Foundations for College Mathematics, Grade 11

Transcription

1 Foudatios for College Mathematics, Grade 11 College Preparatio This course eables studets to broade their uderstadig of mathematics as a problemsolvig tool i the real world. Studets will exted their uderstadig of quadratic relatios; ivestigate situatios ivolvig expoetial growth; solve problems ivolvig compoud iterest; solve fiacial problems coected with vehicle owership; develop their ability to reaso by collectig, aalysig, ad evaluatig data ivolvig oe variable; coect probability ad statistics; ad solve problems i geometry ad trigoometry. Studets will cosolidate their mathematical skills as they solve problems ad commuicate their thikig. Prerequisite: Foudatios of Mathematics, Grade 10, Applied 67

2 MATHEMATICAL PROCESS EXPECTATIONS The mathematical processes are to be itegrated ito studet learig i all areas of this course. Throughout this course, studets will: Problem Solvig Reasoig ad Provig Reflectig develop, select, apply, compare, ad adapt a variety of problem-solvig strategies as they pose ad solve problems ad coduct ivestigatios, to help deepe their mathematical uderstadig; develop ad apply reasoig skills (e.g., use of iductive reasoig, deductive reasoig, ad couter-examples; costructio of proofs) to make mathematical cojectures, assess cojectures, ad justify coclusios, ad pla ad costruct orgaized mathematical argumets; demostrate that they are reflectig o ad moitorig their thikig to help clarify their uderstadig as they complete a ivestigatio or solve a problem (e.g., by assessig the effectiveess of strategies ad processes used, by proposig alterative approaches, by judgig the reasoableess of results, by verifyig solutios); Selectig Tools ad Computatioal Strategies select ad use a variety of cocrete, visual, ad electroic learig tools ad appropriate computatioal strategies to ivestigate mathematical ideas ad to solve problems; Coectig Represetig Commuicatig make coectios amog mathematical cocepts ad procedures, ad relate mathematical ideas to situatios or pheomea draw from other cotexts (e.g., other curriculum areas, daily life, curret evets, art ad culture, sports); create a variety of represetatios of mathematical ideas (e.g., umeric, geometric, algebraic, graphical, pictorial represetatios; oscree dyamic represetatios), coect ad compare them, ad select ad apply the appropriate represetatios to solve problems; commuicate mathematical thikig orally, visually, ad i writig, usig precise mathematical vocabulary ad a variety of appropriate represetatios, ad observig mathematical covetios. 68

3 A. MATHEMATICAL MODELS OVERALL EXPECTATIONS 1. make coectios betwee the umeric, graphical, ad algebraic represetatios of quadratic relatios, ad use the coectios to solve problems;. demostrate a uderstadig of expoets, ad make coectios betwee the umeric, graphical, ad algebraic represetatios of expoetial relatios; 3. describe ad represet expoetial relatios, ad solve problems ivolvig expoetial relatios arisig from real-world applicatios. SPECIFIC EXPECTATIONS 1. Coectig Graphs ad Equatios of Quadratic Relatios 1.4 sketch graphs of quadratic relatios represeted by the equatio y=a(x h) + k (e.g., usig the vertex ad at least oe poit o each side of the vertex; applyig oe or more trasformatios to the graph of y=x ) Foudatios for College Mathematics 1.1 costruct tables of values ad graph quadratic relatios arisig from real-world applicatios (e.g., droppig a ball from a give height; varyig the edge legth of a cube ad observig the effect o the surface area of the cube) 1. determie ad iterpret meaigful values of the variables, give a graph of a quadratic relatio arisig from a real-world applicatio Sample problem: Uder certai coditios, there is a quadratic relatio betwee the profit of a maufacturig compay ad the umber of items it produces. Explai how you could iterpret a graph of the relatio to determie the umbers of items produced for which the compay makes a profit ad to determie the maximum profit the compay ca make. 1.3 determie, through ivestigatio usig techology, the roles of a, h, ad k i quadratic relatios of the form y=a(x h) + k, ad describe these roles i terms of trasformatios o the graph of y=x (i.e., traslatios; reflectios i the x-axis; vertical stretches ad compressios to ad from the x-axis) Sample problem: Ivestigate the graph y=3(x h) + 5 for various values of h, usig techology, ad describe the effects of chagig h i terms of a trasformatio. 1.5 expad ad simplify quadratic expressios i oe variable ivolvig multiplyig biomials 1 [e.g., ( x+1 ) (3x )] or squarig a biomial [e.g., 5(3x 1) ], usig a variety of tools (e.g., paper ad pecil, algebra tiles, computer algebra systems) 1.6 express the equatio of a quadratic relatio i the stadard form y=ax + bx+c, give the vertex form y=a(x h) + k, ad verify, usig graphig techology, that these forms are equivalet represetatios Sample problem: Give the vertex form y=3(x 1) + 4, express the equatio i stadard form. Use techology to compare the graphs of these two forms of the equatio. 1.7 factor triomials of the form ax + bx+c, where a=1 or where a is the commo factor, by various methods 1.8 determie, through ivestigatio, ad describe the coectio betwee the factors of a quadratic expressio ad the x-itercepts of the graph of the correspodig quadratic relatio Sample problem: Ivestigate the relatioship betwee the factored form of 3x + 15x+1 ad the x-itercepts of y=3x + 15x+1. MATHEMATICAL MODELS 69

