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Develop spatial sense and proportional reasoning.

#### A1.

Solve problems that require the manipulation and application of formulas related to: perimeter; area; the Pythagorean theorem; primary trigonometric ratios; income. [C, CN, ME, PS, R]

#### A1.1.

Solve a contextual problem involving the application of a formula that does not require manipulation.

#### A1.10.

Identify a linear equation as having a direct or partial variation relationship.

#### A1.11.

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

#### A1.12.

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

#### A1.13.

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

#### A1.2.

Describe, using examples, how a given formula is used in a trade or occupation.

#### A1.3.

Solve a contextual problem that involves the application of a formula that does not require manipulation.

#### A1.4.

Solve a contextual problem that involves the application of a formula that requires manipulation.

#### A1.5.

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

#### A1.6.

Explain and verify why different forms of the same formula are equivalent.

#### A1.7.

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

#### A1.9.

Relate slope and rate of change to linear relations.

#### A2.

Demonstrate an understanding of slope: as rise over run; as rate of change; by solving problems. [C, CN, PS, V]

#### A2.1.

Describe contexts that involve slope; e.g., ramps, roofs, road grade, flow rates within a tube, skateboard parks, ski hills.

#### A2.2.

Explain, using diagrams, the difference between two given slopes (e.g., a 3:1 and a 1:3 roof pitch), and describe the implications.

#### A2.3.

Describe the conditions under which a slope will be either 0 or undefined.

#### A2.4.

Explain, using examples and illustrations, slope as rise over run.

#### A2.5.

Verify that the slope of an object, such as a ramp or a roof, is constant.

#### A2.6.

Explain, using illustrations, the relationship between slope and angle of elevation; e.g., for a ramp with a slope of 7:100, the angle of elevation is approximately 4".

#### A2.7.

Explain the implications, such as safety and functionality, of different slopes in a given context.

#### A2.8.

Explain, using examples and illustrations, slope as rate of change.

#### A2.9.

Solve a contextual problem that involves slope or rate of change.

#### A3.

Solve problems by applying proportional reasoning and unit analysis. [C, CN, PS, R]

#### A3.1.

Explain the process of unit analysis used to solve a problem.

#### A3.2.

Explain, using an example, how unit analysis and proportional reasoning are related.

#### A3.3.

Solve a problem, using unit analysis.

#### A3.4.

Solve a problem within and between systems, using proportions or tables.

#### AN1.

Demonstrate an understanding of factors of whole numbers by determining the: prime factors; greatest common factor; least common multiple; square root; cube root. [CN, ME, R]

#### AN1.1.

Determine the prime factors of a whole number.

#### AN1.2.

Explain why the numbers 0 and 1 have no prime factors.

#### AN1.3.

Determine, using a variety of strategies, the greatest common factor or lease common multiple of a set of whole numbers, and explain the process.

#### AN1.4.

Solve problems that involve prime factors, greatest common factors, least common multiples, square roots or cube roots.

#### AN1.5.

Determine, concretely, whether a given whole number is a perfect square, a perfect cube or neither.

#### AN1.6.

Determine, using a variety of strategies, the square root of a perfect square, and explain the process.

#### AN1.7.

Determine, using a variety of strategies, the cube root of a perfect cube, and explain the process.

#### AN2.

Demonstrate an understanding of irrational numbers by: representing, identifying and simplifying irrational numbers; ordering irrational numbers. [CN, ME, R, V]

#### AN2.1.

Explain, using examples, the meaning of an index of a radical.

#### AN2.10.

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

#### AN2.2.

Sort a set of numbers into rational and irrational numbers.

#### AN2.3.

Represent using a graphic organizer, the relationship among the subsets of the real numbers (natural, whole, integer, rational, irrational).

#### AN2.4.

Determine an approximate value of a given irrational number.

#### AN2.5.

Approximate the locations or irrational numbers on a number line, using a variety of strategies, and explain the reasoning.

#### AN2.6.

Order a set of irrational number on a number line.

#### AN2.9.

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

#### AN3.

Demonstrate an understanding of powers with integral and rational exponents. [C, CN, PS, R]

#### AN3.1.

Explain, using patterns, why a^(m/n) = (n(""_a))^m, a """ 0.

#### AN3.2.

Express powers with rational exponents as radicals and vice versa.

#### AN3.3.

Solve a problem that involves exponent laws or radicals.

#### AN3.4.

Explain, using patterns, why a^(-n) = 1/(a^n), a """ 0.

#### AN3.5.

Apply the exponent laws: (a^m)(a^n) = a^(m+n); a^m "" (a^n) = a^(m-n); (a^m)^n = a^(mn); (ab)^m = (a^m)(b^m); (a/b)^n = (a^n)/(b^n), b """ 0; to expressions with rational and variable bases and integral and rational exponents, and explain the reasoning.

#### AN3.6.

Identify and correct errors in the simplification of an expression that involves powers.

#### AN4.

Demonstrate an understanding of the multiplication of polynomial expressions (limited to monomials, binomials and trinomials), concretely, pictorially and symbolically. [CN, R, V]

#### AN4.1.

Explain, using examples, the relationship between the multiplication of binomials and the multiplication of two digit numbers.

#### AN4.2.

Model the multiplication of two given binomials, concretely or pictorially, and record the process symbolically.

#### AN4.3.

Relate the multiplication of two binomials expressions to an area model.

#### AN4.4.

Multiply two polynomials symbolically, and combine like terms in the product.

#### AN4.5.

Verify a polynomial by substituting numbers for the variables.

#### AN4.6.

Generalize and explain a strategy for multiplication of polynomials.

