Algebra 1 > Notes > YORKCOUNTY FINAL > YORKCOUNTY > Algebra I A & B

Curriculum Guide

Algebra IA

I. PATTERNS/FUNCTIONSA. Analyzing Data for PatternsB. Determining Functional Relationships

II. STATISTICAL METHODSA. Problem SolvingB. Box-and-Whisker GraphsC. Measures of Central TendencyD. Range

III. VARIABLES AND REAL NUMBERSA. Variables and ExpressionsB. Exponents and PowersC. Properties of Real NumbersD. Operations with Real NumbersE. Justification of Steps in Simplifying Expressions

IV. SOLVING LINEAR EQUATIONS/INEQUALITIESA. Solving Multistep Linear Equations with One or

More TransformationsB. Literal Equations and FormulasC. Problem Solving

V. MATRICESA. Addition and SubtractionB. Scalar Multiplication

VI. LINEAR FUNCTIONS AND GRAPHSA. Graphing Linear FunctionsB. Slope-InterceptC. X- and Y-InterceptsD. Transformations

VII. EQUATIONS OF LINESA. Writing Equations of LinesB. Standard FormC. Problem Solving

Algebra IAContent Outline

SOLsSuggestedTime Frame

A.5

A.17

A.2, A.3

A.1

A.4

A.6, A.7

A.8, A.16

5 blocks/10 single periods







Topic

7/04

Algebra I: Blueprint Summary Table

Reporting Categories

No. of Items

SOLs

Expressions and Operations 12 A.2; A-10; A.11; A.12; A.13

Relations and Functions 12 A.5; A.15; A.18

Equations and Inequalities 18 A.1; A.3; A.6; A.7; A.8; A.9; A.14

Statistics 8 A.4; A.16; A.17 Total Number of Operational Items 50 Field-Test Items** 10 Total Number of Items 60

* These field-test items will not be used to compute students’ scores on the test.

Algebra IAVirginia Standards of Learning

A.1 The student will solve multistep linear equa-tions and inequalities in one variable, solveliteral equations (formulas) for a given vari-able, and apply these skills to solve prac-tical problems. Graphing calculators will beused to confirm algebraic solutions.

A.2 The student will represent verbal quantita-tive situations algebraically and evaluatethese expressions for given replacementvalues of the variables. Students will choosean appropriate computational technique,such as mental mathematics, calculator,or paper and pencil.

A.3 The student will justify steps used in sim-plifying expressions and solving equationsand inequalities. Justifications will includethe use of concrete objects; pictorial rep-resentations; and the properties of realnumbers, equality, and inequality.

A.4 The student will use matrices to organizeand manipulate data, including matrix ad-dition, subtraction, and scalar multiplica-tion. Data will arise from business, indus-trial, and consumer situations.

A.5 The student will create and use tabular,symbolic, graphical, verbal, and physicalrepresentations to analyze a given set ofdata for the existence of a pattern, deter-mine the domain and range of relations,and identify the relations that are functions.

A.6 The student will select, justify, and applyan appropriate technique to graph linearfunctions and linear inequalities in two vari-ables. Techniques will include slope-inter-cept, x- and y-intercepts, graphing by trans-formation, and the use of the graphing cal-culator.

A.7 The student will determine the slope of aline when given an equation of the line, the

graph of the line, or two points on the line.Slope will be described as rate of changeand will be positive, negative, zero, or un-defined. The graphing calculator will beused to investigate the effect of changesin the slope on the graph of the line.

A.8 The student will write an equation of a linewhen given the graph of the line, two pointson the line, or the slope and a point on theline.

A.9 The student will solve systems of two lin-ear equations in two variables both alge-braically and graphically and apply thesetechniques to solve practical problems.Graphing calculators will be used both asa primary tool for solution and to confirman algebraic solution.

A.10 The student will apply the laws of expo-nents to perform operations on expressionswith integral exponents, using scientificnotation when appropriate.

A.11 The student will add, subtract, and multi-ply polynomials and divide polynomials withmonomial divisors, using concrete objects,pictorial and area representations, and al-gebraic manipulations.

A.12 The student will factor completely first- andsecond-degree binomials and trinomials inone or two variables. The graphing calcu-lator will be used as a tool for factoringand for confirming algebraic factorizations.

A.13 The student will express the square root ofa whole number in simplest radical formand approximate square roots to the near-est tenth.

A.14 The student will solve quadratic equationsin one variable both algebraically andgraphically. Graphing calculators will be

used both as a primary tool in solving prob-lems and to verify algebraic solutions.

A.15 The student will, given a rule, find the val-ues of a function for elements in its domainand locate the zeros of the function bothalgebraically and with a graphing calcula-tor. The value of f(x) will be related to theordinate on the graph.

A.16 The student will, given a set of data points,write an equation for a line of best fit anduse the equation to make predictions.

A.17 The student will compare and contrastmultiple one-variable data sets, using sta-tistical techniques that include measuresof central tendency, range, and box-and-whisker graphs.

A.18 The student will analyze a relation to de-termine whether a direct variation existsand represent it algebraically and graphi-cally, if possible.

M9A–1

Curriculum Guide/Math/Algebra IAYork County School Division

Virginia Standard/s of Learning:

Related Standard/s: Technology Standard/s:

Content Component/s:

Assessment Sample/s:

Suggested Time Frame:

C/T 12.1, 12.2,12.3, 12.4

EXPRESSIONS ANDOPERATIONS

RELATIONS ANDFUNCTIONS

EQUATIONS ANDINEQUALITIES STATISTICS

A.5 The student will create and use tabular, symbolic, graphical, verbal, and physical representa-tions to analyze a given set of data for the existence of a pattern; determine the domain andrange of relations; and identify the relations that are functions.

• Analyzing Data for Patterns• Determining Functional Relationships 5 blocks/10 single periods

• Graph and comparison of a set of equations (inequalities) chosen by the teacher• Paragraphs explaining different methods for representing paired data, including one advantage

and disadvantage of using each of the different methods• Unit quizzes/tests

Unit I: Patterns/Functions

S BIO.1, BIO.4, BIO.5, BIO. 8,BIO.9

Essential Knowledge & Skills/MathematicsCurriculum Framework/Algebra I/p. 7:

The student will:• analyze a table of ordered pairs for the exist-

ence of a pattern that defines the change re-lating input and output values.

• write a linear equation to represent a patternin which there is a constant rate of changebetween variables.

• determine from a set of ordered pairs, a table,or a graph whether a relation is a function.

• identify the domain and range for a relation,given a set of ordered pairs, a table, or agraph.

• use physical representations, such as alge-bra manipulatives, to represent quantitativedata.

M9A–2

Instructional Blueprint/s: (Strategies)

The teacher will:• have students, in small groups, work with an

equation of a line, describing all the differentinformation that can be determined about it.The students will discuss the most efficienttechniques of graphing the equation.

• use a motion detector and CBLs to show stu-dents what a graph actually represents.

• use the Wave Lesson and graphing calcula-tor to introduce independent and dependentvariables.

• use the lesson Discovering Rates of Changeand the graphing calculator to show the im-portance of slope.

• provide opportunities for students to investi-gate patterns which arise from various geo-metric shapes to determine such things as,what effects the changing of a dimension willhave on area or perimeter? Students developformulas to illustrate the relations.

• ask students to consider the statement, “Yourshoe size is a function of the size of your foot.”Develop the idea that “is a function of” actu-ally means “depends on.” Have students giveexamples of additional situations in which thevalue of one variable results in only one truevalue for a second variable. Extend this tohave the students use mapping to prove therelationship of data in various examples.

• use cooperative groups. Have a student namea number. Have a second student perform anoperation on that number and give the sec-ond item in the sequence. Each succeedingstudent must decide what operation was usedand give the next sequence item. Rotate andrepeat.

¤ explain and model for students how to deter-mine the domain of a function given its graph.Create a worksheet of the graphs of variousrelations and functions. Guide students indetermining the domain and the range of eachgraph shown.

