Mathematics 20-1



MATHEMATICS 20-1Systems of Equations and InequalitiesHigh School collaborative venture withEdmonton Christian, Harry Ainlay, J. Percy Page, Jasper Place, Millwoods Christian, Ross Sheppard and W. P. Wagner, M. E LaZerte, McNally, Queen Elizabeth, Strathcona and VictoriaEdm Christian High: Aaron TrimbleHarry Ainlay: Ben LuchkowHarry Ainlay: Darwin HoltHarry Ainlay: Lareina RezewskiHarry Ainlay: Mike ShrimptonJ. Percy Page: Debbie YoungerJasper Place: Matt KatesJasper Place: Sue DvorackMillwoods Christian: Patrick YpmaRoss Sheppard: Patricia ElderRoss Sheppard: Dean WallsW. P. Wagner: Amber SteinhauerM. E. LaZerte: Teena WoudstraQueen Elizabeth: David UnderwoodStrathcona: Christian Digout Victoria: Steven Dyck McNally: Neil PetersonFacilitator: John Scammell (Consulting Services)Editor: Jim Reed (Contracted)2010 - 2011TABLE OF CONTENTSSTAGE 1 DESIRED RESULTSPAGEBig IdeaEnduring UnderstandingsEssential Questions444Knowledge Skills56STAGE 2 ASSESSMENT EVIDENCETransfer Task (on a separate page which could be photocopied & handed out to students)Investing in the Future by Looking in the PastTeacher Notes for Transfer Task and RubricTransfer Task and RubricRubricPossible Solution9101417STAGE 3 LEARNING PLANSLesson #1 Solving Systems Graphically21Lesson #2 Solving Systems Algebraically25Lesson #3 Linear Inequalities Two Variables28Lesson #4 Quadratic Inequalities One Variable33Lesson #5 Quadratic Inequalities Two Variables36 Mathematics 20-1 Systems of Equations and InequalitiesSTAGE 1 Desired Results Big Idea: Implementation note:Post the BIG IDEA in a prominentplace in your classroom and refer to it often.The solution to a problem may not be a single value, but a range of values. Using test points and intervals is foundational to further study in mathematics. Enduring Understandings:Students will understand …A system involving quadratic equations can have no solution one solution, two solutions or infinitely many solutions. Quadratic inequalities involve a solution with a range of values.An inequality in one variable has a range of x-values for its solution. (linear axis)An inequality in two variables has a range of coordinates for its solution. (shaded region)Solutions can be found graphically or algebraically.The solution to an inequality does not include the line (or curve) if an equal sign is not included.When graphing inequalities, the line (or curve) should be broken if an equal sign is not included and solid if the equal sign is included. Essential Questions:Where are inequalities used in real life?Does a system involving a quadratic always have a solution?When is it appropriate to have a range of values as a solution?Implementation note: Ask students to consider one of the essential questions every lesson or two.Has their thinking changed or evolved?Will the methods we learned in Math 10C work for systems in involving a quadratic equation? Knowledge:Enduring UnderstandingList enduring understandings (the fewer the better)Specific OutcomesList the reference # from the Alberta Program of StudiesDescription ofKnowledgeThe paraphrased outcome that the group is targetingStudents will understand…A system involving quadratic equations can have no solution one solution, two solutions or infinitely many solutions. *RF 6Students will know …the points of intersection are the solutions of a system of linear-quadratic or quadratic-quadratic equationsa system of linear-quadratic or quadratic-quadratic equations may have zero, one, two or an infinite number of solutionsStudents will understand…Quadratic inequalities involve a solution with a range of values. RF 7, 8Students will know …test points can be used to determine the solution region that satisfies an inequalitya solid line should be used when the boundary line is included in the inequality. A broken line should be used when the boundary line is not included in the inequalityStudents will understand…An inequality in one variable has a range of x-values for its solution. (linear axis) RF 8Students will know …test points can be used to determine the solution region that satisfies an inequalitya solid point indicates that it is part of the solution, whereas an open dot is not part of the solutionStudents will understand…An inequality in two variables has a range of coordinates for its solution. (shaded region) RF 7Students will know …test points can be used to determine the solution region that satisfies an inequalitya solid line should be used when the boundary line is included in the inequality. A broken line should be used when the boundary line is not included in the inequalityStudents will understand…Solutions can be found graphically or algebraically. RF 7Students will know …the graph of a linear or quadratic inequality can be sketched with or without technologylinear or quadratic inequalities can be used to solve some problemsStudents will understand…The solution to an inequality does not include the line (or curve) if an equal sign is not included. RF 7Students will know …test points can be used to determine the solution region that satisfies an inequalitya solid line should be used when the boundary line is included in the inequality. A broken line should be used when the boundary line is not included in the inequalityStudents will understand…When graphing inequalities, the line (or curve) should be broken if an equal sign is not included and solid if the equal