# Tag Info

### 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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### 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 ...

### 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 ...
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### 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, ...

### 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 ...

### 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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### 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 ...

### 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 ...

### 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 ...

### 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 ...

### 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 ...

### Geometrical approaches in algebra

I offer this (community wiki) only to illustrate the OP's 2nd example. From the Archimedes Lab Project: Quite beautiful!
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### 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 ...

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 ...

### 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 ...

### 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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### 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 ...

### 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 ...