Essential Understandings/MathematicsCurriculum Framework/Algebra I/p. 7:

• A set of data may be characterized by pat-terns, and those patterns can be representedin multiple ways.

• Graphs can be used as visual representa-tions to investigate relationships betweenquantitative data.

• Algebra is a tool for describing patterns, mak-ing generalizations, and representing a rela-tionship in which output is related to input.

• A function is a relation for which there is aunique output for each input.

• A relation can be represented by a set ofordered pairs.

• The domain consists of the first coordinatesof the ordered pairs.

• The range consists of the second coordinatesof the ordered pairs.

• A relation is a function if each element in thedomain is paired with a unique element of therange.

Resources:

Text: Algebra 1, Integrations, Applications,Connections, Glencoe, pp. 56-62, 252-314

Technology:• TI-83 Graphing Calculator and CBL• Algeblaster (S)• Graphlink (S)• Algebra I (ProOne) (S)• Equation Editor (Microsoft Office) (S)• Hyperstudio (S)• PowerPoint (S)• Best Grapher

Notes:

• ¤ indicates an instructional blueprint for a topicthat has been included in previous math SOLtests and that is not covered in the math text-book.

M9A–3







C/T 12.1, 12.2,12.3, 12.4




A.17 The student will compare and contrast multiple one-variable data sets, using statistical tech-niques that include: measures of central tendency, range, and box-and-whisker graphs.

• Problem Solving• Box-and-Whisker Plots• Measures of Central Tendency• Range


• Box-and-Whisker Plot with mean, median, and mode displays• Chart describing/providing examples of various measures of central tendency• Unit quizzes/tests

Unit II: Statistical Methods

S BIO.1, BIO.8, BIO.9


The student will:• calculate the measures of central tendency

and range of a set of data with no more than20 data points.

• compare measures of central tendency usingnumerical data from a table with no more than20 data points.

• compare and contrast two sets of data, eachset having no more than 20 data points, usingmeasures of central tendency and the range.

• compare and analyze two sets of data, eachset having no more than 20 data points, usingbox-and-whisker plots.

M9A–4

The teacher will:• give the students a list of data in tabular form

and have cooperative groups of studentsdecide which plot (stem-and-leaf or box-and-whisker) of the data they prefer and providea rationale for their selection.

• give the students the football scores of thehigh school’s team for each of the past fouryears. From this set of data, ask students tocompare and analyze the team’s performanceusing statistics and graphs (e.g., median,mode, box-and-whisker plots). Have studentsuse statistics and graphs to check their analy-sis and make new conclusions. Discuss howsummarizing data is helpful to analyze data.

• give students a list of their previous class testscores. Have students analyze the data aftercalculating the measures of central tendency(e.g., mean, median, mode). Use the infor-mation obtained to make box-and-whiskerplots. Discuss how summarizing data is help-ful to analyze data. Students use their indi-vidual scores over a period of time. This ac-tivity could be adapted to use with spread-sheets.

• provide students with bags of M&M’s. Stu-dents estimate the number in the bag beforeopening. Have students open the bag andrecord the amount on a small post-it-note.Use the post-it-notes to create a stem-and-leaf plot. Students then use the informationfrom this collection of data to create a box-and-whisker plot. Students use measures ofcentral tendency to describe the data.

Instructional Blueprint/s: (Strategies) Essential Understandings/Mathematics Cur-riculum Framework/Algebra I/p.19:

• Measures of central tendency can be used tocharacterize a set of data and to make pre-dictions.

• Statistical techniques can be used to orga-nize, display, and compare sets of data.

• Box-and-whisker plots can be used to ana-lyze data.

Resources:

Text: Algebra 1, Integrations, Applications,Connections, Glencoe, pp. 25-32, 78-83, 339-345, 427-434Technology:• TI-83 Graphing Calculator• Algeblaster (S)• Graphlink (S)• Equation Editor (Microsoft Office) (S)• Hyperstudio (S)• PowerPoint (S)

Notes:

M9A–5







C/T 12.1, 12.2,12.3, 12.4




A.2 The student will represent verbal quantitative situations algebraically and evaluate these ex-pressions for given replacement values of the variable. Students will choose an appropriatecomputational technique, such as mental mathematics, calculator, or paper and pencil.

A.3 The student will justify steps used in simplifying expressions and solving equations and in-equalities. Justifications will include the use of concrete objects, pictorial representations, andthe properties of real numbers, equality and inequality.

• Variables and Expressions• Exponents and Powers• Properties of Real Numbers• Operations with Real Numbers• Justification of Steps in Simplifying Expressions


• Story using 4-8 key words which indicatethe four operations. The students mustdemonstrate comprehension of the wordsand their functions in mathematics. Use arubric to set scoring guidelines for students.

• List of expressions which student hassimplified using pictorial representations

Unit III: Variables and Real Numbers

E 9.8

and numbers and justification of each stepusing the properties of real numbers

• Student-generated expressions to sharewith other groups

• Paragraph explaining how you can showpositive and negative numbers, variablesand variable expressions with the algebra tiles

• Unit quizzes/tests

Essential Knowledge & Skills/MathematicsCurriculum Framework/Algebra I/pp. 2 & 11:

The student will:• translate verbal expressions into algebraic ex-

pressions with three or fewer terms.• relate a polynomial expression with three or

fewer terms to a verbal expression.• evaluate algebraic expressions for a given re-

placement set to include integers and rationalnumbers.

• apply appropriate computational techniquesto evaluate an algebraic expression.

• simplify expressions and solve equations andinequalities, using the commutative, associa-tive, and distributive properties.

• simplify expressions and solve equations andinequalities, using the order of operations.

• solve equations, using the addition, multipli-cation, closure, identity, and inverse proper-ties.

• solve equations, using the reflexive, symmet-ric, transitive, and substitution properties ofequality.

• create and interpret pictorial representationsfor simplifying expressions and solving equa-tions and inequalities.

M9A–6

The teacher will:• prepare the students for the game: “I Have.

You Have.” Create small self-made cards withverbal expressions on one side and an alge-braic expression on the other side. Tell thestudents to read the “I have…” side of thecard. The student with the algebraic formsfor that verbal expression answers “Youhave…”, then he/she reads his/her verbal ex-pression. Play continues in that pattern.

• provide 3 x 5 index cards. Write algebraicexpressions on half of the cards and theequivalent verbal expressions on the other half.After shuffling the cards, distribute them tostudents. The student will then search for amatch to his/her card. This is the “Concentra-tion” format.

• divide students into groups of four or five andask them to make a list of key words thatimply the four basic operations. After com-paring the lists with other groups, each groupwill use the list to translate ten given verbalexpressions to algebraic expressions. Thegroups will develop their own verbal expres-sions to translate.

• place students in groups of three. Have onestudent write a mathematical expression.Have another student write the expression inwords. Next, have a third student translatethe words back to the expression. Comparethe initial and final expressions. If they differ,verbalize each step to determine what wasdone incorrectly.

• provide algebra tiles to verify steps in solvinggiven equations. Students use the tiles to solvetheir own equations.

• pair students. Provide students with a dictio-nary to define terms used in the properties(e.g., commutative). Students use numericalexamples, objects, and pictures to demon-strate understanding of the properties.

• prepare a set of cards. The names of theproperties will be on one set of colored cards.Several examples of each property will be oncards of a different color. The cards shouldbe shuffled. Students match the examples withthe property name.

Instructional Blueprint/s: (Strategies) ¤ explain the identification of the sum of mono-mials represented by a model. Prepare a setof cards for a “Concentration” game with al-gebraic models of polynomial expressions onone card and the matching algebraic expres-sions on the other side. Students are as-signed to find the matching pairs.

¤ provide a variety of examples of solving con-sumer problems using given formulas. Cre-ate a chart of formulas, some within wordproblems, for the students to algebraicallyevaluate for given values. Arrange for stu-dents to work with problems related to cur-rent consumer issues, sports, and music.

