Principles of Mathematics (Grade 10)
31 of 31 topics written so far. Greyed-out topics are coming soon.
Unit 1: Linear Systems
- 1Equations of Lines
The slope formula, zero and undefined slopes, finding a line's equation from a graph, a table, a point and slope, or two points, and switching between y = mx + b, Ax + By + C = 0 and Ax + By = D.
Draft - 2Parallel and Perpendicular Lines
Parallel lines have equal slopes and perpendicular lines have negative reciprocal slopes. Testing pairs of lines and finding equations of parallel and perpendicular lines through a point.
Draft - 3Solving Linear Systems by Graphing
What a solution of a linear system is, solving by graphing, the three possible outcomes (one, none, or infinitely many solutions), predicting them from slopes and intercepts, and the limits of graphing.
Draft - 4Solving Linear Systems by Substitution
Solving a system of two linear equations exactly by isolating one variable and substituting it into the other equation, including systems with fractions and decimals.
Draft - 5Solving Linear Systems by Elimination
Adding or subtracting equations to eliminate a variable, multiplying one or both equations first, and recognizing systems with no solution or infinitely many solutions.
Draft - 6Applications of Linear Systems
Setting up and solving linear systems from word problems — break-even, investments with simple interest, mixtures, and rates with a current or wind — and checking that answers make sense.
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Unit 2: Analytic Geometry
- 1Midpoint of a Line Segment
Developing the midpoint formula by averaging coordinates, finding midpoints and missing endpoints, and finding the equation of a median of a triangle.
Draft - 2Length of a Line Segment
Developing the length (distance) formula from the Pythagorean theorem, finding lengths and perimeters on a grid, and classifying triangles by their side lengths.
Draft - 3Circles Centred at the Origin
Developing the equation x² + y² = r² from the length formula, finding the radius, writing and sketching the equation, testing points, and working with diameters and chords.
Draft - 4Right Bisectors and Distance to a Line
Finding the equation of the right bisector of a segment, the shortest distance from a point to a line, and the circumcentre of a triangle.
Draft - 5Verifying Geometric Properties
Using slope, length and midpoint to classify triangles and quadrilaterals and to verify properties such as the midsegment of a triangle and the diagonals of a rectangle.
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Unit 3: Quadratic Expressions
- 1Exponent Laws
The product, quotient and power laws, what zero and negative exponents mean, and how the graph of y = 2^x compares with y = x².
Draft - 2Adding, Subtracting, and Multiplying Polynomials
Combining like terms, subtracting carefully, and expanding products of polynomials — including the special products.
Draft - 3Common Factoring
Taking out the greatest common factor, factoring out a common binomial, and factoring four-term expressions by grouping.
Draft - 4Factoring Trinomials (x² + bx + c)
Factoring trinomials like x² + 7x + 12 by finding two numbers with the right product and sum, including all the sign cases and common factors.
Draft - 5Factoring Complex Trinomials (ax² + bx + c)
Factoring trinomials like 6x² − x − 12, where the coefficient of x² isn't 1, by decomposition and by inspection.
Draft - 6Factoring Special Cases
Difference of squares and perfect-square trinomials, expressions that take two steps to factor, and a strategy for choosing the right method.
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Unit 4: Quadratic Relations
- 1Introduction to Quadratic Relations
Recognizing quadratic relations from equations and tables (constant second differences), the key features of a parabola, and drawing a curve of best fit for data.
Draft - 2Vertex Form and Transformations
Graphing y = a(x − h)² + k as a transformation of y = x², sketching by hand with the step pattern, and finding the equation of a parabola from its graph.
Draft - 3Factored Form of a Quadratic
Using y = a(x − r)(x − s) to find zeros, the axis of symmetry and the vertex, sketching from factored form, and finding an equation from its zeros.
Draft - 4Completing the Square
Rewriting a quadratic from standard form into vertex form, so you can read off its vertex and maximum or minimum.
Draft - 5Solving Quadratic Equations by Factoring
Solving ax² + bx + c = 0 by rearranging, factoring, and using the zero product property, including double roots, checking roots, and connecting roots to x-intercepts.
Draft - 6The Quadratic Formula
Where the quadratic formula comes from, solving any quadratic equation with it (exact and decimal answers), choosing a solving method, and what a negative b² − 4ac means.
Draft - 7Zeros of Quadratics and the Discriminant
Finding the zeros of a quadratic function by factoring or the quadratic formula, and using the discriminant to count them.
Draft - 8Maximum and Minimum of a Quadratic
Finding the vertex of a quadratic by completing the square or averaging the zeros, and solving real-world optimization problems.
Draft - 9Solving Problems with Quadratic Models
Using quadratic equations and graphs to answer real questions about projectiles, areas and numbers, and interpreting the answers in context.
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Unit 5: Trigonometry
- 1Similar Triangles
Same shape, different size — matching angles, proportional sides, scale factors, and using similar triangles to measure things you can't reach.
Draft - 2The Primary Trigonometric Ratios
Sine, cosine and tangent in right triangles — naming the sides, SOH CAH TOA, and finding missing sides and angles.
Draft - 3Right Triangle Problems
Angles of elevation and depression, clinometers, ladders, ramps, navigation, and problems with two right triangles.
Draft - 4The Sine Law
Solving any triangle when you know two angles and a side, using a/sin A = b/sin B = c/sin C.
Draft - 5The Cosine Law
Solving triangles from two sides and the angle between them, or from three sides, using c² = a² + b² − 2ab cos C.
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