G r a d e 1 2 A p p l i e d M a t h e m a t i c s ( 4 0 S ) Grade 12 Applied Mathematics (40S) is iteded for studets cosiderig post-secodary studies that do ot require a study of theoretical calculus. It is cotext-drive ad promotes the learig of umerical ad geometrical problem-solvig techiques as they relate to the world aroud us. The primary goals of Applied Mathematics are to have studets develop critical-thikig skills through problem solvig ad through modellig real-world situatios mathematically to make predictios. These goals may be attaied i a umber of ways. Studets may collect data i experimets ad activities to develop mathematical cocepts by aalyzig that data. They are ecouraged to lear ad demostrate effective commuicatio skills through a variety of media. Studets are expected to become proficiet i both oral ad writte commuicatio skills. Applied Mathematics is desiged to promote studet flexibility ad resposibility. Flexibility is ecouraged by havig studets work o o-routie problems ad projects. Resposibility is ecouraged as studets work idividually ad i cooperative groups to explore coectios with other mathematical areas, school subjects, ad real-life applicatios. Techology is a itegral part of both learig ad assessmet i Applied Mathematics. Graphig calculators, spreadsheets, or other computer software will be used by studets for mathematical exploratios, modellig, ad problem solvig. Assessmet of Grade 12 Applied Mathematics should be a balace of assessmet for learig, assessmet as learig, ad assessmet of learig. Assessmet tools used i Grade 12 Applied 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 Grade 12 Applied Mathematics course icludes the followig topics: Fiacial Mathematics, Logical Reasoig, Probability, Relatios ad Fuctios, ad Desig ad Measuremet. Additioally, studets will complete a Mathematics Research Project. Learig outcomes from various topic areas may be combied i ay order whe cosiderig learig activities. The suggested time icludes istructioal ad assessmet time. Two possible teachig sequeces ad suggested hours are preseted i the followig tables: Geeral ad Specific Learig Outcomes 155
Possibility 1 Possibility 2 Uit Learig Outcomes Suggested Hours Uit Learig Outcomes Suggested Hours Siusoidal Fuctios L1, R3 12 Aalysis of Games ad Numbers L1 5 (itegrated throughout course) Compoud Iterest L1, FM1, FM2 12 Probability P1, P2, P3 15 Probability L1, L2, L3, P1, P2, P3 20 Persoal Fiace FM1, FM2, FM3 23 Polyomial Fuctios L1, R1 12 Fuctios R1, R2 20 Permutatios ad Combiatios L1, P4, P5, P6 12 Logic L2, L3 7 Ivestmets L1, FM3 12 Permutatios ad Combiatios P4, P5, P6 10 Desig ad Measuremet L1, D1 10 Siusoidal Fuctios R3 10 Research Project L1, RP1 10 Desig ad Measuremet D1 10 Expoetial ad Logarithmic Fuctios L1, R2 10 Research Project RP1 10 Total 110 Total 110 156 Grades 9 to 12 Mathematics: Maitoba Curriculum Framework of Outcomes (2014)
Geeral ad Specific Learig Outcomes with Achievemet Idicators by Course Grade 12 Applied Mathematics [C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 12A.FM.1. Strad: Fiacial Mathematics Specific Learig Outcomes It is expected that studets will: Solve problems that ivolve compoud iterest i fiacial decisio makig. [C, CN, PS, T, V] Geeral Learig Outcome: Develop umber sese i fiacial applicatios. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Explai the advatages ad disadvatages of compoud iterest ad simple iterest. Idetify situatios ivolvig compoud iterest. Graph ad compare the total iterest paid or eared for differet compoudig periods. Determie, give the pricipal, iterest rate, ad umber of compoudig periods, the total iterest of a loa. Graph ad describe the effects of chagig the value of oe of the variables i a situatio ivolvig compoud iterest. Determie, usig techology, the total cost of a loa uder a variety of coditios, such as differet amortizatio periods, iterest rates, ad compoudig periods or terms. Compare ad explai, usig techology, differet credit optios ivolvig compoud iterest, icludig bak or store credit cards or special promotios. Solve a cotextual problem ivolvig compoud iterest. Geeral ad Specific Learig Outcomes 157
