Principles of Mathematics, Grade 9, Academic (MPM1D)

This course enables students to develop generalizations of mathematical ideas and methods through the exploration of applications, the effective use of technology, and abstract reasoning. Students will investigate relationships to develop equations of straight lines in analytic geometry, explore relationships between volume and surface area of objects in measurement, and apply extended algebraic skills in problem solving. Students will engage in abstract extensions of core learning that will deepen their mathematical knowledge and enrich their understanding.

Number Sense and Algebra

Overall Expectations

By the end of this course, students will:

• solve multi-step problems requiring numerical answers, using a variety of strategies and tools;
• demonstrate understanding of the three basic exponent rules and apply them to simplify expressions;
• manipulate first-degree polynomial expressions to solve first-degree equations;
• solve problems, using the strategy of algebraic modelling.

Specific Expectations

By the end of this course, students will:

• demonstrate facility with critical numerical skills, including mental mathematics, estimation, operations with integers (as necessary for working with equations and analytic geometry), and operations with rational numbers (as necessary in analytic geometry, measurement, and equation solving);
• distinguish between exact and approximate representations of the same quantity and choose appropriately between them in given situations (e.g., use the symbol pi instead of 3.14 in determining the effect on the volume of a sphere of doubling its diameter; determine the perimeter of a square having an area of 2);
• solve multi-step problems involving applications of percent, ratio, and rate as they arise throughout the course;
• use a scientific calculator effectively for applications that arise throughout the course;
• judge the reasonableness of answers to problems by considering likely results within the situation described in the problem;
• judge the reasonableness of answers produced by a calculator, a computer, or pencil and paper, using mental mathematics and estimation.

By the end of this course, students will:

• evaluate numerical expressions involving natural-number exponents with rational-number bases;
• substitute into and evaluate algebraic expressions involving exponents, to support other topics of the course (e.g., measurement, analytic geometry);
• determine the meaning of negative exponents and of zero as an exponent from activities involving graphing, using technology, and from activities involving patterning;
• represent very large and very small numbers, using scientific notation;
• enter and interpret exponential notation on a scientific calculator, as necessary in calculations involving very large and very small numbers;
• determine, from the examination of patterns, the exponent rules for multiplying and dividing monomials and the exponent rule for the power of a power, and apply these rules in expressions involving one and two variables.

By the end of this course, students will:

• add and subtract polynomials;
• multiply a polynomial by a monomial, and factor a polynomial by removing a common factor;
• expand and simplify polynomial expressions involving one variable;
• solve first-degree equations, including equations with fractional coefficients, using an algebraic method;
• calculate sides in right triangles, using the Pythagorean theorem, as required in topics throughout the course (e.g., measurement);
• rearrange formulas involving variables in the first degree, with and without substitution, as they arise in topics throughout the course (e.g., analytic geometry, measurement).

By the end of this course, students will:

• use algebraic modelling as one of several problem-solving strategies in various topics of the course (e.g., relations, measurement, direct and partial variation, the Pythagorean theorem, percent);
• compare algebraic modelling with other strategies used for solving the same problem;
• communicate solutions to problems in appropriate mathematical forms (e.g., written explanations, formulas, charts, tables, graphs) and justify the reasoning used in solving the problems.

Relationships

Overall Expectations

By the end of this course, students will:

• determine relationships between two variables by collecting and analysing data;
• compare the graphs and formulas of linear and non-linear relations;
• describe the connections between various representations of relations.

