
Secondary Math Learning Overview
Secondary Math Learning Overview

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Mathematical Processes

"The mathematical processes cannot be separated from the knowledge and skills that students acquire throughout the course. Students must problem solve, communicate, reason, reflect, and so on, as they develop the knowledge, the understanding of concepts, and the skills required in the course."
The Ontario Curriculum, Grades 9 and 10: Mathematics
The mathematical processes are:

Grade 9 Academic, Principles of Mathematics (MPM1D)

"This course enables students to develop an understanding of mathematical concepts related to algebra, analytic geometry, and measurement and geometry through investigation, the effective use of technology, and abstract reasoning. Students will investigate relationships, which they will then generalize as equations of lines, and will determine the connections between different representations of a linear relation. They will also explore relationships that emerge from the measurement of threedimensional figures and twodimensional shapes. Students will reason mathematically and communicate their thinking as they solve multistep problems."
Taken and adapted from The Ontario Curriculum, Grades 9 and 10: Mathematics
Number Sense and Algebra
Linear Relations
use data management to investigate relationships
understand characteristics of linear relations4
connect various representations of linear relations
Analytic Geometry
investigate the relationship between the equation of a relation and the shape of its graph
investigate the properties of slope5
use the properties of linear relations to solve problems
Measurement and Geometry
investigate the optimal values6 of measurements
solve problems involve perimeter, area, surface area, and volume
investigate and apply geometric relationships
1 Exponent. A special use of a superscript in mathematics. For example, in 32, the exponent is 2. An exponent is used to denote repeated multiplication. For example, 54 means 5 x 5 x 5 x 5. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 61)
2 Polynomial expression. An algebraic expression taking the form a + bx + cx2 + . . . , where a, b, and c are numbers. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 63)
3 Firstdegree equation. An equation in which the variable has the exponent 1; for example, 5(3x – 1) + 6 = –20 + 7x + 5. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 61)
4 Linear relation. A relation between two variables that appears as a straight line when graphed on a coordinate system. May also be referred to as a linear function. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 62)
5 Slope. A measure of the steepness of a line, calculated as the ratio of the rise (vertical change between two points) to the run (horizontal change between the same two points). (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 64)
6 Optimal value. The maximum or minimum value of a variable. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 62)

Grade 9 Applied, Foundations of Mathematics (MFM1P)

"This course enables students to develop an understanding of mathematical concepts related to introductory algebra, proportional reasoning, and measurement and geometry through investigation, the effective use of technology, and handson activities. Students will investigate reallife examples to develop various representations of linear relations, and will determine the connections between the representations. They will also explore certain relationships that emerge from the measurement of threedimensional figures and twodimensional shapes. Students will consolidate their mathematical skills as they solve problems and communicate their thinking."
The Ontario Curriculum, Grades 9 and 10: Mathematics
Number Sense and Algebra
Linear Relations
use data management to investigate relationships
determine characteristics of linear relations3
investigate constant rate of change4
connect various representations of linear relations and solving problems using the representations
Measurement and Geometry
investigate the optimal values5 of measurements of rectangles
solve problems involving perimeter, area, and volume
investigate and apply geometric relationships
1 Proportional Reasoning. Reasoning or problem solving based on the examination of equal ratios. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 63)
2 Polynomial expression. An algebraic expression taking the form a + bx + cx2 + . . . , where a, b, and c are numbers. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 63)
3 Linear relation. A relation between two variables that appears as a straight line when graphed on a coordinate system. May also be referred to as a linear function. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 62)
4 Rate of Change.The change in one variable relative to the change in another.The slope of a line represents rate of change. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 63)
5 Optimal value. The maximum or minimum value of a variable. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 62)

Grade 10 Academic, Principles of Mathematics (MPM2D)

