algebra concepts

Algebra ConceptsAlgebra ConceptsAn Interactive Learning System...

Ventura Educational Systems 1999 All Rights Reserved

www.venturaes.com

for Macintosh and Windows Computers

Additional support materials foreducators are available online at:http://www.venturaes.com

Tools for Active Teaching and Active Learning -2- 1999 Ventura Educational Systems

Teachers GuideAlgebra ConceptsCopyright Notice

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ISBN 1-57116-034-5

Copyright 1999 All Rights Reserved

Ventura Educational Systems910 Ramona Avenue Suite E

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www.venturaes.com

1999 Ventura Educational Systems -3- Tools for Active Teaching and Active Learning

Teachers Guide Algebra Concepts

Table of Contents

Overview............................................................................................4

Credits................................................................................................5A Note to Teachers.............................................................................6Introduction........................................................................................7Curriculum and Content.....................................................................8An Overview of the Algebra Concepts Learning System.................11A Conceptual Framework for Algebra Concepts...............................11Materials...........................................................................................12System Requirements......................................................................12Getting Started.................................................................................12File Menu.........................................................................................15Activities Menu.................................................................................16Topic Menu.......................................................................................17Options Menu...................................................................................19

Instructions for Algebra Concepts Activities.....................................20Overview.................................................................................20Lesson....................................................................................21Comprehension Check...........................................................23Identification Game.................................................................24Scrambler................................................................................25Probe......................................................................................26Quiz Machine..........................................................................28Glossary..................................................................................29Integer Practice.......................................................................30Prime Factorization.................................................................31Algebra Tool Kit.......................................................................32Solving for X............................................................................37Scoreboard.............................................................................38

Terms Listed by Topic and Sequence..............................................39

Activity Pages...................................................................................40


Teachers GuideAlgebra Concepts

Overview

This educational software product provides an introduction to computers and computersystems:

Topics: Activities:

Variables and Expressions OverviewReal Numbers LessonsSolving Equations Comprehension CheckPolynomials Identification GameFactoring Scrambler

ProbeMaterials: Quiz Machine

Glossary1 Mac/Win CD-ROM Integer Addition PracticeTeachers Guide and Prime FactorizationSupplementary Worksheets Algebra Tiles

Number LineFeatures: Function Plot

Solving for XMenu Driven FormatSelf-Paced LessonsInteractive Database of Essential Algebra VocabularyChallenging GamesScholastic QuizzesAlgebra TilesNumber Line ActivitiesFunction Plotting Tool

A menubar provides easy access to the main features of AlgebraConcepts.

Menubar


Teachers Guide Algebra ConceptsCredits

Software Design Ventura Educational Systems

Instructional Technology Fred Ventura, Ph.D.and Programming

Editor Marne Ventura, M.A.

Dr. Fred Ventura is an experienced classroom teacher and has taught elementary, secondaryand college levels. He holds a doctorate in education from the University of California, andpresents workshops for educators on the instructional uses of microcomputers.

Marne Ventura is also an experienced classroom teacher and holds a masters degree inreading and language development from the University of California. As a seminar leader,Marne Ventura has assisted many teachers in learning about the educational opportunitiesthat can be derived from the use of microcomputers in the classroom.

Our publications include:Algebra Concepts GraphPowerAll About Light & Sound Hands-On Math SeriesAll About Matter Holidays & SeasonsAll About the Solar System HyperCard Projects for KidsAlphaSmart Projects HyperCard Projects for Language ArtsAnatomy of a Fish HyperCard Projects for Math TeachersAnatomy of a Shark HyperCard Projects for Multicultural EdBalancing Act HyperCard Projects for TeachersBorders, Certificates & Awards Insect WorldCharts, Grids & Forms Internet Projects for TeachersChemAid Kooky CharactersClassroom Critters Life Cycle of a Sea LampreyClip-Art for Elementary Teachers Marine InvertebratesClip-Art for Math Teachers Probability ToolkitClip-Art for Science Teachers ProtozoaCoordinate Geometry Puzzle LogicEarthworm School DaysFetal Pig SensesGeometry Concepts StatesGeometry Toolkit VisiFrog

Additional CD-ROMs

Many schools have more than one computer and to effectively use educational softwarerequire additional legal copies of a program. Additional program disks are available for use ina computer lab. For information on additional program disks contact Ventura EducationalSystems at (800) 336-1022. There is a 30 day warranty on original program disks. If for anyreason a program disk becomes defective within 30 days of the date of purchase, VenturaEducational Systems will replace it at no charge.


Teachers GuideAlgebra ConceptsA Note To Teachers...

The secondary mathematics curriculum traditionally has been designed to provide studentswith the opportunity to develop skills and acquire knowledge that would be important inadulthood and to prepare students to enter careers that would not require a high level ofmathematical ability. Today the context of mathematics instruction needs to be much broader.The accelerated pace at which modern society produces technological change requires thathigh school graduates be prepared for careers in which mathematics will be very important.In order to be successful in many careers students will need to be confident in theirmathematical abilities. They will need to be able to solve problems, to communicatemathematical ideas and to think logically.

The National Council of Teachers of Mathematics (NCTM) in the Curriculum and EvaluationStandards for School Mathematics calls for the establishment of a framework for a corecurriculum in grades 9-12 that reflects the needs of all students. The second standard of theNCTM document addresses the need for the mathematics curriculum to...

q include the continued development of the language and symbolism tocommunicate mathematical ideas so that all students can

q reflect upon and clarify their thinking about mathematical ideas andrelationships;

q formulate mathematical definitions and express generalizations discoveredthrough investigations;

q express mathematical ideas orally and in writing;

q read written presentations of mathematics with understanding;

q ask clarifying and extending questions related to mathematics they haveread or heard about;

q appreciate the economy, power and elegance of mathematical notation andits role in the development of mathematical ideas.

NCTM Standards 1989 National Council of Teachers of Mathematics

Algebra Concepts is a computer software product that is intended to help students developtheir ability to understand algebra and communicate algebraic concepts. It is specificallydesigned to create a learning environment where the vocabulary of algebra can be learned inan enjoyable way. Algebra Tiles give students the ability to use a computer as a tool forexploring algebraic ideas. These programs are most effective if the students work in groupsof two so that the discussion of mathematical ideas is facilitated. Comments and feedbackon the effectiveness of this program are always welcome.


Teachers Guide Algebra ConceptsIntroduction

Algebra Concepts is an interactive learning system which has been designed to provide 7thgrade enrichment through adult level instruction in mathematics. Several approaches to algebrainstruction are combined in the design of this educational software package. Algebra Conceptscontains Lessons with a Comprehension Check, an Identification Game, a Probe, a ScramblerGame, a Probe, a Quiz Machine and a Glossary which is used with each topic. Severalexploratory modules including Integer Addition Practice, Prime Factorization, Algebra Tiles,Number Line, Function Plot, and Solving for X, which all serve to extend the usefulness of theprogram.

With the Algebra Concepts program each topic is studied by reading a lesson, taking acomprehension test, using a game where the object is to identify a term or key concept ofintroductory algebra, a Probe where terms, key concepts and descriptions are accessed, or aquiz where practice matching terms and key concepts is provided. With the IdentificationGame students practice recognizing representations of basic algebraic concepts usingdiagrams and symbolic notation as clues. It can be used as a discovery learning type ofactivity for students who are unfamiliar with the algebraic terms or for reinforcement after theconcepts have been presented in class. The Probe is a computerized reference system thatallows students to review terms and concepts and to read detailed descriptions that explainthe algebraic significance of a term or concept. The quiz presents a term and asks the studentto select the key concept associated with that term.

The main instructional goals of Algebra Concepts are given in these educational objectives:

1. To provide practice identifying and matching terminology related to introductoryalgebra. This computer based instruction unit graphically represents terms andkey concepts and motivates students to learn the term or key concept associatedwith an algebraic expression or process.

2. To incrementally build an understanding of fundamental concepts by providingan easy-to-use format for exploring algebra.

