Mathematics

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Grade 12, College PreparationSample problem: Why is it possible todetermine log10(100) but not log10(0) orlog10(– 100)? Explain your reasoning.2.4 determine, with technology, the approximatelogarithm of a number to any base, includingbase 10 [e.g., by recognizing that log10(0.372)can be determined using the LOG key ona calculator; by reasoning that log329 isbetween 3 and 4 and using systematic trial todetermine that log 29 is approximately 3.07]32.5 make connections between related logarithmicand exponential equations (e.g., log5125 = 33can also be expressed as 5 = 125), and solvesimple exponential equations by rewritingxthem in logarithmic form (e.g., solving 3 = 10by rewriting the equation as log310 = x)2.6 pose problems based on real-world applicationsthat can be modelled with given exponentialequations, and solve these and othersuch problems algebraically by rewritingthem in logarithmic formSample problem: When a potato whose temperatureis 20°C is placed in an oven maintainedat 200°C, the relationship between thecore temperature of the potato T, in degreesCelsius, and the cooking time t, in minutes, istmodelled by the equation 200 – T = 180(0.96) .Use logarithms to determine the time whenthe potato’s core temperature reaches 160°C.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | <strong>Mathematics</strong>126
B. POLYNOMIAL FUNCTIONSOVERALL EXPECTATIONSBy the end of this course, students will:1. recognize and evaluate polynomial functions, describe key features of their graphs, and solveproblems using graphs of polynomial functions;2. make connections between the numeric, graphical, and algebraic representations of polynomialfunctions;3. solve polynomial equations by factoring, make connections between functions and formulas, andsolve problems involving polynomial expressions arising from a variety of applications.SPECIFIC EXPECTATIONS<strong>Mathematics</strong> for College Technology1. Investigating Graphs of PolynomialFunctionsBy the end of this course, students will:1.1 recognize a polynomial expression (i.e., aseries of terms where each term is the productof a constant and a power of x with a nonnegativeintegral exponent, such as3 2x – 5x + 2x – 1); recognize the equation ofa polynomial function and give reasons whyit is a function, and identify linear and quadraticfunctions as examples of polynomialfunctions1.2 compare, through investigation using graphingtechnology, the graphical and algebraicrepresentations of polynomial (i.e., linear,quadratic, cubic, quartic) functions (e.g., investigatethe effect of the degree of a polynomialfunction on the shape of its graph and themaximum number of x-intercepts; investigatethe effect of varying the sign of the leadingcoefficient on the end behaviour of thefunction for very large positive or negativex-values)Sample problem: Investigate the maximumnumber of x-intercepts for linear, quadratic,cubic, and quartic functions using graphingtechnology.1.3 describe key features of the graphs of polynomialfunctions (e.g., the domain and range,the shape of the graphs, the end behaviourof the functions for very large positive ornegative x-values)Sample problem: Describe and compare thekey features of the graphs of the functions234f(x) = x, f(x) = x , f(x) = x , and f(x) = x .1.4 distinguish polynomial functions fromsinusoidal and exponential functions [e.g.,xf(x) = sin x, f(x) = 2 )], and compare andcontrast the graphs of various polynomialfunctions with the graphs of other typesof functions1.5 substitute into and evaluate polynomial functionsexpressed in function notation, includingfunctions arising from real-world applicationsSample problem: A box with no top is beingmade out of a 20-cm by 30-cm piece ofcardboard by cutting equal squares ofside length x from the corners and foldingup the sides. The volume of the box isV = x(20 – 2x)(30 – 2x). Determine the volumeif the side length of each square is 6 cm. Usethe graph of the polynomial function V(x) todetermine the size of square that should becut from the corners if the required volume3of the box is 1000 cm .1.6 pose problems based on real-world applicationsthat can be modelled with polynomialfunctions, and solve these and other suchproblems by using a given graph or a graphgenerated with technology from a table ofvalues or from its equation1.7 recognize, using graphs, the limitations ofmodelling a real-world relationship using apolynomial function, and identify and explainany restrictions on the domain and range(e.g., restrictions on the height and time for aMCT4CPOLYNOMIAL FUNCTIONS127
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CONTENTSINTRODUCTION 3Secondary Sch
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INTRODUCTIONThis document replaces
