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10.3 TeachingIf students move rapidly to formulas without adequate conceptual foundation in area and volume, many could have underlying confusions . . . For example, some students may hold the misconception that if the volume of a three-dimensional shape is known, then its surface area can be determined. NCTM Standards 2000, page 242Write an activity for school students to help dispel the misconception that the volume of a figure determines its surface area. Design this activity for students to use centimeter cubes to form three-dimensional figures, determine their volumes, and determine their surface areas. Give a few examples and explain how your activity and examples help to resolve the issue noted in the preceding statement from the Standards. 10.3 Teaching Whenever possible students should develop formulas and procedures meaningfully through investigation rather than memorize them. Even formulas that are difficult to justify rigorously in the middle grades should be treated in ways that help students develop an intuitive sense of their reasonableness. NCTM Standards 2000, page 244Design activities for school students which involve pouring, filling, submerging, etc., model and concrete objects to illustrate the following formulas. Volume of a rectangular prism (box) = (length)(width)(height) Volume of a pyramid = 1/3(area of the base)(height of pyramid) Volume of cone = (area of base)(height of cone) 10.3 Concepts Archimedes discovered that the volume of a sphere is 2/3 the volume of the circumscribed cylinder and that the surface area of a sphere is 2/3 the surface area of the circumscribed cylinder. Explain ways to demonstrate each of these relationships by using physical materials and a model of a sphere and its circumscribed cylinder. Carry out your demonstration to approximate both of these two-thirds relationships and discuss any difficulties you encountered. |