transparency: Least Squares

2

- 2

- 2 2 4

P6

P5

P4

P3

P 2

P1

y int

Slope

}} }

}

The differences betweenthe y-values of the pointsand the function values

of the line .

This square representsy f x1 1

2− ( )( ) for

point P1 (x1, y1)

II. Linear Functions 2.2 Linear Regression: Activity 1


Activity 1: Sums of Squares

Write a program that will, step by step, find the sum of squaresbetween a linear function and a data set.

The program will:

• Find the differences between the line and they-values of the data points.

• Square the differences.

• Sum the squares.

• Display the sum.



Activity 2: Lines of Best Fit

Using your program and the following data, find a line of bestfit.

1. Enter the data into lists in your calculator.x y

10 12020 12930 14640 16350 18760 19970 212

2. Enter a guess for a trend line into the function grapher inyour calculator.

3. Use your program to find a line of best fit. Record your trendlines and the corresponding sums of squares:

Equation Sum of Squares

4. Compare your equation and your least sum of squares withyour group.



5. In your group, refer to the data collection activities in 2.1Out For a Stretch. Enter a data set from one of the activities.Use your program to find a line of best fit. Compare that linewith those from your group members. Also compare that linewith the trend line you found when you first completed theactivity.



Activity 3: The Correlation Coefficient

1. Construct a data set given the first differences shown.a. b. c. d.x y x y x y x y0 0 0 01 1 1 12 2 2 23 3 3 34 4 4 45

29-403513-18

5

-152115-48

5

20851732

5

44444

2. Sketch a scatter plot of each data set above.a. b. c. d.

3. What do you notice about the above scatter plots? What isthe big visual picture in each of the above graphs?

4. Using linear regression on your calculator, find a line of bestfit for each data set above. Record the equation of the lineand the value of the correlation coefficient, r.

a. b. c. d.

5. What is true about the value of r as the data becomes morelinear?



Is the converse true?

6. Using the linear regression on your calculator, find a line ofbest fit for the data below. Note the value of the correlationcoefficient, r.

x y0 01 12 43 94 165 25

7. Find first differences and then second differences for the dataabove. What do you find?

8. If r is close to 1, have you necessarily found the mostappropriate model? Why or why not?

II. Linear Functions 2.2 Linear Regression: Reflect and Apply


Reflect and Apply

1. What is the difference between a trend line and a line of bestfit?

2. When do you believe students should find trend lines andwhen should they find lines of best fit?

3. How can you use technology to enhance studentunderstanding, without allowing the student to rely on thetechnology as a crutch with little understanding of what thetechnology is doing?

4. How can you use linear regression on your calculator to findthe equation of the line between two points?

II. Linear Functions 3.1 Solving Linear Equations: Leaders’ Notes


3.1 Solving Linear EquationsOverview: Participants solve linear equations with concrete models and make

connections between the concrete model, abstract, and symbolicrepresentations.

Objective: Algebra I TEKS(c.3.B) The student investigates methods for solving linear equations andinequalities using concrete models, graphs, and the properties of equality,selects a method, and solves the equations and inequalities.

Terms: concrete model, addition property of equality, subtraction property of equality

Materials: algebra tiles, overhead algebra tiles

Procedures: Participants should be seated at tables with plenty of elbow room in groups of3—4.

“Why use algebra tiles? Manipulating algebra tiles combines an algebraic anda geometric approach to algebraic concepts using an array-multiplicationmodel similar to that employed in many elementary school classrooms. Ourexperience leads us to believe that students benefit from seeing algebraconcepts developed from such a geometric perspective.Furthermore, we believe that we reach a broader group of students bysequencing instruction from the concrete level, through the pictorial level, andfinally to the abstract—or symbolic—level. Such sequencing gives studentsseveral modes, in addition to just abstract manipulations, that help themunderstand and solve algebraic problems. The algebra tiles give a frame ofreference to students who are not abstract thinkers.” Leitze, Annette Ricksand Kitt, Nancy A., “Using Homemade Algebra Tiles to Develop Algebra andPrealgebra Concepts,” Mathematics Teacher, September, 2000, 462.

Before learning to solve linear equations with algebra tiles, students should befamiliar with the tiles, what the tiles represent, and the relationships that existamong the tile pieces. Students should recognize that color represents positiveor negative quantities and that the shape of the tile determines the value itrepresents (unit, x, x2 .) Students should also have experiences adding andsubtracting integers and should understand the concept of “zero pairs.” Azero pair consists of a negative tile and a positive tile pair. Together their sumis zero. For example, a negative unit tile and a positive unit tile form a zeropair, and a positive x tile and a negative x tile form a zero pair, etc.

Begin by explaining zero pairs. Lead participants through the followingexamples, emphasizing that to maintain equality, manipulations made on oneside must also be made on the other side.



Have participants solve x + =5 7 on their own and then discuss the twodifferent algebra tile solution methods on Transparency 1.

• Which method looks easier? Do you think that it always will be?[Participants may suggest that using the subtraction property ofequality is easier. They may change their minds on the next example.]

Have participants solve x + −( ) =4 3 on their own and then discuss the twodifferent algebra tile solution methods on Transparency 1.

Discuss the differences between using the addition property of equality andusing the subtraction property of equality. The technique must match thesymbolic representation of solutions. The goal is to help participants gainfacility in seeing the connection between the two methods, choosing the mostexpedient method, and being able to perform either whenever called for.

Math note: The addition property of equality: if a b= , then a c b c+ = + .The subtraction property of equality: if a b= , then a c b c− = − .

Activity 1: Concrete ModelsWork through one Exercise with participants, demonstrating the process ofsolving using tiles, drawing a sketch showing the manipulations, and writingthe symbolic version. Sketches should not be tedious, but should be viewedas a bridge from the concrete to the symbolic. Eventually sketches will beused as mental representations to assist students in understanding the symbolicmanipulation.

Make sure that participants write the symbolic representation that matchestheir concrete manipulation. For example, if they remove 2 negative unit tilesthen they should write − −( )2 . If they add two positive unit tiles, they shouldwrite +2 .

Have participants work in pairs to complete the activity. One participantmanipulates the tiles and the other participant records the actions on paper.Participants should switch roles periodically.

One possible sequence is shown for each Exercise. Note that in the exampleanswers, the addition property of equality is used. This is primarily forconsistency and is not the only or necessarily the most expedient way.Discuss both ways with participants.



1. Sample solution:

4 8 5 4x x+ = +

4 8 4 5 4 4x x x x+ + −( ) = + + −( )

8 4= +x8 4 4 4+ −( ) = + + −( )x

4 = x

2. Sample solution:

x x x x+ − − = + −20 9 7 2 4

11 3 2= +x

11 2 3 2 2+ −( ) = + + −( )x 9 3= x

3 = x



3. Sample solution:

x x x+ + −( ) = + − −7 7 4 3 5 2

x x= − +1

x x x x+ = − + +1 2 1x =

x = 12

Activity 2: Using Concrete ModelsHave the participants work through the activity. Have a participant workthrough an exercise using overhead tiles. Ask another participant to quicklydemonstrate a different way to manipulate the tiles to solve the same problem.Continue with the other exercises.

1. Sample solution:

8 12 3 13x x− = + 8 12 3 3 13 3x x x x− + −( ) = + + −( )

5 12 12 13 12x − + = + 5 25x =

x = 5



2. Sample solution:

x x x− = + − −6 2 4 3 10

x x− = − −6 6

x x− + = − − +6 6 6 6x x= −

At this point in the problem, x x= − , you can ask,• What number equals its opposite? [Zero, 0 0= − , therefore x = 0.]

An alternative method is to add a positive x tile to both sides, resulting in2 0x = . Therefore each x tile is equal to zero.

3. Sample solution:

3 7 3 2x x x x+ − −( ) = +( ) + 4 4 2 2x x+ = +

4 4 2 2 2 2x x x x+ + −( ) = + + −( )

2 4 2x + =

2 4 4 2 4x + + −( ) = + −( ) 2 2x = −

x = −1



4. Sample solution:

2 9 3 1 5 4 2x x x x− − +( ) = − −( )− − = +x x10 2

− − + = + +x x x x10 2− = +10 2 2x

− + −( ) = + + −( )10 2 2 2 2x − =12 2x

− =6 x

You may need some additional examples, depending on the level of yourparticipants.

• What are some ways that you used to record the algebra tiles? [Someteachers use dots and lines, some use circles and ovals, etc. Haveparticipants share their recording strategies. Suggest that participantshelp students use the drawings as a bridge from the concrete tiles tothe abstract algebraic notation.]

Answers to Reflect and Apply1. See Transparency 2 for an example.2. See Transparency 1 for an example.3. Answers will vary.

Summary: Students come to algebra classes with varied backgrounds and learning styles.Using concrete models to introduce and support algebraic solution strategiesbridges the gap between student informal understanding to abstractunderstanding. Sequencing instruction from the concrete, through thepictorial, to the abstract gives students several ways to understand algebraicproblems.

II. Linear Functions 3.1 Solving Linear Equations: Transparency 1


Transparency 1: x + =5 7

Algebra tile solution ofx + =5 7, using the addition

property of equality

Algebra tile solution ofx + =5 7, using the

subtraction property ofequality

x + =5 7 x + =5 7

x + + −( ) = + −( )5 5 7 5 x + − = −5 5 7 5

x = 2 x = 2

II. Linear Functions 3.1 Solving Linear Equations: Transparency 2


Transparency 2: x + −( ) =4 3

Algebra tile solution ofx + −( ) =4 3, using the

addition property of equality

Algebra tile solution ofx + −( ) =4 3, using the

subtraction property ofequality

x + −( ) =4 3 x + −( ) =4 3

x + −( ) + = +4 4 3 4

You do not have –4 tosubtract from both sides, so

add 4 zero pairs.

x + −( ) = + + −( )4 3 4 4

x = 7 x + −( ) − −( ) = + −( ) − −( )4 4 7 4 4

x = 7

II. Linear Functions 3.1 Solving Linear Equations: Activity 1


Activity 1: Concrete Models

Use algebra tiles to solve each equation. Sketch each step andrecord the symbolic representation for each step.1.

____________________original equation



2.




3.




Activity 2: Using Concrete Models

Build each equation and solve with algebra tiles. Record theintermediate steps and the solution.

1. 8 12 3 13x x− = +

2. − + = + − −6 2 4 3 10x x x



3. 3 7 3 2x x x x+ − −( ) = +( ) +

4. 2 9 3 1 5 4 2x x x x− − +( ) = − −( )

II. Linear Functions 3.1 Solving Linear Equations: Reflect and Apply


Reflect and Apply

Write an equation that is readily solved with an algebra tile set,and:1. is more easily solved using the addition property of equality.

2. is more easily solved using the subtraction property ofequality.

3. What are your classroom goals for using tiles?

II. Linear Functions 3.2 Stays the Same: Leaders’ Notes


3.2 Stays the SameOverview: Participants solve linear equations in one variable, making connections

between algebraic solution steps, algebra tile solution steps, and graphicalsolution steps.

Objective: Algebra I TEKS(c.3.B) The student investigates methods for solving linear equations andinequalities using concrete models, graphs, and the properties of equality,selects a method, and solves the equations and inequalities.

Terms: algebraic solution method, algebra tile solution method

Materials: algebra tiles, overhead algebra tiles, graphing calculators, 1” grid paper,markers


The big idea in this activity is that just as each algebraic step in solving alinear equation can be modeled with algebra tiles, each algebraic step can bemodeled with a graphic representation. As you graph each side of a resultingequation in the solving process, the x-value of the intersection point remainsthe same. This is because the solution to each resulting equation in thesolving process has the same solution. In other words, equivalent equationshave the same solution. This is an important connection.

Work through the following example with participants on the overheadprojector with the algebra tiles and on the overhead calculator.

Example: 5 3 1+ = +x x

5 3 1+ = +x x[0, 4.7] [0, 10]

WINDOW



5 3 1+ + −( ) = + + −( )x x x x

5 2 1= +x

• What remains the same? [The x-coordinate of the intersection points.This is the solution, x = 2, to bothequations: the original and theequivalent equation.]

5 1 2 1 1+ −( ) = + + −( )x

4 2= x

• What remains the same? [The x-coordinate of the intersection points,which is the solution to each equivalentequation, x = 2.]

2 = x

(You can use avertical line to showthat each intersectionpoint has the same x-value.)

• What remained the same throughout the entire solving process? [The x-coordinate of the intersection points, which is the solution to eachequivalent equation, x = 2.]

• Could we have solved the equation in a different way? [Yes, in a fewdifferent ways. For example, you could have added –5 to both sides first.]

• Would solving the equation differently change the solution to theequation? [No.]

• Would it change the resulting graphs? [Yes.]Work through the following example to demonstrate how solving the problemin a different way changes the way the graph looks, but the x-coordinate of theresulting intersection points remain the same, x = 2.]



5 3 1+ = +x x[0, 4.7] [-8, 10]

WINDOW

5 5 3 1 5+ + −( ) = + + −( )x x

x x= −3 4

x x x x+ −( ) = − + −( )3 3 4 3

− = −2 4x

− = −x 2

x = 2

• If the opposite ofx is –2, − = −x 2,then what is x?x = 2.

Activity: Stays the SameHave participants work on the activity in their small group. Use the modeljust completed by everyone.Assign one of the Exercises to each group, and have each group make a postersize presentation on 1” grid paper of the graphical results of their Exercise.They should list the algebraic steps and sketch the corresponding lines in oneappropriate window. You might suggest that they use a different color for



each step and sketch the corresponding lines in the same color. Then havethem make a second poster size presentation, showing the same Exercise, butwith the equation solved in a different order, thus obtaining different graphs,all with the same x-coordinate for the relevant intersection points.

Have participants present their work to the rest of the group.

• Did anyone solve this Exercise different from the two shown by thisgroup? [If so, briefly describe the results.]

• What connections does this activity build?• What are some of the important concepts or ideas that you want students

to understand as a result of this activity?• How might students view the algebraic solution method differently after

completing this activity?• How do you think this activity might impact how you teach solving one

variable linear equations?

One way of completing each exercise follows. There are many other correctways.1. A sample solution:

7 3 8 2+ = − −x x

[-4.7, 4.7] [-20, 3.1]

−( ) + + = −( ) − −7 7 3 7 8 2x x

3 15 2x x= − −



3 2 15 2 2x x x x+ = − − +

5 15x = −

x = −3

2. A sample solution:

x x+ = −5 1

[-4.7, 4.7] [-7, 5]

x x x x+ + = − +5 1

2 5 1x + =



2 5 5 1 5x + + −( ) = + −( )

2 4x = −

x = −2


3 2 6− = −x x

[-4.7, 4.7] [-6, 10]

3 2 6 6 6− + = − +x x

− + =2 9x x



2 2 9 2x x x x− + = +

9 3= x

3 = x


− + = − +2 4 5 4x x

[-4.7, 4.7] [-3.1, 11]

− + + = − + +2 4 2 5 4 2x x x x

4 5 6= − + x



4 5 5 5 6+ = − + + x

9 6= x

32= x

Answers to Reflect and Apply:1. The original equation is − − = −2 6 3x x .

− − = −2 6 3x x

− − + = − +2 6 2 3 2x x



− = −6 5x x

− + = − +6 5x x x x

− =5 5x

x = −1

Summary: Building on the work for solving one variable equations with concrete models(algebra tiles), participants make connections between the concrete model, thealgebraic solution method, and a graphical look at the steps involved in each.Just as the solution to equivalent equations is the same, so too is the x-coordinate of the intersection points when the equivalent equations aregraphed. This gives teachers one more way to meet all of the learning stylespresent in their classrooms.

II. Linear Functions 3.2 Stays the Same: Activity


Activity: Stays the Same

Solve each of the following problems, showing each step, in thethree ways below. Sketch the algebra tile solution and thegraphical solution.

1. 7 3 8 2+ = − −x x

Algebra TileSolution:

AlgebraicSolution:

GraphicSolution:



2. x x+ = −5 1


AlgebraicSolution:

GraphicSolution:



3. 3 2 6− = −x x


AlgebraicSolution:

GraphicSolution:



4. − + = − +2 4 5 4x x


AlgebraicSolution:

GraphicSolution:

II. Linear Functions 3.2 Stays the Same: Reflect and Apply


Reflect and Apply

1. Fill in the missing steps below:Algebra TileSolution:

AlgebraicSolution:

GraphicSolution:

2. Write a linear equation that:• has an integer solution,• can be solved with one algebra tile set, and• whose solution can be found graphically in the

window [-10,10] [-10, 10].

II. Linear Functions 3.3 Solving Linear Inequalities: Leaders’ Notes


3.3 Solving Linear InequalitiesOverview: Participants use problem situations and technology to explore linear

inequalities.

Objective: Algebra I TEKS(c.3.A) The student analyzes situations involving linear functions andformulates linear equations or inequalities to solve problems.(c.3.B) The student investigates methods for solving linear equations andinequalities using concrete models, graphs, and the properties of equality,selects a method, and solves the equations and inequalities.(c.3.C) For given contexts, the student interprets and determines thereasonableness of solutions to linear equations and inequalities.

Terms: linear inequality, strict inequality

Materials: transparencies of the Student Activity: Age Estimates from 2.1.1 The LinearParent Function, graphing calculators


Activity 1: Linear Inequalities in One VariableMany of the problem situations explored throughout the institute have focusedon situations in which linear equations were solved. Many of those linearequations, however, could very well have been considered linear inequalities.Ask participants to recall some of these problem situations that could havebeen considered as linear inequalities.Have participants work through the activity. Circulate and ask guidingquestions. Discuss their results, using the following answers as a guide.1. 5 4+ x2. 5 4 50+ ≤x3. A tabular approach:

For these x-valuesand less,

the y-values areless than or equal to 50.

so, x ≤ 11 25.



4. Algebraic: 5 4 50+ ≤xGraphic:

When is this line, y x= +5 4 ,less than , or under , this line, y = 50?

In other words, forwhat x-values is thediagonal line under the horizontal line?

Answer: for these x-values

4 45x ≤

x ≤ 11 25.

11.25

Note that the graphing calculator makes no distinction between inequalitiesand strict inequalities. Participants will use the same graph on the graphingcalculator to find or confirm solutions for both inequalities and strictinequalities.

5. 15 3+ x

6. 5 4 15 3+ ≤ +x x



7. A sample solution.

Algebraic:5 4 15 3+ ≤ +x x

Graphic:

5 15+ ≤x 4 10 3x x≤ +

x ≤ 10 x ≤ 10

10

Discuss how using multiple representations for solving linear inequalitiesmakes connections and develops understanding.

• How does the solution to a linear equation differ from the solution to alinear inequality? [The solution to a linear equation is one value, onelocation on the number line. The solution to a linear inequality is aninfinite set of values.]



Extension: Boolean Algebra and InequalitiesMany graphing calculators have the capability to use Boolean algebra on thehome screen and/or with graphing to support solution sets of linearinequalities.On the home screen, inequalities produce either a True = 1 or a False = 0 asfollows.

You can demonstrate what happens when you add to or multiply both sides ofan inequality as follows:

Using the Boolean and graphing features, have participant investigate4 5 50x + ≤ , first on the home screen as follows:

You can also investigate 4 5 50x + ≤ by graphing. The calculator graphsy = 1 when 4 5 50x + ≤ and it graphs y = 0 when 4 5 50x + ≤ .

• What are the implications for assessment when students understandhow to graph solution sets as shown above?



Activity 2: Linear Inequalities in Two VariablesShow participants the transparencies of the Student Activity: Age Estimatesfrom 2.1.1 The Linear Parent Function.

• What were the big ideas in the activity? [Guessed famous people’sages, graphed the scatter plot of (guess, actual age), discussed the liney x= , over-guessing, and under-guessing.]

Have participants work on Exercises 1 – 4. Circulate and ask guidingquestions. Discuss results.1. All ordered pairs representing over-guesses are under and not including

the line y x= and do not include the line itself.

2. Represent the shaded set with y x< .• How can you graph y x< on your graphing calculator? [Help

participants with individual calculators.]

3. All of the ordered pairs representing under-guesses are above, notincluding, the line y x= .

4. Represent the shaded set with y x> .• How can you graph y x> on your graphing calculator? [Help

participants with individual calculators.]



Work through the rest of the activity with participants. Discuss the scenario,demonstrating free-throws, 2-point field goals, and 3-point field goals, ifnecessary. Emphasize that the situation will only consider 2- and 3-pointshots and the total combined score for Friday’s game.

5. Ask participants to fill in the table. After participants have filled in mostof their table, ask for some suggestions from the group and write them onthe Activity transparency. Make sure to include some examples that givea combined score of more than 63.Sample entries

Score for 2-pointShots


Total Score

24(2) 5(3) 6320(2) 10(3) 700(2) 21(3) 63

31(2) 1(3) 6340(2) 15(3) 125

• How did you get the total score? [Multiply 2 times the number of 2-point shots and add to the product of 3 times the number of 3-pointshots, 2 3x y+ .]

• How did you know if your choices for numbers of shots fit theproblem situation ? [If the total score was greater than or equal to 63.]

6. 2 3 63x y+ ≥

7. a. y x≥ −21 23

b. Graph y x= −21 23

c. Use a friendly window for your calculator so that cursor has integervalues.

Each ordered pair listed in the table lies above the line y x= −21 23

.



d. Randomly choose some non-integer points above and below the line.Use mental math to decide if the point satisfies the inequality or checkon the home screen as shown:

This ordered pair works because 82.5 > 63.

This is a non-example because 39.5 < 63.

This ordered pair works because 71.5 > 63.