3. To support the development of a students sense of confidence in his/hermathematical ability by measuring and reporting the students progress towardunderstanding the topics presented in each unit using points and a percentagecorrect rating.

The computer can be used to assist the learning process by providing a wealth of informationand by providing a structure for experimentation.


Teachers GuideAlgebra ConceptsCurriculum and Content

Algebra Concepts covers the content recommend by national and state instructional resources.It supports the curriculum frameworks for California and other states. The content specificallyfollows the recomend guidelines provided by most of the state departments of education.Because an active learning approach is used, Algebra Concepts also supports performancestandards specific to the secondary math curriculum. The main topics covered by this learningsystem are listed below:

Variables and ExpressionsReal Numbers and Their PropertiesSolving EquationsPolynomialsFactoring Polynomials

Constructivist Experiences: Algebra Concepts is designed to provide a variety of learningexperiences using different approaches and creative teaching methods. One key purpose ofthe Algebra Tiles component of the program is to provide students with constructivistexperiences related to fundamental algebra concepts.

Cooperative Learning: The Algebra Tiles experience is appropriate for individuals or smallcooperative learning groups. Students can work together to share and compare their approachto finding the solution to a specific set of algebra problems that can be provided by the teacheror accessed on the internet.

Internet Support: http://www.venturaes.com/site/algebra_concepts/index.html.

Learning Styles: Research in education has found that students have a variety of learningstyles and modalities which must be considered when designing instructional approaches.Algebra Concepts uses many techniques which are well-suited to visual learners. For example,the Algebra Tiles module is an extremely effective way to make abstract algebraic ideas muchmore understandable. In addition, because interactive sound is used is several of the learningmodules, students who are auditory learners will be more engaged. Terms are pronouncedand algebra expressions are read. Kinesthetic learners are engaged through the use of thekeyboard and the mouse. Also, several of the activities suggested in the teachers guide andon the website involve a confluence of visual, auditory and kinesthetic learning approaches.

Independent Investigations: The lesson ideas and instructional activities provided in theteachers guide and through the internet support site encourage students into independentinvestigations of topics related to the application of the algebra concepts which are targetedby the scope of the program.


Teachers Guide Algebra ConceptsAn Open-Ended Approach: Providing for open-ended questioning in math may seem likean oxymoron when we consider that most problems have a single answer. However, givingstudents an opportunity to think divergently in math class is important as a way to sparkcreativity and stimulate insight. Algebra Concepts attempts to provide open-endedexperiences whenever possible. One example is the Solve for X activity where studentspick a number (1-10), a sign (+ or -) and an operation (+, -, x, ) in an attempt to solve for xin a randomly generated equation. The solution can be found several ways and studentsare encouraged to share the approach with others in a cooperative learning group.

Critical Thinking: Algebra Concepts engages the student in critical thinking and decisionmaking through a series of related activities. A key feature in the design of the Algebra Tilesmodule is the empowerment of the student to make connections between representations ofalgebraic expressions in three forms: algebra tiles, number lines, and graphs or functions.Critical thinking in a mathematical context is achieved when teachers pose open-endedquestions and encourage the students to use representations of their own design in theAlgebra Tiles work area to justify the answer they have given. An example of such a questionmight be: Prove that (3x+2)-(2x-4)=x+6

Steps in proof:

1.) Show 3x+2.

2.) Add zero (0) in theform of (+4) and (-4).


Teachers GuideAlgebra Concepts3.) Remove 2x-4 from the first set. The result is x+6.

This type of instructional approach will help students who are first learning to work withoperations with polynomials to internalize the underlying concepts and will better enablethese students to generalize and make inferences about algebra problems that they willencounter later in their high school and college academic careers. Internalizing fundamentalconcepts will facilitate a students ability to apply principles of algebra to new learningexperiences.

Educators will find that Algebra Concepts can be used in combination with other instructionalprograms and closely parallels the algebra curriculum taught in most secondary schools. Theprogram provides students with the opportunity to review and explore the concepts learned inmathematics classes. The format of the system is designed to make learning fun. Each topiccan be studied in a variety of ways. A quiz for each topic assesses the students mastery ofthe material and helps to provide reinforcement. The system maintains a record of the studentsperformance on a disk and can be utilized by the teacher during instructional planning.

Algebra Concepts is useful as a supplement for most introductory level algebra students.Much of the confusion that students feel in algebra is due to an inability to understand theterminology used to describe the fundamental rules of algebra. Algebra Concepts is an effectivetool for mastering the vocabulary of algebra. In a similar way, the program is effective forstudents who are learning English. The key words are read aloud by the computer, colorimages are used and several other instructional strategies are employed to help all studentsimprove.

With Algebra Concepts, students enjoy learning the ideas that are essential for success inhigher level math courses. The program is an effective way to introduce students to algebraicterms and concepts and can also be used as a tool to provide reinforcement.

Supplementary materials are provided in this manual and are designed to be used in conjunctionwith the computer activities. The supplementary worksheets may be duplicated for classroomuse and lab packs with multiple copies of the program disks are available from the publisher.


Teachers Guide Algebra ConceptsAn Overview of the Algebra Concepts Learning System

Algebra Concepts combines traditional and innovative instructional techniques in an easy-to-use learning system. Each topic can be studied in a variety of ways. For example, first timeusers can opt to begin with the Identification Game and learn the terminology and relatedconcepts in a discovery learning mode or may decide to use the Probe first. Computergraphics are used to illustrate key concepts and in many cases the graphics are interactive inways that help to illustrate a fundamental concepts.

The program is designed so that the students name, class period and performance on thecomprehension checks, identification games and quizzes can be saved. An unlimited numberof student scores can be saved. Student performance records can be printed. Supplementaryworksheets and other materials available on the internet are reproducible and can also helpteachers monitor each students progress and assess student achievement.

Text and graphics are used to convey to students an understanding of essential algebraconcepts. Activity pages are provided in this manual and are designed to be used in conjunctionwith the use of the computer and as follow-up. Teachers are encouraged to duplicate thesupplementary materials for classroom use.

In the quiz mode the computers random number generating ability is employed to generate aunique mix of possible answers. The quiz can be set up so that students match terms withconcepts or concepts with terms.

A Conceptual Framework for Algebra Concepts

Learning the terminology of algebra is an important goal because vocabulary is an essentialelement in building a foundation for mathematical knowledge. A strong foundation is necessaryfor success in the study of advanced mathematics where mathematical knowledge and skillsare put to use in a variety ways.

The study of mathematics is most exciting when students are able to proceed at their ownpace, taking time to explore concepts and experiment with the ideas that are being learned.

The philosophical approach taken in the design of Algebra Concepts is to provide a computerbased learning environment for studying terms and their related key concepts. In the AlgebraTiles module links can be made to a number line or function plot. This ability provides studentswith an opportunity to experiment with algebraic concepts and discover important relationships.After completing an activity using the correlated supplementary materials, the student has theopportunity to play a game or take a quiz. The results of the game and quiz are stored in thestudents file and can be reported in a window on the screen or on the printer.


Teachers GuideAlgebra ConceptsMaterials

The Algebra Concepts learning system includes these components:

q Algebra Concepts Mac/Win CD-ROMq Documentation: A Teachers Guide (soft cover book)q Reproducible Black Line Masters and Activity Sheetsq Documentation in Adobe Acrobat Page Description Format (.PDF)q Extended support materials on the internet

at http://www.venturaes.com/algebraconcepts/index.html

System Requirements

The minimum computer system configuration needed to use this program is given below:

q Macintosh LC or Higher, PowerMac, G3, iMac, iBook or equivalent Applecomputer, System 7.5 or 8.0 or higher operating systems. CD-ROM drivewith approximately 10 Mb of free space, color Macintosh monitor capable of256 colors.

q Intel 386, 486, Pentium or equivalent processor, Windows 95, 98 or higher,16Mb or RAM, CD-ROM, hard drive with approximately 60 Mb of free spaceto run from hard drive,color VGA or bettermonitor.

