Mathematics 20-1



MATHEMATICS 20-1Quadratic Functions and EquationsHigh School collaborative venture withEdmonton Christian, Harry Ainlay, J. Percy Page, Jasper Place, Millwoods Christian, Ross Sheppard and W. P. WagnerEdm 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 ElderW. P. Wagner: Amber SteinhauerFacilitator: John Scammell (Consulting Services)Editor: Jim Reed (Contracted)2010 – 2011TABLE OF CONTENTSSTAGE 1 DESIRED RESULTSPAGEBig IdeaEnduring UnderstandingsEssential Questions444KnowledgeSkills56STAGE 2 ASSESSMENT EVIDENCETransfer Task (on a separate page which could be photocopied & handed out to students)Mars Rover Bouncing AirbagsTeacher Notes for Transfer TaskTransfer Task Glossary/RubricPossible Solution7101719STAGE 3 LEARNING PLANSLesson #1 Characteristics of Quadratic Functions 24Lesson #2 Transformations28Lesson #3 Factoring32Lesson #4 Solving/Graphing Quadratics by Factoring or Quadratic Formula35Lesson #5 Discriminant38Lesson #6 Completing the Square41Lesson #7 Determining Quadratic Equations45Lesson #8 Applications of Quadratics48 Mathematics 20-1 Quadratic Functions and EquationsSTAGE 1 Desired Results Big Idea: Implementation note:Post the BIG IDEA in a prominentplace in your classroom and refer to it rmation from quadratic equations and the graphs of quadratic functions can be used to solve problems in areas like physics, calculus, engineering, architecture, sports and business. Enduring Understandings:Students will understand …The basic shape and characteristics of the graph of a quadratic function.Quadratic functions and equations relate to real life situations.Solving an equation means to find the root(s). Essential Questions:To be given early in the unit:Find two examples of real-life situations that could be represented by a quadratic function. Implementation note: Ask students to consider one of the essential questions every lesson or two.Has their thinking changed or evolved?What factors must you consider when deciding how much to charge for a product? To be given before studying vertex form:Get as close as you can to the equation of a parabola with vertex (-2,5).Other:Given the graph of the parabola shown, find the equation.Discuss the similarities and differences between linear and quadratic pare and contrast y = (x + 2) to y = (x + 2)(x – 3)?How does this graph describe the revenue of a school dance?In what situations is one method of solving a quadratic equation preferable to another? Given y?=?a(x - p)2 + q and the values of a and q, how can you determine the number of x-intercepts on the corresponding graph?Compare and contrast y = x 2 and y = 2x.How can algebra tiles be used to model completing the square? Knowledge:Enduring UnderstandingSpecific OutcomesDescription ofKnowledgeStudents will understand…The basic shape and characteristics of the graph of a quadratic function. *RF3 RF4Students will know…that an equation in the form y = x2 is a quadratic function where the graph is a parabolathe effects of changing each parameter?in?the function?y?=?a(x?-?p)2?+?qthat (p, q) is the vertexthat x = p is the axis of symmetrythat a determines the direction of opening of the parabolathat the x-intercept is found where y = 0 and the y-intercept is found where x = 0the domain is the set of permissible values of x in the function and range is all the possible values of y in the functionStudents will understand…Quadratic functions and equations relate to real life situations. RF4 RF5Students will know…what a, p, q means in the context of a problemthere are many methods available to solve problems involving quadraticsStudents will understand…Solving an equation means to find the root(s). RF1 RF5Students will know…the zeros of a function correspond to the x-intercepts of a graphthe discriminant (b2- 4ac) can be used to determine the nature of the roots8888I*RF = Relations and Functions Skills: Enduring UnderstandingSpecific OutcomesDescription of SkillsStudents will understand…The basic shape and characteristics of the graph of a quadratic function. *RF3 RF4Students will be able to…identify/determine the coordinates of the vertexsketch the graph of y?=?a(x - p)2 + q. with or without technologywrite a quadratic function in the form y?=?a(x?-?p)2?+?q. or y?=?ax2 + bx + c given a graph or a set of characteristics of a graphchange the form of a quadratic equation by completing the square or expandingStudents will understand…Quadratic functions and equations relate to real life situations. RF4 RF5Students will be able to…solve a quadratic equation of the form ax2 + bx + c = 0 algebraically and graphicallydetermine an appropriate domain and rangeuse the discriminate to determine the nature and number of rootsanalyze a problem and then determine and solve the quadratic equationStudents will understand…Solving an equation means to find the root(s). RF1 RF5Students will be able to…solve a quadratic equation of the form ax2?+?bx?+?c?