sign is included. RF 7Students will know …a solid line should be used when the boundary line is included in the inequality. A broken line should be used when the boundary line is not included in the inequality8888I*RF = Relations and Functions Skills:Enduring UnderstandingList enduring understandings (the fewer the better)Specific OutcomesList the reference # from the Alberta Program of StudiesDescription of SkillsThe paraphrased outcome that the group is targetingStudents will understand…A system involving quadratic equations can have no solution one solution, two solutions or infinitely many solutions. *RF 6Students will be able to…explain, using examples, why a system of linear-quadratic or quadratic-quadratic equations may have zero, one, two or an infinite number of solutionsStudents will understand…Quadratic inequalities involve a solution with a range of values. RF 7, 8Students will be able to…explain, using examples, how test points can be used to determine the solution region that satisfies an inequalitysketch, with or without technology, the graph of a linear or quadratic inequalitydetermine the solution of a quadratic inequality in one variable, using strategies such as case analysis, graphing, roots and test points, or sign analysis; and explain the strategy usedinterpret the solution to a problem that involves a quadratic inequality in one variableStudents will understand…An inequality in one variable has a range of x-values for its solution. (linear axis) RF 8Students will be able to…determine the solution of a quadratic inequality in one variable, using strategies such as case analysis, graphing, roots and test points, or sign analysis; and explain the strategy usedrepresent and solve a problem that involves a quadratic inequality in one variableinterpret the solution to a problem that involves a quadratic inequality in one variableStudents will understand…An inequality in two variables has a range of coordinates for its solution. (shaded region) RF 7Students will be able to…explain, using examples, how test points can be used to determine the solution region that satisfies an inequalityexplain, using examples, when a solid or broken line should be used in the solution for an inequalitysketch, with or without technology, the graph of a linear or quadratic inequalitysolve a problem that involves a linear or quadratic inequalityStudents will understand…Solutions can be found graphically or algebraically. RF 6, 7, 8Students will be able to…relate a system of linear-quadratic or quadratic-quadratic equations to the context of a given problemdetermine and verify the solution of a system of linear-quadratic or quadratic-quadratic equations graphically, with technologydetermine and verify the solution of a system of linear-quadratic or quadratic-quadratic equations algebraicallyexplain the meaning of the points of intersection of a system of linear-quadratic or quadratic-quadratic equationsexplain, using examples, why a system of linear-quadratic or quadratic-quadratic equations may have zero, one, two or an infinite number of solutionssolve a problem that involves a system of linear-quadratic or quadratic-quadratic equations, and explain the strategy usedexplain, using examples, how test points can be used to determine the solution region that satisfies an inequality.explain, using examples, when a solid or broken line should be used in the solution for an inequalitysketch, with or without technology, the graph of a linear or quadratic inequalitysolve a problem that involves a linear or quadratic inequalitydetermine the solution of a quadratic inequality in one variable, using strategies such as case analysis, graphing, roots and test points, or sign analysis; and explain the strategy usedStudents will understand…The solution to an inequality does not include the line (or curve) if an equal sign is not included. RF 7Students will be able to…explain, using examples, when a solid or broken line should be used in the solution for an inequalityStudents will understand…When graphing inequalities, the line (or curve) should be broken if an equal sign is not included and solid if the equal sign is included. RF 7Students will be able to…explain, using examples, when a solid or broken line should be used in the solution for an inequalityImplementation note:Teachers need to continually askthemselves, if their students are acquiring the knowledge and skills needed for the unit.* RF = Relations and FunctionsSTAGE 2 Assessment Evidencesired Results Desired ResultsResults Desired Results Investing in the Future by Looking in the Past Teacher NotesThere is one transfer task to evaluate student understanding of the concepts relating to systems of equations and inequalities. A photocopy-ready version of the transfer task is included in this section.Implementation note:Students must be given the transfer task & rubric at the beginning of the unit. They need to know how they will be assessed and what they are working toward.Each student will:Analyze the given dataCreate equations that match the data by performing a regression on each set of data using technologyCreate graphs to compare the values of each investmentInterpret the graphs and make decisions based on the interpretationImplementation note:Teachers need to consider what performances and products will reveal evidence of understanding?What other evidence will be collected to