Again, each ordered pair that works lies above the line

y x= −21 23

.

e. To satisfy both the problem situation and the inequality, the points

must lie above the line y x= −21 23

.

f. Help participants with individual calculators.

g. The solution set for y x≥ −21 23

is the set of ordered pairs that lie

above the line y x= −21 23

.

h. Use the free-floating cursor to demonstrate that for that specific x-value, the y-values of the ordered pairs in the solution set are allgreater than the y-value of the line (Use trace to get the y-value of theline.)



8.

a. The solution set for y x≥ −21 23

is the set of ordered pairs that lie

above the line y x= −21 23

.

b. Then use the free-floating cursor to demonstrate that for that specificx-value, the y-values of the ordered pairs in the solution set are allgreater than the y-value of the line (Use trace to get the y-value of theline.)

Answers to Reflect and ApplyHave participants look for general descriptions of the solution(s) for theproblems, not specific solution(s). See the answers for below for examples.1. 13 3 5= −x : The solution to a linear equation in one variable is x a= ,

which is represented graphically by one location on the number line.



2. 13 3 5> −x : The solution to this strict linear inequality in one variable isx a< , which is represented graphically by a subset of the number line.The solution is all x-values where the line y = 13 is above the liney x= −3 5.

3. 13 3 5< −x : The solution to this strict linear inequality in one variable isx a> , which is represented graphically by a subset of the number line.The solution is all x-values where the line y = 13 is under the liney x= −3 5.

4. y x> −3 5: The solution to this strict linear inequality in two variables isthe infinite set of ordered pairs that comprise the section of the planeabove the line y x= −3 5.

5. y x< −3 5: The solution to this strict linear inequality in two variables isthe infinite set of ordered pairs that comprise the section of the planebelow the line y x= −3 5.

Summary: Just as linear functions and linear equations can be developed using problemsituations from life, so can linear inequalities. Linear inequalities in onevariable have solutions in one dimension, which are sets of numbers, andlinear inequalities in two variables have solutions in two dimensions, whichare sets of ordered pairs.

II. Linear Functions 3.3 Solving Linear Inequalities: Activity 1


Activity 1: Linear Inequalities in One Variable

Bianca and Joe are starting their own pet groomingbusiness called Bianca and Joe’s. They havefigured that they can spend no more than $50 amonth on flea shampoo.Joe has found a local dealer of pet shampoo, The PetPantry, who sells quart bottles for $4.00 a bottle plusa $5.00 handling fee per order.

1. Write an expression that represents the amount of moneycharged by The Pet Pantry for an order of shampoo.

2. Write an inequality that represents the amount Bianca andJoe’s is willing to pay per month for The Pet Pantry’sshampoo.

3. Solve the inequality using the table on your calculator.

4. Solve the inequality algebraically step by step. Next to eachstep, show the graphical solution.



Bianca is wondering if they can save money by shoppingaround. She found another local dealer, The Canine Corner,who sells the shampoo for $3.00 a quart bottle plus a $15.00handling charge per order.

5. Write an expression that represents the amount of moneycharged by The Canine Corner for an order of shampoo.

6. Write an inequality that represents when it is less expensiveto buy an order with The Pet Pantry compared to The CanineCorner.

7. Solve the inequality algebraically step by step. Next to eachstep, show the graphical solution.



Activity 2: Linear Inequalities in Two Variables

Recall the Student Activity “Age Estimates” from 2.1.1 TheLinear Parent Function. You estimated the age of famouspeople and graphed the scatter plot (my guess, actual age).

1. On the graph below, shade the region that contains theordered pairs representing the over-guesses possible for theactivity.

2. How can you represent the shaded set with an inequality?

3. On the graph below, shade the region that contains theordered pairs representing the under-guesses possible for theactivity.

4. How can you represent the shaded set with an inequality?

Guess

Act

ual

Act

ual

Guess



The Stars basketball team never makes a free-throw.Considering only 2-point and 3-point shots, whatpossible combination of shots could they score inFriday’s basketball game to meet or beat their seasonaverage score of 63 points?

5. Investigate numerically some combinations of shots thatwould meet or beat their season average of 63.



Total Score

6. Write an inequality in two variables that represents thesituation.

7. Solve the inequality graphically:a. Solve for y.

b. On your graphing calculator graph the liney = ____________.



c. Use the free-floating cursor to find the ordered pairs listedin your table above. Where do these ordered pairs lie onthe graph?

d. Use the free floating cursor to find points that satisfyy x≥ −21 2

3 and the problem situation. Where are these

points?

e. Use the free floating cursor to find other points that satisfy

y x≥ −21 23

, but not the problem situation. Where are

these points?

f. Shade on your graphing calculator y x≥ −21 23

. Sketch.

g. Use the word “over” or “under” to describe the solutionset for y x≥ −21 2

3 in terms of the line y x= −21 2

3.

h. For a specific x-value, how do the y-values of the orderedpairs in the solution set compare to the y-value of the line?



8. Suppose the problem situation had called for the inequality,

y x≤ −21 23

. Predict the graph and produce it on your

graphing calculator. Sketch.

a. Use the word “over” or “under” to describe the solution

set for y x≤ −21 23

in terms of the line y x= −21 23

.

b. For a specific x-value, how do the y-values of the orderedpairs in the solution set compare to the y-value on theline?

II. Linear Functions 3.3 Solving Linear Inequalities: Reflect and Apply


Reflect and Apply

Describe the general solution(s) to the following. Do not solvefor specific solutions.

1. 13 3 5= −x

2. 13 3 5> −x

3. 13 3 5< −x

4. y x> −3 5

5. y x< −3 5

II. Linear Functions 3.4 Systems of Linear Equations and Inequalities: Leaders’ Notes


3.4 Systems of Linear Equations and InequalitiesOverview: Participants use a table to develop a system of linear inequalities. They solve

the system using various methods and make connections between a system ofinequalities and a system of equations.

Objective: Algebra I TEKS(c.4.A) The student analyzes situations and formulates systems of linearequations to solve problems.(c.4.B) The student solves systems of linear equations using concrete models,graphs, tables, and algebraic methods.(c.4.C) For given contexts, the student interprets and determines thereasonableness of solutions to systems of linear equations.

Terms: system of linear equations, system of linear inequalities



Note: Systems of linear inequalities is not an Algebra I TEKS , but is anAlgebra II TEKS. We use a system of inequalities in this activity to stretchteacher understanding. Using a situation that describes a system of linearinequalities lends itself well to developing the system in a table.

Activity 1: Using a TableDescribe the scenario of the scout group going to the movies. Haveparticipants fill in a few rows of the table. Some participants may interpretthe problem to mean that exactly 10 people will go to the show or that theymust spend exactly $45. Clarify that they can take any number of people upto and including 10 as long as they spend $45 or less.

• What factors might the troop consider when deciding how many adultsand children will go to the movies? [Sample Answers. The troop maywant to consider the best adult-child ratio, or they may want the mostnumber of children to go to the movie. Perhaps the adults want to getout of the heat and see the movie.]

Ask for some examples and fill in the table on the transparency. Ask for atleast one example that someone tried that did not work. Record it and draw aline through it as shown below.



Numberof Adults

Number ofchildren

Total numberof people

Cost forAdults

Cost forChildren

Total Cost

2 4 6 2(6) 4(3.50) 265 5 10 5(6) 5(3.50) 47.501 9 10 1(6) 9(3.50) 37.502 8 10 2(6) 8(3.50) 403 7 10 3(6) 7(3.50) 42.50x y x y+ ≤ 10 6x 3.5y 6 3 5 45x y+ ≤.

Ask participants to look for patterns in the table. Help them use their patternsto develop the inequalities in the last row of the table, using the questionsbelow.

• How did you find the total number of people? [Add the number ofadults and the number of children.]

• How can we write this symbolically? [If x is the number of adults (fillin the first box with x) and y is the number of children (fill in thesecond box with y) then the total number of people is x y+ ≤ 10.]

• How did you find the cost for the adults? [Multiply the number ofadults times 6, or 6x .]

• How did you find the cost for the children? [Multiply the number ofchildren times 3.5, or 3 5. y .]

• How did you find the total cost? [Add the cost for the adults, 6x, andthe cost for the children , 3.5y.]

• How did you know if that met the requirement? [It had to be less than$45.]

• How can you write this as an inequality? [The sum of the cost for theadults, 6x, and the cost for the children, 3.5y, must be less than 45,6 3 5 45x y+ ≤. .]

Have participants write the system of inequalities below the table.x y+ ≤ 10

6 3 5 45x y+ ≤.

Activity 2: Solve the System GraphicallyLead the participants with the overhead calculator through the activity usingthe following suggestions.

1. To graph x y+ ≤ 10, solve for y and graph y x= −10 .



Use the free floating cursor to find points that satisfy y x≤ −10 . Look forpoints under the line y x= −10 . Use mental math to check that sum ofthe x-coordinate and the y-coordinate is less than or equal to 10.• Where do the solutions to the equality, y x= −10 , exist? [The

solutions are the ordered pairs, the set of which is the line y x= −10 .In other words, the solutions lie on the line y x= −10 .]

• How can you verify your answer? [Trace to points on the line toconfirm that the sum of the x-coordinate and the y-coordinate is equalto 10.]

Move the cursor around and discuss.

• Which of the above satisfy the problem situation? [Only the firstordered pair found above. The second screen, which shows theordered pair (1, 5.2) satisfies the inequality but not the problemsituation as you cannot take 5.2 people to the movie. The third screen,which shows the ordered pair (-2.2, 2.2) again satisfies the inequality,but not the problem situation as you cannot have negative people.]

Shade y x≤ −10

• What do the shaded points represent? [Each ordered pair represents(number of adults, number of children) such that no more than 10people go to the movie.]

2. To graph 6 3 5 45x y+ =. , solve for y and graph y x= −453 5

63 5. .

.



Use the free floating cursor to find points that satisfy y x≤ −453 5

63 5. .

.

Look for points under the line y x= −453 5

63 5. .

.

• Which of the above satisfy the problem situation? [Only the firstordered pair found above because 5 adults and 2 children satisfy bothconditions of having less than 10 people and costing less than $45.The second screen, which shows the ordered pair (4, 0.6) satisfies theinequality but not the problem situation as you cannot take 0.6 peopleto the movie. The third screen, which shows the ordered pair 4, -1)does satisfy the inequality, but not the problem situation as you cannothave negative people.]

Shade for y x≤ −453 5

63 5. .

• What do the shaded points represent? [Each ordered pair represents(number of adults, number of children) such that the total cost is nomore than $45.]

3. Graph thesystem ofequations andthe solution.

• What does this solution represent? [The solution (4, 6) represents thenumber of adults, 4, and the number of children, 6, to go to the moviesuch that the price will be exactly $45.]

Graph the system of inequalities. The solution is the double shadedsection.



• Where is the solution to the system of inequalities? [Thedouble shaded region.]

• What does the solution to the system of inequalities, the pointsin the double shaded section, represent? [The solution (x, y)represents the number of adults, x, and the number of children,y, to go to the movie such that no more than 10 people will goand the price will be no more than $45.]

Use the free floating cursor to discuss various points as follows:• Do the points satisfy the system and/ or the problem situation and

why?

This point (2, 5) means that2 adults and 5 children go. Itsatisfies the system and theproblem situation because2 5 10+ < and2 6 5 3 5 45( ) + ( ) <. .

This point (1, 10) means that1 adult and 10 children go. Itdoes not satisfy the problemsituation because you cannottake 11 people. It does notsatisfy the system because1 10 10+ < .

This point (-3, 2) does notsatisfy the problem situationbecause you cannot have anegative number of adults. Itdoes satisfy the systembecause − + <3 2 10 and− ( ) + ( ) <3 6 2 3 5 45. .

This point (2, -1) does notsatisfy the problem situationbecause you cannot have anegative number of scouts. Itdoes satisfy the systembecause 2 1 10+ −( ) < and2 6 1 3 5 45( ) + −( )( ) <. .

• What is the relationship between the sets, the set that satisfies theproblem situation and the set that is the solution to the system of



inequalities? [All of the points that satisfy the problem situationare in the set that is the solution to the system of inequalities. Thesolution to the system of inequalities contains many other points.The solution to the system includes points in the second and thirdquadrants, none of which satisfy the problem situation. Also thesolution to the system includes all of the non-integer ordered pairsbounded by the two inequalities. The problem situation includesonly natural number values in the ordered pairs.]

Activity 3: Solve the System SymbolicallyHave participants solve the system using any algebraic methods they choose.Ask participant to consider the following question while they are solving.

• How does the algebraic solution to the system of equations relate tothe graphical solution of the system of inequalities?

1.Solve using substitution Solve using linear combination

(elimination)x y+ = 10

6 3 5 45x y+ =.x y+ = 10

6 3 5 45x y+ =.Solve for y: y x= −10 Multiply − + =( )6 10x ySubstitute: 6 3 5 10 45x x+ −( ) =. Add to 6 3 5 45x y+ =.Solve for x: 6 35 3 5 45x x+ − =.

2 5 10. x =x = 4

Substitute to find y: y = −10 4y = 6

− − = −

+ + =

− = −

6 6 606 3 5 45

2 5 15

x yx y

y

.

.y = 6

The solution is (4, 6) Substitute to find x: x = −10 6x = 4

The solution is (4, 6)

2. The solution to the system of linear equations is an ordered pair, theintersection of the two lines. The solution is written as an ordered pair(x, y).The solution to the system of linear inequalities is a set of ordered pairs.The solution is drawn as a graph that represents all of the ordered pairscontained in the intersection of the shaded regions.

Answers to Reflect and ApplyExplain to participants that we are looking for general descriptions of thesolution(s) for the problems, not specific solution(s). See the answers forbelow for examples.1. 13 3 5= −x : The solution to a linear equation in one variable is x a= ,

which is represented graphically by one location on the number line.



2. 13 3 5> −x : The solution to this strict linear inequality in one variable isx a< , which is represented graphically by a subset of the number line.The solution is all of the x-values where the line y = 13 is above the liney x= −3 5.

3. y x> −3 5: The solution to this strict linear inequality in two variables isthe set of ordered pairs that comprise the section of the plane above theline y x= −3 5.

4. 3 13x y− =x y+ = 11:The solution to a system of linear equations can be the intersection point ifthe lines intersect, a line if the equations of the lines represent the sameline, or there can be no solution if the lines are parallel.

5. 3 13x y− >x y+ < 11:The solution to a system of linear inequalities is the intersection of the twoinequalities. It is a set of ordered pairs if the intersection set is not empty.It is represented by the double shaded area.

6. Answers will vary. An example:Here is a question from a recent End of Course Algebra I test:A truck is carrying 1500 pounds of cargo that occupies 138 cubic feet ofspace. A television weighs 50 pounds and occupies a space of 4 cubicfeet. A microwave oven weighs 30 pounds and occupies a space of 3 cubicfeet. Which system of equations can be used to find the total number oftelevisions, t, and microwaves, m, that are in the truck?

Changed to inequality problem:

A truck is carrying at most 1500 pounds of cargo that occupies at least138 cubic feet of space. A television weighs 50 pounds and occupies aspace of 4 cubic feet. A microwave oven weighs 30 pounds and occupies aspace of 3 cubic feet. Which system of equations can be used to find thetotal number of televisions, t, and microwaves, m, that are in the truck?59 30 1500t m+ ≤4 3 138t m+ ≥

A good window: [0, 60] [0, 40]

Summary: Building on the work of solving equations and inequalities, we end the sectionwith solving systems of equations and systems of inequalities. A table is usedto build the linear inequalities from the written situation. Solving the systemstep by step graphically builds understanding of systems of equations andinequalities. We make the distinction between the solution to a system oflinear equations as being an ordered pair, the intersection point, and thesolution to a system of linear inequalities as being a set of ordered pairs.

II. Linear Functions 3.4 Systems of Linear Equations and Inequalities: Activity 1


Activity 1: Using a Table

A local scout troop and leaders are going to amovie.

• A maximum of 10 people can go.

• They can spend $45 or less for the admission price.

• The movie theater charges $6.00 per adult and $3.50 perchild.

Use the table to investigate possible combinations of people thatsatisfy the conditions.

Numberof Adults

Numberof

children

Totalnumber

of people

Cost forAdults

Cost forChildren

TotalCost



Activity 2: Solve the System Graphically

x y+ ≤106 3 5 45x y+ ≤.

1. Solve x y+ ≤10 graphically.

2. Solve 6 3 5 45x y+ ≤. graphically.



3. Consider the system x y+ ≤10 6 3 5 45x y+ ≤.

• Graph the system of equations:x y+ = 10

6 3 5 45x y+ =.• Find the solution to the system of equations graphically.• Graph the solution to the system of inequalities.• What part of the solution to the system of inequalities

makes sense in the problem situation?



Activity 3: Solve the System Symbolically

1. Solve the system of equations symbolically using at least 2different methods.

x y+ = 106 3 5 45x y+ =.

2. How does the solution for the system of equations compareto the solution for the system of inequalities?

II. Linear Functions 3.4 Systems of Linear Equations and Inequalities: Reflect and Apply


Reflect and Apply

Describe in general the solution(s) to the following.

1. 13 3 5= −x

2. 13 3 5> −x

3. y x> −3 5

4. 3 13x y− =x y+ = 11

5. 3 13x y− >x y+ <11

6. Create a problem that results in a system of inequalities.

II. Linear Functions 3.4 Systems of Linear Equations and Inequalities: Student Activity


Student Activity: Concrete Models and Systems of LinearEquations

Overview: Students use concrete models to solve a system of linear equations.

Objective: Algebra I TEKS (c.4.B) The student solves systems of linear equations using concretemodels, graphs, tables, and algebraic methods.

Terms: system of linear equations

Materials: algebra tiles

Procedures: Students should be seated with plenty of elbow room to work with the algebratiles.Work through the following example with students.Explain that you need to choose one tile shape to represent the variable x, adifferent tile shape to represent the variable y, and a different tile shape torepresent one unit.

x= y=

unit =

Example:3 4 2x y+ =x y− =4 6

3 4 2x y+ =

x y− =4 6



Add the two models together to eliminate the y terms.

3 4 24 6

x yx y+ =

+ − =( )

4 8x =

x = 2

Replace x with 2 in one of the original equations.

3 4 2x y+ = ,3 2 4 2( ) + =y

Solve for y.

6 4 2+ =y

−( ) + +

= + −( )6 6 42 6

y

4 4y = −



y = −1So thesolution isx y= = −2 1,

Now solve the system by substitution.

3 4 2x y+ =

x y− =4 6

Solve for x.

x y yy

− + =

+

4 46 4

x y= +6 4

Substitute 6 4+ y for x in 3 4 2x y+ = .3 6 4 4

2+( ) +

=

y y



16 18 182 18

y + + −( ) =+ −( )

16 16y = −

y = −1

Substitute y = −1 in 3 4 2x y+ = .

3 4 1 2x + −( ) =

3 4 2x − =

3 4 42 4

x − + =

+

3 6x =



x = 2So thesolution isx y= = −2 1,

Have students complete the student activity and discuss.

Activity: Concrete Models and Systems1. An example of solving the system:

x y+ =2 1

− + =x y3 9

Add the two models together to eliminate the x terms.

x yx y+ =

+ − + =( )2 13 9

5 10y =

y = 2



Replace y with 2 units in one of the original equations.

x y+ =2 1,x + ( ) =2 2 1

Solve for x.

x + =4 1

x + + −( ) =+ −( )

4 41 4

x = −3So thesolution isx y= − =3 2,

2. 2 3 1

2 4x yx y+ =

− + = −In this case, the student solves by linear combination.

2 3 1x y+ =

− + = −x y2 4



Multiply − + = −x y2 4 by 2.2 2 4− + = −( )x y

− + = −2 4 8x y

Add the two models together to eliminate the x terms.

2 3 1

2 4 8

x y

x y

+ =

+ − + = −( )

7 7y = −

y = −1

Replace y with -1 in one of the original equations.

2 3 1x y+ = ,2 3 1 1x + −( ) =



Solve for x.

2 3 1x − =

2 3 31 3

x − + =

+

2 4x =x = 2So thesolution isx y= = −2 1,

2. 2 3 1

2 4x yx y+ =

− + = −In this case, the student solves by substitution.

2 3 1x y+ =

− + = −x y2 4

Solve − + = −x y2 4 for x.− + + + =

− + +

x x yx

2 44 4

2 4y x+ =



Substitute 2 4y + for x in 2 3 1x y+ = .

2 2 43 1

yy

+( ) +=

7 8 1y + =

7 8 81 8y + + −( )= + −( )

7 7y = −soy = −1

Replace y with –1 in 2 3 1x y+ =2 3 1 1x + −( ) =

2 3 1x − =

2 3 3 1 3x − + = +



2 4x =x = 2So thesolution isx y= = −2 1,

• How does solving with tiles help you understand the algebraic steps?

Summary: Using concrete models to solve systems of linear equations helps studentsunderstand and make connections when using the linear combination(elimination) algebraic solution method.



Student Activity: Concrete Models and Systems of LinearEquations

1. Solve the following system using algebra. Sketch each stepand write the algebraic representation for each step.

3 4 2x y+ = x y− =4 6



2. Solve the following system using algebra tiles in two ways,by substitution and by linear combination. Sketch each stepand write the algebraic representation for each step.

2 3 1x y+ =− + = −x y2 4

III. Nonlinear Functions 1.1 Quadratic Relationships: Leaders’ Notes


1.1 Quadratic RelationshipsOverview: Participants use lists to develop a quadratic function representing the volume

of a sandbox with a fixed depth. Using the quadratic function, participantssolve quadratic equations numerically and graphically.

Objective: Algebra I TEKS(d.2) The student understands there is more than one way to solve a quadraticequation and solves them using appropriate methods.(d.2.A) The student solves quadratic equations using concrete models, tables,graphs, and algebraic methods.(d.2.B) The student relates the solutions of quadratic equations to the roots oftheir functions.

