Children's ideas about measuring area tend to develop in a predictable way. First we will review key ideas about measurement which were covered in the newsletter on length. These ideas also help children understand the measurement of area. Then we will describe strategies that children use to measure area. Finally, we will also discuss several common misconceptions that children have about this topic.
Key ideas about measurement of area.
If children are simply told to measure area in a unit like a square inch, they develop very little understanding of the big ideas of measurement. Children need the opportunity to build key ideas about measurement of area.
These ideas include:
1) Appropriate units - units for measuring area are not the same as those for measuring length.
2) Identical units - To say that the measure of a triangle is 14 square inches or 9 rectangles means that all the squares are the same (squares where each side is 1 inch long).
3) Completeness of cover - Units are set so as to compleely cover a region for area and then counted.
Strategies children use to measure area
A good way to start having children consider these big ideas is to ask them to compare the amount of space covered by two different figures. Comparison of irregularly shaped figures or of figures which differ in shape are problems which often raise such issues. To figure out which hand covers more space, children usually rely on simple strategies. For example, children often use the superposition strategy - placing one hand on the other and looking for "leftover" area. This approach relies on direct perception. It works very well for similar figures (figures of the same shape, like two rectangles), but it is not as satisfactory for shapes like hands because two hands are not always similar (e.g. one person may have fatter thumbs and shorter fingers).
A second strategy that children use to determine which figure has a larger area is to decompose the region covered by one figure and to try to fit the parts into the other. If the parts fit exactly, the two figures cover the same area. This strategy also works well for familiar shapes like the two rectangles displayed below, but it is often difficult to decompose a shape like a hand into units that can be easily perceived.
Through experience with strategies like superposition or decomposition, children develop ideas about measuring areas. Measuring area allows children to say which hand has the larger area, and also to know how much larger it is. Children often start systematically measuring with a direct modeling strategy.
They will use cut out squares (or rectangles or triangles) to fill a region and then count the number of squares (or rectangles or triangles). From here children progress to less perceptually dependent strategies, perhaps by visualizing a figure as being composed of rows and columns of unit squares (or some other appropriate unit of measure) and then by adding the rows.
As a final step some children invent rules for finding the area of familiar shapes, like length X width = area for squares or rectangles. They can then apply these rules with understanding.
Teaching children these rules before they have experiences leads to rote measurement with little understanding of why the rule works. For example, fifth grade children sometimes tell us that the area of a 3 by 3 square is 9 square inches. They then tell us that the area of this right triangle is 14 square inches 3X3 = 9 + 4 for the diagonal.
Here children are simply applying rote formulas without understanding that the
area of the square cannot be less than the area of the triangle. As children
build their ideas about measurement of area, they often entertain a variety of
ideas that capture some of the key ideas of measurement but violate others.
Nevertheless it is important for them to try out their ideas so that they see
why some won't work.
1. Everything is length.
Children often believe that they can use rulers to measure area. Consequently, they often measure the perimeter (the path around the figure).
|Another mistake is to measure one side of the figure, move the ruler over "a little bit", take another length measure, move the ruler over a little bit, take another length measure, doing this until they believe it is time to "add up all the numbers."|
Both of these approaches are attempts to apply the single dimension of length to area which is two dimensional. These approaches violate the first idea of measurement we noted before (Appropriate units).
2. Units can be different.
Children often believe that it doesn't matter if the units are all identical. They believe that if they can fill a region (like a hand) with units of measure (like beans), then it doesn't matter if some of the units of measure (the beans) are of a different size - children simply count the number of objects contained within the region (the hand). This approach violates the second idea of measurement we noted before (Identical units).
3. Cover need not be complete.
Children often believe that although the units of measure should be identical, it doesn't matter if they don't completely cover a region. For example, nine beans of the same size are used to cover the square depicted to the right. Just as long as the beans do not "spill over" or otherwise violate the boundaries of the figure, some children will report the area of the square as 9 beans. This violates the third idea of measurement noted before (Completeness of cover).