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 ...
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 ...
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.
(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). ...
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.
Other examples with licensing such that I ...
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 ...
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 ...
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 $\...
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 ...
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) ...
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.)
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 ...
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.
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 ...
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 ...
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 ...
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.
A corkscrew (for a helix):
A donut (for a torus):
A football (for a spheroid)
And then, there's also the atomium (for which I am not sure exists a geometric name)
cooling towers (for a hyperboloid)
and the pentagon (well, for a pentagon):
A pyramid is, of course, a pyramid.
Lastly, a soccer ball is a truncated icosahedron
(Images by wiki, pedia)
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.
An article in ...
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.
I'd say different proofs usually employ different techniques, which in turn might be applicable to different sets of other theorems. So the more proofs I know for one theorem, the higher the chances that I'll be able to adapt at least one of them to a similar (or maybe not so similar) theorem I'm trying to prove.
Furthermore, seeing several techniques ...
One angle you could look at is molecular geometry. Not really my subject area but a couple of examples:
Organic molecules can have different chiralities. That means that while one is the mirror image of another you cant rotate one molecule to the other. The reasons for this are pretty deep mathematically, but chemically give rise to interesting things as ...
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 ...
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 ...