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 ...


38

"The whole numbers" is not a term that professional mathematicians use to describe a certain set of numbers. The term is used in elementary education when fractions are introduced, so that one can distinguish between numbers that have a fractional part and numbers that don't. In the US, this happens in grades 3 or 4. As far as I can tell from the ...


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 ...


25

I’m more curious about incorrect things in them. Yet, this is the first thing I found. There's absolutely nothing "incorrect" about this. As Dave L Renfro noted in a comment: and whole number is rarely used as a precise designation outside of school mathematics there is no agreed-upon rigorous definition of the term, and in fact it's largely ...


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.


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

This is "left involution". ("left" because it doesn't work when you try it on the right.) \begin{align*} x \circ y &= z & \\ x \circ (x\circ y) &= x \circ z & [\text{apply $x \circ -$}] \\ y &= x \circ z & [\text{simplify the involution}] \text{.} \end{align*} I would be shocked to see anyone use that term ...


19

I don't think that "textbooks" decided this, usage did. The term "integer" covers positive and negative, so it would be redundant for whole numbers to refer to that category. And there is an argument to be made for the term linguistically: a negative number is sort of the opposite of having a whole thing. But ultimately, there's not much ...


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 =...


11

I have never seen a name for this property specifically. When I was in grade school, I recall learning about Fact Families, which are generated by this property. The idea is that a fact family is all of the arithmetic equations generated by the same numbers. This property in particular is really just a consequence that subtraction is the inverse of addition ...


11

You seem to describe "whole numbers" in this American usage as describing $\mathbb {N}$, the set of natural numbers, whereas you expected it to describe $\mathbb{Z}$, the set of integers. As others have pointed out there is nothing "incorrect" about them, it's a language difference. Although it is worth knowing those language differences ...


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


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

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

As James mentioned, to fully appreciate the term "conjugate" here we need to know a little about Galois theory. The field $\mathbb{Q}[\sqrt{2}] =\{a+b\sqrt{2} \;|\; a,b \in \mathbb{Q}\}$ contains the field of rational numbers $\mathbb{Q}$. If we look for all automorphisms of $\mathbb{Q}[\sqrt{2}]$ that leave $\mathbb{Q}$ fixed pointwise, we get the ...


8

Calling both $a+ib \mapsto a-ib$ and $a+b \sqrt{2} \mapsto a-b \sqrt{2}$ conjugations is reasonable because of the general understanding given by Galois Theory (which has been known much longer than our lifetimes even if the educational establishment can't see fit to require it in our education). Using the same name is in part reasonable beyond the abstract ...


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

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

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 ...


6

Example: $(x+1)(x−1)=x^2+2x+1$ is an linear equation. I agree with all above answers. I wouldn't say that it is a linear equation, but it reduces to, or it is solved by or any other phrase you want to use instead. Because, I mean, that's the idea right? I feel that the relaxed terminology is not quite accurate, and it might generate confusion when they see ...


5

A helpful way to rewrite that statement would be (assuming subtraction for simplicity): $x - y - z ⇔ x - z - y$ We are observing how swapping y and z does not change the value of the expression. While it may initially look like there is a useful property behind this, the example is showing an easy case of what you are allowed to swap. Here is a visual ...


5

The main reason we have the word "critical point" is for the first derivative test. The statement is slightly easier if you only have to say "critical point" instead of "critical point or singular point". However, students might actually remember to check for singular points if they had a different word for this case. So I ...


5

The term Hergert Numbers is sometimes used in my specific region of the US for the values of $x$ where $f''(x) = 0$ or $f''(x)$ is undefined. This is in reference to Rodger Hergert, an Illinois community college professor who sometime in the 1990s became very frustrated with the fact that these numbers had no good name. But it doesn't really matter what you ...


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 ...


5

I think I would tend to disagree with your terminology's bent. The label of the equation speaks to the type of the expression which forms the equation. The type of solution set does not label the equation. Quadratic equations are not differently labeled when they have different solution sets. Furthermore, since there are literally infinitely many equivalent ...


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"...


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