British Columbia Prescribed Learning Outcomes — Grade 9


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12.AWM.1.1.

Demonstrate an understanding of the limitations of measuring instruments, including: precision; accuracy; uncertainty; tolerance and solve problems. [C, PS, R, T, V]

12.AWM.1.1.1.

Explain why, in a given context, a certain degree of precision is required.

12.AWM.1.1.2.

Explain why, in a given context, a certain degree of accuracy is required.

12.AWM.1.1.3.

Explain, using examples, the difference between precision and accuracy.

12.AWM.1.1.4.

Compare the degree of accuracy of two given instruments used to measure the same attribute.

12.AWM.1.1.5.

Relate the degree of accuracy to the uncertainty of a given measure.

12.AWM.1.1.6.

Analyze precision and accuracy in a contextual problem.

12.AWM.1.1.8.

Describe, using examples, the limitations of measuring instruments used in a specific trade or industry; e.g., tape measure versus Vernier caliper.

12.AWM.1.1.9.

Solve a problem that involves precision, accuracy or tolerance.

12.AWM.2.2.

Solve problems that involve: triangles; quadrilaterals; regular polygons. [C, CN, PS, V]

12.AWM.2.2.1.

Describe and illustrate properties of triangles, including isosceles and equilateral.

12.AWM.2.2.2.

Describe and illustrate properties of quadrilaterals in terms of angle measures, side lengths, diagonal lengths and angles of intersection.

12.AWM.2.2.3.

Describe and illustrate properties of regular polygons.

12.AWM.2.2.4.

Explain, using examples, why a given property does or does not apply to certain polygons.

12.AWM.2.2.5.

Identify and explain an application of the properties of polygons in construction, industrial, commercial, domestic and artistic contexts.

12.AWM.2.2.6.

Solve a contextual problem that involves the application of the properties of polygons.

12.AWM.2.3.

Demonstrate an understanding of transformations on a 2-D shape or a 3-D object, including: translations; rotations; reflections; dilations. [C, CN, R, T, V]

12.AWM.2.3.2.

Draw the image of a 2-D shape that results from a given single transformation.

12.AWM.2.3.4.

Create, analyze and describe designs, using translations, rotations and reflections in all four quadrants of a coordinate grid.

12.AWM.2.3.5.

Identify and describe applications of transformations in construction, industrial, commercial, domestic and artistic contexts.

12.AWM.2.3.7.

Determine and explain whether a given image is a dilation of another given shape, using the concept of similarity.

12.AWM.2.3.8.

Draw, with or without technology, a dilation image for a given 2-D shape or 3-D object, and explain how the original 2-D shape or 3-D object and its image are proportional.

12.AWM.2.3.9.

Solve a contextual problem that involves transformations.

12.AWM.3.1.

Analyze puzzles and games that involve logical reasoning, using problem-solving strategies. [C, CN, PS, R]

12.AWM.3.1.1.

Determine, explain and verify a strategy to solve a puzzle or to win a game; e.g.,

12.AWM.3.1.1.f.

Simplify the original problem.

12.AWM.3.1.1.h.

Develop alternative approaches.

12.AWM.3.2.

Solve problems that involve the acquisition of a vehicle by: buying; leasing; leasing to buy. [C, CN, PS, R, T]

12.AWM.3.2.1.

Describe and explain various options for buying, leasing and leasing to buy a vehicle.

12.AWM.3.2.2.

Solve, with or without technology, problems that involve the purchase, lease or lease to purchase of a vehicle.

12.AWM.3.2.3.

Justify a decision related to buying, leasing or leasing to buy a vehicle, based on factors such as personal finances, intended use, maintenance, warranties, mileage and insurance.

12.AWM.3.3.

Critique the viability of small business options by considering: expenses; sales; profit or loss. [C, CN, R]

12.AWM.3.3.1.

Identify expenses in operating a small business, such as a hot dog stand.

12.AWM.3.3.2.

Identify feasible small business options for a given community.

12.AWM.3.3.3.

Generate options that might improve the profitability of a small business.

12.AWM.3.3.4.

Determine the break-even point for a small business.

12.AWM.3.3.5.

Explain factors, such as seasonal variations and hours of operation, that might impact the profitability of a small business.

12.AWM.4.1.

Demonstrate an understanding of linear relations by: recognizing patterns and trends; graphing; creating tables of values; writing equations; interpolating and extrapolating; solving problems. [CN, PS, R, T, V]

12.AWM.4.1.1.

Identify and describe the characteristics of a linear relation represented in a graph, table of values, number pattern or equation.

12.AWM.4.1.11.

Relate slope and rate of change to linear relations.

12.AWM.4.1.12.

Match given contexts with their corresponding graphs, and explain the reasoning.

12.AWM.4.1.13.

Solve a contextual problem that involves the application of a formula for a linear relation.

12.AWM.4.1.2.

Sort a set of graphs, tables of values, number patterns and/or equations into linear and nonlinear relations.

12.AWM.4.1.3.

Write an equation for a given context, including direct or partial variation.

12.AWM.4.1.4.

Create a table of values for a given equation of a linear relation.

12.AWM.4.1.5.

Sketch the graph for a given table of values.

12.AWM.4.1.7.

Create, with or without technology, a graph to represent a data set, including scatterplots.

12.AWM.4.1.8.

Describe the trends in the graph of a data set, including scatterplots.

12.AWM.4.1.9.

Sort a set of scatterplots according to the trends represented (linear, nonlinear or no trend).

12.AWM.5.1.

Solve problems that involve measures of central tendency, including: mean; median; mode; weighted mean; trimmed mean. [C, CN, PS, R]

12.AWM.5.1.1.

Explain, using examples, the advantages and disadvantages of each measure of central tendency.

12.AWM.5.1.10.

Solve a contextual problem that involves measures of central tendency.

12.AWM.5.1.2.

Determine the mean, median and mode for a set of data.

12.AWM.5.1.3.

Identify and correct errors in a calculation of a measure of central tendency.

12.AWM.5.1.4.

Identify the outlier(s) in a set of data.

12.AWM.5.1.5.

Explain the effect of outliers on mean, median and mode.

12.AWM.5.1.6.

Calculate the trimmed mean for a set of data, and justify the removal of the outliers.

12.AWM.5.1.7.

