#### MCT.A: Exponential Functions

MCT.A.1: solve problems involving exponential equations graphically, including problems arising from real-world applications;

MCT.A.1.1: determine, through investigation with technology, and describe the impact of changing the base and changing the sign of the exponent on the graph of an exponential function

MCT.A.1.3: determine, through investigation using graphing technology, the point of intersection of the graphs of two exponential functions (e.g., y = 4 to the -x power and y = 8 to the (x + 3) power), recognize the x-coordinate of this point to be the solution to the corresponding exponential equation (e.g., 4 to the -x power = 8 to the (x + 3) power), and solve exponential equations graphically (e.g., solve 2 to the (x + 2) power = (2 to the x power) + 12 by using the intersection of the graphs of y = 2 to the (x + 2) power and y = (2 to the x power) + 12)

MCT.A.1.4: pose problems based on real-world applications (e.g., compound interest, population growth) that can be modelled with exponential equations, 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

MCT.A.2: solve problems involving exponential equations algebraically using common bases and logarithms, including problems arising from real-world applications.

MCT.A.2.1: simplify algebraic expressions containing integer and rational exponents using the laws of exponents (e.g., x³ ÷ x to the ½ power, square root of (x to the 6th power times y to the 12th power))

MCT.A.2.6: pose problems based on real-world applications that can be modelled with given exponential equations, and solve these and other such problems algebraically by rewriting them in logarithmic form

#### MCT.B: Polynomial Functions

MCT.B.1: recognize and evaluate polynomial functions, describe key features of their graphs, and solve problems using graphs of polynomial functions;

MCT.B.1.1: recognize a polynomial expression (i.e., a series of terms where each term is the product of a constant and a power of x with a nonnegative integral exponent, such as x³ - 5x² + 2x - 1); recognize the equation of a polynomial function and give reasons why it is a function, and identify linear and quadratic functions as examples of polynomial functions

MCT.B.1.2: compare, through investigation using graphing technology, the graphical and algebraic representations of polynomial (i.e., linear, quadratic, cubic, quartic) functions (e.g., investigate the effect of the degree of a polynomial function on the shape of its graph and the maximum number of x-intercepts; investigate the effect of varying the sign of the leading coefficient on the end behaviour of the function for very large positive or negative x-values)

MCT.B.1.3: describe key features of the graphs of polynomial functions (e.g., the domain and range, the shape of the graphs, the end behaviour of the functions for very large positive or negative x-values)

MCT.B.1.4: distinguish polynomial functions from sinusoidal and exponential functions [e.g., f(x) = sin x, f(x) = 2 to the x power)], and compare and contrast the graphs of various polynomial functions with the graphs of other types of functions

MCT.B.1.5: substitute into and evaluate polynomial functions expressed in function notation, including functions arising from real-world applications

MCT.B.1.6: pose problems based on real-world applications that can be modelled with polynomial functions, 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

MCT.B.1.7: recognize, using graphs, the limitations of modelling a real-world relationship using a polynomial function, and identify and explain any restrictions on the domain and range (e.g., restrictions on the height and time for a polynomial function that models the relationship between height above the ground and time for a falling object)

MCT.B.2: make connections between the numeric, graphical, and algebraic representations of polynomial functions;

MCT.B.2.1: factor polynomial expressions in one variable, of degree no higher than four, by selecting and applying strategies (i.e., common factoring, difference of squares, trinomial factoring)

MCT.B.2.2: make connections, through investigation using graphing technology (e.g., dynamic geometry software), between a polynomial function given in factored form [e.g., f(x) = x(x - 1)(x + 1)] and the x-intercepts of its graph, and sketch the graph of a polynomial function given in factored form using its key features (e.g., by determining intercepts and end behaviour; by locating positive and negative regions using test values between and on either side of the x-intercepts)

MCT.B.2.3: determine, through investigation using technology (e.g., graphing calculator, computer algebra systems), and describe the connection between the real roots of a polynomial equation and the x-intercepts of the graph of the corresponding polynomial function [e.g., the real roots of the equation (x to the 4th power) - 13x² + 36 = 0 are the x-intercepts of the graph of f(x) = (x to the 4th power) - 13x² + 36]

MCT.B.3: solve polynomial equations by factoring, make connections between functions and formulas, and solve problems involving polynomial expressions arising from a variety of applications.

