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Colloquially, there's a lot of conceptual overlap between all of these terms, but "sameness" is not a well-defined mathematical property. Congruent shapes need not be "the same" or "equal" in all respects - they can be rotated differently, or be in different positions, or be different colors, or have different names, or differ ...

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I'm confused. Are you really going to try to make this sort of distinction when teaching geometric figures to students "around 9-13 years old"? Students that age (and engineers my age -- much, much older) think that a triangle is a triangle. It's a polygon formed by three non-colinear points. A triangle has many ways you can think about it. ...

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The smart-aleck answer is that most congruent triangles, or congruent figures more generally, aren't actually "the same" or "equal". Usually when we say two things are "the same", we mean that they are not just indistinguishable, but that they are literally the same exact thing. "Equality" means two numbers are the ...

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One encounters exactly the same issue teaching multivariable calculus when one treats integrals over three-dimensional regions and integrals over the surfaces that are their boundaries. In particular the word sphere is particularly confusing in this context. (Mathematicians use sphere to mean the the two-dimensional surface; colloquial speech and some ...

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I think the distinction you are raising is not natural to students at this age. I teach undergraduates and graduate students, not elementary schoolers, but I find that it is not natural for undergraduates who have not had a theoretical math course. In my experience, students do not naturally think of geometric figures as sets of points. If $P = (-1,-1)$, $Q =... 8 I confess, when you start talking about 1-D triangles, my own first thought is "how can you have non-colinear points in 1-D?". So, I imagine most students that age will have a far more difficult time with that. Keep in mind age appropriateness. For 9-12 year old children, you are generally looking at a level of psychological development ... 7 Many of the geometric figures are so elementary that they are deeply rooted in daily language, and there seems to be no great solution. I agree with you here, and I think this is the key point. To me they are clearly well-defined: "Triangle", "square", and polygons in general, are bounded regions on the Euclidean plane, i.e., 2D figures. ... 7 Educator here who has worked with many students in the aforementioned age range (9-13) on triangles and squares. In my experience, it has never come up that a confusion between the boundary and the interior of a plane region was relevant to problem solving at that grade level. For these types of elementary shapes, the boundary and the interior completely ... 6 Historically (and by historically I mean "in Euclid's Elements") the word "equal", when applied to geometric figures, meant "equal in magnitude". So for example: Euclid refers to two segments as equal if they have the same length Two triangles are equal if they have the same area Two solids are equal if they have the same ... 5 I agree with the point that a lot of the answers here are making - some distinctions, while correct and important, are not accessible to the age group you're talking about - I also want to point out an important benefit of not making the distinction for them. While it is crucial in higher mathematics to be able to be extremely precise, it's also important to ... 4 Programmers consider the naming of things to be one of the three leading problems in our field. For the cases you describe, we do already have a well-established and widely-used set of terms that even non-computer users should recognize and be familiar with. A circle can be called a solid or filled circle, contrasted with wireframe or outlined circle. ... 4 A triangle is born from three non-collinear points and the axiom that two points determine a line. In the context of neutral geometry, a triangle has no structure other than three lines and three points. In particular, there is no notion of the interior of a triangle without more axioms. In the real projective plane, one cannot define the "interior"... 4 Well, to start, "usually enough" is ambiguous enough to allow a bit of wiggle room. That said, there are plenty of times where knowing how dimensions 1, 2, and 3 act to start to see the special cases break down and see the general pattern. For example, Jordan is hinting at vector spaces in the cited section. Dimension 1 is a special case where ... 3 The problem is that there are different kinds of 'samenesses' (equivalence relations). One of these is congruency, but another is similarity. If you don't teach your students the word congruent, and use the word same instead, what word are you going to use when introducing similarity, since similar triangles can also be thought of as being the same, but in a ... 2 I would like to add a few of my favourites: Apollonian constructions (construct a circle tangent to three objects, where the objects can be points, lines or circles). There are 10 of these problems, some are standard, some are tricky. Construct the golden ratio (this is related to the regular pentagon mentioned above). Solve a quadratic equation using ... 2 Assuming that by "Geometric constructions", you mean "ruler and compass constructions", here is a sequence of progressively-more-difficult constructions that begin at the trivial and end at the impossible: The "standard" constructions: Perpendicular bisector of a line segment. Angle bisector. Perpendicular to a line through a ... 2 Yes, it is always good when students independently discover mathematical facts, in contrast to being told that some old Greek dude figured it all out 2,500 years ago. It makes students feel useful and participatory, it prepares them for a future where they will be investigating phenomena where formulas aren't as well established, and the experience forms ... 2 I agree with the point in the other answers, that the difference is in the approach, how rigorous we would like to be. However, I would like to also mention that this rigour in discussing elementary geometry is not extended to some other parts of teaching mathematics in high school. I do not think that too many high school teachers discuss vectors in a ... 2 Part of learning to do higher dimensional geometry is learning which aspects of your low dimensional intuition are good or not. I can't visualize two smooth$2$-planes meeting transversely at a point, which I know happens in$\mathbb{R}^4\$. But I am experienced enough to know that, when thinking about dimensions and transversality, I need to compute, because ...

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The question of when and why less guidance works or doesn't work is far from obvious. Instead of providing my personal opinion, as many have already done in other answers, I will try to collect some pointers to research that has already tried to answer this question, which you can use to critically review your own approach. Andrew Blair in his article ...

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Over all I think the idea, of hands on for something like C/D is an invariant is laudable, but there are a few caveats. First, students at that age are very aware of precision. The answer "the ratio is 3 and a bit" is fine, but they will not accept it is a constant if one group measures 3.1, the second group 3.2, a third group 3.15. Indeed, in that ...

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A good think is to students pick up several rounded objects (wheels, cans, glasses, etc.) and mesure with rule the length and the diameter and simply divide. And observe if something happens. You can see this idea here (in Catalan - automatic English translation). It is original from Anton Aubanell. Just a note: for measuring the length I think the better is ...

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I can tell about how it went in Russia, presumably, that might have gone the same way in other countries as well. Traditionally, congruent shapes were called "equal". Then, in the 1960's, it was felt that some set-theoretic notions should be introduced into the school geometry, a charge led by leading mathematicians such as Kolmogorov. Since there'...

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I don't really know what's the big problem. Terms need not be ambiguous; it's up to you to define and use them in clear and unambiguous manner. For example you can use "triangular region" or "cylindrical volume" to clearly differentiate from "triangle" and "cylindrical surface", and of course you have to define whether ...

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