Grade 11. Mathematical Process

Objectives: Ontario Curriculum: Grade 11 Functions, University Preparation

11.A Characteristics of Functions

·        11.A.1 demonstrate an understanding of functions, their representations, and their inverses, and make connections between the algebraic and graphical representations of functions using transformations;

o   11.A.1.1 explain the meaning of the term function, and distinguish a function from a relation that is not a function, through investigation of linear and quadratic relations using a variety of representations (i.e., tables of values, mapping diagrams, graphs, function machines, equations) and strategies (e.g., identifying a one-to-one or many-to-one mapping; using the vertical-line test)

  • Identify functions (11-D.2)

o   11.A.1.2 represent linear and quadratic functions using function notation, given their equations, tables of values, or graphs, and substitute into and evaluate functions [e.g., evaluate f(½), given f(x) = 2x² + 3x – 1]

  • Slope-intercept form: write an equation from a table (10-L.9)
  • Evaluate functions (11-D.3)
  • Complete a function table: quadratic functions (11-H.2)
  • Find the slope of a linear function (12-A.3)

o   11.A.1.3 explain the meanings of the terms domain and range, through investigation using numeric, graphical, and algebraic representations of the functions f (x) = x, f (x) = x², f(x) = the square root of x, and f(x) = 1/x; describe the domain and range of a function appropriately (e.g., for y = x² +1, the domain is the set of real numbers, and the range is y is greater than or equal to 1); and explain any restrictions on the domain and range in contexts arising from real-world applications

  • Domain and range (11-D.1)
  • Domain and range of radical functions (11-K.12)
  • Rational functions: asymptotes and excluded values (11-M.1)

o   11.A.1.4 relate the process of determining the inverse of a function to their understanding of reverse processes (e.g., applying inverse operations)

o   11.A.1.5 determine the numeric or graphical representation of the inverse of a linear or quadratic function, given the numeric, graphical, or algebraic representation of the function, and make connections, through investigation using a variety of tools (e.g., graphing technology, Mira, tracing paper), between the graph of a function and the graph of its inverse (e.g., the graph of the inverse is the reflection of the graph of the function in the line y = x)

o   11.A.1.6 determine, through investigation, the relationship between the domain and range of a function and the domain and range of the inverse relation, and determine whether or not the inverse relation is a function

o   11.A.1.7 determine, using function notation when appropriate, the algebraic representation of the inverse of a linear or quadratic function, given the algebraic representation of the function [e.g., f (x) = (x – 2)² – 5], and make connections, through investigation using a variety of tools (e.g., graphing technology, Mira, tracing paper), between the algebraic representations of a function and its inverse (e.g., the inverse of a linear function involves applying the inverse operations in the reverse order)

o   11.A.1.8 determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y = af (k(x – d)) + c, and describe these roles in terms of transformations on the graphs of f(x) = x, f(x) = x , f(x) = the square root of x, and f(x) = 1/x (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)

o   11.A.1.9 sketch graphs of y = af(k(x – d)) + c by applying one or more transformations to the graphs of f(x) = x, f(x) = x², f(x) = the square root of x, and f(x) = 1/x, and state the domain and range of the transformed functions

  • Graph a linear function (11-D.7)
  • Graph a quadratic function (11-H.4)
  • Translations of functions (12-B.1)
  • Reflections of functions (12-B.2)
  • Dilations of functions (12-B.3)
  • Transformations of functions (12-B.4)
  • Function transformation rules (12-B.5)
  • Describe function transformations (12-B.6)

·        11.A.2 determine the zeros and the maximum or minimum of a quadratic function, and solve problems involving quadratic functions, including problems arising from real-world applications;

o   11.A.2.1 determine the number of zeros (i.e., x-intercepts) of a quadratic function, using a variety of strategies (e.g., inspecting graphs; factoring; calculating the discriminant)

  • Match quadratic functions and graphs (11-H.5)
  • Solve a quadratic equation using the zero product property (11-H.7)
  • Solve a quadratic equation by factoring (11-H.8)
  • Using the discriminant (11-H.11)

o   11.A.2.2 determine the maximum or minimum value of a quadratic function whose equation is given in the form f (x) = ax2 + bx+ c, using an algebraic method (e.g., completing the square; factoring to determine the zeros and averaging the zeros)

  • Find the maximum or minimum value of a quadratic function (12-C.2)

o   11.A.2.3 solve problems involving quadratic functions arising from real-world applications and represented using function notation

o   11.A.2.4 determine, through investigation, the transformational relationship among the family of quadratic functions that have the same zeros, and determine the algebraic representation of a quadratic function, given the real roots of the corresponding quadratic equation and a point on the function

  • Find a quadratic function (11-H.3)
  • Write equations of parabolas in vertex form from graphs (11-I.4)
  • Write equations of parabolas in vertex form using properties (11-I.5)

o   11.A.2.5 solve problems involving the intersection of a linear function and a quadratic function graphically and algebraically (e.g., determining the time when two identical cylindrical water tanks contain equal volumes of water, if one tank is being filled at a constant rate and the other is being emptied through a hole in the bottom)

  • Solve a non-linear system of equations (11-E.12)

·        11.A.3 demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and rational expressions.

o   11.A.3.1 simplify polynomial expressions by adding, subtracting, and multiplying

  • Add and subtract polynomials (11-J.2)
  • Multiply a polynomial by a monomial (11-J.3)
  • Multiply two binomials (11-J.4)
  • Multiply two binomials: special cases (11-J.5)
  • Multiply polynomials (11-J.6)

o   11.A.3.2 verify, through investigation with and without technology, that (the square root of ab) = (the square root of a) x (the square root of b), a is greater than or equal to 0, b is greater than or equal to 0, and use this relationship to simplify radicals (e.g., the square root of 24) and radical expressions obtained by adding, subtracting, and multiplying [e.g., (2 + the square root of 6)(3 – the square root of 12)]