Getting Started

Macintosh: Insert the CD-ROM into thedrive. Double-click the icon labeledAlgebra Concepts.Installer if you wantto load the program on your hard drive.Drag the icon labeled RecommendedFiles to the appropriate hard drive. Afterthe installation process is complete,locate the folder labeled AlgebraConcepts on your computers harddrive, open it and double click theapplication icon to start the program.

Please consult your Macintosh Users Guide for more complete system operating instructions


Teachers Guide Algebra ConceptsWindows: Insert the CD-ROM into the drive. Windows 95 will autorun the installationapplication. Follow the on-screen instructions and choose to either install the application orrun if from the CD.

For more information on how to install and operate Windows programs see the informationthat was supplied with your computer and your operating system manual.

Legal Information: The purchaser of this program is entitled to use one copy of this programfor each license. Purchasing a single copy and installing it on multiple machines forsimultaneous use is prohibited by federal and international copyright laws. (Using multiplecopies on other computers or using more than one copy at one time is considered a copyrightinfringement. Additional licenses for this product are available and may be purchased byschools with computer labs.)

In both the Macintosh and Windows versions of Algebra Concepts, once the program hasbeen launched, a copyright notice screen opens and the title screen sequence is displayed.

Click on the last screen of the Title window sequence and the Student Identification windowappears. Enter the appropriate information and then click on the OK button to close theStudent Identification window and proceed to use the learning system.



Open can be selected toretrieve a score file that hasbeen previously saved. Scorefiles can be kept on a floppydisk, any of the computershard drives or the networkserver.

When a new student isentered, the scores currentlyin memory, if any, are erased.

If the Open button is selected from the New Student screen a standard dialog box will beshown for opening files. The Macintosh version is shown here. For Windows the CommonDialog Box is used. In eiither case opening a score file is just like opening a file in any otherapplication. Teachers may find it convenient to organize score files by class period. It isalso possible to use identification numbers rather than names.


Teachers Guide Algebra ConceptsFile Menu

The File menu contains the options New Student, Open, Save, Print and Quit. Open the Filemenu to reveal these choices and operations:

New Student...

The New Student option is used to enter the nameof the student using the program, the period, ifdesired, and the date. Selecting this option,entering student identification information andcompleting the operation by clicking OK, resetsthe scores for the activities. Cancel can be usedto stop this operation.

Open...

A record of each students performance can bestored on a disk in a file. The name of the file isentered prior to saving. To continue using theprogram and add new scores to an existing file,select Open and then select the appropriate filename from the list presented. A separate disk canbe used for storing student scores. There is no limitto how many students can use the program.

SaveThe save option writes the current scores to a file. The student may specify the file name.

PrintDuring the use of the program there are times when the Print option becomes available. Forexample, a certificate of completion can be printed at the end of the game and the scoreboardcan also be printed. To print a window select the Print option when the window is shown. If theScoreboard or Certificate windows are not opened and Print is selected, the program will printan activity sheet for the selected topic. Use the activity sheet to encourage students to write adescription of the algebra concept shown in the selected view.

QuitThe Quit option is used to exit from the program and return to the operating systems desktop.If scores have not been saved a warning dialog box will be presented.


Teachers GuideAlgebra ConceptsActivities Menu

The Activities Menu provides a variety of options. Some of the options are controlled by theselection that is made in the Topic Menu. The first option, Overview, presents a brief descriptionof the various activities in a window. The Identification Game, Probe and Quiz Machine presentinformation based on which topic has been chosen.

Overview Provides basic information aboutthe program.

Lesson Read about anatomical informationrelated to the selected topic.

ComprehensionCheck Test students ability to understand

and remember informationpresented in the lesson.

IdentificationGame Score points by accurately

identifying algebra term and relatedconcepts.

Scrambler This game challenges students tounscramble the letters in terms fromthe glossary.

Quiz MachineIn this activity students test theirability to match algebra terms andrelated concepts.

Probe The Probe links algebra terminologywith points on on a diagram for eachtopic.

Glossary The Glossary contains a list ofalgebra terms and definitions.

Integer Addition PracticeSelect this item to provide practiceadding and subtracting integers.

Prime FactorizationFind the prime factors for anynumber between 1 to 100.

Algebra Tool KitSelect this item to work with a set ofAlgebra Tiles.

Solving for XSelect this item to practice simplifyingequations to solve for X.


Teachers Guide Algebra ConceptsTopic Menu

The Topic Menu determines the content of the activities. The Algebra Concepts learning systemis divided into 5 topics which represent the important topics discussed in most introductoryalgebra courses. Each of the topics can be studied in a variety of ways. To select a topic, usethe mouse to open the Topics Menu and drag the pointer to the desired selection.

Variables and Expressions

This lesson presents basic information about how variables are represented by symbols inalgebraic expressions. The lesson demonstrates to students the use of Algebra Tiles. Themeaning of each tile is explained and examples show the expressions in algebraic terms andalso with tiles.

Here is one example:

x+4Real Numbers

In the lesson on real numbers, set theory is used to explain to students the different ways inwhich numbers can be classified. First the set of natural numbers are compared to the set ofwhole numbers. Later students learn that rational numbers including whole numbers andnatural numbers.

NaturalNumbers

WholeNumbers

RationalNumbers


Teachers GuideAlgebra ConceptsSolving Equations

In solving algebraic equations knowing which steps to take and when to take them is a criticalskill. The third lesson introduces students to all of the common techniques for solving equations.A number line is used to show that every number has an opposite.

Polynomials

The fourth lesson explains the differences between monomials, binomials, and trinomials.Algebra Tiles are used to help students realize how to recognize the number of terms in apolynomial by determining the number of different algebra tile shapes used to make thepolynomial.

2x-1

Two shapes - binomal

-x2 + 3x + 2

Three shapes - trinomal

6x2 + 3x = 3x(2x+1)

Factoring using Algebra tilesinvolves the process of arrangingthe tiles in a rectangle anddetermining the factors from theedges of the rectangle.

Factoring

The ability to factor polynomials iscritical for success in algebra. Thefifth lesson focuses on techniquesthat students can use to factorpolynomials. Emphasis is placedon finding the greatest possiblecoefficent of a number, as in thisexample:



The Descriptions option is implemented in two activities. When the Descriptions option isoff the Descriptions window is not shown while using the Identification Game and Quiz Machine.A check mark is shown to the left of the word Descriptions in the Options Menu. TheDescriptions and Sound options flip from on to off each time the item is selected from themenu. The default status for the Descriptions and Sound option is off. The fourth item inthe Options Menu is Term/Key Concept. This option has an effect in the Identification Gameand the Quiz Machine. Each time this option is selected it changes from Term to Key Conceptor from Key Concept to Term.

Scoreboard Choosing the Scoreboard option causesthe system to display the current scoresfor the Identification Game and the QuizMachine.

Sound At various points in the program soundsand beeps are used to enhance theprogram. In the Probe, terms arepronounced. When the sound option isactive, the sound option in the menuappears checked.

Options Menu

The Options Menu provides for several special features. To view the current level of performancechoose "Scoreboard. Sound is used in various parts of the program and the computer termsare pronounced when using the Probe. Sound can be turned on and off by selecting Soundfrom the Options Menu. When the sound is on a check mark appears to the left ofthe word Sound in the menu.

Descriptions Descriptions are the concise informationscreens which are presented when usingthe Probe. If this item is checked, theDescriptions are shown after each correctidentification in the Identification Game.



Snap to Grid The Snap to Grid option automatically aligns tiles as they are placedto an invisible grid.

Overview

The Overview option opens a window that explains the basic features of the program.Clicking anywhere on the window closes the Overview Window. Click the MoreInformation button to learn about the pedagogical features of Algebra Concepts.

The More Information button links to information especially forteachers about the instructional applications of the program. Inthis area the curriculum and content, constructivism, cooperativelearning, internet support, learning styles, the value of an open-ended approach, and critical thinking are discussed.

