9.MPM1H.1.2.1 derive, through the investigation and examination of patterns, the exponent rules for multiplying and dividing monomials, and apply these rules in expressions involving one and two variables with positive exponents;
9.MPM1H.1.3.3 expand and simplify polynomial expressions involving one variable [e.g., 2x(4x + 1) – 3x(x + 2)], using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil);
9.MPM1H.1.3.4 solve first-degree equations, including equations with fractional coefficients, using a variety of tools (e.g., computer algebra systems, paper and pencil) and strategies (e.g., the balance analogy, algebraic strategies);
9.MPM1H.1.3.5 rearrange formulas involving variables in the first degree, with and without substitution (e.g., in analytic geometry, in measurement) (Sample problem: A circular garden has a circumference of 30m. Write an expression for the length of a straight path that goes through the centre of this garden.);
9.MPM1H.1.3.6 solve problems that can be modelled with first-degree equations, and compare algebraic methods to other solution methods (Sample problem: Solve the following problem in more than one way: Jonah is involved in a walkathon. His goal is to walk 25 km. He begins at 9:00 a.m. and walks at a steady rate of 4 km/h. How many kilometres does he still have left to walk at 1:15 p.m. if he is to achieve his goal?).
9.MPM1H.2.1 Most of the expectations in the Linear Relations strand of the Grade 9 applied and academic courses are common to the two courses. Some of these expectations may require review since the concepts and skills in this strand are critical for student success in the Analytic Geometry strand of this transfer course.
9.MPM1H.3 Analytic Geometry
9.MPM1H.3.1 Overall Expectations
9.MPM1H.3.1.1 demonstrate an understanding of the characteristics of a linear relation;
9.MPM1H.3.2 Understanding Characteristics of Linear Relations
9.MPM1H.3.2.1 design and carry out an investigation or experiment involving relationships between two variables, including the collection and organization of data, using appropriate methods, equipment, and/or technology (e.g., surveying; using measuring tools, scientific probes, the Internet) and techniques (e.g., making tables, drawing graphs) (Sample problem: Design and perform an experiment to measure and record the temperature of ice water in a plastic cup and ice water in a thermal mug over a 30 min period, for the purpose of comparison. What factors might affect the outcome of this experiment? How could you design the experiment to account for them?);
9.MPM1H.3.2.2 construct equations to represent linear relations derived from descriptions of realistic situations, and connect the equations to tables of values and graphs, using a variety of tools (e.g., graphing calculators, spreadsheets, graphing software, paper and pencil) (Sample problem: Construct a table of values, a graph, and an equation to represent a monthly cellphone plan that costs $25, plus $0.10 per minute of airtime.);
9.MPM1H.3.2.3 determine the equation of a line of best fit for a scatter plot, using an informal process (e.g., using a movable line in dynamic statistical software; using a process of trial and error on a graphing calculator; determining the equation of the line joining two carefully chosen points on the scatter plot).
9.MPM1H.3.3 Investigating the Relationship Between the Equation of a Relation and the Shape of Its Graph
9.MPM1H.3.3.1 determine, through investigation, the characteristics that distinguish the equation of a straight line from the equations of nonlinear relations (e.g., use a graphing calculator or graphing software to graph a variety of linear and non-linear relations from their equations; classify the relations according to the shapes of their graphs; connect an equation of degree one to a linear relation);
9.MPM1H.3.4.1 determine, through investigation, various formulas for the slope of a line segment or a line (e.g., m = rise/run, m = the change in y/the change in x or m = delta y/delta x, m = (y2 - y1)/(x2 - x1), and use the formulas to determine the slope of a line segment or a line;
9.MPM1H.3.4.3 determine, through investigation, connections between slope and other representations of a constant rate of change of linear relation (e.g., if the cost of producing a book of photographs is $50, plus $5 per book, then the slope of the line that represents the cost versus the number of books produced has a value of 5, which is also the rate of change; the value of the slope is the value of the coefficient of the independent variable in the equation of the line, C = 50 + 5p, and the value of the first difference in a table of values);
9.MPM1H.3.4.4 identify, through investigation, properties of the slopes of lines and line segments (e.g., direction, positive or negative rate of change, steepness, parallelism, perpendicularity), using graphing technology to facilitate investigations, where appropriate.