Explain, using examples such as course marks, why some data in a set would be given a greater weighting in determining the mean.

12.AWM.5.1.9.

Explain, using examples from print and other media, how measures of central tendency and outliers are used to provide different interpretations of data.

12.AWM.5.2.

Analyze and describe percentiles. [C, CN, PS, R]

12.AWM.5.2.1.

Explain, using examples, percentile ranks in a context.

12.AWM.5.2.2.

Explain decisions based on a given percentile rank.

12.AWM.5.2.3.

Explain, using examples, the difference between percent and percentile rank.

12.AWM.5.2.4.

Explain the relationship between median and percentile.

12.AWM.5.2.5.

Solve a contextual problem that involves percentiles.

12.AWM.6.1.

Analyze and interpret problems that involve probability. [C, CN, PS, R]

12.AWM.6.1.1.

Describe and explain the applications of probability; e.g., medication, warranties, insurance, lotteries, weather prediction, 100-year flood, failure of a design, failure of a product, vehicle recalls, approximation of area.

12.AWM.6.1.2.

Calculate the probability of an event based on a data set; e.g., determine the probability of a randomly chosen light bulb being defective.

12.AWM.6.1.3.

Express a given probability as a fraction, decimal and percent and in a statement.

12.AWM.6.1.5.

Determine the probability of an event, given the odds for or against.

12.AWM.6.1.6.

Explain, using examples, how decisions may be based on a combination of theoretical probability calculations, experimental results and subjective judgements.

12.AWM.6.1.7.

Solve a contextual problem that involves a given probability.

12.FMP.1.1.

Solve problems that involve compound interest in financial decision making. [C, CN, PS, T, V]

12.FMP.1.1.1.

Explain the advantages and disadvantages of compound interest and simple interest.

12.FMP.1.1.2.

Identify situations that involve compound interest.

12.FMP.1.1.3.

Graph and compare, in a given situation, the total interest paid or earned for different compounding periods.

12.FMP.1.1.4.

Determine, given the principal, interest rate and number of compounding periods, the total interest of a loan.

12.FMP.1.1.5.

Graph and describe the effects of changing the value of one of the variables in a situation that involves compound interest.

12.FMP.1.1.6.

Determine, using technology, the total cost of a loan under a variety of conditions; e.g., different amortization periods, interest rates, compounding periods and terms.

12.FMP.1.1.7.

Compare and explain, using technology, different credit options that involve compound interest, including bank and store credit cards and special promotions.

12.FMP.1.1.8.

Solve a contextual problem that involves compound interest.

12.FMP.1.2.

Analyze costs and benefits of renting, leasing and buying. [CN, PS, R, T]

12.FMP.1.2.2.

Compare, using examples, renting, leasing and buying.

12.FMP.1.2.3.

Justify, for a specific set of circumstances, if renting, buying or leasing would be advantageous.

12.FMP.1.2.4.

Solve a problem involving renting, leasing or buying that requires the manipulation of a formula.

12.FMP.1.2.5.

Solve, using technology, a contextual problem that involves cost-and-benefit analysis.

12.FMP.1.3.

Analyze an investment portfolio in terms of: interest rate; rate of return; total return. [ME, PS, R, T]

12.FMP.1.3.1.

Determine and compare the strengths and weaknesses of two or more portfolios.

12.FMP.1.3.2.

Determine, using technology, the total value of an investment when there are regular contributions to the principal.

12.FMP.1.3.3.

Graph and compare the total value of an investment with and without regular contributions.

12.FMP.1.3.4.

Apply the Rule of 72 to solve investment problems, and explain the limitations of the rule.

12.FMP.1.3.5.

Determine, using technology, possible investment strategies to achieve a financial goal.

12.FMP.1.3.6.

Explain the advantages and disadvantages of long-term and short-term investment options.

12.FMP.1.3.7.

Explain, using examples, why smaller investments over a longer term may be better than larger investments over a shorter term.

12.FMP.1.3.8.

Solve an investment problem.

12.FMP.2.1.

Analyze puzzles and games that involve numerical and logical reasoning, using problem-solving strategies. [CN, ME, PS, R]

12.FMP.2.1.1.

Determine, explain and verify a strategy to solve a puzzle or to win a game; e.g.,

12.FMP.2.1.1.f.

Simplify the original problem.

12.FMP.2.1.1.h.

Develop alternative approaches.

12.FMP.2.2.

Solve problems that involve the application of set theory. [CN, PS, R, V]

12.FMP.2.2.1.

Provide examples of the empty set, disjoint sets, subsets and universal sets in context, and explain the reasoning.

12.FMP.2.2.2.

Organize information such as collected data and number properties, using graphic organizers, and explain the reasoning.

12.FMP.2.2.3.

Explain what a specified region in a Venn diagram represents, using connecting words (and, or, not) or set notation.

12.FMP.2.2.4.

Determine the elements in the complement, the intersection or the union of two sets.

12.FMP.2.2.5.

Explain how set theory is used in applications such as Internet searches, database queries, data analysis, games and puzzles.

12.FMP.2.2.6.

Identify and correct errors in a given solution to a problem that involves sets.

12.FMP.2.2.7.

Solve a contextual problem that involves sets, and record the solution, using set notation.

12.FMP.2.3.

Solve problems that involve conditional statements. [C, CN, PS, R]

12.FMP.2.3.1.

Analyze an if-then statement, make a conclusion, and explain the reasoning.

12.FMP.2.3.2.

Analyze an if-then" statement, make a conclusion, and explain the reasoning.

12.FMP.2.3.3.

Determine the converse, inverse and contrapositive of an if-then statement; determine its veracity; and, if it is false, provide a counterexample.

12.FMP.2.3.4.

Demonstrate, using examples, that the veracity of any statement does not imply the veracity of its converse or inverse.

12.FMP.2.3.5.

Demonstrate, using examples, that the veracity of any statement does imply the veracity of its contrapositive.

12.FMP.2.3.6.

Identify and describe contexts in which a biconditional statement can be justified.

12.FMP.2.3.7.

Analyze and summarize, using a graphic organizer such as a truth table or Venn diagram, the possible results of given logical arguments that involve biconditional, converse, inverse or contrapositive statements.

12.FMP.3.1.