reflectthe desired results? Investing in the Future by Looking at the Past - Assessment TaskA year ago, after a few months of hard work, you and your friends found yourselves with some money to invest. You decided to invest a portion of this money in the stock market and found a broker to manage your investments. You chose companies that were familiar to you and hoped for the best. The following table shows the amount of money each person invested in stocks at the end of last summer and the value of that investment after a year.NameAmount of money investedValue of investment after 12 monthsAli$800$464Ben$1100$956Khalid$900$1212Dimitro$1000$1144Pauline$900$588Sam$1000$1144Investing in the Future by Looking at the PastThough some people in the group had some good fortune, overall the group lost money. You’ve decided to analyze last year’s market activity to see how you could’ve better managed your investments. Below is a table showing the activity of the six pany invested in?(and name of investor)MonthShare Value (unit price)Number of SharesMarket Value of InvestmentCoach (Ali)023.4534.12800.00?336.0234.121229.00?638.5734.121316.00?931.1034.121061.00?1213.6034.12464.00?????Google (Ben)0683.231.611100.00?3468.011.61753.50?6381.371.61614.00?9423.291.61681.50?12593.791.61956.00?????McDonald's (Khalid)046.5619.33900.00?350.5919.33978.00?654.6319.331056.00?958.6719.331134.00?1262.7019.331212.00?????Nike (Dimitro)044.0722.691000.00?352.8022.691198.00?656.7722.691288.00?955.9722.691270.00?1250.4222.691144.00Investing in the Future by Looking at the PastAnalysisAn excellent way to assess the changes in each investment is through graphical representation. Using the values given in the table above, perform a regression to determine the quadratic equation of each investment except McDonald’s, for which you will find a linear equation. Each of these equations should describe the market value of the investment as a function of time. Round all values to the nearest tenth, where necessary.Once you have determined the equation for each of the investments, you will be creating graphs to compare the value of different investments at any given time. For each of the following pairs of investments, respond to the indicated pany invested in?(and name of investor)MonthShare Value (unit price)Number of SharesMarket Value of InvestmentRoyal Bank (Pauline)029.2030.82900.00?346.8230.821443.00?651.0130.821572.00?941.7630.821287.00?1219.0830.82588.00?????Telus (Sam)039.1025.581000.00?334.0025.58874.00?632.6325.58856.00?936.0325.58946.00?1242.6125.581144.00Investing in the Future by Looking at the PastSituation #1 – Dimitro vs PaulineWhen is their investments equal in value?When is Dimitro’s investment greater in value than Pauline’s investment?After one year, how much greater was Dimitro’s investment in value than Pauline’s?If the trend continues for both investments, will the market value of each investment ever be equal again?Situation #2 – Khalid vs AliWhen is their investments equal in value?When is Ali’s investment greater in value than Khalid’s investment?After one year, how much greater was Khalid’s investment in value than Ali’s?If the trend continues for both investments, will the market value of each investment ever be equal again?Situation #3 – Khalid vs SamWhen is their investments equal in value?When is Sam’s investment greater in value than Khalid’s investment?After one year, how much greater was Khalid’s investment in value than Sam’s?If the trend continues for both investments, will the market value of each investment ever be equal again?Situation #4 – Ben vs SamWhen is their investments equal in value?When is Sam’s investment greater in value than Ben’s investment?After one year, how much greater was Sam’s investment in value than Ben’s?If the trend continues for both investments, will the market value of each investment ever be equal again?ExtensionIf you could go back and choose one of these investments, which would you choose?If you were allowed to sell your shares one time during the year and buy into a different stock, which stocks would you choose and when would you trade? How much profit would you make that year?If you were allowed to stocks as many times as you would like, which choices would you make? Discuss and justify your investment decisions AssessmentMathematics 20-1Systems of Equations and InequalitiesInvesting in Stocks Rubric 1ComponentDescription of Requirements AssessmentMathematical ContentAnalyzes information and performs the appropriate regression.IN 1 2 3 4Creates appropriate equations based on the data with and/or without technologyIN 1 2 3 4- Creates accurate graphsIN 1 2 3 4Presentation of DataCommunicates findings using graphsIncludes mathematical vocabulary, notation and symbolismIN 1 2 3 4Interpretation of DataExplains significance of key findings in the graphsShows clear understanding of graphical data by accurately answering the assigned questionsIN 1 2 3 4Investing in Stocks Rubric 2LevelExcellentProficientAdequateLimitedInsufficientCriteria4321BlankMath ContentComparison 1All required elements are present and correctAll required elements are present but may contain minor errorsSome required elements are missing, or contain major errorsMost required elements are missing or incorrectNo score is awarded as there is no evidence givenMath ContentComparison 2All required elements are present and correctAll required elements