"This course enables students to broaden their understanding of relationships and extend their problemsolving and algebraic skills through investigation, the effective use of technology, and abstract reasoning. Students will explore quadratic relations and their applications; solve and apply linear systems; verify properties of geometric figures using analytic geometry; and investigate the trigonometry of right and acute triangles. Students will reason mathematically and communicate their thinking as they solve multistep problems"
The Ontario Curriculum, Grades 9 and 10: Mathematics
Quadratic Relations of the Form y = ax2 + bx + c
investigate the basic properties of quadratic relations1
relate the graph of y = x2 and its transformations2
solve quadratic equations3
solve problems involving quadratic relations
Analytic Geometry
use linear systems4 to solve problems
solve problems involving properties of line segments
use analytic5 geometry to verify geometric properties
Trigonometry
investigate similarity and solving problems involving similar triangles6
solve problems involving the trigonometry of right7 triangles
solve problems involving the trigonometry of acute8 triangles
1 Quadratic Relation. A relation whose equation is in quadratic form; for example, y = x2 + 7x + 10. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 63)
2 Transformation. A change in a figure that results in a different position, orientation, or size.The transformations include translation, reflection, rotation, compression,and stretch. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 64
3 Quadratic Equation. An equation that contains at least one term whose exponent is 2, and no term with an exponent greater than 2; for example, x2 + 7x + 10 = 0. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 63)
4 System of Linear Equations.Two or more linear equations involving two or more variables.The solution to a system of linear equations involving two variables is the point of intersection of two straight lines. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p.64)
5 Analytic Geometry. A geometry that uses the xyplane to determine equations that represent lines and curves. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 60)
6 Similar Triangles. Triangles in which corresponding sides are proportional. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 64)
7 Right Triangle. A triangle containing one 90º angle. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 63)
8 Acute triangle. A triangle in which each of the three interior angles measures less than 90º. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 60)

Grade 10 Applied, Foundations of Mathematics (MFM2P)

"This course enables students to consolidate their understanding of linear relations and extend their problemsolving and algebraic skills through investigation, the effective use of technology, and handson activities. Students will develop and graph equations in analytic geometry; solve and apply linear systems, using reallife examples; and explore and interpret graphs of quadratic relations. Students will investigate similar triangles, the trigonometry of right triangles, and the measurement of threedimensional figures. Students will consolidate their mathematical skills as they solve problems and communicate their thinking."
The Ontario Curriculum, Grades 9 and 10: Mathematics
Measurement and Trigonometry
solve problems involving similar triangles1
solve problems involving the trigonometry of right triangles2
solve problems involving surface area and volume, using imperial and metric systems of measurement
Modelling Linear Relations
manipulate and solving algebraic equations
graph and writing equations of lines
solve and interpreting systems of linear equations3
Quadratic Relations of the Form y = ax2 + bx + c
manipulate quadratic expressions
identify characteristics of quadratic relations4
solve problems by interpreting graphs of quadratic relations
1 Similar Triangles. Triangles in which corresponding sides are proportional. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 64)
2 Right Triangle. A triangle containing one 90º angle. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 63)
3 System of Linear Equations.Two or more linear equations involving two or more variables.The solution to a system of linear equations involving two variables is the point of intersection of two straight lines. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p.64)
4 Quadratic Relation. A relation whose equation is in quadratic form; for example, y = x2 + 7x + 10. (The Ontario Curriculum Grades 9 and 10, Mathematics, 2005, p. 63)

Grade 11 Functions, University Preparation (MCR3U)

"The Grade 11 university preparation course, Functions, builds on the concepts and skills
developed in the Grade 9 and 10 academic mathematics courses. The course is designed
to prepare students for Grade 12 mathematics courses that lead to one of many university
programs, including science, engineering, social sciences, liberal arts, and education.
The concept of functions is introduced in the Characteristics of Functions strand of this
course and extended through the investigation of two new types of relationships in the
Exponential Functions and Trigonometric Functions strands. The Discrete Functions
strand allows students, through the study of different representations of sequences and
series, to revisit patterning and algebra concepts introduced in elementary school and
make connections to financial applications involving compound interest and ordinary
simple annuities."
Prerequisite: Principles of Mathematics, Grade 10, Academic
Taken and adapted from The Ontario Curriculum, Grades 11 and 12, Mathematics

Grade 11 Functions and Applications, University/College Preparation (MCF3M)