¤ explain the identification of a property thatjustifies a given algebraic manipulation. Dis-tribute a solution key for a variety of algebra-ically-evaluated expressions. Place studentsin groups of 2-3. Tell the students to state theproperty that justifies each step.

Essential Understandings/Math CurriculumFramework/Algebra 1/p. 2 & 11:

• Algebra is a tool for reasoning about quanti-tative situations so that relationships becomeapparent.

• Algebra is a tool for describing and repre-senting patterns and relationships.

• The numerical values of an expression aredependent upon the values of the replace-ment set for the variables.

• There are a variety of ways to compute thevalue of a numerical expression and evaluatean algebraic expression.

• The operations and the magnitude of the num-bers in an expression impact the choice of anappropriate method of computation.

• The representation and manipulation of ex-pressions, equations, and inequalities can bemodeled in a variety of ways, using concrete,pictorial, and symbolic representations.

• Properties of real numbers and properties ofequations and inequalities can be used tosolve equations and inequalities and simplifyexpressions.

M9A–7







C/T 12.1, 12.2,12.3, 12.4

Resources:

Text: Algebra 1, Integrations, Applications,Connections, Glencoe, pp. 6-69• Algebra tiles

Technology:• TI-83 Graphing Calculator• Algeblaster (S)• Graphlink (S)• Algebra I (ProOne) (S)• Jasper Woodbury–”The General Is Missing

(LD)• Equation Editor (Microsoft Office) (S)• Hyperstudio (S)• PowerPoint (S)

Notes:

Real Numbers - some history - A history ofreal numbers.ht tp: / /www.rbjones.com/rbjpub/maths/math008.htm

Algebra I, Real Numbers - A short test on realnumbers. Student scores are reported back tostudents.http://library.advanced.org/11771/english/hi/math/tests/alg/1.html


M9A–8

M9A–9







C/T 12.1, 12.2,12.3, 12.4




A.1 The student will solve multistep linear equations and inequalities in one variable, solve literalequations (formulas) for a given variable, and apply these skills to solve practical problems.Graphing calculators will be used to confirm algebraic solutions.

• Solving Multistep Linear Equations with One or MoreTransformations

• Literal Equations and Formulas• Problem Solving


• Identification of property that justifiessteps in a list of 5-6 solved equationsand/or simplified expressions

• Demonstration of how to find thedistance, rate or time, given two of theunknowns, using the graphing calculator.Determination of each answer and

Unit IV: Solving Linear Equations/Inequalities

E 9.8; M A.2

recording the literal equation(e.g., d = rt; d/r = t; d/t = r)

• Poster designed to represent a real-lifesituation that can be translated into anumerical or algebraic word phrase toinclude order of operations.



The student will:• translate verbal sentences to algebraic equa-

tions and inequalities in one variable.• solve multistep linear equations and

noncompound inequalities in one variable withthe variable in both sides of the equation orinequality.

• solve multistep linear equations and inequali-ties in one variable with grouping symbols inone or both sides of the equation or inequal-ity.

• solve multistep equations and inequalities inone variable with rational coefficients and con-stants.

• solve a literal equation (formula) for a speci-fied variable.

• apply skills for solving linear equations to prac-tical situations.

• confirm algebraic solutions to linear equationsand inequalities, using a graphing calculator.

M9A–10

The teacher will:• make a set of “Property Rummy” cards,

which will include the properties of equalityand a number of expressions which illustratethe properties. Have students work in smallgroups of three or four. The object of thiscard game is to form sets of three cardseach. Each set must contain one propertycard and two expression cards that illustratethat property. To begin, each player is dealteight cards. The remaining cards are placedface down and the top card is turned faceup. Next, turns are taken by either drawingthe top face up or face down card and thendiscarding one of the cards. When two setscan be formed from the cards in hand, theyare placed in front of the player. The firstplayer to do this wins the game.

• arrange students in groups of four. Modelnon-numerical processes using a balancescale or a scale made using a meter stickand a triangular prism. Students find differ-ent ways to keep the scales balanced. Stu-dents are asked to add or subtract to findthe “balance” for an unknown object.

• provide students with red and white squares,which represent positive and negative inte-gers, respectively, to model a one-step equa-tion. A white square “cancels” a red one.Students solve equations by manipulating thesquares until one side is empty.

• divide the class into four or five small groups.Have each group create five word problemsinvolving either an equation or inequality. Theword problems should be practical (e.g., in-volving area, interest, deriving a minimum testscore needed for a particular test average,the menu of a fast food restaurant). Havegroups switch and solve each other’s prob-lems. Answers are verified by the group thatoriginally created the problems.

• discuss the use of balance scales. Studentsare asked to find five different ways to keepthe scales balanced. They are also asked toadd or subtract to find the balance weight ofan unknown object.

• provide Quiz on a Card. Each group of 5

Instructional Blueprint/s: (Strategies) students will be given a set of 5 index cardswith a numerical or algebraic word phraseprinted on each card. Each group memberwill choose a card and write the phrase as anexpression and evaluate the expression.

¤ provide examples of equations and how theyare used to solve word problems. Split theclass into groups of 2-3. Write several equa-tions on the board, such as 3x + 5 = 17 or5(x-1) + 10 = 40. Ask the groups to create aword problem based on student interest tomatch the equations, then solve the problemand answer the question. Have the groupsexchange problems to verify that the solutionand the created problem match the equation.

Essential Understandings/Mathematics Cur-riculum Framework/Algebra I/p.10:

• A solution to an equation or inequality is thevalue or set of values that can be substitutedto make the equation or inequality true.

• Equations and inequalities can be solved in avariety of ways.

• The solution of an equation in one variablecan be found by graphing each side of theequation separately and finding the x-coordi-nate of the point of intersection.

• Practical problems can be interpreted, rep-resented, and solved using linear equationsand inequalities in one variable

M9A–11







C/T 12.1, 12.2,12.3, 12.4

Resources:

Text: Algebra 1, Integrations, Applications,Connections, Glencoe, pp. 140-191• Algebra tilesTechnology:• TI-83 Graphing Calculator• Algeblaster (S)• Graphlink (S)• Algebra I (ProOne) (S)• Jasper Woodbury–”The General Is Missing


Notes:

A.1 Strategies• Lesson 5: Linear Equations - The index to

linear algebra and how to graph linear equa-tions using the “Internet Academy.” Excellentsource of problems and how to solve them inlittle increments.http:// iasec.fwsd.wednet.edu/iamath/page2d.htm#Lesson5

• Linear Inequalities - This is a complete lec-ture on slides. This address takes you to thetext version from there you can go to thegraphic version, but it takes a long time todownload.http://mesa7.mesa.colorado.edu/~bornmann/c lasses /math091 / lec tu re /chap_02 /tsld014.htm

• Assignment 10: More solving formulas -This site offers a good explanation on how towork through solving formulas and what youcan and cannot combine.http://198.85.210.30/~stonere/oph101/lesn10c.htm

A.1 Resources• Algebra Through Modeling - Tutorial on how

to use the TI-82 to enter and organize data.http://www-cm.math.uniuc.edu/MathLink/al-gebra-module/ALG.index.html

• Algebra Online - Free service designed toallow students, parents, and educators tocommunicate including free tutoring, chatroom and message board.http://www.algebra-online.com

• Algebra I - Linear Equations - Graphing ca-pabilities for linear equations and quizzes.http://www.bremenbraves.com/algebra/index.html

A.2 Strategies• The next generation of problem solving -

This Pacific Tech site provides various prob-lems and lesson plans about solving wordproblems and how to solve them step-by-step. Not just for algebra, either!h t tp : / /www2.hawa i i .edu /su remath /intro_algebra.html

• Algebra word problems - A question for Dr.Math asked by a student about where to putthe number and the letters when convertingword problems into algebraic equations.http://forum.swarthmore.edu/dr.math/prob-lems/mike6.6.96.html

