50

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 ...


42

Because the aim of geometry is to study properties related to shape, size and length. Therefore, in the context of geometry, we cannot deform our objects because deformation changes these properties. On the other hand, the properties in which graph theory is interested are independent of size, shape and length. Therefore, we do not care about them. (However, ...


37

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. ...


28

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 ...


24

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 ...


22

Generally, this math falls under the scope of what is commonly called Taxicab Geometry. I would use taxicab path as a noun to describe the specific paths illustrated in the original question; whereas taxicab geometry would be a term I'd use for the subset of mathematics covering these types of scenarios.


22

As an answer so that I can paste in a picture: I think that the problem with the one on the left is that it is possibly "good enough" that someone might think that it is to scale and that apparent regularities are real. (Well, maybe not one of those "right angles".) In particular, someone might become under the impression that the length ...


21

I would use this to help students understand three "meta" ideas: (1) Math is not about memorizing lots of random trivia. In the real world, if you go up to a mathematician and ask them which definition of a trapezoid is right, they will just smile indulgently. They don't know or care. (2) There is not always a consensus about definitions. Get over ...


19

There is no value in drawing the figure exactly to scale, but the left-hand figure is inaccurate to the point where it is positively misleading. Since the angle marked 30 degrees is actually drawn greater than 45 degrees, it gives the impression that a is less than 3 (or maybe equal to it, if the student takes the "30 degree" angle as being shown ...


19

Common knowledge The formula $a^2+b^2 = c^2$ is common knowledge and the words for hypotenuse and leg (is "cathetus" not used in English?) are basic mathematical vocabulary. Including these seems a good idea. Connections to other mathematics The notation with AB, CA and BC might be something the students have used or will use in less analytical ...


17

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 =...


12

I really like the idea of a "discovery fiction" -- it gives a name to something I often try to use when teaching. Here is one suggestion. I will try to come back and write a more elaborated version of this answer later, with diagrams and proper notation, but briefly: (a) Don't let on that you are going to prove the Pythagorean Theorem -- don't ...


11

I would split the difference by creating an accurate diagram in Desmos and then giving the students a hand drawing based on that diagram. That way, students could estimate their answer before solving and check their answer against that original estimate. But they couldn't conclude that $b=6$ by measuring the length of the known side. Pedagogically, the two ...


10

In the field of micro/nano-lithography, such geometry would be called Manhattan geometry; containing only two directions of edges orthogonal to each other. If diagonals were included, it would be classified as having skew edges. Finally, if it were truly freeform, it would be called curvilinear. Example: https://doi.org/10.1117/12.2243030


10

In Olympiad geometry, $a$, $b$, $c$ is the so-called standard notation for the sides of a triangle, so it makes sense to use it consistently when referring to a triangle (in isolation). However, in any case the general principle is introducing all your notation. Writing Pythagoras' theorem states that $a^2+b^2=c^2$. or Pythagoras' theorem states that $...


8

These are pretty much universally called lattice paths, which Will Orrick comments on the question. A quick search on the arXiv reveals a large number of papers from combinatorics and computer science using the term in this way. (The main competition is from papers about path integrals on lattices, which gets unfortunately concatenated to "lattice path ...


8

Unfortunately, we don't have a set of universally agreed upon definitions in mathematics. It might seem like we do (or should), especially in Geometry with its long history and so much agreement, but the truth is that we use different definitions frequently. One proof of this is the differing definitions in your textbooks! That is just the nature of a ...


8

I recommend either Excursions in Geometry by Ogilvy or Geometry Revisited by Coxeter and Greitzer. Both are cheap too.


8

Two alternatives I have seen used (am not necessarily recommending them, but will list some pros and cons) which don't seem to have been mentioned yet. Don't denote it algebraically at all! Draw a picture instead For the lower end of the 12-16 age range, I've seen this work really well. You literally draw the squares sticking out from the triangle. Write the ...


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

A mathematics teacher once told me, "at a ratio of 4:5, a rectangle becomes a square." While this is obviously not true in a strict sense, it holds the deeper truth that a rectangle at 4:5 or worse is not a good example to represent a generic rectangle to young students, unless the goal is to teach or test that squares are also rectangles. That ...


7

Note: as in the originally referenced article, I'll take $a=1$ to keep it less complicated. I think it's important to first establish two key issues about the graphical setting: What does it mean to "be a square" in a graph. What information are we assuming we can "read" from the graph. For (1), in the geometric setting this notion is ...


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

This paper calls a horizontal/vertical path a rook path, for the movement of a rook on a chess board. If the paths connect the lower-left corner to the upper-right corner, this would be a North-East Lattice Path. Edit: If the path didn't need to go through points with integer coordinates, I would submit the name "Cardinal Paths" in reference to the ...


6

This is a great question. Love it love it love it!! The following is just what occurs to me off the top of my head. Show a graph paper grid with a dot at the origin and a dot at (3,4). Say we want to find the distance between the dots. We could guess that it would be 3+4=7. Well, that would be right if these were city blocks, but it's not the right answer as ...


6

Since your job is helping the teacher with technology and the teacher should be making the pedagogical decisions, ask her her preference and do it that way. Maybe she meant for the pictures not to be to scale, or maybe they came out that way due to her poor technology skills. Just ask.


6

Finland is a small country with a centralized political system and a university system that is essentially public and free, and highly centralized on a couple of big schools. The US is a big country with a federal system of government and a higher education system that is a mix of public and private, spread across an entire continent. In the US, the federal ...


6

The only one of these that looks objectionable to me is the one that calls the hypotenuse $h$, since in a triangle the letter $h$ usually refers to the triangle's height (which could be either one of the legs but could not be the hypotenuse).


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 ...


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