Interpret and assess the validity of odds and probability statements. [C, CN, ME]

12.FMP.3.1.1.

Provide examples of statements of probability and odds found in fields such as media, biology, sports, medicine, sociology and psychology.

12.FMP.3.1.5.

Explain, using examples, how decisions may be based on probability or odds and on subjective judgments.

12.FMP.3.2.

Solve problems that involve the probability of mutually exclusive and nonmutually exclusive events. [CN, PS, R, V]

12.FMP.3.2.1.

Classify events as mutually exclusive or non mutually exclusive, and explain the reasoning.

12.FMP.3.2.2.

Determine if two events are complementary, and explain the reasoning.

12.FMP.3.2.3.

Represent, using set notation or graphic organizers, mutually exclusive (including complementary) and nonmutually exclusive events.

12.FMP.3.2.4.

Solve a contextual problem that involves the probability of mutually exclusive or nonmutually exclusive events.

12.FMP.3.2.5.

Solve a contextual problem that involves the probability of complementary events.

12.FMP.3.2.6.

Create and solve a problem that involves mutually exclusive or nonmutually exclusive events.

12.FMP.3.3.

Solve problems that involve the probability of two events. [CN, PS, R]

12.FMP.3.3.1.

Compare, using examples, dependent and independent events.

12.FMP.3.3.2.

Determine the probability of an event, given the occurrence of a previous event.

12.FMP.3.3.3.

Determine the probability of two dependent or two independent events.

12.FMP.3.3.4.

Create and solve a contextual problem that involves determining the probability of dependent or independent events.

12.FMP.3.4.

Solve problems that involve the fundamental counting principle. [PS, R, V]

12.FMP.3.4.1.

Represent and solve counting problems, using a graphic organizer.

12.FMP.3.4.2.

Generalize the fundamental counting principle, using inductive reasoning.

12.FMP.3.4.3.

Identify and explain assumptions made in solving a counting problem.

12.FMP.3.4.4.

Solve a contextual counting problem, using the fundamental counting principle, and explain the reasoning.

12.FMP.3.5.

Solve problems that involve permutations. [ME, PS, R, T, V]

12.FMP.3.5.1.

Represent the number of arrangements of n elements taken n at a time, using factorial notation.

12.FMP.3.5.2.

Determine, with or without technology, the value of a factorial.

12.FMP.3.5.3.

Simplify a numeric or algebraic fraction containing factorials in both the numerator and denominator.

12.FMP.3.5.4.

Solve an equation that involves factorials.

12.FMP.3.5.5.

Determine the number of permutations of n elements taken r at a time.

12.FMP.3.5.6.

Determine the number of permutations of n elements taken n at a time where some elements are not distinct.

12.FMP.3.5.7.

Explain, using examples, the effect on the total number of permutations of n elements when two or more elements are identical.

12.FMP.3.5.8.

Generalize strategies for determining the number of permutations of n elements taken r at a time.

12.FMP.3.5.9.

Solve a contextual problem that involves probability and permutations.

12.FMP.3.6.

Solve problems that involve combinations. [ME, PS, R, T, V]

12.FMP.3.6.1.

Explain, using examples, why order is or is not important when solving problems that involve permutations or combinations.

12.FMP.4.1.

Represent data, using polynomial functions (of degree < 3), to solve problems. [C, CN, PS, T, V]

12.FMP.4.1.1.

Describe, orally and in written form, the characteristics of polynomial functions by analyzing their graphs.

12.FMP.4.1.2.

Describe, orally and in written form, the characteristics of polynomial functions by analyzing their equations.

12.FMP.4.1.3.

Match equations in a given set to their corresponding graphs.

12.FMP.4.1.5.

Interpret the graph of a polynomial function that models a situation, and explain the reasoning.

12.FMP.4.1.6.

Solve, using technology, a contextual problem that involves data that is best represented by graphs of polynomial functions, and explain the reasoning.

12.FMP.4.2.

Represent data, using exponential and logarithmic functions, to solve problems. [C, CN, PS, T, V]

12.FMP.4.2.1.

Describe, orally and in written form, the characteristics of exponential or logarithmic functions by analyzing their graphs.

12.FMP.4.2.3.

Match equations in a given set to their corresponding graphs.

12.FMP.4.2.4.

Graph data and determine the exponential or logarithmic function that best approximates the data.

12.FMP.4.2.5.

Interpret the graph of an exponential or logarithmic function that models a situation, and explain the reasoning.

12.FMP.4.2.6.

Solve, using technology, a contextual problem that involves data that is best represented by graphs of exponential or logarithmic functions, and explain the reasoning.

12.FMP.4.3.

Represent data, using sinusoidal functions, to solve problems. [C, CN, PS, T, V]

12.FMP.4.3.1.

Describe, orally and in written form, the characteristics of sinusoidal functions by analyzing their graphs.

12.FMP.4.3.2.

Describe, orally and in written form, the characteristics of sinusoidal functions by analyzing their equations.

12.FMP.4.3.3.

Match equations in a given set to their corresponding graphs.

12.FMP.4.3.4.

Graph data and determine the sinusoidal function that best approximates the data.

12.FMP.4.3.5.

Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning.

12.FMP.4.3.6.

Solve, using technology, a contextual problem that involves data that is best represented by graphs of sinusoidal functions, and explain the reasoning.

12.FMP.5.1.

Research and give a presentation on a current event or an area of interest that involves mathematics. [C, CN, ME, PS, R, T, V]

12.FMP.5.1.1.

Collect primary or secondary data (statistical or informational) related to the topic.

12.FMP.5.1.2.

Assess the accuracy, reliability and relevance of the primary or secondary data collected by:

12.FMP.5.1.2.a.

Identifying examples of bias and points of view.

12.FMP.5.1.2.b.

Identifying and describing the data collection methods.

12.FMP.5.1.2.c.

Determining if the data is relevant.

12.FMP.5.1.2.d.

Determining if the data is consistent with information obtained from other sources on the same topic.

12.FMP.5.1.3.

Interpret data, using statistical methods if applicable.

12.FMP.5.1.4.

Identify controversial issues, if any, and present multiple sides of the issues with supporting data.

12.PC.1.1.

Demonstrate an understanding of angles in standard position, expressed in degrees and radians. [CN, ME, R, V]

12.PC.1.1.2.

