# Course Description

This course enables students to develop an understanding of mathematical concepts related to algebra, analytic geometry, and measurement and geometry through investigation, the effective use of technology, and abstract reasoning. Students will investigate relationships, which they will then generalize as equations of lines, and will determine the connections between different representations of a linear relation. They will also explore relationships that emerge from the measurement of three-dimensional figures and two-dimensional shapes. Students will reason mathematically and communicate their thinking as they solve multi-step problems.

# Overall Provincial Curriculum Expectations

See Ontario math curriculum for grades 9 and 10 for details. See also this news article for possible changes to the grade 9 math curriculum.

### A: Number Sense and Algebra

1. demonstrate an understanding of the exponent rules of multiplication and division, and apply them to simplify expressions;
2. manipulate numerical and polynomial expressions, and solve first-degree equations.

### B: Linear Relations

1. apply data-management techniques to investigate relationships between two variables;
2. demonstrate an understanding of the characteristics of a linear relation;
3. connect various representations of a linear relation.

### C: Analytic Geometry

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

### D: Measurement and Geometry

1. determine, through investigation, the optimal values of various measurements;
2. solve problems involving the measurements of two-dimensional shapes and the surface areas and volumes of three-dimensional figures;
3. verify, through investigation facilitated by dynamic geometry software, geometric properties and relationships involving two-dimensional shapes, and apply the results to solving problems.

## Specific Provincial Curriculum Expectations

### A1: Operating with Exponents

• – substitute into and evaluate algebraic expressions involving exponents (i.e., evaluate expressions involving natural-number exponents with rational-number bases
• – describe the relationship between the algebraic and geometric representations of a single-variable term up to degree three [i.e., length, which is one dimensional, can be represented by x; area, which is two dimensional, can be represented by (x)(x) or $x^2$; volume, which is three dimensional, can be represented by (x)(x)(x), $(x^2)(x)$, or $x^3$];
• – 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;
• – extend the multiplication rule to derive and understand the power of a power rule, and apply it to simplify expressions involving one and two variables with positive exponents.

### A2: Manipulating Expressions and Solving Equations

• – simplify numerical expressions involving integers and rational numbers, with and without the use of technology;
• – solve problems requiring the manipulation of expressions arising from applications of percent, ratio, rate, and proportion;
• – relate their understanding of inverse operations to squaring and taking the square root, and apply inverse operations to simplify expressions and solve equations;
• – add and subtract polynomials with up to two variables [e.g., (2x – 5) + (3x + 1), $(3x^2y + 2xy^2) + \\ (4x^2y – 6xy^2)$], using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil);
• – multiply a polynomial by a monomial involving the same variable [e.g., 2x(x + 4), $2x^2(3x^2 – 2x + 1)$], using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil);
• – 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);
• – 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);
• – rearrange formulas involving variables in the first degree, with and without substitution (e.g., in analytic geometry, in measurement)
• – solve problems that can be modelled with first-degree equations, and compare algebraic methods to other solution methods

### B1: Using Data Management to Investigate Relationships

• – interpret the meanings of points on scatter plots or graphs that represent linear relations, including scatter plots or graphs in more than one quadrant [e.g., on a scatter plot of height versus age, interpret the point (13, 150) as representing a student who is 13 years old and 150 cm tall; identify points on the graph that represent students who are taller and younger than this student]
• – pose problems, identify variables, and formulate hypotheses associated with relationships between two variables
• – 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)
• – describe trends and relationships observed in data, make inferences from data, compare the inferences with hypotheses about the data, and explain any differences between the inferences and the hypotheses (e.g., describe the trend observed in the data. Does a relationship seem to exist? Of what sort? Is the outcome consistent with your hypothesis? Identify and explain any outlying pieces of data. Suggest a formula that relates the variables. How might you vary this experiment to examine other relationships?)