  • Simplify radical expressions (10-D.1)
  • Multiply radical expressions (11-K.7)
  • Divide radical expressions (11-K.8)
  • Add and subtract radical expressions (11-K.9)
  • Simplify radical expressions using the distributive property (11-K.10)

o   11.A.3.3 simplify rational expressions by adding, subtracting, multiplying, and dividing, and state the restrictions on the variable values

  • Simplify rational expressions (11-M.4)
  • Multiply and divide rational expressions (11-M.5)
  • Add and subtract rational expressions (11-M.6)

o   11.A.3.4 determine if two given algebraic expressions are equivalent (i.e., by simplifying; by substituting values)

11.B Exponential Functions

·        11.B.1 evaluate powers with rational exponents, simplify expressions containing exponents, and describe properties of exponential functions represented in a variety of ways;

o   11.B.1.1 graph, with and without technology, an exponential relation, given its equation in the form y = a to the x power (a > 0, a is not equal to 1), define this relation as the function f(x) = a to the x power, and explain why it is a function

o   11.B.1.2 determine, through investigation using a variety of tools (e.g., calculator, paper and pencil, graphing technology) and strategies (e.g., patterning; finding values from a graph; interpreting the exponent laws), the value of a power with a rational exponent (i.e., x to the (m/n) power, where x > 0 and m and n are integers)

  • Evaluate rational exponents (11-L.1)

o   11.B.1.3 simplify algebraic expressions containing integer and rational exponents [e.g., (x³) ÷ (x to the ½ power), ((x to the 6th power)y³) to the 1/3 power], and evaluate numeric expressions containing integer and rational exponents and rational bases [e.g., 2 to the –3 power, (–6)³, 4 to the ½ power, 1.01 to the 120th power]

  • Multiplication with rational exponents (11-L.2)
  • Division with rational exponents (11-L.3)
  • Power rule (11-L.4)
  • Simplify expressions involving rational exponents I (11-L.5)
  • Simplify expressions involving rational exponents II (11-L.6)
  • Evaluate exponential functions (11-N.1)

o   11.B.1.4 determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes (e.g., the domain is the set of real numbers; the range is the set of positive real numbers; the function either increases or decreases throughout its domain) for exponential functions represented in a variety of ways [e.g., tables of values, mapping diagrams, graphs, equations of the form f(x) = a to the x power (a > 0, a is not equal to 1), function machines]

  • Match exponential functions and graphs (11-N.2)
  • Exponential functions over unit intervals (11-N.5)
  • Describe linear and exponential growth and decay (11-N.6)

·        11.B.2 make connections between the numeric, graphical, and algebraic representations of exponential functions;

o   11.B.2.1 distinguish exponential functions from linear and quadratic functions by making comparisons in a variety of ways (e.g., comparing rates of change using finite differences in tables of values; identifying a constant ratio in a table of values; inspecting graphs; comparing equations)

  • Identify linear, quadratic and exponential functions from graphs (10-S.1)
  • Identify linear, quadratic and exponential functions from tables (10-S.2)
  • Linear functions over unit intervals (11-D.9)
  • Identify linear and exponential functions (11-N.4)
  • Exponential functions over unit intervals (11-N.5)
  • Describe linear and exponential growth and decay (11-N.6)

o   11.B.2.2 determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y = af (k(x – d)) + c, and describe these roles in terms of transformations on the graph of f(x) = a to the x power (a > 0, a is not equal to 1) (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)

o   11.B.2.3 sketch graphs of y = af(k(x – d)) + c by applying one or more transformations to the graph of f(x) = a to the x power (a > 0, a is not equal to 1), and state the domain and range of the transformed functions

o   11.B.2.4 determine, through investigation using technology, that the equation of a given exponential function can be expressed using different bases [e.g., f(x) = 9 to the x power can be expressed as f(x) = 3 to the 2x power], and explain the connections between the equivalent forms in a variety of ways (e.g., comparing graphs; using transformations; using the exponent laws)

o   11.B.2.5 represent an exponential function with an equation, given its graph or its properties

  • Match exponential functions and graphs (11-N.2)

·        11.B.3 identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications.

o   11.B.3.1 collect data that can be modelled as an exponential function, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials such as number cubes, coins; measurement tools such as electronic probes), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data

o   11.B.3.2 identify exponential functions, including those that arise from real-world applications involving growth and decay (e.g., radioactive decay, population growth, cooling rates, pressure in a leaking tire), given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range (e.g., ambient temperature limits the range for a cooling curve)

o   11.B.3.3 solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications (e.g., radioactive decay, population growth, height of a bouncing ball, compound interest) by interpreting the graphs or by substituting values for the exponent into the equations

  • Exponential growth and decay: word problems (11-N.7)
  • Compound interest: word problems (11-N.8)
  • Continuously compounded interest: word problems (11-N.9)

11.C Discrete Functions

·        11.C.1 demonstrate an understanding of recursive sequences, represent recursive sequences in a variety of ways, and make connections to Pascal’s triangle;

o   11.C.1.1 make connections between sequences and discrete functions, represent sequences using function notation, and distinguish between a discrete function and a continuous function [e.g., f(x) = 2x, where the domain is the set of natural numbers, is a discrete linear function, and its graph is a set of equally spaced points; f(x) = 2x, where the domain is the set of real numbers, is a continuous linear function and its graph is a straight line]

o   11.C.1.2 determine and describe (e.g., in words; using flow charts) a recursive procedure for generating a sequence, given the initial terms (e.g., 1, 3, 6, 10, 15, 21, …), and represent sequences as discrete functions in a variety of ways (e.g., tables of values, graphs)

  • Find terms of a recursive sequence (11-X.4)
  • Write a formula for a recursive sequence (11-X.8)

o   11.C.1.3 connect the formula for the nth term of a sequence to the representation in function notation, and write terms of a sequence given one of these representations or a recursion formula