Describe the relationship among different systems of angle measurement, with emphasis on radians and degrees.

12.PC.1.1.5.

Express the measure of an angle in radians (exact value or decimal approximation), given its measure in degrees.

12.PC.1.1.6.

Express the measure of an angle in degrees, given its measure in radians (exact value or decimal approximation).

12.PC.1.1.9.

Explain the relationship between the radian measure of an angle in standard position and the length of the arc cut on a circle of radius r, and solve problems based upon that relationship.

12.PC.1.2.

Develop and apply the equation of the unit circle. [CN, R, V]

12.PC.1.2.1.

Derive the equation of the unit circle from the Pythagorean theorem.

12.PC.1.2.2.

Describe the six trigonometric ratios, using a point P (x, y) that is the intersection of the terminal arm of an angle and the unit circle.

12.PC.1.2.3.

Generalize the equation of a circle with centre (0, 0) and radius r.

12.PC.1.3.

Solve problems, using the six trigonometric ratios for angles expressed in radians and degrees. [ME, PS, R, T, V]

12.PC.1.3.1.

Determine, with technology, the approximate value of a trigonometric ratio for any angle with a measure expressed in either degrees or radians.

12.PC.1.3.2.

Determine, using a unit circle or reference triangle, the exact value of a trigonometric ratio for angles expressed in degrees that are multiples of 0, 30, 45, 60 or 90, or for angles expressed in radians that are multiples of 0, 6/, 4/, 3/ or 2/, and explain the strategy.

12.PC.1.3.3.

Determine, with or without technology, the measures, in degrees or radians, of the angles in a specified domain, given the value of a trigonometric ratio.

12.PC.1.3.4.

Explain how to determine the exact values of the six trigonometric ratios, given the coordinates of a point on the terminal arm of an angle in standard position.

12.PC.1.3.5.

Determine the measures of the angles in a specified domain in degrees or radians, given a point on the terminal arm of an angle in standard position.

12.PC.1.3.6.

Determine the exact values of the other trigonometric ratios, given the value of one trigonometric ratio in a specified domain.

12.PC.1.3.7.

Sketch a diagram to represent a problem that involves trigonometric ratios.

12.PC.1.3.8.

Solve a problem, using trigonometric ratios.

12.PC.1.4.

Graph and analyze the trigonometric functions sine, cosine and tangent to solve problems. [CN, PS, T, V]

12.PC.1.4.1.

Sketch, with or without technology, the graph of y = sin x, y = cos x or y = tan x.

12.PC.1.4.10.

Determine a trigonometric function that models a situation to solve a problem.

12.PC.1.4.11.

Explain how the characteristics of the graph of a trigonometric function relate to the conditions in a problem situation.

12.PC.1.4.12.

Solve a problem by analyzing the graph of a trigonometric function.

12.PC.1.4.2.

Determine the characteristics (amplitude, asymptotes, domain, period, range and zeros) of the graph of y = sin x, y = cos x or y = tan x.

12.PC.1.4.3.

Determine how varying the value of a affects the graphs of y = a sin x and y = a cos x.

12.PC.1.4.4.

Determine how varying the value of d affects the graphs of y = sin x + d and y = cos x + d.

12.PC.1.4.5.

Determine how varying the value of c affects the graphs of y = sin (x + c) and y = cos (x + c).

12.PC.1.4.6.

Determine how varying the value of b affects the graphs of y = sin bx and y = cos bx.

12.PC.1.4.7.

Sketch, without technology, graphs of the form y = a sin b(x - c) + d or y = a cos b(x - c) + d, using transformations, and explain the strategies.

12.PC.1.4.8.

Determine the characteristics (amplitude, asymptotes, domain, period, phase shift, range and zeros) of the graph of a trigonometric function of the form y = a sin b(x - c) + d or y = a cos b(x - c) + d.

12.PC.1.4.9.

Determine the values of a, b, c and d for functions of the form y = a sin b(x - c) + d or y = a cos b(x - c) + d that correspond to a given graph, and write the equation of the function.

12.PC.1.5.

Solve, algebraically and graphically, first and second degree trigonometric equations with the domain expressed in degrees and radians. [CN, PS, R, T, V]

12.PC.1.5.1.

Verify, with or without technology, that a given value is a solution to a trigonometric equation.

12.PC.1.5.2.

Determine, algebraically, the solution of a trigonometric equation, stating the solution in exact form when possible.

12.PC.1.5.3.

Determine, using technology, the approximate solution of a trigonometric equation in a restricted domain.

12.PC.1.5.4.

Relate the general solution of a trigonometric equation to the zeros of the corresponding trigonometric function (restricted to sine and cosine functions).

12.PC.1.5.5.

Determine, using technology, the general solution of a given trigonometric equation.

12.PC.1.5.6.

Identify and correct errors in a solution for a trigonometric equation.

12.PC.1.6.

Prove trigonometric identities, using: reciprocal identities; quotient identities; Pythagorean identities; sum or difference identities (restricted to sine, cosine and tangent); double-angle identities (restricted to sine, cosine and tangent). [R, T, V]

12.PC.1.6.1.

Explain the difference between a trigonometric identity and a trigonometric equation.

12.PC.1.6.2.

Verify a trigonometric identity numerically for a given value in either degrees or radians.

12.PC.1.6.3.

Explain why verifying that the two sides of a trigonometric identity are equal for given values is insufficient to conclude that the identity is valid.

12.PC.1.6.4.

Determine, graphically, the potential validity of a trigonometric identity, using technology.

12.PC.1.6.5.

Determine the non-permissible values of a trigonometric identity.

12.PC.1.6.6.

Prove, algebraically, that a trigonometric identity is valid.

12.PC.1.6.7.

Determine, using the sum, difference and double-angle identities, the exact value of a trigonometric ratio.

12.PC.2.1.

Demonstrate an understanding of operations on, and compositions of, functions. [CN, R, T, V]

12.PC.2.1.2.

Write the equation of a function that is the sum, difference, product or quotient of two or more functions, given their equations.

12.PC.2.1.3.

Determine the domain and range of a function that is the sum, difference, product or quotient of two functions.

12.PC.2.1.4.

Write a function h(x) as the sum, difference, product or quotient of two or more functions.