Terms: quadratic function, zero of a function, root of a function, solution of anequation

Materials: graphing calculators, pieces of lumber or cardboard to simulate lumber


Activity 1: Building a SandboxWork through the activity with participants, using the overhead graphingcalculator to demonstrate. Begin by discussing the situation of building arectangular sandbox. Use the 1 foot wide lumber or cardboard to simulate asandbox.

1. Have participants roughly sketch some possible sandboxes from a bird’seye view. Examples:

141

87

4 11

• What is fixed in this situation? [Two things are fixed. The depth ofthe sandbox is 1 foot deep and the perimeter of the sandbox is 30.]

• How will you fill in the depth column? [The depth is fixed. It willalways be 1 foot.}

• If the perimeter is 30, what kind of widths make sense for thesituation? [Widths ranging from more than 0 feet to less than 15 feet.]

• How does the length relate to the width? [The length is always15 − width .]

• How does the volume relate to the width and length? [The volume isthe product of the width, length, and depth, v w l d= ∗ ∗ . In this casesince the depth is always 1, v w l w l= ∗ ∗ = ∗1 ]



2. Sample dimensions:Width Length Depth Volume

2 13 1 264 11 1 446 9 1 548 7 1 5610 5 1 5012 3 1 3614 1 1 14

3. Encourage participants to predict the general shape of the graph.4. Use the following to help participants enter the data into lists.

Ask participants to put the table values for width into a list in thecalculator.• How is the length related to the width? [The sum of the width and the

length is always 15.]• How can you use an expression for the length in terms of the width to

fill in the lengths into the list in your calculator. [15 1− list .]• What is the depth of each sandbox? [Depth is always fixed at 1 foot.]• What expression can you use for volume? [ list list1 2 1( )( ) • .]Sample scatter plot:

Note: Some calculators allow you to name lists. For this situation, youcould name lists WIDTH, LENTH, and VOLUM.

5. list list1 2 1( )( ) •

6. Sample:

Extension: Ask participants to predict how the situation and the graph wouldchange if the depth of the sandbox is 1.5 feet instead of 1 foot. Change thevolume function to be V x x= −( )15 1 5* . and graph. How would the situation



and the graph change if the depth of the sandbox is 0.75 feet. Again changethe volume function to be V x x= −( )15 0 75* . and graph. This is anintroduction to transformations, which will be explored in depth in 1.2Transformations.

Note: The purpose of the following questions, solving quadratic equations, isto familiarize participants with the different types of equations that arise fromquadratic functions. One of the common struggles that students have isdifferentiating between questions that ask for an input value and questions thatask for an output value. We use some non-algebraic solution methods here tointroduce quadratic equations. We want to build confidence in readingquestions and solving equations with power of technology and students willthen be able to solve symbolically with more understanding.

Briefly discuss each of the solution methods shown below, emphasizing thepower of multiple representations in promoting understanding.

7. Solution:V ft= 38 1875 3.

Graph: Table:

Home screen: Using function notation onthe home screen:

• Does this question give an input value and ask for an output value ordoes the question give an output value and ask for an input value?[The question gives an input value, 3.25 feet, and asks for an outputvalue, the volume. You could also use the terms domain and range inthis question and answer.]

8. Solution: width ft= 7 5.You can do some work on the home screen to get a feel for the answer:



Graph, using trace to get anapproximation:

Graph, using the calculatorto find the maximum value:

Table, looking for the x-value that yields the highest y-value:

• Does this question give an input value and ask for an output value ordoes the question give an output value and ask for an input value?[The question gives an output value, the maximum volume, and asksfor an input value, the width.]

• Where do we usually see a maximum question like this?[Traditionally, maximum and minimum problems have usually beenreserved for calculus, but can readily be examined using technology inearlier courses. This kind of question naturally arises when studyingquadratic functions.]

9. Solution:width ft ft= 5 10,

Graph, using trace to get anapproximation for onesolution:

Graph, using trace to get anapproximation for the othersolution:



Graph, using the calculatorto get an exact answer forone solution, by finding theintersection ofy x x= −( ) •15 1 andy = 50:

Graph, using the calculatorto find the other solution,by finding the intersectionof y x x= −( ) •15 1 andy = 50:

Graph, using the calculatorto get an exact answer forone solution, by finding thezero (root) ofy x x= −( ) • −15 1 50:

Graph, using the calculatorto find one solution, byfinding the (root) zero ofy x x= −( ) • −15 1 50:

Table: “Trace” to the x-value where the volume is 50. Two solutions.

Find the intersection where y y1 2= . Two solutions.

Find the zero (root) of y y1 2 0− = . Two solutions.



• Does this question give an input value and ask for an output value ordoes the question give an output value and ask for an input value?[The question gives an output value, 50 ft3, and asks for an input value,the width.]

An important discussion to have with participants is to compare the 3different table methods and 3 different graph methods – trace, intersection,zero.• How does solving quadratic equations in many ways add to your

understanding?• Why might one be more inclined to use the zero method when solving

a quadratic equation and not a linear equation. [Linear equations aresolved algebraically by getting all of the variables on one side and thenumbers on the other side and solving for the variable. Quadraticequations are often solved algebraically by getting everything on oneside equal to zero and then solving using factoring, completing thesquare, or with the quadratic equation.]

Note: When solving linear equations, there is one solution. Often studentsmistakenly find only one solution to a quadratic equation when solvingsymbolically. Now with a picture in their heads of a quadratic equation beinga parabola intersecting a line, they will be more apt to consider how manysolutions they are looking for.

• If you think of the solution to a quadratic equation as the intersectionbetween a parabola and a line, how many solutions are possible?Make a sketch of each to justify your answer. [Two, one, or no realsolutions.]

* * *Two solutions One solution No real solutions

• If you use the “zero” method, that is setting everything equal to zeroand solving, how will your sketch of the possible solutions change?[The sketch is essentially the same except the line is now the x-axis,y = 0 .]

Note: In this problem, we used a quadratic function to represent the volumeof a sandbox as the width and length varied with a fixed perimeter and a fixeddepth. The units of measure of volume are cubic, in this case, cubic feet.Some discussion may arise that area can be modeled with a quadratic functionand the unit of measure is square units, while volume can be modeled with acubic function, the unit of measure is cubic units. In the particular scenario ofthe sandbox, the volume can be modeled by a quadratic function because one



of the three dimensions, depth, is a fixed quantity. Hence,V l w d x x= ⋅ ⋅ = −( ) ⋅15 1, which is a quadratic function.

Activity 2: Projectile MotionHave participants work through the activity, using the following to discuss.

1.

• What are a reasonable domain and range for the situation? [See theabove window.]

2. Solution:h ft= 96

Graph: Table:

Home screen: Using function notation onthe home screen:

• Does this question give an input value and ask for an output value ordoes the question give an output value and ask for an input value?[The question gives an input value, 2 feet. and asks for an outputvalue, the height.]

3. Solution:width ft ft= 1 4,

Graph, using trace to get anapproximation for onesolution:

Graph, using trace to get anapproximation for the othersolution:



Graph, using the calculatorto find one solution, byfinding the intersection ofy x x= − +16 802 andy = 64:

Graph, using the calculatorto find the other solution,by finding the intersectionof y x x= − +16 802 andy = 64:

Graph, using the calculatorto find one solution, byfinding the zero (root) ofy x x= − + −16 80 642 :

Graph, using the calculatorto find the other solution,by finding the zero (root)of y x x= − + −16 80 642 :

Table: “Trace” to the x-value where the volume is 64. Two solutions.

Find the intersection where y y1 2= . Two solutions.

Find the zero (root) of y y1 2 0− = . Two solutions.



• Does this question give an input value and ask for an output value ordoes the question give an output value and ask for an input value?[The question gives an output value, 64 ft, and asks for an input value,the time.]

• How does solving quadratic equations in many ways add to yourunderstanding?

4. Solution:t = 5 sec

Graph: Table:

• Does this question give an input value and ask for an output value ordoes the question give an output value and ask for an input value?[The question gives an output value, 0 feet. and asks for an inputvalue, the time.]

5. Solution:height ft= 100

Graph, using trace to get anapproximation:

Graph, using the calculatorto find the maximum value:

Table, looking for the x-value that yields the highest y-value:

6. The big idea of this question is to use the symmetry of a parabola to findthe vertex. Once you know the roots of a parabola, you can find the x-coordinate of the vertex by finding the average of the roots. Then you can



find the y-coordinate of the vertex by substituting the x-coordinate into theequation.

Answers to Reflect and Apply1. a. h 2 5 16 2 5 64 2 52. . .( ) = − ( ) + ( ) , h = 60. Participants may also answer,

− + =16 64 602x x , x = 2 5. . If they do, discuss that there is anothersolution to the equation in addition to the one that is shown in the graph.b. − + =16 64 482x x , x = 1 3,c. − + =16 64 602x x , x = 1 5 2 5. , .

2. v ft0 96=

sec

Summary: Using the natural quadratic relationship of volume where the depth is fixed,participants build a quadratic function. They solve arising quadratic equationsin several non-algebraic ways, making connections and buildinggeneralizations. Participants further their study by solving equations that arisefrom a projectile motion situation.

III. Nonlinear Functions 1.1 Quadratic Relationships: Activity 1


Activity 1: Building a Sandbox

The Cano family is building a rectangular sandbox one footdeep. Diana has decided to use lumber that is one foot wide.She collected 30 feet of lumber to enclose the sandbox.

1. Sketch a few possible sandboxes.

2. Fill in the table with some possible dimensions:

Width Length Depth Volume

3. Predict: what do you think a scatter plot of (width, volume)will look like?



4. Enter the table values into lists in your calculator, usingexpressions where appropriate. Create a scatter plot of(width, volume) in your calculator in an appropriate windowand sketch:

5. Write the expression in the lists that you used for volume.

6. Enter the expression for volume into the function grapher.Sketch the graph over the scatter plot above.

Using your function for volume:7. If the width of the sandbox is 3.25 feet, find the volume of

sand necessary to fill the box. Solve, using your calculator:Graphically With a Table



8. What dimensions would allow for the greatest volume ofsand? Solve, using your calculator:

Graphically With a Table

9. The family decides they can afford to buy 50 ft3 of sand.What dimensions should they build the sandbox? Solve,using your calculator:

Graphically With a Table



Activity 2: Projectile Motion

It can be shown that after being thrown straight up into the airwith a velocity of 80 ft/sec, a ball’s height t seconds after beingthrown can be represented by h t t= − +16 802 (ignoring airresistance).

1. Find an appropriate viewing window for h t t= − +16 802 forthis problem situation. Sketch the graph. Justify yourwindow choice.

2. How high is the ball after 2 seconds?Graphically With a Table



3. When was the ball 64 feet above the ground?Graphically With a Table

4. When did the ball hit the ground?Graphically With a Table



5. What is the maximum height that the ball reached?Graphically With a Table

6. The ball was thrown from a height of 0 ft. In Exercise 4, youfound that the ball hit the ground, height = 0, at ______ sec.a. Based on this information, how can you find the time at

which the ball reached its maximum height? Explain yourstrategy.

b. Evaluate the function to find the maximum height.c. What is this point (time, maximum height) called on the

parabola, h t t= − +16 802 ?

III. Nonlinear Functions 1.1 Quadratic Relationships: Reflect and Apply


Reflect and Apply

1. Write an equation and its solution for the following screens.

a. __________________ __________________equation solution

b. __________________ __________________equation solution

c. __________________ __________________equation solution

2. The following equation represents the height of an objectafter t seconds when thrown straight up from the ground:h t v t= − +16 2

0 . At what initial velocity, v0, would you haveto throw the ball to get it to a maximum height of 144 feet?(Hint: Use your graphing calculator to find v0 such thatmaximum height the ball reaches is about 144 feet. Graphh t v t= − +16 2

0 in the window [0, 6] [0, 150], guessing andchecking values for v0 until the maximum height is 144 feet.)

III. Nonlinear Functions 1.2 Transformations: Leaders’ Notes


1.2 TransformationsOverview: Participants investigate the effects of changing the parameters of quadratic

function of the form y ax c= +2 . They apply this understanding by fitting aquadratic to real data. Participants extend their understanding and investigatethe effects of changing the parameter h in quadratic functions of the formy x h= −( )2.

Objective: Algebra I TEKS(d.1.B) The student investigates, describes, and predicts the effects ofchanges in a on the graph of y ax= 2.(d.1.C) The student investigates, describes, and predicts the effects ofchanges in c on the graph of y x c= +2 .

Terms: parameter, transformation, scale factor, translation

Materials: graphing calculators, patty paper


Transformations of functions is an important concept to aid students ingraphing various functions and understanding the behavior of variousfunctions. In these activities, participants explore the effects of changingparameters of quadratic functions. They use the power of graphing calculatorsto find many examples quickly, make and check conjectures, and apply whatthey have learned. Exploring transformations with parabolas is a naturalstarting place as participants can watch the vertex “travel” around thecoordinate system. In later courses, students will apply the lessons learned toother parent functions, and they will add other transformations to theirgraphing toolkit.

Have participants work together in groups, comparing observations onActivities 1 – 3. Discuss their answers to Exercise 5 for each activity. Also,look at table values. See the notes for each activity for an example.

Activity 1: Investigating the Role of a

1. 2.



3. 4.

5. The value a is a vertical scale factor. For a > 1, the parabola is verticallystretched. As x increases, the y-values increase faster than for y x= 2 .For 0 1< <a , the parabola is vertically compressed. As x increases,the y-values decrease faster than for y x= 2 . For a < 0, the graph is areflection over the x-axis.

• Does it change the shape of the graph? [For a = −1, the shape of thegraph does not change. It is a reflection over the x-axis. For a > 1, theshape does change because the parabola is vertically stretched. For0 1< <a , the shape also changes because the parabola is verticallycompressed.

Choose an Exercise and look at table values, both on the graphs and in thetable as shown. Use the questions below to discuss.

• How do the y-values (function values) of y x= 2 2 compare with thoseof the parent function y x= 2 ? [The y-values are twice as much.]

• How do the y-values (function values) of y x= 0 5 2. compare withthose of the parent function y x= 2 ? [The y-values are half as much.]

• Why did the vertex remain the same? [Any number times zero is stillzero, x ⋅ =0 0 .]

This process of looking at y-values to compare functions may seemunnecessary because it is so obvious, but it lays important groundwork forstudents. In later courses, students will be expected to discern when aquestion is referring to function values (y-values) and when a question isreferring to x-values.



Activity 2: Investigating the Role of c

1. 2.

3. 4.

[4.7, 4.7] [-250, 50] 15

WINDOW

5. For c > 0 , the graph is vertically translated (shifted) up c units. For c < 0 ,the graph is vertically translated (shifted) down c units.• Does the shape change? [No, vertical translations are shape preserving

transformations.]Put a piece of patty paper over the graph for Exercise 1 and trace the parentfunction y x= 2 . Slide or shift the patty paper up and down until the parentfunction is directly over the translated functions to show that indeed the shapedoes not change.

• Why did we not use patty paper to look at the transformed functionsy ax= 2? [Dilations are shape changing transformations. If you takethe parent function traced on the patty paper and try to make it “fit”one of the stretched or compressed functions from Activity 1, it willnot work. The shapes are different.]




• How do the y-values (function values) of y x= +2 2 compare withthose of the parent function y x= 2 ? [The y-values are 2 more.]

• How do the y-values (function values) of y x= −2 3 compare withthose of the parent function y x= 2 ? [The y-values are 3 less.]

• Why did the vertex change? [ 0 2 2+ = , 0 3 3− = − .]

Activity 3: Investigating the Role of hHorizontal translations are not listed in the Algebra I TEKS. This activity isintended to enhance teachers understanding of transformations and is notintended for an average algebra I class.

1. 2.

3. 4.

[-47, 47] [-31, 31] WINDOW

5. For h > 0, the graph is horizontally translated (shifted) left h units. Forh < 0, the graph is horizontally translated (shifted) right h units.• Does the shape change? [No, horizontal translations are shape

preserving transformations.]Put a piece of patty paper over the graph for Exercise 1 and trace the parentfunction y x= 2 . Slide or shift the patty paper left and right until the parentfunction is directly over the translated functions to show that indeed the shapedoes not change.




• How do the y-values (function values) of y x= +2 2 compare withthose of the parent function y x= 2 ? [The y-values have all beenshifted up two in the table, left two on the graph.]

• How do the y-values (function values) of y x= −2 1 compare withthose of the parent function y x= 2 ? [The y-values have all beenshifted down one in the table, right one on the graph.]

• Why did the vertex change? [The y-value of zero has been shiftedtoo.]

Activity 4: Transformations

Change New Equation Change in GraphAdd 3 to the function y x= +2 3 Vertical translation up 3Multiply by 1/3 y x=

13

2 Scale change of 1/3

Multiply by 3 y x= 3 2 Scale change of 3Replace x with x −( )2 y x= −( )2 2 Horizontal translation right 2Multiply by – 1 y x= − 2 Reflection over the x-axisSubtract 2 from thefunction

y x= −2 2 Vertical translation down 2

Replace x with x +( )1 y x= +( )1 2 Horizontal translation left 1Multiply by 2 y x= 2 2 Scale change of 2Replace x with x −( )3 y x= −( )3 2 Horizontal translation right 3

2. a. Horizontal translation left 5, vertical translation down 1b. Scale change of 3, vertical translation up 2c. Reflection across the x-axis, scale change of 1/3, horizontal translation

left 1.3. a. y x= −( )2 1 2

b. y x= − +( )2 2

c. y x= −( ) −3 22



d. y x= +( ) +3 12

4. 5.

6. 7.

8.

a.

b. As the elapsed time increases, the distance from the motion detectorincreases.

c.

d. The choice for c is the minimum data value.

e.

• What is the significance of a = 16 in this problem? [The formula forthe distance of an object dropped from an initial height of d0 at an



initial velocity of zero is d at d= +12

20 . The acceleration due to the

force of gravity, a, is −32 2ft

sec, so d t d t d= −( ) + = − +

12

32 1620

20 .

(The proof of this statement is dependent on an understanding ofderivatives that are studied in calculus.) The 16 in the problemsituation is positive because the motion detector is measuring thedistance from the motion detector to the book instead of the distancefrom the book to the ground.

Extension:Ask participants to find first and second differences for the data.

• What do the second differences imply about the choice of a quadraticfunction for a model for the data? [Since second differences areconstant, the data can be modeled with a quadratic function.]

• How are the second differences and your value for a related? [Thesecond differences as shown below are 0.32 feet per 0.1 sec per 0.1

sec, 0 32 320 10 1

2...

ft ftsecsec

sec

= . The acceleration due to the force of gravity,

the force pulling the book down to the ground, is −32 2ft

sec. The

second difference, 32, is positive because the motion detector ismeasuring the distance from the motion detector to the book instead ofthe distance from the book to the ground.

Answers to Reflect and ApplyDiscuss with participants the term “appropriate viewing windows,” especiallywith respect to the graph of quadratic functions.

• What should an appropriate viewing window show about a quadraticfunction? [A complete graph]

• What is a complete graph of a quadratic function? [The windowwould include x-intercepts, if any, the direction the parabola opens,and the vertex. Some participants may want to show the y-intercept.]

Sample answers. Window may vary, but should show similar graphs.

1.



2. A windowwithout the y-intercept.

2. A window thatshows the y-intercept.

3.

4. A window thatshows the y-intercept.

4. A windowwithout the y-intercept.

5. y4, A 9. y2, D6. y3, D 10. y1, A7. y2, C 11. y4, B8. y1, B 12. y3, C

Summary: Using technology to see many examples quickly, participants connecttransformations of quadratic functions with the vertex form of the equation ofa parabola y a x h b= −( ) +2 . Participants use transformations to fit aquadratic function to data, i.e. the distance of a dropped object from a motiondetector.

III. Nonlinear Functions 1.2 Transformations: Activity 1


Activity 1: Investigating the Role of a

Sketch a graph of the following using a graphing calculator.Your observations should include some table values.

1. Function: y x= 2 2. y x= 2 2, y x=12

2, y x= 0 2 2.

Observations: Observations:

3. y x= 5 2, y x= − 2 4. y x= −1

102, y x= −25 2


5. In general, what effects do different values of a have on thegraph of y ax= 2?



Activity 2: Investigating the Role of c

Sketch a graph of the following using a graphing calculator.Your observations should include some table values.1. y x= 2 , y x= +2 2, y x= +2 3 2. y x= −2 0 5. , y x= −2 1


3. y x= +2 1 5. , y x= −2 2 5. 4. y x= +2 15, y x= −2 200Observations: Observations:

5. In general, what effects do different values of c have on thegraph of y x c= +2 ?



Activity 3: Investigating the Role of h

Sketch a graph of the following using a graphing calculator.Your observations should include some table values.

1. y x= 2 , y x= +( )2 2, y x= +( )1 2 2. y x= −( )1 2, y x= −

12

2


3. y x= −( )3 2, y x= +( )2 2 4. y x= +( )22 2, y x= −( )15 2


5. In general, what is the effect on the graph of y x= 2 , whenyou replace x with ( )x h+ ?



Activity 4: Transformations

1. Fill in the blanks.Change from theparent function,

y x= 2

New Equation Change in Graph

Add 3 to the function y x= +2 3 Vertical translation up 3Scale change of 1/3

y x= 3 2

Replace x with x −( )2y x= − 2

Vertical translation down 2Horizontal translation left 1

Multiply by 2Horizontal translation right 3

2. Describe the transformations on y x= 2 that will produce thegraph for each function below. Verify on your calculator.a. y x= +( ) −5 12

b. y x= +3 22

c. y x= − +( )13

1 2

3. Write an equation for each graph below. Each graph is arelative of the parent function y x= 2 (shown in bold).

a. b. c. d.



