📐 MPM2D – Principles of Mathematics
Grade 10 Academic · Toronto District School Board · Complete Video Resource Guide
📋 Table of Contents
- Unit 1 – Functions (12 classes) · Introduction · Function Notation · Domain & Range · Graphical Relationships
- Unit 2 – Coordinate Geometry (12 classes) · Midpoint & Length · Slope & Lines · Perpendicular Bisector & Median · Verifying Geometric Properties · Circle Properties
- Unit 3 – Linear Systems (8 classes) · Solving by Graphing · Substitution & Elimination · Word Problems
- Unit 4 – Quadratic Equations & Functions (16 classes) · Expanding & Special Products · Factoring · Vertex Form & Transformations · Completing the Square · Quadratic Formula · Applications
- Unit 5 – Trigonometry (12 classes) · Similar Triangles · SOH CAH TOA · Sine Law · Cosine Law · Word Problems
Functions are the language of mathematics. In this opening unit, students learn to interpret, describe, and analyse relationships between variables — both graphically and algebraically — building the essential foundation for everything that follows in MPM2D.
📈 1.1 · Introduction to Functions
What exactly is a function, and how does it differ from a general relation? This video introduces the core concept: a function is a relation where every input has exactly one output. Students learn the vertical line test, explore mapping diagrams, and see real-life examples — building the intuition they'll rely on for the entire course.
- What is a function?
- Vertical line test
- Mapping diagrams
- Relations vs. functions
Source: MathWithMrMichael
🔣 1.2 · Function Notation
Once you know something is a function, f(x) notation is the standard way to write and communicate about it. This dedicated lesson introduces function notation from scratch for Grades 10–12, explains how to read f(x) = ..., evaluate functions at specific values, and substitute expressions — skills applied constantly throughout the rest of the course.
- f(x) notation
- Evaluating f(a)
- Substituting expressions
- Multiple representations
Source: Math with Mr. M
📊 1.3 · Domain & Range
Domain is the set of all valid inputs; range is the set of all possible outputs. This comprehensive Math 10 lesson covers identifying domain and range from graphs, tables, and equations for linear and quadratic functions, using both interval notation and set notation — a skill tested repeatedly across all five units and on the EQAO.
- Domain from a graph
- Range from a graph
- Interval notation
- Set notation
Source: The Auxi Project – Math 10C
🔁 1.4 · Graphical Relationships & Real-World Functions
Functions show up everywhere — from a ball's flight path to a cell phone plan's cost. This video focuses on interpreting graphical relationships in real-world contexts: reading intercepts, identifying increasing and decreasing intervals, understanding rate of change, and connecting the shape of a graph to the story it tells. A critical thinking skill for word problems.
- Interpreting graphs
- x- and y-intercepts
- Rate of change
- Real-world modelling
Source: Ontario Math
Analytic geometry brings algebra and geometry together on the Cartesian plane. Students use the midpoint formula, distance formula, and slope to find equations of special lines and verify properties of triangles, quadrilaterals, and circles — turning classical geometric proofs into elegant algebraic calculations.
📍 2.1 · Midpoint & Length of a Line Segment
Two of the most fundamental tools in coordinate geometry: the midpoint formula locates the exact centre of any segment, while the distance formula (built from the Pythagorean theorem) calculates its length. This full JensenMath lesson develops both formulas from first principles and works through multiple example problems at the Grade 10 Ontario pace.
- Midpoint formula
- Distance formula
- Pythagorean theorem connection
- Multiple worked examples
Source: JensenMath
📐 2.2 · Slope & Equations of Lines
Slope is the numerical backbone of coordinate geometry. This video covers calculating slope from two points, writing line equations in slope-intercept form (y = mx + b) and standard form (Ax + By + C = 0), identifying parallel lines (equal slopes) and perpendicular lines (negative reciprocal slopes), and sketching lines — skills that tie directly into Units 2 and 3.
- Rise over run
- y = mx + b
- Parallel & perpendicular slopes
- Standard form Ax + By + C = 0
Source: Grade 10 Academic Math
⊥ 2.3 · Perpendicular Bisector, Median & Altitude
Three special line segments appear in every triangle problem: the median (from vertex to midpoint of opposite side), the altitude (perpendicular from vertex to opposite side), and the perpendicular bisector (perpendicular at the midpoint of a side). This MPM2D video defines all three clearly, explains when each is used, and shows how to find their equations — a very common exam topic.
- Median of a triangle
- Altitude of a triangle
- Perpendicular bisector
- Equations of all three lines
Source: MPM2D Video Lessons
🔷 2.4 · Verifying Properties of Triangles & Quadrilaterals
Using midpoint, length, and slope together, students prove algebraically that a given shape is a parallelogram, rectangle, rhombus, square, or right triangle — without needing a ruler or protractor. This video shows exactly how to set up and complete these coordinate geometry proofs, one of the highest-value question types in the course.
