Northwest Territories

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Skills available for Northwest Territories grade 11 math curriculum

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AN Algebra and Number

  • AN.1 Develop algebraic reasoning and number sense.

    • AN.1.1 Demonstrate an understanding of the absolute value of real numbers.

      • AN.1.1.1 Determine the distance of two real numbers of the form ±a, a ∈ R, from 0 on a number line, and relate this to the absolute value of a (|a|).

      • AN.1.1.2 Determine the absolute value of a positive or negative real number.

      • AN.1.1.3 Explain, using examples, how distance between two points on a number line can be expressed in terms of absolute value.

      • AN.1.1.4 Determine the absolute value of a numerical expression.

      • AN.1.1.5 Compare and order the absolute values of real numbers in a given set.

    • AN.1.2 Solve problems that involve operations on radicals and radical expressions with numerical and variable radicands.

      • AN.1.2.1 Compare and order radical expressions with numerical radicands in a given set.

      • AN.1.2.2 Express an entire radical with a numerical radicand as a mixed radical.

      • AN.1.2.3 Express a mixed radical with a numerical radicand as an entire radical.

      • AN.1.2.4 Perform one or more operations to simplify radical expressions with numerical or variable radicands.

      • AN.1.2.5 Rationalize the denominator of a rational expression with monomial or binomial denominators.

      • AN.1.2.6 Describe the relationship between rationalizing a binomial denominator of a rational expression and the product of the factors of a difference of squares expression.

      • AN.1.2.7 Explain, using examples, that (–x)² = x², √x² = |x| and √x² ≠ ±x; e.g., √9 ≠ ±3.

      • AN.1.2.8 Identify the values of the variable for which a given radical expression is defined.

      • AN.1.2.9 Solve a problem that involves radical expressions.

    • AN.1.3 Solve problems that involve radical equations (limited to square roots).

      • AN.1.3.1 Determine any restrictions on values for the variable in a radical equation.

      • AN.1.3.2 Determine the roots of a radical equation algebraically, and explain the process used to solve the equation.

      • AN.1.3.3 Verify, by substitution, that the values determined in solving a radical equation algebraically are roots of the equation.

      • AN.1.3.4 Explain why some roots determined in solving a radical equation algebraically are extraneous.

      • AN.1.3.5 Solve problems by modelling a situation using a radical equation.

    • AN.1.4 Determine equivalent forms of rational expressions (limited to numerators and denominators that are monomials, binomials or trinomials).

      • AN.1.4.1 Compare the strategies for writing equivalent forms of rational expressions to the strategies for writing equivalent forms of rational numbers.

      • AN.1.4.2 Explain why a given value is non-permissible for a given rational expression.

      • AN.1.4.3 Determine the non-permissible values for a rational expression.

      • AN.1.4.4 Determine a rational expression that is equivalent to a given rational expression by multiplying the numerator and denominator by the same factor (limited to a monomial or a binomial), and state the non-permissible values of the equivalent rational expression.

      • AN.1.4.5 Simplify a rational expression.

      • AN.1.4.6 Explain why the non-permissible values of a given rational expression and its simplified form are the same.

      • AN.1.4.7 Identify and correct errors in a simplification of a rational expression, and explain the reasoning.

    • AN.1.5 Perform operations on rational expressions (limited to numerators and denominators that are monomials, binomials or trinomials).

      • AN.1.5.1 Compare the strategies for performing a given operation on rational expressions to the strategies for performing the same operation on rational numbers.

      • AN.1.5.2 Determine the non-permissible values when performing operations on rational expressions.

      • AN.1.5.3 Determine, in simplified form, the sum or difference of rational expressions with the same denominator.

      • AN.1.5.4 Determine, in simplified form, the sum or difference of rational expressions in which the denominators are not the same and which may or may not contain common factors.

      • AN.1.5.5 Determine, in simplified form, the product or quotient of rational expressions.

      • AN.1.5.6 Simplify an expression that involves two or more operations on rational expressions.

    • AN.1.6 Solve problems that involve rational equations (limited to numerators and denominators that are monomials, binomials or trinomials).

