High School Math Ontario Curriculum Standards

Mathematics

Measurement and Trigonometry

Solving Problems Involving Similar Triangles

verify, through investigation (e.g., using dynamic geometry software, concrete materials), properties of similar triangles (e.g., given similar triangles, verify the equality of corresponding angles and the proportionality of corresponding sides);

determine the lengths of sides of similar triangles, using proportional reasoning;

solve problems involving similar triangles in realistic situations (e.g., shadows, reflections, scale models, surveying)

Solving Problems Involving the Trigonometry of Right Triangles

determine, through investigation (e.g., using dynamic geometry software, concrete materials), the relationship between the ratio of two sides in a right triangle and the ratio of the two corresponding sides in a similar right triangle, and define the sine, cosine, and tangent ratios (e.g., sin A = opposite/hypotenuse);

determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem;

solve problems involving the measures of sides and angles in right triangles in reallife applications (e.g., in surveying, in navigation, in determining the height of an inaccessible object around the school), using the primary trigonometric ratios and the Pythagorean theorem

describe, through participation in an activity, the application of trigonometry in an occupation (e.g., research and report on how trigonometry is applied in astronomy; attend a career fair that includes a surveyor, and describe how a surveyor applies trigonometry to calculate distances; job shadow a carpenter for a few hours, and describe how a carpenter uses trigonometry).

Solving Problems Involving Surface Area and Volume, Using the Imperial and Metric Systems of Measurement

use the imperial system when solving measurement problems (e.g., problems involving dimensions of lumber, areas of carpets, and volumes of soil or concrete);

perform everyday conversions between the imperial system and the metric system (e.g., millilitres to cups, centimetres to inches) and within these systems (e.g., cubic metres to cubic centimetres, square feet to square yards), as necessary to solve problems involving measurement

determine, through investigation, the relationship for calculating the surface area of a pyramid (e.g., use the net of a squarebased pyramid to determine that the surface area is the area of the square base plus the areas of the four congruent triangles);

solve problems involving the surface areas of prisms, pyramids, and cylinders, and the volumes of prisms, pyramids, cylinders, cones, and spheres, including problems involving combinations of these figures, using the metric system or the imperial system, as appropriate

develop the formula for the volume of a sphere, using concrete materials and the volume relationships between cylinders, cones, and spheres

Modelling Linear Relations

Manipulating and Solving Algebraic Equations

solve first-degree equations involving one variable, including equations with fractional coefficients (e.g. using the balance analogy, computer algebra systems, paper and pencil)

determine the value of a variable in the first degree, using a formula (i.e., by isolating the variable and then substituting known values; by substituting known values and then solving for the variable) (e.g., in analytic geometry, in measurement)

express the equation of a line in the form y = mx + b, given the form Ax + By + C = 0

Graphing and Writing Equations of Lines

connect the rate of change of a linear relation to the slope of the line, and define the slope as the ratio m = rise/run

identify, through investigation, y = mx + b as a common form for the equation of a straight line, and identify the special cases x = a, y = b;

identify, through investigation with technology, the geometric significance of m and b in the equation y = mx + b;

identify, through investigation, properties of the slopes of lines and line segments (e.g., direction, positive or negative rate of change, steepness, parallelism), using graphing technology to facilitate investigations, where appropriate;

graph lines by hand, using a variety of techniques (e.g., graph y = 2/3 x - 4 using the y-intercept and slope; graph 2x + 3y = 6 using the x- and y-intercepts)

determine the equation of a line, given its graph, the slope and y-intercept, the slope and a point on the line, or two points on the line

Solving and Interpreting Systems of Linear Equations

determine graphically the point of intersection of two linear relations (e.g., using graph paper, using technology)

solve systems of two linear equations involving two variables with integral coefficients, using the algebraic method of substitution or elimination

solve problems that arise from realistic situations described in words or represented by given linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method

Quadratic Relations of the Form y ax2 = + bx + c

Manipulating Quadratic Expressions

expand and simplify second-degree polynomial expressions involving one variable that consist of the product of two binomials [e.g., (2x + 3)(x + 4)] or the square of a binomial [e.g., (x + 3)2], using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil) and strategies (e.g. patterning);

factor binomials (e.g., 4x2 + 8x) and trinomials (e.g., 3x2 + 9x - 15) involving one variable up to degree two, by determining a common factor using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil) and strategies (e.g., patterning);

factor simple trinomials of the form x2 + bx + c (e.g., x2 + 7x + 10, x2 + 2x - 8), using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil) and strategies (e.g., patterning);

factor the difference of squares of the form x2 - a2 (e.g., x2 - 16).

Identifying Characteristics of Quadratic Relations

collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology (e.g., concrete materials, scientific probes, graphing calculators), or from secondary sources (e.g., the Internet, Statistics Canada); graph the data and draw a curve of best fit if appropriate, with or without the use of technology

determine, through investigation using technology, that a quadratic relation of the form y = ax2 + bx + c (a ? 0) can be graphically represented as a parabola, and determine that the table of values yields a constant second difference

identify the key features of a graph of a parabola (i.e., the equation of the axis of symmetry, the coordinates of the vertex, the y-intercept, the zeros, and the maximum or minimum value), using a given graph or a graph generated with technology from its equation, and use the appropriate terminology to describe the features;

compare, through investigation using technology, the graphical representations of a quadratic relation in the form y = x2 + bx + c and the same relation in the factored form y = (x - r)(x - s) (i.e., the graphs are the same), and describe the connections between each algebraic representation and the graph [e.g., the y-intercept is c in the form y = x2 + bx + c; the x-intercepts are r and s in the form y = (x - r)(x - s)] (

Solving Problems by Interpreting Graphs of Quadratic Relations

solve problems involving a quadratic relation by interpreting a given graph or a graph generated with technology from its equation (e.g., given an equation representing the height of a ball over elapsed time, use a graphing calculator or graphing software to graph the relation, and answer questions such as the following:What is the maximum height of the ball? After what length of time will the ball hit the ground? Over what time interval is the height of the ball greater than 3 m?);

solve problems by interpreting the significance of the key features of graphs obtained by collecting experimental data involving quadratic relations

Mathematics

Quadratic Relations of the Form y ax2 = + bx + c

Investigating the Basic Properties of Quadratic Relations

collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology (e.g., concrete materials, scientific probes, graphing calculators), or from secondary sources (e.g., the Internet, Statistics Canada); graph the data and draw a curve of best fit, if appropriate, with or without the use of technology

determine, through investigation with and without the use of technology, that a quadratic relation of the form y = ax2 + bx + c (a ? 0) can be graphically represented as a parabola, and that the table of values yields a constant second difference

identify the key features of a graph of a parabola (i.e., the equation of the axis of symmetry, the coordinates of the vertex, the y-intercept, the zeros, and the maximum or minimum value), and use the appropriate terminology to describe them;

compare, through investigation using technology, the features of the graph of y = x2 and the graph of y = 2x, and determine the meaning of a negative exponent and of zero as an exponent (e.g., by examining patterns in a table of values for y = 2x; by applying the exponent rules for multiplication and division).

Relating the Graph of y = x2 and its Transformations

identify, through investigation using technology, the effect on the graph of y = x2 of transformations (i.e., translations, reflections in the x-axis, vertical stretches or compressions) by considering separately each parameter a, h, and k [i.e., investigate the effect on the graph of y = x2 of a, h, and k in y = x2 + k, y = (x - h) 2, and y = ax2];

explain the roles of a, h, and k in y = a(x - h ) 2 + k, using the appropriate terminology to describe the transformations, and identify the vertex and the equation of the axis of symmetry

sketch, by hand, the graph of y = a(x - h ) 2 + k by applying transformations to the graph of y = x2

determine the equation, in the form y = a(x – h)2 + k, of a given graph of a parabola

Solving Quadratic Equations

expand and simplify second-degree polynomial expressions [e.g., (2x + 5)2, (2x – y)(x + 3y)], using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil) and strategies (e.g., patterning);

factor polynomial expressions involving common factors, trinomials, and differences of squares [e.g., 2x2 + 4x, 2x – 2y + ax – ay, x2 – x – 6, 2a2 + 11a + 5, 4x2 – 25], using a variety of tools (e.g., concrete materials, computer algebra systems, paper and pencil) and strategies (e.g., patterning);

determine, through investigation, and describe the connection between the factors of a quadratic expression and the x-intercepts (i.e., the zeros) of the graph of the corresponding quadratic relation, expressed in the form y = a(x – r)(x – s);

interpret real and non-real roots of quadratic equations, through investigation using graphing technology, and relate the roots to the x-intercepts of the corresponding relations;

express y = ax2 + bx + c in the form y = a(x – h) 2 + k by completing the square in situations involving no fractions, using a variety of tools (e.g. concrete materials, diagrams, paper and pencil);

sketch or graph a quadratic relation whose equation is given in the form y = ax2 + bx + c, using a variety of methods (e.g., sketching y = x2 – 2x – 8 using intercepts and symmetry; sketching y = 3x2 – 12x + 1 by completing the square and applying transformations; graphing h = –4.9t 2 + 50t + 1.5 using technology);

explore the algebraic development of the quadratic formula (e.g., given the algebraic development, connect the steps to a numerical example; follow a demonstration of the algebraic development [student reproduction of the development of the general case is not required]);

solve quadratic equations that have real roots, using a variety of methods (i.e., factoring, using the quadratic formula, graphing)

Solving Problems Involving Quadratic Relations

determine the zeros and the maximum or minimum value of a quadratic relation from its graph (i.e., using graphing calculators or graphing software) or from its defining equation (i.e., by applying algebraic techniques);

solve problems arising from a realistic situation represented by a graph or an equation of a quadratic relation, with and without the use of technology (e.g., given the graph or the equation of a quadratic relation representing the height of a ball over elapsed time, answer questions such as the following: What is the maximum height of the ball? After what length oftime will the ball hit the ground? Over what time interval is the height of the ball greater than 3 m?)

Analytic Geometry

Using Linear Systems to Solve Problems

solve systems of two linear equations involving two variables, using the algebraic method of substitution or elimination

solve problems that arise from realistic situations described in words or represented by linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method

Solving Problems Involving Properties of Line Segments

develop the formula for the midpoint of a line segment, and use this formula to solve problems (e.g., determine the coordinates of the midpoints of the sides of a triangle, given the coordinates of the vertices, and verify concretely or by using dynamic geometry software);

develop the formula for the length of a line segment, and use this formula to solve problems (e.g., determine the lengths of the line segments joining the midpoints of the sides of a triangle, given the coordinates of the vertices of the triangle, and verify using dynamic geometry software);

develop the equation for a circle with centre (0, 0) and radius r, by applying the formula for the length of a line segment;

determine the radius of a circle with centre (0, 0), given its equation; write the equation of a circle with centre (0, 0), given the radius; and sketch the circle, given the equation in the form x2 + y2 = r 2;

solve problems involving the slope, length, and midpoint of a line segment (e.g., determine the equation of the right bisector of a line segment, given the coordinates of the endpoints; determine the distance from a given point to a line whose equation is given, and verify using dynamic geometry software).

identify the relationship between the slopes of parallel and perpendicular lines, and use this relationship to solve related problems

develop the formula for the slope of a line (i.e., ), and use this formula to determine the equations of lines, given information about the lines (e.g., a graph of a line, a table of values, the coordinates of two points)

represent the equations of lines in different forms (e.g., y = mx + b, Ax + By + C = 0, Ax + By = D) and translate between these forms, as appropriate for the context

Using Analytic Geometry to Verify Geometric Properties

determine, through investigation (e.g., using dynamic geometry software, by paper folding), some characteristics and properties of geometric figures (e.g., medians in a triangle, similar figures constructed on the sides of a right triangle);

verify, using algebraic techniques and analytic geometry, some characteristics of geometric figures (e.g., verify that two lines are perpendicular, given the coordinates of two points on each line; verify, by determining side length, that a triangle is equilateral, given the coordinates of the vertices);

plan and implement a multi-step strategy that uses analytic geometry and algebraic techniques to verify a geometric property (e.g., given the coordinates of the vertices of a triangle, verify that the line segment joining the midpoints of two sides of the triangle is parallel to the third side and half its length, and check using dynamic geometry software; given the coordinates of the vertices of a rectangle, verify that the diagonals of the rectangle bisect each other)

Trigonometry

Investigating Similarity and Solving Problems Involving Similar Triangles

verify, through investigation (e.g., using dynamic geometry software, concrete materials), the properties of similar triangles (e.g., given similar triangles, verify the equality of corresponding angles and the proportionality of corresponding sides);

describe and compare the concepts of similarity and congruence;

solve problems involving similar triangles in realistic situations (e.g., shadows, reflections, scale models, surveying)

Solving Problems Involving the Trigonometry of Right Triangles

determine, through investigation (e.g., using dynamic geometry software, concrete materials), the relationship between the ratio of two sides in a right triangle and the ratio of the two corresponding sides in a similar right triangle, and define the sine, cosine, and tangent ratios (e.g., sin A = opposite /hypotenuse);

determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem;

solve problems involving the measures of sides and angles in right triangles in reallife applications (e.g., in surveying, in navigating, in determining the height of an inaccessible object around the school), using the primary trigonometric ratios and the Pythagorean theorem

Solving Problems Involving the Trigonometry of Acute Triangles

explore the development of the sine law within acute triangles (e.g., use dynamic geometry software to determine that the ratio of the side lengths equals the ratio of the sines of the opposite angles; follow the algebraic development of the sine law and identify the application of solving systems of equations [student reproduction of the development of the formula is not required]);

explore the development of the cosine law within acute triangles (e.g., use dynamic geometry software to verify the cosine law; follow the algebraic development of the cosine law and identify its relationship to the Pythagorean theorem and the cosine ratio [student reproduction of the development of the formula is not required]);

determine the measures of sides and angles in acute triangles, using the sine law and the cosine law

solve problems involving the measures of sides and angles in acute triangles.

