# Course Description

This course enables students to broaden their understanding of relationships and extend their problem-solving and algebraic skills through investigation, the effective use of technology, and abstract reasoning. Students will explore quadratic relations and their applications; solve and apply linear systems; verify properties of geometric figures using analytic geometry; and investigate the trigonometry of right and acute triangles. 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.

### A: Quadratic Relations of the Form $y=ax^2+bx+c$

1. determine the basic properties of quadratic relations;
2. relate transformations of the graph of $y = x^2$ to the algebraic representation $y = a(x - h)^2 + k$;
3. solve quadratic equations and interpret the solutions with respect to the corresponding relations;

### B: Analytic Geometry

1. model and solve problems involving the intersection of two straight lines;
2. solve problems using analytic geometry involving properties of lines and line segments;
3. verify geometric properties of triangles and quadrilaterals, using analytic geometry.

### C: Trigonometry

1. use their knowledge of ratio and proportion to investigate similar triangles and solve problems related to similarity;
2. solve problems involving right triangles, using the primary trigonometric ratios and the Pythagorean theorem;
3. solve problems involving acute triangles, using the sine law and the cosine law.

## Specific Provincial Curriculum Expectations

### A1: Investigating the Basic Properties of Quadratic Relations

• – collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology (e.g., concrete materials, scientific probes, graphing calculators), or from secondary sources (e.g., the Internet, Statistics Canada); graph the data and draw a curve of best fit, if appropriate, with or without the use of technology
• – determine, through investigation with and without the use of technology, that a quadratic relation of the form $y = ax^2 + bx + c, a\neq 0$ can be graphically represented as a parabola, and that the table of values yields a constant second difference
• – identify the key features of a graph of a parabola (i.e., the equation of the axis of symmetry, the coordinates of the vertex, the y-intercept, the zeros, and the maximum or minimum value), and use the appropriate terminology to describe them;
• – compare, through investigation using technology, the features of the graph of $y = x^2$ and the graph of $y = 2^x$, 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 $y = 2^x$; by applying the exponent rules for multiplication and division)

### A2: Relating the Graph of $y=x^2$ And Its Transformations

• – identify, through investigation using technology, the effect on the graph of $y = x^2$ of transformations (i.e., translations, reflections in the x-axis, vertical stretches or compressions) by considering separately each parameter $a, h, k$ [i.e., investigate the effect on the graph of $y = x^2$ of $a, h, k$ in $y = x^2 + k, y = (x – h)^2$ and $y = ax^2$];
• – explain the roles of $a, h, k$ in $y = a(x – h )^2 + k$, using the appropriate terminology to describe the transformations, and identify the vertex and the equation of the axis of symmetry;
• – sketch, by hand, the graph of $y = a(x – h )^2 + k$ by applying transformations to the graph of $y = x^2$
• – determine the equation, in the form $y = a(x – h)^2 + k$, of a given graph of a parabola

• – expand and simplify second-degree polynomial expressions [e.g., $(2x + 5)^2$, (2x – y)(x + 3y)], using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil) and strategies (e.g., patterning);
• – factor polynomial expressions involving common factors, trinomials, and differences of squares [e.g., $2x^2 + 4x$, $2x – 2y + ax – ay$, $x2 – x – 6$, $2a^2 + 11a + 5$, $4x^2 – 25$], using a variety of tools (e.g., concrete materials, computer algebra systems, paper and pencil) and strategies (e.g., patterning);
• – 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);
• – 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;
• – express $y = ax^2 + bx + c$ in the form $y = a(x – h)^2 + k$ by completing the square in situations involving no fractions, using a variety of tools (e.g. concrete materials, diagrams, paper and pencil);
• – sketch or graph a quadratic relation whose equation is given in the form $y = ax^2 + bx + c$, using a variety of methods (e.g., sketching $y = x^2 – 2x – 8$ using intercepts and symmetry; sketching $y = 3x^2 – 12x + 1$ by completing the square and applying transformations; graphing $h = –4.9t^2 + 50t + 1.5$ using technology);
• – 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]);
• – solve quadratic equations that have real roots, using a variety of methods (i.e., factoring, using the quadratic formula, graphing)

### A4: Solving Problems Involving Quadratic Equations

• – 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);
• – solve problems arising from a realistic situation represented by a graph or an equation of a quadratic relation, with and without the use of technology (e.g., given the graph or the equation of a quadratic relation representing the height of a ball over elapsed time, 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?)

### B1: Using Linear Systems to Solve Problems

• – solve systems of two linear equations involving two variables, using the algebraic method of substitution or elimination
• – 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

### B2: Solving Problems Involving Properties of Line Segments

• – develop the formula for the midpoint of a line segment, and use this formula to solve problems (e.g., determine the coordinates of the midpoints of the sides of a triangle, given the coordinates of the vertices, and verify concretely or by using dynamic geometry software);
• – 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);
• – develop the equation for a circle with centre (0, 0) and radius r, by applying the formula for the length of a line segment;
• – 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 $x^2 + y^2 = r^2$;
• – solve problems involving the slope, length, and midpoint of a line segment (e.g., determine the equation of the right bisector of a line segment, given the coordinates of the endpoints; determine the distance from a given point to a line whose equation is given, and verify using dynamic geometry software)

### B3: Using Analytic Geometry to Verify Geometric Properties

• – determine, through investigation (e.g., using dynamic geometry software, by paper folding), some characteristics and properties of geometric figures (e.g., medians in a triangle, similar figures constructed on the sides of a right triangle);
• – verify, using algebraic techniques and analytic geometry, some characteristics of geometric figures (e.g., verify that two lines are perpendicular, given the coordinates of two points on each line; verify, by determining side length, that a triangle is equilateral, given the coordinates of the vertices);
• – plan and implement a multi-step strategy that uses analytic geometry and algebraic techniques to verify a geometric property (e.g., given the coordinates of the vertices of a triangle, verify that the line segment joining the midpoints of two sides of the triangle is parallel to the third side and half its length, and check using dynamic geometry software; given the coordinates of the vertices of a rectangle, verify that the diagonals of the rectangle bisect each other)

### C1: Investigating Similarity and Solving Problems Involving Similar Triangles

• – 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);
• – describe and compare the concepts of similarity and congruence;
• – solve problems involving similar triangles in realistic situations (e.g., shadows, reflections, scale models, surveying)

### C2: Solving Problems Involving the Trigonometry of Right Triangles

• – 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
• – determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem;
• – solve problems involving the measures of sides and angles in right triangles in reallife 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

### C3: Solving Problems Involving the Trigonometry of Acute Triangles

• – explore the development of the sine law within acute triangles (e.g., use dynamic geometry software to determine that the ratio of the side lengths equals the ratio of the sines of the opposite angles; follow the algebraic development of the sine law and identify the application of solving systems of equations [student reproduction of the development of the formula is not required]);
• – explore the development of the cosine law within acute triangles (e.g., use dynamic geometry software to verify the cosine law; follow the algebraic development of the cosine law and identify its relationship to the Pythagorean theorem and the
• cosine ratio [student reproduction of the development of the formula is not required]);
• – determine the measures of sides and angles in acute triangles, using the sine law and the cosine law
• – solve problems involving the measures of sides and angles in acute triangles.