Curriculum Comparison Alberta Pure Mathematics 20/30 (2002) and Alberta Mathematics 20-1/30-1(2008)

Transcription

1 Curriculum Comparison Alberta Pure Mathematics 20/30 (2002) and Alberta Mathematics 20-1/30-1(2008) When looking at these two courses as a whole there is a minimal amount of change between the outcomes of the 2002 versions of these courses when compared to the 2008 versions. In 2002 there were 83 general and specific outcomes in these courses. Of these, some outcomes have been deleted outright from this stream; others have been moved to grade 9 or Foundations 10, while others have been moved between the 2008 versions of these two courses. The 2008 versions of these courses now contain a total of 50 outcomes where: 14/50 outcomes or 28% of all outcomes are NEW to this stream 36/50 outcomes or 72% of all outcomes REMAIN from the previous courses either within the same course or moved from one course to the other. A detailed comparison of each of these courses follows. 1

2 Curriculum Comparison: Alberta Pure Mathematics 20 (2002) and Alberta Mathematics 20-1(2008) KEY: Highlighted text in yellow NO LONGER PART OF ALBERTA MATHEMATICS 20-1 Highlighted text in green NEW CONTENT IN ALBERTA MATHEMATICS : 36 outcomes 28/36 outcomes in the old course have been deleted or significantly changed. 2008: 23 outcomes 12/23 outcomes or 52% of the outcomes in the new course are new or significantly changed. Some outcomes moved from Pure 30 to 20-1 Pure Mathematics 20 Mathematics 20-1 Topic 1: Linear and Nonlinear Systems Represent and analyze situations that involve expressions, equations and inequalities. 1.1 Graph linear inequalities, in two variables. [PS, V] 1.2 Solve systems of linear equations, in two variables: algebraically (elimination and substitution) graphically. [CN, PS, T, V] 1.3 Determine the solution to a system of nonlinear equations, using technology as appropriate. [PS, T, V] 1.4 Solve systems of linear equations, in three variables: algebraically with technology. [CN, PS, T, V] Algebra and Number Develop algebraic reasoning and number sense 6. Solve, algebraically and graphically, problems that involve systems of linear-quadratic and quadraticquadratic equations in two variables. [CN, PS, R, T, V] 9. Analyze arithmetic sequences and series to solve problems. [CN, PS, R] 10. Analyze geometric sequences and series to solve problems. [PS, R] Moved from Math 30 Topic 2: Quadratic Functions and Equations Represent and analyze quadratic, polynomial and rational functions, using technology as appropriate. 11. Graph and analyze reciprocal functions (limited to the reciprocal of linear and quadratic functions). [CN, R, T, V] Moved from Math 30 Relations and Functions Develop algebraic and graphical reasoning through the study of relations. 3. Analyze quadratic functions of the form 2

3 2.1 Determine the following characteristics of the graph of a quadratic function: vertex domain and range axis of symmetry intercepts. [C, PS, T, V] 2.2 Connect algebraic and graphical transformations of quadratic functions, using completing the square as required. [CN, T, V] 2.3 Model real-world situations, using quadratic functions. [CN, PS] 2.4 Solve quadratic equations, and relate the solutions to the zeros of a corresponding quadratic function, using: factoring the quadratic formula graphing. [CN, E, T, V] 2.5 Determine the character of the real and non-real roots of a quadratic equation, using: the discriminant in the quadratic formula graphing. [C, R, T, V] y = a( x p) 2 + q and determine the: vertex domain and range direction of opening axis of symmetry x- and y-intercepts. [CN, R, T, V] 4. Analyze quadratic functions of the form y = ax 2 + bx + c to identify characteristics of the corresponding graph, including: vertex domain and range direction of opening axis of symmetry x- and y-intercepts and to solve problems. [CN, PS, R, T, V] 5. Solve problems that involve quadratic equations. [C, CN, PS, R, T, V] 1. Factor polynomial expressions of the form: o ax 2 + bx + c, a 0 o a 2 x 2 b 2 y 2, a 0, b 0 o a( f() x) 2 + b( f() x)+ c, a 0 o a 2 f x ( ()) 2 b 2 ( g() y) 2, a 0, b 0 where a, b and c are rational numbers. [CN, ME, R] Topic 3: Polynomial and Other Nonlinear Equations and Functions Represent and analyze situations that involve expressions, equations and inequalities. 3.1 Solve nonlinear equations: by factoring graphically. [CN, T, V] 3.2 Use the remainder theorem to evaluate 3