#### AN4.7.

Identify and explain errors in a solution for a polynomial multiplication.

#### AN5.

Demonstrate an understanding of common factors and trinomial factoring, concretely, pictorially and symbolically. [C, CN, R, V]

#### AN5.1.

Explain, using examples, the relationship between multiplication and factoring of polynomials.

#### AN5.2.

Express a polynomial as a product of its factors.

#### AN5.3.

Determine the common factors in the terms of a polynomial, and express the polynomial in factored form, concretely, pictorially and symbolically.

#### AN5.4.

Factor a polynomial and verify by multiplying the factors.

#### AN5.5.

Model the factoring of a trinomial, concretely or pictorially, and record the process symbolically.

#### AN5.6.

Identify and explain the errors in a polynomial factorization.

#### AN5.7.

Factor a polynomial that is a difference of squares, and explain why it is a special case of trinomial factoring where b = 0.

#### AN6.

Solve problems that involve rational equations (limited to numerators and denominators that are monomials, binomials or trinomials). [C, PS, R]

#### AN6.2.

Determine the solution to a rational equation algebraically, and explain the strategy used to solve the equation.

#### AN6.3.

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

#### AN6.4.

Solve problems by modeling a situation using a rational equation.

#### C1.

Demonstrate an understanding of the concept of limit and evaluate the limit of a function. [C, CN, R, T, V]

#### C1.1.

Using informal methods, explore the concept of a limit including one sided limits.

#### C1.10.

Evaluate limits of functions as x approaches infinity (limits at infinity).

#### C1.11.

Investigate the end behaviour of the function using limits to identify possible horizontal asymptotes.

#### C1.12.

Investigate the end behaviour of the function using limits to identify possible oblique asymptotes.

#### C1.2.

Using informal methods, establish that the limit of 1/x as x approaches infinity is zero.

#### C1.3.

Explore the concept of limit and the notation used in expressing the limit of a function: lim(x"""a^+) f(x); lim(x"""a^-) f(x); lim(x"""a) f(x).

#### C1.31.

Explain the meaning of the phrase """F(x) is an antiderivative of f (x).

#### C1.4.

Determine the value of the limit of a function as the variable approaches a real number by using a provided graph, including piecewise functions; by using a table of values.

#### C1.7.

Determine the value of the limit of a function as the variable approaches a real number by substitution; by algebraic manipulation.

#### C1.8.

Determine limits that result in infinity (infinite limits).

#### C1.9.

Investigate the behaviour of the function at a vertical asymptote using limits.

#### C11.

Apply derivatives of trigonometric functions. [CN, ME, R, V]

#### C11.1.

Derive the derivatives of the six basic trigonometric functions.

#### C11.2.

Determine the derivative of expressions involving trigonometric functions.

#### C11.3.

Solve problems involving the derivative of a trigonometric function.

#### C12.

Solve problems involving inverse trigonometric functions. [CN, ME, R, V]

#### C12.5.

Derive the inverse trigonometric derivatives.

#### C12.6.

Determine the derivative of an inverse trigonometric function.

#### C12.7.

Solve problems involving the derivative of an inverse trigonometric function.

#### C13.

Determine the indefinite integral of a polynomial and radical function. [C, CN, PS, R]

#### C13.2.

Determine the general antiderivative of functions.

#### C13.3.

Use antiderivatives notation appropriately (i.e., """f(x)dx for the antiderivative of f (x)).

#### C13.4.

Identify the properties of antidifferentiation.

#### C13.5.

Determine the indefinite integral of a function given extra conditions.

#### C13.6.

Use anti-differentiation to solve problems about motion of a particle along a line that involves: computing the displacement given the initial position and velocity as a function of time; computing velocity and/or displacement given the suitable initial conditions and acceleration as a function of time.

#### C14.

Determine the definite integral of a polynomial function. [C, CN, PS, R]

#### C14.1.

Estimate an area using a finite sum.

#### C14.4.

Using definite integrals, determine the area under a polynomial function from x = a to x = b.

#### C14.5.

Calculate the definite integral of a function over an interval [a, b].

#### C14.6.

Determine the area between two polynomial functions.

#### C3.

Demonstrate an understanding of the concept of a derivative and evaluate derivatives of functions using the definition of derivative. [CN, ME, R, V]

#### C3.1.

Describe geometrically a secant line and a tangent line for the graph of a function at x = a.

#### C3.10.

Determine all values for which a function is differentiable, given the graph.

#### C3.11.

Sketch a graph of the derivative of a function, given the graph of a function.

#### C3.12.

Sketch a graph of the function, given the graph of the derivative of a function.

#### C3.3.

Identify the instantaneous rate of change of a function at a point as the limiting value of a sequence of average rates of change.

#### C3.4.

Define and evaluate the derivative at x = a as: lim(h"""0) (f(a+h)-f(a))/h and lim(x"""a) (f(x)-f(a))/x-a.

#### C3.6.

Define and determine the derivative of a function using f"(x) = lim(h"""0) (f(x+h)-f(x))/h (limited to polynomials of degree 3, square root and rational functions with linear terms).

#### C3.7.

Use alternate notation interchangeably to express derivatives (i.e., f"(x), dy/dx, y" etc.).

#### C3.8.

Determine whether a function is differentiable at a given point.

#### C3.9.

Explain why a function is not differentiable at a given point, and distinguish between corners, cusps, discontinuities, and vertical tangents.