12.PC.2.1.5.

Determine the value of the composition of functions when evaluated at a point, including: f(f(a)), f(g(a)), g(f(a))

12.PC.2.1.6.

Determine, given the equations of two functions f(x) and g(x), the equation of the composite function:

12.PC.2.1.6.a.

f(f(x)) and explain any restrictions.

12.PC.2.1.6.b.

f(g(x)) and explain any restrictions.

12.PC.2.1.6.c.

g(f(x)) and explain any restrictions.

12.PC.2.1.7.

Sketch, given the equations of two functions f(x) and g(x), the graph of the composite function:

12.PC.2.1.8.

Write a function h(x) as the composition of two or more functions.

12.PC.2.1.9.

Write a function h(x) by combining two or more functions through operations on, and compositions of, functions.

12.PC.2.10.

Solve problems that involve exponential and logarithmic equations. [C, CN, PS, R]

12.PC.2.10.1.

Determine the solution of an exponential equation in which the bases are powers of one another.

12.PC.2.10.2.

Determine the solution of an exponential equation in which the bases are not powers of one another, using a variety of strategies.

12.PC.2.10.5.

Solve a problem that involves exponential growth or decay.

12.PC.2.10.6.

Solve a problem that involves the application of exponential equations to loans, mortgages and investments.

12.PC.2.10.8.

Solve a problem by modelling a situation with an exponential or a logarithmic equation.

12.PC.2.11.

Demonstrate an understanding of factoring polynomials of degree greater than 2 (limited to polynomials of degree < 5 with integral coefficients). [C, CN, ME]

12.PC.2.11.1.

Explain how long division of a polynomial expression by a binomial expression of the form x-a, aI, is related to synthetic division.

12.PC.2.11.2.

Divide a polynomial expression by a binomial expression of the form x-a, aI, using long division or synthetic division.

12.PC.2.11.3.

Explain the relationship between the linear factors of a polynomial expression and the zeros of the corresponding polynomial function.

12.PC.2.11.4.

Explain the relationship between the remainder when a polynomial expression is divided by x-a, aI, and the value of the polynomial expression at x=a (remainder theorem).

12.PC.2.11.5.

Explain and apply the factor theorem to express a polynomial expression as a product of factors.

12.PC.2.12.

Graph and analyze polynomial functions (limited to polynomial functions of degree < 5 ). [C, CN, T, V]

12.PC.2.12.1.

Identify the polynomial functions in a set of functions, and explain the reasoning.

12.PC.2.12.2.

Explain the role of the constant term and leading coefficient in the equation of a polynomial function with respect to the graph of the function.

12.PC.2.12.3.

Generalize rules for graphing polynomial functions of odd or even degree.

12.PC.2.12.4.

Explain the relationship between:

12.PC.2.12.4.a.

The zeros of a polynomial function.

12.PC.2.12.4.b.

The roots of the corresponding polynomial equation.

12.PC.2.12.4.c.

The x-intercepts of the graph of the polynomial function.

12.PC.2.12.6.

Sketch, with or without technology, the graph of a polynomial function.

12.PC.2.12.7.

Solve a problem by modelling a given situation with a polynomial function and analyzing the graph of the function.

12.PC.2.13.

Graph and analyze radical functions (limited to functions involving one radical). [CN, R, T, V]

12.PC.2.13.1.

Sketch the graph of the function y = _x, using a table of values, and state the domain and range.

12.PC.2.13.2.

Sketch the graph of the function y-k = a_(b(x - h)) by applying transformations to the graph of the function y = _x, and state the domain and range.

12.PC.2.13.3.

Sketch the graph of the function y = _(f(x)), given the graph of the function y = f(x), and explain the strategies used.

12.PC.2.13.6.

Determine, graphically, an approximate solution of a radical equation.

12.PC.2.14.

Graph and analyze rational functions (limited to numerators and denominators that are monomials, binomials or trinomials). [CN, R, T, V]

12.PC.2.14.1.

Graph, with or without technology, a rational function.

12.PC.2.14.3.

Explain the behaviour of the graph of a rational function for values of the variable near a non-permissible value.

12.PC.2.14.4.

Determine if the graph of a rational function will have an asymptote or a hole for a non-permissible value.

12.PC.2.14.5.

Match a set of rational functions to their graphs, and explain the reasoning.

12.PC.2.14.6.

Describe the relationship between the roots of a rational equation and the x-intercepts of the graph of the corresponding rational function.

12.PC.2.14.7.

Determine, graphically, an approximate solution of a rational equation.

12.PC.2.2.

Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of functions and their related equations. [C, CN, R, V]

12.PC.2.2.1.

Compare the graphs of a set of functions of the form y k = f (x) to the graph of y = f (x), and generalize, using inductive reasoning, a rule about the effect of k.

12.PC.2.2.2.

Compare the graphs of a set of functions of the form y = f (x - h) to the graph of y = f (x), and generalize, using inductive reasoning, a rule about the effect of h.

12.PC.2.2.3.

Compare the graphs of a set of functions of the form y - k = f (x - h) to the graph of y = f (x), and generalize, using inductive reasoning, a rule about the effects of h and k.

12.PC.2.2.4.

Sketch the graph of y - k = f (x), y = f (x - h) or y - k = f (x - h) for given values of h and k, given a sketch of the function y = f (x), where the equation of y = f (x) is not given.

12.PC.2.2.5.

Write the equation of a function whose graph is a vertical and/or horizontal translation of the graph of the function y = f (x).

12.PC.2.3.

Demonstrate an understanding of the effects of horizontal and vertical stretches on the graphs of functions and their related equations. [C, CN, R, V]

12.PC.2.3.1.

Compare the graphs of a set of functions of the form y = af(x) to the graph of y = f(x), and generalize, using inductive reasoning, a rule about the effect of a.

12.PC.2.3.2.

Compare the graphs of a set of functions of the form y = f(bx) to the graph of y = f(x), and generalize, using inductive reasoning, a rule about the effect of b.

12.PC.2.3.3.

Compare the graphs of a set of functions of the form y =af(bx) to the graph of y = f(x), and generalize, using inductive reasoning, a rule about the effects of a and b.

12.PC.2.3.4.

