Students in Ms. Lyons's class studied surface area, first by exploring 4 rectangular prisms with which they are very familiar: the one, ten, hundred and thousand blocks. They thought about surface area as the skin that covers any geometric solid.
Students see that the area of each face is 1 sq. cm. Because there are six faces, the surface area of the entire cube is 6 sq. cm. |
4 faces have the same area: 10 sq. cm. 2 other faces have the same area: 1 sq. cm. The surface area of the entire rectangular prism is:
10 + 10 + 10 + 10 + 1 + 1 = 42 sq. cm.
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Four faces have the same area (10 sq. cm) and 2 other faces have the same area (100 sq. cm). Students can practice using parentheses to express the surface area:
(4 x 10) + (2 x 100) = 240 sq. cm
OR
4(1 x 10) + 2(10 x 10) = 240 sq. cm. |
Because the thousand block is a cube, all 6 faces have the same area: 100 sq. cm. The surface area of the cube is:
6(10 x 10) = 600 sq. cm
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Next, students designed nets for geometric solids. A net is a flat shape that can be folded up into a geometric solid. The class focused on designing nets for different cubes.
This student first designed the net for a cube, numbering the 6 faces. She then made another net, and constructed the cube. This particular cube has faces that are 6 x 6. She determines the surface area by calculating:
6(6 x 6) = 216 sq. cm. |
This sample shows 5 different nets that were designed. The student then constructed a cube with dimensions 7 x 7 x 7. The surface area she worked out is 294 sq. cm. |
Here are cubes of different sizes constructed by students. Each has a different surface area.
Dimensions |
Surface Area |
2 x 2 x 2 |
24 sq. cm |
4 x 4 x 4 |
96 sq. cm |
6 x 6 x 6 |
216 sq. cm |
7 x 7 x 7 |
296 sq. cm |
8 x 8 x 8 |
384 sq. cm |
10 x 10 x 10 |
600 sq. cm |
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Students in Ms. Lyons's class also used isometic grid paper to create 2-dimensional drawings of rectangular prisms. This student created a rectangular prism with dimensions of 6 x 3 x 4 cm. He calculated the surface area by grouping the areas of the faces that were congruent:
4(3 x 6) + 2 (4 x 3) = 96 sq cm
The student can generalize the expression to find the surface area as:
4(l x w) + 2 (h x w)
The volume is l x w x h OR
6 x 3 x 4 = 72 cm cubed
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Indicators:
3.6.3.1 develop and use formulas, using related formulas and models, to determine areas of polygons such as triangles, parallelograms, trapezoids, and circles.
2.6.3.3 make a model of a threedimensional figure from a two-dimensional drawing.
2.6.3.4 make a two-dimensional drawing of a three-dimensional figure.
3.7.3.2 use formulas to find thesurface area and volume of basic three-dimensional figures, including prisms and cylinders.
3.7.3.1 use models to find and derive a formula for surface area and volume of prisms and cylinders.
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