Northwest Territories

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Skills available for Northwest Territories grade 12 math curriculum

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T Trigonometry

  • T.1 Develop trigonometric reasoning.

    • T.1.1 Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

      • T.1.1.1 Sketch, in standard position, an angle (positive or negative) when the measure is given in degrees.

      • T.1.1.2 Describe the relationship among different systems of angle measurement, with emphasis on radians and degrees.

      • T.1.1.3 Sketch, in standard position, an angle with a measure of 1 radian.

      • T.1.1.4 Sketch, in standard position, an angle with a measure expressed in the form kπ radians, where k ∈ Q.

      • T.1.1.5 Express the measure of an angle in radians (exact value or decimal approximation), given its measure in degrees.

      • T.1.1.6 Express the measure of an angle in degrees, given its measure in radians (exact value or decimal approximation).

      • T.1.1.7 Determine the measures, in degrees or radians, of all angles in a given domain that are coterminal with a given angle in standard position.

      • T.1.1.8 Determine the general form of the measures, in degrees or radians, of all angles that are coterminal with a given angle in standard position.

      • T.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.

    • T.1.2 Develop and apply the equation of the unit circle.

      • T.1.2.1 Derive the equation of the unit circle from the Pythagorean theorem.

      • T.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.

      • T.1.2.3 Generalize the equation of a circle with centre (0, 0) and radius r.

    • T.1.3 Solve problems, using the six trigonometric ratios for angles expressed in radians and degrees.

      • T.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.

      • T.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, π/6, π/4, π/3 or π/2, and explain the strategy.

      • T.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.

      • T.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.

      • T.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.

      • T.1.3.6 Determine the exact values of the other trigonometric ratios, given the value of one trigonometric ratio in a specified domain.

      • T.1.3.7 Sketch a diagram to represent a problem that involves trigonometric ratios.

      • T.1.3.8 Solve a problem, using trigonometric ratios.

    • T.1.4 Graph and analyze the trigonometric functions sine, cosine and tangent to solve problems.

      • T.1.4.1 Sketch, with or without technology, the graph of y = sin x, y = cos x or y = tan x.

      • T.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.

      • T.1.4.3 Determine how varying the value of a affects the graphs of y = a sin x and y = a cos x.

      • T.1.4.4 Determine how varying the value of d affects the graphs of y = sin x + d and y = cos x + d.

      • T.1.4.5 Determine how varying the value of c affects the graphs of y = sin (x + c) and y = cos (x + c).

      • T.1.4.6 Determine how varying the value of b affects the graphs of y = sin bx and y = cos bx.

      • T.1.4.7 Sketch, without technology, graphs of the form y = a sin b(x – c) + d and y = a cos b(x – c) + d, using transformations, and explain the strategies.

      • T.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.

      • T.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.

      • T.1.4.10 Determine a trigonometric function that models a situation to solve a problem.

      • T.1.4.11 Explain how the characteristics of the graph of a trigonometric function relate to the conditions in a problem situation.

      • T.1.4.12 Solve a problem by analyzing the graph of a trigonometric function.

    • T.1.5 Solve, algebraically and graphically, first and second degree trigonometric equations with the domain expressed in degrees and radians.

      • T.1.5.1 Verify, with or without technology, that a given value is a solution to a trigonometric equation.

      • T.1.5.2 Determine, algebraically, the solution of a trigonometric equation, stating the solution in exact form when possible.

      • T.1.5.3 Determine, using technology, the approximate solution of a trigonometric equation in a restricted domain.

      • T.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).

      • T.1.5.5 Determine, using technology, the general solution of a given trigonometric equation.

      • T.1.5.6 Identify and correct errors in a solution for a trigonometric equation.

    • T.1.6 Prove trigonometric identities, using:

      • T.1.6.a reciprocal identities

      • T.1.6.b quotient identities

      • T.1.6.c Pythagorean identities

      • T.1.6.d sum or difference identities (restricted to sine, cosine and tangent)

      • T.1.6.e double-angle identities (restricted to sine, cosine and tangent).

