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66 votes

Is this homework problem on counting triangles within a 4x4 grid too vague?

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 ...
NiloCK's user avatar
  • 5,020
57 votes

Why do we introduce the notion that triangles are "congruent" instead of just saying that they are "the same" or "equal"?

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 ...
Nuclear Hoagie's user avatar
46 votes

Why should kids learn how to use a compass and straightedge, and not rely on a drawing program?

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'...
Daniel Hast's user avatar
  • 4,893
42 votes
Accepted

Why we have to be so precise in Geometry?

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. ...
Pedro's user avatar
  • 1,800
41 votes

What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

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 -- ...
Flydog57's user avatar
  • 595
39 votes
Accepted

Beautiful planar geometry theorems not encountered in high school

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 ...
Jyrki Lahtonen's user avatar
36 votes

In what curricula are "rectangles" defined so as to exclude squares?

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"...
mweiss's user avatar
  • 17.4k
30 votes

Why do we introduce the notion that triangles are "congruent" instead of just saying that they are "the same" or "equal"?

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 ...
Rivers McForge's user avatar
30 votes
Accepted

Explaining why volume of cone is a third of cylinder

This is an experiment which can lead you to guess that the volume of a cone is approximately $\frac{1}{3}$ the volume of a cylinder with the same base and height. It is not a proof in any sense of ...
Steven Gubkin's user avatar
27 votes

How to properly define volume for beginner calculus students?

It depends somewhat on the style of the course, but the majority of calculus students do not need a formal definition of volume or area, in my experience. They have studied geometry and (usually) ...
user22788's user avatar
  • 854
26 votes

What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

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 ...
Dan Fox's user avatar
  • 5,869
25 votes
Accepted

What are some common errors and misconceptions about the Pythagorean Theorem?

Here's a list of mistakes that I've seen students make. Conceptual Mistakes Applying the Pythagorean Theorem on non-right triangles. (They may also think that the word "hypotenuse" means ...
Justin Skycak's user avatar
23 votes

What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

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 ...
David E Speyer's user avatar
23 votes

What benefit is there to obfuscate the geometry with algebra?

Tests seek to measure ability. Math ability, like most other forms of ability (including athletic ability), isn't solely dependent on one's ability to execute individual skills in isolation -- it also ...
Justin Skycak's user avatar
22 votes
Accepted

Is there a name for paths that follow gridlines?

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 ...
Andrew Sanfratello's user avatar
22 votes
Accepted

Should figures be presented to scale?

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 ...
Adam's user avatar
  • 5,733
21 votes

Are kindergartners supposed to be steered from squares being rectangles?

Kindergartners are generally at an early stage of geometric development, in which shapes are recognized by how well they resemble prototypical images, rather than by whether or not they conform to a ...
mweiss's user avatar
  • 17.4k
21 votes
Accepted

How do I sketch a good gaussian curve freehanded, or by using only common sketching tools?

I would put dots where I want 1 standard deviation to be, because I know that's where the inflection points are. (I just graphed $y=e^{-x^{2}/2}$ on desmos, and I see that the inflection points are at ...
Sue VanHattum's user avatar
  • 20.8k
21 votes
Accepted

What benefit is there to obfuscate the geometry with algebra?

Is your ultimate goal really just to teach cofunctions? Or are you trying to teach cofunctions so that the students can apply them later? I am speaking as a student rather than an educator, but math, ...
PC Luddite's user avatar
20 votes

Why do we care about multiple proofs of the same theorem?

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 ...
MvG's user avatar
  • 369
19 votes

Why should kids learn how to use a compass and straightedge, and not rely on a drawing program?

Of course there is the deep (~2000 yrs) history of compass/straightedge constructions. Another more modern alternative is origami constructions: Robert Lang, "Origami and Geometric Constructions,"...
Joseph O'Rourke's user avatar
19 votes
Accepted

Co-curricular lessons between geometry and chemistry?

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 ...
Nate Bade's user avatar
  • 1,941
19 votes

Should figures be presented to scale?

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 ...
alephzero's user avatar
  • 889
19 votes

What is the preferred way to denote the Pythagorean theorem equation?

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 ...
Tommi's user avatar
  • 7,202
18 votes
Accepted

How to give my students a straightedge instead of a ruler

I have lots of ideas, which, it turns out, aren't going to fit into a comment. If you don't need something too long, but want something that is pretty uniform and consistent, popsicle sticks (also ...
Xander Henderson's user avatar
  • 8,225
18 votes

What benefit is there to obfuscate the geometry with algebra?

This multi-step question requires students to understand and apply multiple concepts or strategies to solve the problem. The goal of a standardized test is not to provide a correctly-sequenced list of ...
Steve's user avatar
  • 1,594
17 votes

How to give my students a straightedge instead of a ruler

A cheap Venetian blind with thin plastic slats can be cut up with ordinary scissors and will yield many straightedges of any length you choose. The slight bow in the blade will flatten out when ...
MTA's user avatar
  • 271
17 votes

How can I build a protractor without a protractor?

As Will Orrick says in the comments under user20315's answer, it is possible, with straightedge and compass, to construct a regular 120-gon, and therefore it is possible to mark off every 3 degrees on ...
mweiss's user avatar
  • 17.4k

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