=?0 by:determining square rootsfactoringcompleting the squareapplying the quadratic formulagraphing its corresponding functionverifying the solutionidentifying and correcting errors in a solution to a quadratic equationanalyzing a problem and then determining and solving the quadratic equationI*RF = Relations and FunctionsImplementation note:Teachers need to continually askthemselves, if their students are acquiring the knowledge and skills needed for the unit.STAGE 2 Assessment Evidence1 Desired Results Desired Results Mars Rover Bouncing Airbags Teacher NotesThere is one transfer task to evaluate student understanding of the concepts relating to quadratic equations and functions. Photocopy-ready version of the transfer task and rubric are included in this section.Each student will . . .create data for each participantanalyze data and plot graphs of the datainterpret the graphsidentify restricted domain and range for each simulationidentify intercepts and vertices and explain their significancePart A: Students will bounce a ball in front of a scale (meter stick) and record the initial coordinates, the vertex and the coordinates of the endpoint. Students may film the bouncing of the ball and analyze video to get more accurate measurements.Part B: Teachers should encourage students to use a variety of different methods including completing the square, factoring, using symmetry or quadratic formula.Part C: Any range of appropriate shifts should be accepted.Part D: A c value has been added (y?=?ax2 + bx + c) indicating that the initial point of impact is above the surface (ie. rock, landing pad, etc.) or the initial point of impact has been moved backwardsPart E: Any reasonable answer accepted.Implementation note:Teachers need to consider what performances and products will reveal evidence of understanding?What other evidence will be collected to reflectthe desired results?Teacher Notes for Mars Rover Bouncing Airbags Transfer TaskGlossary/Memory Aides axis of symmetry – An axis of symmetry is a line about which a figure is symmetrical. If a figure can be folded such that the two parts exactly match, the fold line would be an axis of symmetry.direction of opening – domain – All the x-values of the points of the graph of the functionmaximum value – Greatest y-value of the graphminimum value – Lowest y-value of the graphrange – All the y-values of the points of the graph of the functionvertex – The maximum/minimum point of a parabolax-intercept – The x-coordinate of the intersection point of a graph and the horizontal axisGlossary 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.y-intercept – The y-coordinate of the intersection point of a graph and the vertical axisThe following applet includes most of the glossary terms. Understanding the display will help students remember quadratic terms.Source: Mars Rover Bouncing Airbags - Student Assessment TaskYou are employed by the Canadian Space Agency and are involved in the testing of new high-tech polymer rubber. CSA is hoping to sell the idea to NASA for the Mars Rover Mission planned for 2019. You will have to do some research before you will be able to present your findings to NASA.The high-tech polymers will be used in the airbags, which are released from the rover when it reaches the surface of Mars. See the photos below to get an idea of what it looks like:Mars Rover Bouncing Rover Landed: Canadian Space Agency Tester for New Polymer RubbersAudience: NASA Format: Present using a poster, PowerPoint or booklet,Topic: You are presenting findings from Super-Polymer Rubber research to sell this product to NASA for their 2019 mission to Mars.Materials: ballmetre sticks or metre tape measuresSuggestion: Complete in groups of 3.Mars Rover Bouncing AirbagsHeightInitial Impact PointHorizontal distance travelled between first & second bounceYour experimental research is comprised of the following parts:Part AIn a simple ball-bouncing experiment, measure the distance between the first and second bounces, as well as the maximum height of the bounce. Write an equation in y = a(x – p)2 + q form and then in y = ax2 + bx + c form which describes this experiment. Use the space below to sketch and label your graphLet x represent the horizontal distance, and y represent the vertical height.Distance between first & second bounce: ________cmMaximum height: ________cmIn y = a(x – p)2 + q form: _______________________In y = ax2 + bx + c form: _______________________n form:___________________________Analyze the function you created by stating the following:Vertex _________________________Axis of symmetry ________________Direction of opening ______________ Domain ________________________ Range _________________x-intercepts _____________________ y-intercept _________________Part BYou need to see how 3 different polymers would react on Mars, where the gravity is 38% of that on Earth. A lab is created where this gravity is simulated and the results show the following for the high-tech polymers. Polymer A: y = -0.005x2 + 0.2x Polymer B: y = -0.0625x2 + 0.5x Polymer C: y = -0.375x2 + 3.75x Using algebra, determine which polymer will travel the furthest between the first bounce (initial impact point) and the second bounce, and which polymer attains the maximum height. Show your work in the space below.Part COn a satellite image, it is noticed that there is a 1.5 m high landform, 38 m from the first impact point. Choose one polymer and determine where the first impact point needs to be moved to, in order for that polymer to clear the object between the first and second bounce. Sketch the original graph and the new one you have determined on the grid below. What is the new domain of the graph?State all possible positions for the first impact point.Part DIn the simulation lab, a new experiment was performed on Polymer C with a new set of initial conditions. This was the result:y = -0.375x2 + 3.75x + 6Explain, in the context of this problem, what initial condition was changed?Algebraically, determine the horizontal distance