      • T.1.6.1 Explain the difference between a trigonometric identity and a trigonometric equation.

      • T.1.6.2 Verify a trigonometric identity numerically for a given value in either degrees or radians.

      • T.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.

      • T.1.6.4 Determine, graphically, the potential validity of a trigonometric identity, using technology.

      • T.1.6.5 Determine the non-permissible values of a trigonometric identity.

      • T.1.6.6 Prove, algebraically, that a trigonometric identity is valid.

      • T.1.6.7 Determine, using the sum, difference and double-angle identities, the exact value of a trigonometric ratio.

RF Relations and Functions

  • RF.1 Develop algebraic and graphical reasoning through the study of relations.

    • RF.1.1 Demonstrate an understanding of operations on, and compositions of, functions.

      • RF.1.1.1 Sketch the graph of a function that is the sum, difference, product or quotient of two functions, given their graphs.

      • RF.1.1.2 Write the equation of a function that is the sum, difference, product or quotient of two or more functions, given their equations.

      • RF.1.1.3 Determine the domain and range of a function that is the sum, difference, product or quotient of two functions.

      • RF.1.1.4 Write a function h(x) as the sum, difference, product or quotient of two or more functions.

      • RF.1.1.5 Determine the value of the composition of functions when evaluated at a point, including:

        • RF.1.1.5.a f(f(a))

        • RF.1.1.5.b f(g(a))

        • RF.1.1.5.c g(f(a)).

      • RF.1.1.6 Determine, given the equations of two functions f(x) and g(x), the equation of the composite function:

      • RF.1.1.7 Sketch, given the equations of two functions f(x) and g(x), the graph of the composite function:

        • RF.1.1.7.a f(f(x))

        • RF.1.1.7.b f(g(x))

        • RF.1.1.7.c g(f(x)).

      • RF.1.1.8 Write a function h(x) as the composition of two or more functions.

      • RF.1.1.9 Write a function h(x) by combining two or more functions through operations on, and compositions of, functions.

    • RF.1.2 Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of functions and their related equations.

      • RF.1.2.1 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.

      • RF.1.2.2 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.

      • RF.1.2.3 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.

      • RF.1.2.4 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.

      • RF.1.2.5 Write the equation of a function whose graph is a vertical and/or horizontal translation of the graph of the function y = f(x).

    • RF.1.3 Demonstrate an understanding of the effects of horizontal and vertical stretches on the graphs of functions and their related equations.

      • RF.1.3.1 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.

      • RF.1.3.2 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.

      • RF.1.3.3 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.

      • RF.1.3.4 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.

      • RF.1.3.5 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).

    • RF.1.4 Apply translations and stretches to the graphs and equations of functions.

    • RF.1.5 Demonstrate an understanding of the effects of reflections on the graphs of functions and their related equations, including reflections through the:

      • RF.1.5.a x-axis

      • RF.1.5.b y-axis

      • RF.1.5.c line y = x.

      • RF.1.5.1 Generalize the relationship between the coordinates of an ordered pair and the coordinates of the corresponding ordered pair that results from a reflection through the x-axis, the y-axis or the line y = x.

      • RF.1.5.2 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.

      • RF.1.5.3 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.

      • RF.1.5.4 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.

      • RF.1.5.5 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.

    • RF.1.6 Demonstrate an understanding of inverses of relations.

      • RF.1.6.1 Explain how the graph of the line y = x can be used to sketch the inverse of a relation.

      • RF.1.6.2 Explain how the transformation (x,y) ⇒ (y,x) can be used to sketch the inverse of a relation.

      • RF.1.6.3 Sketch the graph of the inverse relation, given the graph of a relation.

      • RF.1.6.4 Determine if a relation and its inverse are functions.

      • RF.1.6.5 Determine restrictions on the domain of a function in order for its inverse to be a function.

      • RF.1.6.6 Determine the equation and sketch the graph of the inverse relation, given the equation of a linear or quadratic relation.

      • RF.1.6.7 Explain the relationship between the domains and ranges of a relation and its inverse.