      • AN.1.6.1 Determine the non-permissible values for the variable in a rational equation.

      • AN.1.6.2 Determine the solution to a rational equation algebraically, and explain the process used to solve the equation.

      • AN.1.6.3 Explain why a value obtained in solving a rational equation may not be a solution of the equation.

      • AN.1.6.4 Solve problems by modelling a situation using a rational equation.

T Trigonometry

  • T.1 Develop trigonometric reasoning.

    • T.1.1 Demonstrate an understanding of angles in standard position [0° to 360°].

      • T.1.1.1 Sketch an angle in standard position, given the measure of the angle.

      • T.1.1.2 Determine the reference angle for an angle in standard position.

      • T.1.1.3 Explain, using examples, how to determine the angles from 0° to 360° that have the same reference angle as a given angle.

      • T.1.1.4 Illustrate, using examples, that any angle from 90° to 360° is the reflection in the x-axis and/or the y-axis of its reference angle.

      • T.1.1.5 Determine the quadrant in which a given angle in standard position terminates.

      • T.1.1.6 Draw an angle in standard position given any point P (x,y) on the terminal arm of the angle.

      • T.1.1.7 Illustrate, using examples, that the points P(x,y), P(–x,y), P(–x,–y) and P(x,–y) are points on the terminal sides of angles in standard position that have the same reference angle.

    • T.1.2 Solve problems, using the three primary trigonometric ratios for angles from 0° to 360° in standard position.

      • T.1.2.1 Determine, using the Pythagorean theorem or the distance formula, the distance from the origin to a point P(x,y) on the terminal arm of an angle.

      • T.1.2.2 Determine the value of sin θ, cos θ or tan θ, given any point P(x,y) on the terminal arm of angle θ.

      • T.1.2.3 Determine, without the use of technology, the value of sin θ, cos θ or tan θ, given any point P(x,y) on the terminal arm of angle θ, where θ = 0&sup0;, 90&sup0;, 180&sup0;, 270&sup0; or 360&sup0;.

      • T.1.2.4 Determine the sign of a given trigonometric ratio for a given angle, without the use of technology, and explain.

      • T.1.2.5 Solve, for all values of θ, an equation of the form sin θ = a or cos θ = a, where –1 ≤ a ≤ 1, and an equation of the form tan θ = a, where a is a real number.

      • T.1.2.6 Determine the exact value of the sine, cosine or tangent of a given angle with a reference angle of 30&sup0;, 45&sup0; or 60&sup0;.

      • T.1.2.7 Describe patterns in and among the values of the sine, cosine and tangent ratios for angles from 0&sup0; to 360&sup0;.

      • T.1.2.8 Sketch a diagram to represent a problem.

      • T.1.2.9 Solve a contextual problem, using trigonometric ratios.

    • T.1.3 Solve problems, using the cosine law and sine law, including the ambiguous case.

      • T.1.3.1 Sketch a diagram to represent a problem that involves a triangle without a right angle.

      • T.1.3.2 Solve, using primary trigonometric ratios, a triangle that is not a right triangle.

      • T.1.3.3 Explain the steps in a given proof of the sine law or cosine law.

      • T.1.3.4 Sketch a diagram and solve a problem, using the cosine law.

      • T.1.3.5 Sketch a diagram and solve a problem, using the sine law.

      • T.1.3.6 Describe and explain situations in which a problem may have no solution, one solution or two solutions.

RF Relations and Functions

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

    • RF.1.1 Factor polynomial expressions of the form:

      • RF.1.1.a ax² + bx + c, a ≠ 0

      • RF.1.1.b a²x² – b²y², a ≠ 0, b ≠ 0

      • RF.1.1.c a(f(x))² + b(f(x)) + c, a ≠ 0

      • RF.1.1.d a²(f(x))² – b²(g(y))², a ≠ 0, b ≠ 0,

      • RF.1.1.1 Factor a given polynomial expression that requires the identification of common factors.

      • RF.1.1.2 Determine whether a given binomial is a factor for a given polynomial expression, and explain why or why not.