  • Evaluate formulas for sequences (11-X.5)

o   11.C.1.4 represent a sequence algebraically using a recursion formula, function notation, or the formula for the nth term [e.g., represent 2, 4, 8,16, 32, 64, … as t base 1 = 2; t base n = 2(t base (n–1), as f(n) = 2 to the n power, or as t base n = 2 to the n power, or represent ½, 2/3, 3/4, 4/5, 5/6, 6/7,… as t base 1 = ½; t base n = t base (n-1) + (1/(n(n + 1)), as f(n) = n/(n+1), or as t base n = n/(n + 1), where n is a natural number], and describe the information that can be obtained by inspecting each representation (e.g., function notation or the formula for the nth term may show the type of function; a recursion formula shows the relationship between terms)

  • Find terms of a recursive sequence (11-X.4)
  • Write a formula for an arithmetic sequence (11-X.6)
  • Write a formula for a geometric sequence (11-X.7)
  • Write a formula for a recursive sequence (11-X.8)

o   11.C.1.5 determine, through investigation, recursive patterns in the Fibonacci sequence, in related sequences, and in Pascal’s triangle, and represent the patterns in a variety of ways (e.g., tables of values, algebraic notation)

o   11.C.1.6 determine, through investigation, and describe the relationship between Pascal’s triangle and the expansion of binomials, and apply the relationship to expand binomials raised to whole-number exponents [e.g., (1 + x) to the 4th power, (2x –1) to the 5th power, (2x – y) to the 6th power, (x² + 1) to the 5th power].

  • Pascal’s triangle (12-D.14)
  • Pascal’s triangle and the Binomial Theorem (12-D.15)

·        11.C.2 demonstrate an understanding of the relationships involved in arithmetic and geometric sequences and series, and solve related problems;

o   11.C.2.1 identify sequences as arithmetic, geometric, or neither, given a numeric or algebraic representation

  • Classify formulas and sequences (11-X.1)
  • Identify arithmetic and geometric series (11-X.11)

o   11.C.2.2 determine the formula for the general term of an arithmetic sequence [i.e., t base n = a + (n –1)d] or geometric sequence (i.e., t base n = a(r to the (n–1) power)), through investigation using a variety of tools (e.g., linking cubes, algebra tiles, diagrams, calculators) and strategies (e.g., patterning; connecting the steps in a numerical example to the steps in the algebraic development), and apply the formula to calculate any term in a sequence

  • Find terms of an arithmetic sequence (11-X.2)
  • Find terms of a geometric sequence (11-X.3)
  • Write a formula for an arithmetic sequence (11-X.6)
  • Write a formula for a geometric sequence (11-X.7)

o   11.C.2.3 determine the formula for the sum of an arithmetic or geometric series, through investigation using a variety of tools (e.g., linking cubes, algebra tiles, diagrams, calculators) and strategies (e.g., patterning; connecting the steps in a numerical example to the steps in the algebraic development), and apply the formula to calculate the sum of a given number of consecutive terms

  • Find the sum of a finite arithmetic or geometric series (11-X.12)
  • Partial sums of arithmetic series (11-X.14)
  • Partial sums of geometric series (11-X.15)
  • Convergent and divergent geometric series (11-X.17)
  • Find the value of an infinite geometric series (11-X.18)

o   11.C.2.4 solve problems involving arithmetic and geometric sequences and series, including those arising from real-world applications

·        11.C.3 make connections between sequences, series, and financial applications, and solve problems involving compound interest and ordinary annuities.

o   11.C.3.1 make and describe connections between simple interest, arithmetic sequences, and linear growth, through investigation with technology (e.g., use a spreadsheet or graphing calculator to make simple interest calculations, determine first differences in the amounts over time, and graph amount versus time)

o   11.C.3.2 make and describe connections between compound interest, geometric sequences, and exponential growth, through investigation with technology (e.g., use a spreadsheet to make compound interest calculations, determine finite differences in the amounts over time, and graph amount versus time)

o   11.C.3.3 solve problems, using a scientific calculator, that involve the calculation of the amount, A (also referred to as future value, FV), the principal, P (also referred to as present value, PV), or the interest rate per compounding period, i, using the compound interest formula in the form A = P ((1 + i) to the n power) [or FV = PV((1 + i) to the n power)]

  • Compound interest: word problems (11-N.8)

o   11.C.3.4 determine, through investigation using technology (e.g., scientific calculator, the TVM Solver on a graphing calculator, online tools), the number of compounding periods, n, using the compound interest formula in the form A = P((1 + i) to the n power) [or FV = PV((1 + i) to the n power)]; describe strategies (e.g., guessing and checking; using the power of a power rule for exponents; using graphs) for calculating this number; and solve related problems

  • Compound interest: word problems (11-N.8)
  • Continuously compounded interest: word problems (11-N.9)

o   11.C.3.5 explain the meaning of the term annuity, and determine the relationships between ordinary simple annuities (i.e., annuities in which payments are made at the end of each period, and compounding and payment periods are the same), geometric series, and exponential growth, through investigation with technology (e.g., use a spreadsheet to determine and graph the future value of an ordinary simple annuity for varying numbers of compounding periods; investigate how the contributions of each payment to the future value of an ordinary simple annuity are related to the terms of a geometric series)

o   11.C.3.6 determine, through investigation using technology (e.g., the TVM Solver on a graphing calculator, online tools), the effects of changing the conditions (i.e., the payments, the frequency of the payments, the interest rate, the compounding period) of ordinary simple annuities (e.g., long-term savings plans, loans)

o   11.C.3.7 solve problems, using technology (e.g., scientific calculator, spreadsheet, graphing calculator), that involve the amount, the present value, and the regular payment of an ordinary simple annuity (e.g., calculate the total interest paid over the life of a loan, using a spreadsheet, and compare the total interest with the original principal of the loan)

11.D Trigonometric Functions

·        11.D.1 determine the values of the trigonometric ratios for angles less than 360°; prove simple trigonometric identities; and solve problems using the primary trigonometric ratios, the sine law, and the cosine law;

o   11.D.1.1 determine the exact values of the sine, cosine, and tangent of the special angles: 0°, 30°, 45°, 60°, and 90°