[C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 12A.FM.2. 12A.FM.3. Strad: Fiacial Mathematics (cotiued) Specific Learig Outcomes It is expected that studets will: Aalyze costs ad beefits of retig, leasig, ad buyig. [CN, PS, R, T] Aalyze a ivestmet portfolio i terms of iterest rate rate of retur total retur [ME, PS, R, T] Studets should be able to solve a ivestmet problem usig tools such as debt-to-equity ratio, gross debt service ratio, or et worth. Geeral Learig Outcome: Develop umber sese i fiacial applicatios. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Idetify ad describe examples of assets that appreciate or depreciate. Compare, usig examples, retig, leasig, ad buyig. Justify, for a specific set of circumstaces, whether retig, buyig, or leasig would be advatageous. Solve, usig techology, a cotextual problem ivolvig retig, leasig, or buyig. Solve, usig techology, a cotextual problem ivolvig cost-ad-beefit aalysis. Determie, usig techology, the total value of a ivestmet whe there are regular cotributios to the pricipal. Graph ad compare the total value of a ivestmet with or without regular cotributios. Apply the Rule of 72 to solve ivestmet problems, ad explai the limitatios of the rule. Determie, usig techology, possible ivestmet strategies to achieve a fiacial goal. Explai the advatages ad disadvatages of log-term or short-term ivestmet optios. Explai, usig examples, why smaller ivestmets over a loger term may be better tha larger ivestmets over a shorter term. Determie ad compare the stregths ad weakesses of two or more portfolios. Solve a ivestmet problem. 158 Grades 9 to 12 Mathematics: Maitoba Curriculum Framework of Outcomes (2014)
[C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 12A.L.1. Strad: Logical Reasoig Specific Learig Outcomes It is expected that studets will: Aalyze puzzles ad games that ivolve umerical ad logical reasoig, usig problem-solvig strategies. [CN, ME, PS, R, T] It is iteded that this learig outcome be itegrated throughout the course by usig games ad puzzles such as chess, Sudoku, Nim, logic puzzles, magic squares, Kakuro, ad cribbage. Geeral Learig Outcome: Develop logical reasoig. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Determie, explai, ad verify a strategy to solve a puzzle or to wi a game such as guess ad check look for a patter make a systematic list draw or model elimiate possibilities simplify the origial problem work backward develop alterative approaches Idetify ad correct errors i a solutio to a puzzle or i a strategy for wiig a game. Create a variatio o a puzzle or a game, ad describe a strategy for solvig the puzzle or wiig the game. Geeral ad Specific Learig Outcomes 159
[C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 12A.L.2. 12A.L.3. Strad: Logical Reasoig (cotiued) Specific Learig Outcomes It is expected that studets will: Solve problems that ivolve the applicatio of set theory. [CN, PS, R, T, V] Solve problems that ivolve coditioal statemets. [C, CN, PS, R, T] Geeral Learig Outcome: Develop logical reasoig. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Explai how set theory is used i applicatios such as Iteret searches, database queries, data aalysis, games, ad puzzles. Provide examples of the empty set, disjoit sets, subsets, ad uiversal sets i cotext, ad explai the reasoig. Orgaize iformatio such as collected data ad umber properties, usig graphic orgaizers, ad explai the reasoig. Explai what a specified regio i a Ve diagram represets. Determie the elemets i the complemet, the itersectio, or the uio of two sets. Idetify ad correct errors i a solutio to a problem that ivolves sets. Solve a cotextual problem that ivolves sets, ad record the solutio. Aalyze a if-the statemet, make a coclusio, ad explai the reasoig. Make ad justify a decisio, usig what if? questios, i cotexts such as probability, fiace, sports, games, or puzzles, with or without techology. Determie the coverse, iverse, ad cotrapositive of a if-the statemet, determie its truth, ad, if it is false, provide a couter-example. Demostrate, usig examples, that the truth of ay statemet does ot imply the truth of its coverse or iverse. Demostrate, usig examples, that the truth of ay statemet does imply the truth of its cotrapositive. Idetify ad describe cotexts i which a bicoditioal statemet ca be justified. Aalyze ad summarize, usig a graphic orgaizer such as a truth table or Ve diagram, the possible results of give logical argumets that ivolve bicoditioal, coverse, iverse, or cotrapositive statemets. 160 Grades 9 to 12 Mathematics: Maitoba Curriculum Framework of Outcomes (2014)
