MCT4C

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**MCT4C - Mathematics for College Technology** Grade 12, College Preparation

=** __Course Description__ **= This course enables students to extend their knowledge of functions. Students will investigate and apply properties of polynomial, exponential, and trigonometric functions; continue to represent functions numerically, graphically, and algebraically; develop facility in simplifying expressions and solving equations; and solve problems that address applications of algebra, trigonometry, vectors, and geometry. Students will reason mathematically and communicate their thinking as they solve multi-step problems. This course prepares students for a variety of college technology programs. = __Prerequisites__ = Functions and Applications (MCF3M), Grade 11, University/College Preparation, or Functions, Grade 11 (MCR3U ), University Preparation

Mathematical Processes will be integrated into all lessons:
 * Problem Solving
 * Reasoning and Proving
 * Reflecting
 * Selecting Tools and Computational Strategies
 * Connecting
 * Representing
 * Communicating

Adapted from []

=** __Units__ **=

Unit 1. Exponential Functions
Notes: This unit focuses on functions and their graphs. Students coming to this course from Grade 11 University Preparation and Grade 11 University/College Preparation have different prior experiences with exponential functions; this is an opportunity to level the playing field and present logarithms which is new material for all of the students.

Goals:
 * Explore graphs of exponential functions to identify prior learning of students.
 * Using a graphing calculator, determine what happens when the base or the sign of the exponent changes.
 * Solve exponential equations numerically and graphically (guess and check, tracing graph, or using POI)
 * Find the point of intersections of graphs of 2 exponential functions and connect to solving the corresponding exponential equation.
 * Make connection to real-world issues (ex. population growth) by modelling with exponential equations and their graphs, given or derived.
 * Review laws of exponents, integer and rational and use to simplify and evaluate exponential equations with a common base.
 * Solve problems in bases other than 10 graphically or by systematic trial and error using the LOG key on a calculator.
 * Recognize the logarithm of a number to a given base as the exponent to which the base must be raised to get the number.
 * Connect finding the logarithm as the inverse of exponentiation, evaluate logarithmic expressions and rewrite exponential equations in logarithmic form to solve.
 * Pose real-world problems (ex. food temperature in an oven) and collect data that is exponential to model with exponential equations and solve by rewriting in logarithmic form.

Unit 2. Polynomial Functions
Notes: Focuses on functions and their graphs, like Units 1 and 4, so students will be able to connect and compare the characteristics of exponential, polynomial and trigonometric functions.

Goals:
 * Review features of linear and quadratic equations, what they look like, how to describe, determine if it's a function.
 * Define end behaviours and leading coefficient, look at domain and range.
 * Investigate cubic and quartic functions and explain why they are functions.
 * Graph the equations of cubic and quartic functions and investigate end behaviours, domain and range.
 * Describe end behaviours and the impact of the leading coefficient, describe each function with the maximum number of zeros.
 * Categorize curves as cubic, quartic, exponential, linear and quadratic.
 * Analyze graphs of data from real-world situations. Connect restrictions on domain and range to these situations.
 * Compare points on a graph of a function with answers from substitution into a function. Connect restrictions on domain and range.
 * Review finding zeros with quadratics and see the need for factored form.
 * Extend to find zeros for cubic and quartic functions using common factoring, difference of squares, trinomial factoring. Verify using graphs.

Unit 3. Polynomial Equations
Notes: This unit connects the graphical and algebraic representations and allows some time for students to develop fluency with algebra.

Goals:
 * Make connections between polynomial functions in factored form, graphical form, and numeric form
 * Consolidate that the zeros of a function correspond to the solutions or roots of the corresponding equation when f(x) = 0
 * Solve (and verify using technology) equations up to degree four - factor by common factoring, difference of squares, trinomial factoring, and quadratic formula
 * Solve equations of the form xn = a to compare to polynomials
 * Expand and simplify polynomial expressions
 * Solve multi-step problems from real-world applications
 * Rearrange and equation for a given variable and substitute to find the value
 * Make connections between the formula and the variables to determine what type of function it is

Unit 4. Trigonemetric Functions
Notes: This unit focuses on functions and their graphs, like Units 1 and 3, so students will be able to connect and compare the characteristics of exponential, polynomial and trigonometric functions.