Term The object of the Identification Game can be either terms or keyconcepts. In the Quiz Machine the object can be either to matchterms to key concepts or key concepts to terms. Choosing this optionswitches (toggles) these options.

Speech When speech is checked the program will pronounce terms andevaluate expressions aloud.


Teachers Guide Algebra ConceptsLesson

The Lesson option starts a tutorial sequence for the selected topic. There are five uniquelesson sequences: Variables and Expressions, Real Numbers, Solving Equations,Polynomials, and Factoring.

Four buttons are used to control actions while in the lesson:

The Stop button terminates the lesson.

The Back button is used to back up one page.

The Forward button advances the lesson one page.

The Glossary button opens the Glossary Window so that the definition of a

term can be found.



Students should be encouraged to carefully read the lesson and to use the Glossary to findthe definitions of new math vocabulary terms. The general purpose of this software is toenable students to master the terminology and to understand the fundamental concepts relatedto algebra, so it should be appropriate for teachers to allow students to repeatedly access thelessons.

After completing each lesson students should measure their understanding by doing theComprehension Check activity. If a satisfactory score is acheived using the ComprehensionCheck, the student should then go on to the next activity as assigned by the teacher. If thestudents score is unstatisfactory, the student should read the lesson again and then repeatthe Comprehension Check.

Many teachers require a specific level of performance on the activities before allowing studentsto participate in a related math project. Some teachers will want to use the software as apreparatory unit before a long term project is started and will set a performance standardbased on the instructional purpose.


Teachers Guide Algebra ConceptsComprehension Check

The Comprehension Check can be used after reading the lesson. In the ComprehensionCheck Window, a question is presented at the top of the screen, and two or more possibleanswers are shown at the center of the screen. To select an answer, move the mouse pointerto the answer and click the button. The system will check the answer, report a score anddisplay the forward button. Click the forward button to continue the Comprehension Check

Questions in the Comprehension Check are presented in the standard multiple choice format.Students select an answer by clicking on the answer.

Click on the marker to select an answer.

As students work through thequiz, the scoreboard continuallyreports their progress.

When the quiz is completed, the Comprehension Check Scoreboard is presented. From thiswindow the students can choose to try the quiz again, or to read the lesson again. To quit andmove on to a different activity is also an option.

The main Scoreboard will also report the students final score on the Comprehension Checkfor each topic. It is reasonable to expect students to get a perfect score in this area since theycan retake the quiz.



Identification Game

The Identification Game is an activity designed to provide practice in identifying thealgebra terms and related key concepts for the selected topic. The program uses a

detailed graphic display and a pointer which marks an icon or graphic related to the term. Theobject of the game is to choose letters to spell the name of the term indicated by the pointer.The Option Menu can be used to change the object of the game and the way that the gameoperates.

A letter is chosen by selecting the letter buttons on the right side of the screen or by typing. Aseach letter is chosen it is evaluated by the system. If the letter is correct the position of theletter is shown in the word at the bottom of the screen. If the letter occurs more than one timein the term, each occurance is shown. Correct vowels earn 1 point and correct consonantsearn 3 points.

Incorrect letters result in the loss of 1 point. A bonus of 10 points can be earned by identifyingthe term before any letter guesses are made. If the bonus is attempted, one chance is givento enter the correct term. The scores earned in this activity are reported on the scoreboardand can be saved in a file using the students name.


Teachers Guide Algebra ConceptsThe indicator shows the location of the term to be identified.

Letter buttons are used to identify the term.

A prompt shows letters in the term as they are identified.

Scrambler

The Scrambler, as the name suggests, scrambles the letters in a term. The object ofthis game is to unscramble letters as quickly as possible to spell a algebra term,when given a definition of the term. Words are randomly selected from the glossary.

To unscramble a word, use the hand to grab a letter and drag it to the correct position in theset of blanks below the scrambled letters. If the letter is correct it will stay in the position. If itis incorrect the letter will return to its original position. As words are correctly unscrambledpoints are earned.

The Stop button is used to terminate this activity.



The Probe presents a graphic screen and links parts of the graphic to completedescriptions of the computer concept. Four buttons are used to control activitywhile in the Probe.

Probe

Control buttons are used to move forward andbackward through the list of terms and to showdetailed descriptions of key concepts.

To reinforce the study of algebra, themarked term and its related key conceptare shown on the screen.

After a topic has been selected from the Topic Menu access to related information on the topicis given in the Probe. Teachers can use the Probe to present information on the selectedtopic. The data and graphic used in the Probe changes when a different item is selected in theTopics Menu. Topics can be changed at any time during the use of the Probe. The Probe canbe used in a variety of ways to enhance the learning experience.


Teachers Guide Algebra ConceptsWhen the Sound option is checked in the Options Menu, the term shown in the Probe ispronounced. As students use the forward and backward arrows to move through the list ofterms, the indicator moves on the graphic, the term and key concept change and the term ispronounced.

The control buttons in the Probe are explained below:

The Back button is used to return to a previous term.

The Forward button is used to advance to the next term.

The Description button is used to display the informationscreen for the particular algebra term that is currentlyselected.

The Stop button is used to terminate use of the Probe.

Accessing a Description

The student may select theDescription button to opena window containing a morecomplete description of theterm and key concept. Clickthe close box on theDescription Window toresume your activity.


Teachers GuideAlgebra ConceptsQuiz Machine

The Quiz Machine presents a matching quiz for terms and key concepts. The way the Quiz ispresented can be changed by making choices in the Options menu. The object of this activityis to match a key concept with the algebra term. As in the example below, the challenge isto select the term for the key concept yields opposite of number. The correct answer isproperty of negative one, since multiplying a number by negative one yields the opposite ofthe number. Select this answer by clicking the mouse on the term.

After selecting an answer by clicking on one of the three choices,click on the check mark to confirm the selected choice.

If students are unsure of an answer, click the question mark to see thelocation of the term in the associated diagram.

To stop the quiz, click on the stop sign at any time. If students stop thequiz before completing it, the score is reset and they will need to startthe quiz from the beginning to get a score.


Teachers Guide Algebra ConceptsGlossary

The Glossary gives students the opportunity to read a definition of one of the mathematicalterms found in other parts of the program.

Click an arrow indicator to change the term by moving forward orbackward in the list.

Click on the Descriptions button to bring up a description of the term.

The Stop button is used to close the Glossary window and continuewith other activities.


Teachers GuideAlgebra ConceptsInteger Practice

The Integer Practice activity presents a set of problems for the students to answer. The goalis to get 10 problems right. Each time a problem is answered correctly, a star appears in therow at the bottom of the screen.

+1 +2 + 3 +4

-4 -3 -2 -1

The arrow controls are used to select aninteger for the answer. The differentpositions on the arrow increase ordecrease the answer.

The check button is used to enter theanswer. If the answer is correct, a star willappear. The object of the activity is to get10 stars.


Teachers Guide Algebra ConceptsPrime Factorization

The Prime Factorization of a number can be found by selecting a number and then clicking onthe Go button. The expanded version of the prime factorization is shown at the bottom of thescreen. Students should use this information to write the prime factorization as the product ofprimes raised to a power. In this example the prime factorization would be 26.

Use the number line to select a number between 0 and 100. Use thearrow keys to move one number at a time or click on the number line toselect a number.

Find the prime factorization by clicking on the Go button.


Teachers GuideAlgebra ConceptsAlgebra Tool Kit

Algebra Tool Kit provides an open-ended learning environment for exploring algebraconcepts. There are three main tools available in this exploratory work area: Algebra Tiles,Grids for Plotting Functions and Number Lines. A palette of tools at the top and left edge ofthe window provides access to the learning devices. The main purpose of these tools is toprovide teachers with a powerful way to help students develop a lasting understanding offundamental algebra concepts.

A single click on the eraser selects the eraser tool and changes the mouse toan eraser icon. When this tool is selected objects which have been placed inthe work area can be removed by clicking on them.

A double click on the eraser icon clears the entire work area of all icons.