Mathematics

Mathematical Models

Connecting Graphs and Equations of Quadratic Relations: make connections between the numeric, graphical, and algebraic representations of quadratic relations, and use the connections to solve problems;

construct tables of values and graph quadratic relations arising from real-world applications (e.g., dropping a ball from a given height; varying the edge length of a cube and observing the effect on the surface area of the cube)

determine and interpret meaningful values of the variables, given a graph of a quadratic relation arising from a real-world application

determine, through investigation using technology, the roles of a, h, and k in quadratic relations of the form y = a(x - h) + k, and describe these roles in terms of transformations on the graph of y = x (i.e., translations; reflections in the x-axis; vertical stretches and compressions to and from the x-axis)

sketch graphs of quadratic relations represented by the equation y = a(x - h) + k (e.g., using the vertex and at least one point on each side of the vertex; applying one or more transformations to the graph of y = x )

expand and simplify quadratic expressions in one variable involving multiplying binomials [e.g., (1/2x + 1)(3x - 2)] or squaring a binomial [e.g., 5(3x - 1)2], using a variety of tools (e.g., paper and pencil, algebra tiles, computer algebra systems)

express the equation of a quadratic relation in the standard form y = ax + bx + c, given the vertex form y = a(x - h) + k, and verify, using graphing technology, that these forms are equivalent representations

factor trinomials of the form ax + bx + c, where a = 1 or where a is the common factor, by various methods

determine, through investigation, and describe the connection between the factors of a quadratic expression and the x-intercepts of the graph of the corresponding quadratic relation

solve problems, using an appropriate strategy (i.e., factoring, graphing), given equations of quadratic relations, including those that arise from real-world applications (e.g., break-even point)

Connecting Graphs and Equations of Exponential Relations: demonstrate an understanding of exponents, and make connections between the numeric, graphical, and algebraic representations of exponential relations;

determine, through investigation using a variety of tools and strategies (e.g., graphing with technology; looking for patterns in tables of values), and describe the meaning of negative exponents and of zero as an exponent

evaluate, with and without technology, numeric expressions containing integer exponents and rational bases (e.g., 2-3, 63, 34560, 1.0310)

determine, through investigation (e.g., by patterning with and without a calculator), the exponent rules for multiplying and dividing numerical expressions involving exponents [e.g., (1/2)3 x (1/2)2], and the exponent rule for simplifying numerical expressions involving a power of a power [e.g., (53)2]

graph simple exponential relations, using paper and pencil, given their equations [e.g., y = 2x, y = 10x, y = (1/2)x]

make and describe connections between representations of an exponential relation (i.e., numeric in a table of values; graphical; algebraic)

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

Solving Problems Involving Exponential Relations: describe and represent exponential relations, and solve problems involving exponential relations arising from real-world applications.

collect data that can be modelled as an exponential relation, 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

describe some characteristics of exponential relations arising from real-world applications (e.g., bacterial growth, drug absorption) by using tables of values (e.g., to show a constant ratio, or multiplicative growth or decay) and graphs (e.g., to show, with technology, that there is no maximum or minimum value)

pose problems involving exponential relations arising from a variety of real-world applications (e.g., population growth, radioactive decay, compound interest), and solve these and other such problems by using a given graph or a graph generated with technology from a given table of values or a given equation

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

Personal Finance

Solving Problems Involving Compound Interest: compare simple and compound interest, relate compound interest to exponential growth, and solve problems involving compound interest;

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

determine, through investigation (e.g., using spreadsheets and graphs), and describe the relationship between compound interest and exponential growth

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)n [or FV = PV (1 + i)n]

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 )]

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)n [or FV = PV (1 + i)n

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 ttime, the interest rate, or the compounding period

Comparing Financial Services: compare services available from financial institutions, and solve problems involving the cost of making purchases on credit;

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)

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

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

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

solve problems involving applications of the compound interest formula to determine the cost of making a purchase on credit

Owning and Operating a Vehicle: . interpret information about owning and operating a vehicle, and solve problems involving the associated costs.

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

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

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

Geometry and Trigonometry

Representing Two-Dimensional Shapes and Three-Dimensional Figures: represent, in a variety of ways, two-dimensional shapes and three-dimensional figures arising from real-world applications, and solve design problems;

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)

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)

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

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

Applying the Sine Law and the Cosine Law in Acute Triangles: solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications.

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

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/sin A, b/sin B, and in c/sin C triangle ABC while dragging one of the vertices)

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

solve problems that arise from real-world applications involving metric and imperial measurements and that require the use of the sine law or the cosine law in acute triangles

Data Management

Working With One-Variable Data: solve problems involving one-variable data by collecting, organizing, analysing, and evaluating data;

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

solve problems by interpreting and analysing one-variable data collected from secondary sources

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)

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)

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

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)

identify and describe properties associated with common distributions of data (e.g., normal, bimodal, skewed)

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)

explain the appropriate use of measures of central tendency (i.e., mean, median, mode) and measures of spread (i.e., range, standard deviation)

compare two or more sets of one-variable data, using measures of central tendency and measures of spread

Applying Probability: determine and represent probability, and identify and interpret its applications.

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)

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)

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

compare, through investigation, the theoretical probability of an event with the experimental probability, and explain why they might differ

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 tossing tails is likely to be closer to the theoretical probability than if I simulate tossing the coin only 10 times")

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)

Mathematics

Characteristics of Functions

Representing Functions: demonstrate an understanding of functions, their representations, and their inverses, and make connections between the algebraic and graphical representations of functions using transformations;

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 verticalline test)

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(1/2), given f(x) = 2x² + 3x - 1]

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) = √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 ≥ 1); and explain any restrictions on the domain and range in contexts arising from real-world applications

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

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)

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

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) – 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)

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) = x², f(x) = √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)

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) = √x, and f(x) = 1/x, and state the domain and range of the transformed functions

Solving Problems Involving Quadratic Functions: 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;

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)

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

solve problems involving quadratic functions arising from real-world applications and represented using function notation

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

solve problems involving the intersection of a linear function and a quadratic function graphically and algebraically (e.g., determine 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)

Determining Equivalent Algebraic Expressions: demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and rational expressions.

simplify polynomial expressions by adding, subtracting, and multiplying

verify, through investigation with and without technology, that sqrt(ab) = √a × √b, a ≥ 0, b ≥ 0, and use this relationship to simplify radicals (e.g., sqrt(24)) and radical expressions obtained by adding, subtracting, and multiplying [e.g., (2 + √6)(3 - sqrt(12))]

simplify rational expressions by adding, subtracting, multiplying, and dividing, and state the restrictions on the variable values

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

Exponential Functions

Representing Exponential Functions: evaluate powers with rational exponents, simplify expressions containing exponents, and describe properties of exponential functions represented in a variety of ways;

graph, with and without technology, an exponential relation, given its equation in the form y = a (a > 0, a ? 1), define this relation as the function f(x) = a , and explain why it is a function

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)

simplify algebraic expressions containing integer and rational exponents [e.g., (x³) ÷ (x(1/2)), (x6)y³)1/3], and evaluate numeric expressions containing integer and rational exponents and rational bases [e.g., 2-3, (-6)³, 41/2, 1.01120]

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 (a > 0, a ? 1), function machines]

Connecting Graphs and Equations of Exponential Functions: make connections between the numeric, graphical, and algebraic representations of exponential functions;

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)

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 (a > 0, a ? 1) (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)

sketch graphs of y = af(k(x - d)) + c by applying one or more transformations to the graph of f(x) = a (a > 0, a ? 1), and state the domain and range of the transformed functions

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

represent an exponential function with an equation, given its graph or its properties

Solving Problems Involving Exponential Functions: . identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications.

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

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)

solve problems using given graphs or equations of exponential functions arisin 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

Discrete Function

Representing Sequences: demonstrate an understanding of recursive sequences, represent recursive sequences in a variety of ways, and make connections to Pascal's triangle;

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]

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)

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

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(1) = 2; t(n) = 2[t(n - 1)], as f(n) = 2(n), or as t(n) = 2(n), or represent 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, …as t(1) = 1/2; t(n) = t(n - 1) + 1/[n(n + 1)], as f(n) = n/[n + 1], or as t(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

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)

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)4, (2x - 1)5 , (2x - y)6, (x(2) + 1)5]

Investigating Arithmetic and Geometric Sequences and Series: demonstrate an understanding of the relationships involved in arithmetic and geometric sequences and series, and solve related problems

identify sequences as arithmetic, geometric, or neither, given a numeric or algebraic representation

determine the formula for the general term of an arithmetic sequence [i.e., tn = a + (n - 1)d ] or geometric sequence (i.e., tn = ar n - 1), 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

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

solve problems involving arithmetic and geometric sequences and series, including those arising from real-world applications

Solving Problems Involving Financial Applications: make connections between sequences, series, and financial applications, and solve problems involving compound interest and ordinary annuities.

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)

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)

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)n [or FV = PV(1 + i)n]

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) [or FV = PV(1 + i) ]; 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

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)

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)

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)

Trigonometric Functions

Determining and Applying Trigonometric Ratios: determine the values of the trigonometric ratios for angles less than 360º; prove simple trigonometric identities; and solve problems using the primary trigonometric ratios, the sine law, and the cosine law;

determine the exact values of the sine, cosine, and tangent of the special angles: 0º, 30º, 45º, 60º, and 90º

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 special angles)

determine the measures of two angles from 0ºto 360º for which the value of a given trigonometric ratio is the same

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)

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

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)

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

Connecting Graphs and Equations of Sinusoidal Functions: demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions;

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

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)

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) =sinx or f(x) =cosx, and explaining why the relationship is a function

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)

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)

determine the amplitude, period, phase shift, domain, and range of sinusoidal functions whose equations are given in the form f(x) = asin(k(x - d)) + c or f(x) = acos(k(x - d)) + c

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

represent a sinusoidal function with an equation, given its graph or its properties

Solving Problems Involving Sinusoidal Functions: identify and represent sinusoidal functions, and solve problems involving sinusoidal functions, including problems arising from real-world applications.

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

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

determine, through investigation, how sinusoidal functions can be used to model periodic phenomena that do not involve angles

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)

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

Mathematics

Quadratic Functions

Solving Quadratic Equations: expand and simplify quadratic expressions, solve quadratic equations, and relate the roots of a quadratic equation to the corresponding graph;

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?”)

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) ]

factor quadratic expressions in one variable, including those for which a ≠ 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

solve quadratic equations by selecting and applying a factoring strategy

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

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

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 b - 4ac < 0)

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

Connecting Graphs and Equations of Quadratic Functions: demonstrate an understanding of functions, and make connections between the numeric, graphical, and algebraic representations of quadratic functions;

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 verticalline test)

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

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)

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

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 = x + 1, the domain is the set of real numbers, and the range is y ? 1)

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

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

sketch graphs of g(x) = a(x - h) + k by applying one or more transformations to the graph of f(x) = x

express the equation of a quadratic function in the standard form f(x) = ax + bx + c, given the vertex form f(x) = a(x - h) + k, and verify, using graphing technology, that these forms are equivalent representations

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

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

Solving Problems Involving Quadratic Functions: solve problems involving quadratic functions, including problems arising from real-world applications

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

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)

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)

Exponential Functions

Connecting Graphs and Equations of Exponential Functions: simplify and evaluate numerical expressions involving exponents, and make connections between the numeric, graphical, and algebraic representations of exponential functions;

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)

graph, with and without technology, an exponential relation, given its equation in the form y = ax (a > 0, a ? 1), define this relation as the function f(x) = ax, and explain why it is a function

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 (a > 0, a ? 1), function machines]

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., (1/2)3 × (1/2)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)(25) = 28]

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)

Solving Problems Involving Exponential Functions: identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications;

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

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)

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

Solving Financial Problems Involving Exponential Functions: demonstrate an understanding of compound interest and annuities, and solve related problems.

