Course Syllabus

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St. George's School

 Pre-Calculus Math 12

COURSE OUTLINE

2020-2021

THIS COURSE RELIES HEAVILY ON BOTH ALGEBRAIC AND GRAPHICAL METHODS: STUDENTS ARE REQUIRED TO HAVE A GRAPHICAL CALCULATOR. HOWEVER, this must not be capable of accessing the internet etc, nor can it have a “CAS” Computer Algebra System. A TI-84 or equivalent is ideal.

The suggested timing below is approximate. There is some degree of flexibility by the teacher, in the time allocated to each topic, depending upon circumstances, but the program will be delivered broadly as indicated. This is a very tight curriculum, and students should ensure their recall of earlier material and units remains strong, throughout the course.

Each term consists of 25 or 26 classes.                                                                                                                                                                                 Specific outcomes

TERM 1

 

Unit 1:       Function Transformations          7 classes              RF18, RF19

         Unit 2:       The Unit Circle                             6 classes              M14, M16

 

Unit 3:       Radical Functions                        6 classes              RF18, RF 24

Unit 4:       Geometric series and Sigma       4 classes              RF26

        

TERM 2

 

Unit 5:       Polynomial Functions                  6 classes              RF23

Winter break somewhere about here

Unit 6:       Trigonometric Functions             7 classes              M15, M16

Unit 7:       Exponential Functions                5 classes               RF20, RF21

Unit 8:       Logarithmic Functions               8 classes      RF20, RF21, RF27

 

TERM  3:

 

Unit 9:       Rational Functions                       5 classes             RF 18, RF25

Spring Break somewhere about here

Unit 10:     Function Operations                    5 classes              RF17, RF19

Unit 11:     Trigonometric Identities              8 classes              M16, M17 

 

This is a very tight curriculum, with very little opportunity for final review. Students must ensure that their recall and understanding of earlier material remains strong throughout the course.

 

 

 

 

Pre-Calculus 12 : Textbook Detail and Learning Outcomes.

SEQUENCES AND SERIES

            Develop algebraic reasoning through the study of geometric sequences and series, and Sigma notation

TRANSFORMATIONS AND FUNCTIONS

Develop algebraic and graphical reasoning through the study of relations.

TRIGONOMETRY             

Develop Trigonometric reasoning

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Develop algebraic reasoning through the study of functions and relations

EQUATIONS AND FUNCTIONS

Develop algebraic and graphical reasoning through the study of relations and functions

 

 

 

Unit 1          Function Transformations (Chapter1)                                  7 classes

 

RF18   Demonstrate an understanding of the effects of transformations of functions.

RF19   Demonstrate an understanding of inverses of relations

           

 

Specific Outcome

Curriculum Focus

# of classes

1.1  Horizontal and Vertical Translations

 RF18

Determine the effects of h and k in y-k=f(x-h)

Sketching the graph  for given values of h and k

Writing the equation of a function whose graph is a vertical and/or horizontal translation of y=f(x)

1

1.2 Reflections and Stretches

RF18

Developing an understanding of the effects of reflections, and their related equations.

Developing an understanding of the effects of horizontal and vertical stretches, and their related equations.

To include circle transformed to ellipse

2

1.3 Combining Transformations

RF18

RF18

Sketching the graph of a transformed function by applying translations, reflections and stretches. Writing the equation of a function that has been transformed from y=f(x).

2

1.4 Inverse of a Relation

RF19

Sketching the graph of the inverse of a relation.

Determining whether a relation and its inverse are functions.

Determining the equation of an inverse

1

Chapter test

 

 

1

           

 

 

Unit 2          Trigonometry and the Unit Circle (Chapter 4)            6 classes

 

M14   Determine an understanding of angles in standard position, expressed in degrees and radians and properties of CAST in the unit circle

M16   Solve algebraically and graphically, up to second degree, trig equations and problems in degrees and radians

 

 

Specific Outcome

Curriculum Focus

# of classes

4.1       Angles and Angle Measure

M14

Sketching angles in standard position, both in degrees and radians.

Converting between degrees and radians.

Coterminal angles.

Solving problems involving arclength, central angle and radius

1

4.2 The Unit Circle

M14

The equation of the unit circle.

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

1

4.3 Trigonometric Ratios

M14

Relating the trig ratios to co-ords of points  on the unit circle.

Determining exact and approximate values for trig ratios.

Identifying the measures of angles that generate specific trig values.