- Parallelogram proof
- Rectangle verification
- Right angle check
- Equal diagonals & bisection
Source: Grade 10 Analytic Geometry
⭕ 2.5 · Circle Properties in the Coordinate Plane
Circles bring a satisfying challenge to analytic geometry. This MPM2D video covers the equation of a circle centred at the origin and at (h, k), finding the centre using perpendicular bisectors of chords, and verifying whether a point lies on a circle — all using the algebraic tools built across Unit 2. A classic thinking question on TDSB tests.
- Equation of a circle
- Chords & right bisectors
- Finding the centre
- Points on a circle
Source: All Things Mathematics (MPM2D)
A system of linear equations models two conditions at the same time. Students explore three solution methods — graphing, substitution, and elimination — and apply them to real-world problems ranging from break-even analysis to mixture and rate problems.
📉 3.1 · Solving Linear Systems by Graphing
The most visual method: graph both lines on the same grid and identify the point of intersection — that point satisfies both equations simultaneously. This MPM2D lesson teaches how to set up the graph, identify the solution, verify it algebraically, and recognise the special cases where systems have no solution (parallel lines) or infinite solutions (the same line).
- Graph both lines
- Point of intersection
- No solution (parallel)
- Infinite solutions (same line)
Source: All Things Mathematics (MPM2D)
🔄 3.2 · Substitution & Elimination Methods
When exact answers matter, algebra beats graphing. This dedicated Ontario MPM2D video covers both the substitution method (isolate one variable and plug it in) and the elimination method (add or subtract equations to cancel a variable), compares when each strategy is most efficient, and includes full verification steps — essential for achieving full marks on tests.
- Substitution method
- Elimination method
- Multiplying before eliminating
- Verifying the solution
Source: All Things Mathematics (MPM2D)
📝 3.3 · Linear Systems Word Problems
The real power of linear systems is modelling real-world situations — two people saving money at different rates, mixing two solutions to hit a target concentration, or finding the break-even point of a business. This Grade 10 Ontario video focuses entirely on translating word problems into systems of equations and solving them using elimination — one of the most exam-relevant skills in Unit 3.
- Setting up the system
- Break-even problems
- Mixture problems
- Distance-rate-time problems
Source: All Things Mathematics (MPM2D)
The largest and most central unit of MPM2D. Students explore quadratic functions in three equivalent forms — standard, factored, and vertex — connect every algebraic move to the shape of a parabola, and solve quadratic equations using factoring, completing the square, and the quadratic formula.
✖️ 4.1 · Expanding & Special Products
Before factoring, students must master expanding. This Ontario MPM2D video covers expanding binomial products using the distributive property (FOIL), and three special product patterns that save time and appear throughout the unit: difference of squares (a + b)(a − b), perfect square trinomials (a + b)², and squaring a binomial. These patterns re-appear in reverse during factoring.
- FOIL method
- Difference of squares
- Perfect square trinomials
- Simplifying expressions
Source: All Things Mathematics (MPM2D)
➗ 4.2 · Factoring Quadratics
Factoring is the engine that powers quadratic algebra. This video systematically covers every factoring technique in the MPM2D course: common factoring (GCF), factoring simple trinomials (x² + bx + c), factoring complex trinomials (ax² + bx + c) by decomposition, difference of squares, and perfect square trinomials. Mastering factoring unlocks every other subtopic in Unit 4.
- Common factoring (GCF)
- Simple trinomials x² + bx + c
- Complex trinomials (decomposition)
- Difference of squares
Source: All Things Mathematics (MPM2D)
🔺 4.3 · Vertex Form: y = a(x − h)² + k
Vertex form is the most revealing way to write a quadratic because it directly exposes the vertex (h, k), the axis of symmetry (x = h), the direction of opening, and the stretch factor. This video gives a complete overview of vertex form for MPM2D, explains what each parameter does to the graph, and demonstrates how to read vertex form from a parabola — a very common test question type.
- Vertex (h, k)
- Axis of symmetry x = h
- Stretch factor a
- Direction of opening
Source: All Things Mathematics (MPM2D)
📉 4.4 · Transformations of the Parabola
Every parameter in y = a(x − h)² + k transforms the base parabola y = x² in a precise, predictable way. This MPM2D video walks through each transformation individually — vertical stretch and compression (|a| > 1 or |a| < 1), reflection in the x-axis (a < 0), horizontal translation (h), and vertical translation (k) — and combines them to sketch any parabola from its equation.
- Vertical stretch / compression
- Reflection in x-axis (a < 0)
- Horizontal translation h
- Vertical translation k
Source: All Things Mathematics (MPM2D)
□ 4.5 · Completing the Square
Completing the square is the algebraic process that converts a quadratic from standard form (y = ax² + bx + c) to vertex form (y = a(x − h)² + k), revealing the vertex without graphing. This Ontario MPM2D tutorial covers completing the square step by step for both simple (a = 1) and complex (a ≠ 1) cases, and shows how to use it to find the maximum or minimum value of a quadratic — a key application on tests.