travelled by Polymer C between the first and second bounce to the nearest tenth of a metre.What changes have there been in the domain and range compared to the first experiment with Polymer C? You may use your graphing calculator. Answer in the box below:Part EWhen you approach NASA with your polymers, which polymer might you recommend, and why? Defend your choice by using the results of the lab simulation testing. HYPERLINK "" \ of symmetry – An axis of symmetry is a line about which a figure is symmetrical. If a figure can be folded such that the two parts exactly match, the fold line would be an axis of symmetry.direction of opening – domain – All the x-values of the points of the graph of the functionmaximum value – Greatest y-value of the graphminimum value – Lowest y-value of the graphrange – All the y-values of the points of the graph of the functionvertex – The maximum/minimum point of a parabolax-intercept – The x-coordinate of the intersection point of a graph and the horizontal axisGlossary 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.y-intercept – The y-coordinate of the intersection point of a graph and the vertical axisThe following applet includes most of the glossary terms. Understanding the display will help students remember quadratic terms.Source: AssessmentMathematics 20-1Quadratic Functions and Equations Rubric Level CriteriaExcellent4Proficient3Adequate2Limited*1Insufficient / Blank*Analyze Quadratic Functions of the form y?=?a(x - p)2 + q. (Parts A-D)Student is able to determine all features of all required functionsStudent is able to determine most features of all required functionsStudent is able to determine some features of all required functionsStudent is able to determine few features of all required functionsStudent is unable to determine any features of quadratic functionsPerforms Algebraic Operations On Quadratic Functions In the Form (Part B & D)Student is able to correctly perform a variety of algebraic operations including factoring, completing the square, quadratic formula and/or symmetryStudent is able to correctly perform most of the algebraic operations including factoring, completing the square, quadratic formula and/or symmetryStudent demonstrates limited knowledge of algebra or combination of algebra and use of graphing calculatorStudent only used graphing calculator or algebra contained several major errorsStudent demonstrated no significant progress towards correct responsesSolve Problems That Involve Quadratic Equations(Part A,C,D,E)Student derives equation correctly using his/her dataStudent finds all possible values in Part C Student determines correct solution in Part DStudent provides logical justification in Part EStudent work shows up to 3 minor errors and/or one major errorStudent demonstrates correct approach on Parts A, C & D but makes significant errors in calculationsORAt least two solutions are correctly demonstrated with minor errors acceptedStudent demonstrates some knowledge of quadratics in their workNo knowledge of quadratics is apparentMath CommunicationExplains work clearly with all details present and correct answer where applicable.Explains work, with most details present and may or may not have the correct????? answer.Student demonstrates algebraic steps and graphs clearlyStudents demonstrate some algebraic steps and/or graphs incompleteStudent demonstrates poor communication of math.PresentationUses clear and effective diagrams,tables, charts and graphs.Appropriate but incomplete use ofdiagrams, tables, charts and graphs.Student’s work must be legible, neat and organized in a logical mannerStudent’s work is difficult to followStudent work illustrates poor presentationPossible Solution to Mars Rover Bouncing AirbagsYou are employed by the Canadian Space Agency and are involved in the testing of new high-tech polymer rubber. CSA is hoping to sell the idea to NASA for the Mars Rover Mission planned for 2019. You will have to do some research before you will be able to present your findings to NASA.The high-tech polymers will be used in the airbags, which are released from the rover when it reaches the surface of Mars. See the photos below to get an idea of what it looks like:Mars Rover Bouncing Rover Landed: Canadian Space Agency Tester for New Polymer RubbersAudience: NASA Format: Present using a poster, PowerPoint, bookletTopic: You are presenting findings from Super-Polymer Rubber research to sell this product to NASA for their 2019 mission to Mars.Materials: A ball and metre stick or tape measureSuggestion: To be done in groups of 3Mars Rover Bouncing AirbagsHeightInitial Impact PointHorizontal distance travelled between first & second bounceYour experimental research is comprised of the following parts:Part A (Sample solution, as all student solutions will differ based on their findings)In a simple ball-bouncing experiment, measure the distance between the first and second bounces, as well as the maximum height of the bounce. Write an equation in y = a(x – p)2 + q form and then in y?=?ax2 + bx + c form which describes this experiment. Use the space below to sketch and label your graphLet x represent the horizontal distance, and y represent the vertical height.Distance between first & second bounce: ___32____cmMaximum height:____64___cmIn y = a(x – p)2 + q form:__ y = -0.25(x – 16)2 + 16 __In y = ax2 + bx + c form: ___y = -0.25x2 + 8x _______n form:___________________________Sample solution:Use the x-intercept to determine a. Write the equation in y = a(x – p)2 + q form and then in y?