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

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)n [or FV = PV(1 + i)n]

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)n , and present value, PV = A(1 + i)?n, are exponential functions, where the number of compounding periods, n, varies]

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) [or FV = PV(1 + i) ]

explain the meaning of the term annuity, through investigation of numeric and graphical representations using technology

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)

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)

Trigonometric Functions

Applying the Sine Law and the Cosine Law in Acute Triangles: solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications;

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

solve problems involving two right triangles in two dimensions

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/sin A, b/sin B, and c/sin C in triangle ABC while dragging one of the vertices)

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

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)

Connecting Graphs and Equations of Sine Functions: demonstrate an understanding of periodic relationships and the sine function, and make connections between the numeric, graphical, and algebraic representations of sine functions;

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

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)

make connections between the sine ratio and the sine function by graphing the relationship between angles from 0º to 360º 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

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)

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)

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

sketch graphs of f(x) = a sin x, f(x) = sin x + 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

Solving Problems Involving Sine Functions: identify and represent sine functions, and solve problems involving sine functions, including problems arising from real-world applications

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

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

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

Mathematics

Earning and Purchasing

Earning: . interpret information about different types of remuneration, and solve problems and make decisions involving different remuneration methods;

gather, interpret, and compare information about the components of total earnings (e.g., salary, benefits, vacation pay, profit-sharing) in different occupations

gather, interpret, and describe information about different remuneration methods (e.g., hourly rate, overtime rate, job or project rate, commission, salary, gratuities) and remuneration schedules (e.g., weekly, biweekly, semimonthly, monthly)

describe the effects of different remuneration methods and schedules on decisions related to personal spending habits (e.g., the timing of a major purchase, the scheduling of mortgage payments and other bill payments)

solve problems, using technology (e.g., calculator, spreadsheet), and make decisions involving different remuneration methods and schedules

Describing Purchasing Power: demonstrate an understanding of payroll deductions and their impact on purchasing power;

gather, interpret, and describe information about government payroll deductions (i.e., CPP, EI, income tax) and other payroll deductions (e.g., contributions to pension plans other than CPP; union dues; charitable donations; benefit-plan contributions)

estimate and compare, using current secondary data (e.g., federal tax tables), the percent of total earnings deducted through government payroll deductions for various benchmarks (e.g., $15 000, $20 000, $25 000)

describe the relationship between gross pay, net pay, and payroll deductions (i.e., net pay is gross pay less government payroll deductions and any other payroll deductions), and estimate net pay in various situations

describe and compare the purchasing power and living standards associated with relevant occupations of interest

Purchasing: demonstrate an understanding of the factors and methods involved in making and justifying informed purchasing decisions

identify and describe various incentives in making purchasing decisions (e.g., 20% off; 1/3 off; buy 3 get 1 free; loyalty rewards; coupons; 0% financing)

make and justify a decision regarding the purchase of an item, using various criteria (e.g., extra costs, such as shipping costs and transaction fees; quality and quantity of the item; shelf life of the item; method of purchase, such as online versus local) under various circumstances (e.g., not having access to a vehicle; living in a remote community; having limited storage space)

estimate the sale price before taxes when making a purchase (e.g., estimate 25% off 1/4 of $38.99 as 25% or off of $40, giving a discount of about $10 and a sale price of approximately $30; alternatively, estimate the same sale price as about 3/4 of $40)

describe and compare a variety of strategies for estimating sales tax (e.g., estimate the sales tax on most purchases in Ontario by estimating 10% of the purchase price and adding about a third of this estimate, rather than estimating the PST and GST separately), and use a chosen strategy to estimate the after-tax cost of common items

calculate discounts, sale prices, and after-tax costs, using technology

identify forms of taxation built into the cost of an item or service (e.g., gasoline tax, tire tax)

estimate the change from an amount offered to pay a charge

make the correct change from an amount offered to pay a charge, using currency manipulatives

compare the unit prices of related items to help determine the best buy

describe and compare, for different types of transactions, the extra costs that may be associated with making purchases (e.g., interest costs, exchange rates, shipping and handling costs, customs duty, insurance)

Saving, Investing, and Borrowing

Comparing Financial Services: describe and compare services available from financial institutions;

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)

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

read and interpret transaction codes and entries from various financial statements (e.g., bank statement, credit card statement, passbook, automated banking machine printout, online banking statement, account activity report), and explain ways of using the information to manage personal finances

Saving and Investing: demonstrate an understanding of simple and compound interest, and solve problems involving related applications;

determine, through investigation using technology (e.g., calculator, spreadsheet), the effect on simple interest of changes in the principal, interest rate, or time, and solve problems involving applications of simple interest

determine, through investigation using technology, the compound interest for a given investment, using repeated calculations of simple interest for no more than 6 compounding periods

describe the relationship between simple interest and compound interest in various ways (i.e., orally, in writing, using tables and graphs)

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 of changing the total length of time, the interest rate, or the compounding period

solve problems, using technology, that involve applications of compound interest to saving and investing

Borrowing: . interpret information about different ways of borrowing and their associated costs, and make and justify informed borrowing decisions.

gather, interpret, and compare information about the effects of carrying an outstanding balance on a credit card at current interest rates

gather, interpret, and compare information describing the features (e.g., interest rates, flexibility) and conditions (e.g., eligibility, required collateral) of various personal loans (e.g., student loan, car loan, "no interest" deferred-payment loan, loan to consolidate debt, loan drawn on a line of credit, payday or bridging loan)

calculate, using technology (e.g., calculator, spreadsheet), the total interest paid over the life of a personal loan, given the principal, the length of the loan, and the periodic payments, and use the calculations to justify the choice of a personal loan

determine, using a variety of tools (e.g., spreadsheet template, online amortization tables), the effect of the length of time taken to repay a loan on the principal and interest components of a personal loan repayment

compare, using a variety of tools (e.g., spreadsheet template, online amortization tables), the effects of various payment periods (e.g., monthly, biweekly) on the length of time taken to repay a loan and on the total interest paid

gather and interpret information about credit ratings, and describe the factors used to determine credit ratings and the consequences of a good or bad rating

make and justify a decision to borrow, using various criteria (e.g., income, cost of borrowing, availability of an item, need for an item) under various circumstances (e.g., having a large existing debt, wanting to pursue an education or training opportunity, needing transportation to a new job, wanting to set up a business)

Transportation and Travel

Owning and Operating a Vehicle: interpret information about owning and operating a vehicle, and solve problems involving the associated costs;

gather and interpret information about the procedures (e.g., in the graduated licensing system) and costs (e.g., driver training; licensing fees) involved in obtaining an Ontario driver's licence, and the privileges and restrictions associated with having a driver's licence

gather and describe information about the procedures involved in buying or leasing a new vehicle or buying a used vehicle

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

gather and interpret information about the costs (e.g., monthly payments, insurance, depreciation, maintenance, miscellaneous expenses) of purchasing or leasing a new vehicle or purchasing a used vehicle, and describe the conditions that favour each alternative

describe ways of failing to operate a vehicle responsibly (e.g., lack of maintenance, careless driving) and possible financial and non-financial consequences (e.g., legal costs, fines, higher insurance rates, demerit points, loss of driving privileges)

identify and describe costs (e.g., gas consumption, depreciation, insurance, maintenance) and benefits (e.g., convenience, increased profit) of owning and operating a vehicle for business

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

Travelling by Automobile: plan and justify a route for a trip by automobile, and solve problems involving the associated costs;

determine distances represented on maps (e.g., provincial road map, local street map, Web-based maps), using given scales

plan and justify, orally or in writing, a route for a trip by automobile on the basis of a variety of factors (e.g., distances involved, the purpose of the trip, the time of year, the time of day, probable road conditions, personal priorities)

report, orally or in writing, on the estimated costs (e.g., gasoline, accommodation, food, entertainment, tolls, car rental) involved in a trip by automobile, using information from available sources (e.g., automobile association travel books, travel guides, the Internet)

solve problems involving the cost of travelling by automobile for personal or business purposes

Comparing Modes of Transportation: interpret information about different modes of transportation, and solve related problems.

gather, interpret, and describe information about the impact (e.g., monetary, health, environmental) of daily travel (e.g., to work and/or school), using available means (e.g., car, taxi, motorcycle, public transportation, bicycle, walking)

gather, interpret, and compare information about the costs (e.g., insurance, extra charges based on distance travelled) and conditions (e.g., one-way or return, drop-off time and location, age of the driver, required type of driver's licence) involved in renting a car, truck, or trailer, and use the information to justify a choice of rental vehicle

gather, interpret, and describe information regarding routes, schedules, and fares for travel by airplane, train, or bus

solve problems involving the comparison of information concerning transportation by airplane, train, bus, and automobile in terms of various factors (e.g., cost, time, convenience)

Mathematics

Exponential and Logarithmic Functions

Evaluating Logarithmic Expressions: demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;

recognize the logarithm of a number to a given base as the exponent to which the base must be raised to get the number, recognize the operation of finding the logarithm to be the inverse operation (i.e., the undoing or reversing) of exponentiation, and evaluate simple logarithmic expressions

determine, with technology, the approximate logarithm of a number to any base, including base 10 (e.g., by reasoning that log 29 is between 3 and 4 and using systematic trial to determine that log 29 is approximately 3.07)

make connections between related logarithmic and exponential equations (e.g., log5125 = 3 can also be expressed as 53 = 125), and solve simple exponential equations by rewriting them in logarithmic form (e.g., solving 3x = 10 by rewriting the equation as log3 10 = x)

make connections between the laws of exponents and the laws of logarithms [e.g., use the statement 10a + b = 10a 10b to deduce that log10 x + log10 y = log10 (xy)], verify the laws of logarithms with or without technology (e.g., use patterning to verify the quotient law for logarithms by evaluating expressions such as log101000 - log10100 and then rewriting the answer as a logarithmic term to the same base), and use the laws of logarithms to simplify and evaluate numerical expressions

Connecting Graphs and Equations of Logarithmic Functions: identify and describe some key features of the graphs of logarithmic functions, make connections among the numeric, graphical, and algebraic representations of logarithmic functions, and solve related problems graphically;

determine, through investigation with technology (e.g., graphing calculator, spreadsheet) and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, increasing/decreasing behaviour) of the graphs of logarithmic functions of the form f(x) = log x, and make connections between the algebraic and graphical representations of these logarithmic functions

recognize the relationship between an exponential function and the corresponding logarithmic function to be that of a function and its inverse, deduce that the graph of a logarithmic function is the reflection of the graph of the corresponding exponential function in the line y = x, and verify the deduction using technology

determine, through investigation using technology, the roles of the parameters d and c in functions of the form y = log10(x - d) + c and the roles of the parameters a and k in functions of the form y = alog10(kx), and describe these roles in terms of transformations on the graph of f(x) = log10x (i.e., vertical and horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x-and y-axes)

pose problems based on real-world applications of exponential and logarithmic functions (e.g., exponential growth and decay, the Richter scale, the pH scale, the decibel scale), 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

Solving Exponential and Logarithmic Equations: solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.

recognize equivalent algebraic expressions involving logarithms and exponents, and simplify expressions of these types

solve exponential equations in one variable by determining a common base (e.g., solve 4x = 8x + 3 by expressing each side as a power of 2) and by using logarithms (e.g., solve 4x = 8x + 3 by taking the logarithm base 2 of both sides), recognizing that logarithms base 10 are commonly used (e.g., solving 3x = 7 by taking the logarithm base 10 of both sides)

solve simple logarithmic equations in one variable algebraically [e.g., log (5x + 6) = 2, log (x + 1) = 1]

solve problems involving exponential and logarithmic equations algebraically, including problems arising from real-world applications

Trigonometric Functions

Understanding and Applying Radian Measure: demonstrate an understanding of the meaning and application of radian measure;

recognize the radian as an alternative unit to the degree for angle measurement, define the radian measure of an angle as the length of the arc that subtends this angle at the centre of a unit circle, and develop and apply the relationship between radian and degree measure

represent radian measure in terms of ? (e.g., ?/3 radians, 2? radians) and as a rational number (e.g., 1.05 radians, 6.28 radians)

determine, with technology, the primary trigonometric ratios (i.e., sine, cosine, tangent) and the reciprocal trigonometric ratios (i.e., cosecant, secant, cotangent) of angles expressed in radian measure

determine, without technology, the exact values of the primary trigonometric ratios and the reciprocal trigonometric ratios for the special angles 0,?/6, ?/4, ?/3, ?/2, and their multiples less than or equal to 2?

Connecting Graphs and Equations of Trigonometric Functions: make connections between trigonometric ratios and the graphical and algebraic representations of the corresponding trigonometric functions and between trigonometric functions and their reciprocals, and use these connections to solve problems;

sketch the graphs of f(x) = sin x and f(x) = cos x for angle measures expressed in radians, and determine and describe some key properties (e.g., period of 2?, amplitude of 1) in terms of radians

make connections between the tangent ratio and the tangent function by using technology to graph the relationship between angles in radians and their tangent ratios and defining this relationship as the function f(x) = tan x, and describe key properties of the tangent function

graph, with technology and using the primary trigonometric functions, the reciprocal trigonometric functions (i.e., cosecant, secant, cotangent) for angle measures expressed in radians, determine and describe key properties of the reciprocal functions (e.g., state the domain, range, and period, and identify and explain the occurrence of asymptotes), and recognize notations used to represent the reciprocal functions (e.g., the reciprocal of f(x) = sinx can be represented using csc x, 1/f(x), or 1/sinx, but not using f-1(x) or sin-1x, which represent the inverse function)

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, with angles expressed in radians

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 with angles expressed in radians, and state the period, amplitude, and phase shift of the transformed functions

represent a sinusoidal function with an equation, given its graph or its properties, with angles expressed in radians

pose problems based on applications involving a trigonometric function with domain expressed in radians (e.g., seasonal changes in temperature, heights of tides, hours of daylight, displacements for oscillating springs), and solve these and other such problems by using a given graph or a graph generated with or without technology from a table of values or from its equation

Solving Trigonometric Equations: solve problems involving trigonometric equations and prove trigonometric identities.

recognize equivalent trigonometric expressions [e.g., by using the angles in a right triangle to recognize that sin x and cos (?/2 - x) are equivalent; by using transformations to recognize that cos (x + ?/2) and - sin x are equivalent], and verify equivalence using graphing technology

explore the algebraic development of the compound angle formulas (e.g., verify the formulas in numerical examples, using technology; follow a demonstration of the algebraic development [student reproduction of the development of the general case is not required]), and use the formulas to determine exact values of trigonometric ratios [e.g., determining the exact value of sin (?/12) by first rewriting it in terms of special angles as sin (?/4 - ?/6)]

recognize that trigonometric identities are equations that are true for every value in the domain (i.e., a counter-example can be used to show that an equation is not an identity), prove trigonometric identities through the application of reasoning skills, using a variety of relationships (e.g., tan x = sin x/cos x; sin2x + cos2x = 1; the reciprocal identities; the compound angle formulas), and verify identities using technology

solve linear and quadratic trigonometric equations, with and without graphing technology, for the domain of real values from 0 to 2?, and solve related problems

Polynomialand Rational Functions

Connecting Graphs and Equations of Polynomial Functions: identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions;