are present but may contain minor errorsSome required elements are missing, or contain major errorsMost required elements are missing or incorrectNo score is awarded as there is no evidence givenMath Content Comparison 3All required elements are present and correctAll required elements are present but may contain minor errorsSome required elements are missing, or contain major errorsMost required elements are missing or incorrectNo score is awarded as there is no evidence givenMath Content Comparison 4All required elements are present and correctAll required elements are present but may contain minor errorsSome required elements are missing, or contain major errorsMost required elements are missing or incorrectNo score is awarded as there is no evidence givenGraphsPresentation of data is clear, precise and accuratePresentation of data is complete and unambiguousPresentation of data is simplistic and plausiblePresentation of data is vague and inaccuratePresentation of data is incompre-hensibleExplains ChoicesProvides insightful explanationsProvides logical explanationsProvides explanations that are complete but vagueProvides explanations that are incomplete or confusing.No explanation is providedWhen work is judged to be limited or insufficient, the teacher makes decisions about appropriate intervention to help the student improveGlossaryboundary line – A line on the coordinate plane that separates the plane into two regions compound inequality – A compound statement expressing two inequalities at the same timeinequality – A mathematical statement comparing expressions that may not be equal. These can be written using the symbols less than ( QUOTE ), greater than ( QUOTE ), less than or equal to (), greater than or equal to (), and not equal to (). [Math 20-1 (McGraw-Hill, page 588)]model - Method of simulating real-life situations with mathematical equations to forecast their future behavior (source)solution region – All the points in the Cartesian plane that satisfy an inequality. Also known as the solution set. [Math 20-1 (McGraw-Hill, page 590)]system of equations – A group of equations that are to be considered at the same timesystems of linear-quadratic equations – A linear equation and a quadratic equation involving the same variables. A graph of the system involves a line and a parabola. [Math 20-1 (McGraw-Hill, page 591)]systems of quadratic-quadratic equations – Two quadratic equations involving the same variables. A graph of the system involves two parabolas. [Math 20-1 (McGraw-Hill, page 591)]test point – A point not on the boundary of the graph of an inequality that is representative of all the points in a region. A point that is used to determine whether the points in a region satisfy the inequality. [Math 20-1 (McGraw-Hill, page 591)]verify - Verifying a solution ensures the solution satisfies any equation or inequality by using substitutionGlossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.Possible Solution to Transfer Task TitleNike vs Royal BankRoyal Bank (Pauline): Nike (Dimitro): The two plans are equal at 0.65 months and 9.12 months.The market value of Dimitro’s investment is greater than Pauline’s from 0.65 months to 9.12 months. After one year:The market value of Pauline’s investment in Royal Bank is $588.The market value of Dimitro’s investment in Nike is $1144.Dimitro’s investment is worth $556 more than Pauline’s.If the trend continues for both investments, Pauline’s and Dimitro’s investments will never be equal again.McDonalds vs CoachMcDonalds (Khalid): Coach (Ali): The two plans are equal at 0.62 months and 8.54 months.The market value of Ali’s investment is greater than Khalid’s from 0.62 months to 8.54 months. After one year:The market value of Ali’s investment in Coach is $464.The market value of Khalid’s investment in McDonalds is $1212.Khalid’s investment is worth $748 more than Ali’s.If the trend continues for both investments, Ali’s and Khalid’s investments will never be equal again.Google vs TelusGoogle TelusSolve for points where there meet:They are equal at 1.20 months with a value of $288.64If the trends continue they will meet again at 15.16 months and their value would be:They are again equal at 15.16 months with a value of $1469.35In the first year Google is worth less then Telus from 1.20 months till the end of the year.McDonalds vs TelusMcDonalds TelusSolve for points where there meet:They are equal at 1.28 months with a value of $933.28If the trends continue they will meet again at 13.06 months and their value would be:They are again equal at 13.06 months with a value of $1239.56In the first year McDonalds is worth more then Telus from 1.28 months till the end of the year.STAGE 3 Learning PlansLesson 1Solving Systems GraphicallySTAGE 1BIG IDEA: The solution to a problem may not be a single value, but a range of values. Using test points and intervals is foundational to further study in mathematics.ENDURING UNDERSTANDINGS:Students will understand …A system involving quadratic equations can have no solution one solution, two solutions or infinitely many solutions. ESSENTIAL QUESTIONS:Does a system involving a quadratic always have a solution?When is it appropriate to have a range of values as a solution?Will the methods we learned in Math 10C work for systems in involving a quadratic equation?KNOWLEDGE:Students will know …the points of intersection are the solutions of a system