The eraser icon:


Teachers Guide Algebra ConceptsThe hand tool is used to pick up and move tiles which have been placed in thework area.

For some procedures, espcially multiplication of binomials, it is convenient toorientate some tiles vertically and some horizontally. The flip tool changes theorientation of the tile.

The light bulb tool is used to make a frame around a set of algebra tiles. Thetiles are evaluated and the polynomial is displayed at the top of the frame.

Color for function ifframe is linked to agrid.

algebra tiles

polynomial

The cursor will change to this symbol when the light bulb tool is selected:

The flip tool mouse cursor shows a tile placed vertically:

There are six types of algebra tiles that can be placed anywhere in the workarea. Each tile has a related value. This chart shows the tool palette symbol,mouse cursor and value of the 6 tiles. The students should learn to associatered with negative values and black with positive as is commonly used inaccounting.

-1Tool Bar Icon Cursor

An open hand is used for the mouse cursor:

Algebraic Value

+1Algebra Tile

-x+x

-x2+x2


Teachers GuideAlgebra ConceptsThe Link Tool allows the student to link frames containing algebra tiles tonumber lines and grids.

The cursor for the Link Tool is a small blue square:

The Grid Tool allows the student to draw a Cartesian grid where an algebraicfunction can be plotted.

The icon for the Grid Tool is an x/y grid:

Grid lines can be shown or notshown using these icons.

The color of the line drawn on a number line or grid can be set by clicking onone of these four colors.

The cursor will change to a drop of color: Click on the frame display area to set the color.

The Number Line Tool allows the student to draw a number line in the workarea.


Teachers Guide Algebra ConceptsExample #1: In this example links have been made between two frames and a numberline.

The number line shows a jumpfrom 0 to +3 and from +3 to +1,thus the result.

+3+(-2)=+1

Example #2: A link has been drawn between a set of algebra tiles and a grid. The gridshows the plot of the function represented by the algebra tiles contained by the linkedframe.

The color of the plot of the functionis defined by the small box at thetop left of the algebra tile frame. Itcan be changed by click it with adifferent color drop.


Teachers GuideAlgebra ConceptsExample #3: Here two links have been drawn between two sets of algebra tiles and asingle grid. The color drop has been used to change the colors of the lines to red andgreen. The grid shows the plot of the function represented by the algebra tiles containedby the linked frame.

Example #4: In this example, the product of (x+1) and (x-1) has been found using tiles.First (x+1) is placed horizontally above an empty frame and then (x-1) is placed vertically.The product is found by multiplying each horizontal piece times each vertical piece.

(x+1)


Teachers Guide Algebra ConceptsSolving for X

Solving for X is an activity that is designed to help students learn the step involved insolving for x in common equations. When the window for this activity is opened a randomproblem appears at the top of the notepad. The student selects a positive or negativeinteger by clicking on buttons and then choses an operaton (+,i,x or ) to perform.

Step #1: Positive 2 is added to both sides of the equation.

Step #2: The next step is to divide both sides of the equation by 6.

Step #3: When x is found, a happy face appears in the window. Any subsequent mouseclick automatically generates a new problem.


Teachers GuideAlgebra ConceptsScoreboard

The Scoreboard choice in the Options menu opens a window showing the scores for thecurrent student. Note the Y2K compliant date field.

The Scoreboard window can be opened whenever theoption Scoreboard is active in the Options Menu. TheScoreboard reports the number of points earned using theComprehension Check, Identification Game and thepercentage score achieved on the Quiz Machine for eachtopic. When the Scoreboard window is displayed, Printcan be selected from the File menu to print a copy of theScoreboard on the printer.

When save is selected from the File menu, a file containingthe scores shown on the Scoreboard are written to the harddisk or floppy disk. If New Student is selected from the FileMenu, all the scores will be initialized. Open can be usedto retrieve a previously saved score file.



Variables and Expressions1. variable2. less than3. greater than4. equal5. not equal6. less than equal to7. greater than equal to8. value of a variable9. parentheses

10. brackets11. fraction bar12. simplifying an expression13. replacement set14. equation15. solution set

Real Numbers1. graph of a number2. origin3. negative integers4. positive integers5. absolute value6. opposites in products7. opposite of a sum8. property of negative one9. property of zero

10. closure property11. addition identity property12. multiplication identity13. distributive property14. opposites15. commutative property16. associative property

Terms Listed by Topic and Sequence

Solving Equations1. property of equality2. adding to both sides3. negative integers4. multiplication of equals5. multiplying both sides6. addition of the opposite7. rule for subtraction8. multiplying by a reciprocal9. rule for division

Polynomials1. monomial2. binomial3. trinomial4. base5. exponent6. coefficient7. exponents in multiplication8. power of a power rule9. power of a product rule

10. simplification11. order of operations12. factored form13. exponential form



Activity PagesTable of Contents

Activity pages to be used with the Probe:

Variables and Expressions......................................................42Real Numbers.........................................................................43Solving Equations...................................................................44Polynomials.............................................................................45Factoring.................................................................................46

Practice PagesReview of Real Number Properties.........................................47Evaluating Expressions...........................................................48Using the Commutative Property............................................49Evaluating Expressions...........................................................50Problem Solving: Solving Algebraic Expressions....................51Comparing Expressions..........................................................54Using Number Properties to Simplify Expressions..................55Working with Parentheses, Brackets and Fraction Bars.........57Locating Points on a Number Line..........................................58Adding Positive and Negative Numbers..................................59Working with the Distributive Property....................................60Reflecting on Reciprocals.......................................................61

Algebra Tiles: Introduction.......................................................62A Powerful Set of Tools in the Algebra Tool Kit........................63Algebra Tiles: Addition of Integers...........................................64Expressing Binomials with Algebra Tiles.................................65Evaluating Groups of Algebra Tiles.........................................66

Addition of Polynomials...........................................................68Subtraction of Polynomials......................................................70Multiplying Polynomials...........................................................71Factoring Polynimals...............................................................73


Teachers Guide Algebra ConceptsA First Look at Plotting Functions............................................74Which Quadrant?....................................................................75The Graph of a Linear Equation..............................................76Understanding the Slope.........................................................77Looking at the Effect of the b-term..........................................78

Quadratic Functions................................................................79Operations with Integers.........................................................80Table Mania.............................................................................81Graphing Linear Equations.....................................................82Leap into Graphing..................................................................83Equations with Two Variables..................................................84Solving for X............................................................................85Prime Factorization.................................................................88Multiplying Prime Factors........................................................89Finding the GCF......................................................................90Finding the LCM......................................................................91Systems of Equations.............................................................92Solving Problems Using Systems of Equations......................93Writing Your Own Examples of Problems................................95Graph Paper............................................................................96Glossary..................................................................................98Answers..................................................................................99Online Activities.....................................................................109

Name: Date:

Algebra Concepts - Activity Pages -42- 1999 Ventura Educational Systems

Select Variables and Expressions from the Topic Menu. From the Activity Menu select Probeto find out about each of the algebra terms for this topic. Write each term on one of the linesbelow. Put a number on the graphic to show the location of the indicator for each term.

1. ____________________

2. ____________________

3. ____________________

4. ____________________

5. ____________________

6. ____________________

7. ____________________

8. ____________________

9. ____________________

10. ____________________

11. ____________________


Name: Date:

Select Real Numbers from the Topic Menu. From the Activity Menu select Probe to find outabout each of the algebra terms for this topic. Write each term on one of the lines below. Puta number on the graphic to show the location of the indicator for each term.

1. ____________________

2. ____________________

3. ____________________

4. ____________________

5. ____________________

6. ____________________

7. ____________________

8. ____________________

9. ____________________

10. ____________________

11. ____________________

12. ____________________

13. ____________________

14. ____________________

15. ____________________

16. ____________________

Name: Date:


Select Solving Equations from the Topic Menu. From the Activity Menu select Probe to findout about each of the algebra terms for this topic. Write each term on one of the lines below.Put a number on the graphic to show the location of the indicator for each term.