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 x - 5x + 2x - 1); recognize the equation of a polynomial function, give reasons why it is a function, and identify linear and quadratic functions as examples of polynomial functions

compare, through investigation using graphing technology, the numeric, graphical, and algebraic representations of polynomial (i.e., linear, quadratic, cubic, quartic) functions (e.g., compare finite differences in tables of values; 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)

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)

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

make connections, through investigation using graphing technology (e.g., dynamic geometry software), between a polynomial function given in factored form [e.g., f(x) = 2(x - 3)(x + 2)(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)

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) = x3 and f(x) = x4 (i.e., vertical and horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x-and y-axes)

determine an equation of a polynomial function that satisfies a given set of conditions (e.g., degree of the polynomial, intercepts, points on the function), using methods appropriate to the situation (e.g., using the x-intercepts of the function; using a trial-and-error process with a graphing calculator or graphing software; using finite differences), and recognize that there may be more than one polynomial function that can satisfy a given set of conditions (e.g., an infinite number of polynomial functions satisfy the condition that they have three given x-intercepts)

determine the equation of the family of polynomial functions with a given set of zeros and of the member of the family that passes through another given point [e.g., a family of polynomial functions of degree 3 with zeros 5, -3, and -2 is defined by the equation f(x) = k(x - 5)(x + 3)(x + 2), where k is a real number, k ? 0; the member of the family that passes through (-1, 24) is f(x) = -2(x - 5)(x + 3)(x + 2)

determine, through investigation, and compare the properties of even and odd polynomial functions [e.g., symmetry about the y-axis or the origin; the power of each term; the number of x-intercepts; f(x) = f(– x) or ff(– x) = – f(x)], and determine whether a given polynomial function is even, odd, or neither

Connecting Graphs and Equations of Rational Functions: identify and describe some key features of the graphs of rational functions, and represent rational functions graphically;

determine, through investigation with and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals) of the graphs of rational functions that are the reciprocals of linear and quadratic functions, and make connections between the algebraic and graphical representations of these rational functions [e.g., make connections between f(x) = 1/[x2 - 4] and its graph by using graphing technology and by reasoning that there are vertical asymptotes at x = 2 and x = -2 and a horizontal asymptote at y = 0 and that the function maintains the same sign as f(x) = x2 - 4]

determine, through investigation with and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals) of the graphs of rational functions that have linear expressions in the numerator and denominator [e.g., f(x) = 2x/[x - 3], h(x) = x - 2/(3x + 4)], and make connections between the algebraic and graphical representations of these rational functions

sketch the graph of a simple rational function using its key features, given the algebraic representation of the function

Solving Polynomial and Rational Equations: solve problems involving polynomial and simple rational equations graphically and algebraically;

make connections, through investigation using technology (e.g., computer algebra systems), between the polynomial function f(x), the divisor x - a, the remainder from the division f(x)/[x - a], and f(a) to verify the remainder theorem and the factor theorem

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, factoring by grouping, remainder theorem, factor theorem)

determine, through investigation using technology (e.g., graphing calculator, computer algebra systems), the connection between the real roots of a polynomial equation and the x-intercepts of the graph of the corresponding polynomial function, and describe this connection [e.g., the real roots of the equation x4 - 13x2 + 36 = 0 are the x-intercepts of the graph of f(x) = x4 - 13x2 + 36]

solve polynomial equations in one variable, of degree no higher than four (e.g., 2x - 3x + 8x - 12 = 0), by selecting and applying strategies (i.e., common factoring, difference of squares, trinomial factoring, factoring by grouping, remainder theorem, factor theorem), and verify solutions using technology (e.g., using computer algebra systems to determine the roots; using graphing technology to determine the x intercepts of the graph of the corresponding polynomial function)

determine, through investigation using technology (e.g., graphing calculator, computer algebra systems), the connection between the real roots of a rational equation and the x-intercepts of the graph of the corresponding rational function, and describe this connection (e.g., the real root of the equation [x - 2]/[x - 3] = 0 is 2, which is the x-intercept of the function f(x) = [x - 2]/[x - 3]; the equation 1/[x - 3] = 0 has no real roots, and the function f(x) = 1/[x - 3] does not intersect the x-axis)

solve simple rational equations in one variable algebraically, and verify solutions using technology (e.g., using computer algebra systems to determine the roots; using graphing technology to determine the x-intercepts of the graph of the corresponding rational function)

solve problems involving applications of polynomial and simple rational functions and equations [e.g., problems involving the factor theorem or remainder theorem, such as determining the values of k for which the function f(x) = x3 + 6x2 + kx - 4 gives the same remainder when divided by x - 1 and x + 2]

Solving Inequalities: demonstrate an understanding of solving polynomial and simple rational inequalities.

explain, for polynomial and simple rational functions, the difference between the solution to an equation in one variable and the solution to an inequality in one variable, and demonstrate that given solutions satisfy an inequality (e.g., demonstrate numerically and graphically that the solution to 1/x + 1 < 5 is x < -1 or x > -4/5)

determine solutions to polynomial inequalities in one variable [e.g., solve f(x) ≥ 0, where f(x) = x – x + 3x – 9] and to simple rational inequalities in one variable by graphing the corresponding functions, using graphing technology, and identifying intervals for which x satisfies the inequalities

solve linear inequalities and factorable polynomial inequalities in one variable (e.g., x3 + x2 > 0) in a variety of ways (e.g., by determining intervals using x-intercepts and evaluating the corresponding function for a single x-value within each interval; by factoring the polynomial and identifying the conditions for which the product satisfies the inequality), and represent the solutions on a number line or algebraically (e.g., for the inequality x4 - 5x2 + 4 < 0, the solution represented algebraically is - 2 < x < -1 or 1 < x < 2)

Characteristics of Functions

Understanding Rates of Change: demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate of change of a function at a given point;

gather, interpret, and describe information about real-world applications of rates of change, and recognize different ways of representing rates of change (e.g., in words, numerically, graphically, algebraically)

recognize that the rate of change for a function is a comparison of changes in the dependent variable to changes in the independent variable, and distinguish situations in which the rate of change is zero, constant, or changing by examining applications, including those arising from real-world situations (e.g., rate of change of the area of a circle as the radius increases, inflation rates, the rising trend in graduation rates among Aboriginal youth, speed of a cruising aircraft, speed of a cyclist climbing a hill, infection rates)

sketch a graph that represents a relationship involving rate of change, as described in words, and verify with technology (e.g., motion sensor) when possible

calculate and interpret average rates of change of functions (e.g., linear, quadratic, exponential, sinusoidal) arising from real-world applications (e.g., in the natural, physical, and social sciences), given various representations of the functions (e.g., tables of values, graphs, equations)

recognize examples of instantaneous rates of change arising from real-world situations, and make connections between instantaneous rates of change and average rates of change (e.g., an average rate of change can be used to approximate an instantaneous rate of change)

determine, through investigation using various representations of relationships (e.g., tables of values, graphs, equations), approximate instantaneous rates of change arising from real-world applications (e.g., in the natural, physical, and social sciences) by using average rates of change and reducing the interval over which the average rate of change is determined

make connections, through investigation, between the slope of a secant on the graph of a function (e.g., quadratic, exponential, sinusoidal) and the average rate of change of the function over an interval, and between the slope of the tangent to a point on the graph of a function and the instantaneous rate of change of the function at that point Sample problem: Use tangents to investigate the behaviour of a function when the instantaneous rate of change is zero, positive, or negative.

determine, through investigation using a variety of tools and strategies (e.g., using a table of values to calculate slopes of secants or graphing secants and measuring their slopes with technology), the approximate slope of the tangent to a given point on the graph of a function (e.g., quadratic, exponential, sinusoidal) by using the slopes of secants through the given point (e.g., investigating the slopes of secants that approach the tangent at that point more and more closely), and make connections to average and instantaneous rates of change

solve problems involving average and instantaneous rates of change, including problems arising from real-world applications, by using numerical and graphical methods (e.g., by using graphing technology to graph a tangent and measure its slope)

Combining Functions: determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, describe some properties of the resulting functions, and solve related problems;

determine, through investigation using graphing technology, key features (e.g., domain, range, maximum/minimum points, number of zeros) of the graphs of functions created by adding, subtracting, multiplying, or dividing functions [e.g., f(x) = 2-x sin4x, g(x) = x2 + 2x, h(x) = sin x/cos x], and describe factors that affect these properties

recognize real-world applications of combinations of functions (e.g., the motion of a damped pendulum can be represented by a function that is the product of a trigonometric function and an exponential function; the frequencies of tones associated with the numbers on a telephone involve the addition of two trigonometric functions), and solve related problems graphically

determine, through investigation, and explain some properties (i.e., odd, even, or neither; increasing/decreasing behaviours) of functions formed by adding, subtracting, multiplying, and dividing general functions [e.g., f(x) + g(x), f(x)g(x)]

determine the composition of two functions [i.e., f(g(x))] numerically (i.e., by using a table of values) and graphically, with technology, for functions represented in a variety of ways (e.g., function machines, graphs, equations), and interpret the composition of two functions in real-world applications

determine algebraically the composition of two functions [i.e., f(g(x))], verify that f(g(x)) is not always equal to g( f(x)) [e.g., by determining f(g(x)) and g( f(x)), given f(x) = x + 1 and g(x) = 2x], and state the domain [i.e., by defining f(g(x)) for those x-values for which g(x) is defined and for which it is included in the domain of f(x)] and the range of the composition of two functions

solve problems involving the composition of two functions, including problems arising from real-world applications

demonstrate, by giving examples for functions represented in a variety of ways (e.g., function machines, graphs, equations), the property that the composition of a function and its inverse function maps a number onto itself (i.e., f-1(f(x)) = x and f(f-1(x)) = x demonstrate that the inverse function is the reverse process of the original function and that it undoes what the function does)

make connections, through investigation using technology, between transformations (i.e., vertical and horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes) of simple functions f(x) [e.g., f(x) = x + 20, f(x) = sin x, f(x) = log x] and the composition of these functions with a linear function of the form g(x) = A(x + B)

Using Function Models to Solve Problems: compare the characteristics of functions, and solve problems by modelling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques.

compare, through investigation using a variety of tools and strategies (e.g., graphing with technology; comparing algebraic representations; comparing finite differences in tables of values) the characteristics (e.g., key features of the graphs, forms of the equations) of various functions (i.e., polynomial, rational, trigonometric, exponential, logarithmic)

solve graphically and numerically equations and inequalities whose solutions are not accessible by standard algebraic techniques

solve problems, using a variety of tools and strategies, including problems arising from real-world applications, by reasoning with functions and by applying concepts and procedures involving functions (e.g., by constructing a function model from data, using the model to determine mathematical results, and interpreting and communicating the results within the context of the problem)

Mathematics

Rate of Change

Investigating Instantaneous Rate of Change at a Point: demonstrate an understanding of rate of change by making connections between average rate of change over an interval and instantaneous rate of change at a point, using the slopes of secants and tangents and the concept of the limit;

describe examples of real-world applications of rates of change, represented in a variety of ways (e.g., in words, numerically, graphically, algebraically)

describe connections between the average rate of change of a function that is smooth (i.e., continuous with no corners) over an interval and the slope of the corresponding secant, and between the instantaneous rate of change of a smooth function at a point and the slope of the tangent at that point

make connections, with or without graphing technology, between an approximate value of the instantaneous rate of change at a given point on the graph of a smooth function and average rates of change over intervals containing the point (i.e., by using secants through the given point on a smooth curve to approach the tangent at that point, and determining the slopes of the approaching secants to approximate the slope of the tangent)

recognize, through investigation with or without technology, graphical and numerical examples of limits, and explain the reasoning involved (e.g., the value of a function approaching an asymptote, the value of the ratio of successive terms in the Fibonacci sequence)

make connections, for a function that is smooth over the interval a ? x ? a + h, between the average rate of change of the function over this interval and the value of the expression f(a + h) - f(a)/h , and between the instantaneous rate of change of the function at x = a and the value of the limit lim[h -> 0] f(a + h) - f(a)/h

compare, through investigation, the calculation of instantaneous rates of change at a point (a, f(a)) for polynomial functions [e.g., f(x) = x2, f(x) = x3], with and without simplifying the expression []f(a + h) - f(a)]/h before substituting values of h that approach zero (e.g., for f(x) = x2 at x = 3, by determining [f(3 + 1) - f(3)]/1 = 7, [f(3 + 0.1) - f(3)]/0.1 = 6.1, [f(3 + 0.01) - f(3)]/0.01 = 6.01, and [f(3 + 0.001) - f(3)]/0.001 = 6.001, and by first simplifying [f(3 + h) - f(3)]/h as [(3 + h)2 - 32]/h = 6 + h and then substituting the same values of h to give the same results)