Goals:
 * Find the primary trigonometric ratios for angles less than 90º
 * Use the special angles, less than 90º to obtain their coordinates on the unit circle for quadrant 1, ie. (x, y) = (cosθ, sinθ)
 * Use the coordinates from quadrant 1 to generate the coordinates on the unit circle for the related rotation angles (quadrants 2,3 and 4)
 * Use the coordinates on the unit circle generated from the related rotation angles, 90 to 360, to make connections between the sine ratio and the sine function and between the cosine ratio and the cosine functions by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios, with or without technology
 * Determine the measures of two angles from 0º to 360º for which the value of a given trigonometric ratio is the same
 * Sketch the graphs of //f(x)// = sinx and //f(x)// = cos x for angle measures expressed in degrees
 * Determine and describe key properties of both functions
 * Collect and graph data that can be modeled as a sinusoidal function from primary or secondary sources
 * Discuss the behaviour of these graphs, review their properties
 * Determine, through investigation using technology, and describe the roles of the parameters d and c in functions of the form y =sin(x-d) + c and y = cos(x-d) + c for angles expressed in degrees
 * Sketch graphs of y = asin(k(x-d))=c or y = acos (k(x-d))=c by applying transformations to the graphs y = sinx or y = cosx
 * Discuss the domain and range of the transformed function
 * Represent a sinusoidal function with an equation given a graph or its properties
 * Pose and solve problems based on applications involving sinusoidal functions using graphs and graphing technology
 * Collect data that would show sinusoidal behaviour and model with sinusoidal functions with and without technology
 * Describe how sinusoidal graphs change given changes in the context

Unit 5. Applications to Trig Ratios and Vectors
Notes: This unit applies the sine law and cosine law to application problems and vectors.

Goals:
 * Solve problems that will activate prior knowledge about primary trigonometric ratios, the sine law and the cosine law
 * Solve multi-step problems in two and three dimensions, including those that arise from real-world applications by determining the measures of the sides and angles of right triangles using the primary trigonometric ratios
 * Make connections between the graphical solution of sin θ = k, 0 < //k<1//, for 0 < θ <180, and the geometric representation of ambiguous case using a variety of tools and strategies (e.g. dynamic geometry software, graphing calculator, graph paper and string)
 * Solve problems involving oblique triangles, including those that arise from real-world applications, using the sine law (including the ambiguous case) and cosine law relate to the values of sine and cosine for angles 90 < x <180
 * Recognize the properties of a vector (magnitude, direction)
 * Represent a vector as a directed line segment geometrically
 * Understand the equality of vectors
 * Identify, gather, and interpret information about real-world applications of vectors
 * Resolve a vector represented as a directed line segment into its vertical and horizontal components in context
 * Represent a vector as a directed line segment, given its vertical and horizontal components
 * Determine, through investigation using a variety of tools and strategies, the sum and difference of two vectors
 * Solve problems involving the addition and subtraction of vectors, including problems arising from real-world applications

Unit 6. Solving Problems Involving Geometry
Notes: This unit focuses on applications that allow students to end the course with an applied mathematics experience.

Goals:
 * Solve problems involving 2D and 3D figures arising from real-world applications
 * Research and explain applications of geometric shapes in real-world technology-related fields
 * Convert between imperial and metric systems using a variety of tools
 * Solve problems involving areas, volumes, and surface areas in situations arising from real-world applications
 * Determine circular properties and solve related problems, including those arising from real-world applications

= __Resources__ =
 * Textbook: **Mathematics or College Technology 12**  || Pearson Education Canada, 2002, Robert Alexander et. al.   ||
 * OAME Website, TIPS book: ||  @http://www.oame.on.ca/main/staging9.php?code=grspecres&ph=12&sp=MCT4C  ||
 * Ontario Educational Resource Bank: || @http://resources.elearningontario.ca/ ||
 * Ministry Exemplars: || [] ||
 * Math Helper Website: || [] ||
 * Technology: || Graphing calculator and/or Fathom software ||
 * TI83 Tutorials: || <span style="color: blue; font-family: 'Times New Roman','serif';">[] ||
 * TI NSpire Tutorials: || <span style="color: blue; font-family: 'Times New Roman','serif';">[] ||
 * Fathom tutorials: || <span style="font-family: 'Times New Roman','serif';">Included in software. ||
 * Online Function Grapher: || <span style="color: blue; font-family: 'Times New Roman','serif';">[] ||
 * ExploreLearning Gizmos: || <span style="color: blue; font-family: 'Times New Roman','serif';">[] ||
 * National Library of Virtual Manipulatives: || <span style="color: blue; font-family: 'Times New Roman','serif';">[] ||
 * Sir Robert Borden HS Manipulative Tutorial Videos**:** || <span style="color: blue; font-family: 'Times New Roman','serif';">[] ||
 * <span style="font-family: 'Times New Roman','serif';">Sample Problems and Solutions PowerPoints || <span style="color: blue; font-family: 'Times New Roman','serif';">[] ||