=?ax2 + bx + c form. Analyze the function you created by stating the following:Vertex _______(16, 64)___________ Axis of symmetry ______x = 16_____Direction of opening __downward___ Domain __0 < x < 32_____________ Range ___0 < y < 64_______x-intercepts ___(0, 0) & (32, 0)_____ y-intercept _______(0, 0)_______Part BYou need to see how 3 different polymers would react on Mars, where the gravity is 38% of the Earth. A lab is created where this gravity is simulated and the results show the following for the high-tech polymers. Polymer A: y = -0.005x2 + 0.2x Polymer B: y = -0.0625x2 + 0.5x Polymer C: y = -0.375x2 + 3.75x Using algebra, determine which polymer will travel the furthest between the first and second bounce, and which polymer attains the maximum height. Show your work in the space below.Polymer A: y = -0.005x(x - 40) Factor to determine the x-intercepts0 = -0.005x(x - 40)x = 0, 40y = -0.005x2 + 0.2x Complete the square to determine the vertexy = -0.005(x2 - 40x) y = -0.005(x2 - 40x + 400 - 400) y = -0.005(x – 20) 2 + (-0,005)(-400)y = -0.005(x – 20) 2 + 2The x-intercepts are (0, 0) and (40, 0), therefore the distance travelled is 40 m. The vertex is at (20, 2) and therefore maximum height is 2 m.Polymer B: Use the quadratic formula to determine the x-interceptsx = 0, 8y = -0.0625(x2 - 8x + 16 - 16) Complete the square to determine the vertexy = -0.0625(x – 4) 2 + (-0,0625)(-16)y = -0.0625(x – 4) 2 + 1The x-intercept is (8, 0) and distance travelled is 8 m. The vertex is at (4, 1) and the maximum height is 1 m.Polymer C:y = -0.375x2 + 3.75xy = -0.375x(x – 10)0 = -0.375x(x – 10) Use factored form to determine the x-interceptsx = 0, 10y = -0.375(x2 - 10x + 25 - 25) Complete the square to determine the vertexy = -0.375(x – 5) 2 + (-0.375)(-25)y = -0.375(x – 5) 2 + 9.375The x-intercept is (10, 0) and distance travelled is 10 m. The vertex is at (5, 9.375) and the maximum height is 9.375 m.Polymer A travels the furthest (40 m).Polymer C attains the highest point (9.375 m).Part CGraphs will vary and domains will vary.A typical solution for Polymer A might be to move the vertex to (38,2), shifting the graph 18 units right. Thus, the first impact point would be at (18,0). The domain of this graph would be 18 m < x < 58 m. However, moving the initial impact point anywhere between, but not including, 10 m and 30 m to the right would lead to a correct solution. Polymer B will not clear.A typical solution for Polymer C would be to move the vertex to (38, 9.375), shifting the graph 33 units right. Thus, the first impact point would be at (33,0). The domain of this graph would be 33 m < x < 43 m. However, moving the initial impact point anywhere between, but not including, 28.4 m and 37.6 m to the right would lead to a correct solution. On a satellite image, it is noticed that there is a 1.5 m high landform, 38 m from the first impact point. Choose one polymer and determine where the first impact point needs to be moved to, in order for that polymer to clear the object between the first and second bounce. Sketch the original graph and the new one you have determined on the grid below. What is the new domain of the graph?State all possible positions for the first impact point.See to the right.Part DIn the simulation lab, a new experiment was performed on Polymer C with a new set of initial conditions. This was the result:y = -0.375x2 + 3.75x + 6Explain, in the context of this problem, what initial condition was changed?Algebraically, determine the horizontal distance travelled by Polymer C between the first and second bounce to the nearest tenth of a metre.What changes have there been in the domain and range compared to the first experiment with Polymer C? You may use their graphing calculator. Answer in the box below:Solution 1:The initial impact point height has changed to 6 m. (Possibly it bounced off an object in the lab during the experiment)x = 11.4 m, -1.4 mx = 11.4 m, -1.4 mNew Domain: New Range: Note: This range calculation does not have to be done algebraically, and can be done using the GDC.Solution 2:The landing point was moved 1.4 m to the left./The polymer landed to the left of the original landing point in the experiment.x = 11.4 m, -1.4 mNew Domain: New Range: Note: This range calculation does not have to be done algebraically. You can get the same result be using the GDC.Part EWhen you approach NASA with your polymers, which polymer might you recommend, and why? Defend your choice by using the results of the lab simulation testing.Answers will vary.STAGE 3 Learning PlansLesson 1Characteristics of Quadratic FunctionsSTAGE 1BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to solve problems in areas like physics, calculus, engineering, architecture, sports and business.ENDURING UNDERSTANDINGS:Students will understand …The basic shape and characteristics of the graph of a quadratic function.ESSENTIAL QUESTIONS:Find two examples of real-life situations that could be represented by a quadratic function. Get as close as you can to the equation of a parabola with vertex (-2,5). Given the graph of the parabola shown, find the equation.Discuss the similarities and differences between linear and quadratic functions.Given y?=?a(x - p)2 + q and the values of a and q, how can you determine the number of x-intercepts on the corresponding graph?KNOWLEDGE:Students will know …that an equation in the form y = x2 is a quadratic function where the graph is a parabolathe effects of changing each parameter?in?the function?y?