4 THE ONTARIO CURRICULUM, GRADES 11 AND 1 Mathematics Grade 11, College Preparatio solve problems, usig a appropriate strategy (i.e., factorig, graphig), give equatios of quadratic relatios, icludig those that arise from real-world applicatios (e.g., break-eve poit) Sample problem: O plaet X, the height, h metres, of a object fired upward from the groud at 48 m/s is described by the equatio h=48t 16t, where t secods is the time sice the object was fired upward. Determie the maximum height of the object, the times at which the object is 3 m above the groud, ad the time at which the object hits the groud.. Coectig Graphs ad Equatios of Expoetial Relatios.1 determie, through ivestigatio usig a variety of tools ad strategies (e.g., graphig with techology; lookig for patters i tables of values), ad describe the meaig of egative expoets ad of zero as a expoet. evaluate, with ad without techology, umeric expressios cotaiig iteger expoets ad ratioal bases (e.g.,, 6, , 1.03 ).3 determie, through ivestigatio (e.g., by patterig with ad without a calculator), the expoet rules for multiplyig ad dividig umerical expressios ivolvig expoets [e.g., ( 1 3 ) x ( 1 ) ], ad the expoet rule for simplifyig umerical expressios ivolvig a power of a power 3 [e.g., ( 5 ) ] 10.4 graph simple expoetial relatios, usig paper ad pecil, give their equatios x x x [e.g., y=, y=10, y= ( 1 ) ].5 make ad describe coectios betwee represetatios of a expoetial relatio (i.e., umeric i a table of values; graphical; algebraic).6 distiguish expoetial relatios from liear ad quadratic relatios by makig comparisos i a variety of ways (e.g., comparig rates of chage usig fiite differeces i tables of values; ispectig graphs; comparig equatios), withi the same cotext whe possible (e.g., simple iterest ad compoud iterest, populatio growth) 3 3 Sample problem: Explai i a variety of ways how you ca distiguish expoetial growth x represeted by y = from quadratic growth represeted by y=x ad liear growth represeted by y=x. 3. Solvig Problems Ivolvig Expoetial Relatios 3.1 collect data that ca be modelled as a expoetial relatio, through ivestigatio with ad without techology, from primary sources, usig a variety of tools (e.g., cocrete materials such as umber cubes, cois; measuremet tools such as electroic probes), or from secodary sources (e.g., websites such as Statistics Caada, E-STAT), ad graph the data Sample problem: Collect data ad graph the coolig curve represetig the relatioship betwee temperature ad time for hot water coolig i a porcelai mug. Predict the shape of the coolig curve whe hot water cools i a isulated mug. Test your predictio. 3. describe some characteristics of expoetial relatios arisig from real-world applicatios (e.g., bacterial growth, drug absorptio) by usig tables of values (e.g., to show a costat ratio, or multiplicative growth or decay) ad graphs (e.g., to show, with techology, that there is o maximum or miimum value) 3.3 pose problems ivolvig expoetial relatios arisig from a variety of real-world applicatios (e.g., populatio growth, radioactive decay, compoud iterest), ad solve these ad other such problems by usig a give graph or a graph geerated with techology from a give table of values or a give equatio Sample problem: Give a graph of the populatio of a bacterial coloy versus time, determie the chage i populatio i the first hour. 3.4 solve problems usig give equatios of expoetial relatios arisig from a variety of real-world applicatios (e.g., radioactive decay, populatio growth, height of a boucig ball, compoud iterest) by substitutig values for the expoet ito the equatios Sample problem: The height, h metres, of a ball after bouces is give by the equatio h=(0.6). Determie the height of the ball after 3 bouces.

