user52817
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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 ...

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 ...

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 $\frac{p}{q}$ ...

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 ...

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 ...

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 ...

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 ...

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 ...

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, $(f\circ g)\circ h=f\... View answer 0 votes This is a good question, and brings up some subtle issues. A clue is to observe the tacit assumption, made when drawing the two graphs in the original post, that the axes are perpendicular. We draw ... View answer 10 votes I highly recommend "Flatland The Movie." Your institution should be able to purchase it. You can find a free trailer on the internet. When I was young, I read the book "Flatland: A Romance of Many ... View answer 3 votes The general quadric surface in projective space is ruled by two families of lines. These are smooth surfaces. Think about a hyperboloid of one sheet, where is is easy to visualize the two families of ... View answer 5 votes I think you are being harsh in your criticism of the classical notation. Of course, at the mathematician's end of the spectrum, the notation you promote towards the end of your question has merit. But ... View answer 8 votes After teaching Calculus 3 for many semesters, your question certainly resonates. I think it is empowering to coach students in how to sketch surfaces in 3D. Their tendency from curves in 2D is to ... View answer 2 votes Most elementary school teachers in the USA get their teaching certification by majoring in "elementary education" as an undergraduate. Curriculum for such programs typically includes a sequence of ... View answer 8 votes Your approach is in fact geometrical, if we see all this happening on the Riemann sphere, i.e., the one-point compactification of${\bf C}={\bf R}^2$. Briefly, we have the usual coordinate$z$on${\...
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 $... View answer 59 votes Expansion of mathematical knowledge does not unfold in Bourbaki progression. This is true at the level of both societal and individual knowledge. Just as the invention and significant applications of ... View answer 6 votes When I introduce parametric curves in Calculus 3, I like to bring my Etch A Sketch to class. It is a drawing toy from about 1960. The knob on the left encodes$x(t)$, and the knob on the right$y(t)$. ... View answer 7 votes It seems unlikely that the Cardano formula has even been of serious analytic use, i.e., used to approximate roots of a cubic. At least since the inception of calculus, Newton's method can be used to ... View answer 18 votes One observation is that (sum of numerators) divided by (sum of denominators) is not well defined. For example, let's work with the two ratios$a=\frac01$and$b=\frac11$. The ratio of the sum of ... View answer 6 votes If you know what you are doing, then you are wasting your time. Anonymous View answer 2 votes I think that a linear algebra course is the perfect venue to have students develop their understanding of the mathematical concept of "uniqueness." A formative assignment such as the one you suggest ... View answer 4 votes Since log base 2 appears, maybe look at the case$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 ... View answer 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, ... View answer 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 ... View answer 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 ... View answer 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 ... View answer 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 ... View answer 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)...