MPM2D: Principles of Mathematics (Academic)

MPM2D.1: Mathematical process expectations.

MPM2D.1.2: Reasoning and Proving

MPM2D.1.2.1: develop and apply reasoning skills (e.g., recognition of relationships, generalization through inductive reasoning, use of counter-examples) to make mathematical conjectures, assess conjectures, and justify conclusions, and plan and construct organized mathematical arguments;

Biconditional Statements
Conditional Statements

MPM2D.1.5: Connecting

MPM2D.1.5.1: make connections among mathematical concepts and procedures, and relate mathematical ideas to situations or phenomena drawn from other contexts (e.g., other curriculum areas, daily life, current events, art and culture, sports);

Earthquakes 1 - Recording Station
Estimating Population Size

MPM2D.2: Quadratic Relations of the Form y = ax² + bx + c

MPM2D.2.1: determine the basic properties of quadratic relations;

MPM2D.2.1.4: compare, through investigation using technology, the features of the graph of y = x² and the graph of y = 2 to the power of x, and determine the meaning of a negative exponent and of zero as an exponent (e.g., by examining patterns in a table of values for y = 2 to the power of x; by applying the exponent rules for multiplication and division).

Dividing Exponential Expressions
Exponential Functions
Exponents and Power Rules
Multiplying Exponential Expressions
Parabolas
Quadratics in Polynomial Form
Quadratics in Vertex Form
Roots of a Quadratic

MPM2D.2.2: relate transformations of the graph of y = x² to the algebraic representation y = a(x – h)² + k;

MPM2D.2.2.4: determine the equation, in the form y = a(x – h)² + k, of a given graph of a parabola.

Parabolas
Zap It! Game

MPM2D.2.3: solve quadratic equations and interpret the solutions with respect to the corresponding relations;

MPM2D.2.3.2: factor polynomial expressions involving common factors, trinomials, and differences of squares [e.g., 2x² + 4x, 2x – 2y + ax – ay, x² – x – 6, 2a² + 11a + 5, 4x² – 25], using a variety of tools (e.g., concrete materials, computer algebra systems, paper and pencil) and strategies (e.g., patterning);

Factoring Special Products
Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c

MPM2D.2.3.3: 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);

Polynomials and Linear Factors
Quadratics in Factored Form

MPM2D.2.3.4: interpret real and non-real roots of quadratic equations, through investigation using graphing technology, and relate the roots to the x-intercepts of the corresponding relations;

Roots of a Quadratic

MPM2D.2.3.7: explore the algebraic development of the quadratic formula (e.g., given the algebraic development, connect the steps to a numerical example; follow a demonstration of the algebraic development [student reproduction of the development of the general case is not required]);

Roots of a Quadratic

MPM2D.2.3.8: solve quadratic equations that have real roots, using a variety of methods (i.e., factoring, using the quadratic formula, graphing) (Sample problem: Solve x² + 10x + 16 = 0 by factoring, and verify algebraically. Solve x² + x – 4 = 0 using the quadratic formula, and verify graphically using technology. Solve –4.9t² + 50t + 1.5 = 0 by graphing h = –4.9t² + 50t + 1.5 using technology.).

Modeling the Factorization of x²+bx+c
Quadratics in Factored Form
Roots of a Quadratic

MPM2D.2.4: solve problems involving quadratic relations.

MPM2D.2.4.1: determine the zeros and the maximum or minimum value of a quadratic relation from its graph (i.e., using graphing calculators or graphing software) or from its defining equation (i.e., by applying algebraic techniques);

Polynomials and Linear Factors
Roots of a Quadratic
Zap It! Game

MPM2D.3: Analytic Geometry

MPM2D.3.1: model and solve problems involving the intersection of two straight lines;

MPM2D.3.1.1: solve systems of two linear equations involving two variables, using the algebraic method of substitution or elimination (Sample problem: Solve y = 1/2x – 5, 3x + 2y = –2 for x and y algebraically, and verify algebraically and graphically);

Solving Equations by Graphing Each Side
Solving Linear Systems (Slope-Intercept Form)
Solving Linear Systems (Standard Form)

MPM2D.3.1.2: solve problems that arise from realistic situations described in words or represented by linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method (Sample problem: The Robotics Club raised $5000 to build a robot for a future competition. The club invested part of the money in an account that paid 4% annual interest, and the rest in a government bond that paid 3.5% simple interest per year. After one year, the club earned a total of $190 in interest. How much was invested at each rate? Verify your result.).