• Problem Solving Methods - This site, (au-thor - Alan M. Selby, of Montreal, Canada)provides different strategies for attacking aword problem.http://www.cam.org/~aselby/prob.html

A.2 Resources• Math Archive - Lots of word problems within

various categories. Solutions are included,unless they haven’t found one yet.http://bsuvc.bsu.edu/~d004ucslabs/

• Word problems galore from the Mathemat-ics Problem Solving Task Centre of TheMathematical Association of Victoria.http://www.srl.rmit.edu.au/mav/PSTC/gen-eral/index.html

• The Problem Solving Corner - posesweekly problems for Virginia schools. Stu-dents get personal responses and theirnames up on the Web if their solution is cor-rect.http://www.wm.edu/education/Faculty/Ma-son/pscmain.html

• Math Forum Student Center - Weekly prob-lems and the famous Ask Dr. Math corner.http://forum.swarthmore.edu/students/

• Words Before Symbols - Background in-formation on the rationale of algebra and al-gebraic thinking.http://www.cam.org/~aselby/volumee/html/bk1bch03.html

• Best Grapher


M9A–12

M9A–13







C/T 12.1, 12.2,12.3, 12.4




A.4 The student will use matrices to organize and manipulate data, including matrix addition,subtraction, and scalar multiplication. Data will arise from business, industrial, and con-sumer situations.

• Addition and Subtraction• Scalar Multiplication


• Matrix to organize the following information:On the first day of ticket sales, the yearbookstaff sold 60 tickets to 6th graders, 112 ticketsto 7th graders, and 120 tickets to 8th graders.On the second day of ticket sales, 85 ticketswere sold to 6th graders, 86 tickets to 7th grad-ers, and 126 tickets to 8th graders. Whichmatrix best organizes the ticket sales for the

Unit V: Matrices

S BIO.1, BIO.6

two days. Give 4choices in the following form:

Day 1 Day 26th7th8th



The student will:• represent data from practical problems in ma-

trix form.• calculate the sum or difference of two given

matrices that are no larger than 4 x 4.• calculate the product of a scalar and a matrix

that is no larger than 4 x 4.• solve practical problems involving matrix ad-

dition, subtraction, and scalar multiplication,using matrices that are no larger than 4 x 4.

• read and interpret the data in a matrix repre-senting the solution to a practical problem.

M9A–14

The teacher will:• have students obtain data from the class (e.g.,

males, females vs. favorite television showbroken into types) and display this in matrixform. Students perform operations on thematrix. Follow-up by using the graphing cal-culator to enter the date into matrix form andto perform the arithmetic with the matrices.

• give students a stadium or auditorium seat-ing chart and a make-believe ticket for a seat.They will determine their place in the seatingmatrix.

• give the students a set of data that describesthe number of boys and girls for each grade(9, 10, 11, 12) at “East H.S.” and “West H.S.”Students represent the data in matrix formusing sex (male or female) as the rows andgrade (9, 10, 11, and 12) as the columns.The School Board decides to combine thetwo schools into one. Using a matrix additiondescribe the total enrollment by sex and gradefor the new school.

Notes:

A.4 ResourcesJava Script Linear Algebra I - Computer pro-gram that will help with matrices as well as lin-ear equations. Has explanations on how the pro-gram works, too.http://www.csus.edu/org/mathsoc/line_alg.html

Peanut Software for Windows - Freeware fromplotting, exploring geometry, statistics, fractals,discrete math, to matrices.http://www.exeter.edu/~rparris/

Math 45 - Linear Algebra - Perhaps a littleadvanced for Algebra I, but it may have generalideas on how to use every day material in linearand matrix algebra.http://www.redwoods.cc.ca.us/sciweb/instruct/darnold/LinAlg/activity.htm

Linear Algebra - Lots of modules about matrixoperations. Has worksheets for “helper appli-cations such as Mathcad, Maple, andMathematica (which are quite necessary).http://www.math.duke.edu/modules/materials/linalg/

Mathematics Archives: Industrial Mathemat-ics - Links to sites involving industrial math.http://archives.math.utk.edu/topics/industrial/Math.html


Essential Understandings/Math CurriculumFramework/Algebra 1/p. 17:

• Matrices are a tool for organizing and dis-playing data.

• A relationship exists between arithmetic op-erations and operations with matrices.

• Matrices can be used to solve practical prob-lems.

• Only matrices of the same dimensions canbe added or subtracted.

Resources:

Text: Algebra 1, Integrations, Applications,Connections, Glencoe, pp. 70-138Technology:• TI-83 Graphing Calculator• Algeblaster (S)• Graphlink (S)• Algebra I (ProOne) (S)• Equation Editor (Microsoft Office) (S)• Hyperstudio (S)• PowerPoint (S)

M9A–15







C/T 12.1, 12.2,12.3, 12.4




A.6 The student will select, justify, and apply an appropriate technique to graph a linear functionand linear inequalities in two variables. Techniques will include slope-intercept, x- and y-inter-cepts, graphing by transformation, and the use of the graphing calculator.

A.7 The student will determine the slope of a line when given an equation of the line, the graph ofthe line, or two points on the line. Slope will be described as rate of change and will bepositive, negative, zero, or undefined. The graphing calculator will be used to investigate theeffect of changes in the slope on the graph of the line.

• Graphing Linear Functions• Slope-Intercept• X- and Y-Intercepts• Transformations


• Examples of slopes in real situations• Examples of solutions to linear equations• Unit quizzes/tests

Unit VI: Linear Functions and Graphs

S BIO.1, BIO.2, BIO.8, BIO.9

Essential Knowledge & Skills/MathematicsCurriculum Framework/Algebra I/pp. 12–13:

The student will:• graph linear equations and inequalities in two

variables that arise from a variety of practicalsituations.

• use the line y = x as a reference, and applytransformations defined by changes in theslope of y-intercept.

• express linear functions or inequalities inslope-intercept form, and use the graphingcalculator to display the relationship.

• explain why a given technique is appropriatefor graphing a linear function.

• recognize that m represents the slope in the equa-tion of the form y = mx + b.

• find the slope of the line, given the equationof a linear function.

• calculate the slope of a line, given the coordi-nates of two points on the line.

• find the slope of a line, given the graph of aline.

• recognize and describe a line with a slopethat is positive, negative, zero, or undefined.

• describe slope as a constant rate of changebetween two variables.

• compare the slopes of graphs or linear func-tions, using the graphing calculator.

M9A–16

The teacher will:• have students calculate the slope of several

staircases in the school, as well as deliveryramps or handicap ramps. Students write theequation to draw the graph of the line thatwould represent ramps or stairs. Use theequation to draw the graph of the lines on thegraphing calculator. Discuss how the changesin slope affect the “steepness” of a line.

• use graphing calculators. Have students in-vestigate the changes in the graph caused bychanging the values of the constant and coef-ficient. This will allow the student to visuallycompare several equations at the same time.

¤ exhibit examples of tables and graphs of aline, showing students how to identify the tablethat matches the graph of a line. Create anactivity related to a current topic (e.g., music,arts, sports) in which students graph variouslines, by first making a table of values. Em-phasize graphing of horizontal and vertical linesusing this method.

¤ show students how to identify a graph with atranslated equation of line. Pair students andtell them to translate graphs by a given amount.For example, “translate the graph of y = 3x +4 up 2 units. Students will then provide theequation of the new graph.


• Linear functions and inequalities can bewritten in a variety of forms.

• Linear functions and inequalities can begraphed, using a variety of techniques.

• An appropriate technique for graphinglinear functions and inequalities can bedetermined by the given information and/orthe tools available.

• Justification of an appropriate technique forgraphing linear equations and inequalities isdependent upon the application of slope, x-and y-intercepts, and graphing by transfor-mations.

• Linear equations and inequalities arise froma variety of practical situations.

Essential Understandings/Math Curricu-lum Framework/Algebra I/P.12-13:

• The slope of a linear function represents aconstant rate of change in the dependentvariable when the independent variablechanges by a fixed amount.

• The slope of a line determines its relativesteepness.