of linear-quadratic or quadratic-quadratic equationsa system of linear-quadratic or quadratic-quadratic equations may have zero, one, two or an infinite number of solutionsSKILLS:Students will be able to …relate a system of linear-quadratic or quadratic-quadratic equations to the context of a given problemdetermine and verify the solution of a system of linear-quadratic or quadratic-quadratic equations graphically, with technologyexplain the meaning of the points of intersection of a system of linear-quadratic or quadratic-quadratic equationsexplain, using examples, why a system of linear-quadratic or quadratic-quadratic equations may have zero, one, two or an infinite number of solutionssolve a problem that involves a system of linear-quadratic or quadratic-quadratic equations, and explain the strategy usedImplementation note:Each lesson is a conceptual unit and is not intended to be taught on a one lesson per block basis. Each represents a concept to be covered and can take anywhere from part of a class to several classes to complete.Lesson SummaryStudents will solve linear-quadratic and quadratic-quadratic systems graphically. Lesson PlanHookScenario on page 432 in the McGraw Hill TextbookLesson GoalSolve linear-quadratic and quadratic-quadratic systems Activate Prior KnowledgeReview the three possible solutions of a linear-linear system of equations.Solve the following system graphically and algebraically. Determine the numbers of solutions in each case.x + y = -5y = -x + 5x + y = -5y = -x + 3x + y = -53x + 2y = 4LessonStudents will sketch a scenario, which would involve: no solution, one solution, or many solutions, between a linear-quadratic system.Repeat for a quadratic-quadratic system.Discuss each scenario with technology (students need to generate equations that go with the scenarios created above). Going Beyond Discuss coincident parabolas and systems of other functions (i.e. rational, radical, absolute value, etc.)Graph systems of equations without technology.ResourcesMath 20-1 (McGraw-Hill Ryerson: sec 8.1)Pre –Calculus 11 (Pearson: 5.4) Supporting (Ron Blond’s applet)AssessmentExit slip- students will determine their own linear-quadratic system and sketch their solution and explain their solution.Glossaryboundary line – A line on the coordinate plane that separates the plane into two regions compound inequality – A compound statement expressing two inequalities at the same timemodel - Method of simulating real-life situations with mathematical equations to forecast their future behavior (source)system of equations – A group of equations that are to be considered at the same timesystems of linear-quadratic equations – A linear equation and a quadratic equation involving the same variables. A graph of the system involves a line and a parabola. [Math 20-1 (McGraw-Hill, page 591)]systems of quadratic-quadratic equations – Two quadratic equations involving the same variables. A graph of the system involves two parabolas. [Math 20-1 (McGraw-Hill, page 591)]verify - Verifying a solution ensures the solution satisfies any equation or inequality by using substitutionGlossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.Other Lesson 2Solving Systems AlgebraicallySTAGE 1BIG IDEA: The solution to a problem may not be a single value, but a range of values. Using test points and intervals is foundational to further study in mathematics.ENDURING UNDERSTANDINGS:Students will understand …A system involving quadratic equations can have no solution one solution, two solutions or infinitely many solutions. can be found graphically or algebraically.ESSENTIAL QUESTIONS:When is it appropriate to have a range of values as a solution?Will the methods we learned in Math 10C work for systems in involving a quadratic equation?KNOWLEDGE:Students will know …the points of intersection are the solutions of a system of linear-quadratic or quadratic-quadratic equationsa system of linear-quadratic or quadratic-quadratic equations may have zero, one, two or an infinite number of solutionsSKILLS:Students will be able to …relate a system of linear-quadratic or quadratic-quadratic equations to the context of a given problemdetermine and verify the solution of a system of linear-quadratic or quadratic-quadratic equations graphically, with technologydetermine and verify the solution of a system of linear-quadratic or quadratic-quadratic equations algebraicallyexplain the meaning of the points of intersection of a system of linear-quadratic or quadratic-quadratic equationsexplain, using examples, why a system of linear-quadratic or quadratic-quadratic equations may have zero, one, two or an infinite number of solutionssolve a problem that involves a system of linear-quadratic or quadratic-quadratic equations, and explain the strategy usedLesson SummaryStudents will solve linear-quadratic and quadratic-quadratic systems algebraically and through problem solving.Hook – Show video clips of basketball players doing an “alley-oop”. the relevance of the intersection point of the quadratic equations representing the trajectory of the ball and the player doing the “dunk”. Lesson Goal In this lesson students will model, solve algebraically, verify and interpret solutions of a system of linear-quadratics and quadratic-quadratic equations.Activate Prior KnowledgeSolve and verify the following linear system algebraically (using both substitution & elimination).2x – 3y = -24x + y = 24The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many children and how many adults attended?LessonReview the Quadratic Formula and how it is used to find roots. Emphasize that the roots are the solution to the system.Solve and verify linear-quadratic system and a word problem by:substitutionelimination. Explain the meaning of the solution. If the quadratic formula is used to solve, emphasize the meaning of the roots of the resultant equation that is created from the substitution/elimination.Solve and verify quadratic-quadratic system and a word problem by substitutioneliminationExplain the meaning of the solution. If the quadratic formula is used to solve, emphasize the meaning of the roots of the resultant equation that is created from the substitution/elimination.Describe a real life application that would represent this system. Going BeyondSolving other systems algebraically involving different types of functions (cubic, polynomial, radical, absolute value etc.)ResourcesMath 20-1 (McGraw-Hill Ryerson: sec 8.2)YouTube videos: Supporting Provide all equations in the modelling situations.Solve graphically first and explain the solutions in the context of the question.AssessmentExit SlipThink Pair Share Glossarymodel - Method of simulating real-life situations with mathematical equations to forecast their future behavior (source)system of equations – A group of equations that are to be considered at the same timeverify - Verifying a solution ensures the solution satisfies any equation or inequality by using substitutionOtherLesson 3Linear Inequalities Two VariablesSTAGE 1BIG IDEA: The solution to a problem may not be a single value, but a range of values. Using test points and intervals is foundational to further study in mathematics.ENDURING UNDERSTANDINGS:Students will understand …An inequality in one variable has a range of x-values for its solution. (linear axis)An inequality in two variables has a range of coordinates for its solution. (shaded region)Solutions can be found graphically or algebraically.When graphing inequalities, the line (or curve) should be broken if an equal sign is not included and solid if the equal sign is included.The solution to an inequality does not include the line (or curve) if an equal sign is not included.ESSENTIAL QUESTIONS:Where are inequalities used in real life?When is it appropriate to have a range of values as a solution?KNOWLEDGE:Students will know …test points can be used to determine the solution region that satisfies an inequalitya solid line should be used when the boundary line is included in the inequality. A broken line should be used when the boundary line is not included in the inequality.the graph of a linear or quadratic inequality can be sketched with or without technologylinear or quadratic inequalities can be used to solve some problemsSKILLS:Students will be able to …explain, using examples, how test points can be used to determine the solution region that satisfies an inequalityexplain, using examples, when a solid or broken line should be used in the solution for an inequalitysketch, with or without technology, the graph of a linear or quadratic inequalitysolve a problem that involves a linear or quadratic inequalityLesson SummaryStudents will learn to solve linear inequalities. Lesson PlanHookGo to a shopping website of interest to purchase two items of your choice. Determine how many of each item can be purchased for at most $500.Website possibilities:American Eagle: : Amazon: many of each of the two items can you afford to buy? Discuss. You have begun the process of solving a linear inequality.How could this be represented algebraically?Discuss how this might be represented graphically. (This meant for discussion only at this point)4. Discuss the relevance of the domain and range in this scenario ().Lesson GoalStudents will solve linear inequalities algebraically and graphically.Activate Prior KnowledgeReview rearranging equations, graphing linear equations from slope-intercept form and general form, multiplying or dividing by a negative number causes the inequality sign to change direction. Review discrete vs. continuous data.LessonStudent lead lesson:Sketch the following on a horizontal number line:x = 2x ≥ 2x ≤ 2Sketch the following on a vertical number line:y = 2y ≥ 2y ≤ 2Sketch the following on a coordinate plane. Show proof of why your answer is correct.y = xy ≥ xy ≤ xSketch the following on a coordinate plane. Show proof of why your answer is correct.y = 2x + 1y ≥ 2x + 1y ≤ 2x + 1What is different about the last equation and how does it affect your graph. In the discussion about “proof of how we know the answer is correct” and test point needs to be addressed.Teacher - Led lesson:Have students graph a line. Example: .Label the line “C”, the region above the line “A”, and below the line, label it “B”.The line separates the plane into 3 regions, A, B, and C.Which inequality below best describes region A? Discuss.a. Sketch . What would it look like? Which regions from example one would be included in my graph? Discuss. Use shading. (Teacher note: Since the line is included, the solution would include both region A and the line C)b. Sketch . What would it look like? Which regions from example one would be included in my graph? Discuss. Use shading. (Teacher note: Since it includes only region B, the line C would be represented as a broken