#### C4.

Apply derivative rules including: Constant Rule; Constant Multiple Rule; Sum Rule; Difference Rule; Product Rule; Quotient Rule; Power Rule; Chain Rule to determine the derivative of functions. [C, CN, PS, R]

#### C4.3.

Determine second and higher-order derivatives of functions.

#### C4.5.

Solve problems involving derivatives drawn from a variety of applications, limited to tangent and normal lines, straight line motion and rates of change.

#### C5.

Determine the derivative of a relation, using implicit differentiation. [C, CN, PS, R, V]

#### C5.3.

Determine the second derivative of a relation, using implicit differentiation.

#### C6.

Use derivatives to sketch the graph of a polynomial function. [C, CN, PS, R, T, V]

#### C6.2.

Use f""(x) to identify the hypercritical numbers, points of inflection and intervals of concavity.

#### C6.3.

Sketch the graph of f (x) using information obtained from the function and its derivatives.

#### C6.4.

Use the given function f (x) to determine its features such as intercepts and the domain.

#### C7.

Use derivatives to sketch the graph of a rational function. [C, CN, PS, R, T, V]

#### C7.2.

Use f""(x) to identify the hypercritical numbers, points of inflection and intervals of concavity.

#### C7.3.

Sketch the graph of f (x) using information obtained from the function and its derivatives.

#### C7.4.

Use the given function f (x) to determine its features such as intercepts, asymptotes, points of discontinuity and the domain.

#### F1.

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

#### F1.2.

Identify situations that involve compound interest.

#### F1.3.

Solve a contextual problem that involves compound interest.

#### F1.4.

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

#### F1.5.

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

#### F1.6.

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.

#### F1.7.

Determine, using technology, the unknown variable in compound interest loan situations.

#### F1.8.

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

#### F2.

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

#### F2.2.

Compare, using examples, renting, leasing and buying.

#### F2.3.

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

#### F2.4.

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

#### G1.

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

#### G1.1.

Determine, explain and verify a strategy to solve a puzzle or to win a game; e.g., guess and check; look for a pattern; make a systematic list; draw or model; eliminate possibilities; simplify the original problem; work backward; develop alternative approaches.

#### G1.2.

Identify and correct errors in the solution to a puzzle or in a strategy for winning a game.

#### G1.3.

Create a variation on a puzzle or a game, and describe a strategy for solving the puzzle or winning the game.

#### G1.4.

Identify all of the right triangles in a given illustration for a context.

#### G1.5.

Generalize, using inductive reasoning, a rule for the relationship between the sum of the interior angles and the number of sides (n) in a polygon, with or without technology.

#### G2.

Demonstrate an understanding of the Pythagorean theorem by: identifying situations that involve right triangles; verifying the formula; applying the formula; solving problems. [C, CN, PS, V]

#### G2.1.

Describe historical and contemporary applications of the Pythagorean theorem.

#### G2.10.

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

#### G2.2.

Explain, using illustrations, why the Pythagorean theorem only applies to right triangles.

#### G2.3.

Verify the Pythagorean theorem, using examples and counterexamples, including drawings, concrete materials and technology.

#### G2.4.

Determine if a given triangle is a right triangle, using the Pythagorean theorem.

#### G2.5.

Explain why a triangle with the side length ratio of 3:4:5 is a right triangle.

#### G2.6.

Explain how the ratio of 3:4:5 can be used to determine if a corner of a given 3-D object is square (90 degrees) or if a given parallelogram is a rectangle.

#### G2.7.

Solve a problem, using the Pythagorean theorem.

#### G2.8.

Describe and illustrate line symmetry in triangles, quadrilaterals and regular polygons.

#### G2.9.

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

#### G3.

Demonstrate an understanding of similarity of convex polygons, including regular and irregular polygons. [C, CN, PS, V]

#### G3.1.

Determine if two or more regular or irregular polygons are similar.

#### G3.2.

Explain why two or more right triangles with a shared acute angle are similar.

#### G3.3.

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

#### G3.4.

Explain why two given polygons are not similar.

#### G3.5.

Solve a contextual problem that involves the similarity of polygons.

#### G3.6.

Draw, using isometric dot paper, a given 3-D object.

#### G3.7.

Draw a one-point perspective view of a given 3-D object.

#### G3.8.

Identify the point of perspective of a given one-point perspective drawing of a 3-D object.

#### G4.

Demonstrate an understanding of primary trigonometric ratios (sine, cosine, tangent) by: applying similarity to right triangles; generalizing patterns from similar right triangles; applying the primary trigonometric ratios; solving problems. [CN, PS, R, T, V]

#### G4.1.

Show, for a specified acute angle in a set of right triangles, that the ratios of length of the side opposite to the length of the side adjacent are equal, and generalize a formula for the tangent ratio.

#### G4.2.

Show, for a specified acute angle in a set of right triangles, that the ratios of length of the side opposite to the length of the hypotenuse are equal, and generalize a formula for the sine ratio.

#### G4.3.

Show, for a specified acute angle in a set of right triangles, that the ratios of length of the side adjacent to the length of the hypotenuse are equal, and generalize a formula for the cosine ratio.

#### G4.4.

Identify situations where the trigonometric ratios are used for indirect measurement of angles and lengths.

#### G4.5.

Solve a contextual problem that involves right angles, using the primary trigonometric ratios.

#### G4.6.

Determine if a solution to a problem that involves primary trigonometric ratios is reasonable.

#### G5.

Solve problems that involve parallel, perpendicular and transversal lines, and pairs of angles formed between them. [C, CN, PS, V]

#### G5.1.

Sort a set of lines as perpendicular, parallel or neither, and justify this sorting.

#### G5.2.

Illustrate and describe complementary and supplementary angles.

