Newfoundland and Labrador Curriculum — Grade 10


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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.

AN2.7.

Express a radical as a mixed radical in simplest form (limited to numerical radicands).

AN2.8.

Express a mixed radical as an entire radical (limited to numerical radicands).

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]

F1.1.

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

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]

N4.6.

Describe the advantages and disadvantages of debit card purchases.

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]

NL3.1.

Express a mixed radical with a numerical radicand as an entire radical.

NL3.3.

Express an entire radical with a numerical radicand as a mixed radical.

NL3.4.

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

NL3.5.

Express an entire radical with a variable radicand as a mixed radical.

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 > 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 > 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.