Using your knowledge of transformations on the parent functiony x= 2 , graph the following relatives. Verify with yourcalculator.4. Function: y x= +( )2 3 2

5. Function: y x= −( )12

2 2

Describe transformations: Describe transformations:

6. Function: y x= − −2 1 7. Function: y x= +( ) +2 12

Describe transformations: Describe transformations:



8. A book was dropped under a motion detector. The followingdata was collected (elapsed time, distance from the motiondetector).

Elapsed Time(sec)

Distance from theMotion Detector

(feet)0 1.38

0.1 1.540.2 2.020.3 2.820.4 3.940.5 5.38

a. Set up a scatter plot in an appropriate window.b. Why is the graph increasing?c. Graph y x= 2 over the scatter plot.d. Choose a value for c and graph y x c= +2 over the scatter

plot. Explain your choice for c.e. Guess and check a value for a and graph y ax c= +2 over

the scatter plot.

III. Nonlinear Functions 1.2 Transformations: Reflect and Apply


Reflect and Apply

Using your knowledge of transformations on the parent functiony x= 2 , find an appropriate viewing window for the followingfunctions. Sketch each function in the window and note thewindow.

1. y x= − −2 50 2. y x= −( ) +2000 50002

3. y x= 0 001 2. 4. y x= − +( )500 5 2

Match the function with a table: Match with a graph:x y1 y2 y3 y4-2 4 0.8 0.2 20-1 1 0.2 0.05 50 0 0 0 01 1 0.2 0.05 52 4 0.8 0.2 20

___ 5. y x= 5 2 ______ 6. y x= 0 05 2. ______ 7. y x= 0 2 2. ______ 8. y x= 2 ___

AB

CD

x y1 y2 y3 y4-2 1 5 4.5 2-1 -2 2 1.5 -10 -3 1 0.5 -21 -2 2 1.5 -12 1 5 4.5 2

____ 9. y x= +2 1 _______ 10. y x= −2 3_______ 11. y x= −2 2___

____ 12. y x= +2 12

___AB

C

D

III. Nonlinear Functions 1.3 Lines Do It Too: Leaders’ Notes


1.3 Lines Do It TooOverview: Participants connect their knowledge of transformations with quadratic

functions with the equations of lines. The point-slope form of a line is lookedat from a transformational perspective.

Objective: Algebra I TEKS(c.2.C) The student investigates, describes, and predicts the effects of changesin m and b on the graph of y mx b= + .(c.2.D) The student graphs and writes equations of lines given characteristicssuch as two points, a point and a slope, or a slope and y-intercept.

Terms: transformation, translation, dilation, increasing function, decreasing function,rate of change, slope, y-intercept

Materials: graphing calculators, patty paper or blank transparencies


In these activities, participants explore the effects of changing parameters oflinear functions. We have done similar work in previous activities in theinstitute, but always in context at more concrete level. Here participants moveout of context to a more abstract level, using the power of graphing calculatorsto find many examples quickly, to make and check conjectures, and to applywhat they have learned. After their work with transformations of parabolas,participants can connect those lessons with quadratic functions now to linearfunctions. In later courses, students will apply transformations to other parentfunctions, and they will add other transformations to their graphing toolkit.

Activity 1: Exploring SlopeHave participants work through the Exercises. Then discuss Exercise 5.

5. Some generalizations about rate of change, a, in y ax= : The larger a is,steeper the line is and the higher the rate is. The smaller a is, the moreshallow the line is and the lower the rate is. If a is positive, then thefunction is increasing (rising from left to right). If a is negative, thefunction is decreasing (falling from left to right).



On the transparency of Exercise 1, plot the point (1, _) for each of the graphedlines as shown.

(1, 1)(1, 2)

(1, 3)

(1, 5)

• Compare the y-coordinates of these points with the a in y ax= . [The y-coordinates of these points are the same as the a in y ax= .]

Tell participants that we can think of y ax= as a transformation of y x= .• What kind of transformation? [Dilation, stretch or compression by a scale

factor of a.]

Take the point (1,1) and stretch it to (1, a). Some participants may mistakenlythink that the line is rotated by a. Looking at table values may help to showthat all of the y-values of the line y ax= are a times the y-values of the liney x= . On a blank transparency, trace the line y x= and the point (1, 1) andplace it on the transparency of Activity 1, Exercise 1. Rotate the transparencyabout the origin to show that the point (1, 1) does not rotate to the point (1, 2).

The big idea of this activity is to help participants look at y ax= as atransformation of y x= , as a dilation. For a > 1, the line is verticallystretched by a scale factor of a. As x increases, the y-values increase fasterthan for y x= . For 0 1< <a , the line is vertically compressed by a scalefactor of a. As x increases, the y-values decrease faster than for y x= . Fora < 0, the graph is a reflection over the x-axis.

Activity 2: Exploring Vertical ShiftsHave participants work through the Exercises. Then discuss Exercise 3.

3. Some generalizations about the y-intercept, b, in y x b= + : [The larger bis, the higher the y-intercept. The smaller b is, the lower the y-intercept.For b > 0, the y-intercept is above x-axis. . For b < 0, the y-intercept isbelow the x-axis.



In each of the graphs, plot the point (0, _) as shown.

(0, 0)(0, 0.5)(0, 1)

(0, 3)

• Compare the y-coordinates with the point (0,0) from y x= and with the bin y x b= + . [The y-coordinates of these points are the same as the b iny x b= + .]

Tell participants that we can think of y x b= + as a transformation of y x= .• What kind of transformation? [Translation, vertical shift by b.]

Put a piece of patty paper over the graph for Exercise 1 and trace the parentfunction y x= and the point (0, 0). Slide or shift the patty paper up and downso that the point (0,0) shifts to (0,b) in all of the rest of the graphs.

The big idea of this activity is to help participants look at y x b= + as atransformation of y x= , as a translation. For b > 0, the graph is verticallytranslated (shifted) up b units. For b < 0, the graph is vertically translated(shifted) down b units.

Before moving to the next Activity, tell participants that now we are going tolook at these translations in yet another way.On the transparency, put parentheses in Exercise 1 as follows.

1a. y x= b. y x= +( )1 c. y x= +( )3 d. y x= +

12

Ask participants to graph the following points along with you.For Exercise 1b, graph the point (-1, 0) on the line y x= +( )1 .For Exercise 1c, graph the point (-3, 0) on the line y x= +( )3 .

For Exercise 1d, graph the point (-0.5, 0) on the line y x= +

12

.

(0, 0)(0.5, 0)

(-1, 0)(-3, 0)



• What do these points suggest? [That the lines have been translatedhorizontally.]

• Where else have you seen horizontal translations? [In the vertex formof parabola, y a x h k= −( ) +2 , the parent function y x= 2 ishorizontally translated left h units if h < 0 and right h units if h > 0.]

• Looking at Exercise 2b, what does writing the function as y x= −( )2suggest? [It suggests that y x= −( )2 can be graphed as a horizontaltranslation right 2 units.]

• If you horizontally translate the point (0, 0) on the parent functiony x= right 2 units, what point to you get? [(2, 0)]

• Is (2, 0) on the line y x= − 2? [Yes.]

With the functions written y x b= + , you can think of the graph of y x b= +as a vertical translation of y x= by b units. With the functions writteny x b= −( ), you can think of the graph of y x b= −( ) as a horizontaltranslation of y x= by b units .

Activity 3: Exploring Horizontal ShiftsWork through a few of the Exercises with participants, using the notes below.Then have participants work through the rest of the Exercises. Ask groups topresent their strategies for a few of the Exercises, demonstrating how they hadgraphed the lines and what connections they had made between the methods.You can also use patty paper to show that both methods graph the same line.

1. y x= + 4 is a vertical translation of y x= up 4 units and y x= +( )4 is ahorizontal translation of y x= left 4 units.

2. y x= +( ) +2 1 y x= + 3First graph the line y x= +( ) +2 1 bytranslating the line y x= left 2 unitsand up 1 unit.

Simplify y x x= +( ) + = +2 1 3 andgraph y x= + 3 as a line with y-intercept 3 and slope 1.

(-2, 1)(0, 3)

3. y x= −( ) +5 3 y x= − 2



First graph the line y x= −( ) +5 3 bytranslating the line y x= right 5units and up 3 units.

Simplify y x x= −( ) + = −5 3 2 andgraph y x= − 2 as a line with y-intercept -2 and slope 1.

(5, 3)

(0, -2)

4. y x= +( ) −1 4 y x= − 3First graph the line y x= +( ) −1 4 bytranslating the line y x= left 1 unitand down 4 units.

Simplify y x x= +( ) − = −1 4 3 andgraph y x= − 3 as a line with y-intercept -3 and slope 1.

(-1, -4) (0, -3)

5. y x= +( ) +2 2 1 y x= +2 5First graph the line y x= +( ) +2 2 1by translating the line y x= left 2units, up 1 unit, and then verticallystretching the line by a scale factorof 2.

Simplify y x x= +( ) + = +2 2 1 2 5and graph y x= +2 5 as a line withy-intercept 5 and slope 2.

(-2, 1)(0, 5)

6. y x= −( ) +12

4 3 y x= +12

1

First graph the line y x= −( ) +12

4 3

by translating the line y x= right 4units, up 3 units, and then verticallycompressing the line by a scale

factor of 12

.

Simplify y x x= −( ) + = +12

4 3 12

1

and graph y x= +12

1 as a line with

y-intercept 1 and slope 12

.

(4, 3)(0, 1)

7. y x= +( ) −3 1 4 y x= −3 1



First graph the line y x= +( ) −3 1 4by translating the line y x= left 1unit, down 4 units, and thenvertically stretching the line by ascale factor of 3.

Simplify y x x= +( ) − = −3 1 4 3 1and graph y x= −3 1 as a line withy-intercept -1 and slope3.

(-1, -4)

(0, -1)

8. y x= − +( ) +2 1 2 y x= −2First graph the line y x= − +( ) +2 1 2by reflecting the line y x= acrossthe x-axis , translating the line y x=left 1 unit, up 2 units, and thenvertically stretching the line by ascale factor of 2.

Simplify y x x= − +( ) + = −2 1 2 2and graph y x= −2 as a line with y-intercept 0 and slope -2.

(-1, 2) (0, 0)

9. y x= − −( ) +2 1 y x= − + 3First graph the line y x= − −( ) +2 1by reflecting the line y x= acrossthe x-axis , translating the line y x=right 2 units, and then up 1 unit.

Simplify y x x= − −( ) + = − +2 1 3and graph y x= − + 3 as a line withy-intercept 3 and slope -1.

(2, 1)(0, 3)



10. y x= − +( )13

3 y x= − −13

1

First graph the line y x= − +( )13

3 by

reflecting the line y x= across thex-axis , translating the line y x=left 3 units and then verticallycompressing the line by a scale

factor of 13

.

Simplify y x x= − +( ) = − −13

3 13

1

and graph y x= − −13

1 as a line

with y-intercept -1 and slope − 13

.

(-3, 0) (0, -1)

• Which method of graphing is better? [“Which is better?” is the wrongquestion to ask. They are both valuable methods. The emphasis is onunderstanding linear functions in different ways, with differentrepresentations. We want to connect different representations to buildunderstanding. Some say that either method is valid, but to say thismay imply that it is sufficient to use either exclusively, thereforelimiting students’ understanding. However, the big idea is not to leteveryone use what ever way they prefer, but to teach many ways oflooking at something, to make connections between methods, and thusto build understanding of the concept from a wider, broader, moreinclusive prospective.]

Activity 4: A Different Perspective1. Prompt participants to ask: What transformations do you need to

transform the line y x= to contain the points (2, 10) and (5, 4)?You still need to find the slope.

TableTime Distance

2 103

5 4

-6

So the slope is 4 105 2

2−−

= −ft

sec. Now sketch the line y x= and reflect it over

the x-axis to get the line y x= − .



y x= − y x=

Now sketch one of the given points, say (5, 4). The goal is to shift the liney x= so that the transformed line contains (5, 4). So, one way to do this is tofirst shift y x= right 5. This is written y x= − −( )5 .

y x= −

right 5

y x= − −( )5

Next, translate y x= up 4 units. This is written y x= − −( ) +5 4.

up 4

y x= − −( )5

y x= − −( ) +5 4

Next, vertically stretch the line by a scale factor of 2, writteny x= − −( ) +2 5 4 .

y x= − −( ) +5 4

y x= − −( ) +2 5 4



Of course, you could have chosen to do all of the above so that the linecontained the other point (2, 10).The equation of the line above, y x= − −( ) +2 5 4 , simplifies to y x= − +2 14 ,which is the result using the previous method of counting back in the tableafter finding the rate of change.

2. Find the rate of change, which is about 23 cm/block. Choose a point, say(1, 25). So the line y x= will be translated right 1, y x= −( )1 , translatedup 25, y x= − +( )1 25, and vertically stretched by a scale factor of 23,y x= − +23 1 25( ) .

Answers to Reflect and Apply1. The point-slope form of the equation of a line from a transformational

perspective can be seen as transforming the line y x= by translating theline y x= horizontally x1 units, translating the line y x= vertically y1

units, and vertically stretching or compressing the line y x= by a scalefactor of a.The vertex form of the equation of a parabola from a transformationalperspective can be seen as transforming the quadratic function y x= 2 bytranslating y x= 2 horizontally h units, translating y x= 2 vertically kunits, and vertically stretching or compressing y x= 2 by a scale factor ofa.

2. The equation y af x b c= −( ) + is a general way of describingtransformations of any function, f x( ) . The variable a is the same verticalscale factor as in the two specific equations above. The variable b ishorizontal translation as is the x1 in the point-slope form of the equation ofa line and the h in the vertex form of the equation of a parabola.Replacing x with x b−( ) in the function f x( ) has the effect of translatingthe graph of f x( ) horizontally by b units just as the x1 and the h did forthe line and the parabola respectively. The variable c a verticaltranslation, up if c > 0 and down if c < 0 .

3. Participants should recognize that y f x= ⋅ ( ) −2 1 means to verticallystretch the original function by a scale factor of 2 and vertically translatethe original function down 1 unit. Therefore the graph of y x= −2 1sin is



4. Participants should recognize that y f x= − ( ) + 3 means to reflect theoriginal function over the x-axis and vertically translate the function up 3units. Therefore the graph of y x= − + 3 is

Summary: Using technology to explore changing the parameters of the equation of a line,participants make connections between transformations of parabolastransformations of lines. Understanding transformations of the line y x=builds deeper understanding of the point-slope form of the equation of a line.

III. Nonlinear Functions 1.3 Lines Do It Too: Activity 1


Activity 1: Exploring Slope

Sketch a graph of the following in the same viewing window.

1. y x=y x= 3y x= 2y x= 5

2. y x=y x= −y x= −2y x= −5

3. y x=y x= 0 5.y x= 0 25.

y x=15

4. y x=y x= −0 5.

y x= −13

y x= 0 2.

5. Summarize the effects of a on the graph of y ax= .



Activity 2: Exploring Vertical Shifts

Sketch a graph of the following in the same viewing window.

1a. y x= b. y x= +1 c. y x= + 3 d. y x= +12

2a. y x= b. y x= − 2 c. y x= −12

d. y x= −1

3. Summarize the effects of b on the graph of y x b= + .



Activity 3: Exploring Horizontal Shifts

1. Describe the following in two ways: y x= + 4.

Graph the following lines in two ways. First, as transformationsof y x= . Then simplify each linear function to y mx b= + ory b mx= + and graph.

2. y x= +( ) +2 1 3. y x= −( ) +5 3 4. y x= +( ) −1 4



5. y x= +( ) +2 2 1 6. y x= −( ) +12

4 3 7. y x= +( ) −3 1 4

8. y x= − +( ) +2 1 2 9. y x= − −( ) +2 1 10. y x= − +( )13

3



Activity 4: A Different Perspective

The following problems are found previously in the institute.Approach them this time from a transformational perspective.

1. You looked up twice during Juan’s walk. You noted that hewas at the 10 foot mark at the 2nd second and that he was atthe 4 foot mark at the 5th second. Complete the table, graph,and find a rule for his walk. (Assume he was walking at aconstant rate.)

Table Graph

Rule:

2. You collected the following data. Find a trend line.Height

(blocks)Distance

(cm)1 252 47.53 73.754 925 117

You were investigating therelationship between theheight of the pipe and thedistance the marble rolls whenreleased in the pipe at thatheight

Measure

Trend Line:

III. Nonlinear Functions 1.3 Lines Do It Too: Reflect and Apply


Reflect and Apply

1. Discuss each of the following from a transformationalperspective. How are the two equations similar?

Point-Slope form of theequation of a line

Vertex form of the equation ofa parabola

y m x x y= −( ) +1 1 y a x h k= −( ) +2

2. For any function, f x( ), how do the above two equationsrelate to y a f x b c= ⋅ −( ) + ?

Given the graphs of the following functions, sketch the indicatedtransformations.3. Given f x x( ) = sin 4. Given f x x( ) =

Sketch2 1 2 1⋅ ( ) − = −f x xsin

Sketch− ( ) + = − +f x x3 3

III. Nonlinear Functions 2.1 Connections: Leaders’ Notes


2.1 ConnectionsOverview: Participants make connections between the roots of quadratic functions and

the solutions to quadratic equations and the factors of quadratic polynomialsand the x-intercepts of a parabola. They connect this understanding to thevertex, polynomial, and factored form of the equation of a parabola. Usingthis understanding, participants model a vertical jump, finding the height ofthe jump.

Objective: Algebra I TEKS(d.2.A) The student solves quadratic equations using concrete models, tables,graphs, and algebraic methods.(d.2.B) The student relates the solutions of quadratic equations to the roots oftheir functions.

Terms: root, zero, x-intercept, factor, solution, vertex, polynomial

Materials: graphing calculators, data collection devices, light sensors, laser pointers orflashlights


The connections between a function’s, y f x= ( ), roots, the zeros of the graphof f x( ) , the solution(s) to the equation f x( ) = 0, and the linear factors of thepolynomial f x( ) (if f x( ) is a polynomial) are all very important connectionsfor students to make about functions. In this activity, participants make theseconnections for quadratic functions. In later courses, students will apply theselessons to higher order polynomials and other functions.

Activity 1: Roots, Factors, x-intercepts, SolutionsHave participants complete the activity. Give a transparency of the table inExercise 8 to a group and have them present their results.Note: The graphs are meant to be sketches using the roots. The importantcriteria to look for are correct roots and direction of the parabola. We are notconcerned with the maximums and minimums of the parabolas in this activity.

1.

2. The x-intercepts are –3 and 2.



3.

4. The x-intercepts are the same.5. x x x x+( ) −( ) = + −3 2 62

Show the algebra tile (area) method for simplifying the expression.

6. The x-intercepts are the same.7. Participants can solve the equation y4 0= with the graph, table,

multiplication property of zero, etc. Solutions are x = −3, x = 2.8. If your participants do not have much experience with algebra tiles to

expand expressions like x x x x+( ) −( ) = + −3 2 62 , then add a column tothe table that includes an area model to obtain the polynomial form.

Graph y4 y1 y2 y3

(factored form)Roots y4

(polynomialform)

Solutionsy4 0=

x + 3 x − 2 x x+( ) −( )3 2 -3 and 2

x x2 6+ −

x = −3 2,

x +1 x + 2 x x+( ) +( )1 2 -2 and -1

x x2 3 2+ +

x = − −2 1,



x − 4 x −1 x x−( ) −( )4 1 1 and 4

x x2 5 4− +

x = 1 4,

x + 3 x + 4 x x+( ) +( )3 4 -4 and -3

x x2 7 12+ +

x = − −4 3,

x x − 3 x x −( )3 0 and 3

x x2 3−

x = 0 3,

x − 4 x + 3 x x−( ) +( )4 3 -3 and 4

x x2 12− −

x = −3 4,

x + 2 x + 4 x x+( ) +( )2 4 -4 and -2

x x2 6 8+ +

x = − −4 2,

x + 2 x − 3 x x+( ) −( )2 3 -2 and 3

x x2 6− −

x = −2 3,



Transparency: Which Form?Ask participants to recall the situations and activities where the three differentforms of a quadratic equation were explored.Sandbox problem, V x x= −( ) ⋅15 1, is the factored form.Projectile problem, h t t= − +16 802 , is in the polynomial form.Transformations, y a x h k= −( ) +2 , are in the vertex form.

• What different kinds of information can you readily see from eachform? [In the sandbox volume problem, V x x= −( ) ⋅15 1, you canquickly see the roots of the function, 0 and –15, where the volume ofthe sandbox is zero.In the projectile motion problem, h t t= − +16 802 , you can easily seethe acceleration due to the force of gravity because –16 ft/sec2 is halfof –32 ft/sec2. You can see the initial velocity of the ball, 80 ft/sec,and the initial height of the ball at zero feet.In 1.2 Transformations, y a x h k= −( ) +2 , you can see the scale factorand the vertex.]

In Activity 2, we will work more with these different forms of a quadratic equation.

Activity 2: Which Form?Work through Exercise 1 with participants. Then have them complete the restof the activity, circulating and asking guiding questions.

1. Since the x-intercepts of the graph are –3 and 5, factors are x x+( ) −( )3 5 .

Average the x-intercepts − +=

3 52

1 so the x-coordinate of the vertex is 1.

Evaluate 1 3 1 5 4 4 16+( ) −( ) = ( ) −( ) = − . But the y-coordinate of the vertex

shown is –8, so there must be a vertical scale factor of 12

. Thus the

factored form of the equation is y x x= +( ) −( )12

3 5 .

2. Using the distributive property or an algebra tile area model, the

polynomial form is y x x= − −12

152

2 .