line.)In general, if an equal sign is included, the line will be solid to show that every point on the line is included in the solution. If the equal sign is not included, the line is shown as a broken line as it is not part of the solution.Teacher may want to check students’ prior knowledge of solving a linear inequality in one variable. Ex. -2x < 10Sketch 2x - 3y < 6. (Note to teacher: use a table of values, x and y intercepts, or rearrange for y)Step 1: Sketch the line 2x - 3y = 6, but should it be solid or broken? (broken)Step 2: Which region should be shaded? Method 1: If you chose to rearrange for y, you would shade above if there is a ____ sign and below if there is a _____ sign.Method 2: Try the test-point method. Pick any point that is not on the line. (Hint: 0, 0 is a good one to use). Substitute the values into the original equation. If the resulting statement is true, then shade the region containing that point. If it is false, shade the region not containing that point.Hence, the region that should be shaded is above the line.Method 3: Using the GDC, graph the line by rearranging for y, and shade above the line. (Teacher: demonstrate this)Apply knowledge to revisit the questions posed in the hook. Determine how many of each item can be purchased for at most $500.Reiterate the discussion of continuous vs. discrete data. Going BeyondCould do a system of linear inequalities for fun. (See Math 20 Applied)ResourcesMath 20-1 (McGraw-Hill Ryerson: sec 9.1) Supporting AssessmentExit Slip. Solve graphically without the use of technology:3x - 5y > 15 Glossaryboundary line – A line on the coordinate plane that separates the plane into two regions compound inequality – A compound statement expressing two inequalities at the same timeinequality – A mathematical statement comparing expressions that may not be equal. These can be written using the symbols less than (<), greater than (>), less than or equal to (≤), greater than or equal to (≥), and not equal to (≠). [Math 20-1 (McGraw-Hill, page 588)]solution region – All the points in the Cartesian plane that satisfy an inequality. Also known as the solution set. [Math 20-1 (McGraw-Hill, page 590)]system of equations – A group of equations that are to be considered at the same timetest point – A point not on the boundary of the graph of an inequality that is representative of all the points in a region. A point that is used to determine whether the points in a region satisfy the inequality. [Math 20-1 (McGraw-Hill, page 591)]verify - Verifying a solution ensures the solution satisfies any equation or inequality by using substitutionGlossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.OtherLesson 4Quadratic Inequalities One VariableSTAGE 1BIG IDEA: The solution to a problem may not be a single value, but a range of values. Using test points and intervals is foundational to further study in mathematics.ENDURING UNDERSTANDINGS:Students will understand …Quadratic inequalities involve a solution with a range of values.An inequality in one variable has a range of x-values for its solution. (linear axis)Solutions can be found graphically or algebraically.When graphing inequalities, the line (or curve) should be broken if an equal sign is not included and solid if the equal sign is included.The solution to an inequality does not include the line (or curve) if an equal sign is not included.ESSENTIAL QUESTIONS:Where are inequalities used in real life?When is it appropriate to have a range of values as a solution?KNOWLEDGE:Students will know …test points can be used to determine the solution region that satisfies an inequalitya solid line should be used when the boundary line is included in the inequality. A broken line should be used when the boundary line is not included in the inequality.the graph of a linear or quadratic inequality can be sketched with or without technologylinear or quadratic inequalities can be used to solve some problemsSKILLS:Students will be able to …explain, using examples, how test points can be used to determine the solution region that satisfies an inequalityexplain, using examples, when a solid or broken line should be used in the solution for an inequalitysketch, with or without technology, the graph of a linear or quadratic inequalitysolve a problem that involves a linear or quadratic inequalitydetermine the solution of a quadratic inequality in one variable, using strategies such as case analysis, graphing, roots and test points, or sign analysis; and explain the strategy usedrepresent and solve a problem that involves a quadratic inequality in one variableinterpret the solution to a problem that involves a quadratic inequality in one variableLesson SummaryStudents will learn to solve quadratic inequalities in two variables. Lesson PlanLesson GoalStudents will be able to solve quadratic inequalities in two variables.Activate Prior KnowledgeHave students sketch the following and indicate a test point to verify solution.y ≥ x + 1y < -2x + 1Review quadratic function graphing including the vertex and direction of opening. Note: This is assuming that the Quadratic Function unit has been completed.y = x2 - 9LessonSketch the following inequalities: y ≥ x2 y ≥ -x2y < x2 y < -x2Use a test point to verify solution and explain why you would shade above or below the graph.Try the following inequalities:y ≥ (x - 2)2 y ≤ (x + 3)2 - 2 y < (x - 4)2 + 2When given in the ax2 + bx + c = 0 form, have students use the quadratic formula to find the roots.Example:Solve 2x2 - 7x > 12 (see example 3 on p. 482 of McGraw Hill) Going BeyondGiven the following functions:y = x2 – 1y = -x2Sketch the graph of –x2 ≥ x2 – 1.ResourcesMath 20-1 (McGraw-Hill Ryerson: sec 9.1) Supporting Ron Blond’s Applet: Slip: Solve the following inequality both graphically and algebraically. 