#### G5.3.

Identify, in a set of angles, adjacent angles that are not complementary or supplementary.

#### G5.4.

Identify and name pairs of angles formed by parallel lines and a transversal, including corresponding angles, vertically opposite angles, alternate interior angles, alternate exterior angles, interior angles on the same side of transversal, and exterior angles on the same side of the transversal.

#### G5.5.

Explain and illustrate the relationships of angles formed by parallel lines and a transversal.

#### G5.6.

Determine the measures of angles involving parallel lines and a transversal.

#### G5.7.

Explain, using examples, why the angle relationships do not apply when the lines are not parallel.

#### G5.8.

Determine if lines or planes are perpendicular or parallel, e.g., wall perpendicular to the floor, and describe the strategy used.

#### G5.9.

Solve a contextual problem that involves angles formed by parallel lines and a transversal (including perpendicular transversals).

#### G6.

Demonstrate an understanding of angles, including acute, right, obtuse, straight and reflex, by: drawing; replicating and constructing; bisecting; solving problems. [C, CN, ME, PS, R]

#### G6.1.

Measure, using a protractor, angles in various orientations.

#### G6.2.

Draw and describe angles with various measures, including acute, right, straight, obtuse and reflex angles.

#### G6.3.

Identify referents for angles.

#### G6.4.

Sketch a given angle.

#### G6.5.

Estimate the measure of a given angle, using 22.5", 30", 45", 60", 90" and 180" as referent angles.

#### G6.6.

Solve a contextual problem that involves angles.

#### G6.7.

Explain and illustrate how angles can be replicated in a variety of ways; e.g., Mira, protractor, compass and straightedge, carpenter's square, dynamic geometry software.

#### G6.8.

Replicate angles in a variety of ways, with and without technology.

#### G6.9.

Bisect an angle, using a variety of methods.

#### LR1.

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

#### LR1.1.

Determine, explain and verify a strategy to solve a puzzle or to win a game; e.g., guess and check; look for a pattern; make a systematic list; draw or model; eliminate possibilities; simplify the original problem; work backward; develop alternative approaches.

#### LR1.2.

Identify and correct errors in a solution to a puzzle or in a strategy for winning a game.

#### LR1.3.

Create a variation on a puzzle or a game, and describe a strategy for solving the puzzle or winning the game.

#### LR2.

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

#### LR2.1.

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

#### LR2.2.

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

#### LR2.3.

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

#### LR2.4.

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

#### LR2.5.

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

#### LR2.6.

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

#### M1.

Solve problems that involve linear measurement, using: SI and imperial units of measure; estimation strategies; measurement strategies. [ME, PS, V]

#### M1.1.

Interpret rates in a given context, such as the arts, commerce, the environment, medicine or recreation.

#### M1.2.

Determine and compare rates and unit rates.

#### M1.3.

Estimate a linear measure, using a referent, and explain the process used.

#### M1.4.

Justify the choice of units used for determining a measurement in a problem solving context.

#### M1.5.

Solve problems that involve linear measure, using instruments such as rulers, calipers or tape measures.

#### M1.6.

Describe and explain a personal strategy used to determine a linear measurement; e.g., circumference of a bottle, length of a curve, perimeter of the base of an irregular 3-D object.

#### M1.7.

Identify and explain factors that influence a rate in a given context.

#### M1.8.

Solve a contextual problem that involves rates or unit rates.

#### M1.9.

Solve a rate problem that requires the isolation of a variable.

#### M2.

Solve problems that involve scale diagrams, using proportional reasoning. [CN, PS, R, V]

#### M2.1.

Explain, using examples, how scale diagrams are used to model a 2-D shape.

#### M2.12.

Solve a contextual problem that involves the volume of a 3-D object, including composite 3-D objects, or the capacity of a container.

#### M2.2.

Determine, using proportional reasoning, an unknown dimension of a 2-D shape, given a scale diagram or model.

#### M2.3.

Determine, using proportional reasoning, the scale factor, given one dimension of a 2-D shape, and its representation.

#### M2.4.

Draw, with or without technology, a scale diagram of a given 2-D shape, according to a specified scale factor (enlargement or reduction).

#### M2.5.

Solve a contextual problem that involves a scale diagram.

#### M2.6.

Explain, using examples, how scale diagrams are used to model a 3-D object.

#### M2.7.

Write a given capacity expressed in one unit as another unit in the same measurement system.

#### M2.8.

Determine the volume of prisms, cones, cylinders, pyramids, spheres and composite 3-D objects, using a variety of measuring tools such as rulers, tape measures, callipers, micrometers, and displacement.

#### M2.9.

Determine the capacity of prisms, cones, pyramids, spheres and cylinders, using a variety of measuring tools and methods such as graduated cylinders, measuring cups, and measuring spoons.

#### M3.

Solve problems, using SI and imperial units, that involve the surface area and volume of 3-D objects, including: right cones; right cylinders; right prisms; right pyramids; spheres. [CN, PS, R, V]

#### M3.1.

Sketch a diagram to represent a problem that involves surface area or volume.

#### M3.2.

Determine the surface area of a right cone, right cylinder, right prism, or a right pyramid, using an object or its labelled diagram.

#### M3.3.

Determine an unknown dimension of a right cone, right cylinder, right prism, or right pyramid, given the object's surface area and the remaining dimensions.

#### M3.4.

Determine the volume of a right cone, right cylinder, right prism, or right pyramid using an object or its labelled diagram.

#### M3.5.

Estimate the dimensions of a given regular 3-D object or 2-D shape, using a referent; e.g., the height of the desk is about three rulers long, so the desk is approximately three feet height.