MCT.B.3.1: solve polynomial equations in one variable, of degree no higher than four (e.g., x² - 4x = 0, (x to the 4th power) - 16 = 0, 3x² + 5x + 2 = 0), by selecting and applying strategies (i.e., common factoring; difference of squares; trinomial factoring), and verify solutions using technology (e.g., using computer algebra systems to determine the roots of the equation; using graphing technology to determine the x-intercepts of the corresponding polynomial function)

MCT.B.3.2: solve problems algebraically that involve polynomial functions and equations of degree no higher than four, including those arising from real-world applications

MCT.B.3.4: expand and simplify polynomial expressions involving more than one variable [e.g., simplify -2xy(3x²y³ - 5x³y²)], including expressions arising from real-world applications

MCT.B.3.7: make connections between formulas and linear, quadratic, and exponential functions [e.g., recognize that the compound interest formula, A = P((1 + i) to the n power), is an example of an exponential function A(n) when P and i are constant, and of a linear function A(P) when i and n are constant], using a variety of tools and strategies (e.g., comparing the graphs generated with technology when different variables in a formula are set as constants)

#### MCT.C: Trigonometric Functions

MCT.C.1: determine the values of the trigonometric ratios for angles less than 360º, and solve problems using the primary trigonometric ratios, the sine law, and the cosine law;

MCT.C.1.1: determine the exact values of the sine, cosine, and tangent of the special angles 0°, 30°, 45°, 60°, 90°, and their multiples

MCT.C.1.2: determine the values of the sine, cosine, and tangent of angles from 0º to 360º, 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 the special angles)

MCT.C.1.3: determine the measures of two angles from 0º to 360º for which the value of a given trigonometric ratio is the same (e.g., determine one angle using a calculator and infer the other angle)

MCT.C.1.4: solve multi-step problems in two and three dimensions, 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

MCT.C.2: make connections between the numeric, graphical, and algebraic representations of sinusoidal functions;

MCT.C.2.1: 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 0º to 360º 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) = sin x or f(x) = cos x, and explaining why the relationship is a function

MCT.C.2.2: sketch the graphs of f(x) = sin x and f(x) = cos x 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)

MCT.C.2.3: determine, through investigation using technology, the roles of the parameters d and c in functions of the form y = sin (x - d) + c and y = cos (x - d) + c, and describe these roles in terms of transformations on the graphs of f(x) = sin x and f(x) = cos x with angles expressed in degrees (i.e., vertical and horizontal translations)

MCT.C.2.4: determine, through investigation using technology, the roles of the parameters a and k in functions of the form y = a sin kx and y = a cos kx, and describe these roles in terms of transformations on the graphs of f(x) = sin x and f(x) = cos x with angles expressed in degrees (i.e., reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)

MCT.C.2.5: determine the amplitude, period, and phase shift 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, and sketch graphs of y = a sin (k(x - d)) + c and y = a cos (k(x - d)) + c by applying transformations to the graphs of f(x) = sin x and f(x) = cos x

MCT.C.2.6: represent a sinusoidal function with an equation, given its graph or its properties

MCT.C.3: demonstrate an understanding that sinusoidal functions can be used to model some periodic phenomena, and solve related problems, including those arising from real-world applications.

MCT.C.3.1: collect data that can be modelled as a sinusoidal function (e.g., voltage in an AC circuit, pressure in sound waves, the height of a tack on a bicycle wheel that is rotating at a fixed speed), 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

MCT.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

MCT.C.3.3: 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, in degree mode, from a table of values or from its equation

#### MCT.D: Applications of Geometry

MCT.D.1: represent vectors, add and subtract vectors, and solve problems using vector models, including those arising from real-world applications;

MCT.D.1.1: recognize a vector as a quantity with both magnitude and direction, and identify, gather, and interpret information about real-world applications of vectors (e.g., displacement; forces involved in structural design; simple animation of computer graphics; velocity determined using GPS)

MCT.D.1.2: represent a vector as a directed line segment, with directions expressed in different ways (e.g., 320°; N 40° W), and recognize vectors with the same magnitude and direction but different positions as equal vectors

MCT.D.1.4: represent a vector as a directed line segment, given its vertical and horizontal components (e.g., the displacement of a ship that travels 3 km east and 4 km north can be represented by the vector with a magnitude of 5 km and a direction of N 36.9° E)

MCT.D.1.5: determine, through investigation using a variety of tools (e.g., graph paper, technology) and strategies (i.e., head-to-tail method; parallelogram method; resolving vectors into their vertical and horizontal components), the sum (i.e., resultant) or difference of two vectors

MCT.D.1.6: solve problems involving the addition and subtraction of vectors, including problems arising from real-world applications (e.g., surveying, statics, orienteering)

MCT.D.2: solve problems involving two-dimensional shapes and three-dimensional figures and arising from real-world applications;

MCT.D.2.3: solve problems involving the areas of rectangles, parallelograms, trapezoids, triangles, and circles, and of related composite shapes, in situations arising from real-world applications

MCT.D.2.4: solve problems involving the volumes and surface areas of spheres, right prisms, and cylinders, and of related composite figures, in situations arising from real-world applications

MCT.D.3: determine circle properties and solve related problems, including those arising from real-world applications.