=?a(x - p)2 + q.that (p, q) is the vertexthat x = p is the axis of symmetrythat a determines the direction of opening of the parabolaImplementation 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.the domain is the set of permissible values of x in the function and range is all the possible values of y in the functionSKILLS:Students will be able to …identify/determine the coordinates of the vertexsketch the graph of y = a(x – p)2 + q with or without technologyLesson SummaryStudents will be able to sketch the graph in the form y = a(x – p)2 + q (with and without technology) and be able to identify graph characteristics. Lesson PlanHookIf you are looking for a Characteristics of Quadratic Functions video, consider:: The axis of symmetry is calculated using .Lesson GoalStudents will be able to sketch the graph in the form y = a(x – p)2 + q (with and without technology) and be able to identify graph characteristics.LessonHave students work on quadratic functions (with graphing calculator or applet – see below), starting with the basic quadratic function and then progressing in complexity; all in the form y = a(x – p)2 + q. Use y = ax2 and change values of a. Have students come up with a generalization of how a effects the graph. Then, keeping a constant, give examples with only the p value changing. Then proceed with changing q.Identify axis of symmetryvertexmax/min valuedomainrangedirection of openingx-interceptsy-intercepts Going BeyondHave students develop their own understanding of how the equation relates to certain characteristics of the graph.Could discuss parameter a as it relates to a horizontal stretch.ResourcesMath 20-1 (McGraw-Hill Ryerson: sec 3.1) SupportingUse a quadratic function applet: quick quiz could be given at the end of the class. No calculators. Match 2 equations to the correct 2 of 4 graphs drawn beside.Glossary/Memory Aidesaxis of symmetry – An axis of symmetry is a line about which a figure is symmetrical. If a figure can be folded such that the two parts exactly match, the fold line would be an axis of symmetry.direction of opening – domain – All the x-values of the points of the graph of the functionmaximum value – Greatest y-value of the graphminimum value – Lowest y-value of the graphrange – All the y-values of the points of the graph of the functionvertex – The maximum/minimum point of a parabolax-intercept – The x-coordinate of the intersection point of a graph and the horizontal axisGlossary 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.y-intercept – The y-coordinate of the intersection point of a graph and the vertical axisOtherNote to teachers: It would be helpful to bring in laptops or go to computer lab to get students using the applet above to draw conclusions and make generalizations about . The generalizations are more immediate with the applet than calculator.One of the applets uses y = a(x – h)2 + k instead of y = a(x – p)2 + q. The slider label can be changed in the HTML code that displays the applet.The other applet uses Standard instead of Vertex and General instead of Standard. The button names can be changed in the HTML code that displays the applet.Lesson 2TransformationsSTAGE 1BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to solve problems in areas like physics, calculus, engineering, architecture, sports and business.ENDURING UNDERSTANDINGS:Students will understand …The basic shape and characteristics of the graph of a quadratic function.ESSENTIAL QUESTIONS:Find two examples of real-life situations that could be represented by a quadratic function. Get as close as you can to the equation of a parabola with vertex (-2,5). Given the graph of the parabola shown, find the equation.Discuss the similarities and differences between linear and quadratic pare and contrast y = (x + 2) to y?=?(x?+?2)(x?–?3)?How does this graph describe the revenue of a school dance?Given y?=?a(x - p)2 + q and the values of a and q, how can you determine the number of x-intercepts on the corresponding graph?KNOWLEDGE:Students will know …that an equation in the form y = x2 is a quadratic function where the graph is a parabolathe effects of changing each parameter?in?the function?y?=?a(x - p)2 + qthat (p, q) is the vertexthat x = p is the axis of symmetrythat a determines the direction of opening of the parabolathat the x-intercept is found where y = 0 and the y-intercept is found where x = 0the domain is the set of permissible values of x in the function and range is all the possible values of y in the functionSKILLS:Students will be able to …identify/determine the coordinates of the vertexsketch the graph of y = a(x – p)2 + q with or without technologywrite a quadratic function in the form y = a(x – p)2 + q or y = ax2 + bx + c given a graph or a set of characteristics of a graphLesson SummaryStudents will translate a quadratic equation both vertically and horizontally as well as determine reflections about the x-axis. Lesson PlanHookShow students the following clip: GoalStudents will translate a quadratic equation both vertically and horizontally as well as determine reflections about the x-axis.Activating Prior KnowledgeReview a question from Lesson 1. Have students apply their knowledge of the characteristics.LessonLet the students come up with an example of each type of translation independently. Have students trade their examples with another student and verify them. Bring