=** __Assessment & Evaluation__ **=

The student will be expected to demonstrate their understanding of these key learnings through their knowledge, thinking, communication and application of the learning.

Emphasizes the ability to recall factual information, recognize fundamental concepts and the foundational skills of the subject/discipline. || 20% || Knowledge of content (e.g., facts, terms, procedural skills, use of tools) and understanding of mathematical concepts. These may be assessed through quizzes, tests, oral questions and answers, practice question assignments, etc. || Emphasizes the thinking skills used in thinking processes to demonstrate the student’s understanding of information they have processed. || 20% || Use of planning skills: understanding the problem (e.g., formulating and interpreting the problem, making conjectures) and making a plan for solving the problem. Use of processing skills: carrying out a plan (e.g., collecting data, questioning, testing, revising, modelling, solving, inferring, forming conclusions) and looking back at the solution (e.g., evaluating reasonableness, making convincing arguments, reasoning, justifying, proving, reflecting). Use of critical/creative thinking processes (e.g., problem solving, inquiry). These may be assessed through open-ended investigations, inquiry tasks, oral interview, projects, verbal defense, observation of process, etc. || Emphasizes the clear, precise and effective use of oral, written and visual language to communicate the student’s understanding of information and ideas || 10% || Expression and organization of mathematical thinking (e.g., clarity of expression, logical organization), using oral, visual, and written forms (e.g., pictorial, graphic, dynamic, numeric, algebraic forms; concrete materials). Communication for different audiences (e.g., peers, teachers) and purposes (e.g., to present data, justify a solution, express a mathematical argument) in oral, visual, and written forms. Use of conventions, vocabulary, and terminology of the discipline (e.g., terms, symbols) in oral, visual, and written forms. These may be assessed through journals, written explanations or reports, teacher-student conferences, solution presentations, problem form scores, etc. || Emphasizes the application and integration of knowledge, skills, processes and techniques to produce evidence of the student’s understanding. || 20% || Application of knowledge and skills in familiar contexts and transfer of knowledge and skills to new contexts. Making connections within and between various contexts (e.g., connections between concepts, representations, and forms within mathematics; connections involving use of prior knowledge and experience; connections between mathematics, other disciplines, and the real world). These may be assessed with rich tasks, open-ended problems, real-world projects and applications, etc. ||
 * ==**Knowledge**==
 * ==**Thinking**==
 * ==**Communication**==
 * ==**Application**==


 * How will you demonstrate your learning?** (what to say, write and do)

10% || **INQUIRY PERFORMANCE TASK** (10%) consisting of a mathematical investigation or contextual, open-ended problematic situation suited to a variety of approaches including use of technology where appropriate. A small task for each strand is recommended. || 20 % || **FINAL EXAMINATION** (20%) consisting of a variety of question types (e.g. short answer, multiple choice, extended tasks) sampling all strands and categories of 2.5 hours duration or less. || <span style="color: blue; font-family: 'Arial','sans-serif'; font-size: 10pt; line-height: 115%;"> Adapted from [|http://chatt.hdsb.ca/~sampsons/]
 * 70% of your learning will be assessed through: || Formative and Summative Evaluations || See previous section for **70% breakdown**. ||
 * 30% of your learning will be assessed at the end of the course (last four weeks of the semester)through: || Final Evaluation
 * ^  || Final Evaluation
 * 100% of your learning will be recorded as: || Final Grade on **Report Card** ||  ||

//<span style="color: gray; font-family: 'Arial','sans-serif'; font-size: 10pt; line-height: 115%;">Page Created By //<span style="color: gray; font-family: 'Arial','sans-serif'; font-size: 10pt; line-height: 115%;">: //<span style="color: gray; font-family: 'Arial','sans-serif'; font-size: 10pt; line-height: 115%;">Rania Abou-Gabal, Gennaro Busa & Susan Carson //