#### M3.6.

Determine an unknown dimension of a right cone, right cylinder, right prism, or right pyramid, given the object's volume and the remaining dimensions.

#### M3.7.

Determine the surface area and volume of a sphere, using an object or its labelled diagram.

#### M3.8.

Determine an unknown dimension of a sphere, given the object's surface area.

#### M3.9.

Solve a problem that involved surface area or volume, using an object or its labelled diagram of a composite 3-D object.

#### M4.

Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles. [C, CN, PS, R, T, V]

#### M4.1.

Identify the hypotenuse of a right triangle, and the opposite and adjacent sides for a given acute angle in the triangle.

#### M4.10.

Explain, using examples, the effect of changing the measurement of one or more dimensions on area and perimeter of rectangles.

#### M4.11.

Solve a problem that involves determining the surface area of 3-D objects, including right cylinders and cones.

#### M4.2.

Explain the relationships between similar right triangles and the definitions of the primary trigonometric ratios.

#### M4.3.

Estimate the area of a given regular, composite or irregular 2-D shape, using an SI square grid and an imperial square grid.

#### M4.4.

Use the primary trigonometric ratios to determine the length of a missing side in a right triangle.

#### M4.5.

Solve a problem that involves indirect and direct measurements, using the trigonometric ratios, the Pythagorean theorem and measurement instruments such as a clinometer or metre stick.

#### M4.6.

Solve right triangles.

#### M4.7.

Solve a problem that involves one or more right triangles by applying the primary trigonometric ratios or the Pythagorean theorem.

#### M4.8.

Solve a contextual problem that involves the area of a regular, a composite or an irregular 2-D shape.

#### M4.9.

Solve a problem, using formulas for determining the areas of regular, composite and irregular 2-D shapes, including circles.

#### N1.

Solve problems that involve unit pricing and currency exchange, using proportional reasoning. [CN, ME, PS, R]

#### N1.1.

Compare the unit price of two or more given items.

#### N1.2.

Compare, using examples, different sales promotion techniques; e.g., deli meat at \$2 per 100 g seems less expensive than \$20 per kilogram.

#### N1.3.

Solve problems that involve determining the best buy, and explain the choice in terms of the cost as well as other factors, such as quality and quantity.

#### N1.4.

Determine the percent increase or decrease for a given original and new price.

#### N2.

Demonstrate an understanding of income, including: wages; salary; contracts; commissions; piecework to calculate gross and net pay. [C, CN, R, T]

#### N2.1.

Identify income and expenses that should be included in a personal budget.

#### N2.10.

Determine gross pay for earnings acquired by: base wage, plus commission; single commission rate.

#### N2.2.

Explain considerations that must be made when developing a budget; e.g., prioritizing, recurring and unexpected expenses.

#### N2.3.

Determine in decimal form, from a time schedule, the total time worked in hours and minutes, including time and a half and/or double time.

#### N2.4.

Determine gross pay from given or calculated hours worked when given: the base hourly wage, with and without tips; the base hourly wage, plus overtime (time and a half, double time).

#### N2.5.

Modify a budget to achieve a set of personal goals.

#### N2.6.

Identify and correct errors in a solution to a problem that involves gross or net pay.

#### N2.7.

Explain why gross pay and net pay are not the same.

#### N2.8.

Determine the Canadian Pension Plan (CPP), Employment Insurance (EI) and income tax deductions for a given gross pay.

#### N2.9.

Determine net pay when given deductions; e.g., health plans, uniforms, union dues, charitable donations, payroll tax.

#### N3.

Demonstrate an understanding of compound interest. [CN, ME, PS, T]

#### N3.1.

Solve a problem that involves simple interest, given three of the four values in the formula I = Prt.

#### N3.2.

Compare simple and compound interest, and explain their relationship.

#### N3.3.

Solve, using a formula, a contextual problem that involves compound interest.

#### N3.4.

Explain, using examples, the effect of different compounding periods on calculations of compound interest.

#### N4.

Demonstrate an understanding of financial institution services used to access and manage finances. [C, CN, R, T]

#### N5.

Demonstrate an understanding of credit options, including: credit cards; loans. [CN, ME, PS, T]

#### N5.1.

Compare advantages and disadvantages of different types of credit options, including bank and store credit cards, personal loans, lines of credit, overdraft.

#### N5.2.

Make informed decisions and plans related to the use of credit, such as service charges, interest, payday loans and sales promotions, and explain the reasoning.

#### N5.3.

Describe strategies to use credit effectively, such as negotiating interest rates, planning payment timelines, reducing accumulated debt and timing purchases.

#### N5.4.

Compare credit card options from various companies and financial institutions.

#### N5.5.

Solve a contextual problem that involves credit cards or loans.

#### N5.6.

Solve a contextual problem that involves credit linked to sales promotions.

#### NL1.

Analyze and prove conjectures, using inductive and deductive reasoning, to solve problems. [C, CN, PS, R]

#### NL1.1.

Make conjectures by observing patterns and identifying properties, and justify the reasoning.

#### NL1.2.

Explain why inductive reasoning may lead to a false conjecture.

#### NL1.3.

Determine if a given argument is valid, and justify the reasoning.

#### NL1.5.

Prove a conjecture, using deductive reasoning (not limited to two column proofs).

#### NL1.6.

Prove algebraic and number relationships such as divisibility rules, number properties, mental mathematics strategies or algebraic number tricks.

#### NL1.7.

Compare, using examples, inductive and deductive reasoning.