5 B. PERSONAL FINANCE OVERALL EXPECTATIONS 1. compare simple ad compoud iterest, relate compoud iterest to expoetial growth, ad solve problems ivolvig compoud iterest;. compare services available from fiacial istitutios, ad solve problems ivolvig the cost of makig purchases o credit; 3. iterpret iformatio about owig ad operatig a vehicle, ad solve problems ivolvig the associated costs. SPECIFIC EXPECTATIONS 1. Solvig Problems Ivolvig Compoud Iterest 1.1 determie, through ivestigatio usig techology, the compoud iterest for a give ivestmet, usig repeated calculatios of simple iterest, ad compare, usig a table of values ad graphs, the simple ad compoud iterest eared for a give pricipal (i.e., ivestmet) ad a fixed iterest rate over time Sample problem: Compare, usig tables of values ad graphs, the amouts after each of the first five years for a $1000 ivestmet at 5% simple iterest per aum ad a $1000 ivestmet at 5% iterest per aum, compouded aually. 1. determie, through ivestigatio (e.g., usig spreadsheets ad graphs), ad describe the relatioship betwee compoud iterest ad expoetial growth 1.3 solve problems, usig a scietific calculator, that ivolve the calculatio of the amout, A (also referred to as future value, FV ), ad the pricipal, P (also referred to as preset value, PV ), usig the compoud iterest formula i the form A=P(1 + i ) [or FV= PV (1 + i ) ] Sample problem: Calculate the amout if $1000 is ivested for 3 years at 6% per aum, compouded quarterly. 1.4 calculate the total iterest eared o a ivestmet or paid o a loa by determiig the differece betwee the amout ad the pricipal [e.g., usig I=A P(or I=FV PV )] 1.5 solve problems, usig a TVM Solver o a graphig calculator or o a website, that ivolve the calculatio of the iterest rate per compoudig period, i, or the umber of compoudig periods,, i the compoud iterest formula A=P(1 + i ) [or FV = PV (1 + i ) ] Sample problem: Use the TVM Solver o a graphig calculator to determie the time it takes to double a ivestmet i a accout that pays iterest of 4% per aum, compouded semi-aually. 1.6 determie, through ivestigatio usig techology (e.g., a TVM Solver o a graphig calculator or o a website), the effect o the future value of a compoud iterest ivestmet or loa of chagig the total legth of time, the iterest rate, or the compoudig period Sample problem: Ivestigate whether doublig the iterest rate will halve the time it takes for a ivestmet to double. Foudatios for College Mathematics PERSONAL FINANCE 71

6 . Comparig Fiacial Services 3. Owig ad Operatig a Vehicle THE ONTARIO CURRICULUM, GRADES 11 AND 1 Mathematics Grade 11, College Preparatio.1 gather, iterpret, ad compare iformatio about the various savigs alteratives commoly available from fiacial istitutios (e.g., savigs ad chequig accouts, term ivestmets), the related costs (e.g., cost of cheques, mothly statemet fees, early withdrawal pealties), ad possible ways of reducig the costs (e.g., maitaiig a miimum balace i a savigs accout; payig a mothly flat fee for a package of services). gather ad iterpret iformatio about ivestmet alteratives (e.g., stocks, mutual fuds, real estate, GICs, savigs accouts), ad compare the alteratives by cosiderig the risk ad the rate of retur.3 gather, iterpret, ad compare iformatio about the costs (e.g., user fees, aual fees, service charges, iterest charges o overdue balaces) ad icetives (e.g., loyalty rewards; philathropic icetives, such as support for Olympic athletes or a Red Cross disaster relief fud) associated with various credit cards ad debit cards.4 gather, iterpret, ad compare iformatio about curret credit card iterest rates ad regulatios, ad determie, through ivestigatio usig techology, the effects of delayed paymets o a credit card balace.5 solve problems ivolvig applicatios of the compoud iterest formula to determie the cost of makig a purchase o credit Sample problem: Usig iformatio gathered about the iterest rates ad regulatios for two differet credit cards, compare the costs of purchasig a $1500 computer with each card if the full amout is paid 55 days later. 3.1 gather ad iterpret iformatio about the procedures ad costs ivolved i isurig a vehicle (e.g., car, motorcycle, sowmobile) ad the factors affectig isurace rates (e.g., geder, age, drivig record, model of vehicle, use of vehicle), ad compare the isurace costs for differet categories of drivers ad for differet vehicles Sample problem: Use automobile isurace websites to ivestigate the degree to which the type of car ad the age ad geder of the driver affect isurace rates. 3. gather, iterpret, ad compare iformatio about the procedures ad costs (e.g., mothly paymets, isurace, depreciatio, maiteace, miscellaeous expeses) ivolved i buyig or leasig a ew vehicle or buyig a used vehicle Sample problem: Compare the costs of buyig a ew car, leasig the same car, ad buyig a older model of the same car. 3.3 solve problems, usig techology (e.g., calculator, spreadsheet), that ivolve the fixed costs (e.g., licece fee, isurace) ad variable costs (e.g., maiteace, fuel) of owig ad operatig a vehicle Sample problem: The rate at which a car cosumes gasolie depeds o the speed of the car. Use a give graph of gasolie cosumptio, i litres per 100 km, versus speed, i kilometres per hour, to determie how much gasolie is used to drive 500 km at speeds of 80 km/h, 100 km/h, ad 10 km/h. Use the curret price of gasolie to calculate the cost of drivig 500 km at each of these speeds. 7