  • Sin, cos and tan of special angles (11-Q.5)

o   11.D.1.2 determine the values of the sine, cosine, and tangent of angles from 0° to 360°, through investigation using a variety of tools (e.g., dynamic geometry software, graphing tools) and strategies (e.g., applying the unit circle; examining angles related to special angles)

  • Find trigonometric ratios using the unit circle (11-Q.4)
  • Sin, cos and tan of special angles (11-Q.5)

o   11.D.1.3 determine the measures of two angles from 0° to 360° for which the value of a given trigonometric ratio is the same

  • Solve trigonometric equations II (11-Q.11)

o   11.D.1.4 define the secant, cosecant, and cotangent ratios for angles in a right triangle in terms of the sides of the triangle (e.g., sec A = hypotenuse/adjacent), and relate these ratios to the cosine, sine, and tangent ratios (e.g., sec A = 1/cos A)

  • Trigonometric ratios: csc, sec and cot (11-Q.2)
  • Csc, sec and cot of special angles (11-Q.6)

o   11.D.1.5 prove simple trigonometric identities, using the Pythagorean identity sin²x + cos²x = 1; the quotient identity tanx = sinx/cosx; and the reciprocal identities secx = 1/cosx, cscx = 1/sinx, and cotx = 1/tanx

o   11.D.1.6 pose problems involving right triangles and oblique triangles in two-dimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law, and the sine law (including the ambiguous case)

o   11.D.1.7 pose problems involving right triangles and oblique triangles in three-dimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law, and the sine law

·        11.D.2 demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions;

o   11.D.2.1 describe key properties (e.g., cycle, amplitude, period) of periodic functions arising from real-world applications (e.g., natural gas consumption in Ontario, tides in the Bay of Fundy), given a numerical or graphical representation

o   11.D.2.2 predict, by extrapolating, the future behaviour of a relationship modelled using a numeric or graphical representation of a periodic function (e.g., predicting hours of daylight on a particular date from previous measurements; predicting natural gas consumption in Ontario from previous consumption)

o   11.D.2.3 make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0° to 360° and the corresponding sine ratios or cosine ratios, with or without technology (e.g., by generating a table of values using a calculator; by unwrapping the unit circle), defining this relationship as the function f(x) = sinx or f(x) = cosx, and explaining why the relationship is a function

o   11.D.2.4 sketch the graphs of f(x) =sinx and f(x) =cosx for angle measures expressed in degrees, and determine and describe their key properties (i.e., cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/decreasing intervals)

o   11.D.2.5 determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y =af (k(x – d)) + c, where f(x) =sinx or f(x) =cosx with angles expressed in degrees, and describe these roles in terms of transformations on the graphs of f(x) =sinx and f(x) =cosx (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)

o   11.D.2.6 determine the amplitude, period, phase shift, domain, and range of sinusoidal functions whose equations are given in the form f(x) = a sin(k(x – d)) + c or f(x) = a cos(k(x – d)) + c

  • Find properties of sine functions (12-O.1)

o   11.D.2.7 sketch graphs of y = af(k(x – d)) + c by applying one or more transformations to the graphs of f(x) = sinx and f (x) = cosx, and state the domain and range of the transformed functions

o   11.D.2.8 represent a sinusoidal function with an equation, given its graph or its properties

  • Write equations of sine functions from graphs (12-O.2)
  • Write equations of sine functions using properties (12-O.3)

·        11.D.3 identify and represent sinusoidal functions, and solve problems involving sinusoidal functions, including problems arising from real-world applications.

o   11.D.3.1 collect data that can be modelled as a sinusoidal function (e.g., voltage in an AC circuit, sound waves), through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials, measurement tools such as motion sensors), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data

o   11.D.3.2 identify periodic and sinusoidal functions, including those that arise from real-world applications involving periodic phenomena, given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range

o   11.D.3.3 determine, through investigation, how sinusoidal functions can be used to model periodic phenomena that do not involve angles

o   11.D.3.4 predict the effects on a mathematical model (i.e., graph, equation) of an application involving periodic phenomena when the conditions in the application are varied (e.g., varying the conditions, such as speed and direction, when walking in a circle in front of a motion sensor)

o   11.D.3.5 pose problems based on applications involving a sinusoidal function, and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation

 

Objectives: Ontario Curriculum: Grade 11 Foundations for College Mathematics, College Preparation

