(D.C. Heath and Company, 1994)

Transcription

1 Course name: AP Calculus AB Revised: Aug 11, 2008 Course Description: In AP Calculus AB, students will explore four main ideas from calculus: limits, derivatives, indefinite integrals, and definite integrals. Problems will be approached through a balance of multiple representations including: graphically, numerically, analytically/algebraically, and verbally. Wherever practical, concepts will be applied to analyze real-world situations. Graphing calculators and/or computers will be used on a regular basis to help solve problems, experiment, interpret results, and support conclusions. Students will learn to communicate mathematics through the use of a math journal and having time each day to work in groups to exchange ideas and approaches and to reflect on homework assignments. A variety of assessments will be used to including paper based tests and quizzes, homework, group and individual laboratory work. In order to prepare students for the AP exam, tests and quizzes will contain questions similar to those that appear on the AP exam in terms of content, difficulty, and structure. Primary text(s) and other major resources: Calculus of a single variable Larson, Hostetler, and Edwards (D.C. Heath and Company, 1994) Page1of6

2 o unit number & title Objectives (specific skills and knowledge students will have) U1 Prerequisites and Analysis of Graphs Objective set 1: 2-3 Weeks Student will: review the real number system review the cartesian plane review graphs of equations review lines in the plane review functions review trigonometric functions U2 Limits and Continuity Objective set: 4-5 weeks Students will: Evaluate Limits at a Point o 1 sided limits o 2 sided limits o Sandwich Theorem Evaluate Limits involving Infinity o Asymptotic Behavior o End Behavior models o Properties of Limits (Algebraic Analysis) o Visualizing Limits (Graphic Analysis Determine Continuity of a function o Continuity at a Point o Continuous Functions o Discontinuous Functions Removable Discontinuity Jump Discontinuity Infinite Discontinuity Determine and Analyze Rates of Change and Tangent Lines o Average rate of change o Tangent line to a curve o Slope of a curve (algebraically and graphically) Page2of6 Essential Concepts Assessment Students must show proficiency with MLR skills and knowledge in assessments marked Essential in order to progress to the next course level. How are real numbers and sets of real numbers be represented, classified, and ordered? What algebraic techniques are commonly used in calculus? How can data and relationships be organized and represented graphically? Homework and quizzes will be given on a regular basis throughout the course Project #1 Test #1 What is a limit? How do you find a limit with a table, graph, or analytically? When does a limit not exist? What is the definition of continuity for a function on an open or closed interval? What is a continuous function? How can I determine if a function is continuous? What is meant by rates of change? How is the rate of change determined? How are tangent lines and a rate of change related? What is a normal line and how can I determine the formula for a normal

3 o unit number & title Objectives (specific skills and knowledge students will have) o Normal line to a curve (algebraically and graphically) o Instantaneous rate of change U3 The Derivative Objective Set 5-6 Weeks Students will: Determine Rates of Change o Average Speed o Instantaneous Speed Use a variety of methods to determine the Derivative of a Function o Definition of the derivative (difference quotient) o Derivative at a Point o Relationships between the graphs of f and f o Graphing a derivative from data o One sided derivatives Determine if a function is differentiable o Cases where f (x) might fail to exist o Local linearity o Derivatives on the calculator (Numerical derivatives using NDERIV) o Symmetric difference quotient o Relationship between differentiability and continuity o Intermediate Value Theorem for Derivatives Use Rules for Differentiation to determine derivatives of functions o Constant, Power, Sum, Difference, Product, Quotient Rules o Higher order derivatives Apply concepts of the derivative to analyze motion problems o Position, velocity, acceleration, and Page3of6 line? Essential Concepts Assessment Students must show proficiency with MLR skills and knowledge in assessments marked Essential in order to progress to the next course level. Project #2 Test #2 How do I determine a rate of change? What is the difference between an average rate of change and an instantaneous rate of change? How I determine the derivative of a function? How do I determine if a function is differentiable? How do I determine if a function is continuous? What facts can we conclude about a continuous function? How can we use derivatives to analyze motion?

