Ontario · Grade 12
Mathematics for College Technology, Grade 12 (College)
- MCT4C
Ontario’s Grade 12 math course, from Exponential Functions to Applications of Geometry.
- Units
- 4
- Lessons
- 12
- Skills
- 57
- Provincial exam
- None
- Official curriculum
- Ontario Ministry of Education
Unit by unit
Every skill in the course, numbered by unit.
Unit 1 · 10 skills
Exponential Functions
Graphing and Comparing Exponential Functions
How changing the base or the sign of the exponent affects an exponential graph
I can describe how changing the base or the exponent's sign affects an exponential graph.
Skill 1.1
Solving Exponential Equations Graphically
Solving simple exponential equations numerically and graphically
I can solve simple exponential equations numerically and graphically.
Skill 1.2
Solving exponential equations by the intersection of two exponential graphs
I can solve exponential equations by finding the intersection of two exponential graphs.
Skill 1.3
Posing and solving real-world exponential problems with a graph
I can pose and solve real-world exponential problems using a graph.
Skill 1.4
Solving Exponential Equations Algebraically
Simplifying expressions with integer and rational exponents
I can simplify expressions with integer and rational exponents.
Skill 1.5
Solving exponential equations using a common base
I can solve exponential equations by finding a common base.
Skill 1.6
The logarithm as the inverse of exponentiation
I can explain the logarithm as the inverse of exponentiation and evaluate simple logarithms.
Skill 1.7
Approximating the logarithm of a number to any base with technology
I can approximate the logarithm of a number to any base using technology.
Skill 1.8
Connecting logarithmic and exponential equations
I can connect logarithmic and exponential equations and solve simple exponential equations using logarithms.
Skill 1.9
Posing and solving real-world exponential problems by rewriting in logarithmic form
I can pose and solve real-world exponential problems by rewriting them in logarithmic form.
Skill 1.10
The official wording · 10 outcomes
- 1.1
- determine, through investigation with tech- nology, and describe the impact of changing the base and changing the sign of the expo- nent on the graph of an exponential function
- 1.2
- solve simple exponential equations numeric- ally and graphically, with technology
- 1.3
- determine, through investigation using graph- ing technology, the point of intersection of the graphs of two exponential functions
- 1.4
- pose problems based on real-world applica- tions (e.g., compound interest, population growth) that can be modelled with exponen- tial equations, and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation
- 1.5
- simplify algebraic expressions containing integer and rational exponents using the laws of exponents
- 1.6
- solve exponential equations in one variable by determining a common base
- 1.7
- recognize the logarithm of a number to a given base as the exponent to which the base must be raised to get the number, recognize the operation of finding the logarithm to be the inverse operation (i.e., the undoing or reversing) of exponentiation, and evaluate simple logarithmic expressions
- 1.8
- determine, with technology, the approximate logarithm of a number to any base, including base 10
- 1.9
- make connections between related logarithmic and exponential equations
- 1.10
- pose problems based on real-world applica- tions that can be modelled with given expo- nential equations, and solve these and other such problems algebraically by rewriting them in logarithmic form
Key ideas
A larger base makes an exponential function grow faster; a base between 0 and 1 makes it decay instead of grow.
Systematic trial with a calculator narrows in on a solution by testing values.
The exponent laws (multiply, divide, power of a power, roots as fractional exponents) combine to simplify longer expressions.
Changing the sign of the exponent flips the graph horizontally, turning growth into decay or the reverse.
A solution may not be a 'nice' exact number — technology gives a close decimal answer.
Watch out for
- Assuming a negative exponent always makes the whole function's output negative.
- Expecting every exponential equation to have a whole-number solution.
- Mixing up rules for integer exponents with rules for rational (root-based) exponents.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 2 · 19 skills
Polynomial Functions
Investigating Graphs of Polynomial Functions
Recognizing polynomial expressions and functions
I can recognize a polynomial expression and function, and identify linear and quadratic functions as examples.
Skill 2.1
Comparing graphical and algebraic representations of polynomial functions
I can compare the graphical and algebraic representations of polynomial functions.
Skill 2.2
Describing key features of polynomial function graphs
I can describe the key features of a polynomial function's graph.
Skill 2.3
Distinguishing polynomial functions from sinusoidal and exponential functions
I can distinguish polynomial functions from sinusoidal and exponential functions.