      • RF.1.1.3 Factor a given polynomial expression of the form:

      • RF.1.1.4 Factor a given polynomial expression that has a quadratic pattern, including:

    • RF.1.2 Graph and analyze absolute value functions (limited to linear and quadratic functions) to solve problems.

      • RF.1.2.1 Create a table of values for y = |f(x)|, given a table of values for y = f(x).

      • RF.1.2.2 Generalize a rule for writing absolute value functions in piecewise notation.

      • RF.1.2.3 Sketch the graph of y = |f(x)|; state the intercepts, domain and range; and explain the strategy used.

      • RF.1.2.4 Solve an absolute value equation graphically, with or without technology.

      • RF.1.2.5 Solve, algebraically, an equation with a single absolute value, and verify the solution.

      • RF.1.2.6 Explain why the absolute value equation |f(x)| < 0 has no solution.

      • RF.1.2.7 Determine and correct errors in a solution to an absolute value equation.

      • RF.1.2.8 Solve a problem that involves an absolute value function.

    • RF.1.3 Analyze quadratic functions of the form y = a(x – p)² + q and determine the:

      • RF.1.3.a vertex

      • RF.1.3.b domain and range

      • RF.1.3.c direction of opening

      • RF.1.3.d axis of symmetry

      • RF.1.3.e x- and y-intercepts.

      • RF.1.3.1 Explain why a function given in the form y = a(x – p)² + q is a quadratic function.

      • RF.1.3.2 Compare the graphs of a set of functions of the form y = ax² to the graph of y = x², and generalize, using inductive reasoning, a rule about the effect of a.

      • RF.1.3.3 Compare the graphs of a set of functions of the form y = x² + q to the graph of y = x², and generalize, using inductive reasoning, a rule about the effect of q.

      • RF.1.3.4 Compare the graphs of a set of functions of the form y = (x – p)² to the graph of y = x², and generalize, using inductive reasoning, a rule about the effect of p.

      • RF.1.3.5 Determine the coordinates of the vertex for a quadratic function of the form y = a(x – p)² + q, and verify with or without technology.

      • RF.1.3.6 Generalize, using inductive reasoning, a rule for determining the coordinates of the vertex for quadratic functions of the form y = a(x – p)² + q.

      • RF.1.3.7 Sketch the graph of y = a(x – p)² + q, using transformations, and identify the vertex, domain and range, direction of opening, axis of symmetry and x- and y-intercepts.

      • RF.1.3.8 Explain, using examples, how the values of a and q may be used to determine whether a quadratic function has zero, one or two x-intercepts.

      • RF.1.3.9 Write a quadratic function in the form y = a(x – p)² + q for a given graph or a set of characteristics of a graph.

    • RF.1.4 Analyze quadratic functions of the form y = ax² + bx + c to identify characteristics of the corresponding graph, including:

      • RF.1.4.a vertex

      • RF.1.4.b domain and range

      • RF.1.4.c direction of opening

      • RF.1.4.d axis of symmetry

      • RF.1.4.e x- and y-intercepts

      • RF.1.4.1 Explain the reasoning for the process of completing the square as shown in a given example.

      • RF.1.4.2 Write a quadratic function given in the form y = ax² + bx + c as a quadratic function in the form y = a(x – p)² + q by completing the square.

      • RF.1.4.3 Identify, explain and correct errors in an example of completing the square.

      • RF.1.4.4 Determine the characteristics of a quadratic function given in the form y = ax² + bx + c, and explain the strategy used.

      • RF.1.4.5 Sketch the graph of a quadratic function given in the form y = ax² + bx + c.

      • RF.1.4.6 Verify, with or without technology, that a quadratic function in the form y = ax² + bx + c represents the same function as a given quadratic function in the form y = a(x – p)² + q.

      • RF.1.4.7 Write a quadratic function that models a given situation, and explain any assumptions made.

      • RF.1.4.8 Solve a problem, with or without technology, by analyzing a quadratic function.

    • RF.1.5 Solve problems that involve quadratic equations.

    • RF.1.6 Solve, algebraically and graphically, problems that involve systems of linear-quadratic and quadratic-quadratic equations in two variables.