7 C. GEOMETRY AND TRIGONOMETRY OVERALL EXPECTATIONS 1. represet, i a variety of ways, two-dimesioal shapes ad three-dimesioal figures arisig from real-world applicatios, ad solve desig problems;. solve problems ivolvig trigoometry i acute triagles usig the sie law ad the cosie law, icludig problems arisig from real-world applicatios. SPECIFIC EXPECTATIONS 1. Represetig Two-Dimesioal Shapes ad Three-Dimesioal Figures 1.1 recogize ad describe real-world applicatios of geometric shapes ad figures, through ivestigatio (e.g., by importig digital photos ito dyamic geometry software), i a variety of cotexts (e.g., product desig, architecture, fashio), ad explai these applicatios (e.g., oe reaso that sewer covers are roud is to prevet them from fallig ito the sewer durig removal ad replacemet) Sample problem: Explai why rectagular prisms are ofte used for packagig. 1. represet three-dimesioal objects, usig cocrete materials ad desig or drawig software, i a variety of ways (e.g., orthographic projectios [i.e., frot, side, ad top views], perspective isometric drawigs, scale models) 1.3 create ets, plas, ad patters from physical models arisig from a variety of real-world applicatios (e.g., fashio desig, iterior decoratig, buildig costructio), by applyig the metric ad imperial systems ad usig desig or drawig software 1.4 solve desig problems that satisfy give costraits (e.g., desig a rectagular berm that would cotai all the oil that could leak from a cylidrical storage tak of a give height ad radius), usig physical models (e.g., built from popsicle sticks, cardboard, duct tape) or drawigs (e.g., made usig desig or drawig software), ad state ay assumptios made Sample problem: Desig ad costruct a model boat that ca carry the most peies, usig oe sheet of 8.5 i. x 11 i. card stock, o more tha five popsicle sticks, ad some adhesive tape or glue.. Applyig the Sie Law ad the Cosie Law i Acute Triagles.1 solve problems, icludig those that arise from real-world applicatios (e.g., surveyig, avigatio), by determiig the measures of the sides ad agles of right triagles usig the primary trigoometric ratios. verify, through ivestigatio usig techology (e.g., dyamic geometry software, spreadsheet), the sie law ad the cosie law (e.g., compare, usig dyamic geometry software, a b c the ratios,, ad i si A sib si C triagle ABC while draggig oe of the vertices);.3 describe coditios that guide whe it is appropriate to use the sie law or the cosie law, ad use these laws to calculate sides ad agles i acute triagles.4 solve problems that arise from real-world applicatios ivolvig metric ad imperial measuremets ad that require the use of the sie law or the cosie law i acute triagles Foudatios for College Mathematics GEOMETRY AND TRIGONOMETRY 73