1. ____________________

2. ____________________

3. ____________________

4. ____________________

5. ____________________

6. ____________________

7. ____________________

8. ____________________


Name: Date:

Select Polynomials from the Topic Menu. From the Activity Menu select Probe to find outabout each of the algebra terms for this topic. Write each term on one of the lines below. Puta number on the graphic to show the location of the indicator for each term.

1. ____________________

2. ____________________

3. ____________________

4. ____________________

5. ____________________

6. ____________________

7. ____________________

8. ____________________

9. ____________________

10. ____________________

11. ____________________

12. ____________________

13. ____________________

Name: Date:


Select Factoring from the Topic Menu. From the Activity Menu select Probe to find out abouteach of the algebra terms for this topic. Write each term on one of the lines below. Put anumber on the graphic to show the location of the indicator for each term.

1. ____________________

2. ____________________

3. ____________________

4. ____________________

5. ____________________

6. ____________________


Name: Date:

Review of Important Properties of Real Numbers

Write an example of each property. Use a, b and c as variables.

Associative Property (Addition)

a+(b+c)=(a+b)+c

Distributive Property

Identity for Multiplication

Commutative Property (Addition)

Addition of Opposites

Multiplying by the Inverse

Name: Date:


Evaluating Expressions

This table shows the values of six variables. Substitute the value of the variable given in thetable in each expression.

Write the value of each expression.

1. 3y+4 2. 4z+3

3. xyz 4. x+y+z

5. 4a-y 6. 3b+z

7. (x+y) 8. 2x+3y+4z

9. 3y+4c+1 10. 4a+xy

11. 2y+4abc 12. 8ab-c

Simplify the expression.

1. (7-3)+3+2 2. 5+(12-3)

3. 8+(3-1)+5 4. 1+2(4+6)

5. 3(5-2) 6. 3(2+6)-3

7. 3(5+2) 8. 6+(2+8)-3


Name: Date:

Using the Commutative Property

Use the commutative property to write an equivalent expression.

1. 3+x 2. 4y+9

3. 10x+3y 4. 2mn

5. 3mx+mn 6. 5a+2b

7. 8bc 8. abc

9. 9m+8c 10. 2x+y

11. xy+z 12. (3)(6)

Write an equivalent expression.

Name: Date:


Evaluating Expressions

Study each example and determine if the expressions are equivalent. Mark Yes if the state-ments are equivalent and No if they are not.

1. 3x+y and 3y+x.

2. mn+1 and 1+mn.

3. a+3b and 3b+3a.

4. 8+y and 8y+1.

5. 5m+y and my+5.

Yes no

Yes no

Yes no

Yes no

Yes no


Name: Date:

Problem Solving: Solving Algebraic Expressions

Solve each problem by substituting the values and evaluating each expression.

l

w a

b

c

The area of a rectangle is the product of the length and width.

1. If l=20 and w=15, find A. 2. If l=4 and w=8, find A.

3. If l=5.6 and w=3.5, find A. 4. If l=8.5 and w=9.0, find A.

5. If l=5 and w=3.5, find A. 6. If l=8.2 and w=9.3, find A.

A=lw

The perimeter of a rectangle is found by multiplying the sum of the length and width by 2.

1. If l=25 and w=19, find P. 2. If l=4 and w=8, find P.

3. If l=4 and w=9, find P. 4. If l=8 and w=2.5, find P.

5. If l=6 and w=3.5, find P. 6. If l=8.5 and w=4.7, find P.

P=2(l+w)

The area of a triangle is found by dividing the product of the length and width by 2.

1. If a=25 and b=10, find A. 2. If a=4 and b=8, find A.

3. If a=5 and b=8, find A. 4. If a=8 and b=2.5, find P.

5. If a=5 and b=3.2, find A. 6. If a=8.5 and b=4.7, find P.

A= ab2


The perimeter of a triangle is the sum of the three sides.

1. If a=5, b=10 and c=3, find P. 2. If a=4, b=0.8 and c=1.5, find P.

3. If a=5.2, b=1 and c=3 find P. 4. If a=8, b=3.4 and c=2.5, find P.

5. If a=1.5, b=3 and c=1 find P. 6. If a=6.5, b=8.4 and c=4.3, find P.

Evaluate each expression. Use these values for the variables.

d=4u=0

e=3v=2

f=5w=9

Variables

1. def. 2. 3d+4u 3. w-(e+v)

4. 3e+4v. 5. (fw)-e 6. 4d-(e+f)

7. u+e. 8. w(d+2e) 9. 3w-(u+e)

Use the values given in each problem to find the area and perimeter.

Ab c+( )

2h=

Find the perimeter of an isosceles trapezoid with these dimensions.

1. a=4 2. a=6.5 3. a=10 4. a=3.5b1=8 b1=10 b1=9 b1=8b2=5 b2=8 b2=6 b2=4

Find the area of an isosceles trapezoid with these dimensions.

5. h=3 6. h=3.5 7. h=10 8. h=3.5b1=9 b1=12 b1=9 b1=8b2=8 b2=10 b2=6 b2=4

P a b c+ +=

b2

a h

b1IsoscelesTrapezoid P=2a+b1+b2

(b1+b2)


Name: Date:

P 2 a b+( )=A b h=

a h

bFind the perimeter of a parallelogram with these dimensions.

9. a=3 10. a=3.5 11. a=10 12. a=2.5b=5 b=4.5 b=6 b=3.8

Find the area of a parallelogram with these dimensions.

13. b=7 14. b=5.5 15. b=10 16. b=2.6h=5 h=4.5 h=5 h=4.8

Write the symbols for the basic operations (addition, subtraction, multiplication and division)in the small circles to make a true statement.

8

7

3

2

7=start end

=

6

4 5 2

4 5 2

= =

start

start

end

end

Name: Date:


TrueF a l s e7.

TrueF a l s e6.

Comparing Expressions

Circle true or false for each expression.

3 1 TrueF a l s e3.

10 3-( )

4 20.875>

TrueF a l s e5.

3

4 1

3 4 3-( )

2 3( )>

3 4 2+( ) 2 1+( ) 7>


Name: Date:

Using Number Properties to Simplify Expressions

The operations of addition and multiplication have several important basic properties that canmake it easier to simplify expressions. These properties are assumed to be true and arecalled axioms or postulates.

The Axiom of Closure: If a and b are real numbers, then a+b is a uniquereal number and ab is a unique real number.

Simplify each expression to find a unique real number. Show your work.

1. 2. 3.

4. 5. 6.

7. 8. 9.

The Commutative Property: If a and b are real numbers, then a+b isequal to b+a and ab is equal to ba.

Usually addition and multiplication are performed in a left to right order.

The Associative Property: If a,b and c are real numbers, then (a+b)+cis equal to a+(b+c) and (ab)c is equal to a(bc).

34 23+( ) 12 10+( ) 23

8x1

3

43.45 12.8+

9 8( ) 17+ 3

8

1

5+ 3

1

4x9

1

2x2.5

1700 2349+ 1934x27 11

42

1

3x3

1

4 +

33 29 14 78 21 98 78+ + + + + +

3s 2t+( ) 5t+ 3s 2t 5t+( )+ 3s 7t+= =

4 x( ) 5a( ) 6 z( ) 4 5 6 a x z 120axz= =

Name: Date:


Some problems are easier to compute if the terms are in a different order or are grouped ina different way. For example:

25 47 25+ +

25 75 47+ +

is easier to compute if it is changed to:

and, 30 50( ) 2is simpler if the terms are grouped as shown:

38 50 2( )

Use the commutative and associative properties to write each of these expression in a waythat is easier to compute. Simplify the expression.