Investigating the Concept of the Derivative Function: graph the derivatives of polynomial, sinusoidal, and exponential functions, and make connections between the numeric, graphical, and algebraic representations of a function and its derivative;

determine numerically and graphically the intervals over which the instantaneous rate of change is positive, negative, or zero for a function that is smooth over these intervals (e.g., by using graphing technology to examine the table of values and the slopes of tangents for a function whose equation is given; by examining a given graph), and describe the behaviour of the instantaneous rate of change at and between local maxima and minima

generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function, f(x), for various values of x (e.g., construct a tangent to the function, measure its slope, and create a slider or animation to move the point of tangency), graph the ordered pairs, recognize that the graph represents a function called the derivative, f'(x) or dy/dx, and make connections between the graphs of f(x) and f'(x) or y and dy/dx [e.g., when f(x) is linear, f'(x) is constant; when f(x) is quadratic, f'(x) is linear; when f(x) is cubic, f'(x) is quadratic]

determine the derivatives of polynomial functions by simplifying the algebraic expression [f(x + h) - f(x)]/h and then taking the limit of the simplified expression as h approaches zero [i.e., determining lim[h -> 0] [f(x + h) - f(x)]/h]

determine, through investigation using technology, the graph of the derivative f'(x)or dy/dx of a given sinusoidal function [i.e., f(x) = sin x, f(x) = cos x] (e.g., by generating a table of values showing the instantaneous rate of change of the function for various values of x and graphing the ordered pairs; by using dynamic geometry software to verify graphically that when f(x) = sin x, f'(x) = cos x, and when f(x) = cos x, f'(x) = -sin x; by using a motion sensor to compare the displacement and velocity of a pendulum)

determine, through investigation using technology, the graph of the derivative f'(x)or dy/dx of a given exponential function [i.e., f(x) = a(x) (a > 0, a ? 1)] [e.g., by generating a table of values showing the instantaneous rate of change of the function for various values of x and graphing the ordered pairs; by using dynamic geometry software to verify that when f(x) = ax, f'(x) = kf(x)], and make connections between the graphs of f(x) and f'(x) or y and dy/dx [e.g., f(x) and f'(x) are both exponential; the ratio f'(x)/f(x) is constant, or f'(x) = kf(x); f'(x) is a vertical stretch from the x-axis of f(x)]

determine, through investigation using technology, the exponential function f(x) = ax (a > 0, a ? 1) for which f'(x) = f(x) (e.g., by using graphing technology to create a slider that varies the value of a in order to determine the exponential function whose graph is the same as the graph of its derivative), identify the number e to be the value of a for which f'(x) = f(x) [i.e., given f(x) = ex, f(x) = ex], and recognize that for the exponential function f(x) = ex the slope of the tangent at any point on the function is equal to the value of the function at that point

recognize that the natural logarithmic function f(x) = logex, also written as f(x) = ln x, is the inverse of the exponential function f(x) = ex, and make connections between f(x) = ln x and f(x) = ex [e.g., f(x) = ln x reverses what f(x) = ex does; their graphs are reflections of each other in the line y = x; the composition of the two functions, elnx or ln ex , maps x onto itself, that is, elnx = x and ln ex = x]

verify, using technology (e.g., calculator, graphing technology), that the derivative of the exponential function f(x) = ax is f'(x) = ax ln a for various values of a (e.g., verifying numerically for f(x) = 2x that f'(x) = 2x ln 2 by using a calculator to show lim[h -> 0] (2h - 1)/h that is ln 2 or by graphing f(x) = 2x, determining the value of the slope and the value of the function for specific x-values, and comparing the ratio f'(x)/f(x) with ln 2)

Investigating the Properties of Derivatives: verify graphically and algebraically the rules for determining derivatives; apply these rules to determine the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions, and simple combinations of functions; and solve related problems.

verify the power rule for functions of the form f(x) = xn, where n is a natural number (e.g., by determining the equations of the derivatives of the functions f(x) = x, f(x) = x2, f(x) = x3, and f(x) = x4 algebraically using lim[h -> 0] [f(x + h) - f(x)]/h and graphically using slopes of tangents)

verify the constant, constant multiple, sum, and difference rules graphically and numerically [e.g., by using the function g(x) = kf(x) and comparing the graphs of g'(x) and kf'(x); by using a table of values to verify that f'(x) + g'(x) = (f + g)'(x), given f(x) = x and g(x) = 3x], and read and interpret proofs involving lim[h -> 0] [f(x + h) - f(x)]/h of the constant, constant multiple, sum, and difference rules (student reproduction of the development of the general case is not required)

determine algebraically the derivatives of polynomial functions, and use these derivatives to determine the instantaneous rate of change at a point and to determine point(s) at which a given rate of change occurs

verify that the power rule applies to functions of the form f(x) = xn, where n is a rational number [e.g., by comparing values of the slopes of tangents to the function f(x) = x1/2 with values of the derivative function determined using the power rule], and verify algebraically the chain rule using monomial functions [e.g., by determining the same derivative for f(x) = [5x3]1/3 by using the chain rule and by differentiating the simplified form, f(x) = 51/3x] and the product rule using polynomial functions (e.g., by determining the same derivative for f(x) = (3x + 2)(2x2 - 1) by using the product rule and by differentiating the expanded form f(x) = 6x3 + 4x2 - 3x - 2)

solve problems, using the product and chain rules, involving the derivatives of polynomial functions, sinusoidal functions, exponential functions, rational functions [e.g., by expressing f(x) = [x2 + 1]/[x - 1] as the product f(x) = (x2 + 1)(x - 1)-1], radical functions [e.g., by expressing f(x) = sqrt(x2 + 5) as the power f(x) = (x2 + 5)1/2], and other simple combinations of functions [e.g., f(x) = x sin x, f(x) = sin x/cos x]

Derivatives and their Applications

Connecting Graphs and Equations of Functions and Their Derivatives: make connections, graphically and algebraically, between the key features of a function and its first and second derivatives, and use the connections in curve sketching;

sketch the graph of a derivative function, given the graph of a function that is continuous over an interval, and recognize points of inflection of the given function (i.e., points at which the concavity changes)

recognize the second derivative as the rate of change of the rate of change (i.e., the rate of change of the slope of the tangent), and sketch the graphs of the first and second derivatives, given the graph of a smooth function

determine algebraically the equation of the second derivative f"(x) of a polynomial or simple rational function f(x), and make connections, through investigation using technology, between the key features of the graph of the function (e.g., increasing/ decreasing intervals, local maxima and minima, points of inflection, intervals of concavity) and corresponding features of the graphs of its first and second derivatives (e.g., for an increasing interval of the function, the first derivative is positive; for a point of inflection of the function, the slopes of tangents change their behaviour from increasing to decreasing or from decreasing to increasing, the first derivative has a maximum or minimum, and the second derivative is zero)

describe key features of a polynomial function, given information about its first and/or second derivatives (e.g., the graph of a derivative, the sign of a derivative over specific intervals, the x-intercepts of a derivative), sketch two or more possible graphs of the function that are consistent with the given information, and explain why an infinite number of graphs is possible

sketch the graph of a polynomial function, given its equation, by using a variety of strategies (e.g., using the sign of the first derivative; using the sign of the second derivative; identifying even or odd functions) to determine its key features (e.g., increasing/ decreasing intervals, intercepts, local maxima and minima, points of inflection, intervals of concavity), and verify using technology

Solving Problems Using Mathematical Models and Derivatives:

make connections between the concept of motion (i.e., displacement, velocity, acceleration) and the concept of the derivative in a variety of ways (e.g., verbally, numerically, graphically, algebraically)

make connections between the graphical or algebraic representations of derivatives and real-world applications (e.g., population and rates of population change, prices and inflation rates, volume and rates of flow, height and growth rates)

solve problems, using the derivative, that involve instantaneous rates of change, including problems arising from real-world applications (e.g., population growth, radioactive decay, temperature changes, hours of daylight, heights of tides), given the equation of a function*

solve optimization problems involving polynomial, simple rational, and exponential functions drawn from a variety of applications, including those arising from real-world situations

solve problems arising from real-world applications by applying a mathematical model and the concepts and procedures associated with the derivative to determine mathematical results, and interpret and communicate the results

Geometry and Algebra of Vectors

Representing Vectors Geometrically and Algebraically: demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications;

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)

represent a vector in two-space geometrically as a directed line segment, with directions expressed in different ways (e.g., 320°; N 40° W), and algebraically (e.g., using Cartesian coordinates; using polar coordinates), and recognize vectors with the same magnitude and direction but different positions as equal vectors

determine, using trigonometric relationships [e.g., x = rcosθ, y = rsinθ, θ = tan-1 (y/x) or tan-1 (y/x) + 180°, r = sqrt(x2 + y2)], the Cartesian representation of a vector in two-space given as a directed line segment, or the representation as a directed line segment of a vector in two-space given in Cartesian form [e.g., representing the vector (8, 6) as a directed line segment]

recognize that points and vectors in three-space can both be represented using Cartesian coordinates, and determine the distance between two points and the magnitude of a vector using their Cartesian representations

Operating With Vectors: perform operations on vectors in two-space and three-space, and use the properties of these operations to solve problems, including those arising from real-world applications;

perform the operations of addition, subtraction, and scalar multiplication on vectors represented as directed line segments in twospace, and on vectors represented in Cartesian form in two-space and three-space

determine, through investigation with and without technology, some properties (e.g., commutative, associative, and distributive properties) of the operations of addition, subtraction, and scalar multiplication of vectors

solve problems involving the addition, subtraction, and scalar multiplication of vectors, including problems arising from real-world applications

perform the operation of dot product on two vectors represented as directed line segments (i.e., using [vector]a [vector]b = [vector]a vector]b cos?) and in Cartesian form (i.e., using [vector]a [vector]b = a1b1 + a2b2 or [vector]a [vector]b = a1b1 + a2b2 + a3b3) in two-space and three-space, and describe applications of the dot product (e.g., determining the angle between two vectors; determining the projection of one vector onto another)

determine, through investigation, properties of the dot product (e.g., investigate whether it is commutative, distributive, or associative; investigate the dot product of a vector with itself and the dot product of orthogonal vectors)

perform the operation of cross product on two vectors represented in Cartesian form in three-space [i.e., using [vector]a x [vector]b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)], determine the magnitude of the cross product (i.e., using [vector]a x [vector]b vector]a [vector]b sin?), and describe applications of the cross product (e.g., determining a vector orthogonal to two given vectors; determining the turning effect [or torque] when a force is applied to a wrench at different angles)

determine, through investigation, properties of the cross product (e.g., investigate whether it is commutative, distributive, or associative; investigate the cross product of collinear vectors)

solve problems involving dot product and cross product (e.g., determining projections, the area of a parallelogram, the volume of a parallelepiped), including problems arising from real-world applications (e.g., determining work, torque, ground speed, velocity, force)

Describing Lines and Planes Using Linear Equations: distinguish between the geometric representations of a single linear equation or a system of two linear equations in two-space and three-space, and determine different geometric configurations of lines and planes in three-space;

recognize that the solution points (x, y) in two-space of a single linear equation in two variables form a line and that the solution points (x, y) in two-space of a system of two linear equations in two variables determine the point of intersection of two lines, if the lines are not coincident or parallel

determine, through investigation with technology (i.e., 3-D graphing software) and without technology, that the solution points (x, y, z) in three-space of a single linear equation in three variables form a plane and that the solution points (x, y, z) in three-space of a system of two linear equations in three variables form the line of intersection of two planes, if the planes are not coincident or parallel

determine, through investigation using a variety of tools and strategies (e.g., modelling with cardboard sheets and drinking straws; sketching on isometric graph paper), different geometric configurations of combinations of up to three lines and/or planes in three-space (e.g., two skew lines, three parallel planes, two intersecting planes, an intersecting line and plane); organize the configurations based on whether they intersect and, if so, how they intersect (i.e., in a point, in a line, in a plane)

Describing Lines and Planes Using Scalar, Vector, and Parametric Equations: represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and intersections.

recognize a scalar equation for a line in two-space to be an equation of the form Ax + By + C = 0, represent a line in two-space using a vector equation (i.e., r = r0 + tm) and parametric equations, and make connections between a scalar equation, a vector equation, and parametric equations of a line in two-space

recognize that a line in three-space cannot be represented by a scalar equation, and represent a line in three-space using the scalar equations of two intersecting planes and using vector and parametric equations (e.g., given a direction vector and a point on the line, or given two points on the line)

recognize a normal to a plane geometrically (i.e., as a vector perpendicular to the plane) and algebraically [e.g., one normal to the plane 3x + 5y – 2z = 6 is (3, 5, –2)], and determine, through investigation, some geometric properties of the plane (e.g., the direction of any normal to a plane is constant; all scalar multiples of a normal to a plane are also normals to that plane; three non-collinear points determine a plane; the resultant, or sum, of any two vectors in a plane also lies in the plane)

recognize a scalar equation for a plane in three-space to be an equation of the form Ax + By + Cz + D = 0 whose solution points make up the plane, determine the intersection of three planes represented using scalar equations by solving a system of three linear equations in three unknowns algebraically (e.g., by using elimination or substitution), and make connections between the algebraic solution and the geometric configuration of the three planes

determine, using properties of a plane, the scalar, vector, and parametric equations of a plane

determine the equation of a plane in its scalar, vector, or parametric form, given another of these forms

solve problems relating to lines and planes in three-space that are represented in a variety of ways (e.g., scalar, vector, parametric equations) and involving distances (e.g., between a point and a plane; between two skew lines) or intersections (e.g., of two lines, of a line and a plane), and interpret the result geometrically

Mathematics

Mathematical Models

Solving Exponential Equations: evaluate powers with rational exponents, simplify algebraic expressions involving exponents, and solve problems involving exponential equations graphically and using common bases;

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/x5] and the exponent law for simplifying algebraic expressions involving a power of a power [e.g. (x6y3)2]

simplify algebraic expressions containing integer exponents using the laws of exponents

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., xm/n, where x > 0 and m and n are integers)

evaluate, with or without technology, numerical expressions involving rational exponents and rational bases [e.g., 2-3, (-6)3, 41/2, 1.01120]

solve simple exponential equations numerically and graphically, with technology (e.g., use systematic trial with a scientific calculator to determine the solution to the equation 1.05 = 1.276), and recognize that the solutions may not be exact

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) , where h represents the height of a bouncing ball and n represents the number of bounces]

solve exponential equations in one variable by determining a common base (e.g., 2x = 32, 45x - 1 = 22(x + 11), 35x + 8 = 27x)

Modelling Graphically: 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;

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

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)

recognize that graphs and tables of values communicate information about rate of change, and use a given graph or table of values for a relation to identify the units used to measure rate of change (e.g., for a distance-time graph, the units of rate of change are kilometres per hour; for a table showing earnings over time, the units of rate of change are dollars per hour)

identify when the rate of change is zero, constant, or changing, given a table of values or a graph of a relation, and compare two graphs by describing rate of change (e.g., compare distance-time graphs for a car that is moving at constant speed and a car that is accelerating)

