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How can we explain to students these ideas?

  1. A square with 4 sides measuring 25 cm each does not have an area of 1 square meter.
  2. A shape which is not a square can have an area of 1 square meter.

Is “square meter” a badly-phrased term? What would be a more appropriate term?

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    $\begingroup$ No. The concept of a "square meter" is about the area inside the boundary of a shape. A shape does not have to be a square for it to have an area of 1 square meter. [Many different shapes could have an area of 1 square meter.] When you mention a square shape whose side lengths are each 25 cm, you are describing a square that has a perimeter of 1 meter. This is the distance around the boundary of the shape. Summary: We should use "square meters" to describe the area inside a shape, not the distance around the shape. $\endgroup$
    – Nick C
    Aug 11, 2023 at 15:56
  • $\begingroup$ I think it does for the definition of the unit of area 1m^2 ,than it is the the area is defined by 1m^2 is the area of a square with the sides 1m. and of ciuse you can then also say the area of 16 squraes of length 25cm $\endgroup$
    – trula
    Aug 11, 2023 at 17:19
  • $\begingroup$ I think it is a question of mah education. and should not be closed. Since the first answer just assumes that something like area is well defined, but in math it has to be defined, How else if you come to integrals? $\endgroup$
    – trula
    Aug 11, 2023 at 18:51
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    $\begingroup$ Hello all, I don't understand why is all the downvotes and the close, all I want is to understand how to access the idea of a "squared meter" for a pupil --- which is I myself... I just seek a primal, simple, intuitive explanation of the term which can be understood by a child. $\endgroup$ Aug 12, 2023 at 12:48
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    $\begingroup$ If the term "squared meter" doesn't necessarily have to do with squares, but also with circles, triangles, etc (as I understand from the first comment by Nick C), isn't that then quite of a badly-phrased term or a confusing term and a longer more precise term is needed?... What would be a longer, more precise term?... $\endgroup$ Aug 12, 2023 at 12:52

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Peter answered how to explain the ideas to students. So I'll answer your second question:

Is "square meter" a badly-phrased term? What would be a more appropriate term?

It's not badly-phrased, but I see what you're getting at: "square meter" is in reference to a particular shape, whereas something like "area meter" would sound more general to other shapes.

However, the problem with a phrase like "area meter" that is intended to be general to other shapes, is that not knowing what specific shape you're talking about actually creates more confusion. Say I ask you to shade in 1 "area meter". What do you do? Do you draw a triangle with sides of length 1 and then shade it in? Or do you do that with a square? A pentagon? A circle with radius 1? The nice thing about "square meter" is that its name tells you exactly how to measure it.

More abstractly, you can view units of area as being a combination of two independent units: "square" tells you the shape, and "meter" tells you the side length. In that view, "square" is the conventional unit for shape, but you could talk about a thing like "1 triangle inch". But using "square" usually makes things way easier for our brains, just like using a decimal number system usually makes things way easier than working in binary.

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    $\begingroup$ Great explanation. Squares may serve as scaffolding on the way to define square units, but after which the latter is typically used in reference to area (area being the desired property abstracted out of the construction and transferred to use in other contexts, and shape the property abstracted away and left behind). $\endgroup$ Nov 18, 2023 at 16:44
  • $\begingroup$ Regarding your point about the square being the conventional reference shape: we still use the word "quadrature" for area finding, relating to the practice that goes back at least as far as Euclid's Elements of transforming complex shapes into squares of the same area. C.f. squaring the circle. $\endgroup$ Nov 27, 2023 at 20:35
  • $\begingroup$ Added: it actually goes back at least to Hippocrates of Chios and to the Sulba Sutras, which are likely somewhat older. $\endgroup$ Nov 27, 2023 at 20:40
  • $\begingroup$ If you ask me to draw 42 square meters, I wont bother calculating the root of 42, but draw a 1m by 42m rectangle. So the square is just a special case IMO. $\endgroup$
    – corvus_192
    Feb 11 at 17:16
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For the first, give them a shape that is 1 meter on a side. Then let them try to fill it with four shapes that are each .25 m on a side.

For the second, get several other shapes that are 1 square meter but not square. Then cut them up so they fit in the first shape.

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    $\begingroup$ In Germany you can use the translation of square meter, or you call it 1 meter squared, wich for example is better if you have a rectangle 3m by 4m its area being 3m*4mor 12m*m=12m^2 $\endgroup$
    – trula
    Nov 17, 2023 at 21:06

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