Solving Equations by Graphing Each Side
Solving Linear Systems (Standard Form)

MPM2D.3.2: solve problems using analytic geometry involving properties of lines and line segments;

MPM2D.3.2.2: develop the formula for the length of a line segment, and use this formula to solve problems (e.g., determine the lengths of the line segments joining the midpoints of the sides of a triangle, given the coordinates of the vertices of the triangle, and verify using dynamic geometry software);

Distance Formula

MPM2D.3.2.3: develop the equation for a circle with centre (0, 0) and radius r, by applying the formula for the length of a line segment;

Circles

MPM2D.3.2.4: determine the radius of a circle with centre (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;

Circles

MPM2D.4: Trigonometry

MPM2D.4.1: use their knowledge of ratio and proportion to investigate similar triangles and solve problems related to similarity;

MPM2D.4.1.1: verify, through investigation (e.g., using dynamic geometry software, concrete materials), the properties of similar triangles (e.g., given similar triangles, verify the equality of corresponding angles and the proportionality of corresponding sides);

Congruence in Right Triangles
Perimeters and Areas of Similar Figures
Similar Figures
Similarity in Right Triangles

MPM2D.4.1.3: solve problems involving similar triangles in realistic situations (e.g., shadows, reflections, scale models, surveying) (Sample problem: Use a metre stick to determine the height of a tree, by means of the similar triangles formed by the tree, the metre stick, and their shadows.).

Beam to Moon (Ratios and Proportions) - Metric
Similar Figures

MPM2D.4.2: solve problems involving right triangles, using the primary trigonometric ratios and the Pythagorean theorem;

MPM2D.4.2.1: determine, through investigation (e.g., using dynamic geometry software, concrete 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 (e.g., sin A = opposite/hypotenuse);

Sine, Cosine, and Tangent Ratios

MPM2D.4.2.2: determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem;

Cosine Function
Pythagorean Theorem
Sine Function
Sine, Cosine, and Tangent Ratios
Tangent Function

MPM2D.4.2.3: solve problems involving the measures of sides and angles in right triangles in real-life applications (e.g., in surveying, in navigating, in determining the height of an inaccessible object around the school), using the primary trigonometric ratios and the Pythagorean theorem.

Pythagorean Theorem with a Geoboard
Sine, Cosine, and Tangent Ratios

MFM2P: Foundations of Mathematics (Applied)

MFM2P.1: Mathematical process expectations

MFM2P.1.2: Reasoning and Proving

MFM2P.1.2.1: develop and apply reasoning skills (e.g., recognition of relationships, generalization through inductive reasoning, use of counter-examples) to make mathematical conjectures, assess conjectures, and justify conclusions, and plan and construct organized mathematical arguments;

Biconditional Statements
Conditional Statements

MFM2P.1.5: Connecting

MFM2P.1.5.1: make connections among mathematical concepts and procedures, and relate mathematical ideas to situations or phenomena drawn from other contexts (e.g., other curriculum areas, daily life, current events, art and culture, sports);

Earthquakes 1 - Recording Station
Estimating Population Size

MFM2P.2: Measurement and Trigonometry

MFM2P.2.1: use their knowledge of ratio and proportion to investigate similar triangles and solve problems related to similarity;

MFM2P.2.1.1: verify, through investigation (e.g., using dynamic geometry software, concrete materials), properties of similar triangles (e.g., given similar triangles, verify the equality of corresponding angles and the proportionality of corresponding sides);

Congruence in Right Triangles
Perimeters and Areas of Similar Figures
Proving Triangles Congruent
Similar Figures
Similarity in Right Triangles

MFM2P.2.1.3: solve problems involving similar triangles in realistic situations (e.g., shadows, reflections, scale models, surveying) (Sample problem: Use a metre stick to determine the height of a tree, by means of the similar triangles formed by the tree, the metre stick, and their shadows.).