Sketch the graph of y = af(x), y = f(bx) or y =af(bx) for given values of a and b, given a sketch of the function y = f(x), where the equation of y = f(x) is not given.

12.PC.2.3.5.

Write the equation of a function, given its graph which is a vertical and/or horizontal stretch of the graph of the function y = f(x).

12.PC.2.4.

Apply translations and stretches to the graphs and equations of functions. [C, CN, R, V]

12.PC.2.4.1.

Sketch the graph of the function y-k = af(b(x-h)) for given values of a, b, h and k, given the graph of the function y = f(x), where the equation of y = f(x) is not given.

12.PC.2.4.2.

Write the equation of a function, given its graph which is a translation and/or stretch of the graph of the function y = f(x).

12.PC.2.5.

Demonstrate an understanding of the effects of reflections on the graphs of functions and their related equations, including reflections through the: x-axis; y-axis; line y = x. [C, CN, R, V]

12.PC.2.5.2.

Sketch the reflection of the graph of a function y = f(x) through the x-axis, the y-axis or the line y = x, given the graph of the function y = f(x) , where the equation of y = f(x) is not given.

12.PC.2.5.3.

Generalize, using inductive reasoning, and explain rules for the reflection of the graph of the function y = f(x) through the x-axis, the y-axis or the line y = x.

12.PC.2.5.4.

Sketch the graphs of the functions y = -f(x), y = f(-x) and x = -f(y), given the graph of the function y = f (x), where the equation of y = f(x) is not given.

12.PC.2.5.5.

Write the equation of a function, given its graph which is a reflection of the graph of the function y = f(x) through the x-axis, the y-axis or the line y = x.

12.PC.2.6.

Demonstrate an understanding of inverses of relations. [C, CN, R, V]

12.PC.2.6.1.

Explain how the graph of the line y = x can be used to sketch the inverse of a relation.

12.PC.2.6.2.

Explain how the transformation (x, y) =&gt; (y, x) can be used to sketch the inverse of a relation.

12.PC.2.6.3.

Sketch the graph of the inverse relation, given the graph of a relation.

12.PC.2.6.4.

Determine if a relation and its inverse are functions.

12.PC.2.6.5.

Determine restrictions on the domain of a function in order for its inverse to be a function.

12.PC.2.6.6.

Determine the equation and sketch the graph of the inverse relation, given the equation of a linear or quadratic relation.

12.PC.2.6.8.

Determine, algebraically or graphically, if two functions are inverses of each other.

12.PC.2.7.

Demonstrate an understanding of logarithms. [CN, ME, R]

12.PC.2.7.1.

Explain the relationship between logarithms and exponents.

12.PC.2.7.2.

Express a logarithmic expression as an exponential expression and vice versa.

12.PC.2.7.3.

Determine, without technology, the exact value of a logarithm, such as log2(8).

12.PC.2.7.4.

Estimate the value of a logarithm, using benchmarks, and explain the reasoning; e.g., since log2(8) = 3 and log2(16) = 4, log2(9) is approximately equal to 3.1.

12.PC.2.8.

Demonstrate an understanding of the product, quotient and power laws of logarithms. [C, CN, R, T]

12.PC.2.8.1.

Develop and generalize the laws for logarithms, using numeric examples and exponent laws.

12.PC.2.8.2.

Derive each law of logarithms.

12.PC.2.8.3.

Determine, using the laws of logarithms, an equivalent expression for a logarithmic expression.

12.PC.2.8.4.

Determine, with technology, the approximate value of a logarithmic expression, such as log2(9).

12.PC.2.9.

Graph and analyze exponential and logarithmic functions. [C, CN, T, V]

12.PC.2.9.1.

Sketch, with or without technology, a graph of an exponential function of the form y = a^x, a &gt; 0.

12.PC.2.9.2.

Identify the characteristics of the graph of an exponential function of the form y = a^x, a &gt; 0, including the domain, range, horizontal asymptote and intercepts, and explain the significance of the horizontal asymptote.

12.PC.2.9.3.

Sketch the graph of an exponential function by applying a set of transformations to the graph of y = a^x, a &gt; 0, and state the characteristics of the graph.

12.PC.2.9.4.

Sketch, with or without technology, the graph of a logarithmic function of the form y = logb(x), b &gt; 1.

12.PC.2.9.5.

Identify the characteristics of the graph of a logarithmic function of the form y = logb(x), b &gt; 1, including the domain, range, vertical asymptote and intercepts, and explain the significance of the vertical asymptote.

12.PC.2.9.6.

Sketch the graph of a logarithmic function by applying a set of transformations to the graph of y = logb(x), b &gt; 1, and state the characteristics of the graph.

12.PC.2.9.7.

Demonstrate, graphically, that a logarithmic function and an exponential function with the same base are inverses of each other.

12.PC.3.1.

Apply the fundamental counting principle to solve problems. [C, PS, R, V]

12.PC.3.1.1.

Count the total number of possible choices that can be made, using graphic organizers such as lists and tree diagrams.

12.PC.3.1.2.

Explain, using examples, why the total number of possible choices is found by multiplying rather than adding the number of ways the individual choices can be made.

12.PC.3.1.3.

Solve a simple counting problem by applying the fundamental counting principle.

12.PC.3.2.

Determine the number of permutations of n elements taken r at a time to solve problems. [C, PS, R, V]

12.PC.3.2.1.

Count, using graphic organizers such as lists and tree diagrams, the number of ways of arranging the elements of a set in a row.

12.PC.3.2.2.

Determine, in factorial notation, the number of permutations of n different elements taken n at a time to solve a problem.

12.PC.3.2.3.

Determine, using a variety of strategies, the number of permutations of n different elements taken r at a time to solve a problem.

12.PC.3.2.5.

Solve an equation that involves P(n,r) notation, such as P(n,2)=30.

12.PC.3.2.6.

Explain, using examples, the effect on the total number of permutations when two or more elements are identical.

12.PC.3.3.

Determine the number of combinations of n different elements taken r at a time to solve problems. [C, PS, R, V]

12.PC.3.3.1.

Explain, using examples, the difference between a permutation and a combination.

12.PC.3.4.

Expand powers of a binomial in a variety of ways, including using the binomial theorem (restricted to exponents that are natural numbers). [CN, R, V]

12.PC.3.4.1.