Solving problems using trigonometric ratios.

2

4.4 Introduction to Trigonometric Equations       

M16

Algebraically solving first-degree and second-degree trig equations.

Verifying that a specific value is a solution to a partic equation.

Identifying exact and approximate solutions in a restricted domain.

Determining the general solution.

1

Chapter test

 

 

1

 

 

 

Unit 3          Radical Functions            (Chapter 2)                                          6 classes

 

RF18   Demonstrate an understanding of the effects of transformations of functions.

RF24   Graph and analyze radical functions

 

 

Specific Outcome

Curriculum Focus

# of classes

2.1 Radical Functions and Transformations

 

RF18

 

R24

Investigating the function y=√x

Graphing radical functions using transformations

Identifying the domain and range of radical functions

2

2.2 Square Root of a Function

RF24

Sketching the graph of y=√f(x)  given the graph of y=f(x).

Explaining strategies

Compare and explain the domain and ranges of the functions y=f(x)  and y=√f(x)  

1

2.3  Solving Radical Equations Graphically

RF24

Relating the roots of radical equations and the x-intercepts of the graphs of radical functions.

Determining approximate solutions graphically

1

Recap and Chapter test

 

 

2

 

Unit 1 Series and Sigma Notation (from former math 11 course)

RF26   Analyze geometric sequences and series to solve problems

 

1.3 Geometric Sequences

RF26

Derive rules for  general nth term, determine the values of a, tn, r  or n, and use them to solve

½

1.4 Geometric Series

RF26

Derive the rule to find the sum of a series, determine the values to use in the formula. Solve problems

½

1.5 Infinite geometric Series

RF26

Convergence, divergence, Generalized formula for sum to infinitely many terms

1

1.x Sigma notation

RF26

Understand and solve problems involving manipulation of general sigma notation

1

Unit Assessment

 

 

1

 

Unit 5          Polynomial Functions     (Chapter 3)                              6 classes

 

RF22   Demonstrate an understanding of factoring polynomials of degree

2<degree <=5, integer coefficients

RF23   Graph and analyze polynomial functions (of degree <=5)

 

 

Specific Outcome

Curriculum Focus

# of classes

3.1 Characteristics of Polynomial Functions

RF23

Identifying polynomial functions.

Analysing polynomial functions.

1

3.2 The Remainder Theorem

RF22

Describe the relationship between polynomial long division and synthetic division.

Diving polynomials by binomials of the form x-a.

Explaining the relationship between the remainder on division by x-a and the value of the polynomial at x=a

2

3.3 The Factor Theorem

RF22

Factoring polynomials.

Explaining the relationship between the linear factors of a polynomial expression and the zeros of the corresponding function.

Modelling and solving problems involving polynomial functions.

 1

3.4 Equations and Graphs of Polynomial Functions

RF23

Describing the relationship between zeros, roots and x-intercepts  of polynomial functions and equations.

Sketching the graph of a polynomial function without technology.

Modelling and solving problems

1

Chapter test

 

 

1

 

 

 

Unit 6          Trigonometric Functions and Graphs (Chapter 5)       7 classes

 

M15   Graph and analyze the functions sine, cosine and tangent to solve problems

M16   Solve algebraically and graphically, up to second degree, trigonometric equations and problems in degrees and radians

 

 

Specific Outcome

Curriculum Focus

# of classes

5.1 Graphing Sine and Cosine Functions       

M15

Sketching the graphs of sine and cosine.

Determining the characteristics of these two graphs.

Demonstrating an understanding of the effects of vertical and horizontal stretches.

Solving a problem by analyzing the graph of a trigonometric function.

1

5.2 Transformations of Sinusoidal Functions 

M15

Graphing and transforming sinusoidal functions.

Identifying domain, range, phase shift, period, amplitude and vertical displacement.

Developing equations, from graphs and descriptions.

Solving problems graphically that can be modeled using sinusoidal functions.

Recognizing that more than one equation can be used to represent a particular graph.

2

5.3  The Tangent Function

M15

M16

Sketching the graph of y=tan(x).

Determining the amplitude, domain, range, and period.

Determining the asymptotes and x-intercepts

Solving a problem by analyzing the graph of the tangent function

1

5.4  Equations and Graphs of Trigonometric Functions

M15

 

M16

Using the graphs of trig functions to solve equations.

Analysing a trig function to solve a problem.

Determining a trig function which models a problem.

Real world situation.