4 polynomial expressions and the factor theorem to determine factors of polynomials. [E, PS, T] Examine the nature of relations with an emphasis on functions. 3.3 Perform operations on functions and compositions of functions. [CN, E, PS] 3.4 Determine the inverse of a function. [CN, R, V] Represent and analyze quadratic, polynomial and rational functions, using technology as appropriate. 3.5 Describe, graph and analyze polynomial and rational functions, using technology. [C, R, T, V] 3.6 Formulate and apply strategies to solve: absolute value equations radical equations rational equations quadratic inequalities polynomial inequalities. [CN, R, T, V] 1. Demonstrate an understanding of the absolute value of real numbers. [R, V] 2. Graph and analyze absolute value functions (limited to linear and quadratic functions) to solve problems. [C, PS, R, T, V] 2. Solve problems that involve operations on radicals and radical expressions with numerical and variable radicands. [CN, ME, PS, R] 3. Solve problems that involve radical equations (limited to square roots). [C, PS, R] 4. Determine equivalent forms of rational expressions (limited to numerators and denominators that are monomials, binomials or trinomials). [C, ME, R] 5. Perform operations on rational expressions (limited to numerators and denominators that are monomials, binomials or trinomials). [CN, ME, R] 6. Solve problems that involve rational equations (limited to numerators and denominators that are monomials, binomials or trinomials). [C, PS, R] Develop algebraic reasoning and number sense 7. Solve problems that involve linear and quadratic inequalities in two variables. 4

5 [C, PS, T, V] 8. Solve problems that involve quadratic inequalities in one variable. [CN, PS, V] Topic 4: Formal Reasoning Apply the principles of mathematical reasoning to solve problems and to justify solutions. 4.1 Differentiate between inductive and deductive reasoning. [CN, R] 4.2 Explain and apply connecting words, such as and, or and not, to solve problems. [C, PS, R, V] 4.3 Use examples and counterexamples to analyze conjectures. [CN, R] 4.4 Distinguish between an if then proposition, its converse and its contrapositive. [CN, R] 4.5 Prove assertions in a variety of settings, using direct reasoning. [R] Topic 5: Circles and Coordinate Geometry Develop and apply the geometric properties of circles and polygons to solve problems. 5.1 Use technology and measurement to confirm and apply the following properties to particular cases: the perpendicular from the centre of a circle to a chord bisects the chord the measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc the inscribed angles subtended by the same arc are congruent the angle inscribed in a semicircle is a right angle the opposite angles of a cyclic quadrilateral are supplementary a tangent to a circle is perpendicular to the radius at the point of tangency the tangent segments to a circle, from any external point, are congruent the angle between a tangent and a chord is Deleted from Mathematics 20-1 Deleted from Mathematics

6 equal to the inscribed angle on the opposite side of the chord the sum of the interior angles of an n-sided polygon is (2n 4) right angles. [PS, R, T, V] 5.2 Prove the following general properties, using established concepts and theorems: the perpendicular bisector of a chord contains the centre of the circle the angle inscribed in a semicircle is a right angle the tangent segments to a circle from any external point are congruent. [PS, R, CN] 5.3 Solve problems, using a variety of circle properties and relevant trigonometric ratios, and justify the solution strategy used. [PS, R, V] Solve coordinate geometry problems involving lines and line segments, and justify the solutions. 5.4 Solve problems involving distances between points and lines. [CN, PS, R] 5.5 Verify and prove assertions in plane geometry, using coordinate geometry and trigonometric ratios as necessary. [C, R, V] Topic 6: Finance Solve consumer problems, using arithmetic operations. 6.1 Solve consumer problems, which may include examples such as: wages earned in various situations property taxation exchange rates. [CN, E, PS, R, T] 6.2 Reconcile financial statements, which may include: cheque books and bank statements credit card statements loan statements. [CN, PS, T] 6.3 Solve budget problems, using graphs and tables to communicate solutions. [C, PS, T, V] 6.4 Plot and describe financial data of Deleted from Mathematics

7 exponential form. [C, T, V] 6.5 Solve investment and credit problems involving simple and compound interest. [CN, PS, T] Trigonometry Develop trigonometric reasoning. 1. Demonstrate an understanding of angles in standard position [0 to 360 ]. [R, V] Moved from Pure Solve problems, using the three primary trigonometric ratios for angles from 0 to 360 in standard position. [C, ME, PS, R, T, V] 3. Solve problems, using the cosine law and sine law, including the ambiguous case. [C, CN, PS, R, T] 7