4 o unit number & title Objectives (specific skills and knowledge students will have) jerk o Particle motion o L HÔpital s Rule Apply concepts of the derivative to analyze economic problems o Marginal Cost o Marginal Revenue o Marginal Profit Determine derivatives of Trigonometric Functions Use the Chain Rule to derive composite functions Determine rate of change using Implicit Differentiation o Differential Method o Y Method Determine derivatives of Inverse Trigonometric Functions Determine derivatives of Exponential and Logarithmic Functions U4 Applications of the Derivative Objective Set 5-6 Weeks Students will: Locate Extreme Values of a function o Relative Extrema o Absolute Extrema o Extreme Value Theorem o Definition of a Critical Point Interpret Implications of the Derivative o Rolle s Theorem o Mean Value Theorem o Increasing and Decreasing functions Produce accurate graphs by using the relationships of f and f with f(x) Page4of6 Essential Concepts Assessment Students must show proficiency with MLR skills and knowledge in assessments marked Essential in order to progress to the next course level. How can we use derivatives to analyze economic problems? How do we determine the derivative of trigonometric functions? What are composite functions and how do we determine the derivative? What are implicit functions and how do we determine the derivative? What are inverse trigonometric functions and how do we determine the derivative? What are exponential functions and logarithmic functions and how do we determine the derivative Project #3 Test #3 What are extrema and where do they occur? What can we interpret about a function from the derivative? How is a function related to its first

5 o unit number & title Objectives (specific skills and knowledge students will have) o First derivative test for relative max/min o Second Derivative Concavity Inflection Points Test for relative max/min Analyze optimization problems using calculus Estimate functions using Linearization models o Local Linearization o Tangent Line approximation o Differentials Analyze Related Rate problems U5 The Definite Integral Objective Set 3-4 Weeks Students will: Approximate Areas between curves using summations o Riemann sums Left Right Midpoint Trapezoidal Compare and contrast definite integrals to Riemann sums Use Properties of Definite Integrals o Power Rule o Mean Value Theorem for Definite Integrals Understand how The Fundamental Theorem of Calculus describes the inverse relationship between the integral and derivative Page5of6 Essential Concepts Assessment Students must show proficiency with MLR skills and knowledge in assessments marked Essential in order to progress to the next course level. and second derivatives? What is an optimization problem and how can we use derivatives to solve them? How can we produce estimates of the value of a curve using linearization models? Project #4 Saving material by improving the design of cereal boxes Test #4 What is a related rate problem and how can we use derivatives to solve them? What methods can be used to approximate areas underneath curves? How can we modify our approximation methods to improve accuracy? What are the basic properties of definite integrals? How are derivatives and integrals related? Project #5 Test #5

6 o unit number & title Objectives (specific skills and knowledge students will have) o Part 1 o Part 2 U6 Differential Equations and Mathematical Modeling Objective Set 3 Weeks Students will: Produce Slope Fields for differential equations Evaluate Antiderivatives using common formulas o Indefinite Integrals o Power Formulas o Trigonometric Formulas o Exponential and Logarithmic formulas Analyze Logistic Growth models Unit 7 Applications of Definite Integrals Objective Set 3-4 Weeks Students will explore the following topics: Evaluate and interpret Integrals as net change o Calculating distance traveled o Consumption over time o Net Change from data Calculate areas between curves o Integrating with respect to x o Integrating with respect to y Calculate areas between intersecting curves o Integrating with respect to x o Integrating with respect to y Calculate Volume of solids o Cross sections o Disc Method o Shell Method Page6of6 Essential Concepts Assessment Students must show proficiency with MLR skills and knowledge in assessments marked Essential in order to progress to the next course level. What is a differential equation? How can we visualize the solutions for a differential equation? What are some common antiderivatives? What are logistic growth models and how do I analyze them using calculus? Project #6 Test #6 How can integrals be used to describe motion or other changing variables? How can we use integrals to calculate areas between curves? How can we use integrals to calculate volumes of solids? Project #7 Determine distance of a trip using speed and time data. Test #7

7 Notes This is a sample of a sheet that we typically use to analyze various calculus problems using a multiple approaches. We use it to compliment our work with the graphing calculators. The "notes" section is used for a written and analytical approach. The "table" section below is used for numerical approach with enough room to examine the function, first derivative, and second derivative. The "graphs" section is for a graphical approach allowing the student to see function and derivative behavior either on the coordinate plane and/or using intervals along a number line. Line Graphs Graph of Function Function 1 st Derivative 2 nd Derivative We use this sheet for selected homework problems or in-class problems along with our Ti graphing calculators Graph of 1 st Derivative Graph of 2 nd Derivative