2

Chapter test

 

 

1

 

 

Unit 7          Exponential Functions (Chapter 7)                                            5 classes

 

RF20   Graph and analyze exponential and logarithmic functions

RF21   Solve problems that involve exponential and logarithmic equations

 

 

Specific Outcome

Curriculum Focus

# of classes

7.1  Characteristics of Exponential Functions

RF20

 

RF21

Analysing graphs of exponential functions.

Solving problems that involve exponential growth or decay.

1

7.2  Transformations of Exponential Functions

RF20

RF21

Applying translations, stretches and reflections to the graphs of exponential functions.

Representing these transformations in the equations.

Solving problems involving growth and decay.

1

7.3  Solving Exponential Equations

RF21

Determining the solution of an exponential equation, in which the bases are powers of one another.

Solving problems that involve their application to loans, mortgages and investments.

2

Chapter test

 

 

1

 

Unit 8          Logarithmic Functions (Chapter 8)                                        8 classes 

RF20   Graph and analyze exponential and logarithmic functions

RF21   Solve problems that involve exponential and logarithmic equations

 

 

Specific Outcome

Cu rriculum Focus

# of classes

8.1 Understanding Logarithms

RF20

RF21

Demonstrating that logarithms are the inverse of exponential functions.

Sketching the graph of y=log(x)

Determining the characteristics of the logarithm function.

Explaining the relationship between logarithms and exponents

Evaluating logarithms using a variety of methods

2

8.2 Transformations of Logarithmic Functions

RF20

Explaining the effects of parameters  a, b, h, k  in y=-alogc(b(x-h))+k.

Sketching  the graph of a log function by applying a series  of transformations to y=log (x)

1

8.3 Laws of Logarithms

RF27

Developing the laws of logarithms

Determining equivalent forms.

Applying the laws of logarithms to logarithmic scales

2

8.4 Logarithmic and Exponential Equations

RF21

RF 27

Solving a logarithmic equation and verifying it.

Explaining why a possible solution may be extraneous.

Solving an exponential equation involving unrelated bases.

Exponential Growth and Decay

Loans and investments etc.

Modelling a situation using logs and/or exponential functions

 

2

Chapter test

 

 

1

 

 

Unit 9          Rational Functions (Chapter 9)                                                            5 classes

 

RF18   Demonstrate an understanding of the effects of transformations of functions.

RF25   Graph and analyze reciprocal and rational functions (limited to numerator and denominator at most a trinomial)

 

 

Specific Outcome

Curriculum Focus

# of classes

7.4  Reciprocal Functions ( from Math 11)

RF18

RF25

Graph the reciprocal of f(x). Understand zeros become vertical asymptotes and vice versa. Invariant points happen at y=1 or -1 Understand the relationship between f(x) and 1/f(x).

1

9.1 Exploring Rational Functions using Transformations

RF18

RF25

Graphing, analyzing and comparing rational functions, using transformations and technology.

Examining the behavior of graphs of rational functions near non-permissible values.

1

9.2 Analysing Rational Functions

RF25

Graphing, analyzing and comparing rational functions, using transformations and technology.

Determining whether a graph of a rational function has an asymptote or a point of discontinuity for a non-permissible value.

1

9.3 Connecting Graphs and Rational Equations

RF25

Relating the roots of rational equations to the x­-intercepts.

Determining approximate solutions to rational equations graphically.

1

Recap and Chapter test

 

 

1

 

 

Unit 10 Function Operations  (Chapter 10)                                           5 classes

 

RF17   Demonstrate an understanding of operations on, and composition of functions

 

 

Specific Outcome

Curriculum Focus

# of classes

10.1 Sums and Differences of Functions

RF17

Sketching the graph of a function that is the sum or difference of two functions.

Determining the domain and range of a function that is the sum or difference of two functions.

Writing the equation of a function that is the sum or difference of two functions.

1

10.2 Products and Quotients of Functions

RF17

Sketching the graph of a function that is the product or quotient of two functions.

Determining the domain and range of a function that is the product or quotient of two functions.

Writing the equation of a function that is the product or quotient of two functions.

2

10.3 Composite Functions

RF17

Determining the values of a composite function.

Writing the equation of a composite function, explaining any restrictions.