2x2 + 12x - 11 > x2 + 2x + 13 Glossaryboundary line – A line on the coordinate plane that separates the plane into two regions compound inequality – A compound statement expressing two inequalities at the same timeinequality – A mathematical statement comparing expressions that may not be equal. These can be written using the symbols less than (<), greater than (>), less than or equal to (≤), greater than or equal to (≥), and not equal to (≠). [Math 20-1 (McGraw-Hill, page 588)]solution region – All the points in the Cartesian plane that satisfy an inequality. Also known as the solution set. [Math 20-1 (McGraw-Hill, page 590)]test point – A point not on the boundary of the graph of an inequality that is representative of all the points in a region. A point that is used to determine whether the points in a region satisfy the inequality. [Math 20-1 (McGraw-Hill, page 591)]verify - Verifying a solution ensures the solution satisfies any equation or inequality by using substitutionOtherLesson 5Quadratic Inequalities Two VariablesSTAGE 1BIG IDEA: The solution to a problem may not be a single value, but a range of values. Using test points and intervals is foundational to further study in mathematics.ENDURING UNDERSTANDINGS:Students will understand …Quadratic inequalities involve a solution with a range of values.An inequality in one variable has a range of x-values for its solution. (linear axis)An inequality in two variables has a range of coordinates for its solution. (shaded region)Solutions can be found graphically or algebraically.When graphing inequalities, the line (or curve) should be broken if an equal sign is not included and solid if the equal sign is included.The solution to an inequality does not include the line (or curve) if an equal sign is not included.ESSENTIAL QUESTIONS:Where are inequalities used in real life?When is it appropriate to have a range of values as a solution?KNOWLEDGE:Students will know …test points can be used to determine the solution region that satisfies an inequalitya solid line should be used when the boundary line is included in the inequality. A broken line should be used when the boundary line is not included in the inequality.the graph of a linear or quadratic inequality can be sketched with or without technologylinear or quadratic inequalities can be used to solve some problemsSKILLS:Students will be able to …explain, using examples, how test points can be used to determine the solution region that satisfies an inequalityexplain, using examples, when a solid or broken line should be used in the solution for an inequalitysketch, with or without technology, the graph of a linear or quadratic inequalitysolve a problem that involves a linear or quadratic inequalitydetermine the solution of a quadratic inequality in one variable, using strategies such as case analysis, graphing, roots and test points, or sign analysis; and explain the strategy usedrepresent and solve a problem that involves a quadratic inequality in one variableinterpret the solution to a problem that involves a quadratic inequality in one variableLesson SummaryStudents will learn how to solve quadratic inequalities in two variables and apply to word problems. Lesson PlanLesson GoalStudents will solve quadratic inequalities in two variables and apply to word problems.Activate Prior KnowledgeExpress the following on a number line x = 0x + 1 > 0x > 0 2x – 1 ≤ 0x ≤ 0Review the use of set notation to express solution.Review the sketching of quadratic function with a focus on the vertex and direction of openings.LessonHave students graph, shade and state the solution space in set notation for the following:y ≤ -3(x - 4)2 + 2y ≥ 2(x + 3)2 - 5y < x2 – 3x - 6y < 2x2 + 3x + 5Example Word Problem:Do an example word problem, such as McGraw Hill p.498 #10 Going BeyondResourcesMath 20-1 (McGraw-Hill Ryerson: sec 9.3) Supporting Ron Blond’s Applet: HYPERLINK "" Slip: Graph the following without the use of technology. y < -2(x – 1)2 - 5 Glossaryboundary line – A line on the coordinate plane that separates the plane into two regions compound inequality – A compound statement expressing two inequalities at the same timeinequality – A mathematical statement comparing expressions that may not be equal. These can be written using the symbols less than (<), greater than (>), less than or equal to (≤), greater than or equal to (≥), and not equal to (≠). [Math 20-1 (McGraw-Hill, page 588)]solution region – All the points in the Cartesian plane that satisfy an inequality. Also known as the solution set. [Math 20-1 (McGraw-Hill, page 590)]test point – A point not on the boundary of the graph of an inequality that is representative of all the points in a region. A point that is used to determine whether the points in a region satisfy the inequality. [Math 20-1 (McGraw-Hill, page 591)]verify - Verifying a solution ensures the solution satisfies any equation or inequality by using substitutionOther ................
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