Ontario · Grade 10
Principles of Mathematics, Grade 10 (Academic)
- MPM2D
Ontario’s Grade 10 math course, from Quadratic Relations of the Form y = ax² + bx + c to Trigonometry.
- Units
- 3
- Lessons
- 11
- Skills
- 45
- Provincial exam
- None
- Official curriculum
- Ontario Ministry of Education
Unit by unit
Every skill in the course, numbered by unit.
Unit 1 · 21 skills
Quadratic Relations of the Form y = ax² + bx + c
Investigating the Basic Properties of Quadratic Relations
Collecting and graphing quadratic data
I can collect data that forms a quadratic relation and graph a curve of best fit.
Skill 1.1
Recognizing a parabola from a table of second differences
I can show that a quadratic relation makes a parabola and has a constant second difference.
Skill 1.2
Identifying the key features of a parabola
I can identify a parabola's axis of symmetry, vertex, y-intercept, zeros, and maximum or minimum value.
Skill 1.3
Comparing y = x² with y = 2ˣ and meaning zero and negative exponents
I can compare y = x² with y = 2ˣ and explain what zero and negative exponents mean.
Skill 1.4
Relating the Graph of y = x² and Its Transformations
Effect of a, h, and k on the graph of y = x²
I can describe how a, h, and k each transform the graph of y = x².
Skill 1.5
Roles of a, h, and k in vertex form
I can explain the roles of a, h, and k in y = a(x − h)² + k and find the vertex and axis of symmetry.
Skill 1.6
Sketching y = a(x − h)² + k by hand
I can sketch the graph of y = a(x − h)² + k by hand.
Skill 1.7
Finding the vertex-form equation from a graph
I can determine the vertex-form equation of a parabola from its graph.
Skill 1.8
Solving Quadratic Equations
Expanding and simplifying second-degree polynomials
I can expand and simplify second-degree polynomial expressions.
Skill 1.9
Factoring common factors and simple trinomials
I can factor expressions with a common factor and simple trinomials (a = 1).
Skill 1.10
Factoring complex trinomials (a ≠ 1)
I can factor complex trinomials where the leading coefficient is not 1.
Skill 1.11
Factoring differences of squares
I can factor differences of squares.
Skill 1.12
Connecting factors of a quadratic to its x-intercepts
I can connect the factors of a quadratic expression to the x-intercepts of its graph.
Skill 1.13
Interpreting real and non-real roots
I can interpret whether a quadratic equation's roots are real or non-real.
Skill 1.14
Completing the square
I can complete the square to rewrite a quadratic in vertex form.
Skill 1.15
Sketching or graphing from standard form
I can sketch or graph a quadratic relation given in standard form.
Skill 1.16
Exploring the algebraic development of the quadratic formula
I can follow the algebraic development of the quadratic formula.
Skill 1.17
Solving quadratic equations by factoring
I can solve quadratic equations by factoring.
Skill 1.18
Solving quadratic equations with the quadratic formula
I can solve quadratic equations using the quadratic formula.
Skill 1.19
Solving Problems Involving Quadratic Relations
Finding the zeros and the maximum or minimum value
I can find the zeros and the maximum or minimum value of a quadratic relation.
Skill 1.20
Solving realistic problems with a quadratic graph or equation
I can solve realistic problems represented by a quadratic graph or equation.
Skill 1.21
The official wording · 21 outcomes
- 1.1
- collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology (e.g., concrete materials, scientific probes, graphing calculators), or from secondary sources (e.g., the Internet, Statistics Canada); graph the data and draw a curve of best fit, if appropriate, with or without the use of technology
- 1.2
- determine, through investigation with and without the use of technology, that a quadratic relation of the form y = ax2 + bx + c (a ≠ 0) can be graphically represented as a parabola, and that the table of values yields a constant second difference
- 1.3
- identify the key features of a graph of a parabola (i.e., the equation of the axis of symmetry, the coordinates of the vertex, the y-intercept, the zeros, and the maxi- mum or minimum value), and use the appropriate terminology to describe them
- 1.4
- compare, through investigation using tech- nology, the features of the graph of y = x2 and the graph of y = 2x, and determine the meaning of a negative exponent and of zero as an exponent
- 1.5
- identify, through investigation using tech- nology, the effect on the graph of y = x2 of transformations (i.e., translations, reflec- tions in the x-axis, vertical stretches or compressions) by considering separately each parameter a, h, and k
- 1.6
- explain the roles of a, h, and k in y = a(x – h )2 + k, using the appropriate terminology to describe the transforma- tions, and identify the vertex and the equa- tion of the axis of symmetry
- 1.7
- sketch, by hand, the graph of y = a(x – h )2 + k by applying transforma- tions to the graph of y = x2
- 1.8
- determine the equation, in the form y = a(x – h)2 + k, of a given graph of a parabola.