[C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 12A.P.1. 12A.P.2. Strad: Probability Specific Learig Outcomes It is expected that studets will: Iterpret ad assess the validity of odds ad probability statemets. [C, CN, ME, T] Solve problems that ivolve the probability of mutually exclusive ad o-mutually exclusive evets. [CN, PS, R, T, V] Geeral Learig Outcome: Develop critical thikig skills related to ucertaity. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Provide examples of statemets of probability ad odds foud i fields such as media, biology, sports, medicie, weather, sociology, or psychology. Explai, usig examples, the relatioship betwee odds (part-part) ad probability (partwhole). Express odds as a probability ad vice versa. Determie the probability of, or the odds for ad agaist, a outcome i a situatio. Explai, usig examples, how decisios may be based o probability or odds ad o subjective judgmets. Solve a cotextual problem that ivolves odds or probability. Classify evets as mutually exclusive or o-mutually exclusive, ad explai the reasoig. Determie whether two evets are complemetary, ad explai the reasoig. Represet, usig set otatio or graphic orgaizers, mutually exclusive (icludig complemetary) ad o-mutually exclusive evets. Solve a cotextual problem that ivolves the probability of mutually exclusive or omutually exclusive evets. Solve a cotextual problem that ivolves the probability of complemetary evets. Create ad solve a problem that ivolves mutually exclusive or o-mutually exclusive evets. Geeral ad Specific Learig Outcomes 161
[C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 12A.P.3. 12A.P.4. 12A.P.5. Strad: Probability (cotiued) Specific Learig Outcomes It is expected that studets will: Solve problems that ivolve the probability of idepedet ad depedet evets. [CN, PS, R, T] Solve problems that ivolve the fudametal coutig priciple. [PS, R, T, V] Solve problems that ivolve permutatios. [ME, PS, R, T, V] Studets should be able to use strategies such as cases or groupig objects together to solve a cotextual problem. Also, it is iteded that circular permutatios ot be icluded. Geeral Learig Outcome: Develop critical thikig skills related to ucertaity. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Compare, usig examples, depedet ad idepedet evets. Determie the probability of a evet, give the occurrece of a previous evet. Determie the probability of two depedet or two idepedet evets. Create ad solve a cotextual problem that ivolves determiig the probability of depedet or idepedet evets. Represet ad solve coutig problems, usig a graphic orgaizer. Geeralize, from examples, the fudametal coutig priciple. Idetify ad explai assumptios made i solvig a coutig problem. Solve a cotextual coutig problem, usig the fudametal coutig priciple, ad explai the reasoig. Represet the umber of arragemets of elemets take at a time, usig factorial otatio. Determie the value of a factorial. Simplify a umeric fractio cotaiig factorials i both the umerator ad deomiator. Determie the umber of permutatios of elemets take r at a time. Determie the umber of permutatios of elemets take at a time where some elemets are ot distict. Explai, usig examples, the effect o the total umber of permutatios of elemets whe two or more elemets are idetical. Geeralize strategies for determiig the umber of permutatios of elemets take r at a time. Solve a cotextual problem that ivolves probability ad permutatios. 162 Grades 9 to 12 Mathematics: Maitoba Curriculum Framework of Outcomes (2014)
[C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 12A.P.6. Strad: Probability (cotiued) Specific Learig Outcomes It is expected that studets will: Solve problems that ivolve combiatios. [ME, PS, R, T, V] Geeral Learig Outcome: Develop critical thikig skills related to ucertaity. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Explai, usig examples, why order is or is ot importat whe solvig problems that ivolve permutatios or combiatios. Determie the umber of combiatios of elemets take r at a time. Geeralize strategies for determiig the umber of combiatios of elemets take r at a time. Solve a cotextual problem that ivolves probability ad combiatios. Geeral ad Specific Learig Outcomes 163