- Standard → vertex form
- Completing the square (a = 1)
- Completing the square (a ≠ 1)
- Finding max / min value
Source: All Things Mathematics (MPM2D)
✏️ 4.6 · Solving Quadratic Equations by Factoring
Once a quadratic is factored, the zero-product property gives the x-intercepts (roots) instantly: if (x − r)(x − s) = 0, then x = r or x = s. This Ontario curriculum video teaches how to set a quadratic equal to zero, factor completely, apply the zero-product property, and verify each root — working through multiple difficulty levels from simple to complex factoring scenarios.
- Zero-product property
- Setting equation = 0
- Roots as x-intercepts
- Verifying solutions
Source: All Things Mathematics (MPM2D)
🧮 4.7 · The Quadratic Formula
When a quadratic can't be factored easily, the quadratic formula (x = [−b ± √(b² − 4ac)] / 2a) always works. This MPM2D overview video introduces the formula, explains where it comes from (completing the square in general form), demonstrates how to substitute coefficients carefully, evaluate the discriminant (b² − 4ac) to predict the number of solutions, and interpret results as exact and decimal values.
- The formula x = [−b ± √(b²−4ac)] / 2a
- Identifying a, b, c
- The discriminant
- Exact & decimal solutions
Source: All Things Mathematics (MPM2D)
🌍 4.8 · Quadratic Applications & Word Problems
Quadratics model some of the most common real-world situations: the arc of a projectile, maximum area of a fenced yard, maximum revenue of a business, and dimensions of geometric shapes. This Ontario MPM2D video focuses on translating word problems into quadratic equations, choosing the best solving strategy, and interpreting roots and vertex values in the context of the original problem.
- Projectile & height problems
- Maximum area applications
- Revenue & profit modelling
- Interpreting vertex & roots
Source: All Things Mathematics (MPM2D)
From the similarity of triangles to the laws of sines and cosines, this unit builds a complete toolkit for solving both right-angled and acute triangles. Students connect ratio reasoning to the primary trig ratios and apply them to real-world measurement and navigation problems.
🔲 5.1 · Similar Triangles
Trigonometry is built on one powerful insight: similar triangles always have the same side ratios, regardless of size. This Grade 10 MPM2D lesson covers the AA (Angle-Angle) similarity condition, how to identify corresponding sides and set up proportions, and how similar triangles are the conceptual foundation of sine, cosine, and tangent — giving the trig ratios their geometric meaning before the formulas appear.
- AA similarity condition
- Corresponding sides
- Setting up proportions
- Foundation of trig ratios
Source: JensenMath (MPM2D)
📐 5.2 · Primary Trig Ratios – SOH CAH TOA
Sine, cosine, and tangent are the three primary tools for solving right triangles. This comprehensive full-lesson from JensenMath develops all three ratios from similar triangles, teaches the SOH CAH TOA memory device, shows how to label Opposite / Adjacent / Hypotenuse correctly for any reference angle, and solves for both missing sides (direct trig) and missing angles (inverse trig with sin⁻¹, cos⁻¹, tan⁻¹).
- sin, cos, tan definitions
- Labelling sides correctly
- Solving for missing sides
- Inverse trig for missing angles
Source: JensenMath
📏 5.3 · The Sine Law
When a triangle isn't right-angled, SOH CAH TOA no longer applies — the Sine Law takes over. This dedicated MPM2D video introduces the Sine Law (a / sin A = b / sin B = c / sin C), explains the two cases where it applies (AAS: two angles and a non-included side; SSA: two sides and a non-included angle), and walks through finding both missing side lengths and missing angles in acute triangles.
- Sine Law formula
- AAS case (two angles + side)
- Finding missing sides
- Finding missing angles
Source: JensenMath (MPM2D)
📐 5.4 · The Cosine Law
The Cosine Law handles the two cases the Sine Law cannot: SAS (two sides and the included angle) and SSS (all three sides known). This video proves the Cosine Law from the Pythagorean theorem, shows how to apply it to find a missing side (c² = a² + b² − 2ab·cosC), rearranges it to find a missing angle, and explains exactly how to decide whether a problem requires the Sine Law or the Cosine Law — a critical exam decision.
- Cosine Law formula
- SAS case (two sides + included angle)
- SSS case (finding angles)
- Sine Law vs. Cosine Law decision
Source: JensenMath (MPM2D)
🏗️ 5.5 · Trigonometry Word Problems & Applications
Angles of elevation and depression, guy wires on poles, surveying distances, and navigation bearings — trigonometry is one of the most practically useful topics in the entire course. This Ontario MPM2D video focuses exclusively on real-world application problems that require sketching a diagram, labelling the triangle, choosing the correct law (SOH CAH TOA, Sine, or Cosine), and interpreting the answer in context. Perfect EQAO and final-exam preparation.
- Angle of elevation & depression
- Sketching & labelling diagrams
- Choosing the right law
- Ontario EQAO exam style
Source: All Things Mathematics (MPM2D)
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