8 Grade 11, Grade Uiversity/College 11, Preparatio Preparatio THE ONTARIO CURRICULUM, GRADES 11 AND 1 Mathematics 74 D. DATA MANAGEMENT OVERALL EXPECTATIONS 1. solve problems ivolvig oe-variable data by collectig, orgaizig, aalysig, ad evaluatig data;. determie ad represet probability, ad idetify ad iterpret its applicatios. SPECIFIC EXPECTATIONS 1. Workig With Oe-Variable Data 1.1 idetify situatios ivolvig oe-variable data (i.e., data about the frequecy of a give occurrece), ad desig questioaires (e.g., for a store to determie which CDs to stock, for a radio statio to choose which music to play) or experimets (e.g., coutig, takig measuremets) for gatherig oe-variable data, givig cosideratio to ethics, privacy, the eed for hoest resposes, ad possible sources of bias Sample problem: Oe lae of a three-lae highway is beig restricted to vehicles with at least two passegers to reduce traffic cogestio. Desig a experimet to collect oe-variable data to decide whether traffic cogestio is actually reduced. 1. collect oe-variable data from secodary sources (e.g., Iteret databases), ad orgaize ad store the data usig a variety of tools (e.g., spreadsheets, dyamic statistical software) 1.3 explai the distictio betwee the terms populatio ad sample, describe the characteristics of a good sample, ad explai why samplig is ecessary (e.g., time, cost, or physical costraits) Sample problem: Explai the terms sample ad populatio by givig examples withi your school ad your commuity. 1.4 describe ad compare samplig techiques (e.g., radom, stratified, clustered, coveiece, volutary); collect oe-variable data from primary sources, usig appropriate samplig techiques i a variety of real-world situatios; ad orgaize ad store the data 1.5 idetify differet types of oe-variable data (i.e., categorical, discrete, cotiuous), ad represet the data, with ad without techology, i appropriate graphical forms (e.g., histograms, bar graphs, circle graphs, pictographs) 1.6 idetify ad describe properties associated with commo distributios of data (e.g., ormal, bimodal, skewed) 1.7 calculate, usig formulas ad/or techology (e.g., dyamic statistical software, spreadsheet, graphig calculator), ad iterpret measures of cetral tedecy (i.e., mea, media, mode) ad measures of spread (i.e., rage, stadard deviatio) 1.8 explai the appropriate use of measures of cetral tedecy (i.e., mea, media, mode) ad measures of spread (i.e., rage, stadard deviatio) Sample problem: Explai whether the mea or the media of your course marks would be the more appropriate represetatio of your achievemet. Describe the additioal iformatio that the stadard deviatio of your course marks would provide. 1.9 compare two or more sets of oe-variable data, usig measures of cetral tedecy ad measures of spread Sample problem: Use measures of cetral tedecy ad measures of spread to compare data that show the lifetime of a ecoomy light bulb with data that show the lifetime of a log-life light bulb solve problems by iterpretig ad aalysig oe-variable data collected from secodary sources

9 . Applyig Probability.1 idetify examples of the use of probability i the media ad various ways i which probability is represeted (e.g., as a fractio, as a percet, as a decimal i the rage 0 to 1). determie the theoretical probability of a evet (i.e., the ratio of the umber of favourable outcomes to the total umber of possible outcomes, where all outcomes are equally likely), ad represet the probability i a variety of ways (e.g., as a fractio, as a percet, as a decimal i the rage 0 to 1).3 perform a probability experimet (e.g., tossig a coi several times), represet the results usig a frequecy distributio, ad use the distributio to determie the experimetal probability of a evet.4 compare, through ivestigatio, the theoretical probability of a evet with the experimetal probability, ad explai why they might differ Sample problem: If you toss 10 cois repeatedly, explai why 5 heads are ulikely to result from every toss..5 determie, through ivestigatio usig classgeerated data ad techology-based simulatio models (e.g., usig a radom-umber geerator o a spreadsheet or o a graphig calculator), the tedecy of experimetal probability to approach theoretical probability as the umber of trials i a experimet icreases (e.g., If I simulate tossig a coi 1000 times usig techology, the experimetal probability that I calculate for tossig tails is likely to be closer to the theoretical probability tha if I simulate tossig the coi oly 10 times ) Sample problem: Calculate the theoretical probability of rollig a o a umber cube. Simulate rollig a umber cube, ad use the simulatio to calculate the experimetal probability of rollig a over 10, 0, 30,, 00 trials. Graph the experimetal probability versus the umber of trials, ad describe ay tred..6 iterpret iformatio ivolvig the use of probability ad statistics i the media, ad make coectios betwee probability ad statistics (e.g., statistics ca be used to geerate probabilities) Foudatios for College Mathematics DATA MANAGEMENT 75