"The Grade 11 university/college preparation course, Functions and Applications, provides
preparation for students who plan to pursue technologyrelated programs in college, while
also leaving the option open for some students to pursue postsecondary programs that
require the Grade 12 university preparation course Mathematics of Data Management.
The Functions and Applications course explores functions by revisiting key concepts from
the Grade 10 mathematics curriculum and by using a more applied approach with less
emphasis on abstract concepts than in the Grade 11 university preparation course,
Functions. The first strand, Quadratic Functions, extends knowledge and skills related
to quadratics for students who completed the Grade 10 applied mathematics course and
reviews this topic for students entering from the Grade 10 academic course. The strand
also introduces some of the properties of functions. The other two strands, Exponential
Functions and Trigonometric Functions, emphasize realworld applications and help
students develop the knowledge and skills needed to solve problems related to these
applications."
Prerequisite: Principles of Mathematics, Grade 10, Academic, or Foundations of
Mathematics, Grade 10, Applied

Grade 11 Foundations for College Mathematics, College Preparation (MBF3C)

"The Grade 11 college preparation course, Foundations for College Mathematics, includes
a blend of topics needed by students who plan to pursue one of a broad range of college
programs. The course has been designed with four strands that address different areas of
mathematics. The Mathematical Models strand uses the concepts connected to linear and
quadratic relations developed in the Grade 9 and 10 applied mathematics courses to
revisit quadratic relations and introduce exponential relations. The Personal Finance
strand focuses on compound interest and applications related to investing and borrowing
money and owning and operating a vehicle. Applications requiring spatial reasoning are
addressed in the Geometry and Trigonometry strand. The fourth strand, Data
Management, explores practical applications of onevariable statistics and probability."
Prerequisite: Foundations of Mathematics, Grade 10, Applied

Grade 11 Mathematics for Work and Everyday Life, Workplace Preparation (MEL3E)

"The Grade 11 workplace preparation course, Mathematics for Work and Everyday Life, is
designed to help students consolidate the basic knowledge and skills of mathematics used
in the workplace and in everyday life. This course is ideal for students who would like to
take the Grade 12 workplace preparation course before graduating from high school and
entering the workplace. The course also meets the needs of students who wish to fulfill
the senior mathematics graduation requirement but do not plan to take any further
courses in mathematics. All three strands, Earning and Purchasing; Saving, Investing,
and Borrowing; and Transportation and Travel, provide students with the opportunity
to use proportional reasoning to solve a variety of problems."
Prerequisite: Principles of Mathematics, Grade 9, Academic, or Foundations of
Mathematics, Grade 9, Applied, or a ministryapproved locally developed
Grade 10 mathematics course

Grade 12 Advanced Functions, University Preparation (MHF4U)

"The Grade 12 university preparation course Advanced Functions satisfies the mathematical
prerequisite for some universities in areas that include business, social science,
and health science programs. The strands in this course help students deepen their
understanding of functions by revisiting the exponential and trigonometric functions
introduced in Grade 11 to address related concepts such as radian measure and logarithmic
functions and by extending prior knowledge of quadratic functions to explore polynomial
and rational functions. The Characteristics of Functions strand addresses some
of the general features of functions through the examination of rates of change and
methods of combining functions."
Prerequisite: Functions, Grade 11, University Preparation, or Mathematics for College
Technology, Grade 12, College Preparation

Grade 12 Calculus and Vectors, University Preparation (MCV4U)

"The Grade 12 university preparation course Calculus and Vectors is designed to prepare
students for university programs, such as science, engineering, and economics, that
include a calculus or linear algebra course in the first year. Calculus is introduced in the
Rate of Change strand by extending the numeric and graphical representation of rates of
change introduced in the Advanced Functions course to include more abstract algebraic
representations. The Derivatives and Their Applications strand provides students with
the opportunity to develop the algebraic and problemsolving skills needed to solve
problems associated with rates of change. Prior knowledge of geometry and trigonometry
is used in the Geometry and Algebra of Vectors strand to develop vector concepts that
can be used to solve interesting problems, including those arising from realworld
applications."
Note: The new Advanced Functions course (MHF4U) must be taken prior to or
concurrently with Calculus and Vectors (MCV4U).

