G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S )

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1 G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Grade 11 Pre-Calculus Mathematics (30S) is desiged for studets who ited to study calculus ad related mathematics as part of post-secodary educatio. It builds o the topics studied i Grade 10 Itroductio to Applied ad Pre-Calculus Mathematics ad provides backgroud kowledge ad skills for Grade 12 Pre-Calculus Mathematics. The course comprises a high-level study of theoretical mathematics with a emphasis o problem solvig ad metal mathematics. The topics iclude study of algebra, quadratic fuctios, reciprocal fuctios, ad trigoometry. Assessmet of Grade 11 Pre-Calculus Mathematics should be a balace of assessmet for learig, assessmet as learig, ad assessmet of learig. Assessmet tools used i Grade 11 Pre-Calculus Mathematics should be varied ad may iclude observatio, homework, learig coversatios or iterviews, summative uit essays, demostratios, presetatios, performace tasks, learig logs, projects, ivestigatios, reflective jourals, portfolios, quizzes, tests, ad examiatios. A appropriately prepared portfolio requires a cosistet effort throughout the school term ad a commitmet to completig quality work o a daily basis. The learig outcomes are divided ito three topics: Algebra ad Number; Trigoometry; ad Relatios ad Fuctios. For istructioal purposes, the learig outcomes could be arraged ito uits. Learig outcomes from differet topics could be taught i the same uit. Some learig outcomes may fit ito multiple uits, ad parts of the learig outcome could be taught i oe uit while the remaiig parts ca be taught later. Two possible sequeces of the learig outcomes ito uits with suggested time allotmets follow. The suggested times iclude time for istructio ad assessmet. These are ot the oly possibilities, but they will provide some directio for teachers for their first time through the course. Regardless of the orgaizatio of the learig outcomes ito uits, studets should costatly be lookig for ad be give opportuities to see coectios betwee the various learig outcomes i Grade 11 Pre-Calculus Mathematics. Geeral ad Specific Learig Outcomes 139

2 Possibility 1 Possibility 2 Uit Learig Outcomes Suggested Hours Uit Learig Outcomes Suggested Hours uadratic Equatios R1, R5 12 Algebra R1, A4, A5 15 Radicals A2, A3 15 Fuctios R3, R4, R11 20 uadratic Fuctios R3, R4 17 Sequeces R9, R10 10 Sequeces R9, R10 10 Absolute Value A1, R2 10 Ratioals A4, A5, A6, R11 15 Equatio Solvig A3, A6, R5, R6 15 Trigoometry T1, T2, T3 20 Trigoometry T1, T2, T3 20 Systems R6 6 Iequalities R7, R8 10 Iequalities A1, R2, R7, R8 15 Radicals A2, A3 10 Total 110 Total Grades 9 to 12 Mathematics: Maitoba Curriculum Framework of Outcomes (2014)

3 Geeral ad Specific Learig Outcomes with Achievemet Idicators by Course Grade 11 Pre-Calculus Mathematics [C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 11P.A.1. 11P.A.2. Strad: Algebra ad Number Specific Learig Outcomes It is expected that studets will: Demostrate a uderstadig of the absolute value of real umbers. [ME, R, V] Solve problems that ivolve operatios o radicals ad radical expressios with umerical ad variable radicads. [CN, ME, PS, R, T] Geeral Learig Outcome: Develop algebraic reasoig ad umber sese. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Determie the distace of two real umbers of the form ±a, a є R, from 0 o a umber lie, ad relate this to the absolute value of a ( a ). Determie the absolute value of a positive or egative real umber. Explai, usig examples, how distace betwee two poits o a umber lie ca be expressed i terms of absolute value. Determie the absolute value of a umerical expressio. Compare ad order the absolute values of real umbers i a set. Compare ad order a set of radical expressios with umerical radicads. Express a etire radical with a umerical radicad as a mixed radical. Express a mixed radical with a umerical radicad as a etire radical. Perform oe or more operatios to simplify radical expressios with umerical or variable radicads. Ratioalize the deomiator of a ratioal expressio with moomial or biomial deomiators. Describe the relatioship betwee ratioalizig a biomial deomiator of a ratioal expressio ad the product of the factors of a differece of squares expressio. Explai, usig examples, that ( x) 2 = x 2, if x 2 = a, the Idetify the values of the variable for which a radical expressio is defied. Solve a problem that ivolves radical expressios. Geeral ad Specific Learig Outcomes 141

