Page 37 - 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,theoptimalvaluesofvariousmeasurements;
• solve problems involving the measurements of two-dimensional shapes and the surface areas and volumes of three-dimensional figures;
• verify,throughinvestigationfacilitatedbydynamicgeometrysoftware,geometricproperties and relationships involving two-dimensional shapes, and apply the results to solving problems.
Specific Expectations
Investigating the Optimal Values
of Measurements
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;
– identify, through investigation with a vari- ety of tools (e.g. concrete materials, com- puter 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 applica- tions (e.g., the minimum amount of pack- aging material; the relationship between surface area and heat loss);
– pose and solve problems involving maxi- mization and minimization of measure- ments 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) (Sample problem: Determine the dimensions of a square-based, open- topped prism with a volume of 24 cm3 and with the minimum surface area.).
Solving Problems Involving Perimeter, Area, Surface Area, and Volume
By the end of this course, students will:
– relate the geometric representation of the Pythagorean theorem and 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?);