Beam to Moon (Ratios and Proportions) - Metric
Similar Figures

MFM2P.2.2: solve problems involving right triangles, using the primary trigonometric ratios and the Pythagorean theorem;

MFM2P.2.2.1: determine, through investigation (e.g., using dynamic geometry software, concrete 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 (e.g., sin A = opposite/hypotenuse);

Sine, Cosine, and Tangent Ratios

MFM2P.2.2.2: determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem;

Cosine Function
Pythagorean Theorem
Sine Function
Sine, Cosine, and Tangent Ratios
Tangent Function

MFM2P.2.2.3: solve problems involving the measures of sides and angles in right triangles in real-life applications (e.g., in surveying, in navigation, in determining the height of an inaccessible object around the school), using the primary trigonometric ratios and the Pythagorean theorem (Sample problem: Build a kite, using imperial measurements, create a clinometer to determine the angle of elevation when the kite is flown, and use the tangent ratio to calculate the height attained.);

Pythagorean Theorem with a Geoboard
Sine, Cosine, and Tangent Ratios

MFM2P.2.2.4: describe, through participation in an activity, the application of trigonometry in an occupation (e.g., research and report on how trigonometry is applied in astronomy; attend a career fair that includes a surveyor, and describe how a surveyor applies trigonometry to calculate distances; job shadow a carpenter for a few hours, and describe how a carpenter uses trigonometry).

Sine, Cosine, and Tangent Ratios

MFM2P.2.3: solve problems involving the surface areas and volumes of three-dimensional figures, and use the imperial and metric systems of measurement.

MFM2P.2.3.2: perform everyday conversions between the imperial system and the metric system (e.g., millilitres to cups, centimetres to inches) and within these systems (e.g., cubic metres to cubic centimetres, square feet to square yards), as necessary to solve problems involving measurement (Sample problem: A vertical post is to be supported by a wooden pole, secured on the ground at an angle of elevation of 60°, and reaching 3 m up the post from its base. If wood is sold by the foot, how many feet of wood are needed to make the pole?);

Unit Conversions

MFM2P.2.3.3: determine, through investigation, the relationship for calculating the surface area of a pyramid (e.g., use the net of a square-based pyramid to determine that the surface area is the area of the square base plus the areas of the four congruent triangles);

Surface and Lateral Areas of Prisms and Cylinders
Surface and Lateral Areas of Pyramids and Cones

MFM2P.2.3.4: solve problems involving the surface areas of prisms, pyramids, and cylinders, and the volumes of prisms, pyramids, cylinders, cones, and spheres, including problems involving combinations of these figures, using the metric system or the imperial system, as appropriate (Sample problem: How many cubic yards of concrete are required to pour a concrete pad measuring 10 feet by 10 feet by 1 foot? If poured concrete costs $110 per cubic yard, how much does it cost to pour a concrete driveway requiring 6 pads?).

Prisms and Cylinders
Surface and Lateral Areas of Prisms and Cylinders
Surface and Lateral Areas of Pyramids and Cones

MFM2P.3: Modelling Linear Relations

MFM2P.3.1: manipulate and solve algebraic equations, as needed to solve problems;

MFM2P.3.1.3: express the equation of a line in the form y = mx + b, given the form Ax + By + C = 0.

Standard Form of a Line

MFM2P.3.2: graph a line and write the equation of a line from given information;

MFM2P.3.2.1: connect the rate of change of a linear relation to the slope of the line, and define the slope as the ratio m = rise/run;

Cat and Mouse (Modeling with Linear Systems) - Metric
Distance-Time and Velocity-Time Graphs - Metric
Point-Slope Form of a Line
Slope
Slope-Intercept Form of a Line

MFM2P.3.2.2: identify, through investigation, y = mx + b as a common form for the equation of a straight line, and identify the special cases x = a, y = b;

Linear Inequalities in Two Variables
Point-Slope Form of a Line
Points, Lines, and Equations
Slope-Intercept Form of a Line
Standard Form of a Line

MFM2P.3.2.4: identify, through investigation, properties of the slopes of lines and line segments (e.g., direction, positive or negative rate of change, steepness, parallelism), using graphing technology to facilitate investigations, where appropriate;

Slope

MFM2P.3.2.5: graph lines by hand, using a variety of techniques (e.g., graph y = 2/3x - 4 using the y-intercept and slope; graph 2x + 3y = 6 using the x- and y-intercepts);

Cat and Mouse (Modeling with Linear Systems) - Metric
Point-Slope Form of a Line
Slope-Intercept Form of a Line
Standard Form of a Line

MFM2P.3.2.6: determine the equation of a line, given its graph, the slope and y-intercept, the slope and a point on the line, or two points on the line.