4 [C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 11P.A.3. 11P.A.4. Strad: Algebra ad Number (cotiued) Specific Learig Outcomes It is expected that studets will: Solve problems that ivolve radical equatios (limited to square roots). [C, CN, PS, R, T] It is iteded that the equatios will have o more tha two radicals. Determie equivalet forms of ratioal expressios (limited to umerators ad deomiators that are moomials, biomials, or triomials). [C, ME, R] Geeral Learig Outcome: Develop algebraic reasoig ad umber sese. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Determie ay restrictios o values for the variable i a radical equatio. Determie the roots of a radical equatio algebraically, ad explai the process used to solve the equatio. Verify, by substitutio, that the values determied i solvig a radical equatio algebraically are roots of the equatio. Demostrate that some roots determied i solvig a radical equatio algebraically are extraeous. Solve a problem by modellig a situatio usig a radical equatio. Compare the strategies for writig equivalet forms of ratioal expressios to the strategies for writig equivalet forms of ratioal umbers. Explai why a value is o-permissible for a ratioal expressio. Determie the o-permissible values for a ratioal expressio. Determie a ratioal expressio that is equivalet to a ratioal expressio by multiplyig the umerator ad deomiator by the same factor (limited to a moomial or a biomial), ad state the o-permissible values of the equivalet ratioal expressio. Simplify a ratioal expressio. Explai why the o-permissible values of a ratioal expressio ad its simplified form are the same. Idetify ad correct errors i a simplificatio of a ratioal expressio, ad explai the reasoig. 142 Grades 9 to 12 Mathematics: Maitoba Curriculum Framework of Outcomes (2014)

5 [C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 11P.A.5. 11P.A.6. Strad: Algebra ad Number (cotiued) Specific Learig Outcomes It is expected that studets will: Perform operatios o ratioal expressios (limited to umerators ad deomiators that are moomials, biomials, or triomials). [C, CN, ME, R] Solve problems that ivolve ratioal equatios (limited to umerators ad deomiators that are moomials, biomials, or triomials). [C, CN, PS, R] It is iteded that the ratioal equatios be those that ca be simplified to liear ad quadratic equatios. Geeral Learig Outcome: Develop algebraic reasoig ad umber sese. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Compare the strategies for performig a operatio o ratioal expressios to the strategies for performig the same operatio o ratioal umbers. Determie the o-permissible values whe performig operatios o ratioal expressios. Determie, i simplified form, the sum or differece of ratioal expressios with the same deomiator. Determie, i simplified form, the sum or differece of ratioal expressios i which the deomiators are ot the same ad which may or may ot cotai commo factors. Determie, i simplified form, the product or quotiet of ratioal expressios. Simplify a expressio that ivolves two or more operatios o ratioal expressios. Determie the o-permissible values for the variable i a ratioal equatio. Determie the solutio to a ratioal equatio algebraically, ad explai the process. Explai why a value obtaied i solvig a ratioal equatio may ot be a solutio of the equatio. Solve a problem by modellig a situatio usig a ratioal equatio. Geeral ad Specific Learig Outcomes 143

6 [C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 11P.T.1. Strad: Trigoometry Specific Learig Outcomes It is expected that studets will: Demostrate a uderstadig of agles i stadard positio [0 to 360 ]. [C, R, V] Geeral Learig Outcome: Develop trigoometric reasoig. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Sketch a agle i stadard positio, give the measure of the agle. Determie the referece agle for a agle i stadard positio. Explai, usig examples, how to determie the agles from 0 to 360 that have the same referece agle as a give agle. Illustrate, usig examples, that ay agle from 90 to 360 is the reflectio i the x-axis ad/ or the y-axis of its referece agle. Determie the quadrat i which a agle i stadard positio termiates. Draw a agle i stadard positio give ay poit P (x, y) o the termial arm of the agle. Illustrate, usig examples, that the poits P (x, y), P ( x, y), P ( x, y), ad P (x, y) are poits o the termial sides of agles i stadard positio that have the same referece agle. 144 Grades 9 to 12 Mathematics: Maitoba Curriculum Framework of Outcomes (2014)