Specific Expectations

By the end of this course, students will:

• pose problems, identify variables, and formulate hypotheses associated with relationships (Sample problem: If you look through a paper tube at a wall, you can see a region of a certain height on the wall. If you move farther from the wall, the height of that region changes. What is the relationship between the height of the visible region and your distance from the wall? Describe the relationship that you think will occur);
• demonstrate an understanding of some principles of sampling and surveying (e.g., randomization, representivity, the use of multiple trials) and apply the principles in designing and carrying out experiments to investigate the relationships between variables (Sample problem: What factors might affect the outcome of this experiment? How could you design the experiment to account for them?);
• collect data, using appropriate equipment and/or technology (e.g., measuring tools, graphing calculators, scientific probes, the Internet) (Sample problem: Acquire or construct a paper tube and work with a partner to measure the heights of visible regions at various distances from a wall);
• organize and analyse data, using appropriate techniques (e.g., making tables and graphs, calculating measures of central tendency) and technology (e.g., graphing calculators, statistical software, spreadsheets) (Sample problem: Enter the data into a spreadsheet. Decide what analysis would be appropriate to examine the relationship between the variables – a graph, measures of central tendency, ratios);
• describe trends and relationships observed in data, make inferences from data, compare the inferences with hypotheses about the data, and explain the differences between the inferences and the hypotheses (Sample problem: Describe any trend observed in the data. Does a relationship seem to exist? Of what sort? Is the outcome consistent with your original hypothesis? Discuss any outlying pieces of data and provide explanations for them. Suggest a formula relating the height of the visible region to the distance from the wall. How might you vary this experiment to examine other relationships?);
• communicate the findings of an experiment clearly and concisely, using appropriate mathematical forms (e.g., written explanations, formulas, charts, tables, graphs), and justify the conclusions reached;
• solve and/or pose problems related to an experiment, using the findings of the experiment.

By the end of this course, students will:

• construct tables of values, graphs, and formulas to represent linear relations derived from descriptions of realistic situations (e.g., the cost of holding a banquet in a rented hall is $25 per person plus $975 for the hall);
• construct tables of values and scatter plots for linearly related data collected from experiments (e.g., the rebound height of a ball versus the height from which it was dropped) or from secondary sources (e.g., the number of calories in fast food versus the number of grams of fat);
• determine the equation of a line of best fit for a scatter plot, using an informal process (e.g., a process of trial and error on a graphing calculator; calculation of the equation of the line joining two carefully chosen points on the scatter plot);
• construct tables of values and graphs to represent non-linear relations derived from descriptions of realistic situations (Sample problem: A triangular prism has a height of 20 cm and a square base. Represent the relationship between the volume of the prism and the side length of its base, as the side length varies);
• construct tables of values and scatter plots for non-linearly related data collected from experiments (e.g., the relationship between height and age) or from secondary sources (e.g., the population of Canada over time); sketch a curve of best fit;
• demonstrate an understanding that straight lines represent linear relations and curves represent non-linear relations.

By the end of this course, students will:

• determine values of a linear relation by using the formula of the relation and by interpolating or extrapolating from the graph of the relation (e.g., if a student earns $5/h caring for children, determine how long he or she must work to earn $143);
• describe, in written form, a situation that would explain the events illustrated by a given graph of a relationship between two variables (e.g., write a story that matches the events shown in the graph);
• identify, by calculating finite differences in its table of values, whether a relation is linear or non-linear;
• describe the effect on the graph and the formula of a relation of varying the conditions of a situation they represent (e.g., if a graph showing partial variation represents the cost of producing a yearbook, describe how the appearance of the graph changes if the cost per book is altered; describe how it changes if the fixed costs are altered).

Analytic Geometry

Overall Expectations

By the end of this course, students will:

• determine, through investigation, the relationships between the form of an equation and the shape of its graph with respect to linearity and non-linearity;
• determine, through investigation, the properties of the slope and y-intercept of a linear relation;
• solve problems, using the properties of linear relations.

Specific Expectations

By the end of this course, students will:

• determine, through investigations, the characteristics that distinguish the equation of a straight line from the equations of non-linear relations (e.g., use graphing software to obtain the graphs of a variety of linear and non-linear relations from their equations; classify the relations according to the shapes of their graphs; focus on the characteristics of the equations of linear relations and how they differ from the characteristics of the equations of non-linear relations);
• select the equations of straight lines from a given set of equations of linear and non-linear relations;
• identify the equation of a line in any of the forms y=mx + b, Ax + By + C=0, x=a, y=b;
• rearrange the equation of a line from the form y=mx + b to the form Ax + By + C=0, and vice versa.