      • RF.1.6.8 Determine, algebraically or graphically, if two functions are inverses of each other.

    • RF.1.7 Demonstrate an understanding of logarithms.

      • RF.1.7.1 Explain the relationship between logarithms and exponents.

      • RF.1.7.2 Express a logarithmic expression as an exponential expression and vice versa.

      • RF.1.7.3 Determine, without technology, the exact value of a logarithm, such as log? 8.

      • RF.1.7.4 Estimate the value of a logarithm, using benchmarks, and explain the reasoning; e.g., since log? 8 = 3 and log? 16 = 4, log? 9 is approximately equal to 3.1.

    • RF.1.8 Demonstrate an understanding of the product, quotient and power laws of logarithms.

    • RF.1.9 Graph and analyze exponential and logarithmic functions.

      • RF.1.9.1 Sketch, with or without technology, a graph of an exponential function of the form y = a?, a > 0.

      • RF.1.9.2 Identify the characteristics of the graph of an exponential function of the form y = a?, a > 0, including the domain, range, horizontal asymptote and intercepts, and explain the significance of the horizontal asymptote.

      • RF.1.9.3 Sketch the graph of an exponential function by applying a set of transformations to the graph y = a?, a > 0, and state the characteristics of the graph.

      • RF.1.9.4 Sketch, with or without technology, the graph of a logarithmic function of the form y = log[subcript b] x, b > 1.

      • RF.1.9.5 Identify the characteristics of the graph of a logarithmic function of the form y = log[subscript b] x, b > 1, including the domain, range, vertical asymptote and intercepts, and explain the significance of the vertical asymptote.

      • RF.1.9.6 Sketch the graph of a logarithmic function by applying a set of transformations to the graph y = log[subscript b] x, b > 1, and state the characteristics of the graph.

      • RF.1.9.7 Demonstrate, graphically, that a logarithmic function and an exponential function with the same base are inverses of each other.

    • RF.1.10 Solve problems that involve exponential and logarithmic equations.

    • RF.1.11 Demonstrate an understanding of factoring polynomials of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients).

      • RF.1.11.1 Explain how long division of a polynomial expression by a binomial expression of the form x – a, a ∈ I, is related to synthetic division.

      • RF.1.11.2 Divide a polynomial expression by a binomial expression of the form x – a, a ∈ I, using long division or synthetic division.

      • RF.1.11.3 Explain the relationship between the linear factors of a polynomial expression and the zeros of the corresponding polynomial function.

      • RF.1.11.4 Explain the relationship between the remainder when a polynomial expression is divided by x – a, a ∈ I, and the value of the polynomial expression at x = a (remainder theorem).

      • RF.1.11.5 Explain and apply the factor theorem to express a polynomial expression as a product of factors.

    • RF.1.12 Graph and analyze polynomial functions (limited to polynomial functions of degree ≤ 5).

      • RF.1.12.1 Identify the polynomial functions in a set of functions, and explain the reasoning.

      • RF.1.12.2 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.

      • RF.1.12.3 Generalize rules for graphing polynomial functions of odd or even degree.

      • RF.1.12.4 Explain the relationship between:

        • RF.1.12.4.a the zeros of a polynomial function

        • RF.1.12.4.b the roots of the corresponding polynomial equation

        • RF.1.12.4.c the x-intercepts of the graph of the polynomial function.

      • RF.1.12.5 Explain how the multiplicity of a zero of a polynomial function affects the graph.

      • RF.1.12.6 Sketch, with or without technology, the graph of a polynomial function.

      • RF.1.12.7 Solve a problem by modelling a given situation with a polynomial function and analyzing the graph of the function.

    • RF.1.13 Graph and analyze radical functions (limited to functions involving one radical).

      • RF.1.13.1 Sketch the graph of the function y = √ x, using a table of values, and state the domain and range.

      • RF.1.13.2 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.

      • RF.1.13.3 Sketch the graph of the function y = √f(x), given the graph of the function y = f(x), and explain the strategies used.

