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 ...
- 4,980
56
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 ...
- 879
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'...
- 4,873
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.
...
- 1,112
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 -- ...
- 595
37
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 ...
- 2,156
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"...
- 17.3k
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 ...
- 540
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 ...
- 24.3k
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) ...
- 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 ...
- 5,728
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 ...
- 6,191
23
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
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 ...
- 390
23
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
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 ...
- 10.3k
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 ...
- 5,243
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 ...
- 6,191
22
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
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 ...
- 345
22
votes
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 ...
- 3,646
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 ...
- 5,192
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 ...
- 17.3k
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 ...
- 20.1k
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, ...
- 326
20
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
Graph paper (or square floor tiling) to the rescue!
Proof by picture for a 3 4 5 triangle:
Because the drawing is on the grid and not the skew tiling of the square on the hypotenuse, determining ...
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 ...
- 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,"...
- 29.5k
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 ...
- 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 ...
- 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 ...
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