a few of the examples up to the board to show the class. The teacher introduces combination example(s):describing transformations from equationsmaking equations from descriptionsmaking an equation from a graphdescribe transformations from a graphThe teacher assigns practice questions for students to complete.The teacher provides reflection examples and checks for understanding (orally).The teacher assigns practice questions for students to complete. Going BeyondThe teacher demonstrates transforming an equation and graph using a coordinate.ResourcesMath 20-1 (McGraw-Hill Ryerson: sec 3.1, 3.2, 3.3)This section will involve jumping around the text. Supporting AssessmentHave students start at the origin on a grid. Then teacher will describe a series of transformations and students will come up with the correct resulting position. Glossaryreflection – The mirror image of a function about an axis or linetransformation – Changing the position, direction or shape of a functiontranslation – The vertical or horizontal shift of a functionOtherLesson 3FactoringSTAGE 1BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to solve problems in areas like physics, calculus, engineering, architecture, sports and business.ENDURING UNDERSTANDINGS:Students will understand …The basic shape and characteristics of the graph of a quadratic function.Quadratic functions and equations relate to real life situations.Solving an equation means to find the root(s).ESSENTIAL QUESTIONS:Find two examples of real-life situations that could be represented by a quadratic function. What factors must you consider when deciding how much to charge for a product? Given the graph of the parabola shown, find the pare and contrast y = (x + 2) to y?=?(x?+?2)(x?–?3)?How does this graph describe the revenue of a school dance?In what situations is one method of solving a quadratic equation preferable to another? Given y?=?a(x - p)2 + q and the values of a and q, how can you determine the number of x-intercepts on the corresponding graph?KNOWLEDGE:Students will know …that an equation in the form y = x2 is a quadratic function where the graph is a parabolathat a determines the direction of opening of the parabolathat the x-intercept is found where y = 0 and the y-intercept is found where x = 0the domain is the set of permissible values of x in the function and range is all the possible values of y in the functionthere are many methods available to solve problems involving quadratics the zeros of a function correspond to the x-intercepts of a graphthe discriminant (b2- 4ac) can be used to determine the nature of the rootsSKILLS:Students will be able to …solve a quadratic equation of the form ax2?+?bx?+?c?=?0 algebraically and graphicallyuse the discriminate to determine the nature and number of rootsanalyze a problem and then determine and solve the quadratic equationsolve a quadratic equation of the form ax2?+?bx?+?c?=?0 by:factoringgraphing its corresponding functionverifying the solutionidentifying and correcting errors in a solution to a quadratic equationanalyzing a problem and then determining and solving the quadraticLesson SummaryStudents will review factoring polynomials of degree 2 (quadratics). Lesson PlanActivating Prior KnowledgeFactor polynomialsLessonReview factoring and expand to more complex quadratics. Going BeyondFactor more complex quadratics (rational coefficients).ResourcesMath 20-1 (McGraw-Hill Ryerson: sec 4.3) Supporting AssessmentCreate a formative assessment on factoring. Glossarybinomial – A two term polynomial expressionfactor – To express an algebraic expression as a producttrinomial – A three term polynomial expressionOtherLesson 4Solving/Graphing Quadratics by Factoring or Quadratic FormulaSTAGE 1BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to solve problems in areas like physics, calculus, engineering, architecture, sports and business.ENDURING UNDERSTANDINGS:Students will understand …Solving an equation means to find the root(s).ESSENTIAL QUESTIONS:Find two examples of real-life situations that could be represented by a quadratic function. In what situations is one method of solving a quadratic equation preferable to another? Given y?=?a(x - p)2 + q and the values of a and q, how can you determine the number of x-intercepts on the corresponding graph?KNOWLEDGE:Students will know …the zeros of a function correspond to the x-intercepts of a graphSKILLS:Students will be able to …solve a quadratic equation of the form ax2?+?bx?+?c?=?0 by:determining square rootsfactoringcompleting the squareapplying the quadratic formulagraphing its corresponding functionverifying the solutionidentifying and correcting errors in a solution to a quadratic equationLesson SummarySolve Quadratics by GraphingSolve Quadratics by FactoringIntroduce the Quadratic Formula to solve Quadratics Lesson PlanHookSolve a linear equation algebraically and graphically. Learning GoalStudents will solve quadratics by graphing, factoring and the quadratic formula.Activating Prior KnowledgeAsk students to factor some quadratic functions to check for understanding of the previous lesson.LessonDiscuss the importance and what it means to solve a quadratic equation.Graphing Provide a set of quadratics to be solved using technology.AlgebraicProvide an increasing more complex set of quadratics to be solved by factoring. (integral, rational and exact value roots)Provide the Quadratic Formula (extension derive the formula) Discuss the uses of the quadratic