Ontario Grade 10 MPM2D Principles of Mathematics (Grade 10, Academic)
Chapters: 3
1. Measurement and Geometry
1 Investigating the Optimal Values of Measurements of Rectangles · 2 Solving Problems Involving Perimeter, Area, and Volume · 3 Investigating and Applying Geometric Relationships
- Surface Area and Volume of Cuboid, Cylinder, Cone, Pyramid and Sphere – Surface area is the total area of the outside skin of a solid, like the paper needed to wrap it. Volume is the space inside, like the water it can hold. A cuboid's volume is the number of 1 cm cubes that fit in it. A cone holds one third of a cylinder with the same base and height, and a pyramid holds one third of the matching prism.
- Geometry Basics: 2D Shapes, 3D Solids, Nets and Scale – Geometry is the maths of shape, size and position. A 2D (flat) shape has length and width only: triangles, squares, hexagons, circles. A 3D (solid) shape also has depth: cubes, prisms, pyramids, cylinders. A solid has faces (flat sides), edges (where two faces meet) and vertices (corners). A net is a flat pattern that folds into a solid. Two shapes are congruent if they have the same shape and size, and similar if they have the same shape but a different size. A tessellation covers a floor with shapes and leaves no gaps. A scale drawing shows a real object smaller or bigger by a fixed scale factor.
2. Quadratic Relations of the Form y = ax2 + bx + c
1 Investigating the Basic Properties of Quadratic Relations · 2 Relating the Graph of y = x2 and Its Transformations · 3 Solving Quadratic Equations · 4 Solving Problems Involving Quadratic Relations
- Quadratic Functions and Their Graphs – A quadratic function is y = ax² + bx + c with a ≠ 0. Its graph is a U-shaped curve called a parabola. If a > 0 it opens up and has a lowest point; if a < 0 it opens down and has a highest point. That turning point is the vertex, at x = −b/2a. In vertex form y = a(x − h)² + k the vertex is (h, k).
- Quadratic Equations – A quadratic equation has x² as its highest power: ax² + bx + c = 0 with a ≠ 0. It has at most two roots. Find them by factorising (split the middle term, then set each bracket to zero) or by the formula x = (−b ± √(b² − 4ac)) / 2a. The discriminant D = b² − 4ac tells the nature of roots: D > 0 two different real roots, D = 0 two equal roots, D < 0 no real roots.
3. Analytic Geometry
1 Using Linear Systems to Solve Problems · 2 Solving Problems Involving Properties of Line Segments · 3 Using Analytic Geometry to Verify Geometric Properties
- Pair of Linear Equations in Two Variables – Two equations like a₁x + b₁y = c₁ and a₂x + b₂y = c₂ each make a straight line. The answer that fits both is the point where the lines meet. Lines that cross give one answer, parallel lines give none, and lines that lie on top of each other give endless answers. We can find the answer by drawing (graph), by substitution or by elimination.
- The Cartesian Plane: Coordinates, Distance and Midpoint – Two number lines that cross at right angles turn a flat surface into a map where every point has an address (x, y). The axes meet at the origin O(0, 0) and cut the plane into four quadrants. Distance between two points comes from Pythagoras: square the x-gap and the y-gap, add, take the square root. The midpoint is the average of the x values and the average of the y values. Three points are collinear when the two short distances add up to the long one, and they make a right angle when the squares of the two short sides add up to the square of the long side.
- Coordinate Geometry: Distance and Section Formula – Every point on a flat plane has an address (x, y). The distance between A(x₁, y₁) and B(x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²]; it is just Pythagoras on the x-gap and the y-gap. A point P that cuts AB inside in the ratio m : n is P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). When m = n, P is the midpoint.