11.A Mathematical Models

11.B Personal Finance

  • B.1 compare simple and compound interest, relate compound interest to exponential growth, and solve problems involving compound interest;
    • B.1.1 determine, through investigation using technology, the compound interest for a given investment, using repeated calculations of simple interest, and compare, using a table of values and graphs, the simple and compound interest earned for a given principal (i.e., investment) and a fixed interest rate over time
    • B.1.2 determine, through investigation (e.g., using spreadsheets and graphs), and describe the relationship between compound interest and exponential growth
    • B.1.3 solve problems, using a scientific calculator, that involve the calculation of the amount, A (also referred to as future value, FV), and the principal, P (also referred to as present value, PV), using the compound interest formula in the form A = P((1 + i) to the n power) [or FV = PV((1 + i) to the n power)]
    • B.1.4 calculate the total interest earned on an investment or paid on a loan by determining the difference between the amount and the principal [e.g., using I = A – P (or I = FV – PV)]
    • B.1.5 solve problems, using a TVM Solver on a graphing calculator or on a website, that involve the calculation of the interest rate per compounding period, i, or the number of compounding periods, n, in the compound interest formula A = P((1 + i) to the n power) [or FV = PV((1 + i) to the n power)]
    • B.1.6 determine, through investigation using technology (e.g., a TVM Solver on a graphing calculator or on a website), the effect on the future value of a compound interest investment or loan of changing the total length of time, the interest rate, or the compounding period
  • B.2 compare services available from financial institutions, and solve problems involving the cost of making purchases on credit;
    • B.2.1 gather, interpret, and compare information about the various savings alternatives commonly available from financial institutions (e.g., savings and chequing accounts, term investments), the related costs (e.g., cost of cheques, monthly statement fees, early withdrawal penalties), and possible ways of reducing the costs (e.g., maintaining a minimum balance in a savings account; paying a monthly flat fee for a package of services)
    • B.2.2 gather and interpret information about investment alternatives (e.g., stocks, mutual funds, real estate, GICs, savings accounts), and compare the alternatives by considering the risk and the rate of return
    • B.2.3 gather, interpret, and compare information about the costs (e.g., user fees, annual fees, service charges, interest charges on overdue balances) and incentives (e.g., loyalty rewards; philanthropic incentives, such as support for Olympic athletes or a Red Cross disaster relief fund) associated with various credit cards and debit cards
    • B.2.4 gather, interpret, and compare information about current credit card interest rates and regulations, and determine, through investigation using technology, the effects of delayed payments on a credit card balance
    • B.2.5 solve problems involving applications of the compound interest formula to determine the cost of making a purchase on credit
    • B.3 interpret information about owning and operating a vehicle, and solve problems involving the associated costs.
      • B.3.1 gather and interpret information about the procedures and costs involved in insuring a vehicle (e.g., car, motorcycle, snowmobile) and the factors affecting insurance rates (e.g., gender, age, driving record, model of vehicle, use of vehicle), and compare the insurance costs for different categories of drivers and for different vehicles
      • B.3.2 gather, interpret, and compare information about the procedures and costs (e.g., monthly payments, insurance, depreciation, maintenance, miscellaneous expenses) involved in buying or leasing a new vehicle or buying a used vehicle
      • B.3.3 solve problems, using technology (e.g., calculator, spreadsheet), that involve the fixed costs (e.g., licence fee, insurance) and variable costs (e.g., maintenance, fuel) of owning and operating a vehicle

11.C Geometry and Trigonometry

  • C.1 represent, in a variety of ways, two-dimensional shapes and three-dimensional figures arising from real-world applications, and solve design problems;
    • C.1.1 recognize and describe real-world applications of geometric shapes and figures, through investigation (e.g., by importing digital photos into dynamic geometry software), in a variety of contexts (e.g., product design, architecture, fashion), and explain these applications (e.g., one reason that sewer covers are round is to prevent them from falling into the sewer during removal and replacement)
    • C.1.2 represent three-dimensional objects, using concrete materials and design or drawing software, in a variety of ways (e.g., orthographic projections [i.e., front, side, and top views], perspective isometric drawings, scale models)
    • C.1.3 create nets, plans, and patterns from physical models arising from a variety of real-world applications (e.g., fashion design, interior decorating, building construction), by applying the metric and imperial systems and using design or drawing software
    • C.1.4 solve design problems that satisfy given constraints (e.g., design a rectangular berm that would contain all the oil that could leak from a cylindrical storage tank of a given height and radius), using physical models (e.g., built from popsicle sticks, cardboard, duct tape) or drawings (e.g., made using design or drawing software), and state any assumptions made
  • C.2 solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications.

11.D Data Management

  • D.1 solve problems involving one-variable data by collecting, organizing, analysing, and evaluating data;
    • D.1.1 identify situations involving one-variable data (i.e., data about the frequency of a given occurrence), and design questionnaires (e.g., for a store to determine which CDs to stock, for a radio station to choose which music to play) or experiments (e.g., counting, taking measurements) for gathering one-variable data, giving consideration to ethics, privacy, the need for honest responses, and possible sources of bias
    • D.1.2 collect one-variable data from secondary sources (e.g., Internet databases), and organize and store the data using a variety of tools (e.g., spreadsheets, dynamic statistical software)
    • D.1.3 explain the distinction between the terms population and sample, describe the characteristics of a good sample, and explain why sampling is necessary (e.g., time, cost, or physical constraints)
    • D.1.4 describe and compare sampling techniques (e.g., random, stratified, clustered, convenience, voluntary); collect one-variable data from primary sources, using appropriate sampling techniques in a variety of real-world situations; and organize and store the data
    • D.1.5 identify different types of one-variable data (i.e., categorical, discrete, continuous), and represent the data, with and without technology, in appropriate graphical forms (e.g., histograms, bar graphs, circle graphs, pictographs)
    • D.1.6 identify and describe properties associated with common distributions of data (e.g., normal, bimodal, skewed)
    • D.1.7 calculate, using formulas and/or technology (e.g., dynamic statistical software, spreadsheet, graphing calculator), and interpret measures of central tendency (i.e., mean, median, mode) and measures of spread (i.e., range, standard deviation)
    • D.1.8 explain the appropriate use of measures of central tendency (i.e., mean, median, mode) and measures of spread (i.e., range, standard deviation)
    • D.1.9 compare two or more sets of one-variable data, using measures of central tendency and measures of spread
    • D.1.10 solve problems by interpreting and analysing one-variable data collected from secondary sources
    • D.2 determine and represent probability, and identify and interpret its applications.
      • D.2.1 identify examples of the use of probability in the media and various ways in which probability is represented (e.g., as a fraction, as a percent, as a decimal in the range 0 to 1)
      • D.2.2 determine the theoretical probability of an event (i.e., the ratio of the number of favourable outcomes to the total number of possible outcomes, where all outcomes are equally likely), and represent the probability in a variety of ways (e.g., as a fraction, as a percent, as a decimal in the range 0 to 1)
      • D.2.3 perform a probability experiment (e.g., tossing a coin several times), represent the results using a frequency distribution, and use the distribution to determine the experimental probability of an event
      • D.2.4 compare, through investigation, the theoretical probability of an event with the experimental probability, and explain why they might differ
      • D.2.5 determine, through investigation using class-generated data and technology-based simulation models (e.g., using a random-number generator on a spreadsheet or on a graphing calculator), the tendency of experimental probability to approach theoretical probability as the number of trials in an experiment increases (e.g., “If I simulate tossing a coin 1000 times using technology, the experimental probability that I calculate for tossing tails is likely to be closer to the theoretical probability than if I simulate tossing the coin only 10 times”)
      • D.2.6 interpret information involving the use of probability and statistics in the media, and make connections between probability and statistics (e.g., statistics can be used to generate probabilities)