Grade 12 Mathematics of Data Management, University Preparation (MDM4U)

"The Grade 12 university preparation course Mathematics of Data Management is
designed to satisfy the prerequisites for a number of university programs that may
include statistics courses, such as those found in the social sciences and the humanities.
The expectations in the strands of this course require students to apply mathematical
process skills developed in prerequisite courses, such as problem solving, reasoning, and
communication, to the study of probability and statistics. The Counting and Probability
strand extends the basic probability concepts learned in the elementary school program
and introduces counting techniques such as the use of permutations and combinations;
these techniques are applied to both counting and probability problems. The Probability
Distributions strand introduces the concept of probability distributions; these include the
normal distribution, which is important in the study of statistics. In the Organization of
Data for Analysis strand, students examine, use, and develop methods for organizing
large amounts of data, while in the Statistical Analysis strand, students investigate and
develop an understanding of powerful concepts used to analyse and interpret large
amounts of data. These concepts are developed with the use of technological tools such
13THE PROGRAM IN MATHEMATICS
as spreadsheets and Fathom, a ministrylicensed dynamic statistical program. The
Culminating Data Management Investigation strand requires students to undertake
a culminating investigation dealing with a significant issue that will require the
application of the skills from the other strands of the course."
Prerequisite: Functions, Grade 11, University Preparation, or Functions and Applications,
Grade 11, University/College Preparation

Grade 12 Mathematics for College Technology, College Preparation (MCT4C)

"The Grade 12 college preparation course Mathematics for College Technology provides
excellent preparation for success in technologyrelated programs at the college level. It
extends the understanding of functions developed in the Grade 11 university/college
preparation course, Functions and Applications, using a more applied approach, and may
help students who decide to pursue certain university programs to prepare for the
Grade 12 university preparation course Advanced Functions. Exponential and trigonometric
functions are revisited, developing algebraic skills needed to solve problems
involving exponential equations and extending the skills associated with graphical representations
of trigonometric functions. The Polynomial Functions strand extends to polynomial
functions concepts that connect graphs and equations of quadratic functions.
Finally, students apply geometric relationships to solve problems involving composite
shapes and figures and investigate the properties of circles and their applications."
Prerequisite: Functions and Applications, Grade 11, University/College Preparation, or
Functions, Grade 11, University Preparation

Grade 12 Foundations for College Mathematics, College Preparation (MAP4C)

"The Grade 12 college preparation course Foundations for College Mathematics satisfies
the mathematical prerequisites for many college programs, including programs in business,
human services, hospitality and tourism, and some of the health sciences. The four
strands of this course focus on the same areas of mathematics addressed in the Grade 11
college preparation course, Foundations for College Mathematics. The Mathematical
Models strand extends the concepts and skills that related to exponential relations
introduced in Grade 11 and provides students with an opportunity to revisit all of the
relations they have studied in the secondary mathematics program by using a graphical
and algebraic approach. The Personal Finance strand focuses on annuities and mortgages,
renting or owning accommodation, and designing budgets. Problem solving in the
Geometry and Trigonometry strand reinforces the application of relationships associated
with a variety of shapes and figures. The fourth strand, Data Management, addresses
practical applications of twovariable statistics and examines applications of data
management."
Prerequisite: Foundations for College Mathematics, Grade 11, College Preparation, or
Functions and Applications, Grade 11, University/College Preparation
Taken and adapted from The Ontario Curriculum, Grades 11 and 12, Mathematics

Grade 12 Mathematics for Work and Everyday Life, Workplace Preparation (MEL4E)

"The Grade 12 workplace preparation course, Mathematics for Work and Everyday Life,
extends the knowledge and skills developed in Grade 11. The gathering, interpretation,
and display of onevariable data and the investigation of probability concepts are the main
components of the Reasoning With Data strand. Topics in the Personal Finance strand
address owning or renting accommodation, designing a budget, and filing an income tax
return. A variety of problems involving metric and imperial measurement are presented
in the Applications of Measurement strand. The expectations support the use of handson
projects and other experiences that make the mathematics more meaningful for students."
Prerequisite: Mathematics for Work and Everyday Life, Grade 11, Workplace Preparation