#### NL1.8.

Identify errors in a given proof; e.g., a proof that ends with 2 = 1.

#### NL1.9.

Solve a contextual problem that involves inductive or deductive reasoning.

#### NL2.

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

#### NL2.1.

Determine, explain and verify a strategy to solve a puzzle or to win a game; e.g., guess and check; look for a pattern; make a systematic list; draw or model; eliminate possibilities; simplify the original problem; work backward; develop alternative approaches

#### NL2.2.

Create a variation on a puzzle or a game, and describe a strategy for solving the puzzle or winning the game.

#### NL2.3.

Identify and correct errors in a solution to a puzzle or in a strategy for winning a game.

#### NL3.

Solve problems that involve operations on radicals and radical expressions with numerical and variable radicands (limited to square roots). [CN, ME, PS, R]

#### NL3.4.

Identify values of the variable for which the radical expression is defined.

#### NL3.6.

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

#### NL3.7.

Rationalize the monomial denominator of a radical expression.

#### NL4.

Solve problems that involve radical equations (limited to square roots or cube roots). [C, PS, R]

#### NL4.1.

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

#### NL4.2.

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

#### NL4.3.

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

#### NL4.4.

Explain why some roots determined in solving a radical equation are extraneous.

#### NL4.5.

Solve problems by modelling a situation with a radical equation and solving the equation.

#### P1.

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

#### P1.1.

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

#### P1.3.

Determine the probability of, or the odds for and against, an outcome in a situation.

#### P1.5.

Solve a contextual problem that involves odds or probability.

#### P1.6.

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

#### P2.

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

#### P2.1.

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

#### P2.2.

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

#### P2.3.

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

#### P2.4.

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

#### P2.5.

Create and solve a problem that involves mutually exclusive or non"""mutually exclusive events

#### P3.

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

#### P3.1.

Compare, using examples, dependent and independent events.

#### P3.2.

Determine the probability of two independent events.

#### P3.3.

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

#### P3.4.

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

#### P4.

Solve problems that involve the Fundamental Counting Principle. [PS, R, V]

#### P4.1.

Represent and solve counting problems, using a graphic organizer.

#### P4.2.

Generalize, using inductive reasoning, the Fundamental Counting Principle.

#### P4.3.

Identify and explain assumptions made in solving a counting problem.

#### P4.4.

Solve a contextual counting problem, using the Fundamental Counting Principle, and explain the reasoning.

#### P5.

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

#### P5.1.

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

#### P5.2.

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

#### P5.3.

Simplify a numeric or an algebraic fraction that contains factorials in both the numerator and denominator.

#### P5.4.

Solve an equation that involves factorials.

#### P5.5.

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

#### P5.6.

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

#### P5.7.

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

#### P5.8.

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

#### P5.9.

Solve a contextual problem that involves probability and permutations.

#### P6.

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

#### P6.1.

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

#### P6.4.

Solve a contextual problem that involves probability and combinations.

#### PCBT1.

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

#### PCBT1.1.

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

#### PCBT1.2.

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.

#### PCBT1.3.

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

#### PCBT2.

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

#### PCBT2.1.

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

#### PCBT2.2.

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

#### PCBT2.3.

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

#### PCBT2.4.

Explain why n must be greater than or equal to r in the notation nPr.

#### PCBT2.5.

Given a value of k, k """ ", solve nPr = k for either n or r.

#### PCBT2.6.

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

#### PCBT2.7.

Solve problems involving permutations with constraints.

#### PCBT3.

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

#### PCBT3.1.

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

#### PCBT4.

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]

#### PCBT4.1.

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

#### PCBT4.4.

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

#### PCBT4.5.

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

#### PCBT4.6.

Determine a specific term in a binomial expansion.

#### RF1.

Interpret and explain the relationships among data, graphs and situations. [C, CN, R, T, V]

#### RF1.1.

Describe a possible situation for a given graph.

#### RF1.2.

Sketch a possible graph for a given situation.

#### RF1.3.

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

#### RF1.4.

Graph, with or without technology, a set of data, and determine the domain and range.

#### RF1.5.

Determine, and express in a variety of ways, the domain and range of a graph, a set of ordered pairs or a table of values.

#### RF1.6.

Determine the domain and range of a quadratic function.

#### RF1.7.

Determine the characteristics of a quadratic function (y = a(x - h)^2 + k, a """0) through manipulation of the parameters h and k.

#### RF1.9.

Solve a contextual problem that involves the characteristics of a quadratic function.

#### RF10.

Analyze geometric sequences and series to solve problems. [PS, R]

#### RF10.1.

Identify assumptions made when identifying a geometric sequence or series.

#### RF10.2.

Provide and justify an example of a geometric sequence.

#### RF10.3.

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

#### RF10.4.

Determine t1, r, n or tn in a problem that involves a geometric sequence.

#### RF10.5.

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

#### RF10.6.

Determine t1, r, n or Sn in a problem that involves a geometric series.

#### RF10.7.

Solve a problem that involves a geometric sequence or series.

#### RF10.8.

Explain why a geometric series is convergent or divergent.

#### RF10.9.

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

#### RF11.

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

#### RF11.1.

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

#### RF11.2.

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.

#### RF11.4.

Explain the relationship among the following: the zeros of a polynomial function; the roots of the corresponding polynomial equation; the x-intercepts of the graph of the polynomial function.

#### RF11.5.

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

#### RF11.6.

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

#### RF11.7.

Solve a problem by modeling a given situation with a polynomial function.

#### RF12.

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

#### RF12.1.

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

#### RF12.3.

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

#### RF12.4.

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.