Skill 2.4
Connecting Graphs and Equations of Polynomial Functions
Evaluating polynomial functions in function notation
I can substitute into and evaluate polynomial functions in function notation.
Skill 2.5
Posing and solving real-world problems using polynomial function graphs
I can pose and solve real-world problems using polynomial function graphs.
Skill 2.6
Recognizing the limitations and domain/range restrictions of polynomial models
I can recognize the limitations of a polynomial model and explain domain/range restrictions.
Skill 2.7
Factoring polynomial expressions up to degree four
I can factor polynomial expressions up to degree four.
Skill 2.8
Connecting a polynomial function's factored form to its x-intercepts and sketching its graph
I can connect a polynomial's factored form to its x-intercepts and sketch its graph.
Skill 2.9
Connecting the real roots of a polynomial equation to its graph's x-intercepts
I can connect the real roots of a polynomial equation to its graph's x-intercepts.
Skill 2.10
Solving Problems Involving Polynomial Equations
Solving polynomial equations up to degree four
I can solve polynomial equations up to degree four.
Skill 2.11
Solving real-world problems involving polynomial functions and equations
I can solve real-world problems involving polynomial functions and equations.
Skill 2.12
Identifying and explaining the roles of constants and variables in a formula
I can identify and explain the roles of constants and variables in a formula.
Skill 2.13
Expanding and simplifying polynomial expressions with more than one variable
I can expand and simplify polynomial expressions with more than one variable.
Skill 2.14
Solving equations of the form xⁿ = a using rational exponents
I can solve equations of the form xⁿ = a using rational exponents.
Skill 2.15
Determining a variable's value from a real-world formula
I can find the value of a variable in a real-world formula.
Skill 2.16
Connecting formulas to linear, quadratic, and exponential functions
I can connect formulas to linear, quadratic, and exponential functions.
Skill 2.17
Solving multi-step real-world problems using formulas
I can solve multi-step real-world problems using formulas.
Skill 2.18
Gathering information about mathematical modelling in occupations and college programs
I can find and describe how occupations and college programs use mathematical modelling.
Skill 2.19
The official wording · 19 outcomes
- 2.1
- recognize a polynomial expression (i.e., a series of terms where each term is the product of a constant and a power of x with a non- negative integral exponent
- 2.2
- compare, through investigation using graph- ing technology, the graphical and algebraic representations of polynomial (i.e., linear, quadratic, cubic, quartic) functions
- 2.3
- describe key features of the graphs of poly- nomial functions (e.g., the domain and range, the shape of the graphs, the end behaviour of the functions for very large positive or negative x-values)
- 2.4
- distinguish polynomial functions from sinusoidal and exponential functions
- 2.5
- substitute into and evaluate polynomial func- tions expressed in function notation, including functions arising from real-world applications
- 2.6
- pose problems based on real-world applica- tions that can be modelled with polynomial functions, and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation
- 2.7
- recognize, using graphs, the limitations of modelling a real-world relationship using a polynomial function, and identify and explain any restrictions on the domain and range
- 2.8
- factor polynomial expressions in one variable, of degree no higher than four, by selecting and applying strategies (i.e., common factor- ing, difference of squares, trinomial factoring)
- 2.9
- make connections, through investigation using graphing technology (e.g., dynamic geometry software), between a polynomial function given in factored form
- 2.10
- determine, through investigation using tech- nology (e.g., graphing calculator, computer algebra systems), and describe the connection between the real roots of a polynomial equa- tion and the x-intercepts of the graph
- 2.11
- solve polynomial equations in one variable, of degree no higher than four
- 2.12
- solve problems algebraically that involve polynomial functions and equations of degree no higher than four, including those arising from real-world applications
- 2.13
- identify and explain the roles of constants and variables in a given formula
- 2.14
- expand and simplify polynomial expressions involving more than one variable
- 2.15
- solve equations of the form x = a using rational exponents
- 2.16
- determine the value of a variable of degree no higher than three, using a formula drawn from an application
- 2.17
- make connections between formulas and lin- ear, quadratic, and exponential functions
- 2.18
- solve multi-step problems requiring formulas arising from real-world applications
- 2.19
- gather, interpret, and describe information about applications of mathematical modelling in occupations, and about college programs that explore these applications
Key ideas
A polynomial expression is a sum of terms, each a constant times a power of x with a whole-number exponent.