[C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 12A.R.1. 12A.R.2. Strad: Relatios ad Fuctios Specific Learig Outcomes It is expected that studets will: Represet data, usig polyomial fuctios (of degree 3), to solve problems. [C, CN, PS, T, V] Represet data, usig expoetial ad logarithmic fuctios, to solve problems. [C, CN, PS, 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. Describe, orally ad i writte form, the characteristics of polyomial fuctios by aalyzig their graphs or their equatios. Match equatios i a set to their correspodig graphs. Graph data ad determie the polyomial fuctio that best approximates the data. Iterpret the graph of a polyomial fuctio that models a situatio, ad explai the reasoig. Solve, usig techology, a cotextual problem that ivolves data that is best represeted by graphs of polyomial fuctios, ad explai the reasoig. Describe, orally ad i writte form, the characteristics of expoetial or logarithmic fuctios by aalyzig their graphs or their equatios. Match equatios i a set to their correspodig graphs. Graph data ad determie the expoetial or logarithmic fuctio that best approximates the data. Iterpret the graph of a expoetial or logarithmic fuctio that models a situatio, ad explai the reasoig. Solve, usig techology, a cotextual problem that ivolves data that is best represeted by graphs of expoetial or logarithmic fuctios, ad explai the reasoig. 164 Grades 9 to 12 Mathematics: Maitoba Curriculum Framework of Outcomes (2014)
[C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 12A.R.3. Strad: Relatios ad Fuctios (cotiued) Specific Learig Outcomes It is expected that studets will: Represet data, usig siusoidal fuctios, to solve problems. [C, CN, PS, T, V] Geeral Learig Outcome: Develop algebraic ad graphical reasoig through the study or relatios. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Describe, orally ad i writte form, the characteristics of siusoidal fuctios by aalyzig their graphs or their equatios. Match equatios i a set to their correspodig graphs. Graph data ad determie the siusoidal fuctio that best approximates the data. Iterpret the graph of a siusoidal fuctio that models a situatio, ad explai the reasoig. Solve, usig techology, a cotextual problem that ivolves data that is best represeted by graphs of siusoidal fuctios, ad explai the reasoig. Geeral ad Specific Learig Outcomes 165
[C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 12A.RP.1. Strad: Mathematics Research Project Specific Learig Outcomes It is expected that studets will: Research ad give a presetatio o a curret evet or a area of iterest that ivolves mathematics. [C, CN, ME, PS, R, T, V] Geeral Learig Outcome: Develop a appreciatio of the role of mathematics i society. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Collect primary or secodary data (statistical or iformatioal) related to the topic. Assess the accuracy, reliability, ad relevace of the primary or secodary data collected by idetifyig examples of bias ad poits of view idetifyig ad describig the data collectio methods determiig whether the data is relevat determiig whether the data is cosistet with iformatio obtaied from other sources o the same topic Iterpret data, usig statistical methods if applicable. Idetify cotroversial issues, if ay, ad preset multiple sides of the issues with supportig data. Orgaize ad preset the research project, with or without techology. 166 Grades 9 to 12 Mathematics: Maitoba Curriculum Framework of Outcomes (2014)
[C] Commuicatio [PS] Problem Solvig [CN] Coectios [R] Reasoig [ME] Metal Mathematics [T] Techology ad Estimatio [V] Visualizatio 12A.D. 1 Strad: Desig ad Measuremet Specific Learig Outcomes It is expected that studets will: Aalyze objects, shapes, ad processes to solve cost ad desig problems. [C, CN, ME, PS, R, T, V] Geeral Learig Outcome: Develop critical thikig skills related to measuremet desig. Achievemet Idicators The followig set of idicators may be used to determie whether studets have met the correspodig specific learig outcome. Solve a problem ivolvig perimeter, area, ad volume usig dimesios ad uit prices. Solve a problem ivolvig estimatio ad costig for objects, shapes, or processes whe a desig is give. Idetify ad correct errors i a solutio to a problem that ivolves costig for objects, shapes, or processes. Estimate the solutios to complex measuremet problems usig simplified models. Desig a object, shape, layout, or process withi a specified budget. Geeral ad Specific Learig Outcomes 167