3. The vertex form is y x= −( ) −12

1 82

4. y x x= −( ) −( )6 95.

a. The x-intercepts are 0 and 2.

b. Since the x-intercepts are 0 and 2, factors are x and x −( )2 and the x-coordinate of the vertex is 1. As the maximum height of the ball is 4.9meters, the y-coordinate of the vertex is 4.9. Therefore the vertex is(1, 4.9).



c. Evaluate y x x= −( )2 at the x-coordinate of the vertex, x = 1,y = −( ) = −1 1 2 1. But the y-coordinate of the vertex is 4.9, so theremust be a vertical scale factor of –4.9. Therefore, y x x= − −( )4 9 2. isthe factored form.

d. The vertex form is y x= − −( ) +4 9 1 4 92. . .e. The polynomial form is y x x= − +4 9 9 82. . .• What information can be readily seen by looking at each form of the

equation? [The factored form shows the x-intercepts, in this situation,the times when the ball was on the ground. The vertex form shows thevertex, in this case the maximum height of the ball. The polynomialform shows the y-intercept, in this case, the height of the ball attime=0.]

6. One way is to work backwards.

x = − 52

and x = 32

2 52

2( ) = − ( )x 2 32

2( ) = ( )x

2 5x = − 2 3x =2 5 0x + = 2 3 0x − =

Therefore y x x= +( ) −( )2 5 2 3 .

Activity 3: Jump!Demonstrate the activity for participants, by having a participant jump whenprompted by the program. Repeat if necessary to get an appropriate graph, asthe example below. Then have participants complete the activity.To add a note of competition, have the three participants with the highestjump found in the experiment come to the front of the group and jump. Ifthey leave the ground at the same time, the last one to land is the winner. Seeif the result matches the predicted result.The program for this activity should record intensity of light. When thejumper is standing in between the flashlight and the sensor, the sensor shouldrecord the ambient light level in the room. When the jumper jumps, thesensor should record the higher intensity level of the flashlight. When thejumper returns to the ground, the sensor should again record the ambient lightlevel in the room. It is obviously important that the jumper takes off and landsin relatively the same place on the floor, in order to effectively block theflashlight’s light.

1. Sample data:



2.

3. The quadratic function to model the jump (time, height) isy x root x root= − −( ) −( )192 1 2 . Recall that the position equation is

h at v t h= − + +12

20 0 . The force of acceleration due to gravity is

− =−

⋅ = −32 32 121

3842 2 2ft

secft

secinft

insec

and 12

384 192 2−( ) = − in

sec.

Participants could also use −490 2cm

sec to find the maximum jump in

centimeters.For our sample data, the model is y x x= − −( ) −( )192 0 06 0 54. . .

4.

Jump heights from a few inches to around 15 inches are reasonable.

Answers to Reflect and Apply1a. c < 4 . The equation f x( ) = 0 has two real solutions.b. c = 4. The equation f x( ) = 0 has one real solution.c. c > 4 . The equation f x( ) = 0 has no real solutions.2. For y Ax Bx C= + +2 , you can see the y-intercept, C. You can also see A

and B which are useful for certain application problems.For y a x x x x= −( ) −( )1 2 , you can see the value of a and the roots, whichare also the x-intercepts and the solutions to y = 0 .For y a x h k= −( ) +2 , you can see the value of a and the vertex (h, k),where k is the maximum or minimum value of the function.

Summary: Roots, solutions, x-intercepts, and factors are often taught as isolated concepts.Bringing them all together helps participants make connections and buildsunderstanding about the different forms of the equation of a quadraticfunction.

III. Nonlinear Functions 2.1 Connections: Transparency


Transparency: Which Form?

Match the quadratic equations with the correct form:

Sandbox problem:V x x= −( ) ⋅15 1

Vertex form

Projectile problem:h t t= − +16 802

Factored form

Transformations:y a x h k= −( ) +2

Polynomial form

III. Nonlinear Functions 2.1 Connections: Activity 1


Activity 1: Roots, Factors, x-intercepts, Solutions

1. Graph the two functions in the same viewing window andsketch.y x1 3= +y x2 2= −

2. What are the x-intercepts of the above equations?

3. Add to your sketch the graph of y y y x x3 1 2 3 2= ⋅ = +( ) −( ).

4. How do the x-intercepts of y x x3 3 2= +( ) −( ) compare to thex-intercept of y x1 3= + and of y x2 2= − ?

5. Using algebra tiles, simplify y x x3 3 2= +( ) −( ) to rewrite inpolynomial form and graph this expression in y4 .

6. How do the x-intercepts of y4 compare to those above?

7. Solve y4 0= .


2.1 Connections: A

ctivity 1

TEXTEA

MS Algebra I: 2000 and Beyond

Spring 2001375

8.C

omplete the table.

Graph y

4y1

y2

y3

(factored form)

Roots

y4

(polynomial form

)Solution(s)to y

40

=

x+

3x

−2

xx

+(

)−

()

32

- 3 and 2x

x2

6+

−x

=−3

2,

x+1

x+

2

xx

−(

)−

()

41

-3 and -4


2.1 Connections: A

ctivity 1

TEXTEA

MS Algebra I: 2000 and Beyond

Spring 2001376

Graph y

4y1

y2

y3

(factored form)

Roots

y4

(polynomial form

)Solution(s)to y

40

=

yx

x=

−−

212

x=

−23,



Activity 2: Which Form?

Write the equation of the graph in three forms:1. Factored form

2. Polynomial form

3. Vertex form

4. Write an equation for a quadratic function that has x-intercepts 6 and 9 and has a vertical scale factor of 1.

5. A soccer goalie kicks the ball from the ground. Ithits the ground after 2 seconds, reaching amaximum height of 4.9 meters.a. Find the x-intercepts for the quadratic function that

models the relationship (time, height).b. Find the vertex of the quadratic function.Write the quadratic function inc. factored form,d. vertex form,e. polynomial form.

6. Explain how you can find the factored form of the equationfor this quadratic function, given it has a vertical scale factorof 1.



Activity 3: Jump!

How high can you jump? You canuse the time that you are in the air tofind your vertical jump.Set up the experiment as shown,creating a photo-gate with theflashlight and the light sensor.Interrupt the signal by standingbetween the laser pointer andthe sensor. Run the program,jumping when prompted

1. Sketch the results.

2. Trace and record the time when you left the floor and thetime when you landed.

3. Use the two times to create a quadratic function that modelsyour jump (time, height).

4. Use your function to find your maximum jump.

III. Nonlinear Functions 2.1 Connections: Reflect and Apply


Reflect and Apply

1. Name values for c such that f x x x c( ) = − +2 4 satisfies each.a. The graph of f has two x-intercepts. What does this imply

about the solution(s) to f x( ) = 0?

b. The graph of f has exactly one x-intercept. What does thisimply about the solution(s) to f x( ) = 0?

c. The graph of f has no x-intercepts. What does this implyabout the solution(s) to f x( ) = 0?

2. What information can you readily see from each form of aquadratic equation?

y Ax Bx C= + +2

y a x x x x= −( ) −( )1 2

y a x h k= −( ) +2

III. Nonlinear Functions 2.2 The Quadratic Formula: Leaders’ Notes


2.2 The Quadratic FormulaOverview: Participants program the quadratic formula into the graphing calculator and

use the program to solve quadratic equations at appropriate times.

Objective: Algebra I TEKS(d.2.A) The student solves quadratic equations using concrete models, tables,graphs, and algebraic methods.

Terms: quadratic equation, quadratic formula

Materials: graphing calculators, 1” graph paper, markers, meter sticks


Activity 1: Programming the Quadratic FormulaWork with participants to write a program for their calculators.A sample program:

Have participants check their programs by trying a simple quadratic equationlike 0 62= − −x x , x = −2 3,

• What result do you get for 0 12= + +x x ? [Since the roots areimaginary, the calculator may return an error message or “non-realanswer” or if the calculator has an imaginary mode, it may return theimaginary roots. Discuss the results with participants.]

• When do you think it is appropriate for students to use the program?

Extension: You could include in the quadratic formula program a conditionalstatement that tests whether the discriminant is negative. If b ac2 4 0− < , thenthe program would return a line “No real solutions.”

Activity 2: Hang TimeHave participants work together on the activity in groups. They should makea poster size presentation on 1” grid paper with their graphs color coded andclearly labeled.Participants will have to deal with the “Pluto problem,” that Pluto’s g is sosmall that it is difficult to clearly show the graphs representing the rest of theplanets and include the graph representing Pluto. One way of dealing with itis the following example.



2.Planet Gravity at the

Surface, gft

sec2

Vertical Height Model HangTime(sec)

7. Mercury 11.84 h t t t( ) = − + +5 92 10 32. 1.955. Venus 28.16 h t t t( ) = − + +14 08 10 32. 0.94

4. Earth 32 h t t t( ) = − + +16 10 32 0.856. Mars 12.16 h t t t( ) = − + +6 08 10 32. 1.901. Jupiter 84.48 h t t t( ) = − + +42 24 10 32. 0.41

2. Saturn 36.80 h t t t( ) = − + +18 4 10 32. 0.762. Uranus 36.80 h t t t( ) = − + +18 4 10 32. 0.763. Neptune 35.84 h t t t( ) = − + +17 92 10 32. 0.77

8. Pluto 1.28 h t t t( ) = − + +0 64 10 32. 15.92

3.

12,3

4

5

67

8

1.95 sec0.94 sec

0.85 sec

1.90 sec0.41 sec

0.76 sec0.77 sec

WINDOW: [0, 2] .1 [-.5, 8] 1



12,3

4

5

67

8

7.22 ft

4.78 ft

4.56 ft

7.11 ft

3.59 ft

4.36 ft4.40 ft

WINDOW: [0, 2] .1 [-.5, 8] 1

15.92 sec

42.06 ft8

[0, 16] 1 [-5, 45] 10

• Is this the kind of problem where it is appropriate to use technology?

Answers to Reflect and ApplyIt makes sense to choose to solve Exercise 3 by completing the square becausethe value of B in Ax Bx C2 + + is an even number, therefore making it easy to

find 12

B . It makes sense to solve Exercise 2 by factoring because you can.

1. Solved with the quadratic formula calculator program, x ≈ −0 134 3 195. , .2. Solved by factoring, x x x x2 12 0 4 3+ − = = +( ) −( ) , x = −4 3, .3. Solved by completing the square, 0 4 12= − +x x .

0 4 2 1 42 2= − + −( ) + −x x0 2 32= −( ) −x3 2 2= −( )x± = −3 2xx = ±2 3



4. Solved with the quadratic formula calculator program, x ≈ −0 25 4 625. , . .5. For Exercise 1, a ball was thrown up from a height of 2.1 meters at an

initial velocity of 15 meters/sec. The 4.9 is one-half the force of gravitymeasured in meters per second per second. The solution to the equationanswers the question, “When does the ball hit the ground?”For Exercise 4, a ball was thrown up from a rooftop 18.5 feet high at aninitial velocity of 70 feet/sec. The 16 is one-half the force of gravitymeasured in feet per second per second The solution to the equationanswers the question, “When does the ball hit the ground?”

6.a. x +( )4 2

x x2 8 16+ +4

x2

16

x

x

4x

4x4

b. x +( )9 2

x x2 18 81+ +

81

x

9x

9

x x2

9

9x

c. x −( )5 2

x x2 10 25− +

x2

x

-5

−5x

25

x

-5

−5x

d. x x+( ) −( )3 2x x2 6+ −

x

-2

3

-6

x2

x

3x

−2x

e. x x−( ) −( )3 4x x2 7 12− +

xx2 −3x

-4 −4x

-3

12

x

f. x x+( ) +( )6 5x x2 11 30+ +

30

x

5

6xx2x

6

5x

Summary: The quadratic formula programmed into a graphing calculator can providestudents with a powerful tool to solve quadratic equations. Care should beexercised about when students should use the program. In the midst of a bigproblem where the program is used as a tool to quickly obtain otherwisecumbersome solutions and students must ascertain at which point to use theprogram, certainly this is an appropriate time to use technology.

III. Nonlinear Functions 2.2 The Quadratic Formula: Activity 1


Activity 1: Programming the Quadratic Formula

1. Write a program that will find the roots of a quadraticequation using the quadratic formula.

The program should:• Ask the user to input A, B, C from y Ax Bx C= + +2

• Find the root(s)• Display the root(s)



Activity 2: Hang Time

If you were to jump around on differentplanets, your motion would differbecause the acceleration due to the forceof gravity is different. Imagine that youjump from a 3 foot high platform with aninitial velocity of 10 ft/sec. How wouldyour hang time compare on eachdifferent planet?

1. Based on the values of g below, make some predictions. Onwhich planet would you land first or last? On which planetwould you jump the highest or lowest?

2. Complete the table.Planet Gravity at the

Surface, gft

sec2

Vertical HeightModel

HangTime

Mercury 11.84Venus 28.16Earth 32Mars 12.16

Jupiter 84.48Saturn 36.80Uranus 36.80Neptune 35.84

Pluto 1.28



3. Make a graph, showing the graphs of the jumps (time,height), each labeled with the maximum height of the jumpand the time you would land on that planet.

III. Nonlinear Functions 2.2 The Quadratic Formula: Reflect and Apply


Reflect and Apply

Solve the equations. Choose one to solve by completing thesquare. Choose one to solve by factoring. Explain your choices.

1. 0 4 9 15 2 12= − + +. .x x2. x x2 12 0+ − =3. 0 4 12= − +x x4. − + + =16 70 18 5 02x x .

5. Choose two of the above equations and give them a real-world context, including what the various numbers representand what question the equation answers.

6. Fill in the missing values. Then write two expressions for thetotal area for each figure.a.

x2

16

x

?

?

?

4

?

b.

81

x

? 9x

?

?

9

?

c.

x2

x

?

?-5

−5x

?

?

d.? 3

? ?

?

-6

x2x

e.x

x2 −3x

-4

?

? ?

?f.

?

?

?

?

30

x

5

6x

III. Nonlinear Functions 2.2 The Quadratic Formula: Student Activity


Student Activity: Investigate Completing the SquareOverview: Students investigate completing the square with algebra tiles.

Objective: Algebra I TEKS(d.2.A) The student solves quadratic equations using concrete models, tables,graphs, and algebraic methods.

Terms: complete the square

Materials: algebra tiles, graphing calculator

Procedures: Have students work through Exercises 1 – 4 in groups. As a whole group,discuss their answers using the following.

Note: This activity assumes that students have prior experience withrepresenting, adding and subtracting polynomials with algebra tiles (areamodel), with using algebra tiles to model monomial and binomialmultiplication and with modeling factoring trinomials with algebra tiles.Note: This activity uses a concrete model to lay the foundation for thealgebraic work of completing the square that students will do in Algebra II.

1. a. You need 9 unit tiles to complete the square.b. The dimensions of the completed square are x +( )3 by x +( )3 .c. x x x2 26 9 3+ + = +( )

2. a. You need 16 unit tiles to complete the square.b. The dimensions of the completed square are x −( )4 by x −( )4 .c. x x x2 28 16 4− + = −( )

3. The number of unit tiles needed to complete the square is the square ofhalf of the coefficient of x. For x Bx2 + , the number of unit tiles needed is

B2

2

.

4. a. x x x22

3 94

32

− + = −

b. x bx b x b22 2

2 2+ +

= −

Do the following 2 examples with students.The first example is to write the equation, y x x= + +2 4 5 in vertex form.Complete the square.y x x= + + − +2 4 4 4 5y x x= + +( ) − +2 4 4 4 5y x= +( ) +2 12



Have students quickly sketch a graph.

The second example is to solve a quadratic equation, 0 10 192= − +x x , bycompleting the square.Complete the square:0 10 192= − +x x0 10 25 25 192= − + − +x x0 10 25 25 192= − +( ) − +x x0 5 62= −( ) −x6 5 2= −( )x± = −6 5xx = ±5 6

Have students complete the rest of the Activity.

5. y x x= + +2 6 4y x x= + + − +2 6 9 9 4y x x= + +( ) − +2 6 9 9 4y x= +( ) −3 52

6. 3 1 6 02x +( ) − =3 1 62x +( ) =x +( ) =1 22

x + = ±1 2x = − ±1 2

7. x x2 4 8 0− − =x x2 4 4 4 8 0− + − − =x x2 4 4 4 8 0− +( ) − − =

x −( ) − =2 12 02

x −( ) =2 122

x − = ±2 12x − = ±2 2 3x = ±2 2 3

Summary: Using algebra tiles to complete the square based on the area model ofmultiplication gives students a geometric approach to understanding thealgebraic steps to complete the square.



Student Activity: Investigate Completing the Square

1. Create a partial square with algebra tiles torepresent x x2 6+ as shown.a. How many unit tiles do you need to

complete the square?b. What are the dimensions of the

completed square?c. x x x2 26+ + = +( )? ?

2. Create a partial square with algebra tilesto represent x x2 8− as shown.a. How many unit tiles do you need to

complete the square?b. What are the dimensions of the

completed square?c. x x x2 28− + = −( )? ?

3. How does the number of unit tiles to complete the squarecompare to each respective coefficient of x?

4. Based on the above, complete the two square diagrams.a. x x x2 23− + = −( )? ? b. x bx x2 2+ + = −( )? ?

x 2 −1 5. x

−1 5. x ?

x 2

?

b x2

b x2



5. Write the function in vertex form by completing the squarey x x= + +2 6 4. Then sketch a graph.

6. Solve the quadratic equation written in vertex form.3 1 6 02x +( ) − =

7. Solve the equation by completing the square.x x2 4 8 0− − =

III. Nonlinear Functions 3.1 Exponential Relationships: Leaders’ Notes


3.1 Exponential RelationshipsOverview: Participants explore exponential growth and decay situations. Participants

develop the ideas of the common multiplier or ratio as the base of anexponential function and the starting point as the y-intercept of an exponentialfunction.

Objective: Algebra I TEKS(b.3.B) Given situations, the student looks for patterns and representsgeneralizations algebraically.(d.3.C) The student analyzes data and represents situations involvingexponential growth and decay using concrete models, tables, graphs, oralgebraic methods.

Terms: growth, decay, recursion,

Materials: graphing calculators, sheets of blank paper


In this section, we explore exponential relationships similar to the way weexplored linear relationships in II. Linear Functions. The connection betweenthe linear starting point and y-intercept is analogous to the connectionbetween the exponential starting point and y-intercept. The connectionbetween the added constant and the slope of linear functions is analogous tothe constant multiplier and the base of exponential functions. Participants willlearn to write exponential functions using a starting point and a commonmultiplier or ratio just as they learned to write linear functions using a startingpoint and a common difference. Encourage participants to make connectionsbetween what is happening in the problem situation and the parameters in theexponential functions.

Do the Student Activity with participants, depending on the level of yourparticipants.

Activity 1: Paper FoldingWork through Activity 1 with participants.

Introduce the scenario and demonstrate a couple of folds. Have participantsfold a piece of paper in half as many times as they can.1. Lead participants in filling in the table, using language similar to the

following:For no folds, you have 1 layer of the piece of paper

Number of folds Process Number of Layers0 1 1



After the 1st fold, you have 2 layers.Number of folds Process Number of Layers

0 1 11 1 2 2⋅ = 2

After the 2nd fold, you have 4 layers. In other words, you have twice as manyas before.

• How can you write 1 2 2⋅ ⋅ with exponents? [1 22⋅ ]Number of

foldsProcess Number of Layers

0 1 11 1 2 2⋅ = 22 1 2 2 1 22⋅ ⋅ = ⋅ 4

When completing the table, one is more apt to operate recursively on theprevious term, continuing to multiply by 2. The emphasis here is onexpressing the number of layers in terms of the number of folds in order todevelop the function rule.Continue to fill in the table, establishing the pattern.

Number offolds

Process Number ofLayers

0 1 11 1 2 2⋅ = 22 1 2 2 1 22⋅ ⋅ = ⋅ 43 1 2 2 2 1 23⋅ ⋅ ⋅ = ⋅ 84 1 2 2 2 2 1 24⋅ ⋅ ⋅ ⋅ = ⋅ 165 1 2 2 2 2 2 1 25⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ 326 1 2 2 2 2 2 2 1 26⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ 64n 1 2 2 2 2 1 2⋅ ⋅ ⋅ ⋅ ⋅ = ⋅. . .

n

n

factors1 244 34 4 1 2⋅ n

2. Write the function as follows.• After n folds, how many 2’s will be multiplied by each other? In other

words, how many factors of 2 are there is the expression? [n]• How can you write 2 2 2 2⋅ ⋅ ⋅ ⋅. . .

n factors1 244 34 4 ? [ 2n ]

Number offolds

Process Number ofLayers

n 1 2 2 2 2 1 2⋅ ⋅ ⋅ ⋅ ⋅ = ⋅. . .n

n

factors1 244 34 4 1 2⋅ n

3. Use questions to lead participants to finding a suitable viewing window.• What does x stand for in this problem? [Number of folds]• What values make sense for x in this problem? [Answers will vary.

Sample answer. Zero folds to 10 folds.]



• What does y stand for in this problem? [Total number of layers ofpaper]

• What values make sense for y in this problem? [Answers will vary.Sample answer. No layers to 2^10, which is 1024.]

4. Sample answer. The variable x stands for number of folds so zero to 10folds shows a few more folds than I could actually fold with my piece ofpaper. The variable y stands for number of layers, so zero to 60 will showabout all of the layers and the x-axis.

5. 1 2 262 14418⋅ =( ) , . You will have 262,144 layers after 18 folds.

6. Because there are 5 reams of paper, you want to know when the value of yis 5 times 500, or 2500. 1 2 2500⋅ =x . You need to fold at least 12 timesto get as thick as a box of paper. At the eleventh fold, the paper wouldonly be as thick as 2048 sheets of paper, so you would need to fold it onemore time, which gets you as thick as 4096 sheets of paper.