formula Going BeyondDerive the quadratic formula.Introduce/discuss non-real roots.ResourcesMath 20-1 (McGraw-Hill Ryerson: sec 4.1, 4.2, 4.4) Supporting AssessmentExit SlipTwo quadratics solved algebraicallyone factorable one non-factorable (exact value) Glossaryexact values – Answers involving non-rounded values, such as whole numbers, terminating or repeating decimals, fractions or radicals. roots – Find the solution(s) to an equationsolve – To find the solutions to an equationx-intercept – The x-coordinate of the intersection point of a graph and the horizontal axiszeros – The values of x that make a function equal to zeroOtherLesson 5DiscriminantSTAGE 1BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to solve problems in areas like physics, calculus, engineering, architecture, sports and business.ENDURING UNDERSTANDINGS:Students will understand …Solving an equation means to find the root(s).ESSENTIAL QUESTIONS:Find two examples of real-life situations that could be represented by a quadratic function. Given y?=?a(x - p)2 + q and the values of a and q, how can you determine the number of x-intercepts on the corresponding graph?KNOWLEDGE:Students will know …the discriminant (b2- 4ac) can be used to determine the nature of the rootsSKILLS:Students will be able to …use the discriminate to determine the nature and number of rootsLesson SummaryStudents will use the discriminant to determine the number and nature of the roots. Lesson PlanHookDiscuss 3 quadratics graphs with 3 different natures of the roots.Lesson GoalStudents will recognize and make the connection between the numbers of roots and the discriminant.Activate Prior KnowledgeKnowledge of the quadratic formulaLessonSolve the initial three introductory graphs using the quadratic formula.Discover and discuss the connection between the radicand in the quadratic formula and the nature of the roots.Identify and discuss the discriminant. Going BeyondLooking at the parameters a, b, c in a quadratic equation and without using the discriminant, determine the nature of the roots.ResourcesMath 20-1 (McGraw-Hill Ryerson: sec 4.4) Supporting Assessment Glossarydiscriminant – (b2 – 4ac) This value tells the nature of the rootsequal real roots – the discriminate equals zeronon-real number – Imaginary number (such as the square root of a negative number)non-real roots – the discriminate (b2 – 4ac) value is less than zerounique real roots – the discriminate (b2 – 4ac) value is greater than zeroOtherThe following applet includes most of the glossary terms. Understanding the display will help students remember quadratic terms.Source: 6Completing the SquareSTAGE 1BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to solve problems in areas like physics, calculus, engineering, architecture, sports and business.ENDURING UNDERSTANDINGS:Students will understand …The basic shape and characteristics of the graph of a quadratic function.Solving an equation means to find the root(s).ESSENTIAL QUESTIONS:Find two examples of real-life situations that could be represented by a quadratic function. Get as close as you can to the equation of a parabola with vertex (-2,5). Given the graph of the parabola shown, find the equation.How does this graph describe the revenue of a school dance?Given y?=?a(x - p)2 + q and the values of a and q, how can you determine the number of x-intercepts on the corresponding graph?How can algebra tiles be used to model completing the square?KNOWLEDGE:Students will know …that (p, q) is the vertexwhat a, p, q means in the context of a problemSKILLS:Students will be able to …identify/determine the coordinates of the vertexchange the form of a quadratic equation by completing the square or expandingLesson SummaryStudents will be able to change the form of a quadratic equation by completing the square and verify. Lesson PlanAdvance PlanningStudents will be required to cut out the paper version of the algebra tiles in order to manipulate the tiles.: The teacher will have to modify the tiles if they choose to use them to work out the example in the Hook on the next page. Alternate tiles resource: the topic by doing a class example with the newly created algebra tiles. Refer to website for examples. GoalStudents will be able to complete the square to change the form of a quadratic equation. Students need to be able to verify their answer.Activating Prior KnowledgeUse the algebra tiles to connect to polynomial terms.LessonGet pairs of students to create their own examples, then exchange with another group. All pairs should attempt to solve the new questions.In a group discussion, come up with the key ideas involved in completing the square. Go through the same example together algebraically. Proceed to an example where:b is an odd numbera is not equal to 1.Allow students time to practice. Check for understanding (orally).At the end of the class, have the students complete an Exit Slip to check their understanding. Going BeyondStudents can expand their results to verify their solutions.ResourcesMath 20-1 (McGraw-Hill Ryerson: sec 3.3) Supporting AssessmentComplete an Exit Slip to check the students’ understanding of the material. This topic will also be formally evaluated at a later date. Glossarycompleting the square – A method used to convert a quadratic equation from standard form (y = ax2 +bx + c) to vertex form (y = a(x - p)2 + q).OtherLesson 7Determining Quadratic EquationsSTAGE 1BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to solve problems in areas like physics, calculus, engineering, architecture, sports and business.ENDURING UNDERSTANDINGS:Students will understand …The basic shape and characteristics of the graph of a quadratic function.Solving an equation means to find the root(s).ESSENTIAL QUESTIONS:Get as close as you can to the equation of a parabola with vertex (-2,5). Given the graph of the parabola shown, find the equation.KNOWLEDGE:Students will know …that an equation in the form y = x2 is a quadratic function where the graph is a parabolathe effects of changing each parameter?in?the function?y?