Appendix A: Principles of Math 12 Prescribed Learning Outcomes

The organizers for Principles of Math 12 are as follows:

A: Problem Solving
B: Patterns and Relations (Patterns)
C: Patterns and Relations (Variables and Equations)
D: Patterns and Relations (Relations and Functions)
E: Shape and Space (3-D Objects and 2-D Shapes)
F: Shape and Space (Transformations)
G: Statistics and Probability (Chance and Uncertainty)

It is expected that students will use a variety of methods to solve real-life, practical, technical, and theoretical problems

It is expected that students will:

A1. solve problems that involve a specific content area such as, geometry, algebra, trigonometry, statistics, probability, etc.
A2. solve problems that involve more than one content area
A3. Solve problems that involve mathematics within other disciplines
A4. analyse problems and identify the significant elements
A5. develop specific skills in selecting and using an appropriate problem-solving strategy or combination of strategies chosen from, but not restricted to, the following:

• guess and check
• look for a pattern
• make a systematic list
• make and use a drawing or model
• eliminate possibilities
• work backward
• simplify the original problem
• develop alternative original approaches
• - analyse keywords

A6. demonstrate the ability to work individually and co-operatively to solve problems
A7. determine that their solutions are correct and reasonable
A8. clearly communicate a solution to a problem and the process used to solve it
A9. use appropriate technology to assist in problem solving

It is expected that students will generate and analyse exponential patterns.

It is expected that students will:

B1. derive and apply expressions to represent general terms and sums for geometric growth and to solve problems
B2. connect geometric sequences to exponential functions over the natural numbers
B3. estimate values of expressions for infinite geometric processes

C: Patterns and Relations (Variables and Equations)

It is expected that students will solve exponential, logarithmic and trigonometric equations and identities.

It is expected that students will:

C1. Solve exponential equations having bases that are powers of one another
C2. Solve and verify exponential and logarithmic equations and identities
C3. distinguish between degree and radian measure, and solve problems using both
C4. Determine the exact and the approximate values of trigonometric ratios for any multiples of 0°, 30°, 45°, 60° and 90° and rad
C5. Solve first and second degree trigonometric equations over a domain of length :

• algebraically
• graphically

C6. Determine the general solutions to trigonometric equations where the domain is the set of real numbers
C7. analyse trigonometric identities:

• graphically
• algebraically for general cases

C8. Use sum, difference,and double angle identities for sine and cosine to verify and simplify trigonometric expressions

D: Patterns and Relations (Relations and Functions)

It is expected that students will represent and analyse exponential and logarithmic functions, using technology as appropriate.

It is expected that students will:

D1. model, graph, and apply exponential functions to solve problems
D2. change functions from exponential form to logarithmic form and vice versa
D3. Model, graph, and apply logarithmic functions to solve problems
D4. explain the relationship between the laws of logarithms and the laws of exponents
D5. describe the three primary trigonometric functions as circular functions with reference to the unit circle and an angle in standard position
D6. draw (using technology), sketch, and analyse the graphs of sine, cosine, and tangent functions, for:

• amplitude, if defined
• period
• domain and range
• asymptotes, if any
• behavior under transformations

D7. Use trigonometric functions to model and solve problems

E: Shape and Space (Measurement)

It is expected that students will classify conic sections, using their shapes and equations.

It is expected that students will:

E1. classify conic sections according to shape
E2. Classify conic sections according to a given equation in general or standard (completed square) form (vertical or horizontal axis of symmetry only)
E3. convert a given equation of a conic section from general to standard form and vice versa

F: Shape and Space (Transformations)

It is expected that students will perform, analyse and create transformations of functions and relations that are described by equations or graphs.

It is expected that students will:

F1. Describe how various translations of functions affect graphs and their related equations:

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F2. Describe how various stretches of functions (compressions and expansions) affect graphs and their related equations:

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F3. Describe how reflections of functions in both axes and in the line affect graphs and their related equations:

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F4. using the graph and/or the equation of describe and sketch
F5. Using the graph and/or the equation of describe and sketch
F6. Describe and perform single transformations and combinations of transformations on functions and relations

G: Statistics and Probability (Chance and Uncertainty)

It is expected that students will:

• use normal and binomial probability distributions to solve problems involving uncertainty.
• Solve problems based on the counting of sets, using techniques such as the fundamental counting principle, permutations and combinations.
• Model the probability of a compound event, and solve problems based on the combining of simpler probabilities.

It is expected that students will:

G1. find the standard deviation of a data set or a probability distribution, using technology
G2. Use z-scores and the normal distribution to solve problems
G3. Use the normal approximation to the binomial distribution to solve problems involving probability calculations for large samples (where npq>10)
G4. Solve pathway problems, interpreting and applying any constraints
G5. Use the fundamental counting principle to determine the number of different ways to perform multistep operations
G6. Determine the number of permutations of n different objects taken r at a time, and use this to solve problems
G7. Determine the number of combinations of n different objects taken r at a time, and use this to solve problems
G8. Solve problems, using the binomial theorem where N belongs to the set of natural numbers
G9. construct a sample space for two or three events
G10. Classify events as independent or dependent
G11. Solve problems, using the probabilities of mutually exclusive and complementary events
G12. Determine the conditional probability of two events
G13. Solve probability problems involving permutations, and combinations and conditional probability

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Maintained by: Mathematics Coordinator
Revised: November 28, 2000

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