Objectives: Ontario Curriculum: Grade 11 Functions and Applications, University/College Preparation

11.A Quadratic Functions

·        11.A.1 expand and simplify quadratic expressions, solve quadratic equations, and relate the roots of a quadratic equation to the corresponding graph;

o   11.A.1.1 pose problems involving quadratic relations arising from real-world applications and represented by tables of values and graphs, and solve these and other such problems (e.g., “From the graph of the height of a ball versus time, can you tell me how high the ball was thrown and the time when it hit the ground?”)

o   11.A.1.2 represent situations (e.g., the area of a picture frame of variable width) using quadratic expressions in one variable, and expand and simplify quadratic expressions in one variable [e.g., 2x(x + 4) – (x + 3)²]

  • Multiply two binomials (11-J.4)
  • Multiply two binomials: special cases (11-J.5)

o   11.A.1.3 factor quadratic expressions in one variable, including those for which a is not equal to 1 (e.g., 3x² + 13x – 10), differences of squares (e.g., 4x² – 25), and perfect square trinomials (e.g., 9x² + 24x + 16), by selecting and applying an appropriate strategy

  • Factor quadratics (11-G.2)
  • Factor quadratics using algebra tiles (11-G.3)
  • Factor by grouping (11-G.5)
  • Factor polynomials (11-G.7)

o   11.A.1.4 solve quadratic equations by selecting and applying a factoring strategy

  • Solve a quadratic equation by factoring (11-H.8)

o   11.A.1.5 determine, through investigation, and describe the connection between the factors used in solving a quadratic equation and the x-intercepts of the graph of the corresponding quadratic relation

  • Match quadratic functions and graphs (11-H.5)

o   11.A.1.6 explore the algebraic development of the quadratic formula (e.g., given the algebraic development, connect the steps to a numeric example; follow a demonstration of the algebraic development, with technology, such as computer algebra systems, or without technology [student reproduction of the development of the general case is not required]), and apply the formula to solve quadratic equations, using technology

  • Solve a quadratic equation using the quadratic formula (11-H.10)

o   11.A.1.7 relate the real roots of a quadratic equation to the x-intercepts of the corresponding graph, and connect the number of real roots to the value of the discriminant (e.g., there are no real roots and no x-intercepts if b² – 4ac < 0)

  • Match quadratic functions and graphs (11-H.5)
  • Using the discriminant (11-H.11)

o   11.A.1.8 determine the real roots of a variety of quadratic equations (e.g., 100x² = 115x + 35), and describe the advantages and disadvantages of each strategy (i.e., graphing; factoring; using the quadratic formula)

  • Solve a quadratic equation by completing the square (10-Q.7)
  • Solve a quadratic equation using square roots (11-H.6)
  • Solve a quadratic equation using the zero product property (11-H.7)
  • Solve a quadratic equation by factoring (11-H.8)
  • Solve a quadratic equation using the quadratic formula (11-H.10)

·        11.A.2 demonstrate an understanding of functions, and make connections between the numeric, graphical, and algebraic representations of quadratic functions;

o   11.A.2.1 explain the meaning of the term function, and distinguish a function from a relation that is not a function, through investigation of linear and quadratic relations using a variety of representations (i.e., tables of values, mapping diagrams, graphs, function machines, equations) and strategies (e.g., using the vertical-line test)

  • Relations: convert between tables, graphs, mappings and lists of points (10-J.1)
  • Identify functions: vertical line test (10-J.5)
  • Identify functions (11-D.2)

o   11.A.2.2 substitute into and evaluate linear and quadratic functions represented using function notation [e.g., evaluate f(½), given f(x) = 2x² + 3x – 1], including functions arising from real-world applications

  • Evaluate a function (10-J.7)
  • Solve linear equations: word problems (11-B.2)
  • Complete a function table: quadratic functions (11-H.2)
  • Evaluate functions (12-A.7)

o   11.A.2.3 explain the meanings of the terms domain and range, through investigation using numeric, graphical, and algebraic representations of linear and quadratic functions, and describe the domain and range of a function appropriately (e.g., for y = x² + 1, the domain is the set of real numbers, and the range is y is greater than or equal to 1)

  • Domain and range (11-D.1)

o   11.A.2.4 explain any restrictions on the domain and the range of a quadratic function in contexts arising from real-world applications

o   11.A.2.5 determine, through investigation using technology, the roles of a, h, and k in quadratic functions of the form f(x) = a(x – h)² + k, and describe these roles in terms of transformations on the graph of f(x) = x² (i.e., translations; reflections in the x-axis; vertical stretches and compressions to and from the x-axis)

o   11.A.2.6 sketch graphs of g(x) = a(x – h)² + k by applying one or more transformations to the graph of f(x) = x²

  • Graph a quadratic function (11-H.4)

o   11.A.2.7 express the equation of a quadratic function in the standard form f (x) = ax² + bx + c, given the vertex form f (x) = a(x – h)² + k, and verify, using graphing technology, that these forms are equivalent representations

o   11.A.2.8 express the equation of a quadratic function in the vertex form f(x) = a(x – h)² + k, given the standard form f(x) = ax² + bx + c, by completing the square (e.g., using algebra tiles or diagrams; algebraically), including cases where b/a is a simple rational number (e.g., ½, 0.75), and verify, using graphing technology, that these forms are equivalent representations

