The trouble is that in everyday life, we usually use the word "rectangle" to refer to non-squares, because if the shape were a square, we would call it that. Then I moved on to the other issue, which may or may not trouble Alisa: that classification uses words in a different way than everyday speech: It sounds less silly with squares and rectangles, but the idea is the same. It's almost silly to ask and answer those questions. Can a dog be an animal? Of course - every dog is an animal! Can an animal be a dog? Yes, if you look at a dog, that is an animal that is a dog. It may help to think about other categories, like dogs and animals. That's more than saying that a square CAN be a rectangle it IS.Īnd since squares are rectangles, you know that some rectangles are squares-namely, the squares are! So I started by clarifying the wording, and the idea of classification: Hi, Alisa.Īctually, EVERY square is a rectangle, since the angles in a square are always right angles. Here is a typical question, from 2004: Are Squares Rectangles? Are Rectangles Squares?Ĭan a rectangle be a square? I just was wondering because a square can be a rectangle, but can a rectangle be a square?Īlisa’s comments suggest that she is confused about the word “can”, as I said above, but has no trouble with the concept of squares being rectangles. At a much more advanced level, we would be talking about “universal and existential quantifiers” but even the basic language is tricky for the uninitiated. On the other hand, “Can an X be a Y?” means, “Is some X a Y?” (or, “Are there any X’s that are Y’s?”), and not “Is it possible that every X is a Y?”. I find that many students struggle with the very form of the question, as much as with the definitions it is asking about: Are we being asked if every square is a rectangle, or just if some particular square is a rectangle? A big part of the difficulty is that kids are not used to technical language, where the details of the wording are essential.They may need to be shown that, for example, “Is an X a Y?” means, “Is every X a Y?” (or, “Is an X always a Y?”), and not “Is some X a Y?”. They often have trouble with the very idea of classifying items by applying definitions, and also with the fact that definitions can vary, both between everyday and technical usages, and from one textbook to another. We get many questions about classifying shapes, from both elementary and high school students (or their parents or teachers).
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