Sketching the graph of a composite function

1

Chapter test

 

 

1

 

 

Unit 11        Trigonometric Identities ( Chapter 6)                               8 classes

 

M16   Solve algebraically and graphically, up to second degree, trigonometric equations and problems in degrees and radians

M17   prove trigonometric identities, including sum/ difference & double angle (limited to sin, cos, tan)

 

 

Specific Outcome

Curriculum Focus

# of classes

6.1 Reciprocal, Quotient and Pythagorean Identities

M17

Verifying an identity numerically and graphically.

Exploring reciprocal, quotient and Pythag identities.

Determining non-permissible values.

Understanding and explaining the difference between an identity and an equation.

1

6.2 Sum, Difference and Double-Angle Identities

M17

Applying sum, difference and double-angle identities to verify the equivalence of trig expressions.

Verifying a trig identity numerically and graphically.

1

6.3 Proving Identities

M17

Proving trigonometric identities algebraically.

Understanding the difference between proving and verifying.

Showing that two sides of an identity are equal for a given value, is insufficient to prove the identity.

2

6.4 Solving Trigonometric Equations Using Identities

M16

 

M17

Solving trig equations algebraically using identities.

Determining exact solutions where possible.

Determining general solutions.

Identifying and correcting errors.

 

2

Recap and Chapter test

 

 

2

 

 

 

LEARNING OUTCOMES for Pre-Calculus 12 2012-21 and suggested weightings

 

 

M14   Determine an understanding of angles in standard position, expressed                       6%

in degrees and radians and properties of CAST in the unit circle

M15   Graph and analyze the functions sine, cosine and tangent to solve problems                      6.5%

M16   Solve algebraically and graphically, up to second degree, trig equations

and problems in degrees and radians                                                                                      6.5%

M17   prove trigonometric identities, including sum/ difference & double angle

(limited to sin, cos, tan)                                                                                                                6%

 

 

RF17   Demonstrate an understanding of operations on, and composition of functions 4%

RF18   Demonstrate an understanding of the effects of transformations of funcs.            7%

RF19   Demonstrate an understanding of inverses of relations                                                            4%

RF20   Graph and analyze exponential and logarithmic functions                                          6%

RF21   Solve problems that involve exponential and logarithmic equations                               6%

RF22   Demonstrate an understanding of factoring polynomials

(of 2< degree <=5, integer coeffs)                                                                                             6%

RF23   Graph and analyze polynomial functions (of degree <=5)                                                    6%

RF24   Graph and analyze radical functions                                                                                        6%

RF25   Graph and analyze rational functions (limited to num and denom <= trinom)                 6.5%

RF26   Analyze geometric sequences and series  and sigma to solve problems                   4.5%

RF27   Understand and solve problems involving the laws of logarithms.                           3%

 

These weights total 84%

Assessment will be aligned against these Learning Outcomes and the following

Curricular Competencies:

 

CC1     Reasoning and Analysing                        4%

CC2     Understanding and Solving                    4%

CC3     Communicating and Representing       4%

CC4     Connecting and Reflecting                     4%

 

 

 

 

CURRICULUM COMPETENCIES:

 

These skills will be emphasized, embraced, valued, supported and encouraged throughout, and will be built into assessment where appropriate and relevant.

 

Reasoning and modelling

  • Develop thinking strategies to solve puzzles and play games
  • Explore, analyze, and apply mathematical ideas using reason, technology, and other tools
  • Estimate reasonably and demonstrate fluent, flexible, and strategic thinking about number
  • Model with mathematics in situational contexts
  • Think creatively and with curiosity and wonder when exploring problems

 

Understanding and solving

  • Develop, demonstrate, and apply mathematical understanding through play, story, inquiry, and problem solving
  • Visualize to explore and illustrate mathematical concepts and relationships
  • Apply flexible and strategic approaches to solve problems
  • Solve problems with persistence and a positive disposition
  • Engage in problem-solving experiences connected with place, story, cultural practices, and perspectives relevant to local First Peoples communities, the local community, and other cultures

 

Communicating and representing

  • Explain and justify mathematical ideas and decisions in many ways
  • Represent mathematical ideas in concrete, pictorial, and symbolic forms
  • Use mathematical vocabulary and language to contribute to discussions in the classroom
  • Take risks when offering ideas in classroom discourse

 

Connecting and reflecting

  • Reflect on mathematical thinking
  • Connect mathematical concepts with each other, other areas , and personal interests
  • Use mistakes as opportunities to advance learning
  • Incorporate First Peoples worldviews, perspectives, knowledge, and practices to make connections with mathematical concept

 

Course Summary:

Date Details Due