Challenge: The Meaning of Area

Shape

Perimeter & Area of Polygons--Challenge: The Meaning of Area

Materials needed: Shape Worksheet; 10mm, 5 mm, and 2mm Grid transparencies

Technologies: The Geometers Sketchpad and model 3_Polygons.gsp

Many children are introduced to the concept of area using arrays similar to that shown in Figure 3.15. In arrays of this sort, the area of the rectangle may be obtained by counting the number of unit cells. Alternatively, the area may be represented in groups of cells, such as “two groups of three equals six.” Introducing multiplication as repeated addition, this representation eventually becomes “two times three equals six.” A generalization of this procedure may then be stated as “Area equals length times width” for rectangles. While this approach is a powerful tool for teaching the concept of multiplication, it leaves many children with the impression that only rectangles “have area.”

Figure 3.15: Area as Array of Unit Cells

Use the grid transparencies to estimate the area each object on the Shapes worksheet. Superimposing each transparency over the Shapes worksheet, count the number of grid cells completely inside each object. Record this data in Table 3.7 and convert each cell count to an area estimate. When the table is complete, use The Geometers Sketchpad and model 3_Polygons.gsp to measure the actual area of each object.

Figure	# of 10mm Grids Area Estimate	# of 5 mm Grids Area Estimate	# of 2 mm Grids Area Estimate	Actual Area
A
B
C
D
E
F
G
H

Table 3.7: Area Estimates

· Which grid transparency produced the best estimate of the area? Why?

· How could you use a grid to produce a better estimate?

· If you changed the procedure to include grid cells that are partially inside the object, how would that change your estimates?

· How could you use this approach to convince a child that all polygons “have area”?