• The slope of a line can be determined in avariety of ways.

• Changes in slope affect the graph of a line.

Resources:

Text: Algebra 1, Integrations, Applications,Connections, Glencoe, pp. 252-316Technology:• TI-83 Graphing Calculator• Algeblaster (S)• Graphlink (S)• Algebra I (ProOne) (S)• Equation Editor (Microsoft Office) (S)• Hyperstudio (S)• PowerPoint (S)• Best Grapher

Notes:

¤ indicates an instructional blueprint for a topicthat has been included in previous math SOLtests and that is not covered in the math text-book.

M9A–17







C/T 12.1, 12.2,12.3, 12.4




A.8 The student will write an equation of a line when given the graph of the line, two points on theline, or the slope and a point on the line.

A.16 The student will, given a set of data points, write an equation for a line of best fit and use theequation to make predictions.

• Writing Equations of Lines• Standard Form• Problem Solving 10 blocks/20 single periods

• Writing of equations given 2 points, or a point and slope• Graph of line of best fit with specified data from teacher• Transformation of a linear equation from standard to slope-intercept form• Unit quizzes/tests

Unit VII: Equations of Lines

S BIO.1, BIO.2, BIO.4, BIO.5

Essential Knowledge & Skills/MathematicsCurriculum Framework/Algebra I/pp. 14 &18:

The student will:• recognize that equations of the form y = mx +

b and Ax + By = C are equations of lines.• write an equation of a line when given the

graph of a line.• write an equation of a line when given two

points on the line whose coordinates are in-tegers.

• write an equation of a line when given theslope and a point on the line whose coordi-nates are integers.

• write an equation of a vertical line as x = c.

• write an equation of a horizontal line as y = c.• write an equation for the line of a best fit,

given a set of six to ten data points in a table,on a graph, or from a practical situation.

• make predictions about unknown outcomes,using the equation of a line of best fit.

M9A–18

The teacher will:• give students graphs of several lines and have

students reproduce the graphs on the graph-ing calculator. Students describe the strat-egy used to find the equation. Sketch a flowchart that shows the steps.

• divide students into pairs. One student is givena card with a graph or line which he/she de-scribes as accurately and precisely as pos-sible to his/her partner. The other studentwrites the equation of the line. Students thenlook at the graphs to check the answers. Thepartners then switch positions and repeat.

• have students use a chalkboard grid to findslope and y-intercept of the line. Have stu-dents use the slope formula and one point tofind the equation. Use the slope formula todiscuss vertical and horizontal lines. Havestudents use graphing calculators to find slopeand intercepts.

• allow students to take on the role of “forensicmathematicians,” trying to determine how talla deceased person was whose femur is 17inches long. Students measure their own fe-murs and their heights, entering this data intoa graphing calculator or computer and creat-ing a scatterplot. They note that the data areapproximately linear, so they use the built-inlinear regression procedures to find the lineof best fit.

• provide activities from Real World Math withthe CBL System and CBL units connected toTI-82 calculators to collect data. Students orgroups of students link with others to haveadditional data. They use the stat calc func-tions to determine the best fit (e.g., model).

• provide students with a board, which repre-sents the first quadrant of a plane, and nailsto represent the location of points to plotdata. Using rubber bands wrapped around arod, students connect the rubber bands tothe nails. The “line” which is formed by therod is a good representation of the “best fitline” for the data.

¤ model how to determine an equation for aline on a graph using technology,manipulatives, and the chalkboard. Provide

Instructional Blueprint/s: (Strategies) students with a variety of graphs related tocurrent topics. Tell students to review thegraphs and find the equations of the linesshown. Students can use manipulatives toportray the graphs and equations.

Essential Understandings/Math CurriculumFramework/Algebra 1/pp. 14 & 18:

• The equation of a line defines the relationshipbetween two variables.

• The graph of a line represents the set of pointsthat satisfies the equation of a line.

• A line can be represented by its graph or byan equation.

• The equation of a line can be determined bytwo points on the line or by the slope and apoint on the line.

• The graphing calculator can be used to de-termine the equation of a line of best fit for aset of data.

• The line of best fit for the relationship amonga set of data points can be used to makepredictions where appropriate.

Resources:

Text: Algebra 1, Integrations, Applications,Connections, Glencoe, pp. 322-380Technology:• TI-83 Graphing Calculator• Algeblaster (S)• Graphlink (S)• Algebra I (ProOne) (S)• Equation Editor (Microsoft Office) (S)• Hyperstudio (S)• PowerPoint (S)• Best Grapher

Notes:

¤ indicates an instructional blueprint for a topicthat has been included in previous math SOLtests and that is not covered in the math text-book.

Curriculum Guide

Algebra IB

I. SYSTEMS OF EQUATIONS AND INEQUALITIESA. Solving Linear Systems Graphically and

AlgebraicallyB. Systems of Linear InequalitiesC. Linear ProgrammingD. Problem Solving

II. POWERS AND EXPONENTSA. Properties of ExponentsB. Scientific NotationC. Problem SolvingD. Square Roots

III. POLYNOMIALSA. Polynomial OperationsB. Factoring Quadratic TrinomialsC. Problem Solving

IV. QUADRATIC EQUATIONSA. Factoring TechniquesB. Graphing Solutions

V. DIRECT VARIATIONA. Writing EquationsB. Problem Solving

VI. FUNCTIONAL RELATIONSHIPSA. Domain and RangeB. ZerosC. Graphing

VII. RATIONAL EXPRESSIONS, EQUATIONS ANDTHEIR APPLICATIONS (AFTER SOL TEST)A. Proportion and PercentB. ProbabilityC. Simplifying Rational ExpressionsD. Operations on Rational ExpressionsE. Solving Rational EquationsF. Problem Solving

Algebra IBContent Outline

SOLsSuggestedTime Frame

A.9

A.10, A.13

A.11, A.12

A.14

A.18

A.15








Topic

7/04

Algebra I: Blueprint Summary Table

Reporting Categories

No. of Items

SOLs

Expressions and Operations 12 A.2; A-10; A.11; A.12; A.13

Relations and Functions 12 A.5; A.15; A.18

Equations and Inequalities 18 A.1; A.3; A.6; A.7; A.8; A.9; A.14

Statistics 8 A.4; A.16; A.17 Total Number of Operational Items 50 Field-Test Items** 10 Total Number of Items 60

* These field-test items will not be used to compute students’ scores on the test.

A.1 The student will solve multistep linear equa-tions and inequalities in one variable, solveliteral equations (formulas) for a given vari-able, and apply these skills to solve prac-tical problems. Graphing calculators will beused to confirm algebraic solutions.

A.2 The student will represent verbal quantita-tive situations algebraically and evaluatethese expressions for given replacementvalues of the variables. Students will choosean appropriate computational technique,such as mental mathematics, calculator,or paper and pencil.

A.3 The student will justify steps used in sim-plifying expressions and solving equationsand inequalities. Justifications will includethe use of concrete objects; pictorial rep-resentations; and the properties of realnumbers, equality, and inequality.

A.4 The student will use matrices to organizeand manipulate data, including matrix ad-dition, subtraction, and scalar multiplica-tion. Data will arise from business, indus-trial, and consumer situations.

A.5 The student will create and use tabular,symbolic, graphical, verbal, and physicalrepresentations to analyze a given set ofdata for the existence of a pattern, deter-mine the domain and range of relations,and identify the relations that are functions.

A.6 The student will select, justify, and applyan appropriate technique to graph linearfunctions and linear inequalities in two vari-ables. Techniques will include slope-inter-cept, x- and y-intercepts, graphing by trans-formation, and the use of the graphing cal-culator.

A.7 The student will determine the slope of aline when given an equation of the line, the

Algebra IBVirginia Standards of Learning

graph of the line, or two points on the line.Slope will be described as rate of changeand will be positive, negative, zero, or un-defined. The graphing calculator will beused to investigate the effect of changesin the slope on the graph of the line.