7 [C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 11P.T.2. 11P.T.3. Strad: Trigoometry (cotiued) Specific Learig Outcomes It is expected that studets will: Solve problems, usig the three primary trigoometric ratios (sie, cosie, ad taget) for agles from 0 to 360 i stadard positio. [C, ME, PS, R, T, V] Solve problems, usig the cosie law ad sie law, icludig the ambiguous case. [C, CN, PS, R, T] Geeral Learig Outcome: Develop trigoometric reasoig. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Determie, usig the Pythagorea theorem or the distace formula, the distace from the origi to a poit P (x, y) o the termial arm of a agle. Determie the value of si θ, cos θ, or ta θ, give ay poit P (x, y) o the termial arm of agle θ. Determie, without the use of techology, the value of si θ, cos θ, or ta θ, give ay poit P (x, y) o the termial arm of agle θ, where θ = 0º, 90º, 180º, 270º, or 360º. Determie the sig of a trigoometric ratio for a agle, without the use of techology, ad explai. Solve a equatio of the form si θ = a or cos θ = a, where 1 a 1, or a equatio of the form ta θ = a, where a is a real umber. Determie the exact value of the sie, cosie, or taget of a agle with a referece agle of 30º, 45º, or 60º. Describe patters i ad amog the values of the sie, cosie, ad taget ratios for agles from 0 to 360. Sketch a diagram to represet a problem ivolvig trigoometric ratios. Solve a cotextual problem, usig trigoometric ratios. Sketch a diagram to represet a problem that ivolves a triagle without a right agle. Solve a o-right triagle usig right triagle methods. Explai the steps i a give proof of the sie law or cosie law. Sketch a diagram ad solve a cotextual problem, usig the cosie law. Sketch a diagram ad solve a cotextual problem, usig the sie law. Describe ad explai ambiguous case problems that may have o solutio, oe solutio, or two solutios. Geeral ad Specific Learig Outcomes 145

8 [C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 11P.R.1. Strad: Relatios ad Fuctios Specific Learig Outcomes It is expected that studets will: Factor polyomial expressios of the form ax 2 + bx + c, a 0 a2 x 2 b 2 y 2, a 0, b 0 a(f(x)) 2 + b(f(x)) + c, a 0 a2 (f(x)) 2 b 2 (g(y)) 2, a 0, b 0 where a, b, ad c are ratioal umbers. [ME, R] Geeral Learig Outcome: Develop algebraic ad graphical reasoig through the study of relatios. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Factor a polyomial expressio that requires the idetificatio of commo factors. Determie whether a biomial is a factor for a polyomial expressio, ad explai why or why ot. Factor a polyomial expressio of the form ax 2 + bx + c, a 0 a 2 x 2 b 2 y 2, a 0, b 0 Factor a polyomial expressio that has a quadratic patter, icludig a(f(x)) 2 + b(f(x)) + c, a 0 a 2 (f(x)) 2 b 2 (g(y)) 2, a 0, b Grades 9 to 12 Mathematics: Maitoba Curriculum Framework of Outcomes (2014)

9 [C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 11P.R.2. Strad: Relatios ad Fuctios (cotiued) Specific Learig Outcomes It is expected that studets will: Graph ad aalyze absolute value fuctios (limited to liear ad quadratic fuctios) to solve problems. [C, PS, R, T, V] Geeral Learig Outcome: Develop algebraic ad graphical reasoig through the study of relatios. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Create a table of values for y = f(x), give a table of values for y = f(x). Geeralize a rule for writig absolute value fuctios i piecewise otatio. Sketch the graph of y = f(x) ; state the itercepts, domai ad rage; ad explai the strategy used. Solve absolute value equatios graphically, with or without techology. Solve, algebraically, equatios with a sigle absolute value, ad verify the solutio. Explai why the absolute value equatio f(x) = a, a < 0 has o solutio. Determie ad correct errors i a solutio to a absolute value equatio. Solve a problem that ivolves absolute value fuctios. Geeral ad Specific Learig Outcomes 147