Explain the patterns found in the expanded form of (x+y)^n, n<4 and nN, by multiplying n factors of (x+y).

12.PC.3.4.4.

Explain, using examples, how the coefficients of the terms in the expansion of (x+y)^n are determined by combinations.

12.PC.3.4.5.

Expand, using the binomial theorem, (x+y)^n.

12.PC.3.4.6.

Determine a specific term in the expansion of (x+y)^n.

9.BI.1.

The principles and processes underlying operations with numbers apply equally to algebraic situations and can be described and analyzed.

9.BI.2.

Computational fluency and flexibility with numbers extend to operations with rational numbers.

9.BI.3.

Continuous linear relationships can be identified and represented in many connected ways to identify regularities and make generalizations.

9.BI.4.

Similar shapes have proportional relationships that can be described, measured, and compared.

9.BI.5.

Analyzing the validity, reliability, and representation of data enables us to compare and interpret.

9.C.1.

Operations with rational numbers (addition, subtraction, multiplication, division, and order of operations)

9.C.1.5.

Elements of visual/graphic texts

9.C.2.

Exponents and exponent laws with whole-number exponents

9.C.2.2.

Human sexual reproduction

9.C.2.3.

Metacognitive strategies

9.C.3.

Operations with polynomials, of degree less than or equal to 2

9.C.3.1.

Features of oral language

9.C.3.10.

Connotation and denotation

9.C.3.6.

Syntax and sentence fluency

9.C.3.8.

Presentation techniques

9.C.4.

Two-variable linear relations, using graphing, interpolation, and extrapolation

9.C.5.

Multi-step one-variable linear equations

9.C.6.

Spatial proportional reasoning

9.C.7.

Statistics in society

9.C.8.

Financial literacy simple budgets and transaction

9.CC.1.

Reasoning and analyzing

9.CC.1.1.

Use logic and patterns to solve puzzles and play games

9.CC.1.10.

Explain how literary elements, techniques, and devices enhance and shape meaning

9.CC.1.11.

Recognize an increasing range of text structures and how they contribute to meaning

9.CC.1.12.

Recognize and appreciate the role of story, narrative, and oral tradition in expressing First Peoples perspectives, values, beliefs, and points of view

9.CC.1.13.

Develop an awareness of the diversity within and across First Peoples societies represented in texts

9.CC.1.14.

Recognize the influence of place in First Peoples and other Canadian texts

9.CC.1.2.

Use reasoning and logic to explore, analyze, and apply mathematical ideas

9.CC.1.3.

Formulate multiple hypotheses and predict multiple outcomes

9.CC.1.4.

Demonstrate and apply mental math strategies

9.CC.1.5.

Use tools or technology to explore and create patterns and relationships, and test conjectures

9.CC.1.6.

Model mathematics in contextualized experiences

9.CC.1.7.

Recognize how language constructs personal, social, and cultural identity

9.CC.1.8.

Construct meaningful personal connections between self, text, and world

9.CC.1.9.

Respond to text in personal, creative, and critical ways

9.CC.2.

Understanding and solving

9.CC.2.1.

Apply multiple strategies to solve problems in both abstract and contextualized situations

9.CC.2.2.

Develop, demonstrate, and apply mathematical understanding through play, inquiry, and problem solving

9.CC.2.3.

Visualize to explore mathematical concepts

9.CC.2.4.

Engage in problem-solving experiences that are connected to place, story, cultural practices, and perspectives relevant to local First Peoples communities, the local community, and other cultures

9.CC.2.5.

Use and experiment with oral storytelling processes

9.CC.2.6.

Select and use appropriate features, forms, and genres according to audience, purpose, and message

9.CC.2.7.

Transform ideas and information to create original texts

9.CC.2.8.

Express an opinion and support it with credible evidence

9.CC.3.

Communicating and representing

9.CC.3.1.

Use mathematical vocabulary and language to contribute to mathematical discussions

9.CC.3.2.

Explain and justify mathematical ideas and decisions

9.CC.3.3.

Communicate mathematical thinking in many ways

9.CC.3.4.

Construct, analyze and interpret graphs (including interpolation and extrapolation), models and/or diagrams

9.CC.3.5.

Use knowledge of scientific concepts to draw conclusions that are consistent with evidence

9.CC.3.6.

Analyze cause-and-effect relationships

9.CC.4.

Connecting and reflecting

9.CC.4.1.

Reflect on mathematical thinking

9.CC.4.2.

Connect mathematical concepts to each other and to other areas and personal interests

9.CC.4.3.

Use mathematical arguments to support personal choices

9.CC.4.4.

Demonstrate an awareness of assumptions, question information given, and identify bias in their own work and secondary sources

9.CC.4.5.

Consider the changes in knowledge over time as tools and technologies have developed

9.CC.4.6.

Connect scientific explorations to careers in science

9.CC.4.7.

Exercise a healthy, informed skepticism, and use scientific knowledge and findings to form their own investigations and to evaluate claims in secondary sources

9.CC.4.8.

Consider social, ethical, and environmental implications of the findings from their own and others' investigations

9.CC.4.9.

Critically analyze the validity of information in secondary sources and evaluate the approaches used to solve problems

9.CC.5.

Assess how prevailing conditions and the actions of individuals or groups affect events, decisions, or developments (cause and consequence)

9.CC.5.1.

Contribute to care for self, others, community, and world through individual or collaborative approaches

9.CC.5.2.

Transfer and apply learning to new situations

9.CC.5.3.

Generate and introduce new or refined ideas when problem solving

9.CC.5.4.

Contribute to finding solutions to problems at a local and/or global level through inquiry

9.CC.5.5.

Consider the role of scientists in innovation

9.CC.6.

Explain and infer different perspectives on past or present people, places, issues, or events by considering prevailing norms, values, worldviews, and beliefs (perspective)

9.CC.6.1.

Formulate physical or mental theoretical models to describe a phenomenon

9.CC.6.2.

Communicate scientific ideas, claims, information, and perhaps a suggested course of action, for a specific purpose and audience, constructing evidence-based arguments and using appropriate scientific language, conventions, and representations

9.CC.7.

Recognize implicit and explicit ethical judgments in a variety of sources (ethical judgment)

9.CC.8.