A.8 The student will write an equation of a linewhen given the graph of the line, two pointson the line, or the slope and a point on theline.

A.9 The student will solve systems of two lin-ear equations in two variables both alge-braically and graphically and apply thesetechniques to solve practical problems.Graphing calculators will be used both asa primary tool for solution and to confirman algebraic solution.

A.10 The student will apply the laws of expo-nents to perform operations on expressionswith integral exponents, using scientificnotation when appropriate.

A.11 The student will add, subtract, and multi-ply polynomials and divide polynomials withmonomial divisors, using concrete objects,pictorial and area representations, and al-gebraic manipulations.

A.12 The student will factor completely first- andsecond-degree binomials and trinomials inone or two variables. The graphing calcu-lator will be used as a tool for factoringand for confirming algebraic factorizations.

A.13 The student will express the square root ofa whole number in simplest radical formand approximate square roots to the near-est tenth.

A.14 The student will solve quadratic equationsin one variable both algebraically andgraphically. Graphing calculators will be

used both as a primary tool in solvingproblems and to verify algebraic solu-tions.

A.15 The student will, given a rule, find thevalues of a function for elements in itsdomain and locate the zeros of the func-tion both algebraically and with a graph-ing calculator. The value of f(x) will berelated to the ordinate on the graph.

A.16 The student will, given a set of datapoints, write an equation for a line ofbest fit and use the equation to makepredictions.

A.17 The student will compare and contrastmultiple one-variable data sets, usingstatistical techniques that include mea-sures of central tendency, range, andbox-and-whisker graphs.

A.18 The student will analyze a relation todetermine whether a direct variation

M9B–1

Curriculum Guide/Math/Algebra IBYork County School Division






C/T 12.1, 12.2,12.3, 12.4




A.9 The student will solve systems of two linear equations in two variables, both algebraically andgraphically, and apply these techniques to solve practical problems. Graphing calculators willbe used both as a primary tool of solution and to confirm an algebraic solution.

• Solving Linear Systems Graphically and Algebraically• Systems of Linear Inequalities• Linear Programming• Problem Solving


• Real-life applications that can be solved by a system• Solution to a system of linear equalities by graphing• Model of real-life situation using linear programming• Unit quizzes/tests

Unit I: Systems of Equations and Inequalities

H/SS WHII.1


The student will:• given a system of two linear equations in two

variables that has a unique solution, solve thesystem by substitution or elimination to findthe ordered pair which satisfies both equa-tions.

• given a system of two linear equations in twovariables that has a unique solution, solve thesystem graphically to find the point of inter-section.

• determine whether a system of two linearequations has one solution, no solution, orinfinite solutions.

• write a system of two linear equations thatdescribes a practical situation.

• interpret and determine the reasonablenessof the algebraic or graphical solution of asystem of two linear equations that describesa practical situation.

M9B–2


The teacher will:• give the students an equation such as 5x + 3y

= 15 to find two or more equations that sat-isfy each of these requirements: (1) Thegraphs of the given equation and a secondequation intersect at a single point; (2) Thegraphs of the given equation and a secondequation intersect at more than one point.

• have students use a chalkboard grid and agraphing calculator to solve systems and tocheck systems solved by elimination and sub-stitution. Have students use systems to solveproblems involving price increases and profitmargins, banking problems, investments, dif-ferent rates, and different principles.

• divide students into groups. Have each groupsolve a system of equations by a pre-de-scribed method. Make sure that all methodsare assigned. Have students display theirsolutions to the class and discuss the mostappropriate method for solving the system.

• play “SYSTO” with the class. Make bingocards (5 x 5) in the squares of which studentswill write ordered pairs. The ordered pairscome from a teacher-generated list of solu-tions to systems of equations. The teachershows the students the systems of equations,one-by-one. After solving, students mark theircards with the solution. The first student to filla row, wins.

• have the students find points of intersectionon the TI-83 calculators and connect this con-cept to systems with “no solutions” or “manysolutions.”

lution is characterized by the graphs of twolines that do not intersect but are parallel.

• A system of two linear equations having infi-nite solutions is characterized by two graphsthat coincide (the graphs will appear to bethe graph of one line), and all the coordi-nates on this one line satisfy both equations.

• Systems of two linear equations can be usedto represent two conditions that must be sat-isfied simultaneously.


• A system of linear equations with exactly onesolution is characterized by the graphs of twolines whose intersection is a single point, andthe coordinates of this point satisfy both equa-tions.

• A point shared by two intersecting graphs andthe ordered pair that satisfies the equationscharacterize a system of equations with onlyone solution.

• A system of two linear equations with no so-

Text: Algebra 1, Integrations, Applications,Connections, Glencoe, pp. 382-444, 450-492Technology:• TI-83 Graphing Calculator• Algeblaster (S)• Graphlink (S)• Algebra I (ProOne) (S)• Jasper Woodbury–”The General Is Missing


Resources:

Notes:A.9 ResourcesMathematics of Cartography - Using the themeof maps, an investigation into the mathematicalconcepts of points, lines, areas, coordinates,and linear algebra.http://math.rice.edu/~lanius/pres/map

Technology:• Best Grapher

M9B–3







C/T 12.1, 12.2,12.3, 12.4




A.10 The student will apply the laws of exponents to perform operations on expressions with inte-gral exponents, using scientific notation when appropriate.

A.13 The student will express the square root of a whole number in simplest radical form andapproximate square roots to the nearest tenth.

• Properties of Exponents• Scientific Notation• Problem Solving• Square Roots


• Model of a paper repeatedly folded into thirds, showing powers of three• Explanation of the use of negative and zero exponents in algebraic expressions• Detailed solution using scientific notation to a specific problem related to a content

area other than math• Unit quizzes/tests

Unit II: Powers and Exponents

E 9.8


The student will:• identify the base, exponent, and coefficient in

a monomial expression.• simplify monomial expressions and ratios of

monomial expressions in which the exponentsare integers, using the laws of exponents.

• express numbers, using scientific notation, andperform operations, using the laws of expo-nents.

• estimate the square root of a non-perfectsquare to the nearest tenth by- identifying the two perfect squares it lies

between;

- finding the square root of those two perfectsquares; and

- using those values to estimate the squareroot of the non-perfect square.

• find the square root of a number, and make areasonable interpretation of the displayedvalue for a given situation, using a calculator.

• express the square root of a whole numberless than 1,000 in simplest radical form

M9B–4


The teacher will:• have students investigate the following situa-

tions involving the equation y = 2x.1. Compare the total money earned if a per-

son makes a penny the first day, anddoubles the amount every day to theamount that a person makes who earnsthe same amount every day.

2. Fold a piece of paper in half repeatedlyand investigate how many sections thereare.

3. Compute population over a period of time(e.g., from two cats, two mosquitoes).

4. Investigate large and small numbers fromthe almanac, a science book, or other ap-propriate data and write in scientific nota-tion.

• have students work in learning groups of stu-dent ecologists. Each group chooses twospecies of animals to research. Each groupprepares a chart to generate population poly-nomials. Each chart should include:1. Average number of offspring per litter.2. Estimated number of females per litter (use

50 total offsprings). (This number will serveas x.)

3. Columns for each generation of females:first (1), second (x), third (x2), fourth (x3).

4. The fifth column will be the total (1 + x + x2

+ x3).Each group finds the number of animals thatcan be expected after four generations.

• give students a list of numbers. Have themidentify which two perfect squares each num-ber is between. Students estimate the squareroot and check estimates with calculators.

• have students work in pairs and write numberon cards. They write the square or squareroot on an equal number of cards. Have stu-dents find matches in a game format like “Con-centration” or “Jeopardy.”

• present “Square Root Price is Right.” If a stu-dent is given a radical expression such as:85 , , , , he/she must give

the nearest integral square root, “withoutgoing over.”

• form a number line using students to repre-sent whole numbers. Give other students

numbers that are not perfect squares. Havethese students determine between which twostudents to stand to represent the squareroot of their number. Students can determinewhere the person should move to get a goodestimate of the square root.