#### RF12.5.

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

#### RF12.6.

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

#### RF2.

Demonstrate an understanding of relations and functions. [C, R, V]

#### RF2.1.

Represent a relation in a variety of ways.

#### RF2.2.

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

#### RF2.3.

Determine if a set of ordered pairs represents a function.

#### RF2.4.

Explain, using examples, why some relations are not functions but all functions are relations.

#### RF2.5.

Sort a set of graphs as functions or non-functions.

#### RF2.6.

Generalize and explain rules for determining whether graphs and sets of ordered pairs represent functions.

#### RF2.7.

Solve a contextual problem by modelling a situation with a quadratic equation and solving the equation.

#### RF3.

Demonstrate an understanding of slope with respect to: rise and run; line segments and lines; rate of change; parallel lines; perpendicular lines. [PS, R,V]

#### RF3.1.

Determine the slope of a line segment by measuring and calculating the rise and run.

#### RF3.2.

Explain, using examples, slope as a rate of change.

#### RF3.3.

Solve a contextual problem involving slope.

#### RF3.4.

Classify lines in a given set as having a positive or negative slopes.

#### RF3.5.

Explain the meaning of the slope of a horizontal or vertical line.

#### RF3.6.

Draw a line, given its slope and a point on the line.

#### RF3.7.

Determine another point on a line, given the slope and a point on the line.

#### RF3.8.

Explain why the slope of a line can determined by using any two points on that line.

#### RF3.9.

Generalize and apply a rule for determining whether two lines are parallel or perpendicular.

#### RF4.

Describe and represent linear relations, using: words; ordered pairs; table of values; graphs; equations. [C, R, V]

#### RF4.1.

Match corresponding representations of linear relations.

#### RF4.2.

Determine whether a table of values or a set of ordered pairs represents a linear relation, and explain why or why not.

#### RF4.3.

Determine whether a graph represents a linear relation and explain why or why not.

#### RF4.4.

Draw a graph given a set of ordered pairs and determine whether the relationship between the variables is linear.

#### RF4.5.

Determine whether an equation represents a linear relation, and explain why or why not.

#### RF4.6.

Verify, with or without technology, that a quadratic function in the form y = ax^2 + bx + c represents the same function as a given quadratic function in the form y = a(x - p)^2 + q.

#### RF4.7.

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

#### RF4.8.

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

#### RF4.9.

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

#### RF5.

Determine the characteristics of the graphs of linear relations, including the: intercepts; rate of change; domain; range. [CN, PS, R, V]

#### RF5.1.

Determine the rate of change of the graph of a linear relation.

#### RF5.2.

Determine the intercepts of the graph of a linear relation, and state the intercepts as values of ordered pairs.

#### RF5.3.

Determine the domain and range of the graph of a linear relation.

#### RF5.4.

Identify the graph that corresponds to a given rate of change and vertical intercept.

#### RF5.5.

Identify the rate of change and vertical intercept that correspond to a given graph.

#### RF5.6.

Solve a contextual problem that involves intercepts, rate of change, domain or range of a linear relation.

#### RF5.7.

Sketch a linear relation that has one intercept, two intercepts or an infinite number of intercepts.

#### RF5.8.

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

#### RF6.

Relate linear relations expressed in: slope-intercept form y = mx + b; general form Ax + By + C = 0; slope-point form; y """ y1 = m(x"""x1) to their graphs. [CN, R, T, V]

#### RF6.1.

Express a linear relation in different forms, and compare their graphs.

#### RF6.10.

Graph data, and determine the logarithmic function that best approximates the data.

#### RF6.11.

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

#### RF6.12.

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

#### RF6.2.

Generalize and explain strategies for graphing a linear relation in slope-intercept, general or slope-point form.

#### RF6.3.

Graph, with and without technology, a linear relation given in slope-intercept, general or slope-point form, and explain the strategy used to create the graph.

#### RF6.4.

Match a set of linear relations to their graphs.

#### RF6.5.

Rewrite a linear relation in either slope-intercept or general form.

#### RF6.6.

Identify equivalent linear relations from a set of linear relations.

#### RF6.7.

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

#### RF6.8.

Describe, orally and in written form, the characteristics of a logarithmic function by analyzing its equation.

#### RF6.9.

Match equations in a given set to their corresponding graphs.

#### RF7.

Determine the equation of a linear relation, given: a graph; a point and the slope; two points; a point and the equation of a parallel or perpendicular line to solve problems. [CN, PS, R, V].

#### RF7.1.

Determine the slope and y-intercept of a given linear relation from its graph, and write the equation in the form y = mx + b.

#### RF7.2.

Write the equation of a linear relation, given its slope and the coordinates of a point on the line, and explain the reasoning.

#### RF7.3.

Write the equation of a linear relation, given the coordinates of two points on the line, and explain the reasoning.

#### RF7.4.

Graph linear data generated from a context, and write the equation of the resulting line.

#### RF7.5.

Solve a problem, using the equation of a linear relation.

#### RF7.6.

Explain, using examples, when a solid or broken line should be used in the solution for a quadratic inequality.

#### RF7.7.

Sketch, with or without technology, the graph of a quadratic inequality.

#### RF7.8.

Solve a problem that involves a quadratic inequality.

#### RF8.

Represent a linear function, using function notation. [CN, ME, V]

#### RF8.1.

Express the equation of a linear function in two variables, using function notation.

#### RF8.2.

Express an equation given in function notation as a linear function in two variables.

#### RF8.3.

Determine the realted range value, given a domain value for a linear function; e.g., If f(x)=3x-2, determine f(-1).