Function notation like V(x) means 'the volume as a function of x' — substituting a value for x gives a specific output.
Factoring a polynomial equation and setting each factor to zero finds all its real solutions.
Linear and quadratic functions are polynomial functions of degree 1 and 2.
Real-world formulas, like a box's volume from cut corners, are often written and evaluated as polynomial functions.
Watch out for
- Thinking an expression with a negative or fractional exponent still counts as polynomial.
- Substituting the value into only part of the function's expression.
- Dividing both sides by x instead of factoring it out, losing the x = 0 solution.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 3 · 14 skills
Trigonometric Functions
Applying Trigonometric Ratios
Exact sine, cosine, and tangent values of special angles
I can determine the exact sine, cosine, and tangent values of special angles.
Skill 3.1
Determining sine, cosine, and tangent for angles from 0° to 360°
I can determine sine, cosine, and tangent for angles from 0 degrees to 360 degrees.
Skill 3.2
Finding two angles from 0° to 360° with the same trig ratio value
I can find two angles from 0 to 360 degrees with the same trigonometric ratio value.
Skill 3.3
Solving multi-step 2D and 3D right-triangle problems
I can solve multi-step 2D and 3D right-triangle problems.
Skill 3.4
Solving oblique triangle problems including the ambiguous case
I can solve oblique triangle problems, including the ambiguous case of the sine law.
Skill 3.5
Connecting Graphs and Equations of Sinusoidal Functions
Connecting the sine and cosine ratios to the sine and cosine functions
I can connect the sine and cosine ratios to the sine and cosine functions.
Skill 3.6
Representing a sinusoidal function with an equation given its graph or properties
I can represent a sinusoidal function with an equation given its graph or properties.
Skill 3.7
Collecting and graphing data that fits a sinusoidal model
I can collect and graph data that fits a sinusoidal model.
Skill 3.8
Identifying periodic and sinusoidal functions and their domain/range restrictions
I can identify periodic and sinusoidal functions and explain any domain/range restrictions.
Skill 3.9
Posing and solving real-world sinusoidal function problems
I can pose and solve real-world sinusoidal function problems.
Skill 3.10
Solving Problems Involving Sinusoidal Functions
Sketching sine and cosine graphs and describing their key properties
I can sketch sine and cosine graphs and describe their key properties.
Skill 3.11
Investigating the roles of d and c in vertical and horizontal sinusoidal translations
I can investigate the roles of d and c as vertical and horizontal translations of sinusoidal graphs.
Skill 3.12
Investigating the roles of a and k in reflections and stretches of sinusoidal graphs
I can investigate the roles of a and k as reflections, stretches, and compressions of sinusoidal graphs.
Skill 3.13
Determining amplitude, period, and phase shift, and sketching transformed sinusoidal graphs
I can determine amplitude, period, and phase shift, and sketch transformed sinusoidal graphs.
Skill 3.14
The official wording · 14 outcomes
- 3.1
- determine the exact values of the sine, cosine, and tangent of the special angles 0°, 30°, 45°, 60°, 90°, and their multiples
- 3.2
- determine the values of the sine, cosine, and tangent of angles from 0º to 360º, through investigation using a variety of tools
- 3.3
- determine the measures of two angles from 0º to 360º for which the value of a given trigono- metric ratio is the same
- 3.4
- solve multi-step problems in two and three dimensions, including those that arise from real-world applications (e.g., surveying, navi- gation), by determining the measures of the sides and angles of right triangles using the primary trigonometric ratios
- 3.5
- solve problems involving oblique triangles, including those that arise from real-world applications, using the sine law (including the ambiguous case) and the cosine law
- 3.6
- make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the rela- tionship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios
- 3.7
- represent a sinusoidal function with an equation, given its graph or its properties
- 3.8
- collect data that can be modelled as a sinu- soidal function (e.g., voltage in an AC circuit, pressure in sound waves, the height of a tack on a bicycle wheel that is rotating at a fixed speed), through investigation with and with- out technology
- 3.9
- identify periodic and sinusoidal functions, including those that arise from real-world applications involving periodic phenomena, given various representations
- 3.10
- pose problems based on applications involv- ing a sinusoidal function, and solve these and other such problems by using a given graph or a graph generated with technology, in degree mode
- 3.11
- sketch the graphs of f(x) = sin x and f(x) = cos x for angle measures expressed in degrees, and determine and describe their key properties
- 3.12
- determine, through investigation using technology, the roles of the parameters d and c in functions of the form y = sin (x – d) + c and y = cos (x – d) + c
- 3.13
- determine, through investigation using technol- ogy, the roles of the parameters a and k in functions of the form y = a sin kx and y = a cos kx
- 3.14
- determine the amplitude, period, and phase shift of sinusoidal functions whose equations are given in the form f(x) = a sin (k(x – d)) + c or f(x) = a cos (k(x – d)) + c, and sketch graphs
Key ideas
Special angles (0°, 30°, 45°, 60°, 90°) and their multiples have exact trig ratio values, not just decimal approximations.