MCT.D.3.1: recognize and describe (i.e., using diagrams and words) arcs, tangents, secants, chords, segments, sectors, central angles, and inscribed angles of circles, and some of their real-world applications (e.g., construction of a medicine wheel)

MCT.D.3.2: determine the length of an arc and the area of a sector or segment of a circle, and solve related problems

MCT.D.3.3: determine, through investigation using a variety of tools (e.g., dynamic geometry software), properties of the circle associated with chords, central angles, inscribed angles, and tangents (e.g., equal chords or equal arcs subtend equal central angles and equal inscribed angles; a radius is perpendicular to a tangent at the point of tangency defined by the radius, and to a chord that the radius bisects)

### FCM: Foundation for College Mathematics

#### FCM.A: Mathematical Models

FCM.A.1: evaluate powers with rational exponents, simplify algebraic expressions involving exponents, and solve problems involving exponential equations graphically and using common bases;

FCM.A.1.1: determine, through investigation (e.g., by expanding terms and patterning), the exponent laws for multiplying and dividing algebraic expressions involving exponents [e.g., (x³)(x²), x³ ÷ (x to the 5th power)] and the exponent law for simplifying algebraic expressions involving a power of a power [e.g. (x to the 6th power times y³)²]

FCM.A.1.2: simplify algebraic expressions containing integer exponents using the laws of exponents

FCM.A.1.6: solve problems involving exponential equations arising from real-world applications by using a graph or table of values generated with technology from a given equation [e.g., h = 2((0.6) to the n power), where h represents the height of a bouncing ball and n represents the number of bounces]

FCM.A.2: describe trends based on the interpretation of graphs, compare graphs using initial conditions and rates of change, and solve problems by modelling relationships graphically and algebraically;

FCM.A.2.1: interpret graphs to describe a relationship (e.g., distance travelled depends on driving time, pollution increases with traffic volume, maximum profit occurs at a certain sales volume), using language and units appropriate to the context

FCM.A.2.2: describe trends based on given graphs, and use the trends to make predictions or justify decisions (e.g., given a graph of the men's 100-m world record versus the year, predict the world record in the year 2050 and state your assumptions; given a graph showing the rising trend in graduation rates among Aboriginal youth, make predictions about future rates)

FCM.A.2.5: compare, through investigation with technology, the graphs of pairs of relations (i.e., linear, quadratic, exponential) by describing the initial conditions and the behaviour of the rates of change (e.g., compare the graphs of amount versus time for equal initial deposits in simple interest and compound interest accounts)

FCM.A.2.6: recognize that a linear model corresponds to a constant increase or decrease over equal intervals and that an exponential model corresponds to a constant percentage increase or decrease over equal intervals, select a model (i.e., linear, quadratic, exponential) to represent the relationship between numerical data graphically and algebraically, using a variety of tools (e.g., graphing technology) and strategies (e.g., finite differences, regression), and solve related problems

FCM.A.3: make connections between formulas and linear, quadratic, and exponential relations, solve problems using formulas arising from real-world applications, and describe applications of mathematical modelling in various occupations.

FCM.A.3.3: make connections between formulas and linear, quadratic, and exponential functions [e.g., recognize that the compound interest formula, A = P((1 + i) to the n power), is an example of an exponential function A(n) when P and i are constant, and of a linear function A(P) when i and n are constant], using a variety of tools and strategies (e.g., comparing the graphs generated with technology when different variables in a formula are set as constants)

FCM.A.3.5: gather, interpret, and describe information about applications of mathematical modelling in occupations, and about college programs that explore these applications

#### FCM.B: Personal Finance

FCM.B.1: demonstrate an understanding of annuities, including mortgages, and solve related problems using technology;

FCM.B.1.2: 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 an ordinary simple annuity (i.e., an annuity in which payments are made at the end of each period, and compounding and payment periods are the same) (e.g., long-term savings plans, loans)

FCM.B.1.5: gather and interpret information about mortgages, describe features associated with mortgages (e.g., mortgages are annuities for which the present value is the amount borrowed to purchase a home; the interest on a mortgage is compounded semi-annually but often paid monthly), and compare different types of mortgages (e.g., open mortgage, closed mortgage, variable-rate mortgage)

