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) ...
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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 ...
- 6,121
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 ...
- 6,121
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, ...
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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 ...
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17
votes
How to properly define volume for beginner calculus students?
$dV$ represents a tiny bit of $V$.
$V = \int dV$ says that you can find the volume by adding up all the tiny bits of volume. This is why it is called an "integral;" you need to integrate all ...
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12
votes
Accepted
Models for spherical geometry
Here are three projections that result in models of spherical geometry. I think the stereographic model is closest to what you're looking for.
Stereographic projection
One possible stereographic ...
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11
votes
Geometrical verifications for Algebraic formulae
In my experience,
geometry can lead you into interesting questions that can be answered geometrically in special cases, and
algebra can help you tighten up the rigor of your answers and generalize ...
- 6,121
10
votes
Models for spherical geometry
I often teach a geometry class like the one you describe and always try to include both elliptic and hyperbolic plane geometry.
As you indicate, elliptic geometry is modeled by spherical geometry, but ...
- 10.2k
9
votes
What benefit is there to obfuscate the geometry with algebra?
I have been on committees that write questions for standardized tests and placement tests. In this role, I have reviewed results of many trigonometry questions that were piloted and then revised for ...
- 10.2k
8
votes
Geometric line: constructing fractions
In Growing Ideas of Number, Crossley provides this diagram, Figure 3.4, in a discussion of the notion of the "geometric line." From the perspective of modern mathematics, all of the points ...
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8
votes
How to convince a student without calculus that great circles are geodesics in a sphere?
Take a physical sphere such as a beach ball, and a string. Pick two points. Hold the string down with one finger at one point then stretch it to the second point. Next, holding the string tight at ...
- 10.2k
7
votes
Geometrical approaches in algebra
I offer this (community wiki) only to illustrate the OP's 2nd example.
From the Archimedes Lab Project:
Quite beautiful!
6
votes
Accepted
Triples or triplets in Pythagoras theorem
The word “triple” is appropriate here because $(3,4,5)$ is a tuple consisting of three elements.
In mathematics, a tuple is a finite ordered list (sequence) of elements... Mathematicians usually ...
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6
votes
Why do standard geometry textbooks not start with trigonometry?
Here is a late but brief answer to the question:
If this is the normal way of teaching geometry, why? Why is the course
focused more on memorizing theorems rather than understanding where
they come ...
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6
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How to convince a student without calculus that great circles are geodesics in a sphere?
Answer inspired by Michał Miśkiewicz's comment on the OP.
To a high school student:
Put an ant on a basketball. Draw a tiny arrow representing the direction it should walk. The ant always just puts ...
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5
votes
Geometric line: constructing fractions
I don't think this diagram would help kids understand fractions. But I do like how it makes me think.
Your problem might be that the 1's are not to scale.
If we are given that the 3 lines (or line ...
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5
votes
Accepted
How to convince a student without calculus that great circles are geodesics in a sphere?
Recall that a great circle is the intersection of the unit sphere and a plane passing through the origin. A key point is that for short arcs of great circles, the ratio of euclidean distance between ...
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5
votes
How to formalize high-school (Euclidean) geometry?
Clark and Pathania might be of interest to you.
"This textbook provides a full and complete axiomatic development of exactly that part of plane Euclidean geometry that forms the standard content ...
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5
votes
Is it correct to state that a cone has no faces?
This isn't a bad question, it's a good opportunity to discuss how mathematical definitions are made and their relative merits.
In convex geometry, a face of a convex body $K$ is the intersection $K \...
- 5,243
5
votes
Geometrical verifications for Algebraic formulae
I agree with the general idea of Justin Skycak, and I think you'd be doing the students a disservice if you don't show them some kind of geometric pictures to go with this stuff.
In your specific ...
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5
votes
Proof that volume of cone is 1/3 that of a cylinder
Here is an answer on Math.SE that proves that the volume of a cone is $\dfrac{1}{3}$ the volume of a cylinder using only algebra and geometry. No calculus required.
Note: The proof relies on Cavalieri'...
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4
votes
Geometrical approaches in algebra
This may not be what you have in mind, but your question reminds me of “proofs without words” aka visual proofs. The three examples you gave have relatively famous visual proofs.
There’s some ...
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4
votes
How to properly define volume for beginner calculus students?
How to properly define volume for beginner calculus students?
I'd say that the Apostol's approach, which I found after looking through several books, is the best. His definition of volume is axiomatic....
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4
votes
Geometrical verifications for Algebraic formulae
I personally had a lot of fun in high school trying to extend the geometric picture for completing the square to include negative numbers. I ended up inventing a system of signed areas which was ...
- 24.3k
3
votes
How to convince a student without calculus that great circles are geodesics in a sphere?
Determine the shortest route from New York City USA to Rome Italy using a piece of string on a globe. Both cities are south of the 45th parallel and yet the the shortest route deviates considerably to ...
- 1,074
3
votes
How to explain square meters?
Peter answered how to explain the ideas to students. So I'll answer your second question:
Is "square meter" a badly-phrased term? What would be a more appropriate term?
It's not badly-...
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3
votes
How to convince a student without calculus that great circles are geodesics in a sphere?
Maybe this would work? The spherical law of cosines says that, on a sphere of radius $R$, if you have a sphercial triangle with edge lengths $a$, $b$, $c$ and angles $A$, $B$, $C$, then
$$\cos \tfrac{...
- 5,243
3
votes
Multiple proofs for the same problem
It just crossed my mind that I can offer you some option you haven't probably considered yourself.
Once I experimented in my calculus course (which also involved some elements of analytic geometry and ...
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3
votes
Accepted
Is there a particular reason why segment addition postulate and partition postulate are two different things?
In the Elements, Euclid did not use lengths. But in contemporary high school geometry, we typically find lengths of segments being represented as real numbers. The "ruler postulate" was ...
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