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)

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

Modelling Algebraically: 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.

solve equations of the form xn = a using rational exponents (e.g., solve x3 = 7 by raising both sides to the exponent 1/3)

determine the value of a variable of degree no higher than three, using a formula drawn from an application, by first substituting known values and then solving for the variable, and by first isolating the variable and then substituting known values

make connections between formulas and linear, quadratic, and exponential functions [e.g., recognize that the compound interest formula, A = P(1 + i) , 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)

solve multi-step problems requiring formulas arising from real-world applications (e.g., determining the cost of two coats of paint for a large cylindrical tank)

gather, interpret, and describe information about applications of mathematical modelling in occupations, and about college programs that explore these applications

Personal Finance

Understanding Annuities: demonstrate an understanding of annuities, including mortgages, and solve related problems using technology

gather and interpret information about annuities, describe the key features of an annuity, and identify real-world applications (e.g., RRSP, mortgage, RRIF, RESP)

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)

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

demonstrate, through investigation using technology (e.g., a TVM Solver), the advantages of starting deposits earlier when investing in annuities used as long-term savings plans

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)

read and interpret an amortization table for a mortgage

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

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

Renting or Owning Accommodation: gather, interpret, and compare information about owning or renting accommodation, and solve problems involving the associated costs;

gather and interpret information about the procedures and costs involved in owning and in renting accommodation (e.g., apartment, condominium, townhouse, detached home) in the local community

compare renting accommodation with owning accommodation by describing the advantages and disadvantages of each

solve problems, using technology (e.g., calculator, spreadsheet), that involve the fixed costs (e.g., mortgage, insurance, property tax) and variable costs (e.g., maintenance, utilities) of owning or renting accommodation

Designing Budgets: design, justify, and adjust budgets for individuals and families described in case studies, and describe applications of the mathematics of personal finance.

gather, interpret, and describe information about living costs, and estimate the living costs of different households (e.g., a family of four, including two young children; a single young person; a single parent with one child) in the local community

design and present a savings plan to facilitate the achievement of a long-term goal (e.g., attending college, purchasing a car, renting or purchasing a house)

design, explain, and justify a monthly budget suitable for an individual or family described in a given case study that provides the specifics of the situation (e.g., income; personal responsibilities; costs such as utilities, food, rent/mortgage, entertainment, transportation, charitable contributions; long-term savings goals), with technology (e.g., using spreadsheets, budgeting software, online tools) and without technology (e.g., using budget templates)

identify and describe the factors to be considered in determining the affordability of accommodation in the local community (e.g., income, long-term savings, number of dependants, non-discretionary expenses), and consider the affordability of accommodation under given circumstances

make adjustments to a budget to accommodate changes in circumstances (e.g., loss of hours at work, change of job, change in personal responsibilities, move to new accommodation, achievement of a long-term goal, major purchase), with technology (e.g., spreadsheet template, budgeting software)

gather, interpret, and describe information about applications of the mathematics of personal finance in occupations (e.g., selling real estate, bookkeeping, managing a restaurant, financial planning, mortgage brokering), and about college programs that explore these applications

Geometry And Trigonometry

Solving Problems Involving: solve problems involving measurement and geometry and arising from real-world applications;

perform required conversions between the imperial system and the metric system using a variety of tools (e.g., tables, calculators, online conversion tools), as necessary within applications

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

solve problems involving the volumes and surface areas of rectangular prisms, triangular prisms, and cylinders, and of related composite figures, in situations arising from realworld applications

Investigating Optimal Dimensions: explain the significance of optimal dimensions in real-world applications, and determine optimal dimensions of two-dimensional shapes and three-dimensional figures;

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)

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 twodimensional shape in metric or imperial units for a given constraint (e.g., the dimensions that give the minimum perimeter for a given area)

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)

Solving Problems Involving Trigonometry: 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.

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

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)

determine the values of the sine, cosine, and tangent of obtuse angles

solve problems involving oblique triangles, including those that arise from real-world applications, using the sine law (in nonambiguous cases only) and the cosine law, and using metric or imperial units

gather, interpret, and describe information about applications of trigonometry in occupations, and about college programs that explore these applications

Data Management

Working With Two-Variable Data: collect, analyse, and summarize two-variable data using a variety of tools and strategies, and interpret and draw conclusions from the data;

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?)

describe characteristics of an effective survey (e.g., by giving consideration to ethics, privacy, the need for honest responses, and possible sources of bias, including cultural bias), and design questionnaires (e.g., for determining if there is a relationship between age and hours per week of Internet use, between marks and hours of study, or between income and years of education) or experiments (e.g., growth of plants under different conditions) for gathering two-variable data

collect two-variable data from primary sources, through experimentation involving observation or measurement, or from secondary sources (e.g., Internet databases, newspapers, magazines), and organize and store the data using a variety of tools (e.g., spreadsheets, dynamic statistical software)

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

determine an algebraic summary of the relationship between two variables that appear to be linearly related (i.e., the equation of the line of best fit of the scatter plot), using a variety of tools (e.g., graphing calculators, graphing software) and strategies (e.g., using systematic trials to determine the slope and y-intercept of the line of best fit; using the regression capabilities of a graphing calculator), and solve related problems (e.g., use the equation of the line of best fit to interpolate or extrapolate from the given data set)

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)

determine whether a linear model (i.e., a line of best fit) is appropriate given a set of twovariable data, by assessing the correlation between the two variables (i.e., by describing the type of correlation as positive, negative, or none; by describing the strength as strong or weak; by examining the context to determine whether a linear relationship is reasonable)

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)

Applying Data Management: demonstrate an understanding of the applications of data management used by the media and the advertising industry and in various occupations.

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)

describe examples of indices used by the media (e.g., consumer price index, S&P/TSX composite index, new housing price index) and solve problems by interpreting and using indices (e.g., by using the consumer price index to calculate the annual inflation rate)

interpret statistics presented in the media (e.g., the UN’s finding that 2% of the world’s population has more than half the world’s wealth, whereas half the world’s population has only 1% of the world’s wealth), and explain how the media, the advertising industry, and others (e.g., marketers, pollsters) use and misuse statistics (e.g., as represented in graphs) to promote a certain point of view (e.g., by making a general statement based on a weak correlation or an assumed causeand-effect relationship; by starting the vertical scale on a graph at a value other than zero; by making statements using general population statistics without reference to data specific to minority groups)

assess the validity of conclusions presented in the media by examining sources of data, including Internet sources (i.e., to determine whether they are authoritative, reliable, unbiased, and current), methods of data collection, and possible sources of bias (e.g., sampling bias, non-response bias, a bias in a survey question), and by questioning the analysis of the data (e.g., whether there is any indication of the sample size in the analysis) and conclusions drawn from the data (e.g., whether any assumptions are made about cause and effect)

gather, interpret, and describe information about applications of data management in occupations, and about college programs that explore these applications

Mathematics

Exponential Functions

Solving Exponential Equations Graphically: solve problems involving exponential equations graphically, including problems arising from real-world applications;

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

solve simple exponential equations numerically and graphically, with technology (e.g., use systematic trial with a scientific calculator to determine the solution to the equation 1.05x = 1,276), and recognize that the solutions may not be exact

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

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

Solving Exponential Equations Algebraically: solve problems involving exponential equations algebraically using common bases and logarithms, including problems arising from real-world applications.

simplify algebraic expressions containing integer and rational exponents using the laws of exponents (e.g., x3/x1/2, sqrt(x6y12)

solve exponential equations in one variable by determining a common base (e.g., 2x = 32, 45x - 1 = 2(2)(x + 11), 35x + 8 = 27x)

recognize the logarithm of a number to a given base as the exponent to which the base must be raised to get the number, recognize the operation of finding the logarithm to be the inverse operation (i.e., the undoing or reversing) of exponentiation, and evaluate simple logarithmic expressions

determine, with technology, the approximate logarithm of a number to any base, including base 10 [e.g., by recognizing that log (0.372) can be determined using the LOG key on a calculator; by reasoning that log 29 is between 3 and 4 and using systematic trial to determine that log 29 is approximately 3.07]

make connections between related logarithmic and exponential equations (e.g., log5125 = 3 can also be expressed as 53 = 125), and solve simple exponential equations by rewriting them in logarithmic form (e.g., solving 3x = 10 by rewriting the equation as log310 = x)

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

Polynomial Functions

Investigating Graphs of Polynomial Functions: . recognize and evaluate polynomial functions, describe key features of their graphs, and solve problems using graphs of polynomial functions;

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 x - 5x + 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

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)

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)

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

substitute into and evaluate polynomial functions expressed in function notation, including functions arising from real-world applications

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

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)

Connecting Graphs and Equations of Polynomial Functions: make connections between the numeric, graphical, and algebraic representations of polynomial functions;

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)

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)

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 x4 − 13x² + 36 = 0 are the x-intercepts of the graph of f(x) = x4 − 13x² + 36)

Solving Problems Involving Polynomial Equations: solve polynomial equations by factoring, make connections between functions and formulas, and solve problems involving polynomial expressions arising from a variety of applications.

solve polynomial equations in one variable, of degree no higher than four (e.g., x² − 4x = 0, x4 − 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)

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

identify and explain the roles of constants and variables in a given formula (e.g., a constant can refer to a known initial value or a known fixed rate; a variable changes with varying conditions)

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

solve equations of the form xn = a using rational exponents (e.g., solve x3 = 7 by raising both sides to the exponent 1/3)

determine the value of a variable of degree no higher than three, using a formula drawn from an application, by first substituting known values and then solving for the variable, and by first isolating the variable and then substituting known values

make connections between formulas and linear, quadratic, and exponential functions [e.g., recognize that the compound interest formula, A = P(1 + i) , 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)

solve multi-step problems requiring formulas arising from real-world applications (e.g., determining the cost of two coats of paint for a large cylindrical tank)

gather, interpret, and describe information about applications of mathematical modelling in occupations, and about college programs that explore these applications

Trigonometric Functions

Applying Trigonometric Ratios: determine the values of the trigonometric ratios for angles less than 360�, and solve problems using the primary trigonometric ratios, the sine law, and the cosine law;

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

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 the special angles)

determine the measures of two angles from 0° to 360° 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)

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

solve problems involving oblique triangles, including those that arise from real-world applications, using the sine law (including the ambiguous case) and the cosine law

Connecting Graphs and Equations of Sinusoidal Functions: make connections between the numeric, graphical, and algebraic representations of sinusoidal functions;

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) = sin x or f(x) = cos x, and explaining why the relationship is a function

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)

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)

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)

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

represent a sinusoidal function with an equation, given its graph or its properties

Solving Problems Involving Sinusoidal Functions: 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.

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

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

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

Applications of Geometry

Modelling With Vectors: represent vectors, add and subtract vectors, and solve problems using vector models, including those arising from real-world applications;

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)

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

resolve a vector represented as a directed line segment into its vertical and horizontal components

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)

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

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

Solving Problems Involving Geometry: solve problems involving two-dimensional shapes and three-dimensional figures and arising from real-world applications;

gather and interpret information about realworld applications of geometric shapes and figures in a variety of contexts in technologyrelated fields (e.g., product design, architecture), 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)

perform required conversions between the imperial system and the metric system using a variety of tools (e.g., tables, calculators, online conversion tools), as necessary within applications

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

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

Solving Problems Involving Circle Properties: determine circle properties and solve related problems, including those arising from real-world applications

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)

determine the length of an arc and the area of a sector or segment of a circle, and solve related problems

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)

solve problems involving properties of circles, including problems arising from real-world applications

Mathematics

Reasoning with Data

Interpreting and Displaying Data: collect, organize, represent, and make inferences from data using a variety of tools and strategies, and describe related applications;

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

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)

collect categorical data from primary sources, through experimentation involving observation (e.g., by tracking food orders in restaurants offering healthy food options) or measurement, or from secondary sources (e.g., Internet databases, newspapers, magazines), and organize and store the data using a variety of tools (e.g., spreadsheets, dynamic statistical software)

represent categorical data by constructing graphs (e.g., bar graph, broken-line graph, circle graph) using a variety of tools (e.g., dynamic statistical software, graphing calculator, spreadsheet)

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)

make and justify conclusions about a topic of personal interest by collecting, organizing (e.g., using spreadsheets), representing (e.g., using graphs), and making inferences from categorical data from primary sources (i.e., collected through measurement or observation) or secondary sources (e.g., electronic data from databases such as E-STAT, data from newspapers or magazines)

explain how the media, the advertising industry, and others (e.g., marketers, pollsters) use and misuse statistics (e.g., as represented in graphs) to promote a certain point of view (e.g., by making general statements based on small samples; by making statements using general population statistics without reference to data specific to minority groups)

gather, interpret, and describe information about applications of data management in the workplace and in everyday life

Investigating Probability: determine and represent probability, and identify and interpret its applications.

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)

identify examples of the use of probability in the media (e.g., the probability of rain, of winning a lottery, of wait times for a service exceeding specified amounts) 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)

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

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?")