FCM.B.1.7: generate an amortization table for a mortgage, using a variety of tools and strategies (e.g., input data into an online mortgage calculator; determine the payments using the TVM Solver on a graphing calculator and generate the amortization table using a spreadsheet), calculate the total interest paid over the life of a mortgage, and compare the total interest with the original principal of the mortgage

FCM.B.1.8: determine, through investigation using technology (e.g., TVM Solver, online tools, financial software), the effects of varying payment periods, regular payments, and interest rates on the length of time needed to pay off a mortgage and on the total interest paid

#### FCM.C: Geometry and Trigonometry

FCM.C.1: solve problems involving measurement and geometry and arising from real-world applications;

FCM.C.1.2: solve problems involving the areas of rectangles, triangles, and circles, and of related composite shapes, in situations arising from real-world applications

FCM.C.1.3: solve problems involving the volumes and surface areas of rectangular prisms, triangular prisms, and cylinders, and of related composite figures, in situations arising from real-world applications

FCM.C.2: explain the significance of optimal dimensions in real-world applications, and determine optimal dimensions of two-dimensional shapes and three-dimensional figures;

FCM.C.2.1: recognize, through investigation using a variety of tools (e.g., calculators; dynamic geometry software; manipulatives such as tiles, geoboards, toothpicks) and strategies (e.g., modelling; making a table of values; graphing), and explain the significance of optimal perimeter, area, surface area, and volume in various applications (e.g., the minimum amount of packaging material, the relationship between surface area and heat loss)

FCM.C.2.2: determine, through investigation using a variety of tools (e.g., calculators, dynamic geometry software, manipulatives) and strategies (e.g., modelling; making a table of values; graphing), the optimal dimensions of a two-dimensional shape in metric or imperial units for a given constraint (e.g., the dimensions that give the minimum perimeter for a given area)

FCM.C.2.3: determine, through investigation using a variety of tools and strategies (e.g., modelling with manipulatives; making a table of values; graphing), the optimal dimensions of a right rectangular prism, a right triangular prism, and a right cylinder in metric or imperial units for a given constraint (e.g., the dimensions that give the maximum volume for a given surface area)

FCM.C.3: solve problems using primary trigonometric ratios of acute and obtuse angles, the sine law, and the cosine law, including problems arising from real-world applications, and describe applications of trigonometry in various occupations.

FCM.C.3.1: solve problems in two dimensions using metric or imperial measurements, including problems that arise from real-world applications (e.g., surveying, navigation, building construction), by determining the measures of the sides and angles of right triangles using the primary trigonometric ratios, and of acute triangles using the sine law and the cosine law

FCM.C.3.2: make connections between primary trigonometric ratios (i.e., sine, cosine, tangent) of obtuse angles and of acute angles, through investigation using a variety of tools and strategies (e.g., using dynamic geometry software to identify an obtuse angle with the same sine as a given acute angle; using a circular geoboard to compare congruent triangles; using a scientific calculator to compare trigonometric ratios for supplementary angles)

FCM.C.3.3: determine the values of the sine, cosine, and tangent of obtuse angles

#### FCM.D: Data Management

FCM.D.1: collect, analyse, and summarize two-variable data using a variety of tools and strategies, and interpret and draw conclusions from the data;

FCM.D.1.1: distinguish situations requiring one-variable and two-variable data analysis, describe the associated numerical summaries (e.g., tally charts, summary tables) and graphical summaries (e.g., bar graphs, scatter plots), and recognize questions that each type of analysis addresses (e.g., What is the frequency of a particular trait in a population? What is the mathematical relationship between two variables?)

FCM.D.1.4: create a graphical summary of two-variable data using a scatter plot (e.g., by identifying and justifying the dependent and independent variables; by drawing the line of best fit, when appropriate), with and without technology

FCM.D.1.6: describe possible interpretations of the line of best fit of a scatter plot (e.g., the variables are linearly related) and reasons for misinterpretations (e.g., using too small a sample; failing to consider the effect of outliers; interpolating from a weak correlation; extrapolating nonlinearly related data)

FCM.D.1.8: make conclusions from the analysis of twovariable data (e.g., by using a correlation to suggest a possible cause-and-effect relationship), and judge the reasonableness of the conclusions (e.g., by assessing the strength of the correlation; by considering if there are enough data)

FCM.D.2: demonstrate an understanding of the applications of data management used by the media and the advertising industry and in various occupations.