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and art. It is important that these
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THE PROGRAM INMATHEMATICSOVERVIEW O
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Courses in Mathematics, Grades 11 a
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Half-Credit CoursesThe courses outl
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The Grade 11 university preparation
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Grade 11FOUNDATIONS FORCOLLEGEMATHE
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THE MATHEMATICALPROCESSESPresented
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REASONING AND PROVINGReasoning help
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Mental computation involves calcula
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ASSESSMENTAND EVALUATIONOF STUDENTA
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THE ACHIEVEMENT CHART FOR MATHEMATI
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course and reflects the correspondi
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Categories 50−59%(Level 1)60−69
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The approaches and strategies used
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If the student requires either acco
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use of a variety of instructional s
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THE ROLE OF INFORMATION AND COMMUNI
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they need to be aware of harassment
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Functions, Grade 11University Prepa
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A. CHARACTERISTICS OF FUNCTIONSOVER
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2.5 solve problems involving the in
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2.2 determine, through investigatio
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2. Investigating Arithmetic andGeom
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D. TRIGONOMETRIC FUNCTIONSOVERALL E
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Sample problem: The relationship be
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MATHEMATICAL PROCESS EXPECTATIONSTh
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Grade 11, University/College Prepar
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MATHEMATICAL PROCESS EXPECTATIONSTh
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THE ONTARIO CURRICULUM, GRADES 11 A
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2. Comparing Financial Services 3.
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Grade 11, Grade University/College
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Page 81 and 82: A. EARNING AND PURCHASINGOVERALL EX
Page 83 and 84: B. SAVING, INVESTING, ANDBORROWINGO
Page 85 and 86: C. TRANSPORTATION AND TRAVELOVERALL
Page 87 and 88: Advanced Functions,Grade 12Universi
Page 89 and 90: A. EXPONENTIAL AND LOGARITHMICFUNCT
Page 91 and 92: B. TRIGONOMETRIC FUNCTIONSOVERALL E
Page 93 and 94: C. POLYNOMIAL AND RATIONALFUNCTIONS
Page 95 and 96: Sample problem: Investigate, using
Page 97 and 98: D. CHARACTERISTICS OF FUNCTIONSGrad
Page 99 and 100: 2.3 determine, through investigatio
Page 101 and 102: Calculus and Vectors,Grade 12Univer
Page 103 and 104: A. RATE OF CHANGEOVERALL EXPECTATIO
Page 107 and 108: B. DERIVATIVES AND THEIRAPPLICATION
Page 109 and 110: joules the bird gets if it spends t
Page 111 and 112: 2.3 solve problems involving the ad
Page 113 and 114: Mathematics of DataManagement, Grad
Page 115 and 116: A. COUNTING AND PROBABILITYOVERALL
Page 117 and 118: B. PROBABILITY DISTRIBUTIONSOVERALL
Page 119 and 120: 2.7 make connections, through inves
Page 121 and 122: 2.3 describe how the use of random
Page 123 and 124: as a measure of the fit of the data
Page 125 and 126: Mathematics for CollegeTechnology,
Page 127: A. EXPONENTIAL FUNCTIONSOVERALL EXP
Page 131 and 132: 3.6 determine the value of a variab
Page 133 and 134: 2.2 sketch the graphs of f(x) = sin
Page 135 and 136: D. APPLICATIONS OF GEOMETRYOVERALL
Page 137 and 138: Foundations for CollegeMathematics,
Page 139 and 140: A. MATHEMATICAL MODELSOVERALL EXPEC
Page 141 and 142: Years after 1955 Population of Gees
Page 143 and 144: payable monthly. Use a given amorti
Page 147: etween the two variables (i.e., by
Page 150 and 151: MATHEMATICAL PROCESS EXPECTATIONSTh
Page 152 and 153: THE ONTARIO CURRICULUM, GRADES 11 A
Page 154 and 155: Grade 12, Workplace Preparationmont
Page 156 and 157: THE ONTARIO CURRICULUM, GRADES 11 A
Page 159 and 160: The Ministry of Education wishes to
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Mathematics

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