65

This is a bit of a can of worms. Let's unpack a little. We were given no additional information other than that stated in the page. This is your daughter's homework, not yours. Be careful with this distinction, because coloring a child's question with adult interpretations can lead to trouble (and does in this case). it is not stated whether we are ...


54

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 ...


43

The point of compass-and-straightedge constructions is for the students to get experience reasoning about axiomatic systems. The accuracy of the actual drawings is basically irrelevant (as long as it's not so sloppy that it impedes visualization) — the point is that they can prove the accuracy of the idealized constructions. This method of axiomatic ...


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, ...


38

(Disclaimer: I peronally really don't care if one uses $\tau$ or $\pi$, both are just numbers for me.) But I would strongly recommend to use $\pi$. Why? In every technical literature, in many popular literature, the people always use $\pi$ (even worse: $\tau$ is used for different things than $\tau=2\pi$ which would confuse when reading those literature). ...


38

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. ...


37

Trapezoid Native Peruvian architecture makes heavy use of the trapezoid for stability in earthquakes. (The Spaniards thought they were primitive as they didn't use arches ... but most of the Spanish buildings have collapsed or had to be rebuilt). It's especially apparent in their doorways and windows. (hi res) Other examples with licensing such that I ...


34

I cannot answer the OP's question about cross-cultural/international perspectives, but here is a historical perspective that may be helpful. The issue here (whether the category "rectangles" includes or excludes the category "squares") is one aspect of a larger question having to do with whether the classification of quadrilaterals ...


34

Given that I had a good time coercing Mathematica to give me the pictures below I might as well promote my comment to an answer. My favorite such a result is the so called Steiner's porism. It is a statement about the relation of two circles, one inside the other. Such as these two: We get a so called Steiner chain of circles from this by first drawing a ...


31

I feel that it is perhaps a little irresponsible to teach $\tau$ instead of $\pi$. As a first introduction, it is the norm which should be taught: teaching a rare alternative to $\pi$ only serves to confuse students, especially when almost all available resources use $\pi$ instead of $\tau$. Imagine a student's confusion when they see $\tau$ in class and $\...


30

The National Library of Belarus, a rhombicuboctahedron:                    


30

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 like the Gateway Arch in St. Louis as an example of a catenary with a formula of the form $y= A \cosh(\frac{C X}{L}) -A$. More information on the wiki: Gateway Arch: Mathematical Elements.


25

Dice You get all Platonic solids, some trapecohedrons and bipyramids, and the tetrahexahedron and the rhombic triacontahedron:


24

I'd like to tackle the question from another point of view than JPBurkes answer: If you accept, that mathematical argumentation (whatever level) is an essential part of mathematics courses in K-12, than Euclidean Geometry is a great way to implement this: Visuality Euclidean Geometry deals with objects that can be easily visualized. It can be properly ...


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 ...


23

One should teach $\pi$. One might discuss that there is a choice that is made that is somewhat arbitrary, and there are also reasons for a different choice but I do not see this as that relevant to make much ado about this. It should perhaps also be noticed that there are two conflicting proposals for $\tau$. Eagle (1958) proposed $\pi/2$ and Palais (2001) ...


22

Here is a simple construction. Adjust to taste. 1. Draw a line $l$ passing through a point O. 2. Construct circles of radius 3 and 4 with centre O. Call them $C_3$ and $C_4$. Let the intersection of $C_4$ and $l$ be A. (Note that OA is 4 units.) 3. Construct an interval of length 5 from A to $C_3$. Call it B. (Note that OB is 3 units, and AB is 5 units.) ...


22

The Chinese came up with the following a long time ago. Probably something better, but this is the gist of it. Let's start with a right triangle with height b=4 and base a=3. We know it has some hypotenuse, c, but we don't know it's length because that's what we're trying to prove. Now we're going to make 3 copies, and rotate them each 90°. Next we're ...


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

There is a simple solution to this debate:


21

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 =...


20

TL;DR: It's not the triangles that are interesting; it's the mathematical concepts that can best be explained by using one of the most primitive geometrical shapes. The reason for intensive use of triangles goes beyond knowledge about triangles per se. It's the act of and steps in proving a theorem that's important to learn at this stage - start with ...


20

There's a fair attempt at a Hypercube with the Grande Arche de la Défense in Paris.


20

The hexagon at the north pole of Saturn:   It is known that "[regular shapes] form in an area of turbulent flow between ... two different rotating fluid bodies with dissimilar speeds." and this has been proposed as an explanation for the phenomenon. Incidentally, the Earth could easily fit inside the pole hexagon. Added (23Sep15). An article in ...


20

The fact that there is a 3-4-5 triangle that is a right triangle is unique to the Euclidean plane. There is no such triangle in the spherical or hyperbolic planes. Since the Pythagorean theorem is equivalent to the parallel postulate, any proof that a 3-4-5 triangle is a right triangle will somehow depend on the Pythagorean theorem/parallel postulate.


19

The best explanation I know comes from this answer by Emanuele Paolini (the only thing I did was to redo the pictures, please go and upvote Emanuele's post). The point is that squares in the usual picture doesn't have to be squares. The only thing is that we need to use the area (so that is scales with a square of the scale), it could have been pentagons. ...


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