Linear Inequalities in Two Variables
Point-Slope Form of a Line
Points, Lines, and Equations
Slope-Intercept Form of a Line
Standard Form of a Line

MFM2P.3.3: solve systems of two linear equations, and solve related problems that arise from realistic situations.

MFM2P.3.3.1: determine graphically the point of intersection of two linear relations (e.g., using graph paper, using technology) (Sample problem: Determine the point of intersection of y + 2x = –5 and y = 2/3x + 3 using an appropriate graphing technique, and verify.);

Cat and Mouse (Modeling with Linear Systems) - Metric
Solving Equations by Graphing Each Side
Solving Linear Systems (Matrices and Special Solutions)
Solving Linear Systems (Slope-Intercept Form)

MFM2P.3.3.2: solve systems of two linear equations involving two variables with integral coefficients, using the algebraic method of substitution or elimination (Sample problem: Solve y = 2x + 1, 3x + 2y = 16 for x and y algebraically, and verify algebraically and graphically.);

Solving Equations by Graphing Each Side
Solving Linear Systems (Slope-Intercept Form)
Solving Linear Systems (Standard Form)

MFM2P.3.3.3: solve problems that arise from realistic situations described in words or represented by given linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method (Sample problem: Maria has been hired by Company A with an annual salary, S dollars, given by S = 32 500 + 500a, where a represents the number of years she has been employed by this company. Ruth has been hired by Company B with an annual salary, S dollars, given by S = 28 000 + 1000a, where a represents the number of years she has been employed by that company. Describe what the solution of this system would represent in terms of Maria’s salary and Ruth’s salary. After how many years will their salaries be the same? What will their salaries be at that time?).

Solving Equations by Graphing Each Side
Solving Linear Systems (Matrices and Special Solutions)
Solving Linear Systems (Slope-Intercept Form)
Solving Linear Systems (Standard Form)

MFM2P.4: Quadratic Relations of the Form y = ax² + bx + c

MFM2P.4.1: manipulate algebraic expressions, as needed to understand quadratic relations;

MFM2P.4.1.2: factor binomials (e.g., 4x² + 8x) and trinomials (e.g., 3x² + 9x – 15) involving one variable up to degree two, by determining a common factor using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil) and strategies (e.g., patterning);

Factoring Special Products

MFM2P.4.1.3: factor simple trinomials of the form x² + bx + c (e.g., x² + 7x + 10, x² + 2x – 8), using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil) and strategies (e.g., patterning);

Factoring Special Products
Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c

MFM2P.4.1.4: factor the difference of squares of the form x² − a² (e.g., x² − 16)

Factoring Special Products

MFM2P.4.2: identify characteristics of quadratic relations;

MFM2P.4.2.4: compare, through investigation using technology, the graphical representations of a quadratic relation in the form y = x² + bx + c and the same relation in the factored form y = (x – r)(x – s) (i.e., the graphs are the same), and describe the connections between each algebraic representation and the graph [e.g., the y-intercept is c in the form y = x² + bx + c; the x-intercepts are r and s in the form y = (x – r)(x – s)] (Sample problem: Use a graphing calculator to compare the graphs of y = x² + 2x – 8 and y = (x + 4)(x – 2). In what way(s) are the equations related? What information about the graph can you identify by looking at each equation? Make some conclusions from your observations, and check your conclusions with a different quadratic equation.).

Polynomials and Linear Factors

MFM2P.4.3: solve problems by interpreting graphs of quadratic relations.

MFM2P.4.3.1: solve problems involving a quadratic relation by interpreting a given graph or a graph generated with technology from its equation (e.g., given an equation representing the height of a ball over elapsed time, use a graphing calculator or graphing software to graph the relation, and answer questions such as the following: What is the maximum height of the ball? After what length of time will the ball hit the ground? Over what time interval is the height of the ball greater than 3 m?);

Quadratics in Vertex Form
Translating and Scaling Functions