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")

interpret information involving the use of probability and statistics in the media, and describe how probability and statistics can help in making informed decisions in a variety of situations (e.g., weighing the risk of injury when considering different occupations; using a weather forecast to plan outdoor activities; using sales data to stock a clothing store with appropriate styles and sizes)

Personal Finance

Renting or Owning Accommodation: gather, interpret, and compare information about owning or renting accommodation and about the associated costs;

identify the financial implications (e.g., responsibility for paying the cost of accommodation and furnishings; greater responsibility for financial decision making) and the nonfinancial implications (e.g., greater freedom to make decisions; the demands of time management or of adapting to a new environment; the possibility of loneliness or of the need to share responsibilities) associated with living independently

gather and compare, through investigation, information about the costs and the advantages and disadvantages of different types of rental accommodation in the local community (e.g., renting a room in someone's house; renting a hotel room; renting or leasing an apartment)

gather and compare, through investigation, information about purchase prices of different types of owned accommodation in the local community (e.g., trailer, condominium, townhouse, detached home)

gather, interpret, and compare information about the different types of ongoing living expenses associated with renting and owning accommodation (e.g., hydro, cable, telephone, internet, heating, parking, laundry, groceries, cleaning supplies, transportation) and related costs

gather, interpret, and describe information about the rights and responsibilities of tenants and landlords

generate a checklist of necessary tasks associated with moving (e.g., change of address, set-up of utilities and services, truck rental), and estimate the total cost involved under various conditions (e.g., moving out of province; hiring a moving company)

Designing Budgets: interpret, design, and adjust budgets for individuals and families described in case studies;

categorize personal expenses as nondiscretionary (e.g., rent, groceries, utilities, loan payments) or discretionary (e.g., entertainment, vacations)

categorize personal non-discretionary expenses as fixed (e.g., rent, cable, car insurance) or variable (e.g., groceries, clothing, vehicle maintenance)

read and interpret prepared individual or family budgets, identify and describe the key components of a budget, and describe how budgets can reflect personal values (e.g., as they relate to shopping, saving for a longterm goal, recreational activities, family, community)

design, with technology (e.g., using spreadsheet templates, budgeting software, online tools) and without technology (e.g., using budget templates), explain, and justify a monthly budget suitable for an individual or family described in a given case study that provides the specifics of the situation (e.g., income; personal responsibilities; expenses such as utilities, food, rent/mortgage, entertainment, transportation, charitable contributions; long-term savings goals)

identify and describe factors to be considered in determining the affordability of accommodation in the local community (e.g., income, long-term savings, number of dependants, non-discretionary expenses)

make adjustments to a budget to accommodate changes in circumstances (e.g., loss of hours at work, change of job, change in personal responsibilities, move to new accommodation, achievement of a long-term goal, major purchase), with technology (e.g., spreadsheet template, budgeting software)

Filing Income Tax: demonstrate an understanding of the process of filing a personal income tax return, and describe applications of the mathematics of personal finance

explain why most Canadians are expected to file a personal income tax return each year, and identify and describe the major parts of a personal income tax return (i.e., identification, total income, net income, taxable income, refund or balance owing)

gather, interpret, and describe the information and documents required for filing a personal income tax return (e.g., CRA guides, forms, and schedules; T4 slips; receipts for charitable donations), and explain why they are required

gather, interpret, and compare information about common tax credits (e.g., tuition fees, medical expenses, charitable donations) and tax deductions (e.g., moving expenses, child care expenses, union dues)

complete a simple personal income tax return (i.e., forms and schedules), with or without tax preparation software

gather, interpret, and describe some additional information that a self-employed individual should provide when filing a personal income tax return (e.g., a statement of business activities that includes business expenses such as insurance, advertising, and motor-vehicle expenses)

gather, interpret, and describe information about services that will complete a personal income tax return (e.g., tax preparation service, chartered accountant, voluntary service in the community) and resources that will help with completing a personal income tax return (e.g., forms and publications available on the Canada Revenue Agency website, tax preparation software for which rebates are available), and compare the services and resources on the basis of the assistance they provide and their cost

gather, interpret, and describe information about applications of the mathematics of personal finance in the workplace (e.g., selling real estate, bookkeeping, managing a restaurant)

Applications of Measurement

Measuring and Estimating: determine and estimate measurements using the metric and imperial systems, and convert measures within and between systems;

measure, using a variety of tools (e.g., measuring tape, metre or yard stick, measuring cups, graduated cylinders), the lengths of common objects and the capacities of common containers, using the metric system and the imperial system

estimate lengths, distances, and capacities in metric units and in imperial units by applying personal referents (e.g., the width of a finger is approximately 1 cm; the length of a piece of standard loose-leaf paper is about 1 ft; the capacity of a pop bottle is 2 L)

estimate quantities (e.g., bricks in a pile, time to complete a job, people in a crowd), and describe the strategies used

convert measures within systems (e.g., centimetres and metres, kilograms and grams, litres and millilitres, feet and inches, ounces and pounds), as required within applications that arise from familiar contexts

convert measures between systems (e.g., centimetres and inches, pounds and kilograms, square feet and square metres, litres and U.S. gallons, kilometres and miles, cups and millilitres, millilitres and teaspoons, degrees Celsius and degrees Fahrenheit), as required within applications that arise from familiar contexts

Applying Measurement and Design: 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;

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

apply the concept of perimeter in familiar contexts (e.g., baseboard, fencing, door and window trim)

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

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

solve problems involving the volumes and surface areas of rectangular prisms, triangular prisms, and cylinders, and of related composite figures, in situations arising from realworld applications

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

construct, with reasonable accuracy, a threedimensional scale model of an object or environment of personal interest (e.g., appliance, room, building, garden, bridge)

investigate, plan, design, and prepare a budget for a household improvement (e.g., landscaping a property; renovating a room), using appropriate technologies (e.g., design or decorating websites, design or drawing software, spreadsheet)

Solving Measurement Problems Using Proportional Reasoning: 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.

identify and describe applications of ratio and rate, and recognize and represent equivalent ratios (e.g., show that 4:6 represents the same ratio as 2:3 by showing that a ramp with a height of 4 m and a base of 6 m and a ramp with a height of 2 m and a base of 3 m are equally steep) and equivalent rates (e.g., recognize that paying $1.25 for 250 mL of tomato sauce is equivalent to paying $3.75 for 750 mL of the same sauce), using a variety of tools (e.g., concrete materials, diagrams, dynamic geometry software)

identify situations in which it is useful to make comparisons using unit rates, and solve problems that involve comparisons of unit rates

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

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)

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)

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)

Mathematics

Counting and Probability

Solving Probability Problems Involving Discrete Sample Spaces: solve problems involving the probability of an event or a combination of events for discrete sample spaces;

recognize and describe how probabilities are used to represent the likelihood of a result of an experiment (e.g., spinning spinners; drawing blocks from a bag that contains differentcoloured blocks; playing a game with number cubes; playing Aboriginal stick-and-stone games) and the likelihood of a real-world event (e.g., that it will rain tomorrow, that an accident will occur, that a product will be defective)

describe a sample space as a set that contains all possible outcomes of an experiment, and distinguish between a discrete sample space as one whose outcomes can be counted (e.g., all possible outcomes of drawing a card or tossing a coin) and a continuous sample space as one whose outcomes can be measured (e.g., all possible outcomes of the time it takes to complete a task or the maximum distance a ball can be thrown)

determine the theoretical probability, Pi (i.e., a value from 0 to 1), of each outcome of a discrete sample space (e.g., in situations in which all outcomes are equally likely), recognize that the sum of the probabilities of the outcomes is 1 (i.e., for n outcomes, P1 + P2 + P3 + ... + Pn = 1), recognize that the probabilities Pi form the probability distribution associated with the sample space, and solve related problems

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; using dynamic statistical software to simulate repeated trials in an experiment), the tendency of experimental probability to approach theoretical probability as the number of trials in an experiment increases (e.g., "If I simulate tossing two coins 1000 times using technology, the experimental probability that I calculate for getting two tails on the two tosses is likely to be closer to the theoretical probability of 1/4 than if I simulate tossing the coins only 10 times")

recognize and describe an event as a set of outcomes and as a subset of a sample space, determine the complement of an event, determine whether two or more events are mutually exclusive or non-mutually exclusive (e.g., the events of getting an even number or getting an odd number of heads from tossing a coin 5 times are mutually exclusive), and solve related probability problems [e.g., calculate P(~A), P(A and B), P(A or B)] using a variety of strategies (e.g., Venn diagrams, lists, formulas)

determine whether two events are independent or dependent and whether one event is conditional on another event, and solve related probability problems [e.g., calculate P(A and B), P(A or B), P(A given B)] using a variety of strategies (e.g., tree diagrams, lists, formulas)

Solving Problems Using Counting Principles: solve problems involving the application of permutations and combinations to determine the probability of an event

recognize the use of permutations and combinations as counting techniques with advantages over other counting techniques (e.g., making a list; using a tree diagram; making a chart; drawing a Venn diagram), distinguish between situations that involve the use of permutations and those that involve the use of combinations (e.g., by considering whether or not order matters), and make connections between, and calculate, permutations and combinations

solve simple problems using techniques for counting permutations and combinations, where all objects are distinct, and express the solutions using standard combinatorial notation [e.g., n!, P(n, r), (n/r )]

solve introductory counting problems involving the additive counting principle (e.g., determining the number of ways of selecting 2 boys or 2 girls from a group of 4 boys and 5 girls) and the multiplicative counting principle (e.g., determining the number of ways of selecting 2 boys and 2 girls from a group of 4 boys and 5 girls)

make connections, through investigation, between combinations (i.e., n choose r) and Pascal's triangle [e.g., between ( 2/r) and row 3 of Pascal's triangle between (n/2) and diagonal 3 of Pascal's triangle]

solve probability problems using counting principles for situations involving equally likely outcomes

Probability Distributions:

Understanding Probability Distributions for Discrete Random Variables: demonstrate an understanding of discrete probability distributions, represent them numerically, graphically, and algebraically, determine expected values, and solve related problems from a variety of applications;

recognize and identify a discrete random variable X (i.e., a variable that assumes a unique value for each outcome of a discrete sample space, such as the value x for the outcome of getting x heads in 10 tosses of a coin), generate a probability distribution [i.e., a function that maps each value x of a random variable X to a corresponding probability, P(X = x)] by calculating the probabilities associated with all values of a random variable, with and without technology, and represent a probability distribution numerically using a table

calculate the expected value for a given probability distribution [i.e., using E(X)= ? xP(X = x)], interpret the expected value in applications, and make connections between the expected value and the weighted mean of the values of the discrete random variable

represent a probability distribution graphically using a probability histogram (i.e., a histogram on which each rectangle has a base of width 1, centred on the value of the discrete random variable, and a height equal to the probability associated with the value of the random variable), and make connections between the frequency histogram and the probability histogram (e.g., by comparing their shapes)

recognize conditions (e.g., independent trials) that give rise to a random variable that follows a binomial probability distribution, calculate the probability associated with each value of the random variable, represent the distribution numerically using a table and graphically using a probability histogram, and make connections to the algebraic representation P(X = x) = (n x)px (1 - p)n - x

recognize conditions (e.g., dependent trials) that give rise to a random variable that follows a hypergeometric probability distribution, calculate the probability associated with each value of the random variable (e.g., by using a tree diagram; by using combinations), and represent the distribution numerically using a table and graphically using a probability histogram

compare, with technology and using numeric and graphical representations, the probability distributions of discrete random variables (e.g., compare binomial distributions with the same probability of success for increasing numbers of trials; compare the shapes of a hypergeometric distribution and a binomial distribution)

solve problems involving probability distributions (e.g., uniform, binomial, hypergeometric), including problems arising from real-world applications

Understanding Probability Distributions for Continuous Random Variables: demonstrate an understanding of continuous probability distributions, make connections to discrete probability distributions, determine standard deviations, describe key features of the normal distribution, and solve related problems from a variety of applications.

recognize and identify a continuous random variable (i.e., a variable that assumes values from the infinite number of possible outcomes in a continuous sample space), and distinguish between situations that give rise to discrete frequency distributions (e.g., counting the number of outcomes for drawing a card or tossing three coins) and situations that give rise to continuous frequency distributions (e.g., measuring the time taken to complete a task or the maximum distance a ball can be thrown)

recognize standard deviation as a measure of the spread of a distribution, and determine, with and without technology, the mean and standard deviation of a sample of values of a continuous random variable

describe challenges associated with determining a continuous frequency distribution (e.g., the inability to capture all values of the variable, resulting in a need to sample; uncertainties in measured values of the variable), and recognize the need for mathematical models to represent continuous frequency distributions

represent, using intervals, a sample of values of a continuous random variable numerically using a frequency table and graphically using a frequency histogram and a frequency polygon, recognize that the frequency polygon approximates the frequency distribution, and determine, through investigation using technology (e.g., dynamic statistical software, graphing calculator), and compare the effectiveness of the frequency polygon as an approximation of the frequency distribution for different sizes of the intervals

recognize that theoretical probability for a continuous random variable is determined over a range of values (e.g., the probability that the life of a lightbulb is between 90 hours and 115 hours), that the probability that a continuous random variable takes any single value is zero, and that the probabilities of ranges of values form the probability distribution associated with the random variable

recognize that the normal distribution is commonly used to model the frequency and probability distributions of continuous random variables, describe some properties of the normal distribution (e.g., the curve has a central peak; the curve is symmetric about the mean; the mean and median are equal; approximately 68% of the data values are within one standard deviation of the mean and approximately 95% of the data values are within two standard deviations of the mean), and recognize and describe situations that can be modelled using the normal distribution (e.g., birth weights of males or of females, household incomes in a neighbourhood, baseball batting averages)

make connections, through investigation using dynamic statistical software, between the normal distribution and the binomial and hypergeometric distributions for increasing numbers of trials of the discrete distributions (e.g., recognizing that the shape of the hypergeometric distribution of the number of males on a 4-person committee selected from a group of people more closely resembles the shape of a normal distribution as the size of the group from which the committee was drawn increases)

recognize a z-score as the positive or negative number of standard deviations from the mean to a value of the continuous random variable, and solve probability problems involving normal distributions using a variety of tools and strategies (e.g., calculating a z-score and reading a probability from a table; using technology to determine a probability), including problems arising from real-world applications