50 98 25 34 25+ + + + (23 x 25) x 4

1

3

5

8

2

3+ + 3

1

42

3

52

3

4+ +

2.5x 6.7x4( ) 60 45 35 40+ + +

Use the commutative and associative properties to simplify these expressions. For example:

3a( ) 4b( ) 3 4( ) ab( ) 12ab= =

4a( ) 9b( ) 5c( ) 3x 6y 9z( ) 2 x+ +

3b( ) 9a( ) 2b( ) 34 2s+( ) 8r( )

5t( ) a 3m( )

6b 4a 3b( )( ) 3 x 4 yy2( )( ) 2 x

1.

3.

5.

2.

4.

6.

1.

3.

5.

2.

4.

6.


Name: Date:

3- 2+( ) 81.2. 1 8-3. 8-

Working with Parentheses, Brackets and Fraction Bars

Parentheses, brackets and fraction bars are used to tell the order in which calculations shouldbe performed. Study these examples:

1. 3 7+( ) 5( )2 6( ) 2-

2.

3. 5010

5=

10 5

12 2-Simplify each expression.Remember to do the operationswithin the parentheses and bracketsfirst. Multiply or divide in a left-to-right order second and do additionor subtraction last.

2 3 x 4 x+( )2

1

23

1

3+

72. 3.

4. 5.

3 4-( ) 5+[ ] 6+

5 4-( )- 3+( )( ) 3 6 2+( )-+ 5x 3x 2x 4x+( )+[ ]

To simplify expressions with a fraction bar, first solve the expression above the bar, then theexpression below the bar. If possible divide the expression above the bar by the expressionbelow the bar.

1. 2. 3.3 x 2+( )x 2 x+( )

1 x 2+( )-

4 x 2-( )

15 12 9-( )

4.55 4.45+( )

4. 2 9+( ) 5-( ) 126 18 15-( )

5. 315

41

10+

5

13

43

1

4+

All over theworld the samerules for mathare used. Thatway everybodygets the same

answer!

1.

Name: Date:


3. 4.

2.1.

Locating Points on a Number Line

Every real number corresponds to a point on a number line. Simplify each expression andlocate the result on this number line.

3- 3-( )-( )

4 2-( )

5-( )- 2+( ) 2

5 3+( ) 9-( ) 69+

5 7 9 4-( )-( )+( )

9 3+( ) 7 9 4-( )-( )+( )

32 2-( )

5- 3( )-( )

1 1 2+( )-( )-

8-( )- 2+

Locate these numbers on this close-up of a sectionof a number line.Number Line

3 2.5-8 3 4+( )-

5 2 1+( )-

0.1 0.3-- 22 3.5-


Name: Date:

Use this numberline to simplifyeach expression.

6- 3- 2-( )+

31

2-

1

4- +

1- 3-( ) 4- 2+ +

4-

5 3-( )+5-( )+

2.

4.

6.

8.

2- 8+( ) 4-( )+

15- 22+( ) 3-( )+

4.5 3.2-( )+

6- 7-+

1.

3.

5.

7.

9. 10. 3- 8+

Adding Positive and Negative Numbers

Adding a positive number can be thought of as moving to the right on a number line. Movingto the left on a number line is the same as adding a negative number. To find the sum of (+3)plus (-9), start at +3 on the number line and move to the left nine units. The result is (-6). Tofind the absolute value of a negative number using the number line locate the positive numberthat is exactly the same distance from 0.

3- 7-+

Name: Date:


Working with the Distributive Property

a(b+c)=ab+ac and (b+c)a=ba+caUse the distributive property to simplify these expressions.

301

3

2

5+

1. 2. 3.

4. 5. 6.

7. 8. 9.

18t- 12t+

3 b 7+( ) 8+

15s 3-( ) s+

8 x 3y+( ) 14-

6a 7a+

10z 5 b z+( )+

10. 11. 12. 3a b 2c+( )

13. 14. 15.2 a 3-( ) 5b+4 2 a b 3c-( )

16. 17.5c c2 ab-( ) 6ab 4a b c-( )-

bc b 3bc c-( )-

b 3 b c c-( )

31

3a b-

3a 5b 8c+( )

a b c+( )

The distributive property is used in simplifying expressions. Therule states that for all real numbers a, b and c.


Name: Date:

Reflecting on Reciprocals

Each number in the set of real numbers hasa reciprocal except for 0.

a1

a 1=

1

aa( ) 1=

and

Study these examples of reciprocals:

5

81

3

5 1=

0.4 2.5( ) 1= xyz1

xyz 1=

1

ab2ab2( ) 1=

31

3 1=

Draw a line to match each expression on the left with its reciprocal on the right.

a

bc2

ab 3c+( )

ab

c

a bc+

c2

ab

ab

c2

1

a bc+

b

2a

bc2

a

c

ab

1

ab 3c+

2a

b

0.5 2-( )- 1=

1.

2.

3.

4.

5.

6.

Name: Date:


Algebra Tiles: Introduction

Algebra tiles are used to represent positive and negative values.

The small white square is used to represent +1 and the shaded square represents -1. Therectangle is used to represent the x term. A white rectangle represents x and a shadedrectangle represents -x. The large square represents x2. As with the other tiles, the whitesquare is positive x2 and the shaded square is -x2. Select Algebra Tiles from the ActivityMenu to open the Algebra Tiles window.

Positive

Negative

1 x x2

Eraser ToolErase a single tile, frame, grid or number line.Double-click erase everything.

Hand Tool Pick up a tile and move it.

Rotate Tool Rotate a tile 90.

Evaluate Tool Frame a group of tiles and do algebraicevaluation.

The evaluate tool is used to make aset in which the tiles within therectangle are evaluated as apolynomial. Evaluation rectangles canbe drawn when the tool is selected andthe cursor is a small light bulb.

A small cursor is used to indicate that tiles can be placed in the work area.


Name: Date:

A Powerful Set of Tools in the Algebra Tool Kit

The Algebra Tool Kit provides teachers and students with a work area that is specificallydesigned for exploring algebra concepts. In addition to the tiles, a grid tool and number linetool make it possible for teachers to creatively investigate mathematical relationships.

LinkToolUse this tool to draw a link between a tile frameand a grid or number line.

Grid ToolErase a single tile, frame, grid or number line.Double-click to erase everything.

Grid LinesClick on this icon to enable grid lines.

No Grid LinesClick on this icon to disable grid lines.

Color Drops Drop a color on a frame to set the color of the gridor number line plot.

NumberLine Tool

Use this tool to draw a number line. The numberline expands from the center.

A Sample of Linking Tiles to a Grid

The Algebra Tool Kit provides teachers and students with a work area that is specificallydesigned for exploring algebra concepts. In addition to the tiles, a grid tool and number linetool make it possible for teachers to creatively investigate mathematical relationships.

Name: Date:


2.

3.

4.

5.

6.

7.

8.

Algebra Tiles: Addition of Integers

In this activity you will build two sets of algebra tiles and find the sum.

This examples shows a method forsimplifying this expression byfinding the sum of two integers.

3 4-( )+

Practice Adding Integers3- 5+

6- 2+

2 2-( )+

4-( ) 3-( )+

5-( ) 1+

8 1-( )+

3-( ) 1-( )+

3-( ) 5+

3-( ) 2 2-( )+ +

2-( ) 3 3-( )+ +

3- 5+( ) 3-( )+

5-( ) 6 3-( )+( )+

4-( ) 5 3-( )+( )+

3-( ) 4 1-( )+( )+

-2-6 -6

-2-6+2

+2-6

-3

+6

-6+7

-2-4

1.


Name: Date:

Expressing Binomials with Algebra Tiles

A binomial is an expression with two terms.

3x2 4x+ 7x 4+

2x 1+ x2 1-

Algebra tiles can be used to represent binomials.

3x2 3x+

2x- 4+ x2- 2x-

2x2- x- x2- 4+

1. 2.

3. 4.

Name: Date:


Evaluating Groups of Algebra TilesAny group of algebra tiles represents a monomial, binomial or trinomial. For each problem,examine the set of algebra tiles shown. Decide if a monomial, binomial or trinomial isrepresented. Write an algebraic expression for the tiles.