10 [C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 11P.R.3. Strad: Relatios ad Fuctios (cotiued) Specific Learig Outcomes It is expected that studets will: Aalyze quadratic fuctios of the form y = a(x p) 2 + q ad determie the vertex domai ad rage directio of opeig axis of symmetry x- ad y-itercepts [C, CN, R, T, V] Geeral Learig Outcome: Develop algebraic ad graphical reasoig through the study of relatios. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Explai why a fuctio give i the form y = a(x p) 2 + q is a quadratic fuctio. Compare the graphs of a set of fuctios of the form y = ax 2 to the graph of y = x 2, ad geeralize, usig iductive reasoig, a rule about the effect of a. Compare the graphs of a set of fuctios of the form y = x 2 + q to the graph of y = x 2, ad geeralize, usig iductive reasoig, a rule about the effect of q. Compare the graphs of a set of fuctios of the form y = (x p) 2 to the graph of y = x 2, ad geeralize, usig iductive reasoig, a rule about the effect of p. Determie the coordiates of the vertex for a quadratic fuctio of the form y = a(x p) 2, ad verify with or without techology. Geeralize, usig iductive reasoig, a rule for determiig the coordiates of the vertex for quadratic fuctios of the form y = a(x p) 2 + q. Sketch the graph of y = a(x p) 2 + q, usig trasformatios, ad idetify the vertex, domai ad rage, directio of opeig, axis of symmetry, ad x- ad y-itercepts. Explai, usig examples, how the values of a ad q may be used to determie whether a quadratic fuctio has zero, oe, or two x-itercepts. Write a quadratic fuctio i the form y = a(x p) 2 + q for a graph or a set of characteristics of a graph. 148 Grades 9 to 12 Mathematics: Maitoba Curriculum Framework of Outcomes (2014)

11 [C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 11P.R.4. Strad: Relatios ad Fuctios (cotiued) Specific Learig Outcomes It is expected that studets will: Aalyze quadratic fuctios of the form y = ax 2 + bx + c to idetify characteristics of the correspodig graph, icludig vertex domai ad rage directio of opeig axis of symmetry x- ad y-itercepts [C, CN, PS, R, T, V] Geeral Learig Outcome: Develop algebraic ad graphical reasoig through the study of relatios. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Explai the reasoig for the process of completig the square, as show i a example. Write a quadratic fuctio give i the form y = ax 2 + bx + c as a quadratic fuctio i the form y = a(x p) 2 q by completig the square. Idetify, explai, ad correct errors i a example of completig the square. Determie the characteristics of a quadratic fuctio give i the form y = ax 2 + bx + c, ad explai the strategy used. Sketch the graph of a quadratic fuctio give i the form y = ax 2 + bx + c. Verify, with or without techology, that a quadratic fuctio i the form y = ax 2 + bx + c represets the same fuctio as the quadratic fuctio i the form y = a(x p) 2 + q. Write a quadratic fuctio that models a situatio, ad explai ay assumptios made. Solve a problem, with or without techology, by aalyzig a quadratic fuctio. Geeral ad Specific Learig Outcomes 149

12 [C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 11P.R.5. Strad: Relatios ad Fuctios (cotiued) Specific Learig Outcomes It is expected that studets will: Solve problems that ivolve quadratic equatios. [C, CN, PS, R, T, V] Geeral Learig Outcome: Develop algebraic ad graphical reasoig through the study of relatios. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Explai, usig examples, the relatioship amog the roots of a quadratic equatio, the zeros of the correspodig quadratic fuctio, ad the x-itercepts of the graph of the quadratic fuctio. Derive the quadratic formula, usig deductive reasoig. Solve a quadratic equatio of the form ax 2 + bx + c = 0 by usig strategies such as determiig square roots factorig completig the square applyig the quadratic formula graphig its correspodig fuctio Select a method for solvig a quadratic equatio, justify the choice, ad verify the solutio. Explai, usig examples, how the discrimiat may be used to determie whether a quadratic equatio has two, oe, or o real roots; ad relate the umber of zeros to the graph of the correspodig quadratic fuctio. Idetify ad correct errors i a solutio to a quadratic equatio. Solve a problem by determiig or aalyzig a quadratic equatio. 150 Grades 9 to 12 Mathematics: Maitoba Curriculum Framework of Outcomes (2014)

