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Correlation of the Expectations for Principles of Mathematics 10 to Nelson Mathematic 10 Page 1 Principles of Mathematics, Grade 10, Academic (MPM2D) This course enables students to broaden their understanding of relations, extend their skills in multi-step problem solving, and continue to develop their abilities in abstract reasoning. Students will pursue investigations of quadratic functions and their applications; solve and apply linear systems; solve multi-step problems in analytic geometry to verify properties of geometric figures; investigate the trigonometry of right and acute triangles; and develop supporting algebraic skills. Quadratic Functions Overall Expectations By the end of this course, students will: Code QFV.01 Expectation solve quadratic equations; QFV.02 determine, through investigation, the relationships between the Throughout Chapters 3 and 4 graphs and the equations of quadratic functions; determine, through investigation, the basic properties of quadratic Throughout Chapters 3 and 4 functions; solve problems involving quadratic functions. Throughout Chapters 3 and 4 QFV.03 QFV.04 Nelson Mathematics 10 Throughout Chapters 3 and 4 Specific Expectations Solving Quadratic Equations By the end of this course, students will: Code Expectation QF1.01 expand and simplify second-degree polynomial expressions; QF1.02 factor polynomial expressions involving common factors, differences of squares, and trinomials; Nelson Mathematics 10 3.5 Exploration: Multiplying Binomials Using Algebra Tiles 3.7 Standard Form of a Quadratic Relation 3.8 Extending Algebra Skills: Factoring Quadratic Expressions Correlation of the Expectations for Principles of Mathematics 10 to Nelson Mathematic 10 QF1.03 QF1.04 QF1.05 solve quadratic equations by factoring and by using graphing calculators or graphing software; solve quadratic equations, using the quadratic formula; interpret real and non-real roots of quadratic equations geometrically as the x-intercepts of the graph of a quadratic function. 3.9 4.7 4.7 4.7 3.4 Page 2 Solving Problems with Quadratic Equations Solving Quadratic Equations: The Quadratic Formula Solving Quadratic Equations: The Quadratic Formula Solving Quadratic Equations: The Quadratic Formula The Role of the Zeros of a Quadratic Relation Investigating the Connection Between the Graphs and the Equations of Quadratic Functions By the end of this course, students will: Code Expectation QF2.01 identify the effect of simple transformations (i.e., translations, reflections, vertical stretch factors) on the graph and the equation of y=x2, using graphing calculators or graphing software; explain the role of a, h, and k in the graph of y=a(x – h)2 + k; QF2.03 QF2.04 express the equation of a quadratic function in the form y=a(x – h)2 + k, given it in the form y=ax2 + bx + c, using the algebraic method of completing the square in situations involving no fractions; sketch, by hand, the graph of a quadratic function whose equation is given in the form y=ax2 + bx + c, using a suitable method [e.g., complete the square; locate the x-intercepts if the equation is factorable; express in the form y=ax(x – s) + t to locate two points and deduce the vertex]. Nelson Mathematics 10 4.3 Using Technology to Investigate Transformations of Quadratics 4.2 The Vertex Form of a Quadratic Relation 4.3 Using Technology to Investigate Transformations of Quadratics 4.6 Determining Maximum and Minimum Values Algebraically: Completing the Square 3.4 The Role of the Zeros of a Quadratic Relation 4.2 The Vertex Form of a Quadratic Relation 4.3 Using Technology to Investigate Transformations of Quadratics 4.4 Using Symmetry to Relate Standard Form to Vertex Form 4.6 Determining Maximum and Minimum Values Algebraically: Completing the Square Investigating the Basic Properties of Quadratic Functions By the end of this course, students will: Code Expectation Nelson Mathematics 10 Correlation of the Expectations for Principles of Mathematics 10 to Nelson Mathematic 10 Page 3 QF3.01 collect data that may be represented by quadratic functions, from secondary sources (e.g., the Internet, Statistics Canada), or from experiments, using appropriate equipment and technology (e.g., scientific probes, graphing calculators); 3.1 Quadratic Relations 4.1 Exploration: Quadratic Dental Models 4.8 Investigating Quadratic Relations QF3.02 fit the equation of a quadratic function to a scatter plot, using an informal process (e.g., a process of trial and error on a graphing calculator), and compare the results with the equation of a curve of best fit produced by using graphing calculators or graphing software; QF3.03 3.6 Technology: Using Quadratic Regression to Find a Curve of Best Fit with a TI-83 Plus Calculator 3.7 Standard Form of a Quadratic Relation 4.1 Exploration: Quadratic Dental Models 4.2 The Vertex Form of a Quadratic Relation 4.4 Using Symmetry to Relate Standard Form to Vertex Form 3.1 Quadratic Relations 3.2 Properties of Quadratic Relations describe the nature of change in a quadratic function, using finite differences in tables of values, and compare the nature of change in a quadratic function with the