=?a(x - p)2 + qthat (p, q) is the vertexthat x = p is the axis of symmetrythat a determines the direction of opening of the parabolathat the x-intercept is found where y = 0 and the y-intercept is found where x = 0the domain is the set of permissible values of x in the function and range is all the possible values of y in the functionSKILLS:Students will be able to …write a quadratic function in the form y?=?a(x - p)2 + q or y = ax2 + bx + c given a graph or a set of characteristics of a graphLesson SummaryStudents will determine equations of quadratic functions, in standard and vertex form, given graphs and word descriptions. Lesson PlanHookConsider the following video on projectile motion. Ask students how this is different than other projectile examples they have looked at. You are looking for them to indicate the projectile is shot out sideways or dropped instead of propelled upward. show them at least one video of a projectile being shot up at an angle. Ask them what information they need in order to draw the graph of the projectile motion and what information is required from the graph to determine the equation of the projectile motion in vertex (y?=?a(x - p)2 + q) form. GoalStudents will determine equations of quadratic functions, in standard and vertex form, given graphs and word descriptions.Activate Prior KnowledgeReview vertex (y?=?a(x - p)2 + q) form transformations when a, p and q are changed (Lesson 1).LessonGive students a mixture of graphing and word problems that include:a graph.a vertex and one point.x-intercepts and range.a value, one point and range.a value, one point and axis of symmetry.a congruent graph, with opposite direction and a vertex. Going BeyondGiven any three points on a parabola, determine the quadratic equation. ResourcesMath 20-1 (McGraw-Hill Ryerson: Chapter 3) Supporting Assessment“Exit Slip” or quiz question… May end the class with a final graph/written description and have students come up with the equation to match.Groups create an appropriate equation in response to a graph or description created by another group. GlossaryGlossary 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.congruent graphs - Two graphs are congruent if they have the same shape and sizeOtherLesson 8Applications of QuadraticsSTAGE 1BIG IDEA: Information from quadratic equations and the graphs of quadratic functions can be used to solve problems in areas like physics, calculus, engineering, architecture, sports and business.ENDURING UNDERSTANDINGS:Students will understand …quadratic functions and equations relate to real life situations ESSENTIAL QUESTIONS:Find two examples of real-life situations that could be represented by a quadratic function. How does this graph describe the revenue of a school dance?In what situations is one method of solving a quadratic equation preferable to another? KNOWLEDGE:Students will know …that (p, q) is the vertexthat x = p is the axis of symmetrythat a determines the direction of opening of the parabolathat the x-intercept is found where y = 0 and the y-intercept is found where x = 0the domain is the set of permissible values of x in the function and range is all the possible values of y in the functionthere are many methods available to solve problems involving quadraticsSKILLS:Students will be able to …identify/determine the coordinates of the vertexdetermine an appropriate domain and rangeanalyze a problem and then determine and solve the quadratic equationLesson SummaryApply real life situations to quadratics and visa versa. Lesson PlanLearning GoalApply real life situations to quadratics and visa versa.HOOKYoutube videos of projectiles (cannonballs, football field goals….)Mythbusters videosActivating Prior KnowledgeThis is a culmination of the entire unit of study.LessonStudents may encounter the following problems:Sportsdiving, hammer throw, football, shot-put, javelin, discus, basketball, volleyball, shooting, golf, etc…..Areamaximum area of an enclosed shape, given certain amount of materialNumbersmaximum or minimum product of two unknown numbers, in one variable, where one number is the variation of the otherPhysicsdistance, speed, time, etc….RevenueFind the maximum revenue for selling lemonade, tickets by studying the effect increase/decrease of price has on the number of sales.Provide an example of any of the preceding and let the students explore all the characteristics of a real life quadratic problem. (ie maximum, minimum, vertex, equation, domain & range). Encourage students discuss the characteristics verbally, visually (graphs) and symbolically (equations).The teacher may follow with a directed lesson on more complex examples. Going BeyondResourcesMath 20-1 (McGraw-Hill: chapters 3 & 4) Supporting AssessmentTransfer Task - RAFT Glossary/Formularevenue = price x number of itemsOther ................
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