Graphing angle (0° to 360°) against its sine or cosine ratio produces a smooth, repeating wave.
Key properties of sine and cosine graphs include cycle length, domain, range, amplitude, period, and max/min values.
These exact values come from geometric relationships in special right triangles.
This graph defines a function because every angle input gives exactly one ratio output.
Watch out for
- Using a calculator's decimal approximation when an exact value (like √2/2) is expected.
- Not recognizing the sine ratio graph as a genuine function because it repeats.
- Mixing up which graph (sine or cosine) starts at its maximum versus at zero.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 4 · 14 skills
Applications of Geometry
Modelling With Vectors
Recognizing vectors and their real-world applications
I can recognize a vector and describe its real-world applications.
Skill 4.1
Representing a vector as a directed line segment
I can represent a vector as a directed line segment and recognize equal vectors.
Skill 4.2
Resolving a vector into vertical and horizontal components
I can resolve a vector into its vertical and horizontal components.
Skill 4.3
Representing a vector as a directed line segment given its components
I can represent a vector as a directed line segment given its components.
Skill 4.4
Determining the sum or difference of two vectors
I can determine the sum or difference of two vectors.
Skill 4.5
Solving real-world problems with vector addition and subtraction
I can solve real-world problems involving vector addition and subtraction.
Skill 4.6
Solving Problems Involving Geometry
Gathering information about real-world geometric shapes in technology-related fields
I can gather and explain information about geometric shapes in technology-related fields.
Skill 4.7
Converting between the imperial and metric systems
I can convert between the imperial and metric systems as needed.
Skill 4.8
Solving real-world problems with areas of rectangles, parallelograms, trapezoids, triangles, circles, and composite shapes
I can solve real-world problems with areas of rectangles, parallelograms, trapezoids, triangles, circles, and composite shapes.
Skill 4.9
Solving real-world problems with volumes and surface areas of spheres, prisms, and cylinders
I can solve real-world problems with volumes and surface areas of spheres, prisms, and cylinders.
Skill 4.10
Solving Problems Involving Circle Properties
Recognizing and describing arcs, tangents, secants, chords, and angles of circles
I can recognize and describe the parts of a circle: arcs, tangents, secants, chords, and angles.
Skill 4.11
Determining arc length and the area of a sector or segment
I can determine arc length and the area of a sector or segment of a circle.
Skill 4.12
Investigating circle properties involving chords, central and inscribed angles, and tangents
I can investigate circle properties involving chords, central and inscribed angles, and tangents.
Skill 4.13
Solving real-world problems involving circle properties
I can solve real-world problems involving circle properties.
Skill 4.14
The official wording · 14 outcomes
- 4.1
- recognize a vector as a quantity with both magnitude and direction, and identify, gather, and interpret information about real-world applications of vectors
- 4.2
- represent a vector as a directed line segment, with directions expressed in different ways
- 4.3
- resolve a vector represented as a directed line segment into its vertical and horizontal components
- 4.4
- represent a vector as a directed line segment, given its vertical and horizontal components
- 4.5
- determine, through investigation using a va- riety of tools (e.g., graph paper, technology) and strategies (i.e., head-to-tail method; paral- lelogram method; resolving vectors into their vertical and horizontal components), the sum (i.e., resultant) or difference of two vectors
- 4.6
- solve problems involving the addition and subtraction of vectors, including problems arising from real-world applications
- 4.7
- gather and interpret information about real- world applications of geometric shapes and figures in a variety of contexts in technology- related fields
- 4.8
- perform required conversions between the imperial system and the metric system using a variety of tools
- 4.9
- solve problems involving the areas of rect- angles, parallelograms, trapezoids, triangles, and circles, and of related composite shapes, in situations arising from real-world applications
- 4.10
- solve problems involving the volumes and surface areas of spheres, right prisms, and cylinders, and of related composite figures, in situations arising from real-world applications
- 4.11
- recognize and describe (i.e., using diagrams and words) arcs, tangents, secants, chords, segments, sectors, central angles, and inscribed angles of circles, and some of their real-world applications
- 4.12
- determine the length of an arc and the area of a sector or segment of a circle, and solve relat- ed problems
- 4.13
- determine, through investigation using a vari- ety of tools (e.g., dynamic geometry software), properties of the circle associated with chords, central angles, inscribed angles, and tangents
- 4.14
- solve problems involving properties of circles, including problems arising from real-world applications
Key ideas
A vector has both magnitude (size) and direction, unlike a scalar which has only size.