Organization of Data for Analysis

Understanding Data Concepts: demonstrate an understanding of the role of data in statistical studies and the variability inherent in data, and distinguish different types of data;

recognize and describe the role of data in statistical studies (e.g., the use of statistical techniques to extract or mine knowledge of relationships from data), describe examples of applications of statistical studies (e.g., in medical research, political decision making, market research), and recognize that conclusions drawn from statistical studies of the same relationship may differ (e.g., conclusions about the effect of increasing jail sentences on crime rates)

recognize and explain reasons why variability is inherent in data (e.g., arising from limited accuracy in measurement or from variations in the conditions of an experiment; arising from differences in samples in a survey), and distinguish between situations that involve one variable and situations that involve more than one variable

distinguish different types of statistical data (i.e., discrete from continuous, qualitative from quantitative, categorical from numerical, nominal from ordinal, primary from secondary, experimental from observational, microdata from aggregate data) and give examples (e.g., distinguish experimental data used to compare the effectiveness of medical treatments from observational data used to examine the relationship between obesity and type 2 diabetes or between ethnicity and type 2 diabetes)

Collecting and Organizing Data

determine and describe principles of primary data collection (e.g., the need for randomization, replication, and control in experimental studies; the need for randomization in sample surveys) and criteria that should be considered in order to collect reliable primary data (e.g., the appropriateness of survey questions; potential sources of bias; sample size)

explain the distinction between the terms population and sample, describe the characteristics of a good sample, explain why sampling is necessary (e.g., time, cost, or physical constraints), and describe and compare some sampling techniques (e.g., simple random, systematic, stratified, convenience, voluntary)

describe how the use of random samples with a bias (e.g., response bias, measurement bias, non-response bias, sampling bias) or the use of non-random samples can affect the results of a study

describe characteristics of an effective survey (e.g., by giving consideration to ethics, privacy, the need for honest responses, and possible sources of bias, including cultural bias), and design questionnaires (e.g., for determining if there is a relationship between a person's age and their hours per week of Internet use, between marks and hours of study, or between income and years of education) or experiments (e.g., growth of plants under different conditions) for gathering data

collect data from primary sources, through experimentation, or from secondary sources (e.g., by using the Internet to access reliable data from a well-organized database such as E-STAT; by using print sources such as newspapers and magazines), and organize data with one or more attributes (e.g., organize data about a music collection classified by artist, date of recording, and type of music using dynamic statistical software or a spreadsheet) to answer a question or solve a problem

Statistical Analysis

Analysing One-Variable Data: analyse, interpret, and draw conclusions from one-variable data using numerical and graphical summaries;

recognize that the analysis of one-variable data involves the frequencies associated with one attribute, and determine, using technology, the relevant numerical summaries (i.e., mean, median, mode, range, interquartile range, variance, and standard deviation)

determine the positions of individual data points within a one-variable data set using quartiles, percentiles, and z-scores, use the normal distribution to model suitable onevariable data sets, and recognize these processes as strategies for one-variable data analysis

generate, using technology, the relevant graphical summaries of one-variable data (e.g., circle graphs, bar graphs, histograms, stem-and-leaf plots, boxplots) based on the type of data provided (e.g., categorical, ordinal, quantitative)

interpret, for a normally distributed population, the meaning of a statistic qualified by a statement describing the margin of error and the confidence level (e.g., the meaning of a statistic that is accurate to within 3 percentage points, 19 times out of 20), and make connections, through investigation using technology (e.g., dynamic statistical software), between the sample size, the margin of error, and the confidence level (e.g., larger sample sizes create higher confidence levels for a given margin of error)

interpret statistical summaries (e.g., graphical, numerical) to describe the characteristics of a one-variable data set and to compare two related one-variable data sets (e.g., compare the lengths of different species of trout; compare annual incomes in Canada and in a third-world country; compare Aboriginal and non-Aboriginal incomes); describe how statistical summaries (e.g., graphs, measures of central tendency) can be used to misrepresent one-variable data; and make inferences, and make and justify conclusions, from statistical summaries of one-variable data orally and in writing, using convincing arguments

Analysing Two-Variable Data: analyse, interpret, and draw conclusions from two-variable data using numerical, graphical, and algebraic summaries;

recognize that the analysis of two-variable data involves the relationship between two attributes, recognize the correlation coefficient as a measure of the fit of the data to a linear model, and determine, using technology, the relevant numerical summaries (e.g., summary tables such as contingency tables; correlation coefficients)

recognize and distinguish different types of relationships between two variables that have a mathematical correlation (e.g., the causeand-effect relationship between the age of a tree and its diameter; the common-cause relationship between ice cream sales and forest fires over the course of a year; the accidental relationship between the consumer price index and the number of known planets in the universe)

generate, using technology, the relevant graphical summaries of two-variable data (e.g., scatter plots, side-by-side boxplots) based on the type of data provided (e.g., categorical, ordinal, quantitative)

determine, by performing a linear regression using technology, the equation of a line that models a suitable two-variable data set, determine the fit of an individual data point to the linear model (e.g., by using residuals to identify outliers), and recognize these processes as strategies for two-variable data analysis

interpret statistical summaries (e.g., scatter plot, equation representing a relationship) to describe the characteristics of a twovariable data set and to compare two related two-variable data sets (e.g., compare the relationship between Grade 12 English and mathematics marks with the relationship between Grade 12 science and mathematics marks); describe how statistical summaries (e.g., graphs, linear models) can be used to misrepresent two-variable data; and make inferences, and make and justify conclusions, from statistical summaries of two-variable data orally and in writing, using convincing arguments

Evaluating Validity: demonstrate an understanding of the applications of data management used by the media and the advertising industry and in various occupations.

interpret statistics presented in the media (e.g., the UN’s finding that 2% of the world’s population has more than half the world’s wealth, whereas half the world’s population has only 1% of the world’s wealth), and explain how the media, the advertising industry, and others (e.g., marketers, pollsters) use and misuse statistics (e.g., as represented in graphs) to promote a certain point of view (e.g., by making a general statement based on a weak correlation or an assumed cause-andeffect relationship; by starting the vertical scale at a value other than zero; by making statements using general population statistics without reference to data specific to minority groups)

assess the validity of conclusions presented in the media by examining sources of data, including Internet sources (i.e., to determine whether they are authoritative, reliable, unbiased, and current), methods of data collection, and possible sources of bias (e.g., sampling bias, non-response bias, cultural bias in a survey question), and by questioning the analysis of the data (e.g., whether there is any indication of the sample size in the analysis) and conclusions drawn from the data (e.g., whether any assumptions are made about cause and effect)

Culminating Data Management Investigation

Designing and Carrying Out a Culminating Investigation: design and carry out a culminating investigation* that requires the integration and application of the knowledge and skills related to the expectations of this course;

pose a significant problem of interest that requires the organization and analysis of a suitable set of primary or secondary quantitative data (e.g., primary data collected from a student-designed game of chance, secondary data from a reliable source such as E-STAT), and conduct appropriate background research related to the topic being studied

design a plan to study the problem (e.g., identify the variables and the population; develop an ethical survey; establish the procedures for gathering, summarizing, and analysing the primary or secondary data; consider the sample size and possible sources of bias)

gather data related to the study of the problem (e.g., by using a survey; by using the Internet; by using a simulation) and organize the data (e.g., by setting up a database; by establishing intervals), with or without technology

interpret, analyse, and summarize data related to the study of the problem (e.g., generate and interpret numerical and graphical statistical summaries; recognize and apply a probability distribution model; calculate the expected value of a probability distribution), with or without technology

draw conclusions from the analysis of the data (e.g., determine whether the analysis solves the problem), evaluate the strength of the evidence (e.g., by considering factors such as sample size or bias, or the number of times a game is played), specify any limitations of the conclusions, and suggest follow-up problems or investigations

Presenting and Critiquing the Culminating Investigation: communicate the findings of a culminating investigation and provide constructive critiques of the investigations of others

compile a clear, well-organized, and detailed report of the investigation

present a summary of the culminating investigation to an audience of their peers within a specified length of time, with technology (e.g. presentation software) or without technology

answer questions about the culminating investigation and respond to critiques (e.g., by elaborating on the procedures; by justifying mathematical reasoning)

critique the mathematical work of others in a constructive manner

Mathematics

Number

Development of Numbers and Number Sets: demonstrate an understanding of the development and use of numbers, and make connections between sets of numbers

research a number concept to tell a story about its development and use in a specific culture, and describe its relevance in a current context

describe how various subsets of a number system are defined, and describe similarities and differences between these subsets

use patterns and number relationships to explain density, infinity, and limit as they relate to number sets

Powers: represent numbers in various ways, evaluate powers, and simplify expressions by using the relationships between powers and their exponents

analyse, through the use of patterning, the relationship between the sign and size of an exponent and the value of a power, and use this relationship to express numbers in scientific notation and evaluate powers

analyse, through the use of patterning, the relationships between the exponents of powers and the operations with powers, and use these relationships to simplify numeric and algebraic expressions

Number Sense and Operations: apply an understanding of rational numbers, ratios, rates, percentages, and proportions, in various mathematical contexts, and to solve problems

apply an understanding of integers to describe location, direction, amount, and changes in any of these, in various contexts

apply an understanding of unit fractions and their relationship to other fractional amounts, in various contexts, including the use of measuring tools

apply an understanding of integers to explain the effects that positive and negative signs have on the values of ratios, rates, fractions, and decimals, in various contexts

solve problems involving operations with positive and negative fractions and mixed numbers, including problems involving formulas, measurements, and linear relations, using technology when appropriate

pose and solve problems involving rates, percentages, and proportions in various contexts, including contexts connected to real-life applications of data, measurement, geometry, linear relations, and financial literacy

Algebra

Algebraic Expressions and Equations: demonstrate an understanding of the development and use of algebraic concepts and of their connection to numbers, using various tools and representations

research an algebraic concept to tell a story about its development and use in a specific culture, and describe its relevance in a current context

create algebraic expressions to generalize relationships expressed in words, numbers, and visual representations, in various contexts

compare algebraic expressions using concrete, numerical, graphical, and algebraic methods to identify those that are equivalent, and justify their choices

simplify algebraic expressions by applying properties of operations of numbers, using various representations and tools, in different contexts

create and solve equations for various contexts, and verify their solutions

Coding: apply coding skills to represent mathematical concepts and relationships dynamically, and to solve problems, in algebra and across the other strands

use coding to demonstrate an understanding of algebraic concepts including variables, parameters, equations, and inequalities

create code by decomposing situations into computational steps in order to represent mathematical concepts and relationships, and to solve problems

read code to predict its outcome, and alter code to adjust constraints, parameters, and outcomes to represent a similar or new mathematical situation

Application of Relations: represent and compare linear and non-linear relations that model real-life situations, and use these representations to make predictions

compare the shapes of graphs of linear and non-linear relations to describe their rates of change, to make connections to growing and shrinking patterns, and to make predictions

represent linear relations using concrete materials, tables of values, graphs, and equations, and make connections between the various representations to demonstrate an understanding of rates of change and initial values

compare two linear relations of the form y = ax + b graphically and algebraically, and interpret the meaning of their point of intersection in terms of a given context

Data

Collection, Representation, and Analysis of Data: describe the collection and use of data, and represent and analyse data involving one and two variables

identify a current context involving a large amount of data, and describe potential implications and consequences of its collection, storage, representation, and use

represent and statistically analyse data from a real-life situation involving a single variable in various ways, including the use of quartile values and box plots

create a scatter plot to represent the relationship between two variables, determine the correlation between these variables by testing different regression models using technology, and use a model to make predictions when appropriate

Mathematical Modelling: apply the process of mathematical modelling, using data and mathematical concepts from other strands, to represent, analyse, make predictions, and provide insight into real-life situations

describe the value of mathematical modelling and how it is used in real life to inform decisions

identify a question of interest requiring the collection and analysis of data, and identify the information needed to answer the question

create a plan to collect the necessary data on the question of interest from an appropriate source, identify assumptions, identify what may vary and what may remain the same in the situation, and then carry out the plan

determine ways to display and analyse the data in order to create a mathematical model to answer the original question of interest, taking into account the nature of the data, the context, and the assumptions made

report how the model can be used to answer the question of interest, how well the model fits the context, potential limitations of the model, and what predictions can be made based on the model

Geometry and Measurement

Geometric and Measurement Relationships: demonstrate an understanding of the development and use of geometric and measurement relationships, and apply these relationships to solve problems, including problems involving real-life situations

research a geometric concept or a measurement system to tell a story about its development and use in a specific culture or community, and describe its relevance in connection to careers and to other disciplines

create and analyse designs involving geometric relationships and circle and triangle properties, using various tools

solve problems involving different units within a measurement system and between measurement systems, including those from various cultures or communities, using various representations and technology, when appropriate

show how changing one or more dimensions of a two-dimensional shape and a three-dimensional object affects perimeter/circumference, area, surface area, and volume, using technology when appropriate

solve problems involving the side-length relationship for right triangles in real-life situations, including problems that involve composite shapes

solve problems using the relationships between the volume of prisms and pyramids and between the volume of cylinders and cones, involving various units of measure

Financial Literacy

Financial Decisions: demonstrate the knowledge and skills needed to make informed financial decisions

identify a past or current financial situation and explain how it can inform financial decisions, by applying an understanding of the context of the situation and related mathematical knowledge

identify financial situations that involve appreciation and depreciation, and use associated graphs to answer related questions

compare the effects that different interest rates, lengths of borrowing time, ways in which interest is calculated, and amounts of down payments have on the overall costs associated with purchasing goods or services, using appropriate tools

modify budgets displayed in various ways to reflect specific changes in circumstances, and provide a rationale for the modifications