      • RF.1.13.4 Compare the domain and range of the function y = √f(x), to the domain and range of the function y = f(x), and explain why the domains and ranges may differ.

      • RF.1.13.5 Describe the relationship between the roots of a radical equation and the x-intercepts of the graph of the corresponding radical function.

      • RF.1.13.6 Determine, graphically, an approximate solution of a radical equation.

    • RF.1.14 Graph and analyze rational functions (limited to numerators and denominators that are monomials, binomials or trinomials).

      • RF.1.14.1 Graph, with or without technology, a rational function.

      • RF.1.14.2 Analyze the graphs of a set of rational functions to identify common characteristics.

      • RF.1.14.3 Explain the behaviour of the graph of a rational function for values of the variable near a non-permissible value.

      • RF.1.14.4 Determine if the graph of a rational function will have an asymptote or a hole for a non-permissible value.

      • RF.1.14.5 Match a set of rational functions to their graphs, and explain the reasoning.

      • RF.1.14.6 Describe the relationship between the roots of a rational equation and the x-intercepts of the graph of the corresponding rational function.

      • RF.1.14.7 Determine, graphically, an approximate solution of a rational equation.

PCB Permutations, Combinations and Binomial Theorem

  • PCB.1 Develop algebraic and numeric reasoning that involves combinatorics.

    • PCB.1.1 Apply the fundamental counting principle to solve problems.

      • PCB.1.1.1 Count the total number of possible choices that can be made, using graphic organizers such as lists and tree diagrams.

      • PCB.1.1.2 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.

      • PCB.1.1.3 Solve a simple counting problem by applying the fundamental counting principle.

    • PCB.1.2 Determine the number of permutations of n elements taken r at a time to solve problems.

      • PCB.1.2.1 Count, using graphic organizers such as lists and tree diagrams, the number of ways of arranging the elements of a set in a row.

      • PCB.1.2.2 Determine, in factorial notation, the number of permutations of n different elements taken n at a time to solve a problem.

      • PCB.1.2.3 Determine, using a variety of strategies, the number of permutations of n different elements taken r at a time to solve a problem.

      • PCB.1.2.4 Explain why n must be greater than or equal to r in the notation ?P?.

      • PCB.1.2.5 Solve an equation that involves ?P? notation, such as ?P? = 30.

      • PCB.1.2.6 Explain, using examples, the effect on the total number of permutations when two or more elements are identical.

    • PCB.1.3 Determine the number of combinations of n different elements taken r at a time to solve problems.

      • PCB.1.3.1 Explain, using examples, the difference between a permutation and a combination.

      • PCB.1.3.2 Determine the number of ways that a subset of k elements can be selected from a set of n different elements.

      • PCB.1.3.3 Determine the number of combinations of n different elements taken r at a time to solve a problem.

      • PCB.1.3.4 Explain why n must be greater than or equal to r in the notation ?C? or (n [choose] r).

      • PCB.1.3.5 Explain, using examples, why ?C? = ?C?-?, or (n [choose] r) = (n [choose] n – r).

      • PCB.1.3.6 Solve an equation that involves ?C? or (n [choose] r) notation, such as ?C? = 15 or (n [choose] 2) = 15.

    • PCB.1.4 Expand powers of a binomial in a variety of ways, including using the binomial theorem (restricted to exponents that are natural numbers).

      • PCB.1.4.1 Explain the patterns found in the expanded form of (x + y)n, n ≤ 4 and n ∈ N, by multiplying n factors of (x + y).

      • PCB.1.4.2 Explain how to determine the subsequent row in Pascal's triangle, given any row.

      • PCB.1.4.3 Relate the coefficients of the terms in the expansion of (x + y)n to the (n + 1) row in Pascal's triangle.

      • PCB.1.4.4 Explain, using examples, how the coefficients of the terms in the expansion of (x + y)n are determined by combinations.

      • PCB.1.4.5 Expand, using the binomial theorem, (x + y)n.

      • PCB.1.4.6 Determine a specific term in the expansion of (x + y)n.