8 CalculusProject DesigninganEnvironmentallyFriendlyPackage Introduction YouhavebeenhiredbytheCEOofamajorcompanytodesignthepackagingforitsnewestcereal.The CEOtakesenvironmentalissuesveryseriouslyandwantstomakesurethattheboxthatyoudesignuses aslittlematerialaspossible. Objectives: Youwillbeabletocalculatethederivativeofafunctionusingtheproductrule Youwillbeabletocalculatethederivativeofafunctionusingthequotientrule Youwillbeableutilizederivativestoanalyzeoptimizationproblems. Youwillbeabletoshowmultiplerepresentationsofamathematicalsolution(algebraic/analytical, numeric,graphical,writtenenglish) Specificationsforthepackaging Thecerealmustbepackagedinarectangularbox. Thebasemustbe timesthewidth.(numbertobeassignedbyteacher) Thevolumeoftheboxmustholdexactly inches 3.(numbertobeassignedbyteacher) Theboxmustusetheleastamountofcardboard(minimizethesurfacearea)tosavemoney. Youmustwriteaclearstepbystepdescriptionofyourdesignprocesssupportedbyfunctions, derivatives,graphs,anddatatablestoconvincetheceothatyourdesignisthemostefficient useofmaterial. Teacher Notes: This Box Project is an example of one of our project that requires the student to apply calculus to solve a real world problem. In this case, the student has to design a cereal box that meets the volume requirements while minimizing the amount of material used in order to help minimize the environmental impact. The students must use an analytical, graphical, numerical, and written approach to describe their solution. Furthermore, to complete the realworld hands-on experience, the students must actually construct their proposed box. The next project (Zoo Project) is also a real-world application of calculus. Students use limited materials to construct a zoo that maximizes living spaces for the animals. Again, the students use multiple approaches (analytical, written, numerical, and graphical) to help gain a deeper understanding of the solution. Consistent with the hands-on experience, students complete the project with a blueprint scaled map of their proposed zoo configuration. Students will explore these, and several other application of calculus throughout the year.

9 Calculus Project1:Designanenvironmentallyfriendlypackage Name,Date,Class,andTitle AllofName,Date,Class,and Projecttitleareclearly writtenattopofthepaper. MostofName,Date,Class, andprojecttitlearewrittenat topofthepaper. SomeofName,Date,Class, andprojecttitlearewrittenat topofthepaper. NoneofName,Date,Class, andprojecttitlearewrittenat topofthepaper. WrittenDescriptionofyour process Averyclearstepbystep descriptionofyourdesign strategyisincluded.allmajor stepsareexplainedclearly andprecisely.key mathematicaltermsareused appropriately. Amostlyclearstepbystep descriptionofyourdesign strategyisincluded.most, butnotall,majorstepsare explainedclearlyand precisely.keymathematical termsareusedappropriately. Astepbystepdescriptionof yourdesignstrategyis included.some,butnotall, majorstepsareexplained. Littleorimproperuseof mathematicalterms. Astepbystepdescriptionof yourdesignstrategyisnot includedorlacksmostmajor steps.littleornouseof mathematicalterms. AlgebraicRepresentationof yourprocess Averyclearalgebraicsolution toyourdesignstrategyis included. Thismeansthatproper formulasandfunctionsare correctlywrittenandsolved. Allstepsareneatlyshown untilaconclusionisfound. Nomistakes. Amostlyclearalgebraic solutiontoyourdesign strategyisincluded. Thismeansthatproper formulasandequationsare correctlywrittenandsolved. Most,butnotall,stepsare neatlyshownuntila conclusionisfound. Onlyoneortwomistakes. Analgebraicsolutiontoyour designstrategyisincluded. Thismeansthatproper formulasandequationsare correctlywrittenandsolved. Somestepsareshown,but someareskippedmakingthe processdifficulttofollow. Threeorfourmistakes. Analgebraicsolutiontoyour designstrategyisnot includedorhasmany mistakes. Veryfewornostepsare shownmakingtheprocess difficulttofollow. Morethanfourmistakes. NumericalRepresentationof yourprocess(datatableof functionsandderivatives) Averycleardatatableshows thebehaviorofallkey functionsandderivativesthat supportyourdesignstrategy. Allofthemostimportant datavaluesarehighlighted andtheirsignificancetoyour solutionisaccurately described.columnsare labeledwithcorrectunits. Amostlycleardatatable showsthebehaviorofallkey functionsandderivativesthat supportyourdesignstrategy. Someofthemostimportant datavaluesarehighlighted andtheirsignificancetoyour solutionisdescribedwithfew mistakes.columnsare labeledwithcorrectunits. Adatatableshowsthe behaviorofsome,butnotall, keyfunctionsandderivatives. Fewornoneofthemost importantdatavaluesare highlighted.incorrectorno useoflabelsandunits. Nodatatableincluded