- 1.9
- expand and simplify second-degree poly- nomial expressions [e.g., (2x + 5)2, (2x – y)(x + 3y)], using a variety of tools
- 1.10
- factor polynomial expressions involving common factors, trinomials, and differ- ences of squares [e.g., 2x2 + 4x, 2x – 2y + ax – ay, x2 – x – 6, 2a2 + 11a + 5, 4x2 – 25], using a variety of tools
- 1.11
- factor polynomial expressions involving common factors, trinomials, and differ- ences of squares [e.g., 2x2 + 4x, 2x – 2y + ax – ay, x2 – x – 6, 2a2 + 11a + 5, 4x2 – 25], using a variety of tools
- 1.12
- factor polynomial expressions involving common factors, trinomials, and differ- ences of squares [e.g., 2x2 + 4x, 2x – 2y + ax – ay, x2 – x – 6, 2a2 + 11a + 5, 4x2 – 25], using a variety of tools
- 1.13
- determine, through investigation, and describe the connection between the factors of a quadratic expression and the x-intercepts (i.e., the zeros) of the graph of the corresponding quadratic relation, expressed in the form y = a(x – r)(x – s)
- 1.14
- interpret real and non-real roots of qua- dratic equations, through investigation using graphing technology, and relate the roots to the x-intercepts of the corre- sponding relations
- 1.15
- express y = ax2 + bx + c in the form y = a(x – h)2 + k by completing the square in situations involving no fractions, using a variety of tools
- 1.16
- sketch or graph a quadratic relation whose equation is given in the form
- 1.17
- explore the algebraic development of the quadratic formula (e.g., given the algebraic development, connect the steps to a numerical example; follow a demonstra- tion of the algebraic development
- 1.18
- solve quadratic equations that have real roots, using a variety of methods (i.e., factoring, using the quadratic formula, graphing)
- 1.19
- solve quadratic equations that have real roots, using a variety of methods (i.e., factoring, using the quadratic formula, graphing)
- 1.20
- determine the zeros and the maximum or minimum value of a quadratic relation from its graph (i.e., using graphing calcu- lators or graphing software) or from its defining equation (i.e., by applying alge- braic techniques)
- 1.21
- solve problems arising from a realistic situ- ation represented by a graph or an equa- tion of a quadratic relation, with and without the use of technology
Key ideas
Real experiments (like a ball rolling or falling) or secondary sources (like Statistics Canada data) can produce data that follows a quadratic pattern.
k shifts the parabola up or down; h shifts it left or right.
Expanding means multiplying out brackets like (2x + 5)² or (2x − y)(x + 3y) into a sum of terms.
The zeros of a quadratic relation can be read from a graph or found algebraically by factoring or the quadratic formula.
A curve of best fit smooths out experimental scatter to reveal the underlying quadratic shape.
Watch out for
- Connecting data points with straight-line segments instead of a smooth curve.
- Thinking y = (x − h)² shifts the graph in the SAME direction as the sign inside the brackets (e.g., expecting (x − 3)² to shift left).
- Forgetting the middle term when squaring a binomial, e.g., treating (2x + 5)² as 4x² + 25.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 2 · 14 skills
Analytic Geometry
The Slope-and-Line Toolkit (2022 Addendum)
Slopes of parallel and perpendicular lines
I can identify how the slopes of parallel and perpendicular lines relate.
Skill 2.1
Developing and using the slope formula
I can develop and use the slope formula.
Skill 2.2
Determining a line's equation from a graph, table, or two points
I can determine the equation of a line from a graph, a table of values, or two points.
Skill 2.3
Translating between forms of a line's equation
I can translate the equation of a line between different forms.
Skill 2.4
Solving Systems of Linear Equations
Solving linear systems by substitution or elimination
I can solve a system of two linear equations algebraically using substitution or elimination.
Skill 2.5
Solving realistic problems with linear systems
I can solve realistic problems represented by a linear system of two equations.
Skill 2.6
Solving Problems Involving Properties of Line Segments
Developing and using the midpoint formula
I can develop and use the midpoint formula.