By the end of this course, students will:

• determine the slope of a line segment, using various formulas

• identify the slope of a linear relation as representing a constant rate of change;
• calculate the finite differences in the table of values of a linear relation and relate the result to the slope of the relation;
• identify the geometric significance of m and b in the equation y=mx + b through investigation;
• identify the properties of the slopes of line segments (e.g., direction, positive or negative rate of change, steepness, parallelism, perpendicularity) through investigations facilitated by graphing technology, where appropriate.

Using the Properties of Linear Relations to Solve Problems

By the end of this course, students will:

• plot points on the xy-plane and use the terminology and notation of the xy-plane correctly;
• graph lines by hand, using a variety of techniques (e.g., making a table of values, using intercepts, using the slope and y-intercept);
• graph lines, using graphing calculators or graphing software;
• determine the equation of a line, given information about the line (e.g., the slope and y-intercept, the slope and a point, two points, a line parallel to a given line and having the same x-intercept as another given line);
• communicate solutions to multi-step problems in established mathematical form, with clear reasons given for the steps taken;
• describe the meaning of the slope and y-intercept for a linear relation arising from a realistic situation, interpolate and extrapolate from the graph and the equation of the relation, and identify and explain any restrictions on the variables in the relation;
• describe a situation that would be modelled by a given linear equation;
• determine the point of intersection of two linear relations, by hand for simple examples, and using graphing calculators or graphing software for more complex examples; interpret the intersection point in the context of an application.

Measurement and Geometry

Overall Expectations

By the end of this course, students will:

• determine the optimal values of various measurements through investigations facilitated, where appropriate, by the use of concrete materials, diagrams, and calculators or computer software;
• solve problems involving the surface area and the volume of three-dimensional objects;
• formulate conjectures and generalizations about geometric relationships involving two-dimensional figures, through investigations facilitated by dynamic geometry software, where appropriate.

Specific Expectations

By the end of this course, students will:

• identify, through investigation, the effect of varying the dimensions of a rectangular prism or cylinder on the volume or surface area of the object;
• identify, through investigation, the relationships between the volume and surface area of a given rectangular prism or cylinder;
• explain the significance of optimal surface area or volume in various applications (e.g., packaging; the relationship between surface area and heat loss);
• pose and solve a problem involving the relationship between the perimeter and the area of a figure when one of the measures is fixed.

By the end of this course, students will:

• solve simple problems, using the formulas for the surface area and the volume of prisms, pyramids, cylinders, cones, and spheres;
• solve multi-step problems involving the volume and the surface area of prisms, cylinders, pyramids, cones, and spheres;
• judge the reasonableness of answers to measurement problems by considering likely results within the situation described in the problem;
• judge the reasonableness of answers produced by a calculator, a computer, or pencil and paper, using mental mathematics and estimation.

By the end of this course, students will:

• illustrate and explain the properties of the interior and the exterior angles of triangles and quadrilaterals, and of angles related to parallel lines;
• determine the properties of angle bisectors, medians, and altitudes in various types of triangles through investigation;
• determine the properties of the sides and the diagonals of polygons (e.g., the diagonals in quadrilaterals, the diagonals of regular pentagons, the figure that results from joining the midpoints of sides of quadrilaterals) through investigation;
• pose questions about geometric relationships, test them, and communicate the findings, using appropriate language and mathematical forms (e.g., written explanations, diagrams, formulas, tables);
• confirm a statement about the relationships between geometric properties by illustrating the statement with examples, or deny the statement on the basis of a counter-example (e.g., confirm or deny the following statement: If a quadrilateral has perpendicular diagonals, then it is a square).