### B2: Understanding Characteristics of Linear Relations

• – construct tables of values, graphs, and equations, using a variety of tools (e.g., graphing calculators, spreadsheets, graphing software, paper and pencil), to represent linear relations derived from descriptions of realistic situations
• – construct tables of values, scatter plots, and lines or curves of best fit as appropriate, using a variety of tools (e.g., spreadsheets, graphing software, graphing calculators, paper and pencil), for linearly related and non-linearly related data collected from a variety of sources (e.g., experiments, electronic secondary sources, patterning with concrete materials)
• – identify, through investigation, some properties of linear relations (i.e., numerically, the first difference is a constant, which represents a constant rate of change; graphically, a straight line represents the relation), and apply these properties to determine whether a relation is linear or non-linear;
• – compare the properties of direct variation and partial variation in applications, and identify the initial value (e.g., for a relation described in words, or represented as a graph or an equation)
• – 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; deterand error on a graphing calculator; determining the equation of the line joining two carefully chosen points on the scatter plot)

### B3: Connecting Various Representations of Linear Relations

• – determine values of a linear relation by using a table of values, by using the equation of the relation, and by interpolating or extrapolating from the graph of the relation
• – describe a situation that would explain the events illustrated by a given graph of a relationship between two variables
• – determine other representations of a linear relation, given one representation (e.g., given a numeric model, determine a graphical model and an algebraic model; given a graph, determine some points on the graph and determine an algebraic model);
• – describe the effects on a linear graph and make the corresponding changes to the linear equation when the conditions of the situation they represent are varied (e.g., given a partial variation graph and an equation representing the cost of producing a yearbook, describe how the graph changes if the cost per book is altered, describe how the graph changes if the fixed costs are altered, and make the corresponding changes to the equation)

### C1: Investigating the Relationship Between the Equation of a Relation and the Shape of Its Graph

• – 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);
• – identify, through investigation, the equation of a line in any of the forms y = mx + b, Ax + By + C = 0, x = a, y = b;
• – express the equation of a line in the form y = mx + b, given the form Ax + By + C = 0.

### C2: Investigating the Properties of Slope

• – determine, through investigation, various formulas for the slope of a line segment or a line, and use the formulas to determine the slope of a line segment or a line;
• – identify, through investigation with technology, the geometric significance of m and b in the equation y = mx + b;
• – determine, through investigation, connections among the representations of a constant rate of change of a linear relation
• – 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

### C3: Using the Properties of Linear Relations to Solve Problems

• – graph lines by hand, using a variety of techniques (e.g., graph $y =\frac{2}{3} x – 4$ using the y-intercept and slope; graph 2x + 3y = 6 using the x- and y-intercepts);
• – 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)
• – describe the meaning of the slope and y-intercept for a linear relation arising from a realistic situation
• – 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);
• – determine graphically the point of intersection of two linear relations, and interpret the intersection point in the context of an application

### D1: Investigating the Optimal Values of Measurements

• – determine the maximum area of a rectangle with a given perimeter by constructing a variety of rectangles, using a variety of tools (e.g., geoboards, graph paper, toothpicks, a pre-made dynamic geometry sketch), and by examining various values of the area as the side lengths change and the perimeter remains constant;
• – determine the minimum perimeter of a rectangle with a given area by constructing a variety of rectangles, using a variety of tools (e.g., geoboards, graph paper, a premade dynamic geometry sketch), and by examining various values of the side lengths and the perimeter as the area stays constant;
• – 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];
• – explain the significance of optimal area, surface area, or volume in various applications (e.g., the minimum amount of packaging material; the relationship between surface area and heat loss);
• – pose and solve problems involving maximization and minimization of measurements of geometric shapes and figures (e.g., determine the dimensions of the rectangular field with the maximum area that can be enclosed by a fixed amount of fencing, if the fencing is required on only three sides)

### D2: Solving Problems Involving Perimeter, Area, Surface Area, and Volume

• – relate the geometric representation of the Pythagorean theorem and the algebraic representation $a^2 + b^2 = c^2$;
• – solve problems using the Pythagorean theorem, as required in applications (e.g., calculate the height of a cone, given the radius and the slant height, in order to determine the volume of the cone);
• – solve problems involving the areas and perimeters of composite two-dimensional shapes (i.e., combinations of rectangles, triangles, parallelograms, trapezoids, and circles)
• – develop, through investigation (e.g., using concrete materials), the formulas for the volume of a pyramid, a cone, and a sphere
• – determine, through investigation, the relationship for calculating the surface area of a pyramid (e.g., use the net of a squarebased pyramid to determine that the surface area is the area of the square base plus the areas of the four congruent triangles);
• – solve problems involving the surface areas and volumes of prisms, pyramids, cylinders, cones, and spheres, including composite figures

### D3: Investigating and Applying Geometric Relationships

• – determine, through investigation using a variety of tools (e.g., dynamic geometry software, concrete materials), and describe the properties and relationships of the interior and exterior angles of triangles, quadrilaterals, and other polygons, and apply the results to problems involving the angles of polygons
• – 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);
• – 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)
• – 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