Skill 2.7
Developing and using the length-of-a-segment formula
I can develop and use the length-of-a-segment formula.
Skill 2.8
Developing the equation of a circle centred at the origin
I can develop the equation of a circle centred at the origin.
Skill 2.9
Finding a circle's radius, equation, and sketch
I can find a circle's radius, write its equation, and sketch it from x² + y² = r².
Skill 2.10
Solving problems with slope, length, and midpoint together
I can solve problems that combine slope, length, and midpoint of a line segment.
Skill 2.11
Using Analytic Geometry to Verify Geometric Properties
Investigating properties of geometric figures
I can investigate characteristics and properties of geometric figures.
Skill 2.12
Verifying geometric properties using algebra and analytic geometry
I can verify geometric properties using algebra and analytic geometry.
Skill 2.13
Planning a multi-step proof of a geometric property
I can plan and carry out a multi-step strategy to verify a geometric property.
Skill 2.14
The official wording · 14 outcomes
- 2.1
- identify the relationship between the slopes of parallel and perpendicular lines, and use this relationship to solve related problems;
- 2.2
- develop the formula for the slope of a line
- 2.3
- use this formula to determine the equations of lines, given information about the lines (e.g., a graph of a line, a table of values, the coordinates of two points)
- 2.4
- represent the equations of lines in different forms (e.g., y = mx + b, Ax + By + C = 0, Ax + By = D) and translate between these forms, as appropriate for the context.
- 2.5
- solve systems of two linear equations involving two variables, using the algebraic method of substitution or elimination
- 2.6
- solve problems that arise from realistic sit- uations described in words or represented by linear systems of two equations involv- ing two variables, by choosing an appro- priate algebraic or graphical method
- 2.7
- develop the formula for the midpoint of a line segment, and use this formula to solve problems
- 2.8
- develop the formula for the length of a line segment, and use this formula to solve problems
- 2.9
- develop the equation for a circle with centre (0, 0) and radius r, by applying the formula for the length of a line segment
- 2.10
- determine the radius of a circle with cen- tre (0, 0), given its equation; write the equation of a circle with centre (0, 0), given the radius; and sketch the circle, given the equation in the form x2 + y2 = r2
- 2.11
- solve problems involving the slope, length, and midpoint of a line segment
- 2.12
- determine, through investigation (e.g., using dynamic geometry software, by paper folding), some characteristics and properties of geometric figures
- 2.13
- verify, using algebraic techniques and analytic geometry, some characteristics of geometric figures
- 2.14
- plan and implement a multi-step strategy that uses analytic geometry and algebraic techniques to verify a geometric property
Key ideas
Parallel lines always have the SAME slope.
Substitution replaces one variable with an equivalent expression from the other equation.
The midpoint of a segment is the average of the two endpoints' x-coordinates and the average of their y-coordinates.
Dynamic geometry software or paper folding can reveal patterns in figures, like where medians of a triangle intersect.
Perpendicular lines have slopes that are negative reciprocals of each other (their product is −1).
Watch out for
- Thinking perpendicular lines just need 'different' slopes, rather than negative reciprocal slopes.
- Substituting into the SAME equation the expression came from, instead of into the other equation.
- Adding the coordinates instead of averaging them (forgetting to divide by 2).
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 3 · 10 skills
Trigonometry
Solving Problems Involving the Trigonometry of Right Triangles
Verifying properties of similar triangles
I can verify the properties of similar triangles.
Skill 3.1
Comparing similarity and congruence
I can describe and compare similarity and congruence.
Skill 3.2
Solving real problems with similar triangles
I can solve real problems involving similar triangles.
Skill 3.3
Right-Triangle Trigonometry
Defining the sine, cosine, and tangent ratios
I can define the sine, cosine, and tangent ratios.
Skill 3.4
Finding sides and angles in right triangles
I can find unknown sides and angles in right triangles using the primary trig ratios and the Pythagorean theorem.
Skill 3.5
Solving real-life right-triangle problems
I can solve real-life right-triangle problems.
Skill 3.6
Solving Problems Involving the Trigonometry of Acute Triangles
Exploring the development of the sine law
I can explore the development of the sine law for acute triangles.
Skill 3.7
Exploring the development of the cosine law
I can explore the development of the cosine law for acute triangles.
Skill 3.8
Using the sine law and cosine law to find sides and angles
I can use the sine law and cosine law to find sides and angles in acute triangles.