A numerically powerful exercise is have participants guess and check amore exact answer to 1 2 2500⋅ =x , using the home screen.

• Explain how this answer applies to this problem situation? [It makesno sense as the domain values must be whole numbers.]

• Usually we solve this kind of an equation how and when? [In a typicalcurriculum this type of exponential equation is solved using logarithmsin Algebra II and Precalculus. With the power of technology, using



graphs, tables, and the home screen, we can get good approximationsof the solution and build understanding about exponentialrelationships.]

Discuss the Activity:• What operation is being repeated in this problem? [multiplication]• What function results from repeated multiplication? [exponential

functions]• Earlier in the institute, we worked with another repeated operation,

repeated addition. What function results from repeated addition?[linear functions]

• What kind of graphs result from repeated addition? [linear]• What kind of graphs result from repeated multiplication?

[exponential]• Do you think this paper folding activity is an example of exponential

growth or decay? [growth]

Activity 2: Measure with PaperWork through the Activity with participants, discussing using the following.

1. Each sheet of paper measures approximately 0.004 inches. Themeasurement can be converted using dimensional analysis.21

1500

2500

0 004inchesream

reamsheets

inches per sheet inches per sheet⋅ = = . .

2. Develop the table in the same way as Activity 1.

Number of folds Process Thickness(inches)

0 0.004 0.0041 0 004 2. ⋅ 0.0082 0 004 2 2 0 004 22. .⋅ ⋅ = ⋅ 0.0163 0 004 2 2 2 0 004 23. .⋅ ⋅ ⋅ = ⋅ 0.0324 0 004 2 2 2 2 0 004 24. .⋅ ⋅ ⋅ ⋅ = ⋅ 0.0645 0 004 2 2 2 2 2 0 004 25. .⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ 0.1286 0 004 2 2 2 2 2 2 0 004 26. .⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ 0.256n 0 004 2 2 2 2 0 004 2. . . . .⋅ ⋅ ⋅ ⋅ ⋅ = ⋅

n

n

factors1 244 34 4 0 004 2. ⋅ n

3. y n= ⋅0 004 2.• How does this pattern and table compare with the table in Activity 1?

[In Activity 1, we were counting layers of paper. Here we aremeasuring how thick the folded paper is. The output (range) values in



both tables are doubling each time. The starting values for each tableare different.]

4. Use questions to lead participants to finding a suitable viewing window.• How might the table help you find a viewing window?

5. Sample answer. The variable x stands for number of folds so zero to 10folds shows a few more folds than I could actually fold with my piece ofpaper. The variable y stands for height of the folded paper in inches, sozero to 4.1 inches ( 0 004 2 4 09610. .⋅ ≈ ) will show most of the graph andthe x-axis.

6. 0 004 2 131 07215. .⋅ =( ) . 131 12 10 92÷ ≈ . so you have approximately 11feet of paper after 15 folds. This is one foot higher than a 10 foot ceiling.It is as high as double a 5.5 foot person.

7. If the Eiffel Tower is 1050 feet tall, its height in inches is1050 12 12 600⋅ = , . The equation you are solving here is0 004 2 12 600. ,⋅ =x . When you fold the paper 21 times, the value is8388.6. When you fold the paper 22 times, the height is 16,777, so it mustbe folded 22 times to reach the top of the tower.

Again, a numerically powerful exercise is to have participants guess andcheck a more exact answer to 0 004 2 12 600. ,⋅ =x , using the home screen.



• Does this approach have meaning in the problem situation? [No, thedomain values must be whole numbers.]

• How does the table and graph compare with the table and graph inActivity 1? [In Activity 1, we were counting layers of paper. Here weare measuring how thick the folded paper is. The output (range)values in both tables are doubling each time. The starting values foreach table are different. Look at both tables.]

• Find a viewing window to compare the graphs of y x= ⋅1 2 andy x= ⋅0 004 2. .

• Compare the two rules, y x= ⋅1 2 and y x= ⋅0 004 2. . [They both havethe same common multiplier. Each function value doubles with eachincrease by 1 of x.]

• Look at the table values again. How do you know that this data is notlinear? [Look at differences. There is no common difference.Remember, if there is not a common difference, the data is not linear.We did not obtain these table values from repeated addition.]

• What operation is being repeated in this problem? [Multiplication]• What kind of function results from repeated multiplication?

[Exponential function]• What is the common multiplier for Activities 1 and 2? [The common

multiplier is 2 because the values double each time.]• How can you find the common multiplier of 2 just by looking at the

values in the table? [Divide each y-value by the previous y-value toobtain the common multiplier.]

• What is the relationship between the two functions? [ y y2 10 004= . .This is a transformation, a vertical compression.]



Activity 3: RegionsFold a piece of paper again and show participants the regions formed. Beforeyou fold the paper, you have one region equal to the entire piece of paper.When you fold the paper once, you have two regions, each of which is one-half the entire sheet of paper. When you fold the paper again, you have 4regions, each of which is one-fourth the entire sheet of paper.Have participants complete the Activity. Then discuss using the following.

1. Develop the table in the same way as Activity 1.

Number offolds

Process Fraction of thePiece of Paper

0 1 11 1 1

2⋅

12

21 1

212

1 12

2

⋅ ⋅ = ⋅14

31 1

212

12

1 12

3

⋅ ⋅ ⋅ = ⋅18

41 1

212

12

12

1 12

4

⋅ ⋅ ⋅ ⋅ = ⋅1

165

1 12

12

12

12

12

1 12

5

⋅ ⋅ ⋅ ⋅ ⋅ = ⋅1

32n

1 12

12

12

12

1 12

⋅ ⋅ ⋅ ⋅ ⋅ = ⋅. . .

n

n

factors1 24 4 34 4

1 12⋅

n

2. yn

= ⋅1 12

• How does this pattern and table compare with the table in Activity 1?[In Activity 1, we were counting layers of paper. Here we are findingthe fraction of the piece of paper for each region formed by the fold.The output (range) values in the first table are being doubled with eachnew fold. In the second table, the fractions are being multiplied byone-half with each new fold. The starting values for each table aredifferent.]

3. Use questions to lead participants to finding a suitable viewing window.Sample:



4. Sample answer. The variable x stands for number of folds so zero to 8folds shows a few more folds than I could actually fold with my piece ofpaper. The variable y stands for fraction of the paper for each region, so–0.1 to 1 inches will show most of the graph and the x-axis.

5. 1 12

1512

0 0019531259

⋅ = = . . After the ninth fold, the region is one-five

hundred twelfth of the original paper. This is comparable to a sheet ofpaper from an entire ream of paper.

Note: Some calculators have the capability to change a decimal to afraction, which may be useful here.

6. The equation you are solving here is 1 12

1400

0 0025⋅ = =

x

. . When you

fold the paper 8 times, you get regions that are 1256

of the paper, which is

not enough. So you must fold it 9 times to get regions that are 1512

of the

paper, smaller than 1400

of the paper.

Again, a numerically powerful exercise is have participants guess and

check a more exact answer to 1 12

1400

0 0025⋅ = =x

. , using the home

screen.



• What meaning does this approach have in this problem situation?{None, the number of folds must be a whole number.]

• How does the table and graph compare with the table and graph inActivity 1? [In Activity 1, we looked at exponential growth, y x= ⋅1 2 .

Here we are looking at exponential decay, yx

= ⋅

1 12

. The output

(range) values in Activity 1 are doubling each time. The output(range) values in Activity 2 are halving each time The starting valuesfor each table are the same, 1.]

• Find a viewing window to compare the graphs of y x= ⋅1 2 and

yx

= ⋅

1 12

. [The graph of exponential growth, y x= ⋅1 2 , is

increasing. The graph of exponential decay, yx

= ⋅

1 12

is

decreasing.]

Activity 4: How Big is a Region?Work through the Activity with participants, discussing using the following.

1. A sheet of typing paper is 93.5 in2.2. Develop the table in the same way as Activity 1.

Number of folds Process Area of a Region0 93.5 93.51 93 5 1

2. ⋅ 46.75



293 5 1

212

93 5 12

2

. .⋅ ⋅ = ⋅23.375

393 5 1

212

12

93 5 12

3

. .⋅ ⋅ ⋅ = ⋅11.6875

493 5 1

212

12

12

93 5 12

4

. .⋅ ⋅ ⋅ ⋅ = ⋅5.84375

593 5 1

212

12

12

12

93 5 12

5

. .⋅ ⋅ ⋅ ⋅ ⋅ = ⋅2.921875

n93 5 1

212

12

12

93 5 12

. . . . .⋅ ⋅ ⋅ ⋅ ⋅ = ⋅

n

n

factors1 24 4 34 4

93 5 12

. ⋅n

3. yn

= ⋅

93 5 12

.

• How does this pattern and table compare with the table in Activity 3?[In Activity 3, we were looking at the fraction of the piece of paper foreach region. Here we are measuring the area of each region. Theoutput (range) values in both tables are halving each time. Thestarting values for each table are different.]

4. Use questions to lead participants to finding a suitable viewing window.Sample:

5. Sample answer. The variable x stands for number of folds so zero to 10folds shows a few more folds than I could actually fold with my piece ofpaper. The variable y stands for the area of a region in inches2, so –10 to100 inches2 will show most of the graph and the x-axis.

6. 93 5 12

0 091310

. .⋅

≈( )

. So you have approximately a tenth of a square

inch of paper after 10 folds. A small pill might measure 0.1 in2.



7. The equation you are solving here is 93 5 12

1 55 10 5. .⋅ = × −

x

. When you

fold the paper 23 times you get an area smaller than a plant cell.

Have participants guess and check a more exact answer to

93 5 12

1 55 10 5. .⋅ = × −

x

, using the home screen.

Discuss the Activity.• How does the table compare with the table in Activity 3? [The output

(range) values in both tables are halving each time. The startingvalues for each table are different. Look at both tables.]

• How does the graph compare with the graph in Activity 3? Find a

viewing window to compare the graphs of yx

= ⋅

1 12

and

yx

= ⋅

93 5 12

. .



• Compare the two rules, yx

= ⋅

1 12

and yx

= ⋅

93 5 12

. . [They both

have the same common multiplier, 12

. Each function value halves

with each increase by 1 of x. The y-intercepts, starting value, aredifferent.]

• How can you find the common multiplier of 12

by looking at the

values in the table? [Divide each y-value by the previous y-value toobtain the common multiplier.]

• What is the relationship between the two functions? [ y y2 193 5= . .This is a transformation, a vertical stretch.]

Discuss all 4 Activities.

Compare each table with the function that represents the table values. Notethat each function follows: y starting point common multiplier x

= ⋅ ( ) ory starting point common ratio x= ⋅ ( ) .• Which functions are increasing? [The functions where the common

multiplier or ratio is greater than 1 are increasing.]• What do we call increasing exponential functions? [exponential

growth]• Which functions are decreasing? [The functions where the common

multiplier or ratio is between 0 and 1.]• What do we call decreasing exponential functions? [exponential

decay]• How can you determine the base of the exponential function b in

y a bx= ⋅ from the table values? [Divide each y-value by the previousy-value.]

• How can you determine the y-intercept (starting point in the problemsituations), a, in y a bx= ⋅ from the table values? [The value of a isthe y-value when x = 0.]

Answers to Reflect and Apply1. y4, b2. y3, d3. y1, c4. y2, a5. ii, b6. i, a7. iii, c



Summary: Building on the work with repeated addition and linear functions, participantslook at repeated multiplication and exponential functions.

III. Nonlinear Functions 3.1 Exponential Relationships: Activity 1


Activity 1: Paper Folding

Fold a piece of paper in half. Fold it in half again. Continuefolding, filling in the table below.1.Numberof folds

Process Number ofLayers of

Paper0

1

2

3

4

5

6

n

2. Write a function for the number of layers of paper you willhave if you fold the paper n times.







Use the home screen, graph, and table to find the following:5. If you fold the paper 18 times, how many layers of paper will

you have? Write the equation. Show how you got yoursolution.

6. A box of paper is 5 reams of paper deep. A ream of paperhas 500 sheets of paper. About how many folds would youneed to be at least as thick as a box of paper? Show how youfound your solution.



Activity 2: Measure with Paper

A ream of paper measures approximately 2 inches thick.

1. If a ream is 500 sheets of paper, approximately how thick is apiece of paper?

2. Folding paper again, build the table and find a model.

Numberof folds

Process Thickness(inches)

0

1

2

3

4

5

6

n

3. Write a function for how thick the stack will be, in inches, ifyou fold the paper n times.







Use the home screen, graph, and table to find the following:6. If you fold the paper 15 times, how many inches of paper will

you have? Compare this measurement to something in theroom that has approximately the same measurement.

7. The Eiffel Tower is approximately 1050 feet tall. If you hada big enough piece of paper, how many folds would you needto match or exceed that height?



Activity 3: Regions

When you fold the piece of paper, you split the paper intoregions, bounded by the fold lines. What fraction of the piece ofpaper is each region formed? Complete the table below.


Numberof folds

Process Fraction of thePiece of Paper

0 1

1

2

3

4

5

6

n

2. Write a function for the fraction of a piece of paper for eachregion, if you fold the paper n times.







Use the home screen, graph, and table to find the following:5. If you fold the paper 9 times, what fraction of the piece of

paper is each region? Write your answer as a fraction. Givean example of a different situation where that fraction mightappear.

6. Your school has a paper confetti machine that cuts 8.5” by11” sheets of paper into about 400 pieces. What is the leastnumber of times you need to fold the paper to get regions thatare no larger than 1

400 of the piece of paper?



Activity 4: How Big is a Region?

A piece of paper typing paper measures 8.5” by 11” inches.

1. What is the area in inches2 of a piece of typing paper?


Numberof folds

Process Area of aRegion

0

1

2

3

4

5

n

3. Write a function for the area of a region, in inches2, if youfold the paper n times.







Use the home screen, graph, and table to find the following:6. If you fold the paper 10 times, what is the area of a region?

Compare this measurement to something in real life that hasapproximately the same measurement.

7. Some plant cells have an area of approximately 1 55 10 5. × −

in2. How many folds do you need to have a region with atleast that small of an area?

III. Nonlinear Functions 3.1 Exponential Relationships: Reflect and Apply


Reflect and Apply

Match each function with a column in the table and with agraph:

x y1 y2 y3 y41. y x= ⋅( )1 3

-2 9 0.56 45 0.11

-1 3 1.67 15 0.332. y

x

= ⋅5 1

30 1 5 5 1

1 0.33 15 1.67 33. y

x

= ⋅1 1

32 0.11 45 0.56 9

4. y x= ⋅( )5 3

a

b

cd

Match each function with a recursive routine and with a graph:

5. yx

= ⋅1024 1

4

6. y x= ⋅( )1000 1 08.

7. yx

= ⋅27 4

3

i. 1000, ENTER ANS*(1+0.08), ENTER . . .

ii. 1024, ENTER ANS*(0.25), ENTER . . .

iii. 27, ENTER ANS*(4/3), ENTER . . .

ab

c

III. Nonlinear Functions 3.1 Exponential Relationships: Student Activity


Student Activity: Recursion AgainOverview: Students use recursive routines on the home screen to explore three

exponential relationships.

Objective: Algebra I TEKS(d.3.C) The student analyzes data and represents situations involvingexponential growth and decay using concrete models, tables, graphs, oralgebraic methods.

Terms: interest, recursion, fractal


Procedures: Work together with students on Exercise 1.

1. Discuss the problem. Find the interest for one year numerically.

• How do we find the interest for one year? [multiply 1000 by 0.08 toget 80.]

• How do we find the total amount after one year? [add 1000 and 80,which can be written 1000 1000 0 08+ ⋅ . ]

• How much money would you have at the end of the first year?[$1080]

• How do we find the interest for year 2? [multiply 1080 by 0.08 to get86.40]

• How do we find the amount after two years? [add 1080 and 86.40,which can be written 1080 1080 0 08+ ⋅ . ]]

• How much money would you have at the end of the second year?[$1166.40]

• Using the distributive property, how can you rewrite 1000 1000 80+ ⋅and 1080 1080 0 08+ ⋅ . ? [1000 1000 0 08 1000 1 0 08+ ⋅ = +( ). . and1080 1080 0 08 1080 1 0 08+ ⋅ = +( ). . .]

• What pattern do you see? [The amount for each year is equal to theamount from the year before times 1 0 08+( ). .]



• Do you think this pattern will continue? Find the amount at the end ofyear three in two ways. [At the end of the second year, you had$1166.40 so the interest for year three is 1166 4 0 08 93 312. . .⋅ = .Therefore, the total amount at the end of year three is1166 4 93 31 1259 712. . .+ = , so you have $1259.71.. Using our pattern,the amount at the end of year three is 1166 4 1 0 08 1259 712. . .+( ) = , soyou have $1259.71.]

We can use the pattern, that the amount for each year is equal to theamount from the year before times 1 1 08+( ). , to investigate the scenariorecursively on the home screen.

Start students on the recursive routine and have them solve the remainingproblems.

a. $1080.00b. $1469.33c. $2158.90d. 10 yearse. 15 yearsf. $46,901.61g. {0, 1000}, {ANS(1)+1, ANS(2)*(1+0.08)}

2. Have students complete the Exercise and then discuss.a. 256 units2

b. 64 units2

c. 0.00390625 units2

d. stage 6e. {0, 1024}, {ANS(1)+1, ANS(2)*(0.25)}



Note: The fractal produced by this recursive process is called the SierpinskiTriangle.

3. Have students complete the Exercise.

In stage 1, you now have 4 segments that are 13

the length of the previous

segment. In other words, you have 43

of 27, which is 36.

In stage 2, you have 43

of each of the previous segments. In other words, you

have 43

of 43

of 27, which is 43

of 36, which is 48.

a. 36 unitsb. 48 unitsc. 359.5939643 unitsd. stage 15e. {0, 27}, {ANS(1)+1, ANS(2)*(4/3)}

Note: The fractal produced by this recursive process is called the KochSnowflake.

Summary: Using repeated multiplication in recursive routines, students gain intuition forthe exponential growth and decay patterns in interest and two fractals.



Student Activity: Recursion Again

1. Suppose you have $1000 in a savings accountearning 8% interest compounded annually. Solve eachproblem using recursion on the home screen of yourcalculator.

a. How much money would you have at the end of the firstyear?

b. How much money would you have at the end of five years?

c. How much money would you have at the end of ten years?

d. How long would it take you to double your money?

e. If you have about $3,172, how long has your money beeninvested?

f. If you wanted to retire in 50 years, how much money wouldyou have then?

g. Write the recursive routine that you used.



2. Given the following sequence of figures, solve each problemusing recursion on the home screen of your calculator.The area of original triangle is 1024 units2.

, , , . . .stage 0 stage 1 stage 2

a. What is the area of one of the smallest triangles in stage 1?

b. What is the area of one of the smallest triangles in stage 2?

c. What is the area of one of the smallest triangles in stage 9?

d. If the area of one of the smallest triangles is 14

, what is the

stage number?

e. Write the recursive routine that you used.



3. The length of the original segment is 27 units long. The

length of each new segment is 13

the length of the previous

segment.Stage 0

Stage 1

Stage 2

a. What is the total length of the figure in stage 1?

b. What is the total length of the figure in stage 2?

c. What is the total length of the figure in stage 9?

d. If the length of the figure is approximately 2020 units, whatis the stage number?

e. Write the recursive routine that you used.

III. Nonlinear Functions 3.2 Exponential Growth and Decay: Leaders’ Notes


3.2 Exponential Growth and DecayOverview: Participants find models for exponential growth and decay situations.


Terms: growth, decay, multiplier, rate, quotient, ratio, percent increase, percentdecrease



Recall the exponential relationships explored in the Student Activity:Recursion Again. If you did not do these problems with participants, divideinto three groups and assign each of the groups one of the problems toinvestigate.Use Transparency 1 to discuss the relationship between the constant multiplierand the percent increase or decrease.You can also use Transparency 2 to develop the relationship live, usingTransparency 1 as a guide.

• What is the relationship between the constant multiplier and thepercent increase or decrease? [The constant multiplier is one plus thepercent increase for exponential growth. It is one minus the percentdecrease for exponential decay.]

• What does this form, writing the constant multiplier as 1+ percentincrease or 1 – percent decrease, emphasize? [The percent increase orpercent decrease.]

• What is the multiplier for the interest earning problem? [1.08]• How can we write the constant multiplier for the interest earning

problem? [1.08 or 1+0.08]• So what is the percent increase? [8%]• What is the multiplier for the triangle problem? [0.25]• How can we write the constant multiplier for the triangle problem?

[1 0 75− . ]• So what is the percent decrease? [75%]

• What is the multiplier for the snowflake problem? [ 43

]

• How can we write the constant multiplier for the snowflake problem?

[1 13

+ ]

• So what is the percent increase? [33.3%]



In the previous 3.1 Exponential Relationships, participants were given asituation and, using a table, they found exponential functions to model thesituation. Now we will give participants percent increase or percent decrease,along with an initial amount and they can find exponential functions to modelsituations without relying on a table.Use Transparency 3 to bridge this gap from the table to the function:y starting amount rate x= ⋅ ( ) , where rate percent increase= +1 forexponential growth or rate percent decrease= −1 for exponential decay.

Activity 1: Exponential GrowthHave participants work together on the Activity and then discuss, using thefollowing.