FCM.D.2.1: recognize and interpret common statistical terms (e.g., percentile, quartile) and expressions (e.g., accurate 19 times out of 20) used in the media (e.g., television, Internet, radio, newspapers)

### MWEL: Mathematics for Work and Everyday Life

#### MWEL.A: Reasoning With Data

MWEL.A.1: collect, organize, represent, and make inferences from data using a variety of tools and strategies, and describe related applications;

MWEL.A.1.1: read and interpret graphs (e.g., bar graph, broken-line graph, histogram) obtained from various sources (e.g., newspapers, magazines, Statistics Canada website)

MWEL.A.1.2: 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)

MWEL.A.1.5: make inferences based on the graphical representation of data (e.g., an inference about a sample from the graphical representation of a population), and justify conclusions orally or in writing using convincing arguments (e.g., by showing that it is reasonable to assume that a sample is representative of a population)

MWEL.A.1.8: gather, interpret, and describe information about applications of data management in the workplace and in everyday life

MWEL.A.2: determine and represent probability, and identify and interpret its applications.

MWEL.A.2.1: 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)

MWEL.A.2.3: perform simple probability experiments (e.g., rolling number cubes, spinning spinners, flipping coins, playing Aboriginal stick-and-stone games), record the results, and determine the experimental probability of an event

MWEL.A.2.4: compare, through investigation, the theoretical probability of an event with the experimental probability, and describe how uncertainty explains why they might differ (e.g., "I know that the theoretical probability of getting tails is 0.5, but that does not mean that I will always obtain 3 tails when I toss the coin 6 times"; "If a lottery has a 1 in 9 chance of winning, am I certain to win if I buy 9 tickets?")

MWEL.A.2.5: determine, through investigation using classgenerated 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 getting tails in any one toss is likely to be closer to the theoretical probability than if I simulate tossing the coin only 10 times")

#### MWEL.C: Applications of Measurement

MWEL.C.2: apply measurement concepts and skills to solve problems in measurement and design, to construct scale drawings and scale models, and to budget for a household improvement;

MWEL.C.2.1: construct accurate right angles in practical contexts (e.g., by using the 3-4-5 triplet to construct a region with right-angled corners on a floor), and explain connections to the Pythagorean theorem

MWEL.C.2.3: estimate the areas and volumes of irregular shapes and figures, using a variety of strategies (e.g., counting grid squares; displacing water)

MWEL.C.2.4: solve problems involving the areas of rectangles, triangles, and circles, and of related composite shapes, in situations arising from real-world applications

MWEL.C.2.5: solve problems involving the volumes and surface areas of rectangular prisms, triangular prisms, and cylinders, and of related composite figures, in situations arising from real-world applications

MWEL.C.2.6: construct a two-dimensional scale drawing of a familiar setting (e.g., classroom, flower bed, playground) on grid paper or using design or drawing software

MWEL.C.2.7: construct, with reasonable accuracy, a three-dimensional scale model of an object or environment of personal interest (e.g., appliance, room, building, garden, bridge)

MWEL.C.3: identify and describe situations that involve proportional relationships and the possible consequences of errors in proportional reasoning, and solve problems involving proportional reasoning, arising in applications from work and everyday life.

MWEL.C.3.3: identify and describe real-world applications of proportional reasoning (e.g., mixing concrete; calculating dosages; converting units; painting walls; calculating fuel consumption; calculating pay; enlarging patterns), distinguish between a situation involving a proportional relationship (e.g., recipes, where doubling the quantity of each ingredient doubles the number of servings; long-distance phone calls billed at a fixed cost per minute, where talking for half as many minutes costs half as much) and a situation involving a non-proportional relationship (e.g., cellular phone packages, where doubling the minutes purchased does not double the cost of the package; food purchases, where it can be less expensive to buy the same quantity of a product in one large package than in two or more small packages; hydro bills, where doubling consumption does not double the cost) in a personal and/or workplace context, and explain their reasoning

MWEL.C.3.4: identify and describe the possible consequences (e.g., overdoses of medication; seized engines; ruined clothing; cracked or crumbling concrete) of errors in proportional reasoning (e.g., not recognizing the importance of maintaining proportionality; not correctly calculating the amount of each component in a mixture)

MWEL.C.3.5: solve problems involving proportional reasoning in everyday life (e.g., applying fertilizers; mixing gasoline and oil for use in small engines; mixing cement; buying plants for flower beds; using pool or laundry chemicals; doubling recipes; estimating cooking time from the time needed per pound; determining the fibre content of different sizes of food servings)

MWEL.C.3.6: solve problems involving proportional reasoning in work-related situations (e.g., calculating overtime pay; calculating pay for piecework; mixing concrete for small or large jobs)

Correlation last revised: 7/15/2014

This correlation lists the recommended Gizmos for this province's curriculum standards. Click any Gizmo title below for more information.