13 [C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 11P.R.6. Strad: Relatios ad Fuctios (cotiued) Specific Learig Outcomes It is expected that studets will: Solve, algebraically ad graphically, problems that ivolve systems of liear-quadratic ad quadratic-quadratic equatios i two variables. [C, CN, PS, R, T, V] It is iteded that the quadratic equatios be limited to those that correspod to quadratic fuctios. Geeral Learig Outcome: Develop algebraic ad graphical reasoig through the study of relatios. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Model a situatio, usig a system of liear-quadratic or quadratic-quadratic equatios. Relate a system of liear-quadratic or quadratic-quadratic equatios to the cotext of a problem. Determie ad verify the solutio of a system of liear-quadratic or quadratic-quadratic equatios graphically, with techology. Determie ad verify the solutio of a system of liear-quadratic or quadratic-quadratic equatios algebraically. Explai the meaig of the poits of itersectio of a system of liear-quadratic or quadratic-quadratic equatios. Explai, usig examples, why a system of liear-quadratic or quadratic-quadratic equatios may have zero, oe, two, or a ifiite umber of solutios. Solve a problem that ivolves a system of liear-quadratic or quadratic-quadratic equatios, ad explai the strategy used. Geeral ad Specific Learig Outcomes 151

14 [C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 11P.R.7. 11P.R.8. 11P.R.9. Strad: Relatios ad Fuctios (cotiued) Specific Learig Outcomes It is expected that studets will: Solve problems that ivolve liear ad quadratic iequalities i two variables. [C, PS, T, V] Solve problems that ivolve quadratic iequalities i oe variable. [CN, PS, V] Aalyze arithmetic sequeces ad series to solve problems. [C, CN, PS, R, T] Geeral Learig Outcome: Develop algebraic ad graphical reasoig through the study of relatios. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Explai, usig examples, how test poits ca be used to determie the solutio regio that satisfies a iequality. Explai, usig examples, whe a solid or broke lie should be used i the solutio for a iequality. Sketch, with or without techology, the graph of a liear or quadratic iequality. Solve a problem that ivolves a liear or quadratic iequality. Determie the solutio of a quadratic iequality i oe variable, usig strategies such as case aalysis, graphig the related fuctio, roots ad test poits, or sig aalysis; ad explai the strategy used. Represet ad solve a problem that ivolves a quadratic iequality i oe variable. Iterpret the solutio to a problem that ivolves a quadratic iequality i oe variable. Idetify the assumptio(s) made whe defiig a arithmetic sequece or series. Provide ad justify a example of a arithmetic sequece. Derive a rule for determiig the geeral term of a arithmetic sequece. Describe the relatioship betwee arithmetic sequeces ad liear fuctios. Determie the first term, the commo differece, the umber of terms, or the value of a specific term i a problem ivolvig a arithmetic sequece. Derive a rule for determiig the sum of terms of a arithmetic series. Determie the first term, the commo differece, the umber of terms, or the value of the sum of specific umbers of terms i a problem ivolvig a arithmetic series. Solve a problem that ivolves a arithmetic sequece or series. 152 Grades 9 to 12 Mathematics: Maitoba Curriculum Framework of Outcomes (2014)

15 [C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 11P.R P.R.11. Strad: Relatios ad Fuctios (cotiued) Specific Learig Outcomes It is expected that studets will: Aalyze geometric sequeces ad series to solve problems. [C, CN, PS, R, T] Graph ad aalyze reciprocal fuctios (limited to the reciprocal of liear ad quadratic fuctios). [CN, R, T, V] Geeral Learig Outcome: Develop algebraic ad graphical reasoig through the study of relatios. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Idetify assumptios made whe idetifyig a geometric sequece or series. Provide ad justify a example of a geometric sequece. Derive a rule for determiig the geeral term of a geometric sequece. Determie the first term, the commo ratio, the umber of terms, or the value of a specific term i a problem ivolvig a geometric sequece. Derive a rule for determiig the sum of terms of a geometric series. Determie the first term, the commo ratio, the umber of terms, or the value of the sum of a specific umber of terms i a problem ivolvig a geometric series. Geeralize, usig iductive reasoig, a rule for determiig the sum of a ifiite geometric series. Explai why a ifiite geometric series is coverget or diverget. Solve a problem that ivolves a geometric sequece or series. Compare the graph of to the graph of y = f(x). Idetify, give a fuctio f(x), values of x for which will have vertical asymptotes; ad describe their relatioship to the o-permissible values of the related ratioal expressio. Graph, with or without techology, explai the strategies used. Graph, with or without techology, y = f(x), give explai the strategies used., give y = f(x) as a fuctio or a graph, ad as a fuctio or a graph, ad Geeral ad Specific Learig Outcomes 153