10 GraphicalRepresentationof yourprocess Verycleargraphsshowthe behaviorofallkeyfunctions andderivativesthatsupport yourdesignstrategy.allof theimportantpointsonthe curvesarehighlightedand theirsignificancetoyour solutionisclearlydescribed. Allaxisareproperlylabeled andhavecorrectunits. Graphsshowthebehaviorof allormostkeyfunctionsand derivativesthatsupportyour designstrategy.mostofthe importantpointsonthe curvesarehighlightedand theirsignificancetoyour solutionisdescribedwithfew mistakes.mostaxisare properlylabeledandhave correctunits. Graphsshowthebehaviorof somefunctionsand derivatives.fewofthe importantpointsonthe curvesarehighlightedand theirsignificancetoyour solutionisnotmadeclear. Axisarenotproperlylabeled. Graphsarenotincludedor havenumerousmajor mistakes. Useofcalculusconcepts Appropriateandaccurate applicationandinterpretation ofderivativesisanimportant partofyourdesignsolution. Theseconceptsareclearly incorporatedinyourwritten, algebraic,numeric,and graphicalrepresentations. Appropriateandaccurate applicationandinterpretation ofderivativesispartofyour designsolution.these conceptsareincorporatedin yourwritten,algebraic, numeric,andgraphical representationswithsome mistakes. Appropriateandaccurate applicationandinterpretation ofderivativesispartofyour designsolution.these conceptsareincorporatedin yoursomeofyourwritten, algebraic,numeric,and graphicalrepresentations withmanymistakes. Appropriateandaccurate applicationandinterpretation ofderivativesisnotpartof yourdesignsolutionor containsmanymistakes. ActualCerealBox Youhaveconstructedthefull sizecerealboxthatis accuratetoyourcalculations. Allkeydimensionsofthebox areclearlylabeledwithunits (length,width,height,volume andsurfacearea) Youhaveconstructedfullsize cerealboxthatismostly accuratetoyourcalculations withoneortwomistakes. Mostkeydimensionsofthe boxarelabeledwithunits (length,width,height,volume andsurfacearea) Youhaveconstructedthe cerealboxthatissomewhat accuratetoyourcalculations withthreeorfourmistakes. Somekeydimensionsofthe boxarelabeledwithunits (length,width,height,volume andsurfacearea) Youhavenotconstructedthe cerealboxoryouhavemore thanfourmistakes. OnTime Thecompletedprojectis deliveredonorbeforethe duedate: StartofclassonTuesday December9. Thecompletedprojectis deliveredwithinonedayafter theduedate: StartofclassonTuesday December9. Thecompletedprojectis deliveredwithinthreedays aftertheduedate: StartofclassonTuesday December9. Thecompletedprojectis deliveredmorethanthree daysaftertheduedate: StartofclassonTuesday December9

11 Calculus Project Optimizeour ZooUsingFunctions(Revised) Introduction Youhavebeenhiredbythelocalcommunitytoadvisethemonhowtobestdesigntheirnewzoo.Theyhave limitedmaterialstouseandtheywanttoensurethattheygetthemostoutofthematerials.youwillneedto examinetheirrequirements,determinethebestwaytomeettherequirements,andmakeascalemodelormap ofyourproposeddesign. Objectives: Youwillbeabletomodelrealsituationswithfunctions. Youwillbeabletorepresentthefunctionsalgebraically,numerically(datatable),andgraphically. Youwillbeabletouseproportionstoproduceamaptoscale. ZooRequirements 1. Gorilla a. Wehave feetoffencetouseforthegorillacage. b. Wewantthegorillatoliveinarectangularspacewiththemaximumareapossibletoswingaround. c. 2. SheepandPigs a. Wehave feetoffencetouseforthesheepandpigs. b. Wewantthesheepandpigstoliveintwoadjacentrectangularpens. c. Onesideofthesheepandpigsareawillbeborderedbyariverfordrinking d. Wewanttogivethesheepandpigsthemaximumareapossible. e. Thepigareashouldbethesamesizeasthesheeparea. f.