9.MPM1H.3.5.2 determine the equation of a line from information about the line (e.g., the slope and y-intercept; the slope and a point; two points) (Sample problem: Compare the equations of the lines parallel to and perpendicular to y = 2x – 4, and with the same x-intercept as 3x – 4y = 12. Verify using dynamic geometry software.);
9.MPM1H.3.5.3 describe the meaning of the slope and y-intercept for a linear relation arising from a realistic situation (e.g., the cost to rent the community gym is $40 per evening, plus $2 per person for equipment rental; the y-intercept, 40, represents the $40 cost of renting the gym; the value of the slope, 2, represents the $2 cost per person);
9.MPM1H.3.5.4 identify and explain any restrictions on the variables in a linear relation arising from a realistic situation (e.g., in the relation C = 50 + 25n, C is the cost of holding a party in a hall and n is the number of guests; n is restricted to whole numbers of 100 or less, because of the size of the hall, and C is consequently restricted to $50 to $2550).
9.MPM1H.4 Measurement and Geometry
9.MPM1H.4.1 Overall Expectations
9.MPM1H.4.1.1 solve problems involving the surface areas and volumes of three-dimensional figures;
9.MPM1H.4.1.2 verify, through investigation facilitated by dynamic geometry software, geometric properties and relationships involving two-dimensional shapes, and apply the results to solving problems.
9.MPM1H.4.2 Solving Problems Involving Surface Area and Volume
9.MPM1H.4.2.1 solve problems involving the volumes of composite figures composed of prisms, pyramids, cylinders, cones, and spheres;
9.MPM1H.4.2.2 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);
9.MPM1H.4.2.3 solve problems involving the surface areas of prisms, pyramids, cylinders, cones, and spheres, including composite figures (Sample problem: Break-bit Cereal is sold in a single-serving size, in a box in the shape of a rectangular prism of dimensions 5 cm by 4 cm by 10 cm. The manufacturer also sells the cereal in a larger size, in a box with dimensions double those of the smaller box. Compare the surface areas and the volumes of the two boxes, and explain the implications of your answers.);
9.MPM1H.4.2.4 identify, through investigation with a variety of tools (e.g. concrete materials, computer software), the effect of varying the dimensions on the surface area [or volume] of square-based prisms and cylinders, given a fixed volume [or surface area];
9.MPM1H.4.2.5 explain the significance of optimal surface area or volume in various applications (e.g., the minimum amount of packaging material; the relationship between surface area and heat loss);
9.MPM1H.4.2.6 pose and solve problems involving maximization and minimization of measurements of geometric figures (i.e., square-based prisms or cylinders) (Sample problem: Determine the dimensions of a square-based, open-topped prism with a volume of 24 cm³ and with the minimum surface area.).
9.MPM1H.4.3 Investigating and Applying Geometric Relationships
9.MPM1H.4.3.1 determine, through investigation using a variety of tools (e.g., dynamic geometry software, paper folding), and describe some properties of polygons (e.g., the figure that results from joining the midpoints of the sides of a quadrilateral is a parallelogram; the diagonals of a rectangle bisect each other; the line segment joining the midpoints of two sides of a triangle is half the length of the third side), and apply the results in problem solving (e.g., given the width of the base of an A-frame tree house, determine the length of a horizontal support beam that is attached half way up the sloping sides);
9.MPM1H.4.3.2 pose questions about geometric relationships, investigate them, and present their findings, using a variety of mathematical forms (e.g., written explanations, diagrams, dynamic sketches, formulas, tables) (Sample problem: How many diagonals can be drawn from one vertex of a 20- sided polygon? How can I find out without counting them?);
9.MPM1H.4.3.3 illustrate a statement about a geometric property by demonstrating the statement with multiple examples, or deny the statement on the basis of a counter-example, with or without the use of dynamic geometry software (Sample problem: Confirm or deny the following statement: If a quadrilateral has perpendicular diagonals, then it is a square.).