Product design, architecture, and other technology fields choose shapes deliberately for functional reasons.
A chord connects two points on a circle; a tangent touches the circle at exactly one point; a secant crosses through it.
Vectors describe real quantities like displacement, force, and velocity.
Explaining WHY a shape is used (not just identifying it) shows understanding of its purpose.
Watch out for
- Treating a vector as just a number (a scalar) without direction.
- Describing a shape's appearance without explaining its functional purpose.
- Confusing a chord with a tangent, since both are line segments related to a circle.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Plan the term
Pick your term and the days you study. Each unit gets its share of the weeks and comes back a week later for a review. The last two weeks go over the whole course.
Which course?
Ontario has 6 math courses for Grade 12. Here they are side by side.
- Advanced Functions, Grade 12 (University)MHF4U4 units · 72 skillsNo provincial examWhat it coversExponential and Logarithmic Functions · Trigonometric Functions · Polynomial and Rational Functions · Characteristics of Functions
- Calculus and Vectors, Grade 12 (University)MCV4U3 units · 54 skillsNo provincial examWhat it coversRate of Change · Derivatives and Their Applications · Geometry and Algebra of Vectors
- Foundations for College Mathematics, Grade 12 (College)MAP4C4 units · 59 skillsNo provincial examWhat it coversMathematical Models · Personal Finance · Geometry and Trigonometry · Data Management
- Mathematics for College Technology, Grade 12 (College)This courseMCT4C4 units · 57 skillsNo provincial examWhat it coversExponential Functions · Polynomial Functions · Trigonometric Functions · Applications of Geometry
- Mathematics for Work and Everyday Life, Grade 12 (Workplace)MEL4E3 units · 52 skillsNo provincial examWhat it coversReasoning With Data · Personal Finance · Applications of Measurement
- Mathematics of Data Management, Grade 12 (University)MDM4U5 units · 57 skillsNo provincial examWhat it coversCounting and Probability · Probability Distributions · Organization of Data for Analysis · Statistical Analysis · Culminating Data Management Investigation
What a university, college or program asks for is theirs to say: check its admission page, and your school’s course guide.
The official curriculum
This page follows Ontario’s own curriculum for Mathematics for College Technology, Grade 12 (College).
Strands
| Strand | Outcomes | Taught in |
|---|---|---|
| AExponential Functions | Outcomes 10 | Taught in Unit 1 |
| BPolynomial Functions | Outcomes 19 | Taught in Unit 2 |
| CTrigonometric Functions | Outcomes 14 | Taught in Unit 3 |
| DApplications of Geometry | Outcomes 14 | Taught in Unit 4 |
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Questions about Mathematics for College Technology, Grade 12 (College)
What is Mathematics for College Technology, Grade 12 (College)?
Ontario’s Grade 12 math course: 4 units and 57 skills, from Exponential Functions to Applications of Geometry.
What units are in Mathematics for College Technology, Grade 12 (College)?
1. Exponential Functions, 2. Polynomial Functions, 3. Trigonometric Functions and 4. Applications of Geometry.
Is there a provincial exam for Mathematics for College Technology, Grade 12 (College)?
No. Ontario gives no provincial exam for Mathematics for College Technology, Grade 12 (College).
Can MapleMind help with Mathematics for College Technology, Grade 12 (College)?
Yes. MapleMind teaches every skill of Mathematics for College Technology, Grade 12 (College), step by step, with practice and checks, and its tutor answers questions about any of them. It’s free with daily limits, on iPhone, Android and the web.
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