user52817
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Question about function notation
6 votes

You are not being pedantic. The name of the process that slices is $g$, and the result of slicing $x$ is $g(x)$. On the other hand, the textbook presentation seems to be for students who are just ...

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What is the motivation for teaching Factoring by Grouping?
3 votes

This is a thought provoking question. Learning to think algebraically is a gradual process. It takes practice to see that $$ax+ay=a(x+y)$$ is the same as $$(x+1)x+(x+1)y=(x+1)(x+y).$$ Eventually, some ...

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Intuition explanation about Lebesgue measure zero of the rational numbers
Accepted answer
5 votes

Let me build on the idea of Steven Gubkin in his comments. One way to visualize this scenario is to use Ford circles. The standard picture is to plot a circle tangent to the $x$-axis at $\frac{p}{q}$ ...

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Missing Step in Most Proofs of the Irrationality of $\sqrt{2}$
3 votes

You are correct in that the coprimality of $a$ and $b$ is not used in its full strength. It is adequate to merely assume that they are not both even. But since people are so used to reducing a ...

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Practical case for solving with system of 2 equations
2 votes

One of the most interesting word problems of all time, which broadened the human intellect in many ways, is the Archimedes cattle problem. There are many excellent books and articles about this--start ...

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When does thinking $(-8)^{1/3} = -2$ result in problems for an undergraduates?
2 votes

The equation $(-8)^{1/3}=-2$ in isolation is taught in early algebra. Later, in precalculus, on learns about the Fundamental Theorem of Algebra. At this point, one starts to understand that this ...

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How to best explain that the sum of conditional probabilities still sum to 1
2 votes

Here is another way to look at this. $P(A)+P(A^c)=1$ aligns with the logical tautology $(A\text{ or }\lnot A)$. The preposition $B\rightarrow(A\text{ or }\lnot A)$ is also a logical tautology, and ...

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Are there direct practical applications of differentiating natural logarithms?
5 votes

Boltzmann's equation for entropy is $S=k\ln W$, and the second law of thermodynamics is all about change in entropy. Maybe this is a place to start with your quest for a practical application of the ...

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How to naturally encounter the properties of identity, commutativity, associativity, and distributivity (to define rings)?
3 votes

I think associativity is the most natural property to encounter. This is because transformations are associative. For transformations $f,g,h$ with compatible domains and ranges, $(f\circ g)\circ h=f\...

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How to define "axes with the same scale" in Secondary/High School?
0 votes

This is a good question, and brings up some subtle issues. A clue is to observe the tacit assumption, made when drawing the two graphs in the original post, that the axes are perpendicular. We draw ...

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Ideas for explaining 4D and higher dimensions
10 votes

I highly recommend "Flatland The Movie." Your institution should be able to purchase it. You can find a free trailer on the internet. When I was young, I read the book "Flatland: A Romance of Many ...

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On a special degenerate conic
3 votes

The general quadric surface in projective space is ruled by two families of lines. These are smooth surfaces. Think about a hyperboloid of one sheet, where is is easy to visualize the two families of ...

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Undergraduate Vector Calculus Notation Mess
5 votes

I think you are being harsh in your criticism of the classical notation. Of course, at the mathematician's end of the spectrum, the notation you promote towards the end of your question has merit. But ...

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Fear of 3-dimensions
8 votes

After teaching Calculus 3 for many semesters, your question certainly resonates. I think it is empowering to coach students in how to sketch surfaces in 3D. Their tendency from curves in 2D is to ...

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Are there standard notations for 'number talks' / ‘math talks?'
2 votes

Most elementary school teachers in the USA get their teaching certification by majoring in "elementary education" as an undergraduate. Curriculum for such programs typically includes a sequence of ...

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An intuitive explanation of l'Hôpital's rule for ∞/∞
8 votes

Your approach is in fact geometrical, if we see all this happening on the Riemann sphere, i.e., the one-point compactification of ${\bf C}={\bf R}^2$. Briefly, we have the usual coordinate $z$ on ${\...

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Are degrees of polynomials illogically defined in elementary algebra, intermediate algebra and college algebra courses?
1 votes

Soon after introducing polynomials, students learn to add and subtract polynomials. Notice that if $f(x)$ and $g(x)$ are polynomials with degrees $n$ and $m$ respectively, and if $f(x)-g(x)=0$, then $...

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Why is it possible to teach real numbers before even rigorously defining them?
59 votes

Expansion of mathematical knowledge does not unfold in Bourbaki progression. This is true at the level of both societal and individual knowledge. Just as the invention and significant applications of ...

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How to teach sketching a parametric curve?
6 votes

When I introduce parametric curves in Calculus 3, I like to bring my Etch A Sketch to class. It is a drawing toy from about 1960. The knob on the left encodes $x(t)$, and the knob on the right $y(t)$. ...

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Does anyone use the cubic formula these days?
7 votes

It seems unlikely that the Cardano formula has even been of serious analytic use, i.e., used to approximate roots of a cubic. At least since the inception of calculus, Newton's method can be used to ...

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How to explain that the sums of numerators over sums of denominators isn't the same as the mean of ratios?
18 votes

One observation is that (sum of numerators) divided by (sum of denominators) is not well defined. For example, let's work with the two ratios $a=\frac01$ and $b=\frac11$. The ratio of the sum of ...

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Quote to show students don't have to fear making mistakes
6 votes

If you know what you are doing, then you are wasting your time. Anonymous

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Asking students to define "unique"
2 votes

I think that a linear algebra course is the perfect venue to have students develop their understanding of the mathematical concept of "uniqueness." A formative assignment such as the one you suggest ...

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How to show $3(\log_2 n)^5 < \sqrt{n}$
4 votes

Since log base 2 appears, maybe look at the case $n=2^k$. Then $3(\log_2 2^k)^5$ simplifies to $3k^5$, which is a polynomial in $k$. For the other quantity, $\sqrt{n}$ simplifies to $2^{k/2}$, which ...

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Why do you need to distinguish between apparently identical objects in probability?
7 votes

This is a very good question. The issue comes up frequently. I explain this using a toy model: throw two regular six-sided die. What is the probability that the sum is 3? With some physical modeling, ...

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Easy examples of correspondence between global and local, as preparation for Gauss's theorem and Stokes's theorem
7 votes

One idea that comes to mind is the fact that a connected graph has a Eulerian circuit if and only if every vertex has even degree. Another idea that comes to mind is just the Fundamental Theorem of ...

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Is there a more telling name for "Calculus 2"?
Accepted answer
18 votes

In my job, I evaluate university math courses for transfer equivalency on a regular basis. In the US, "Calculus 1" typically refers to single variable differential calculus up to the fundamental ...

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How to help new students accept function notation
3 votes

Function notation is a next step in mathematical maturation. In the language of Dubinsky et al., your students are in the process of encapsulating functions as primary objects. At one point in ...

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Practical applications of integration by substitution where integrand is unknown
Accepted answer
5 votes

This is probably too abstract for students just learning about integration by substitution. But convolution is a "real-world application of integration," finding applications in such things as image ...

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How can I explain why numerical integration is easy, but symbolic integration is hard?
9 votes

It might be fun to have your students pretend that the only functions they know are sums of monomials $cx^n$ where $n\in{\bf Z}$, and in particular, play like they know nothing about the function $f(x)...

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