Essential Understandings/Math CurriculumFramework/Algebra 1/pp. 3 & 6.:

• Repeated multiplication can be representedwith exponents.

• The laws of exponents can be investigatedusing patterns.

• The base and the exponent impact the magni-tude of the expression.

• A relationship exists between the laws of ex-ponents and scientific notation.

• The square root of a perfect square is an in-teger.

• The square root of a non-perfect square liesbetween two consecutive integers.

• The inverse of squaring a number is deter-mining the square root.

• A radical in simplest form is one in which theradicand has no perfect square factors otherthan one.

• The square root of a product is the product ofthe square roots.

Resources:

Text: Algebra 1, Integrations, Applications,Connections, Glencoe, pp. 494-519, 556-606Technology:• TI-83 Graphing Calculator• Algeblaster (S)• Graphlink (S)• Algebra I (ProOne) (S)• Equation Editor (Microsoft Office) (S)• Hyperstudio (S)• PowerPoint (S)

Notes:Technology:• Best Grapher

M9B–5







C/T 12.1, 12.2,12.3, 12.4




A.11 The student will add, subtract, and multiply polynomials and divide polynomials with mono-mial divisors, using concrete objects, pictorial, and area representations, and algebraic ma-nipulations.

A.12 The student will factor completely first- and second-degree binomials and trinomials in one ortwo variables. The graphing calculator will be used as a tool for factoring and for confirmingalgebraic factorizations.

• Polynomial Operations• Factoring Quadratic Trinomials• Problem Solving


• Drawing of a rectangle with sides labeled with binomials and explanation of howthe area of the rectangle can be derived

• Chart explaining the factoring of polynomials• Games invented involving polynomials and their factors• Unit quizzes/tests

Unit III: Polynomials

E 9.8

Essential Knowledge & Skills/MathematicsCurriculum Framework/Algebra I/pp. 4–5:

The student will:• model sums, differences, products, and quo-

tients of polynomials with concrete objectsand their related pictorial representations.

• relate concrete and pictorial representationsfor polynomial operations to their correspond-ing algebraic manipulations.

• find sums and differences of polynomials.• multiply polynomials by monomials and bino-

mials by binomials symbolically.• find the quotient of polynomials, using a mo-

nomial divisor.• use the distributive property to “factor out” all

common monomial factors.• factor second-degree polynomials and bino-

mials with integral coefficients and a positiveleading coefficient less than four.

• identify polynomials that cannot be factoredover the set of real numbers.

• use the x-intercepts from the graphical rep-resentation of the polynomial to determine andconfirm its factors.

M9B–6

The teacher will:• have students prepare a chart on factoring

polynomials. Each chart should include: 1)Type of factoring, 2) Number of terms (2, 3,4, or more), 3) Examples of each type, and4) Personal notes.

• have students use area principles andAlgeblocks to show operations with polyno-mials.

• have students make a model block out ofcardboard. The block will be named x anddistances will be measured using x and show-ing remainder in inches (e.g., the length ofthe board could be 7x – 7 inches). Find lengthsand areas in the classroom.

• have students invent a game that involvesmatching polynomials and their factors. Sug-gested format could include Concentration,Jeopardy, Tic Tac Toe, or Old Maid.

• have students use Algeblocks to show thatthe factors they find will give the area de-sired.

• have students, working in pairs, (each groupreceives 16 cards) write two numbers oneach of the eight cards, and the GCF of thetwo numbers on the other eight cards. Groupsthen exchange cards and place them on a flatsurface. Students take turns trying to matchpairs of numbers with the GCFs. If a match ismade, the player keeps that pair of cards.The player with the most sets of cards at theend is the winner.

• provide instruction on polynomials. Have stu-dents work in cooperative groups and haveeach group member write a polynomial on anindex card. Cards should be collected,shuffled, and placed in a pile. Students pick acard from the pile and identify the degree ofeach polynomial.

¤ use models, manipulatives, and the chalk-board to teach students how to determinethe width of a rectangle given its area andlength. Provide a collection of rectangles cutfrom colored paper. Give each student sev-eral different colored rectangles. Have stu-dents find area from dimensions or one di-mension given area and the other dimension.Begin with numeric values and progressthrough polynomial expressions for the areaand dimension

Industrial Blueprint/s: (Strategies) Essential Understandings/Math CurriculumFramework/Algebra 1/p. 4-5:

• A relationship exists between arithmetic op-erations and operations with polynomials.

• Polynomials can be represented in a varietyof forms.

• Operations with polynomials can be repre-sented concretely, pictorially, and algebra-ically.

• Polynomial expressions can be used to modelreal-life situations.

• The distributive property is the unifying con-cept for polynomial operations.

• Factoring reverses polynomial multiplication.• There is a relationship between the factors

of a polynomial and the x-intercepts of itsrelated graph.

• Some polynomials cannot be factored overthe set of real numbers.

• Polynomial expressions in a variable x andtheir factors can be used to define functionsby setting y equal to the polynomial expres-sion or y equal to a factor, and these func-tions can be represented graphically.

Resources:

Text: Algebra 1, Integrations, Applications,Connections, Glencoe, pp. 520-549Technology:• TI-83 Graphing Calculator• Algeblaster (S)• Graphlink (S)• Algebra I (ProOne) (S)• Equation Editor (Microsoft Office) (S)• Hyperstudio (S)• PowerPoint (S)

Notes:

A.11 ResourcesLearning About Algebra Tiles - Algebra tilesprovide a useful way to introduce operations onpolynomials. There are patterns of make-your-own tiles included.http://www.ucs.mun.ca/~mathed/t/rc/alg/tiles/tiles1.htmlTechnology:• Best Grapher• ¤ indicates an instructional blueprint for a topic

that has been included in previous math SOLtests and that is not covered in the math text-book.

M9B–7







C/T 12.1, 12.2,12.3, 12.4




A.14 The student will solve quadratic equations in one variable both algebraically and graphi-cally. Graphing calculators will be used both as a primary tool in solving problems and toverify algebraic solutions.

• Factoring Techniques• Graphing Solutions 4 blocks/8 single periods

• Demonstration of the use of one method to find the roots of a quadratic equation and the use ofother methods (at least one) to check accuracy of roots


Unit IV: Quadratic Equations

S CHEM.1


The student will:• solve quadratic equations algebraically or by

using the graphing calculator. When solutionsare represented in radical form, the decimalapproximation will also be given.

• verify algebraic solutions, using the graphingcalculator.

• identify the x-intercepts of the quadratic func-tion as the solutions(s) to the quadratic equa-tion that is formed by setting the given qua-dratic expression equal to zero.

M9B–8


The teacher will:• have students graph y = x2, y = x2 –4, and y

= (x + 2)2 + 5. Students should identify theminimum point of each graph. Have studentspredict the minimum point of y = x2 + 7. Stu-dents should use the graphing calculator toverify their results.

• present the following problem: The length ofeach side of a baseball diamond is 90 feet.What distance must a catcher throw the ballto pick off a base runner stealing second?Give the students a diagram and supply themwith the Pythagorean Theorem. Solve theequation for “c.”

¤ create an activity in which students learn tocalculate a zero of a given function. Explainusing manipulatives and the chalkboard x-intercepts as “zeros” and “roots.” Give stu-dents one term and tell them to provide thesynonyms for that term (e.g., Teacher says,“Zero;” student responds, “x-intercept.”). Pro-vide practice for students with a variety ofproblems that require them to calculate zeroof a given function.


• The zeros or the x-intercepts of the quadraticfunction are the real root(s) or solution(s) ofthe quadratic equation that is formed by set-ting the given quadratic expression equal tozero.

• Quadratic equations can be solved in a vari-ety of ways.

• A quadratic equation can have two solutions,one solution, or no solution.

• A solution to a quadratic equation is the valueor set of values that can be substituted tomake the equation true.