Skill 3.9
Solving real problems with acute triangles
I can solve real problems involving the measures of sides and angles in acute triangles.
Skill 3.10
The official wording · 10 outcomes
- 3.1
- verify, through investigation (e.g., using dynamic geometry software, concrete materials), the properties of similar trian- gles
- 3.2
- describe and compare the concepts of similarity and congruence;
- 3.3
- solve problems involving similar triangles in realistic situations (e.g., shadows, reflec- tions, scale models, surveying)
- 3.4
- determine, through investigation (e.g., using dynamic geometry software, con- crete materials), the relationship between the ratio of two sides in a right triangle and the ratio of the two corresponding sides in a similar right triangle, and define the sine, cosine, and tangent ratios
- 3.5
- determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem
- 3.6
- solve problems involving the measures of sides and angles in right triangles in real- life applications
- 3.7
- explore the development of the sine law within acute triangles
- 3.8
- explore the development of the cosine law within acute triangles
- 3.9
- determine the measures of sides and angles in acute triangles, using the sine law and the cosine law
- 3.10
- solve problems involving the measures of sides and angles in acute triangles.
Key ideas
Similar triangles have equal corresponding angles.
In similar right triangles, the ratio between any two corresponding sides is always the same for a given angle.
The sine law relates the sides of a triangle to the sines of their opposite angles: a/sin A = b/sin B = c/sin C.
Similar triangles have proportional corresponding sides — the same ratio between matching sides.
Sine, cosine, and tangent are named ratios of specific side pairs relative to a chosen angle (e.g., sin A = opposite/hypotenuse).
Watch out for
- Thinking similar triangles must be the same SIZE, confusing similarity with congruence.
- Mixing up which side is 'opposite' versus 'adjacent' depending on which angle is being used.
- Pairing a side with the WRONG opposite angle when setting up the sine law ratio.
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 2 math courses for Grade 10. Here they are side by side.
- Foundations of Mathematics, Grade 10 (Applied)MFM2P3 units · 34 skillsNo provincial examWhat it coversMeasurement and Trigonometry · Modelling Linear Relations · Quadratic Relations
- Principles of Mathematics, Grade 10 (Academic)This courseMPM2D3 units · 45 skillsNo provincial examWhat it coversQuadratic Relations of the Form y = ax² + bx + c · Analytic Geometry · Trigonometry
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 Principles of Mathematics, Grade 10 (Academic).
Textbooks
The textbooks Ontario Ministry of Education lists for Principles of Mathematics, Grade 10 (Academic):
- Nelson Principles of Mathematics 10
- Principles of Mathematics 10
- Nelson Mathematics 10
- Addison Wesley Principles of Mathematics 10
- MATHPOWER 10™, Ontario Edition
From the Trillium List, read October 27, 2026. ontario.ca
Study it in MapleMind
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Questions about Principles of Mathematics, Grade 10 (Academic)
What is Principles of Mathematics, Grade 10 (Academic)?
Ontario’s Grade 10 math course: 3 units and 45 skills, from Quadratic Relations of the Form y = ax² + bx + c to Trigonometry.
What units are in Principles of Mathematics, Grade 10 (Academic)?
1. Quadratic Relations of the Form y = ax² + bx + c, 2. Analytic Geometry and 3. Trigonometry.
Is there a provincial exam for Principles of Mathematics, Grade 10 (Academic)?
No. Ontario gives no provincial exam for Principles of Mathematics, Grade 10 (Academic).
What textbook does Principles of Mathematics, Grade 10 (Academic) use?
The textbooks Ontario Ministry of Education lists for Principles of Mathematics, Grade 10 (Academic): Nelson Principles of Mathematics 10, Nelson Education Ltd., Principles of Mathematics 10, McGraw-Hill Ryerson, Nelson Mathematics 10, Nelson Thomson Learning, Addison Wesley Principles of Mathematics 10, Pearson Education Canada and MATHPOWER 10™, Ontario Edition, McGraw-Hill Ryerson.
What comes after Principles of Mathematics, Grade 10 (Academic)?
Functions, Grade 11 (University), Functions and Applications, Grade 11 (University/College), Foundations for College Mathematics, Grade 11 (College) and Mathematics for Work and Everyday Life, Grade 11 (Workplace).
Can MapleMind help with Principles of Mathematics, Grade 10 (Academic)?
Yes. MapleMind teaches every skill of Principles of Mathematics, Grade 10 (Academic), 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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