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

2.1: Overall Expectations

2.1.1: determine the basic properties of quadratic relations;

Parabolas - Activity A
Quadratic and Absolute Value Functions
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A
Roots of a Quadratic

2.1.2: relate transformations of the graph of y = x² to the algebraic representation y = a(x - h)² + k;

Parabolas - Activity A

2.1.4: solve problems involving quadratic relations.

Parabolas - Activity A
Quadratic and Absolute Value Functions
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A
Roots of a Quadratic

2.2: Investigating the Basic Properties of Quadratic Relations

2.2.1: 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 (Sample problem: Make a 1 m ramp that makes a 15° angle with the floor. Place a can 30 cm up the ramp. Record the time it takes for the can to roll to the bottom. Repeat by placing the can 40 cm, 50 cm, and 60 cm up the ramp, and so on. Graph the data and draw the curve of best fit.);

Parabolas - Activity A
Quadratic and Absolute Value Functions
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A
Roots of a Quadratic

2.2.2: determine, through investigation with and without the use of technology, that a quadratic relation of the form y = ax² + bx + c (a "not equal to" 0) can be graphically represented as a parabola, and that the table of values yields a constant second difference (Sample problem: Graph the relation y = x² - 4x by developing a table of values and plotting points. Observe the shape of the graph. Calculate first and second differences. Repeat for different quadratic relations. Describe your observations and make conclusions, using the appropriate terminology.);

Parabolas - Activity A
Quadratic and Absolute Value Functions
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A
Roots of a Quadratic
Translating and Scaling Functions
Using Tables, Rules and Graphs

2.2.3: 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;

Parabolas - Activity A
Polynomials and Linear Factors
Quadratics in Factored Form

2.3: Relating the Graph of y = x² and Its Transformations

2.3.1: identify, through investigation using technology, the effect on the graph of y = x² of transformations (i.e., translations, reflections in the x-axis, vertical stretches or compressions) by considering separately each parameter a, h, and k [i.e., investigate the effect on the graph of y = x² of a, h, and k in y = x² + k, y = (x - h)², and y = ax²];

Parabolas - Activity A
Quadratics in Factored Form
Reflections
Rotations, Reflections and Translations
Translations

2.3.2: explain the roles of a, h, and 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;

Defining a Line with Two Points
Dilations
Parabolas - Activity A
Point-Slope Form of a Line - Activity A
Reflections
Slope-Intercept Form of a Line - Activity A
Standard Form of a Line

2.3.3: sketch, by hand, the graph of y = a(x - h)² + k by applying transformations to the graph of y = x² [Sample problem: Sketch the graph of y =- 1/2(x - 3)² + 4, and verify using technology.];

Parabolas - Activity A
Quadratics in Factored Form

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

Parabolas - Activity A
Quadratics in Factored Form

2.4: Solving Quadratic Equations

2.4.1: 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);

Addition of Polynomials - Activity A

2.4.2: 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

2.4.3: 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

2.4.4: 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;

Polynomials and Linear Factors
Quadratic and Absolute Value Functions
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A
Roots of a Quadratic

2.4.5: express y = ax² + 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);

Parabolas - Activity A

2.4.6: sketch or graph a quadratic relation whose equation is given in the form y = ax² + bx + c, using a variety of methods (e.g., sketching y = x² - 2x - 8 using intercepts and symmetry; sketching y = 3x² - 12x + 1 by completing the square and applying transformations; graphing h = -4.9t² + 50t + 1.5 using technology);

Parabolas - Activity A
Quadratic and Absolute Value Functions
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A
Roots of a Quadratic

2.4.7: 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

2.4.8: 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.).

Roots of a Quadratic

2.5: Solving Problems Involving Quadratic Relations

2.5.1: 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);

Parabolas - Activity A
Polynomials and Linear Factors

2.5.2: 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?).

Parabolas - Activity A
Quadratic and Absolute Value Functions
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A
Roots of a Quadratic

3: Analytic Geometry

3.1: Overall Expectations

3.1.3: verify geometric properties of triangles and quadrilaterals, using analytic geometry.

Parallelogram Conditions
Triangle Inequalities

3.2: Using Linear Systems to Solve Problems

3.2.1: 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);

Special Types of Solutions to Linear Systems

3.2.2: 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 Linear Systems by Graphing
Special Types of Solutions to Linear Systems
Systems of Linear Equations - Activity A

3.3: Solving Problems Involving Properties of Line Segments

3.3.2: 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 - Activity A
Geoboard: The Pythagorean Theorem
Pythagorean Theorem - Activity A
Pythagorean Theorem - Activity B

3.3.3: 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
Distance Formula - Activity A

3.3.4: 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

3.3.5: 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).

Distance Formula - Activity A
Geoboard: The Pythagorean Theorem
Pythagorean Theorem - Activity A
Slope - Activity B

3.4: Using Analytic Geometry to Verify Geometric Properties

3.4.1: 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);

Classifying Quadrilaterals - Activity B
Classifying Triangles
Parallelogram Conditions
Perimeters and Areas of Similar Figures
Polygon Angle Sum - Activity A
Similar Figures - Activity A
Similar Polygons
Triangle Angle Sum - Activity A

3.4.2: 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);

Classifying Triangles
Isosceles and Equilateral Triangles

4: Trigonometry

4.1: Overall Expectations

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

Perimeters and Areas of Similar Figures
Similar Figures - Activity A
Similar Polygons

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

Distance Formula - Activity A
Geoboard: The Pythagorean Theorem
Pythagorean Theorem - Activity A
Pythagorean Theorem - Activity B
Sine and Cosine Ratios - Activity A
Sine, Cosine and Tangent
Tangent Ratio

4.2: Investigating Similarity and Solving Problems Involving Similar Triangles

4.2.1: 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);

Perimeters and Areas of Similar Figures
Similar Figures - Activity A
Similar Polygons

4.2.2: describe and compare the concepts of similarity and congruence;

Congruence in Right Triangles
Perimeters and Areas of Similar Figures
Similar Figures - Activity A
Similar Polygons

4.2.3: 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.).

Perimeters and Areas of Similar Figures
Reflections
Rotations, Reflections and Translations
Similar Figures - Activity A
Similar Polygons

4.3: Solving Problems Involving the Trigonometry of Right Triangles

4.3.1: 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 Function
Sine and Cosine Ratios - Activity A
Sine, Cosine and Tangent
Tangent Ratio

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

Distance Formula - Activity A
Geoboard: The Pythagorean Theorem
Pythagorean Theorem - Activity A
Pythagorean Theorem - Activity B
Sine and Cosine Ratios - Activity A
Sine, Cosine and Tangent
Tangent Ratio
Triangle Angle Sum - Activity A

4.3.3: 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.

Geoboard: The Pythagorean Theorem
Pythagorean Theorem - Activity B
Sine and Cosine Ratios - Activity A
Sine, Cosine and Tangent

4.4: Solving Problems Involving the Trigonometry of Acute Triangles

4.4.2: 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]);

Cosine Function
Geoboard: The Pythagorean Theorem

4.4.4: solve problems involving the measures of sides and angles in acute triangles.

Pythagorean Theorem - Activity B
Sine and Cosine Ratios - Activity A