Make reasoned ethical judgments about actions in the past and present, and determine appropriate ways to remember and respond (ethical judgment)

A1.1.

Support and extend the learning of self and others

A1.2.

Explore experiences, ideas, and information

A1.3.

Gain insight into others&apos; perspectives

A1.4.

Respond to and analyse a variety of texts

A1.5.

Create a variety of texts

A10.1.

Personalizing ideas and information

A10.2.

Explaining relationships among ideas and information

A10.3.

Applying new ideas and information

A10.4.

Transforming existing ideas and information

A10.5.

Contextualizing ideas and information

A11.1.

Referring to criteria

A11.2.

Setting goals for improvement

A11.3.

Creating a plan for achieving goals

A11.4.

Evaluating progress and setting new goals

A12.8.

Nonverbal techniques

A12.9.

Idiomatic expressions

A2.1.

Explore and respond

A2.2.

Recall and describe

A2.3.

Narrate and explain

A2.4.

Argue, persuade, and support

A2.5.

Engage and entertain

A3.7.

Context, including historical and cultural influences

A4.1.

Initiating and sharing responsibilities

A4.3.

Contributing ideas and supporting the ideas of others

A4.4.

Acknowledging and discussing diverse points of view

A4.5.

Reaching consensus or agreeing to differ

A5.1.

Interpreting a task and setting a purpose

A5.3.

Considering multiple perspectives

A5.5.

Planning and rehearsing presentations

A6.3.

Nonverbal techniques

A6.5.

Organizational and memory aids

A7.1.

Extending understanding by accessing prior knowledge

A7.2.

Making plausible predictions

A7.3.

Summarizing main points

A7.4.

Generating thoughtful questions

A7.5.

Clarifying and confirming meaning

A8.1.

Making connections with prior knowledge and experiences

A8.2.

Relating reactions and emotions to understanding of the text

A8.3.

Generating thoughtful questions

A8.5.

Explaining opinions using reasons and evidence

A9.1.

Making and supporting reasoned judgments

A9.2.

Examining and comparing ideas and elements among texts

A9.3.

Describing and comparing perspectives

A9.4.

Describing bias, contradictions, and non-represented perspectives

A9.5.

Identifying the importance and impact of historical and cultural contexts

B1.1.

Literature reflecting a variety of times, places, and perspectives

B1.2.

Literature reflecting a variety of prose forms

B1.3.

Poetry in a variety of narrative and lyric forms

B1.4.

Significant works of Canadian literature (e.g., the study of plays, short stories, poetry, or novels)

B1.5.

Traditional forms from Aboriginal and other cultures

B1.6.

Student-generated material

B10.1.

Personalizing ideas and information

B10.2.

Explaining relationships among ideas and information

B10.5.

Contextualizing ideas and information

B11.1.

Referring to criteria

B11.2.

Setting goals for improvement

B11.3.

Creating a plan for achieving goals

B11.4.

Evaluating progress and setting new goals

B12.6.

Non-fiction elements

B13.1.

Analysing the origins and roots of words

B13.2.

Determining meanings and uses of words based on context

B13.3.

Identifying, selecting, and using appropriate academic and technical language

B13.4.

Using vocabulary appropriate to audience and purpose

B13.5.

Discerning nuances in meaning of words considering historical, cultural, and literary contexts

B2.1.

Articles and reports

B2.2.

Biographies and autobiographies

B2.3.

Textbooks, magazines, and newspapers

B2.4.

Print and electronic reference material

B2.5.

Advertising and promotional material

B2.6.

Opinion-based material

B2.7.

Student-generated material

B3.8.

Student-generated material

B5.1.

Interpreting a task

B5.2.

Setting a purpose or multiple purposes

B5.3.

Accessing prior knowledge, including knowledge of genre, form, and context

B5.5.

Generating guiding or speculative questions

B6.1.

Comparing and refining predictions, questions, images, and connections

B6.2.

Making inferences and drawing conclusions

B6.3.

Summarizing and paraphrasing

B6.4.

Using text features

B6.5.

Determining the meaning of unknown words and phrases

B7.1.

Reflecting on predictions, questions, images, and connections made during reading

B7.2.

Reviewing text and purpose for reading

B7.3.

Making inferences and drawing conclusions

B7.4.

Summarizing, synthesizing, and applying ideas:

B7.5.

Identifying stylistic techniques

B8.1.

Making comparisons to other ideas and concepts

B8.2.

Relating reactions and emotions to understanding of the text

B8.3.

Explaining opinions using reasons and evidence

B8.4.

Suggesting contextual influences

B9.1.

Making and supporting reasoned judgments

B9.2.

Comparing ideas and elements among texts

B9.3.

Identifying and describing diverse voices

B9.4.

Describing bias, contradictions, and non-represented perspectives

B9.5.

Identifying the importance and impact of historical and cultural contexts

C1.4.

Reflect and respond

C10.1.

Personalizing ideas and information

C10.2.

Explaining relationships among ideas and information

C12.1.

Syntax and sentence fluency

C13.1.

Organization of ideas and information

C14.2.

Punctuation, capitalization, and Canadian spelling

C14.3.

Copyright and citation of references

C14.4.

Presentation/layout

C2.2.

Record and describe

C2.3.

Analyse and explain

C2.4.

Speculate and consider

C3.1.

Strengthen connections and insights

C3.2.

Explore and adapt literary forms and techniques

C3.3.

Experiment with increasingly sophisticated language and style

C3.4.

Engage and entertain

C4.1.

Explore and respond

C4.2.

Record and describe

C4.3.

Explain and persuade

C5.2.

Setting a purpose and considering audience

C5.3.

Gathering and summarizing ideas from personal interest, knowledge, and inquiry

C5.4.

Analysing writing samples or models

C6.1.

Using a variety of sources to collect ideas and information

C6.4.

Analysing writing samples or models

C7.2.

Enhancing supporting details and examples

C7.3.

Refining specific aspects and features of text

C8.2.

Relating reactions and emotions to understanding of the text

C8.3.

Developing opinions using reasons and evidence

C8.4.

Suggesting contextual influences

C9.1.

Making and supporting reasoned judgments

C9.2.

Describing and comparing perspectives

C9.3.

Describing bias, contradictions, and non-represented perspectives

C9.4.

Identifying the importance and impact of historical and cultural contexts