8 Curriculum Comparison: Alberta Pure Mathematics 30 (2002) and Alberta Mathematics 30-1(2008) KEY: Highlighted text in yellow NO LONGER PART OF ALBERTA MATHEMATICS 30-1 Highlighted text in green NEW CONTENT IN ALBERTA MATHEMATICS : 47 outcomes 13/47 outcomes in the old course have been deleted or significantly changed. 2008: 27 outcomes 2/27 outcomes or 7% of the outcomes in the new course are new or significantly changed. Some outcomes moved from Pure 20 to 30-1 Pure Mathematics 30 Mathematics 30-1 Topic 1: Transformations of Functions Perform, analyze and create transformations of functions and relations that are described by equations or graphs. 1.1 Describe how various translations of functions affect graphs and their related equations: y = f(x h) y k = f(x). [C, T, V] 1.2 Describe how various stretches of functions (compressions and expansions) affect graphs and their related equations: y = af(x) y = f(kx). [C, T, V] 1.3 Describe how reflections of functions in both axes and in the line y = x affect graphs and their related equations: y = f( x) y = f(x) y = f 1 (x). [C, T, V] 1.4 Using the graph and/or the equation of f(x), describe and sketch 1/f (x ) [C, T, V] 1.5 Describe and perform single transformations and combinations of transformations on functions and relations. [C, T, V] Relations and Functions Develop algebraic and graphical reasoning through the study of relations. 1. Demonstrate an understanding of operations on, and compositions of, functions. [CN, R, T, V] Moved From Pure Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of functions and their related equations. [C, CN, R, V] 3. Demonstrate an understanding of the effects of horizontal and vertical stretches on the graphs of functions and their related equations. [C, CN, R, V] 4. Apply translations and stretches to the graphs and equations of functions. [C, CN, R, V] 5. Demonstrate an understanding of the effects of reflections on the graphs of functions and their related equations, including reflections through the: x-axis y-axis line y = x. [C, CN, R, V] 6. Demonstrate an understanding of inverses of relations. 8

9 [C, CN, R, V] Moved From Pure Demonstrate an understanding of factoring polynomials of degree greater than 2 (limited to polynomials of degree 5 with integral coefficients). [C, CN, ME] Moved From Pure Graph and analyze polynomial functions (limited to polynomial functions of degree 5 ). [C, CN, T, V ] Moved From Pure Graph and analyze radical functions (limited to functions involving one radical). [CN, R, T, V] Topic 2: Exponents, Logarithms and Geometric Series Generate and analyze exponential patterns. 2.1 Derive and apply expressions to represent general terms and sums for geometric growth and to solve problems. [CN, R, T] 2.2 Connect geometric sequences to exponential functions over the natural numbers. [E, R, V] Solve exponential, logarithmic and trigonometric equations and identities. 2.3 Solve exponential equations having bases that are powers of one another. [E, R] 2.4 Use the laws of exponents and logarithms to: solve and verify exponential equations and identities solve logarithmic equations simplify logarithmic expressions. Represent and analyze exponential and logarithmic functions, using technology as appropriate. 2.5 Graph and analyze an exponential function, using technology. [R, T, V] 2.6 Model, graph and apply exponential 14. Graph and analyze rational functions (limited to numerators and denominators that are monomials, binomials or trinomials). [CN, R, T, V] Moved From Pure 20 Deleted from Mathematics 30-1 and moved To Demonstrate an understanding of logarithms. [CN, ME, R] 8. Demonstrate an understanding of the product, quotient and power laws of logarithms. [C, CN, ME, R, T] 10. Solve problems that involve exponential and logarithmic equations. [C, CN, PS, R] 9