Expression:

3x2 3x- 3+

Check one:m monomialm binomialm trinomialx

Expression:Check one:m monomialm binomialm trinomial






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Evaluating Groups of Algebra Tiles (page 2)






These groups of algebra tiles representmonomials, binomials or trinomials. Foreach problem, examine the set ofalgebra tiles shown. Decide if amonomial, binomial or trinomial isrepresented. Write an algebraicexpression for the tiles.

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Addition of Polynomials

You can add polynomials by combining two groups of algebra tiles.

2x2 4x- 2+x2- x 1-+

x2 3 x- 1+

Use algebra tiles tofind the sum of twopolynomials.

Use the Algebra Tool Kiit to build the two groups of algebra tiles shown in each problem.Write the polynomial represented in the space at the top of the enclosing frame.

1.

2.


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Addition of Polynomials (page 2)

Continue adding polynomials by combining two groups of algebra tiles.

3.

4.

Use algebra tiles to find the sum of these polynomials.

3x2 2x 1-+2x2 5x 4-+

5x2 2x- 3+3x2 x- 3-

2x2 4x 2+ +4x2 5x 6+ +

x2 2 x 5+ +3x2 x 1+ +

5.

6.

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Subtraction of Polynomials

Algebra tiles can be used to study the process of subtracting polynomials.

Use algebra tiles to find thedifference between twopolynomials.

2x2 4x- 2+

x2- x 1-+

3x2 5x- 3+

Remember that the axiom of opposites states that a positive and a negative equals 0. Inorder to subtract -x2+x-1 from 2x2-4x+2 you must first apply the axiom of opposites. Followthese steps to place as many sets of positive and negative tiles needed to make it possibleto take away -x2+x-1:

1. In the example there are no negative x2's. Place one +x2 tileand one -x2 tile in the work area so that one negative x2 canbe taken away.

2. Again, since there are no +x's, again a set with one +x andone -x is placed in the work area.

3. Also, since there are no negative units, a set of positive andnegative units is placed in the work area.

4. Move the tiles representing -x2+x-1 off to the right side ofthe work area.

5. Use the evaluation tool to find the value of the remainingtiles. Check your answer.


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Multiplying Polynomials

Multiplication of binomials can be illustrated withalgebra tiles. Here is how to set up a problem.

2x 2+( ) x- 1+( )

Rules for MultiplicationA positive integer times a positiveinteger yields a positive integer.

A negative integer times a negativeinteger yields a positive integer.

A positive integer times a negativeinteger yields a negative integer.

How to multiply with algebra tiles. 3x 1+( ) x- 2+( )

2. Place the tiles that represent the secondbinomial in a column along the left side ofthe work area. Use the rotate tool to turn atile 90 and use the evaluate tool to checkthe binomial.

1. Place tiles representingthe first binomial in a rowacross the top of the workarea. Use the evaluatetool to check the binomial.

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3. Apply the rules for multiplication of positive and negative integers to place the tilesfor the product. If necessary use the rotate tool to turn a tile 90.

Use the Algebra Tool Kit to find the product of these binomials. Write the answer in thebox.

1. 2x- 3+( ) x 1-( )

2.

3.

4.

x- 1+( ) x 1-( )

2x- 1-( ) 2x 3-( )

x- 5+( ) x 2-( )


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Factoring Polynomials

Using algebra tiles to represent the process of factoring a polynomial is similar to multiplyingbinomials.

The first step in factoring a polynomialis to find a way to represent thepolynomial with tiles in a rectangleshape. The next step is to think aboutwhich tiles would fit in the factorpositions on the edges of therectangle and finally to use the rulesfor multiplying positive and negativenumbers to determine the sign of thetiles on the edges.

x2 3 x- 2+ x2 3 x- 2+

x- 2+

x2 3 x- 2+

polynomials factorsx2 2 x- 3-x2 2 x+

6x2 5x- 1+2x2 3x- 1+

6x2 3x-8x2 2x+

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A First Look at Plotting Functions

A coordinate plane is drawn from a horizontal and vertical number line. The point where thehorizontal axis intersects with the vertical axis is called the origin. The coordinate plane isdivided into four sections called quadrants. Each quadrant is labeled with a Roman numeral.An ordered pair is used to name the graph of a point in a plane. A coordinate plane is usedto graph points.

An ordered pair gives the position of a point. The first number in the pair is the horizontalposition. The second number is the vertical position. To locate the point (2, -3), first locatethe graph of 2 on the horizontal axis and then find the graph of -3 on the vertical axis. Theintersection is the location of the point (2,-3).

(2, -3)

x-axis

y-axis


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Which Quadrant?

The horizontal and vertical axes divide the coordinate plane into four sections. Every point iseither in one of the four quadrants or the point is on one of the axes. The origin is the pointwhich is at the intersection of the coordinate axes.

A

H

D

B

C

E

F

G

The first coordinate in an ordered pair corresponds to the horizontal axis (abscissa). Thesecond coordinate corresponds to the vertical axis (ordinate). In the first column write theordered pair for each point. In the second column write the name of the quadrant in whichthe point lies. If a point is on an axis, write the name of the axis.

point ordered pair positionA (2,3)BCDEFGH

quadrant I

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The Graph of a Linear Equation

In a coordinate plane, the horizontal axis is customarily called the x-axis. The vertical axisis called the y-axis. The ordered pair which is used to name a point is represented by (x,y).The set of numbers which solve an equation can be used to produce a graph.

The abscissa is the x-coordinateThe ordinate is the y-coordinate

(4,4)(3,3)

(1,1)

(-2,-2)

(-3,-3)(-4,-4)

The set {(-4,-4),(-3,-3),(-2,-2),(1,1),(3,3),(4,4)} solves the equation x-y=0. This equationcan also be written in the form y=x. When these points are located on a coordinate plane,a straight line results.

The equation y=x is a linear equation with two variables. If A, B and C are real numbersand A and B are both not zero, any equation which can be written in the form given belowis called a linear equation. The standard form for a linear equation is:

Another form of the linear equation which is commonly used in algebra is the slope-interceptform. The slope-intercept form expresses the relationship between the x and y as a function.

y m x b+=

slope intercept

ax by c-+


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Understanding the Slope

In the slope-intercept form, the slope is a ratio defined by ther i s e

run

slope=63

run

6

3

1. Complete the table and sketch the graphs.

2. Use the Algebra Tool Kit to build these polynomials with algebra tiles and link eachset to a graph.

The line that results when an equation in the form y=mx is graphed in a coordinate planepasses through the origin. The slop of the line is the coefficient of the x-term in the equation.

Link a set of algebra tiles to a grid to graph each equation given in this exercise.

y x-= y 3 x= y 2x-=

equation slope type of slope

y 4 x=

y 3x-=

y 2 x=

positivenegative

positivenegative

positivenegative

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Looking at the Effect of the b-term

In the slope-intercept form of a linear equation, the b-term is the point where the line crossesthe y-axis.

Each of these lines have the same slope but cross the y-axis at a different point.

y x 3+=

y x 2+=

y x=

y x 3-=

1. Use the Algebra Tool Kit to plot these functions. Complete the chart by writing the slope and y-intercept. Sketch the graph.

equation slope y-intercepty x 2+=

y 3x- 3-( )+=

y 2x- 1+=

2. Use the Algebra Tool Kit to graph each of these functions.

y 3x- 1+= y 2 x 1-= y 3 x 4-( )+=

y 3 x 2-= y x- 2+= y x- 1+=


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Quadratic Functions

A function produces a set of ordered pairs. An equation with two variables can be used todefine a function. A graph is used to show the solution set for the equation. For example:

y 3 x 1-= y x2 3 x- 4-= y x2 9-=

Quadratic functions produce a special typeof curve called a parabola. If the coefficientof x2 is positive the curve opens upward. Ifthe coefficient of x2 is negative, the curvewill open downward.

The form of a quadratic function is:y a x2 bx c+ += (a 0)

All quadratic functions produce parabolas.

Use the Function Plot program to graph these quadratic functions.

algebra concepts

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