      • RF.1.6.1 Model a situation, using a system of linear-quadratic or quadratic-quadratic equations.

      • RF.1.6.2 Relate a system of linear-quadratic or quadratic-quadratic equations to the context of a given problem.

      • RF.1.6.3 Determine and verify the solution of a system of linear-quadratic or quadratic-quadratic equations graphically, with technology.

      • RF.1.6.4 Determine and verify the solution of a system of linear-quadratic or quadratic-quadratic equations algebraically.

      • RF.1.6.5 Explain the meaning of the points of intersection of a system of linear-quadratic or quadratic-quadratic equations.

      • RF.1.6.6 Explain, using examples, why a system of linear-quadratic or quadratic-quadratic equations may have zero, one, two or an infinite number of solutions.

      • RF.1.6.7 Solve a problem that involves a system of linear-quadratic or quadratic-quadratic equations, and explain the strategy used.

    • RF.1.7 Solve problems that involve linear and quadratic inequalities in two variables.

      • RF.1.7.1 Explain, using examples, how test points can be used to determine the solution region that satisfies an inequality.

      • RF.1.7.2 Explain, using examples, when a solid or broken line should be used in the solution for an inequality.

      • RF.1.7.3 Sketch, with or without technology, the graph of a linear or quadratic inequality.

      • RF.1.7.4 Solve a problem that involves a linear or quadratic inequality.

    • RF.1.8 Solve problems that involve quadratic inequalities in one variable.

      • RF.1.8.1 Determine the solution of a quadratic inequality in one variable, using strategies such as case analysis, graphing, roots and test points, or sign analysis; and explain the strategy used.

      • RF.1.8.2 Represent and solve a problem that involves a quadratic inequality in one variable.

      • RF.1.8.3 Interpret the solution to a problem that involves a quadratic inequality in one variable.

    • RF.1.9 Analyze arithmetic sequences and series to solve problems.

      • RF.1.9.1 Identify the assumption(s) made when defining an arithmetic sequence or series.

      • RF.1.9.2 Provide and justify an example of an arithmetic sequence.

      • RF.1.9.3 Derive a rule for determining the general term of an arithmetic sequence.

      • RF.1.9.4 Describe the relationship between arithmetic sequences and linear functions.

      • RF.1.9.5 Determine t?, d, n, or t? in a problem that involves an arithmetic sequence.

      • RF.1.9.6 Derive a rule for determining the sum of n terms of an arithmetic series.

      • RF.1.9.7 Determine t?, d, n, or S? in a problem that involves an arithmetic series.

      • RF.1.9.8 Solve a problem that involves an arithmetic sequence or series.

    • RF.1.10 Analyze geometric sequences and series to solve problems.

      • RF.1.10.1 Identify assumptions made when identifying a geometric sequence or series.

      • RF.1.10.2 Provide and justify an example of a geometric sequence.

      • RF.1.10.3 Derive a rule for determining the general term of a geometric sequence.

      • RF.1.10.4 Determine t?, r, n or t? in a problem that involves a geometric sequence.

      • RF.1.10.5 Derive a rule for determining the sum of n terms of a geometric series.

      • RF.1.10.6 Determine t?, r, n or S? in a problem that involves a geometric series.

      • RF.1.10.7 Generalize, using inductive reasoning, a rule for determining the sum of an infinite geometric series.

      • RF.1.10.8 Explain why a geometric series is convergent or divergent.

      • RF.1.10.9 Solve a problem that involves a geometric sequence or series.

    • RF.1.11 Graph and analyze reciprocal functions (limited to the reciprocal of linear and quadratic functions).

      • RF.1.11.1 Compare the graph of y = 1/f(x) to the graph of y = f(x).

      • RF.1.11.2 Identify, given a function f(x), values of x for which y = 1/f(x) will have vertical asymptotes; and describe their relationship to the non-permissible values of the related rational expression.

      • RF.1.11.3 Graph, with or without technology, y = 1/f(x), given y = f(x) as a function or a graph, and explain the strategies used.

      • RF.1.11.4 Graph, with or without technology, y = f(x), given y = 1/f(x) as a function or a graph, and explain the strategies used.