10 functions to solve problems. [PS, T, V] 2.7 Change functions from exponential form to logarithmic form and vice versa. [CN] 2.8 Use logarithms to model practical problems. [CN, PS, V] 2.9 Explain the relationship between the laws of logarithms and the laws of exponents. [C, T] 2.10 Graph and analyze logarithmic functions with and without technology. [R, T, V] Topic 3: Trigonometry Solve exponential, logarithmic and trigonometric equations and identities. 3.1 Distinguish between degree and radian measure, and solve problems using both. [CN, E] 3.2 Determine the exact and the approximate values of trigonometric ratios for any multiples of 0, 30, 45, 60 and 90 and 0, π/6, π/4, π/3, π/2 [CN, E] 3.3 Solve first and second degree trigonometric equations over the domain 0 θ 2π or the domain 0 o θ 360 o algebraically graphically. [PS, T] 3.4 Determine the general solutions to trigonometric equations where the domain is the set of real numbers. [PS, T] 3.5 Verify trigonometric identities: numerically for any particular case algebraically for general cases graphically. [PS, R, T, V] Represent and analyze trigonometric functions, using technology as appropriate. 9. Graph and analyze exponential and logarithmic functions. [C, CN, T, V] Trigonometry Develop trigonometric reasoning. 5. Solve, algebraically and graphically, first and second degree trigonometric equations with the domain expressed in degrees and radians. [CN, PS, R, T, V] 6. Prove trigonometric identities, using: reciprocal identities quotient identities Pythagorean identities sum or difference identities (restricted to sine, cosine and tangent) double-angle identities (restricted to sine, cosine and tan 10

11 3.6 Describe sine, cosine and tangent as circular functions, with reference to the unit circle and an angle in standard position. [PS, R, V] 3.7 Use sum, difference and double angle identities for sine and cosine to verify and simplify trigonometric expressions. [R, T] 3.8 Draw (using technology), sketch and analyze the graphs of sine, cosine and tangent functions, for: amplitude, if defined period domain and range asymptotes, if any behaviour under transformations. [CN, T, V] 3.9 Draw and sketch (using technology) and analyze the graphs of secant, cosecant and cotangent functions, for: period domain and range asymptotes behaviour under horizontal and vertical stretches. [CN, T, V] 3.10 Use sine and cosine functions to model and solve problems. [PS, R, V] Topic 4: Conic Sections Classify conic sections, using their shapes and equations. 4.1 Classify conic sections according to shape. [C, R, V] 4.2 Classify conic sections according to a given equation in general or standard (completed square) form (vertical or horizontal axis of symmetry only). [CN, T, V] 4.3 Convert a given equation of a conic section from general to standard form and vice versa. [R, T] 1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians. [CN, ME, R, V] 2. Develop and apply the equation of the unit circle. [CN, R, V] 3. Solve problems, using the six trigonometric ratios for angles expressed in radians and degrees. [ME, PS, R, T, V] 4. Graph and analyze the trigonometric functions sine, cosine and tangent to solve problems. [CN, PS, T, V] Deleted from Mathematics

12 Topic 5: Permutations and Combinations Solve problems based on the counting of sets, using techniques such as the fundamental counting principle, permutations and combinations. 5.1 Use the fundamental counting principle to determine the number of different ways to perform multistep operations. [PS, R] 5.2 Determine the number of linear permutations of n objects taken r at a time, and use this to solve problems. [PS, R, V] 5.3 Determine the number of combinations of n distinguishable objects taken r at a time, and use this to solve problems. [PS, R, V] 5.4 Determine the number of pathways in a given simple pathway problem. [CN, PS, R, V] 5.5 Determine the number of pathways in a given compound pathway problem. [CN, PS, R, V] 5.6 Solve problems using the binomial theorem, where the exponent n belongs to the set of natural numbers. [CN, E, PS, V] Model the probability of a compound event, and solve problems based on the combining of simpler probabilities. 5.7 Solve probability problems using either permutations and combinations or the fundamental counting principle. [E, PS, R] Topic 6: Statistics Use normal and binomial probability distributions to solve problems involving uncertainty. 6.1 Find the population standard deviation of a data set, using technology. [CN, E, T, V] 6.2 Solve probability problems, using the binomial distribution. [PS, R, T] 6.3 Use z-scores to solve problems related to the normal distribution. [PS, R, T, V] Permutations, Combinations and Binomial Theorem Develop algebraic and numeric reasoning that involves combinatorics. 1. Apply the fundamental counting principle to solve problems. [C, PS, R, V] 2. Determine the number of permutations of n elements taken r at a time to solve problems. [C, PS, R, V] 3. Determine the number of combinations of n different elements taken r at a time to solve problems. [C, PS, R, V] 4. Expand powers of a binomial in a variety of ways, including using the binomial theorem (restricted to exponents that are natural numbers). [CN, R, V] Deleted from Mathematics