#### RF8.4.

Determine the related domain value, given a range value for a linear function; e.g., If g(t) = 7 + t, determine t so that g(t)=15.

#### RF8.5.

Sketch the graph of a linear function expressed in function notation.

#### RF8.6.

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

#### RF8.7.

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

#### RF9.

Solve problems that involve systems of linear equations in two variables, graphically and algebraically. [CN, PS, R, T, V]

#### RF9.1.

Model a situation, using a system of linear equations.

#### RF9.2.

Relate a system of linear equation to the context of a problem.

#### RF9.3.

Explain the meaning of the point of intersection of a system of linear equations.

#### RF9.4.

Determine and verify the solution of a system of linear equation graphically, with and without technology.

#### RF9.5.

Solve a problem that involves a system of linear equations.

#### RF9.6.

Determine and verify the solution of a system of linear equations algebraically.

#### RF9.7.

Explain a strategy to solve a system of linear equations.

#### RF9.8.

Explain, using examples, why a system of equations may have no solution, one solution or an infinite number of solutions.

#### RP1.

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

#### RP1.1.

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

#### RP1.2.

Assess the accuracy, reliability and relevance of the primary or secondary data collected by: identifying examples of bias and points of view; identifying and describing the data collection methods; determining if the data is relevant; determining if the data is consistent with information obtained from other sources on the same topic.

#### RP1.3.

Interpret data, using statistical methods if applicable.

#### RP1.4.

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

#### S1.

Demonstrate an understanding of normal distribution, including: standard deviation; z"""scores. [CN, PS, T, V]

#### S1.1.

Determine the possible graphs that can be used to represent a given data set, and explain the advantages and disadvantages of each.

#### S1.10.

Determine, with or without technology, and explain the z-score for a given value in a normally distributed data set.

#### S1.11.

Explain, using examples representing multiple perspectives, the application of standard deviation for making decisions in situations such as warranties, insurance or opinion polls.

#### S1.2.

Create frequency tables and graphs from a set of data.

#### S1.3.

Explain, using examples, the meaning of standard deviation.

#### S1.4.

Calculate, using technology, the population standard deviation of a data set.

#### S1.5.

Solve a contextual problem that involves the interpretation of standard deviation.

#### S1.6.

Explain, using examples, the properties of a normal curve, including the mean, median, mode, standard deviation, symmetry and area under the curve.

#### S1.7.

Determine if a data set approximates a normal distribution and explain the reasoning.

#### S1.8.

Compare the properties of two or more normally distributed data sets.

#### S1.9.

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

#### S2.

Interpret statistical data, using: confidence intervals; confidence levels; margin of error. [C, CN, R]

#### S2.1.

Explain, using examples, the significance of a confidence interval, margin of error or confidence level.

#### S2.2.

Explain, using examples, how confidence levels, margin of error and confidence intervals may vary depending on the size of the random sample.

#### S2.3.

Make inferences about a population from sample data, using confidence intervals, and explain the reasoning.

#### S2.4.

Provide examples from print or electronic media in which confidence intervals and confidence levels are used to support a particular position.

#### S2.5.

Interpret and explain confidence intervals and margin of error, using examples found in print or electronic media.

#### T1.

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

#### T1.1.

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

#### T1.2.

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

#### T1.3.

Describe the relationship between radian measure and degree measure.

#### T1.4.

Sketch, in standard position, an angle with a measure expressed in the form k"" radians, where k """ Q.

#### T1.5.

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

#### T1.6.

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

#### T1.9.

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.

#### T2.

Solve problems, using the three primary trigonometric ratios for angles from 0" to 360" in standard position. [C, ME, PS, R, T, V]

#### T2.1.

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

#### T2.2.

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

#### T2.3.

Determine the sign of a given trigonmetric ratio for a given angle, without the use of technology, and explain.

#### T2.4.

Sketch a diagram to represent a problem.

#### T2.6.

Solve, for all values of "", an equation of the form sin "" = a or cos "" = a, where """1 ""_ a ""_ 1, and an equation of the form tan "" = a, where a is a real number.

#### T2.7.

Determine the exact value of the sine, cosine or tangent of a given angle with a reference angle of 30", 45"or 60".

#### T2.8.

Describe patterns in and among the values of the sine, cosine and tangent ratios for angles from to 0" to 360".

#### T2.9.

Solve a contextual problem, using trigonometric ratios.

#### T3.

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

#### T3.1.

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

#### T3.2.

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 multiples of 0, ""/6, ""/4, ""/3 or ""/2 and explain the strategy.

#### T3.3.

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

#### T3.4.

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

#### T3.5.

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.

#### T3.7.

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

#### T3.8.

Solve a problem, using trigonometric ratios.

#### T4.

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

#### T4.1.

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

#### T4.10.

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

#### T4.11.

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

#### T4.12.

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

#### T4.13.

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

#### T4.14.

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

#### T4.2.

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

#### T4.3.

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

#### T4.4.

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

#### T4.5.

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

#### T4.6.

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

#### T4.7.

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.

#### T4.8.

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

#### T4.9.

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

#### T5.

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

#### T5.1.

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

#### T5.2.

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

#### T5.3.

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

#### T5.4.

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

#### T5.5.

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

#### T6.

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]

#### T6.1.

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

#### T6.2.

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

#### T6.3.

Determine the non-permissible values of a trigonometric identity.

#### T6.4.

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

#### T6.5.

Prove, algebraically, that a trigonometric identity is valid.

#### T6.6.

Simplify trigonometric expressions using trigonometric identities.

#### T6.7.

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

#### T6.8.

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.