1. Write a function for each offer.A. y x= +( )1000 1 0 04.B. y x= +( )1000 1 0 03.C. y x= +( )1000 1 0 065.

2. Compare the three offers in a table.Offer A – 4% Offer B – 3% Offer C – 6.5%

1 year $1040.00 $1030.00 $1065.002 years $1081.60 $1060.90 $1134.235 years $1216.65 $1159.27 $1370.09

10 years $1480.24 $1343.91 $1877.1420 years $2191.12 $1806.11 $3523.6530 years $3243.40 $2427.26 $6614.37

3. Compare the three offers graphically.

Ask participants to generate a list of possible questions to ask students aboutthe three offers. These might include:

• How long does it take to double your money with each offer?• How long does it take you to save $XX with each offer?• Compare the three offers over time.

Have participants solve one of their questions in at least three ways and sharetheir strategies.



Ask participants:• What information does the graph give you that the equations do not?

[The relative growth of the money over time.]• How could you find the multiplier for each offer by looking only at the

table of values? [You would divide the amount at the end of year 2 bythe amount at the end of year 1. This ratio is the common multiplieryou use to get the amount for the next year.]

• Could you divide any amount by the previous amount in this particulartable? [No, because the values in the input column do not change by 1consistently. In order to divide an outcome by a previous outcome toget the multiplier, the input values must increase by 1.]

• For example, to find the multiplier for Offer A, can you divide theamount after year 5, $1216.65, by the amount after year 2, $1081.60?

[No, 1216 651081 60

1 12..

.≈ which is the percent increase over 3 years, not the

annual percent increase.]

Activity 2: Exponential DecayExplain the scenario to participants. As you eat some substances, the amountin the bloodstream eventually reaches a maximum amount. Thereafter, thesubstance is flushed from the bloodstream, in these cases by a certain percentevery hour. We are interested in how much of the substance is remaining inthe bloodstream t hours after the substance has reached the maximum level.

1. Write a function for the amount of substance remaining in the bloodstreamt hours after the maximum level is reached.

Encourage participants to write the functions below in both forms shownto help them later recognize both forms.A. y t t= ⋅ −( ) = ⋅ ( )30 1 0 25 30 0 75. .B. y t t= ⋅ −( ) = ⋅ ( )30 1 0 15 30 0 85. .C. y t t= ⋅ −( ) = ⋅ ( )30 1 0 2 30 0 8. .

2. Compare the three situations in a table.Time afterMaximum

Level

Caffeine – Child Caffeine – Adult Vitamin

1 hour 22.5 25.5 242 hours 16.88 21.68 19.23 hours 12.66 18.42 15.364 hours 9.49 15.66 12.295 hours 7.12 13.31 9.83



3. Compare the three situations graphically.

Ask participants to generate a list of possible questions to ask students aboutthe three situations. These might include:

• How long does it take to halve the amount in the bloodstream witheach situation?

• How long does it take you to have XX amount in the bloodstream witheach situation?

• When will you have less than 1 mg of each substance in thebloodstream?

Have participants solve one of their questions in at least three ways and sharetheir strategies.

Ask participants:• How could you find the multiplier for each offer by looking only at the

table of values? [You would divide the amount present at the end ofthe second hour by the amount at the end of the first hour. This ratio isthe common multiplier you use to get the amount for the next hour.]

• Could you divide any amount by the previous amount in this particulartable? [Yes, because the values in the input column change by 1consistently. Any quotient of successive output values in this tablewill give you the multiplier for this situation.]

Answers to Reflect and Apply1. y4, b 5. ii, b2. y3, a 6. iv, c3. y1, c 7. i, a4. y2, d 8. iii, d

Summary: The big idea in these activities is for participants to write exponential modelsfor situations given an initial amount and a percent increase or percentdecrease.

III. Nonlinear Functions 3.2 Exponential Growth and Decay: Transparency 1


Transparency 1: The Common Multiplier

Sometimes it is more useful to write the multiplier as:exponential growth: (1 + percent increase)exponential decay: (1 – percent decrease)

Earning 8% interest on $1000

1 0 08+ .

Area of the smallest triangle


0 25 1 0 75. .= −

Total length of the segment

Stage 2

43

1 13

= +



Transparency 2: The Common Multiplier

Write the multiplier in terms of percent increase or decrease:exponential growth:exponential decay:

Earning 8% interest on $1000

Area of the smallest triangle


Total length of the segment

Stage 2



Transparency 3: Interest

Suppose you have $500 in a savings account earning 6% annualinterest.

We can think of 6% annual interest as a 6% annual growth rate.

Balance Process TotalAmount

0 500 $500.001 500 1 06+( ). $530.002 500 1 06 1 06 500 1 06 2+( ) +( ) = +( ). . . $561.803 500 1 06 1 06 1 06 500 1 06 3+( ) +( ) +( ) = +( ). . . . $595.514t 500 1 06+( ). t

y starting amount rate x= ⋅( )

where rate percent increase= +1 for exponential growth andrate percent decrease= −1 for exponential decay.

III. Nonlinear Functions 3.2 Exponential Growth and Decay: Activity 1


Activity 1: Exponential Growth

Suppose you have $1000 to invest. To simplify thecomparisons, consider only interest compounded annually.

A. Your credit union’s savings account is offering 4% interest.B. The corner bank’s savings account is offering 3% interest.C. A Certificate of Deposit (CD) is offering 6.5% interest.

1. Write a function for each offer.A.B.C.

2. Compare the three offers in a table.

1 year2 years5 years10 years20 years30 years

3. Compare the three offers graphically.

III. Nonlinear Functions 3.2 Exponential Growth and Decay: Activity 2


Activity 2: Exponential Decay

A person drinks a caffeinated soda and takes a vitamin. Thecaffeine and the vitamin in the bloodstream each reach amaximum level of 30 milligrams.A. Caffeine is flushed out of a child’s blood stream at a rate of

about 25% an hour.B. Caffeine is flushed out of an adult’s blood stream at a rate of

about 15% an hour.C. The vitamin is flushed out of a person’s blood stream at a

rate of about 20% an hour.

1. Write a function for the amount of substance remaining in thebloodstream t hours after the maximum level is reached.A.B.C.

2. Compare the three situations in a table.

Time afterMaximum

Level1 hour2 hours3 hours4 hours5 hours

3. Compare the three situations graphically.

III. Nonlinear Functions 3.2 Exponential Growth and Decay: Reflect and Apply


Reflect and Apply

Match each function with a table and a graph:

x y1 y2 y3 y41. y x= ⋅( )20 0 75.

-2 16.5 8.89 24.7 35.6

-1 18.2 13.3 22.2 26.72. y x= ⋅( )20 0 9.

0 20 20 20 20

1 22 30 18 153. y x= ⋅( )20 1 1.

2 24.2 45 16.2 11.3

4. y x= ⋅( )20 1 5.

ab

c

d

Match each function with a recursive routine and a graph:

5. y x= ⋅( )1000 1 09.

6. y x= ⋅( )500 0 7.

7. y x= ⋅( )1000 1 03.

i. 1000, ENTER ANS*(1+0.03), ENTER . . .

ii. 1000, ENTER ANS*(1+0.09), ENTER . . .

iii. 500, ENTER ANS*(0.9), ENTER . . .

iv. 500, ENTER ANS*(0.7), ENTER . . .

8. y x= ⋅( )500 0 9.

abc

d

III. Nonlinear Functions 3.2 Exponential Growth and Decay: Student Activity


Student Activity: On the WallOverview: Students use different sized paper to model exponential growth and decay

graphs.


Terms: exponential function, growth, decay

Materials: sticky notes, poster boards, large blank paper (for coordinate axes), markers,tape

Procedures: Do this as a whole class activity.

One half of the class will split into groups 1 – 4 and construct the respectivegraphs using sticky notes. The other half of the class will split into groups 1 –4 and construct the same graphs using poster board or other similar sizedpaper.

Ask the class to predict the difference in the graphs made with sticky notesand the graphs made with poster boards. Have them discuss their constructionstrategies.

Note: These constructions only simulate exponential growth and decay. Theyare not accurate graphs of exponential functions. The domain for thesephysical models is integers. The domain for exponential functions is all realnumbers. The idea here is an attempt to give students a geometric feel forexponential growth and decay by having students take slips of paper andphysically double them, triple them, halve them, and cut them into thirds.Then they place the slips on a large coordinate axes system to get a concretefeel for exponential graphs.

After the graphs are completed and hanging on the wall, have students do agallery tour and note similarities and differences among the graphs. Thenhave a class discussion about the activity.

Some points to bring out in the discussion include the following.• Why did the graphs of y x= 2 and y x= 3 using sticky notes contain

more in the first quadrant than did the graphs using poster board?[The poster board is so big that it did not allow students to stack manyin the first quadrant. The sticky notes are small enough to allow manymore to be stacked in the first quadrant.]



• Where else did the large size of the poster board compared to the

sticky notes limit the graph? [In the graphs of yx

=

12

and yx

=

13

,

the graphs in the second quadrant were limited.]• Why did the graphs of y x= 2 and y x= 3 using poster board contain

more in the second quadrant than did the graphs using sticky notes?[The poster board is big enough that it allowed students to halve andthird many more times in the second quadrant. The sticky notes are sosmall that students could not split the slips into as many pieces in thesecond quadrant.]

Sample graphs for y x= 2 :

Summary: Students gain intuition about exponential growth and decay as they physicallyproduce simulations of exponential functions using concrete materials.



Student Activity: On the Wall

Use the height of your slip of paper as a unit. Simulate thegraph of the following exponential functions by using pieces ofthe paper and partial pieces of the paper affixed to a large gridon the wall.Sketch your graph here:

1. y x= 2

2. y x= 3



3. yx

=

12

4. yx

=

13

III. Nonlinear Functions 3.3 Exponential Models: Leaders’ Notes


3.3 Exponential ModelsOverview: Participants find exponential models for given data sets.


Terms: growth, decay, ratio, multiplier, quotient



Participants will apply what they have learned about exponential equations tofind exponential models for data.

Activity 1: Population GrowthWork through Exercises 1 – 4 with participants.

1. Determine the growth rate by finding successive quotients as shownbelow:

Year Population (billions) Quotients1980 0 4.46 4.53/4.46=1.01571981 1 4.53 4.61/4.53=1.01771982 2 4.61 4.69/4.61=1.01741983 3 4.69

If you find quotients on the graphing calculator, you can average thequotients.

So the growth rate is about 1.0172, which is to say that the population isgrowing at about 1.72% per year.

2. To find an equation to model the growth using the ruley starting amount rate t= ⋅ ( ) , use years from 0 to 9 for 1980 to 1989. Thenthe starting amount (y-intercept) is 4.46. The rate is 1.0172.Therefore, the equation to model the growth is y x= ⋅ ( )4 46 1 0172. . .



• How can you use the table on your calculator to check the accuracy ofthe model? [See below. Look to see that the model returns close tothe same y-values as in the original data set.]

3. To predict the population in 1999, remember that we substituted years0 – 9 for 1980 – 1989. So the year 1999 is year 19 for our model. Findthe prediction in a few ways.

4. The year 2010 is year 30 for our model. Find the prediction in a fewways.

Have participants work on Exercises 5 – 9. Make sure participants understandthat they are to create a new model, using only the data in the new table.Discuss, using the following. Ask a group to present their work.

5. Determine the growth rate by finding successive quotients as shownbelow:

Year Population (billions) Quotients1980 0 5.28 5.37/5.28=1.01701981 1 5.37 5.45/5.37=1.01491982 2 5.45



If you find quotients on the graphing calculator, you can average thequotients.

So the growth rate is about 1.0143, which is to say that the population isgrowing at about 1.43% per year.

6. To find an equation to model the growth using the ruley starting amount rate t= ⋅ ( ) , use years from 0 to 9 for 1990 to 1999. Thenthe starting amount (y-intercept) is 5.28. The rate is 1.0143.So the equation to model the growth is y x= ⋅ ( )5 28 1 0143. . .

• How can you use the table on your calculator to check the accuracy ofthe model? [See below. Look to see that the model returns close tothe same y-values as in the original data set.]

7. To predict the population in 2010, remember that we substituted years 0 –9 for 1990 – 1999. So the year 2010 is year 20 for our model. Find theprediction in a few ways.

8. Using the year 1998 as year 0 for our model, the year 2010 is year 12 inour model. The population reported in 1998 was 5.92 billion so this is thestarting point. The model is y x= ⋅ ( )5 92 1 0133. . . Find the predictions in afew ways.



• Why do the three predictions for the population in 2010 differ? [Eachprediction was found using a model with a different growth rate.]

• Compare the growth rate in the 80’s with the rate in the 90’s. [Thegrowth rate in the 80’s was about 1.72%. The growth rate in the 90’swas about 1.43%. The growth rate decreased.]

• How does the growth rate reported by the UN in 1998 compare withthe two rates you found? [The growth rate reported in 1998 was 1.33percent, which is less than the two that we found. In fact the world’spopulation growth rate has been declining for a few years. See belowand the Transparency.]

Year Population Growth Rate1962 3,136,197,751 2.191970 3,707,610,112 2.071980 4,456,705,217 1.701990 5,283,755,345 1.561998 5,924,574,901 1.31

• How do the growth rates shown in the Transparency compare to therates you found?

• What might be some factors influencing the UN to predict lowergrowth rates in the future?

Activity 2: Cooling DownHave participants do the Activity. Then discuss, using the following. Ask agroup to present their work.

1. Find the cooling rate by taking successive quotients.

The temperature is decreasing by about 4% each minute. The cooling rateis about 0.96.

2. An equation to model the temperature decrease is y x= ⋅ ( )46 0 96. .



3. Use the model to predict when the temperature will be 5 ˚C.

Answers to Reflect and Apply1. A linear model for 1980-1989 is y x= +4 46 0 0822. . .

2. Compare the model and the data. The model returns slightly higher values(in billions.)

3. A linear model for 1990-1999 is y x= +5 28 0 08. . .

4. Compare the model and the data. The model returns slightly lower values(in billions.)



5. A linear model for the cooling data is y x= −46 1 56. .

6. The model returns slightly higher temperatures.

7. Answers will vary.

Summary: The big idea in these activities is to find the multiplier from data fromexponential situations by taking successive quotients. If successive quotientsare constant, then an exponential model is reasonable. With the multiplier andthe y-intercept, participants can find exponential models for data.

III. Nonlinear Functions 3.3 Exponential Models: Transparency


Transparency: World Population Trends

World Population Size:past estimates and medium-, high- and low fertility variants,

1950-2050 (billions)

*Source: United Nations Population Division, World Population Prospects: The 1998 Revision, forthcoming

.

III. Nonlinear Functions 3.3 Exponential Models: Activity 1


Activity 1: Population Growth

The Population of the World (in billions)

Year Population(billions)

1980 0 4.461981 1 4.531982 2 4.611983 3 4.691984 4 4.771985 5 4.851986 6 4.941987 7 5.021988 8 5.111989 9 5.20

*1998 Revision of the official United Nations world population estimates and projections

1. Determine the growth rate.

2. Find an equation to model the population growth.

3. Use your model to predict the population in 1999.




The Population of the World (in billions)

Year Population(billions)

1990 0 5.281991 1 5.371992 2 5.451993 3 5.531994 4 5.611995 5 5.691996 6 5.771997 7 5.851998 8 5.921999 9 6.00

*1998 Revision of the official United Nations world population estimates and projections

Create a new model, using the data in this table.

5. Determine the growth rate.

6. Find an equation to model the population growth.


8. In 1998, the United Nations reported the current populationgrowth rate as 1.33 per cent. Create a model using thisinformation and the population for 1998 in the table. Usethis model to predict the population in 2010.



Activity 2: Cooling Down

A student placed a hot cup of water in the freezer and recordedthe following temperatures at the indicated times.

Time (min) 0 1 2 3 4 5 6 7 8 9Temperature(C˚)

46 44 42 41 39 37.5 36 34.5 33 32

1. Find the cooling rate.

2. Find an equation to model the temperature decrease.

3. Use your model to predict when the water will be about 5˚C.

III. Nonlinear Functions 3.3 Exponential Models: Reflect and Apply


Reflect and Apply

Although population and cooling data is often modeled withexponential models, a linear model can be quite accurate forshort sections of data.

1. Find a linear model for the world population data for 1980-1989.

2. Use the table on your graphing calculator to check the modelfor accuracy for the years for 1980-1989.

3. Find a linear model for the world population data for 1990-1999.

4. Use the table on your graphing calculator to check the modelfor accuracy for the years for 1990-1999.

5. Find a linear model for the cooling data.

6. Use the table on your graphing calculator to check the model.

7. What do you think about the appropriateness of either model?

III. Nonlinear Functions 4.1 Bounce It: Leaders’ Notes


4.1 Bounce It!Overview: Collecting three sets of data from a bouncing ball experiment, participants

find appropriate models and justify their choices.

Objective: Algebra I TEKS(b.1.B) The student gathers and records data, or uses data sets, to determinefunctional (systematic) relationships between quantities.(b.1.E) The student interprets and makes inferences from functionalrelationships.(c.1.A) The student determines whether or not given situations can berepresented by linear functions. (d.3.C) The student analyzes data and represents situations involvingexponential growth and decay using concrete models, tables, graphs, oralgebraic methods.

Terms: linear function, quadratic function, exponential function, parameter,acceleration

Materials: balls, data collection devices, motion detectors, graphing calculators

Procedures: Participants should be seated at tables in groups of 3 – 4. Do all of theActivities with participants, using the data that you collect. After you haveworked through an Activity with them using your data, have them completethe Activity with the data that they collect.

Activity 1: Collect the DataDescribe the procedure for collecting the data and demonstrate by dropping aball under a motion detector.

1. Encourage participants to predict the graph of the ball’s distance from thefloor versus time.

Math Note: The data collected by the motion detector is actually the distancefrom the motion detector versus the time. The program we used thentransformed the data to the distance from the floor versus the time.

2. It is important that the graph shows at least 5 good bounces.3. This sample data is from a racket ball.



Activity 2: A BounceDo the Activity with participants, using your collected data.

1. A quadratic function is an appropriate model. To answer the questionwhy, you can look at second differences in your data, which should befairly constant. There will probably be some glitches around where theball hits the floor, because you are finding differences between two sets ofquadratic data (2 different bounces).

2. Find a model with participants.• How can we fit a quadratic to the first complete bounce? [A sample

follows.]

Graph y x= 2 over the original graph. Reflect over the x-axis by graphingy x= − 2 .

Trace to the vertex of the first complete bounce. Shift right 0.86 (the x-coordinate of the vertex). Shift up 2.241 (the y-coordinate of the vertex).

Now guess and check the stretch factor.

Does -16 have any particular significance? It is because of theacceleration due to the force of gravity in the physics position equation

d at vt d= + +12

2 , where a is the acceleration due to gravity, which is

equal to -32 ft /sec2 or -9.8 m/sec2.

4. We reflected the parent function y x= 2 over the x-axis, horizontallytranslated it, vertically translated it, and vertically stretched it.



5. Now have participants find quadratics to model some of the other bounces.You may want to graph equations for all the bounces if time permits.

Have participants complete the Activity using the data they collected in theirgroups.

Activity 3: Bounce Height versus Bounce NumberDo the Activity with participants, using your collected data.

1. Do not fill in the maximum height of bounce number 0. We will figurethat height later.Sample data:

BounceNumber

Maximum Height ofBounce

01 2.2412 1.7093 1.3364 1.015 0.7786 0.607

2. Do not erase the original data, as you may need it later. Put the data into 2different lists. If your calculator allows you to name lists, this may be atime to do so.

3. An exponential function is an appropriate model.Have participants look at first and second differences to determine that thedata is neither linear nor quadratic. Note that both the first and seconddifferences decrease, they are not constant.

Take successive quotients. Note that the quotients do not continuallyincrease or decrease, but sort of cluster around 0.77.



4. To find an appropriate exponential model, we need to figure the height ofbounce number 0. By finding the quotients above, we found the commonmultiplier.• How can you use the common multiplier to work backwards to find

the height of bounce number 0? [Findmaximum height of bounce

multiplier# 2 . For our sample data, 2 241

0 772 91.

..= .]

So for our sample data, the starting point, is 2.91 and the commonmultiplier is 0.77, so our model is y x= ⋅ ( )2 91 0 77. .

Have participants complete the Activity using the data they collected in theirgroups.

Activity 4: Bounce Height versus Drop HeightDo the Activity with participants, using your collected data.

1. Use your data from the table in Activity 3. Use the height you found forbounce number 0 for the first drop height.

Drop Height Bounce Height2.91 2.2412.241 1.7091.709 1.3361.336 1.011.01 0.7780.778 0.607

2. Have participants predict what the scatter plot will look like before yougraph it.



3. To find an appropriate model, first note that the domain values do not have

a common difference. Remember, for data to be linear, ∆∆

yx

should be

constant. Find ∆∆

yx

for consecutive points and you should find ∆∆

yx

to be

fairly constant.

Our sample data found ∆∆

yx

in L6.

4. So, since ∆∆

yx

is fairly constant, we use a point (2.91, 2.241) and

∆∆

yx= 0 76. (from mean(L6)) in y m x x y= −( ) +1 1 to get

y x= −( ) +0 76 2 91 2 241. . . . (Stretch the line y x= by 0.76, horizontallytranslate it right 2.91 and vertically translate it up 2.241).

Have participants complete the Activity using the data they collected intheir groups.

Discuss Activities 2 – 4.� How do you determine if a linear model is appropriate for data? [Look

for a constant ∆∆

yx

, by taking first differences.]

� How do you determine if a quadratic model is appropriate for data?[Look for constant second differences.]

� How do you determine if an exponential model is appropriate for data?[Look for constant quotients.]



Note: Our purpose here is for participants to learn more about linear,quadratic, and exponential functions. Participants must know the importantdifferences between linear, quadratic, and exponential data, use thesedifferences to choose a model, and then they adjust parameters in the modelsto approximate good fits for data. This helps participants make connectionsbetween and differentiate among the three functions.Our purpose is not to teach statistical analysis. Do not get bogged down instatistical discussions.