  • Write equations of parabolas in vertex form (12-Q.2)

o   11.A.2.9 sketch graphs of quadratic functions in the factored form f(x) = a(x – r)(x – s) by using the x-intercepts to determine the vertex

o   11.A.2.10 describe the information (e.g., maximum, intercepts) that can be obtained by inspecting the standard form f(x) = ax² + bx + c, the vertex form f(x) = a(x – h)² + k, and the factored form f(x) = a(x – r)(x – s) of a quadratic function

  • Characteristics of quadratic functions (11-H.1)
  • Find the maximum or minimum value of a quadratic function (12-C.2)

o   11.A.2.11 sketch the graph of a quadratic function whose equation is given in the standard form f(x) = ax² + bx + c by using a suitable strategy (e.g., completing the square and finding the vertex; factoring, if possible, to locate the x-intercepts), and identify the key features of the graph (e.g., the vertex, the x- and y-intercepts, the equation of the axis of symmetry, the intervals where the function is positive or negative, the intervals where the function is increasing or decreasing)

  • Graph a quadratic function (11-H.4)
  • Match quadratic functions and graphs (11-H.5)
  • Identify the direction a parabola opens (11-I.1)
  • Find the vertex of a parabola (11-I.2)
  • Find the axis of symmetry of a parabola (11-I.3)
  • Graph parabolas (11-I.6)

·        11.A.3 solve problems involving quadratic functions, including problems arising from real-world applications.

o   11.A.3.1 collect data that can be modelled as a quadratic function, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials; measurement tools such as measuring tapes, electronic probes, motion sensors), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data

o   11.A.3.2 determine, through investigation using a variety of strategies (e.g., applying properties of quadratic functions such as the x-intercepts and the vertex; using transformations), the equation of the quadratic function that best models a suitable data set graphed on a scatter plot, and compare this equation to the equation of a curve of best fit generated with technology (e.g., graphing software, graphing calculator)

o   11.A.3.3 solve problems arising from real-world applications, given the algebraic representation of a quadratic function (e.g., given the equation of a quadratic function representing the height of a ball over elapsed time, answer questions that involve the maximum height of the ball, the length of time needed for the ball to touch the ground, and the time interval when the ball is higher than a given measurement)

11.B Exponential Functions

·        11.B.1 simplify and evaluate numerical expressions involving exponents, and make connections between the numeric, graphical, and algebraic representations of exponential functions;

o   11.B.1.1 determine, through investigation using a variety of tools (e.g., calculator, paper and pencil, graphing technology) and strategies (e.g., patterning; finding values from a graph; interpreting the exponent laws), the value of a power with a rational exponent (i.e., x(m/n), where x > 0 and m and n are integers)

  • Evaluate rational exponents (11-L.1)

o   11.B.1.2 evaluate, with and without technology, numerical expressions containing integer and rational exponents and rational bases [e.g., 2 to the -3rd power, (–6)³, 4 to the ½ power, 1.01 to the 120th power]

  • Evaluate rational exponents (11-L.1)
  • Evaluate rational expressions I (11-M.2)
  • Evaluate rational expressions II (11-M.3)

o   11.B.1.3 graph, with and without technology, an exponential relation, given its equation in the form y = a to the x power (a > 0, a is not equal to 1), define this relation as the function f(x) = a to the x power, and explain why it is a function

  • Match exponential functions and graphs (11-N.2)

o   11.B.1.4 determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes (e.g., the domain is the set of real numbers; the range is the set of positive real numbers; the function either increases or decreases throughout its domain) for exponential functions represented in a variety of ways [e.g., tables of values, mapping diagrams, graphs, equations of the form f(x) = a to the x power (a > 0, a is not equal to 1), function machines]

  • Exponential functions over unit intervals (11-N.5)
  • Describe linear and exponential growth and decay (11-N.6)

o   11.B.1.5 determine, through investigation (e.g., by patterning with and without a calculator), the exponent rules for multiplying and dividing numeric expressions involving exponents [e.g., (½)³ x (½)²], and the exponent rule for simplifying numerical expressions involving a power of a power [e.g., (5³)²], and use the rules to simplify numerical expressions containing integer exponents [e.g., (2³)(2 to the 5th power) = 2 to the 8th power]

  • Multiplication with rational exponents (11-L.2)
  • Division with rational exponents (11-L.3)
  • Power rule (11-L.4)
  • Simplify expressions involving rational exponents I (11-L.5)
  • Simplify expressions involving rational exponents II (11-L.6)

o   11.B.1.6 distinguish exponential functions from linear and quadratic functions by making comparisons in a variety of ways (e.g., comparing rates of change using finite differences in tables of values; identifying a constant ratio in a table of values; inspecting graphs; comparing equations), within the same context when possible (e.g., simple interest and compound interest, population growth)

  • Identify linear, quadratic and exponential functions from graphs (10-S.1)
  • Identify linear, quadratic and exponential functions from tables (10-S.2)
  • Write linear, quadratic and exponential functions (10-S.3)
  • Linear functions over unit intervals (11-D.9)
  • Identify linear and exponential functions (11-N.4)
  • Exponential functions over unit intervals (11-N.5)
  • Describe linear and exponential growth and decay (11-N.6)

·        11.B.2 identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications;

o   11.B.2.1 collect data that can be modelled as an exponential function, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials such as number cubes, coins; measurement tools such as electronic probes), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data

o   11.B.2.2 identify exponential functions, including those that arise from real-world applications involving growth and decay (e.g., radioactive decay, population growth, cooling rates, pressure in a leaking tire), given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range (e.g., ambient temperature limits the range for a cooling curve)

  • Identify linear and exponential functions (11-N.4)

o   11.B.2.3 solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications (e.g., radioactive decay, population growth, height of a bouncing ball, compound interest) by interpreting the graphs or by substituting values for the exponent into the equations

  • Exponential growth and decay: word problems (11-N.7)
  • Compound interest: word problems (11-N.8)
  • Continuously compounded interest: word problems (11-N.9)

·        11.B.3 demonstrate an understanding of compound interest and annuities, and solve related problems.