12 3. Lions,Tigers,andBears a. Wehave feetoffencetouseforthelionstigersandbears. b. Theanimalswillliveinonerectangularareadividedintothreeequalpartsasshownbelow. c. Wewantthelions,tigers,andbearstoliveinthemaximumareapossible. d. 4. SharkAquarium a. Wewantarectangularaquariumwiththeleft,right,front,back,andbottomismadeofclearglass sopeoplecanseethesharks b. Thetopisopen(noglassusedfortop) c. Thelengthshouldbe timesthewidth. d. Thetankshouldholdavolumeof. e. Wewanttominimizetheamountofglassthatweusebecausethematerialisveryexpensive. f.

13 Element 4Mastered 3 Proficient 2 Developing 1 Emerging Labeledpictureof description UseAlgebratocreate theappropriate Function(s) NumericalBehaviorof thefunction GraphicalBehaviorof thefunction Youalwaysdothesewithno errors: drawaclearpictureofeach applicationclearlylabeledwith allknownandunknown variables. Youalwaysdothesenoerrors: writekeyequations/functions thatshowhoweachofthe variablesarerelatedtoone another.(i.e.v=lwh). arrangetheseequationstowrite anappropriatefunction(s)that canbeusedtodescribeeach problem(i.e.surfacearea function) Youalwaysdothesenoerrors: produceacleartable(s)of valuesthatshowthebehaviorof yourfunction(s) Uselabels,variables,andunits onyourtable(s)soit sveryeasy toknowwhatthedatameans. Highlightanydataonthe table(s)importanttoyour solutionandclearlyexplainwhy ithelpsyousolvetheproblem. Youalwaysdothesewithno errors: produceacleargraph(s)that showthebehaviorofyour function(s) Uselabels,variables,andunits onyourgraph(s)soit sveryeasy toknowwhatthecurvemeans Highlightanypoint(s)onthe curvethatisimportanttoyour solutionandclearlyexplainwhy ithelpsyousolvetheproblem. You domostofthesewithfew(1 2)minorerrors: Drawaclearpictureofeach applicationclearlylabeledwith allknownandunknown variables. You domostofthesewithfew(1 2)minorerrors: writekeyequations/functions thatshowhoweachofthe variablesarerelatedtoone another.(i.e.v=lwh). arrangetheseequationstowrite anappropriatefunction(s)that canbeusedtodescribeeach problem(i.e.surfacearea function) You domostofthesewithfew(1 2)minorerrors: produceacleartable(s)of valuesthatshowthebehaviorof yourfunction(s) Uselabels,variables,andunits onyourtable(s)soit sveryeasy toknowwhatthedatameans. Highlightanydataonthe table(s)importanttoyour solutionandclearlyexplainwhy ithelpsyousolvetheproblem. You domostofthesewithfew(1 2)minorerrors: produceacleargraph(s)that showthebehaviorofyour function(s) Uselabels,variables,andunits onyourgraph(s)soit sveryeasy toknowwhatthecurvemeans Highlightanypoint(s)onthe curvethatisimportanttoyour solutionandclearlyexplainwhy ithelpsyousolvetheproblem. Youdomostofthesewithsome(3 6)errors Drawaclearpictureofeach applicationclearlylabeledwith allknownandunknown variables. Youdomostofthesewithsome(3 6)errors writekeyequations/functions thatshowhoweachofthe variablesarerelatedtoone another.(i.e.v=lwh). arrangetheseequationstowrite anappropriatefunction(s)that canbeusedtodescribeeach problem(i.e.surfacearea function) Youdomostofthesewithsome(3 6)errors produceacleartable(s)of valuesthatshowthebehaviorof yourfunction(s) Uselabels,variables,andunits onyourtable(s)soit sveryeasy toknowwhatthedatameans. Highlightanydataonthe table(s)importanttoyour solutionandclearlyexplainwhy ithelpsyousolvetheproblem. Youdomostofthesewithsome(3 6)errors produceacleargraph(s)that showthebehaviorofyour function(s) Uselabels,variables,andunits onyourgraph(s)soit sveryeasy toknowwhatthecurvemeans Highlightanypoint(s)onthe curvethatisimportanttoyour solutionandclearlyexplainwhy ithelpsyousolvetheproblem. You dofewornoneoftheseor havemajorerrors. Drawaclearpictureofeach applicationclearlylabeledwith allknownandunknown variables. You dofewornoneoftheseor havemajorerrors. writekeyequations/functions thatshowhoweachofthe variablesarerelatedtoone another.(i.e.v=lwh). arrangetheseequationstowrite anappropriatefunction(s)that canbeusedtodescribeeach problem(i.e.surfacearea function) You dofewornoneoftheseor havemajorerrors. produceacleartable(s)of valuesthatshowthebehaviorof yourfunction(s) Uselabels,variables,andunits onyourtable(s)soit sveryeasy toknowwhatthedatameans. Highlightanydataonthe table(s)importanttoyour solutionandclearlyexplainwhy ithelpsyousolvetheproblem. You dofewornoneoftheseor havemajorerrors. produceacleargraph(s)that showthebehaviorofyour function(s) Uselabels,variables,andunits onyourgraph(s)soit sveryeasy toknowwhatthecurvemeans Highlightanypoint(s)onthe curvethatisimportanttoyour solutionandclearlyexplainwhy ithelpsyousolvetheproblem.