After you have completed all 4 Activities, split the group into smaller groupsand work on the Student Activities as time allows. Have groups present theirwork.Note: The Student Activities 1 – 2: Pattern Blocks and Throw Up! arefinding quadratic models for data. The Student Activities 3 –4: RadioactiveDecay and Pendulum Decay are finding exponential models for data.

Answers to Reflect and Apply

1. y x= 0 0526 2. . The model is quadratic because, as seen below, the seconddifferences are relatively constant.



2. y x= −( )23000 1 0 15. or y x= ( )23000 0 85. . The model is exponentialbecause the first and second differences are not constant, but the quotientsare, as seen below.

3. y x= −16 7 0 007. . . The model is linear because first differences areconstant.

Summary: Collecting three sets of data from a bouncing ball experiment, participantsfind appropriate models and justify their choices.

III. Nonlinear Functions 4.1 Bounce It: Activity 1


Activity 1: Collect the Data

1. Read the directions below.Predict the graph of the distanceof the ball from the ground versustime.

2. Using a motion detector, a data collection device, and anappropriate program, do the following.

� Hold the motion detector at least 0.5 meters above theball.

� Drop the ball and let it bounce under the motion detector.� Collect distance data for about 5 seconds.� Collect data for a least 5 good bounces.� Repeat if necessary.

3. Sketch the resulting graph:

0.5 m



Activity 2: A Bounce

1. Choose the first complete bounce on the graph. What kind offunction would be an appropriate model for this data? Why?

2. Find a model. Write the function.

3. Sketch the data and the model.

4. List the transformations you used to find the model.

5. Choose another complete bounce and find a model for thatdata.



Activity 3: Bounce Height versus Bounce Number

1. Use the trace feature to find the maximum height for eachfull bounce. Do not fill in the height for bounce number 0.

BounceNumber

Maximum Heightof Bounce

0123456

2. Make a scatter plot of (bounce number, maximum height) inan appropriate viewing window. Sketch it. (Do not lose theoriginal data!)

3. What kind of function would be an appropriate model?Why?

4. Find an appropriate model. Write the function. Sketch itabove.



Activity 4: Bounce Height versus Drop Height

1. Using the data from Activity 3, fill in the table.

Drop Height Bounce Height

2. Make a scatter plot of (drop height, bounce height) in anappropriate viewing window. Sketch it. (Do not lose theoriginal data!)

3. What kind of function would be an appropriate model?Why?

4. Find an appropriate model. Write the function. Sketch itabove.

III. Nonlinear Functions 4.1 Bounce It: Reflect and Apply


Reflect and Apply

Find a model for each data set and justify your choice.

1. When a car stops, thebraking distance dependson the speed of the car.

2. A car loses value eachyear. This is calleddepreciation.

Speed(mph)

BrakingDistance

(ft)

Year Car’sValue

10 5 0 $23,00020 21 1 $19,55030 47 2 $16,61840 84 3 $1412550 132 4 $12,00660 189 5 $10,20570 258

*after the driver has observed anobstacle and has begun braking

3. The air temperature outside of a plane depends on the altitudeof the plane.Altitude

(m)Temperature

(C˚)0 16.7

1000 9.72000 2.73000 −4 3.4000 −11 3.

III. Nonlinear Functions 4.1 Bounce It: Student Activity 1


Student Activity 1: Pattern BlocksOverview: Students identify functional relationships using pattern blocks.

Objective: Algebra I TEKS(b.1.B) The student gathers and records data, or uses data sets, to determinefunctional (systematic) relationships between quantities.

Terms: trapezoid, rhombus, triangle, hexagon

Materials: pattern blocks, graphing calculator


Explain to students that the Transparency of Hexagons is to be used with eachof the patterns.

Build the first Hexagon with trapezoids. Then have students complete the restof the Activity.

Note: The hexagon number is also the unit length of a side.

1. It takes 2 trapezoids to build the first hexagon.2. It takes 8 trapezoids to build the second hexagon.3.

Hexagon Number Number of Trapezoids1 22 83 18n 2 2n

4. Scatter plot:

5. y x= 2 2. To determine the parent function for the model, students can findsecond differences to determine that a quadratic function is an appropriatemodel, because second differences are constant.

Have students use the homescreen, tables, and graphs to find the answers.6. 128 trapezoids7. 14th hexagon



8.Hexagon Number Number of Rhombi

1 32 123 274 485 75n 3 2n

9. Scatter plot:

10. y x= 3 2 . Again, look at second differences to determine that a quadraticfunction is an appropriate model because second differences are constant.

Have students use the homescreen, tables, and graphs to find the answers.11. 243 trapezoids12. 13th hexagon13.

Hexagon Number Number of Triangles1 62 243 544 965 150n 6 2n

14. Scatter plot:

15. y x= 6 2 . Again, look at second differences to determine that a quadraticfunction is an appropriate model because second differences are constant.

Have students use the homescreen, tables, and graphs to find the answers.16. 294 triangles17. 15th hexagon

Summary: Using pattern blocks to build hexagons, students find patterns and writefunction rules to model the situation.



Student Activity: Transparency of Hexagons



Student Activity 1: Pattern Blocks

1. How many trapezoids does it take to build the first hexagon?2. How many trapezoids does it take to build the second

hexagon?

3. Fill in the table.Hexagon Number Number of Trapezoids

123

n

4. Create a scatter plot of (hexagon number, number oftrapezoids) and sketch.

5. Find a model for the data.

Use your model to find the following in at least 3 ways:6. How many trapezoids do you need to build the 8th hexagon?7. If you use 392 trapezoids, what hexagon number did you

build?



How many rhombi does it take to build the hexagons?

8. Fill in the table.

Hexagon Number Number of Rhombi123

n

9. Create a scatter plot of (hexagon number, number of rhombi)and sketch.


Use your model to find the following in at least 3 ways:11. How many rhombi do you need to build the 9th hexagon?

12. If you use 507 rhombi, what hexagon number did you build?



How many triangles does it take to build the hexagons?

13. Fill in the table.Hexagon Number Number of Triangles

123

n

14. Create a scatter plot of (hexagon number, number oftriangles) and sketch.


Use your model to find the following in at least 3 ways:16. How many triangles do you need to build the 7th hexagon?

17. If you use 1350 triangles, what hexagon number did youbuild?



Student Activity 2: Throw Up!Overview: Students explore their own projectile motion problem and find an equation to

model the height of the ball versus time.

Objective: Algebra I TEKS(b.1.B) The student gathers and records data, or uses data sets, to determinefunctional (systematic) relationships between quantities.

Terms: acceleration, velocity, speed,

Materials: balls, stop watches, graphing calculators

Procedures: Students should work in groups of 3 – 4.

Discuss each team member’s responsibilities as you demonstrate the Activity.Throw the ball in the air and note that the timer should start timing when theball leaves the thrower’s hand and should end timing when the ball hits theground.Have students estimate, before they throw:� How long do you think the ball will be in the air? [Students’ estimates

will vary, but they will probably be much too high.]� How high do you think the ball will go?� With what initial velocity do you think you threw the ball?

Send the groups out to collect the data. When they return, help groupsunderstand how to guess and check their initial velocity given the time theycollected. An example follows.

1. When we threw a racket ball, we timed that it was in the air for 2.55seconds.

2. Substituting 5 for v0 , h t v t= − + +16 520

3. We estimated that our thrower threw at 50 ft/sec, so h t t= − + +16 50 52 .This initial guess is just to give students a starting place.

4. Students will need to find an appropriate viewing window and may needto adjust it as well as the initial velocity.



5. The maximum height of our throw was 28.58 feet. Students can use thecalculator to find the maximum of the function, use the graph to zoom inon the maximum, use the table to zoom in on the maximum, or they couldeven use the home screen to guess and check until they found themaximum.

• What is another method of finding the initial velocity, given the time?[The equation to solve is 0 16 52

0= − + +t v t , for t = 2 55. . Solve

0 16 2 55 2 55 520= − ( ) + ( ) +. .v for v0 . v ft

sec0 38 839≈ . ]

Extension Questions:Assuming the initial height is 5 feet, answer the following for other throws.

• How high did the ball go if its airtime was 4.5 seconds?• What was the initial velocity if the maximum height was 90 ft?• What will be the maximum height of a ball that reaches an altitude of

30 feet after half a second?



• How long would the ball be in the air if Mark McGuire hit a ballvertically and it went up to a height of 1200 feet?

• How high would the ball go if Sammy Sosa hit the ball with an initialvelocity of 200 feet per second?

• What would be the initial velocity of a ball hit vertically by JuanGonzalez if it had an airtime of 10 seconds?

• Which ball went higher, Mark’s, Sammy’s, or Juan’s?• What initial velocity would you need if you want the ball to go as high

as the Sears tower (approximately 1454 feet)?• How long would it take for this ball to complete its flight?• Does it take longer for the ball to go up to its maximum height or to

come down from its maximum height?

Collect the times and maximum heights from the various groups. Note thatthe longer the ball was in the air, the higher its maximum height.Congratulate the highest thrower.

Summary: Using technology, students approximate the velocity with which they threw aball. This helps students gain intuition for velocity and also for the quadraticmodel.



Student Activity 2: Throw Up!

Go outside or to a place with a high ceiling.� The thrower throws a ball straight into the air as high as

he/she can.� The timer times how long the ball is in the air, from the

time it leaves the thrower’s hand until it hits the ground.� The recorder records the time.

The position equation for the height of the ball is

h at v t h= − + +12

20 0, where a ft

sec= 32 2 , v0 is the initial velocity,

and h0 is the initial height.

1. How long was the ball in the air?

2. Assuming that the thrower released the ball at a height of 5ft., what is the position equation for the throw?

3. How fast do you think the thrower threw the ball? In otherwords, what was the ball’s initial velocity? Estimate andsubstitute this value into the position equation.

4. Graph your equation in Exercise 3. Does it show the ball inthe air for the correct amount of time? Adjust your estimatefor the ball’s initial velocity until it shows the ball in the airfor approximately the correct amount of time. Write yourequation and sketch the graph.

5. Find the maximum height of the ball. Describe your method.



Student Activity 3: Radioactive DecayOverview: Students use graphing calculators to simulate the radioactive decay of radon

gas. They find a model for the data and use the model to predict.

Objective: Algebra I TEKS(b.1.B) The student gathers and records data, or uses data sets, to determinefunctional (systematic) relationships between quantities.(d.3.C) The student analyzes data and represents situations involvingexponential growth and decay using concrete models, tables, graphs, oralgebraic methods.

Terms: decay, radioactive



Discuss briefly radioactive decay – that radioactive substances decay over aperiod of time. This means that it turns into a different material. Differentradioactive materials decay at different rates over time. Not all of the materialdecays at the same time, but a certain percentage decays in a certain period oftime. Carbon-14, a radioactive isotope found in living material, decays soslowly that scientists use it to date fossils. The half-life of Carbon-14 is 5730years.

Explain to students that they will be simulating the radioactive decay of thegas radon, which decays at a rate of about 16.7% per day. Students will find17 random integers out of 100 to approximate the 16.7% decay rate.

Explain the data collection procedure to students. Have students workthrough the activity in groups.

Activity:

1. Encourage students to predict.

2. Sample data.First find 17 random integers between 0 – 99 for Day 1.



Day Units Remaining0 100

0 1 2 3 4 5 6 7 8 9 1 8510 11 12 13 14 15 16 17 18 19 220 21 22 23 24 25 26 27 28 29 330 31 32 33 34 35 36 37 38 39 440 41 42 43 44 45 46 47 48 49 550 51 52 53 54 55 56 57 58 59 660 61 62 63 64 65 66 67 68 69 770 71 72 73 74 75 76 77 78 79 880 81 82 83 84 85 86 87 88 89 990 91 92 93 94 95 96 97 98 99 10

Next find 17 random integers between 0 – 99 for Day 2. Another way todo this is to store the generated integers into a list and then sort the list.This makes it easier to cross off the numbers because they are in order andalso it makes it easier to spot duplicates, which are ignored.

Day Units Remaining0 100

0 1 2 3 4 5 6 7 8 9 1 8510 11 12 13 14 15 16 17 18 19 2 7220 21 22 23 24 25 26 27 28 29 330 31 32 33 34 35 36 37 38 39 440 41 42 43 44 45 46 47 48 49 550 51 52 53 54 55 56 57 58 59 660 61 62 63 64 65 66 67 68 69 770 71 72 73 74 75 76 77 78 79 880 81 82 83 84 85 86 87 88 89 990 91 92 93 94 95 96 97 98 99 10



After repeating, our sample data looked like:Day Units

Remaining0 1001 852 723 644 565 516 437 348 319 28

10 23

3. For our sample data:

4. For our sample data:

5. For our sample data, the multiplier is 0.8642 and the number of units ofradon at day 0 is 100. So our model is y x= ⋅100 0 8642. . Compare thetable values for the model to the data.

• How does this compare to the 17% that we used to simulate the 16.7%decay rate? [It should be fairly close. Try the model y x= ⋅100 0 17. .Graph and compare table values.]



• What do you think would happen if we combined all of the data in theclass? [The data should get closer to the theoretical model with adecay rate of closer to 17% because of the law of large numbers. Tryit and see.]

6. Have students answer using at least 3 methods.

7. Have students answer using at least 3 methods.

8. Originally we had 100 units of radon. So we are looking for when thereare half of that or 50 units of radon. For our sample data, there are 50units left between the 4th and the 5th days. So the half life is between the4th and the 5th days. The real half life for radon gas is approximately 3.8days.

Summary: Students use the power of technology to simulate radioactive decay. Theyfind a model for the exponential decay data and use the model to answerquestions.



Student Activity 3: Radioactive Decay

Radon is a radioactive gas that decays at a rate of about 16.67%a day. This means that after a day, the amount of radon gaspresent is 83.33% of the original amount.

Simulate the decay of 100 units of radon gas as follows.� Use your graphing calculator to randomly choose 17

integers between 0 and 99. Mark off each listed integer inthe grid. These represent the decayed units of gas. Ignoreany repeated integers.

� Count the remaining units (not crossed off) and record thisin the table.

� Continue to do this until you have completed the table.

1. Predict the graph of (day, units remaining).



2. Fill in the table.Day Units

Remaining0 100

0 1 2 3 4 5 6 7 8 9 110 11 12 13 14 15 16 17 18 19 220 21 22 23 24 25 26 27 28 29 330 31 32 33 34 35 36 37 38 39 440 41 42 43 44 45 46 47 48 49 550 51 52 53 54 55 56 57 58 59 660 61 62 63 64 65 66 67 68 69 770 71 72 73 74 75 76 77 78 79 880 81 82 83 84 85 86 87 88 89 990 91 92 93 94 95 96 97 98 99 10

3. Create a scatter plot of (day, units remaining).

4. Find the multiplier for the decay of radon by takingsuccessive quotients.

5. Find a model for the data using the multiplier and the numberof units of radon at day 0.



Use your model to find the following in at least 3 ways.

6. How much radon is left after 15 days?

7. If there are 5 units of radon left, how many days havepassed?

8. What is the half-life for radon gas? In other words, after howmany days will there be half of the original amount of radonleft?



Student Activity 4: Pendulum DecayOverview: Students collect data of the maximum swing of a pendulum versus time and

find a model for the data.

Objective: Algebra I TEKS(b.1.B) The student gathers and records data, or uses data sets, to determinefunctional (systematic) relationships between quantities.(d.3.C) The student analyzes data and represents situations involvingexponential growth and decay using concrete models, tables, graphs, oralgebraic methods.

Terms: decay, pendulum, exponential function, quotients

Materials: graphing calculators, data collection devices, motion detectors, soda cans,string, meter sticks


Demonstrate the procedure for the experiment.

Points for the experiment:� The soda can should hang straight, not crooked.� The motion detector should be at least 1.5 feet from the maximum

swing of the pendulum.� The motion detector should be at the same height from the floor as the

hanging can.Notes on the program:The program collects data about 30 times for about 4 seconds each. It findsthe maximum distance that the pendulum swings in each of those periods. Itcalls each of the periods “a swing” therefore there are about 30 “swings”graphed.

Have students complete the data collection in their groups. Have them repeatif necessary.1. Encourage students to predict.2. Sample data:

3. The successive quotients cluster around 0.9. For our sample data, themean of the successive quotients is 0.90853 and the maximum distance for



swing number 0 is 0.14857. So the model for our sample data isy x= ⋅ ( )0 1486 0 9085. . .

4. In other words, how many swings would you expect before the distance ishalf of the original distance?For our sample data, the original distance was 0.1486. So,0 1486 2 0 0743. .÷ = . Thus we need to find for what swing number wasthe distance was 0.0743. Solve 0 0743 0 1486 0 9085. . .= ⋅ x graphically, onthe home screen, in a table, etc.The half life occurs between the 7th and the 8th swing.

Summary: As a pendulum swings, its swing decays. After collecting this distance data,students apply their knowledge of exponential functions to find a model forthe data.



Student Activity 4: Pendulum Decay

Construct a pendulum by tying a piece of string about 1 meterlong through the pull-tab of a half-empty soda can and fixing theother end of the string to a solid point 1.5 – 2 meters off thefloor.Place a motion detector about 0.5 meters from the maximumswing of the pendulum at the same height as the pendulum atrest.Run the program PENDULUM, which will collect data and plot(swing number, maximum swing of the pendulum). Gently pullback the pendulum about 50 cm and release it.

1. Predict a graph of the results.

2. Sketch the resulting graph.


4. What is the half-life of the pendulum swing?

Calculator Programs

TEXTEAMS Algebra I: 2000 and Beyond 477

LINETBL

FnOff

PlotsOff

ClrDraw

GridOn:AxesOn

randInt(–5,5)�Y

randInt(–5,5)�X

randInt(–3,3)�B

randInt(–10,10)�TblStart

randInt(1,5)�∆Tbl

(Y/X)�A

"B+AX"�Y⁄

DispTable

LINEGRPH

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PlotsOff

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GridOn:AxesOn

randInt(–5,5)�Y

randInt(–5,5)�X

randInt(–3,3)�B

(Y/X)�A

"B+AX"�Y⁄

ZDecimal

ACTSCRS

FnOff

PlotsOff

ClrDraw

GridOn:AxesOn

randInt(–5,5)�Y

randInt(–5,5)�X

randInt(–3,3)�B

(Y/X)�A

"B+AX"�Y⁄

ZDecimal

PENDULUM

31�N:ClrHome

Send({0})

Send({1,11,2,0,0,0})

Disp "SWING, ENTER"

ClrList L€

Pause

For(J,1,N)

Send({3,.04,99,0})

Get(L⁄)

min(L⁄)�L€(J)

End

1-Var Stats L€

Q⁄-(Med-Q⁄)¯/(Q‹-2Med+Q⁄)�K

K-L€�L€

seq(X,X,0,N-1)�L⁄

Plot1(Scatter,L⁄,L€,␣)

ZoomStat

Calculator Programs


JUMPIT

Full

ClrHome

Disp "CHECKING THE"

Disp "CALCULATOR-CBL"

Disp "LINK CONNECTION."

{1,0}�L⁄

Send(L⁄)

{0}�L€

Lbl M

{7}�L⁄

Send(L⁄)

Get(L€)

If dim(L€)=1 and L€(1)=0

Then

ClrHome

Disp "***LINK ERROR***"

Disp "RE-INSERT THE"

Disp "LINK CORD

Disp "CONNECTORS"

Disp "FIRMLY"

Disp ""

Disp "THEN PUSH ENTER"

Pause

Goto M

End

Disp ""

ClrHome

Output(6,1," STATUS: O.K."

Output(8,10,"[ENTER]")

Pause

Full

ClrHome

GridOff:FnOff :PlotsOff

Disp "TURN ON THE CBL"

Disp ""

Disp "STAND BETWEEN","PROBE AND LIGHT","ON FLOOR."

Disp ""

Disp "PRESS [ENTER]"

Pause

ClrHome

Disp "JUMP WHEN READY"

{1,0}�L⁄

Send(L⁄)

{1,1,1}�L⁄

Send(L⁄)

{3,0.01,88,2,1,.2,10,0,1}�L⁄

Send(L⁄)

Get(L€)

Get(L⁄)

round(L⁄,2)�L⁄


ZoomStat

Calculator Programs


CMOVE

Full

ClrHome

Output(4,1," CBR MOTION")

Output(8,10,"[ENTER]")

Pause

Normal

Connected

Full

Func

Float

RectGC

GridOff

LabelOff

PlotsOff

FnOff

ClrDraw

CoordOn

AxesOn

ClrHome

Menu("COLLECT

DATA?","YES",H,"QUIT",3)

Lbl 3

ClrHome

Stop

Lbl H

1�D

Disp "MOVE IN FRONT OF"

Disp "THE CBR TO MAKE"

Disp "A DISTANCE-TIME"

Disp "PLOT."

Output(8,1," [ENTER]")

Pause

{1,0}�L⁄

Send(L⁄)

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ClrHome

Disp "***LINK ERROR***"

Disp "PUSH IN THE LINK"

Disp "CORD CONNECTORS"

Disp "FIRMLY THEN HIT"

Disp "[ENTER]."

Pause

Goto M

Else

Full

ClrHome

Full

PlotsOff

FnOff

Func

AxesOn

0�Xmin

40�Xmax

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10�Ymax

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Text(4,1,"D(FT)")

Text(51,81,"T(S)")

Text(1,30,"HIT [ENTER]")

Text(7,34,"TO START")

Pause

Text(1,30,"

")

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0�Xmin

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4�Xmax

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PlotsOff

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DispGraph

Text(4,1,"D(FT)")

Text(51,81,"T(S)")

StoreGDB GDB6

0�U

0�V

Stop

2000 and beyond

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