# Parallelogram

## Meaning

### Informal

A four-sided shape whose opposite sides are parallel. Because the definition says nothing about angles or side lengths, "generic" parallelograms are often pictured as long and slanty. But any four-sided figure with parallel sides is a parallelogram. Special cases include rectangles (which also have right angles), rhombuses (whose sides are all the same length), and squares (with same-length sides and right angles).

## Emerging conceptions, misconceptions

Perhaps because examples of parallelograms are so often the generic (long-and-slanty) case, drawn with the longer side horizontal, beginning students often have difficulties recognizing other orientations or special cases (like rectangles) as parallelograms. So, for example, the figures at the right are all parallelograms, but not all readily recognized as parallelograms. The first two may be rejected "because they're not slanty" (but "slantiness" is not required in the definition); the next two may be rejected just because they are not in the familiar position; and the last may be an ambiguous case for some students, rejected because its rectangular shape does not conform to their concept image or accepted, but for the wrong reason (because "it is slanted"). (See also article on examples.)

The request to identify parallelograms in a set of figures and the request to draw a parallelogram call for different responses. When we ask people to identify parallelograms, we expect them to include special as well as generic cases. When we ask people to draw a parallelogram, we tend to expect the generic case.

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We especially seek research on student difficulties, confusions, or emerging conceptions.

## Etymology

The -gram in parallelogram means something drawn (as in diagram), or something written (as in telegram). The word parallelogram suggests a drawing with parallel lines. The formal definition contains other restrictions (four sides, exactly two sets of parallels).