Principles of Mathematics

In this unit you will meet the most important curve in high school math: the parabola, the graph of a quadratic relation. You will start by collecting data that curves (a ball rolling down a ramp, water draining from a bottle) and drawing curves of best fit. You will learn to recognize a quadratic relation from its table (constant second differences), its graph (a parabola) and its equation (y = ax² + bx + c, a ≠ 0), and to name its key features: vertex, axis of symmetry, zeros, y-intercept, and maximum or minimum value. You will use these features to solve problems about heights, areas and profits, and you will compare y = x² with the exponential y = 2ˣ, using its table to explain zero and negative exponents.

In the second half you will study transformations: how k, h and a translate, stretch, compress and reflect the graph of y = x². You will put them together in vertex form, y = a(x − h)² + k, sketch parabolas by hand in under a minute, and find the equation of a parabola from a graph, a table or a real arch.

Remember from Grade 9: you used first differences to tell linear from non-linear relations, met the U-shaped graph of y = x², transformed lines, fitted regression models with technology, and used the exponent laws. Grade 10 studies the quadratic relation in depth. The companion unit Quadratic Expressions and Equations covers expanding, factoring, solving quadratic equations and problem solving with them; it builds on the graphs you learn here. These ideas lead directly to Grade 11 functions (MCR3U).

Every lesson has worked examples with full solutions, diagrams, a hands-on activity, lots of practice with answers and a short quiz. The mathematical processes (problem solving, reasoning and proving, reflecting, selecting tools and strategies, connecting, representing and communicating) are woven through every lesson.

Curriculum expectations covered: Quadratic Relations of the Form y = ax² + bx + c, specific expectations 1.1 to 1.4 (Investigating the Basic Properties of Quadratic Relations) and 2.1 to 2.4 (Relating the Graph of y = x² and Its Transformations) (The Ontario Curriculum, Grades 9 and 10: Mathematics, 2005, Principles of Mathematics, Grade 10, Academic, MPM2D).

This unit gives you the algebra of quadratic relations. In the companion unit, Quadratic Relations and Their Graphs, you studied parabolas: second differences, the vertex, the axis of symmetry, zeros, and transformations of y = x² written as y = a(x − h)² + k. Here you learn to work with the equations themselves.

You will start by expanding and simplifying second-degree expressions, then learn every kind of factoring in the course: common factors, grouping, trinomials x² + bx + c and ax² + bx + c, differences of squares and perfect squares. You will connect the factors of an expression to the zeros of its graph, solve quadratic equations by factoring, by square roots, with the quadratic formula (and see where it comes from) and by graphing, and rewrite relations in vertex form by completing the square. The discriminant will tell you whether an equation has two, one or no real roots. Finally, you will find maximum and minimum values and solve real problems about thrown balls, dives, garden paths, arches and revenue.

Remember from Grade 9: you used the distributive property, multiplied binomials with algebra tiles and area models, factored out common factors, and solved linear equations. Every lesson here has worked examples, a hands-on activity, plenty of practice with full solutions, and a short quiz. The mathematical processes (problem solving, reasoning and proving, reflecting, selecting tools, connecting, representing and communicating) are built into every lesson.

Curriculum expectations covered: the "Solving Quadratic Equations" expectations 3.1 to 3.8 and the "Solving Problems Involving Quadratic Relations" expectations 4.1 and 4.2 of the strand Quadratic Relations of the Form y = ax² + bx + c (The Ontario Curriculum, Grades 9 and 10: Mathematics, 2005, Principles of Mathematics, Grade 10, Academic, MPM2D).

In this unit you will use coordinates and algebra to solve problems about lines, line segments, circles and shapes. You will start with linear systems: two linear equations that must be true at the same time. You will solve them by graphing, substitution and elimination, learn to predict whether a system has one solution, no solution or infinitely many, and use systems to model real situations such as ticket sales, job offers, canoe trips, mixtures and investments.

Next you will develop the formulas for the midpoint and the length of a line segment, and use the length formula to develop the equation of a circle centred at the origin, x² + y² = r². You will combine slopes, midpoints and lengths to find equations of right bisectors, medians and altitudes, and the distance from a point to a line.

Finally, you will investigate properties of triangles and quadrilaterals (medians and the centroid, similar figures on the sides of a right triangle, diagonals of quadrilaterals) and then verify them with analytic geometry, from single checks to multi-step strategies and general proofs using variables.

Remember from Grade 9: you graphed lines in the forms y = mx + b and ax + by = k, found points of intersection, used parallel and perpendicular slopes, worked with the Pythagorean relationship, and met the centroid and circumcentre of a triangle. Grade 10 makes all of this exact with algebra. Every lesson has worked examples, a hands-on activity, lots of practice with full answers and a short quiz. The mathematical processes (problem solving, reasoning and proving, reflecting, selecting tools and strategies, connecting, representing and communicating) are woven through every lesson.

Curriculum expectations covered: Analytic Geometry 1.1, 1.2 (using linear systems to solve problems); 2.1, 2.2, 2.3, 2.4, 2.5 (solving problems involving properties of line segments); 3.1, 3.2, 3.3 (using analytic geometry to verify geometric properties). Source: The Ontario Curriculum, Grades 9 and 10: Mathematics, 2005 (revised), Principles of Mathematics, Grade 10, Academic (MPM2D).

In this unit you will learn to measure things you cannot reach. You will start with similar triangles: what makes two triangles similar or congruent, the AA, SSS and SAS similarity conditions, and how to use proportions to find unknown lengths, including shadows, mirrors, scale models and surveying. Then you will discover why every right triangle with the same acute angle has the same side ratios, and give those ratios their names: sine, cosine and tangent. You will use them, with the Pythagorean theorem, to find sides and angles of right triangles, to work with angles of elevation and depression, and to solve surveying and navigation problems. Finally you will follow how the sine law and the cosine law are developed and use them to solve acute triangles that have no right angle.

Remember from earlier grades: in Grade 8 you made scale drawings, performed dilations and learned the Pythagorean theorem; in Grade 9 you used the Pythagorean theorem in real-life and composite-shape problems and solved equations. Grade 10 builds on all of these. You will need a scientific calculator set to degree mode. Every lesson has worked examples, a hands-on activity, lots of practice and a short quiz.

Curriculum: The Ontario Curriculum, Grades 9 and 10: Mathematics (2005), Principles of Mathematics, Grade 10, Academic (MPM2D), strand Trigonometry: 1.1 to 1.3 (investigating similarity and solving problems involving similar triangles), 2.1 to 2.3 (solving problems involving the trigonometry of right triangles) and 3.1 to 3.4 (solving problems involving the trigonometry of acute triangles). The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools, connecting, representing and communicating) are woven through every lesson.