14 Element 4Mastered 3 Proficient 2 Developing 1 Emerging Solvetheproblem Youalwaysdothesewithno errors: Clearlyinterpretallthe informationfromyourgraphs, tables,andalgebratosolvethe problem. Thesolutioncouldbeone numberorasetofnumbersso labelsandunitsareusedto clarify. Ifpossible,apictureisusedto clarifyyoursolutionandshow thatyouunderstandhowthe numberssolvetheoriginal problem. Checktoseethatyoursolution makessense. ScaleMap Youalwaysdothesewithno errors: Makeacreative,accurately scaledmapthatrepresentsyour proposalfortheconfigurationof thezoo. Showalegendthatindicatesthe scaleofactuallengthstomap lengths. Labelallrequiredareasand clearlyindicateanykey measurements OnTime You reprojectiscompletedand handedinbeforeoratthe beginningofclassontheduedate: January14,2009 You domostofthesewithfew(1 2)minorerrors: Clearlyinterpretallthe informationfromyourgraphs, tables,andalgebratosolvethe problem. Thesolutioncouldbeone numberorasetofnumbersso labelsandunitsareusedto clarify. Ifpossible,apictureisusedto clarifyyoursolutionandshow thatyouunderstandhowthe numberssolvetheoriginal problem. Checktoseethatyoursolution makessense. You domostofthesewithfew(1 2)minorerrors: Makeacreative,accurately scaledmapthatrepresentsyour proposalfortheconfigurationof thezoo. Showalegendthatindicatesthe scaleofactuallengthstomap lengths. Labelallrequiredareasand clearlyindicateanykey measurements You reprojectiscompletedand handedinwithin2dayspastthe duedate: January14,2009 Youdomostofthesewithsome(3 6)errors Clearlyinterpretallthe informationfromyourgraphs, tables,andalgebratosolvethe problem. Thesolutioncouldbeone numberorasetofnumbersso labelsandunitsareusedto clarify. Ifpossible,apictureisusedto clarifyyoursolutionandshow thatyouunderstandhowthe numberssolvetheoriginal problem. Checktoseethatyoursolution makessense. Youdomostofthesewithsome(3 6)errors Makeacreative,accurately scaledmapthatrepresentsyour proposalfortheconfigurationof thezoo. Showalegendthatindicatesthe scaleofactuallengthstomap lengths. Labelallrequiredareasand clearlyindicateanykey measurements You reprojectiscompletedand handedinwithin4dayspastthe duedate: January14,2009 You dofewornoneoftheseor havemajorerrors. Clearlyinterpretallthe informationfromyourgraphs, tables,andalgebratosolvethe problem. Thesolutioncouldbeone numberorasetofnumbersso labelsandunitsareusedto clarify. Ifpossible,apictureisusedto clarifyyoursolutionandshow thatyouunderstandhowthe numberssolvetheoriginal problem. Checktoseethatyoursolution makessense. You dofewornoneoftheseor havemajorerrors. Makeacreative,accurately scaledmapthatrepresentsyour proposalfortheconfigurationof thezoo. Showalegendthatindicatesthe scaleofactuallengthstomap lengths. Labelallrequiredareasand clearlyindicateanykey measurements You reprojectiscompletedand handedinmorethan4dayspast theduedate: January14,2009