Page 38 - Math 9-10 (2009)

PRINCIPLES OF MATHEMATICS, GRADE 9, ACADEMIC (MPM1D)
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– 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 show that the volume of a pyramid
[or cone] is the volume of a prism
[or cylinder] with the same base and height, and therefore that
Vpyramid = or
– determine, through investigation, the relationship for calculating the surface area of a pyramid (e.g., use the net of a square- based pyramid to determine that the surface area is the area of the square base plus the areas of the four congruent triangles);
– solve problems involving the surface areas and volumes of prisms, pyramids, cylinders, cones, and spheres, including composite figures (Sample problem: Break-bit Cereal is sold in a single-serving size, in a box in the shape of a rectangular prism of dimen- sions 5 cm by 4 cm by 10 cm.The manu- facturer also sells the cereal in a larger size, in a box with dimensions double those of the smaller box. Compare the surface areas and the volumes of the two boxes, and explain the implications of your answers.).
Investigating and Applying Geometric Relationships
By the end of this course, students will:
– determine, through investigation using a variety of tools (e.g., dynamic geometry software, concrete materials), and describe the properties and relationships of the interior and exterior angles of triangles, quadrilaterals, and other polygons, and
–
apply the results to problems involving the angles of polygons (Sample problem: With the assistance of dynamic geometry soft- ware, determine the relationship between the sum of the interior angles of a poly- gon and the number of sides. Use your conclusion to determine the sum of the interior angles of a 20-sided polygon.);
determine, through investigation using a variety of tools (e.g., dynamic geometry software, paper folding), and describe some properties of polygons (e.g., the fig- ure that results from joining the midpoints of the sides of a quadrilateral is a parallelo- gram; the diagonals of a rectangle bisect each other; the line segment joining the midpoints of two sides of a triangle is half the length of the third side), and apply the results in problem solving (e.g., given the width of the base of an A-frame tree house, determine the length of a horizon- tal support beam that is attached half way up the sloping sides);
pose questions about geometric relation- ships, investigate them, and present their findings, using a variety of mathematical forms (e.g., written explanations, dia- grams, dynamic sketches, formulas, tables) (Sample problem: How many diagonals can be drawn from one vertex of a 20-sided polygon? How can I find out without counting them?);
illustrate a statement about a geometric property by demonstrating the statement with multiple examples, or deny the state- ment on the basis of a counter-example, with or without the use of dynamic geometry software (Sample problem: Confirm or deny the following statement: If a quadrilateral has perpendicular diago- nals, then it is a square.).
Vpyramid =
Vprism
1 3
3
(area of base)(height)
);
3
–
–