Page 45 - Math 9-10 (2009)

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THE ONTARIO CURRICULUM, GRADES 9 AND 10: MATHEMATICS
Measurement and Geometry
Overall Expectations
By the end of this course, students will:
• determine,throughinvestigation,theoptimalvaluesofvariousmeasurementsofrectangles;
• solve problems involving the measurements of two-dimensional shapes and the volumes of three-dimensional figures;
• determine,throughinvestigationfacilitatedbydynamicgeometrysoftware,geometric properties and relationships involving two-dimensional shapes, and apply the results to solving problems.
Specific Expectations
Investigating the Optimal Values of Measurements of Rectangles
By the end of this course, students will:
– determine the maximum area of a rectan- gle with a given perimeter by constructing a variety of rectangles, using a variety of tools (e.g., geoboards, graph paper, tooth- picks, 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 pre- made dynamic geometry sketch), and by examining various values of the side lengths and the perimeter as the area stays constant;
– solve problems that require maximizing the area of a rectangle for a fixed perimeter or minimizing the perimeter of a rectangle for a fixed area (Sample problem: You have 100 m of fence to enclose a rectangular area to be used for a snow sculpture com- petition. One side of the area is bounded by the school, so the fence is required for only three sides of the rectangle. Deter- mine the dimensions of the maximum area that can be enclosed.).
Solving Problems Involving Perimeter, Area, and Volume
By the end of this course, students will:
– relate the geometric representation of
the Pythagorean theorem to the algebraic representation a2 + b2 = c2;
– 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) (Sample problem: A new park is in the shape of an isosceles trapezoid with a square attached to the shortest side. The side lengths of the trapezoidal section are 200 m,500 m,500 m,and 800 m,and the side length of the square section is 200 m. If the park is to be fully fenced and sodded, how much fencing and sod are required?);
– develop, through investigation (e.g., using concrete materials), the formulas for the volume of a pyramid, a cone, and a sphere (e.g., use three-dimensional figures to