nature of change in a linear function; report the findings of an experiment in a clear and concise manner, 4.1 Exploration: Quadratic Dental Models using appropriate mathematical forms (e.g., written explanations, 4.8 Investigating Quadratic Relations tables, graphs, formulas, calculations), and justify the conclusions reached. QF3.04 Solving Problems Involving Quadratic Functions By the end of this course, students will: Code Expectation QF4.01 determine the zeros and the maximum or minimum value of a quadratic function, using algebraic techniques; Nelson Mathematics 10 3.4 The Role of the Zeros of a Quadratic Relation 3.8 Extending Algebra Skills: Factoring Quadratic Expressions 4.4 Using Symmetry to Relate Standard Form to Vertex Form 4.6 Determining Maximum and Minimum Values Algebraically: Completing the Square Correlation of the Expectations for Principles of Mathematics 10 to Nelson Mathematic 10 QF4.02 determine the zeros and the maximum or minimum value of a quadratic function from its graph, using graphing calculators or graphing software; QF4.03 solve problems related to an application, given the graph or the formula of a quadratic function (e.g., given a quadratic function 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 touch the ground? Over what interval is the height of the ball greater than 3 m?). 3.3 Technology: Finding the Zeros of a Quadratic Relation Using a TI-83 Plus Calculator 3.4 The Role of the Zeros of a Quadratic Relation 4.5 Technology: Finding the Maximum or Minimum Value of a Quadratic Relation 4.6 Determining Maximum and Minimum Values Algebraically: Completing the Square Throughout Chapters 3 and 4 Analytic Geometry Overall Expectations By the end of this course, students will: Code Expectation AGV.01 model and solve problems involving the intersection of two straight lines; Nelson Mathematics 10 Throughout Chapter 1 AGV.02 Throughout Chapter 2 AGV.03 solve problems involving the analytic geometry concepts of line segments; verify geometric properties of triangles and quadrilaterals, using analytic geometry. Throughout Chapter 2 Specific Expectations Using Linear Systems to Solve Problems By the end of this course, students will: Code Expectation Page 4 Nelson Mathematics 10 Correlation of the Expectations for Principles of Mathematics 10 to Nelson Mathematic 10 AG1.01 determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators or graphing software, and interpret the intersection point in the context of a realistic situation; AG1.02 solve systems of two linear equations in two variables by the algebraic methods of substitution and elimination; solve problems represented by linear systems of two equations in two variables arising from realistic situations, by using an algebraic method and by interpreting graphs. AG1.03 Page 5 1.3 Solving Linear Systems: Graphing by Hand 1.5 Investigating the Ways That Two Lines Can Intersect 1.6 Technology: Determining the Point of Intersection 1.7 Solving a Linear System Using Graphing Technology 1.11 Modelling Using Linear Systems 1.8 Solving a Linear System Using Algebra: Substitution 1.9 Solving a Linear System Using Algebra: Elimination Throughout Chapter 1 2.10 Using the Point of Intersection to Solve Problems Solving Problems Involving the Properties of Line Segments By the end of this course, students will: Code Expectation AG2.01 determine formulas for the midpoint and the length of a line segment and use these formulas to solve problems; AG2.02 AG2.03 AG2.04 determine the equation for a circle having centre (0, 0) and radius r, by applying the formula for the length of a line segment; identify the radius of a circle of centre (0, 0), given its equation; and write the equation, given the radius solve multi-step problems, using the concepts of the slope, the length, and the midpoint of line segments (e.g., determine the equation of the right bisector of a line segment, the coordinates of whose end points are given; determine the distance from a given point to a line whose equation is given; show that the centre of a given circle lies on the right bisector of a given chord);identify the radius of a circle of centre (0, 0), given its equation; and write the equation, given the radius; communicate the solutions to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions. Nelson Mathematics 10 2.3 Distance on the Plane—Part I: Distance from the Origin 2.5 Distance on the Plane—Part II 2.7 Finding the Midpoint of a Line Segment 2.4 The Equation of a Circle 2.7 Finding the Midpoint of a Line Segment 2.8 Classifying Shapes on a Coordinate Plane 2.10 Using the Point of Intersection to Solve Problems 2.11 Verifying Geometric Properties Throughout Chapters 1 and 2 Correlation of the Expectations for Principles of Mathematics 10 to Nelson Mathematic 10 Page 6 Using Analytic Geometry to Verify Geometric Properties By the end of this course, students will: Code Expectation AG3.01 determine characteristics of a triangle whose vertex coordinates are given (e.g., the perimeter; the classification by side length; the equations of medians, altitudes, and right bisectors; the location of the circumcentre and the centroid); AG3.02 determine characteristics of a quadrilateral whose vertex coordinates are given (e.g., the perimeter; the classification by side length; the properties of the diagonals; the classification of a quadrilateral as a square, a rectangle, or a parallelogram); AG3.03 verify geometric properties of a triangle or quadrilateral whose vertex coordinates are given (e.g., the line joining the midpoints of two sides of a triangle is parallel to the third side; the diagonals of a rectangle bisect each other). Nelson Mathematics 10 2.7 Finding the Midpoint of a Line Segment 2.8 Classifying Shapes on a Coordinate Plane 2.10 Using the Point of Intersection to Solve Problems 2.8 Classifying Shapes on a Coordinate Plane 2.11 Verifying Geometric Properties Trigonometry Overall Expectations By the end of this course, students will: Code Expectation TRV.01 develop the primary trigonometric ratios, using the properties of similar triangles; Nelson Mathematics 10 First Half of Chapter 5 5.1 Technology: Constructing Triangles with The Geometer’s Sketchpad 5.2 Investigating and Comparing Triangles 5.3 Technology: Enlarging or Reducing a Triangle with The Geometer’s Sketchpad 5.4 Modelling with Similar Triangles Correlation of the Expectations for Principles of Mathematics 10 to Nelson Mathematic 10 TRV.02 solve trigonometric problems involving right triangles; TRV.03 solve trigonometric problems involving acute triangles. Page 7 Second Half of Chapter 5 5.5 Exploration: Slopes and Angles of Ramps 5.6 Investigating the Connection Between Slope and Angle 5.7 The Primary Trigonometric Ratios 5.8 Solving Problems Using Right Triangle Models and Trigonometry Throughout Chapter 6 Specific Expectations Developing the Primary Trigonometric Ratios By the end of this course, students will: Code Expectation TR1.01 determine the properties of similar triangles (e.g., the correspondence and equality of angles, the ratio of corresponding sides, the ratio of areas) through investigation, using dynamic geometry software; TR1.02 describe and compare the concepts of similarity and congruence; TR1.03 solve problems involving similar triangles in realistic situations (e.g., problems involving shadows, reflections, surveying); TR1.04 define the formulas for the sine, the cosine, and the tangent of angles, using the ratios of sides in right triangles. Nelson Mathematics 10 5.2 Investigating and Comparing Triangles 5.4 Modelling with Similar Triangles 5.7 The Primary Trigonometric Ratios 6.2 Similar-Triangle Models 5.2 Investigating and Comparing Triangles 5.2 Investigating and Comparing Triangles 5.4 Modelling with Similar Triangles 5.6 Investigating the Connection Between Slope and Angle 6.2 Similar-Triangle Models 5.6 Investigating the Connection Between Slope and Angle 5.7 The Primary Trigonometric Ratios Solving Problems Involving the Trigonometry of Right Triangles By the end of this course, students will: Code TR2.01 Expectation determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios; Nelson Mathematics 10 5.7 The Primary Trigonometric Ratios 5.8 Solving Problems Using Right Triangle Models and Trigonometry Correlation of the Expectations for Principles of Mathematics 10 to Nelson Mathematic 10 TR2.02 solve problems involving the measures of sides and angles in right triangles (e.g., in surveying, navigation); TR2.03 determine the height of an inaccessible object in the environment around the school, using the trigonometry of right triangles. Page 8 5.7 The Primary Trigonometric Ratios 5.8 Solving Problems Using Right Triangle Models and Trigonometry 6.7 Solving Measurement Problems Modelled by Triangles 5.4 Modelling with Similar Triangles 5.7 The Primary Trigonometric Ratios 5.8 Solving Problems Using Right Triangle Models and Trigonometry 6.2 Similar-Triangle Models Solving Problems Involving the Trigonometry of Acute Triangles By the end of this course, students will: Code TR3.01 TR3.02 Expectation determine, through investigation, the relationships between the angles and sides in acute triangles (e.g., the largest angle is opposite the longest side; the ratio of side lengths is equal to the ratio of the sines of the opposite angles), using dynamic geometry software; calculate the measures of sides and angles in acute triangles, using the sine law and cosine law; TR3.03 describe the conditions under which the sine law or the cosine law should be used in a problem; TR3.04 solve problems involving the measures of sides and angles in acute triangles; TR3.05 describe the application of trigonometry in science or industry. Nelson Mathematics 10 6.2 Similar-Triangle Models 6.3 Investigating the Sine Law 6.4 Proving and Using the Sine Law 6.6 Adjusting the Pythagorean Theorem: The Cosine Law 6.3 Investigating the Sine Law 6.4 Proving and Using the Sine Law 6.6 Adjusting the Pythagorean Theorem: The Cosine Law 6.7 Solving Measurement Problems Modelled by Triangles 6.4 Proving and Using the Sine Law 6.6 Adjusting the Pythagorean Theorem: The Cosine Law 6.7 Solving Measurement Problems Modelled by Triangles 6.3 Investigating the Sine Law 6.4 Proving and Using the Sine Law 6.6 Adjusting the Pythagorean Theorem: The Cosine Law 6.7 Solving Measurement Problems Modelled by Triangles Throughout Chapters 5 and 6