Notes:

Technology:• Best Grapher


Resources:

Text: Algebra 1, Integrations, Applications,Connections, Glencoe, pp. 608-652Technology:• TI-83 Graphing Calculator• Algeblaster (S)• Graphlink (S)• Equation Editor (Microsoft Office) (S)• Hyperstudio (S)• PowerPoint (S)Web Site:• www.pbs.org/wgbh/nova/proof/puzzle/

baseball.html

M9B–9







C/T 12.1, 12.2,12.3, 12.4




A.18 The student will analyze a relation to determine whether a direct variation exists andrepresent it algebraically and graphically, if possible.

• Writing Equations• Problem Solving 4 blocks/8 single periods

Unit V: Direct Variation

E 9.8


The student will:• given a table of values, determine whether a

direct variation exists.• write an equation for a direct variation, given

a set of data.• graph a direct variation from a table of values

or a practical situation.

• Project involving real life situations with direct variation. Cartoon will be drawn for each of thethree situations and as a caption, the variation will be stated in words and algebraically


M9B–11







C/T 12.1, 12.2,12.3, 12.4




A.15 The student will, given a rule, find the values of a function for elements in its domain and locatethe zeros of the function both algebraically and with a graphing calculator. The value of f(x) willbe related to the ordinate on the graph.

• Domain and Range• Zeros• Graphing


Unit VI: Functional Relationships

M A.5; S BIO.1, BIO.2, BIO.8,BIO.9


The student will:• for each x in the domain of f, find f(x).• identify the zeros of the function algebraically

and confirm them, using the graphing calcu-lator.

• Organized pair of related data such as lunch items in school and their prices shown by a mapping or graph of thepicture relationship between the data. Test data to see if a function exists using the vertical line test.

• Representation of the above data in table form. Give four choices of graphs. Ask which graph could be used torepresent the situation.

• Graph depicting which of the following tables is best represented by the line shown in the graph?Give four choices in this form.

xy


M9B–12


The teacher will:• have students provide series of data to place

in a mapping. The domain could be the stu-dents’ names, the range could be theirhomeroom class. Have students provide ad-ditional data that could be represented by amapping or graph.

• ask the students to identify the relationshipbetween consecutive numbers of a sequenceand numbers written on the chalkboard: 5, 3,6, 2, 7, 1, 8, 0.

• have students use the graphing calculator andenter data and equations to determine pat-terns. Ask the students to make up a func-tional formula, generate a data table and bringthe data table on a separate piece of paper toclass. Students are asked to find a rule andexpress it as a formula. Have the studentsgraph some of these formulas on a coordi-nate plane and decide if they are functions(use the vertical line test).

• ask the students to examine a table written onthe chalkboard and discover a pattern in termsof how y changes when x changes. Explain inyour own words how to find y in terms of x.

x y0 51 82 113 144 17

Assume that the pattern continues indefinitely.Use the rule you have found to extend the datato include negative numbers for x. Did you finda unique value for y given a particular valuefor x? Use a formula to describe the patternyou have found.

• tell students that on the first day they will have1 problem for homework, 1 on the secondday, 2 the third day, 3 the fourth day, 5 thefifth day, 8 the sixth day and 13 the seventhday. If this pattern is continued how many prob-lem will be given on the eleventh day? Stu-dents will make a set of ordered pairs andthen graph the set to determine if a function

exists.• present the following problem to the students:

Cosetta built a square pen in the corner of herbackyard. The side of her backyard is 24 feetlonger than the side of the pen. Find the di-mensions of the pen and Cosetta’s backyardif the difference of their areas is 1152 squarefeet.

• give students, in small groups, graphs of vari-ous curves or random points. Have them de-termine if a relationship exists between thefirst and second coordinate, and if so, expressit algebraically. Check for a function using thevertical line test, mapping or on the graphingcalculator. This can be extended to determinethe relationship to data in word problems (e.g.,include a utility bill, renting a car, buying quan-tities instead of single items, service job rates).

• draw a coordinate plane on a flat surface out-side or on the floor. Students pick a numberon the x-axis to be used as a domain element.Give them a rule for which they will find therange value for this number. Students will moveto this point on the plane. “Connect” the stu-dents using yarn or string. Analyze the differ-ent types of graphs obtained.

• divide students into groups. Give each groupa folder. Have them write a relationship on theoutside as well as a list of five numbers to beused as domain elements. They should writethe domain and range elements in set nota-tion on a piece of paper and place it in thefolder. Folders are passed to each group.When the folder is returned to the group withwhich it began, the results will be analyzedand verified.

M9B–13







C/T 12.1, 12.2,12.3, 12.4


• An equation represents the relationship be-tween the independent and dependent vari-ables.

• The object f(x) is the unique object in the rangeof the function f that is associated with theobject x in the domain of f.

• For each x in the domain of f, x is a memberof the input of the function f, f(x) is a memberof the output of f, and the ordered pair [x,f(x)] is a member of f.

• An object x in the domain of f is an x-interceptor a zero of a function f if and only if f(x) = 0.

Resources:

Text: Algebra 1, Integrations, Applications,Connections, Glencoe, pp. 287-302Technology:• TI-83 Graphing Calculator• Algeblaster (S)• Graphlink (S)• Equation Editor (Microsoft Office) (S)• Hyperstudio (S)• PowerPoint (S)

Notes:

A.5 ResourcesFunction Basics - Description and illustrationsof functions and graphing.ht tp: / / tqd.advanced.org/2647/a lgebra/funcbasc.htm

M9B–14

M9B–15







C/T 12.1, 12.2,12.3, 12.4




• Proportion and Percent• Probability• Simplifying Rational Expressions• Operations with Rational Expressions


• Matching activity using a set of 5 word sentences representing real-life situations and a setof 5 equations translated from the word sentences

• Project depicting the use of percents to solve real-life situations• Unit quizzes/tests

Unit VII: Radicals, Expressions, Equations and Their Applications (after SOL Test)

M A.1, A.2

Topics in this unit are designed to provide extension and enrichment in the study of Algebra I.They are to be taught after the administration of the Algebra I SOL Test in the spring. Dependingupon the amount of time remaining in a given school year, topics, required research, and use ofresources may vary.

• Solving Rational Equations• Problem Solving

The student will:• solve equations using proportions.• using percents, solve real-life situations.• find the probability of an event.• simplify rational expressions.• add, subtract, multiply and divide rational ex-

pressions.• solve rational expressions used in real-life set-

tings.

M9B–16


The teacher will:• give sets of real-life, percent-based, situa-

tions. Students use proportions to problemsolve.

• instruct students to use proportions to createscale drawings of their houses.

• provide a pre-unit test for diagnostic purposes.Have students complete the test, using ateacher-made key, and prepare a correctionand error analysis for each error

Math091 - Intermediate Algebra - Chapter 6:Rational expressions: A whole lecture on ratio-nal expressions and how to solve them.http://mesa7.mesa.colorado.edu/~bornmann/classes/math091/lecture/chap_06/tsld001.htm

Review Algebra I - Operations on real numbersand problems to boot.http://www.xnet.com/~fidler/triton/math/review/mat055/mat055.htm

ResourcesLesson 3: Simple Equations - The index tosimple equations lessons. Excellent source ofproblems and how to solve them in little incre-ments.h t tp : / / iasec. fwsd.wednet .edu/ iamath /page2d.htm#Lesson3

Lesson 4: Equations and Inequalities - Theindex to solving equations and inequalities les-sons. Excellent source of problems and how tosolve them in little increments.h t tp : / / iasec. fwsd.wednet .edu/ iamath /page2d.htm#Lesson4

Resources:

Text: Algebra 1, Integrations, Applications,Connections, Glencoe, pp. 192-238, 658-750Technology:• TI-83 Graphing Calculator• Algeblaster (S)• Graphlink (S)• Equation Editor (Microsoft Office) (S)• Hyperstudio (S)• PowerPoint (S)

Notes:

Algebra 1 > Notes > YORKCOUNTY FINAL > YORKCOUNTY > Algebra I A & B

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