o   11.B.3.1 compare, using a table of values and graphs, the simple and compound interest earned for a given principal (i.e., investment) and a fixed interest rate over time

o   11.B.3.2 solve problems, using a scientific calculator, that involve the calculation of the amount, A (also referred to as future value, FV), and the principal, P (also referred to as present value, PV), using the compound interest formula in the form A = P((1 + i) to the n power) [or FV = PV((1 + i) to the n power)]

  • Compound interest: word problems (11-N.8)
  • Continuously compounded interest: word problems (11-N.9)

o   11.B.3.3 determine, through investigation (e.g., using spreadsheets and graphs), that compound interest is an example of exponential growth [e.g., the formulas for compound interest, A = P((1 + i) to the n power), and present value, PV = A((1 + i) to the -n power), are exponential functions, where the number of compounding periods, n, varies]

o   11.B.3.4 solve problems, using a TVM Solver on a graphing calculator or on a website, that involve the calculation of the interest rate per compounding period, i, or the number of compounding periods, n, in the compound interest formula A = P((1 + i) to the n power) [or FV = PV((1 + i) to the n power)]

  • Compound interest: word problems (11-N.8)
  • Continuously compounded interest: word problems (11-N.9)

o   11.B.3.5 explain the meaning of the term annuity, through investigation of numeric and graphical representations using technology

o   11.B.3.6 determine, through investigation using technology (e.g., the TVM Solver on a graphing calculator, online tools), the effects of changing the conditions (i.e., the payments, the frequency of the payments, the interest rate, the compounding period) of ordinary simple annuities (i.e., annuities in which payments are made at the end of each period, and the compounding period and the payment period are the same) (e.g., long-term savings plans, loans)

o   11.B.3.7 solve problems, using technology (e.g., scientific calculator, spreadsheet, graphing calculator), that involve the amount, the present value, and the regular payment of an ordinary simple annuity (e.g., calculate the total interest paid over the life of a loan, using a spreadsheet, and compare the total interest with the original principal of the loan)

  • Compound interest: word problems (11-N.8)
  • Continuously compounded interest: word problems (11-N.9)

11.C Trigonometric Functions

·        11.C.1 solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications;

o   11.C.1.1 solve problems, including those that arise from real-world applications (e.g., surveying, navigation), by determining the measures of the sides and angles of right triangles using the primary trigonometric ratios

  • Trigonometric ratios: find a side length (11-Q.12)
  • Trigonometric ratios: find an angle measure (11-Q.13)
  • Solve a right triangle (11-Q.14)

o   11.C.1.2 solve problems involving two right triangles in two dimensions

  • Trigonometric ratios in similar right triangles (11-Q.3)

o   11.C.1.3 verify, through investigation using technology (e.g., dynamic geometry software, spreadsheet), the sine law and the cosine law (e.g., compare, using dynamic geometry software, the ratios a/sinA, b/sinB, and c/sinC in triangle ABC while dragging one of the vertices)

o   11.C.1.4 describe conditions that guide when it is appropriate to use the sine law or the cosine law, and use these laws to calculate sides and angles in acute triangles

  • Law of Sines (11-Q.15)
  • Law of Cosines (11-Q.16)
  • Solve a triangle (11-Q.17)

o   11.C.1.5 solve problems that require the use of the sine law or the cosine law in acute triangles, including problems arising from real-world applications (e.g., surveying, navigation, building construction)

  • Law of Sines (11-Q.15)
  • Law of Cosines (11-Q.16)
  • Area of a triangle: Law of Sines (11-Q.19)

·        11.C.2 demonstrate an understanding of periodic relationships and the sine function, and make connections between the numeric, graphical, and algebraic representations of sine functions;

o   11.C.2.1 describe key properties (e.g., cycle, amplitude, period) of periodic functions arising from real-world applications (e.g., natural gas consumption in Ontario, tides in the Bay of Fundy), given a numeric or graphical representation

o   11.C.2.2 predict, by extrapolating, the future behaviour of a relationship modelled using a numeric or graphical representation of a periodic function (e.g., predicting hours of daylight on a particular date from previous measurements; predicting natural gas consumption in Ontario from previous consumption)

o   11.C.2.3 make connections between the sine ratio and the sine function by graphing the relationship between angles from 0° to 360° and the corresponding sine ratios, with or without technology (e.g., by generating a table of values using a calculator; by unwrapping the unit circle), defining this relationship as the function f(x) = sinx, and explaining why the relationship is a function

o   11.C.2.4 sketch the graph of f(x) = sinx for angle measures expressed in degrees, and determine and describe its key properties (i.e., cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/ decreasing intervals)

o   11.C.2.5 make connections, through investigation with technology, between changes in a real-world situation that can be modelled using a periodic function and transformations of the corresponding graph (e.g., investigate the connection between variables for a swimmer swimming lengths of a pool and transformations of the graph of distance from the starting point versus time)

o   11.C.2.6 determine, through investigation using technology, the roles of the parameters a, c, and d in functions in the form f(x) = a sinx, f(x) = sinx + c, and f(x) = sin(x – d), and describe these roles in terms of transformations on the graph of f(x) = sinx with angles expressed in degrees (i.e., translations; reflections in the x-axis; vertical stretches and compressions to and from the x-axis)

o   11.C.2.7 sketch graphs of f(x) = a sinx, f(x) = sinx + c, and f(x) = sin(x – d) by applying transformations to the graph of f(x) = sinx, and state the domain and range of the transformed functions

·        11.C.3 identify and represent sine functions, and solve problems involving sine functions, including problems arising from real-world applications.

o   11.C.3.1 collect data that can be modelled as a sine function (e.g., voltage in an AC circuit, sound waves), through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials, measurement tools such as motion sensors), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data

o   11.C.3.2 identify periodic and sinusoidal functions, including those that arise from real-world applications involving periodic phenomena, given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range

o   11.C.3.3 pose problems based on applications involving a sine function, and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation