I like to grade questions like this on a five-point rubric. The main aspect of this problem is distributing the negative sign into the second group. Presumably students had practice problems similar ...

You are integrating a function $z=f(x,y)$ but the units of $z$ do not have to be for length. The units could be for mass density, in which case the units of the double integral would be for mass. Or $... View answer 7 votes The definition of an irrational number as a "number which is not rational" is not without its own difficulties. It presumes that we have a clear definition of a real number. The audience you refer to ... View answer 7 votes The prototypical way for a function to not be continuous is that of a jump discontinuity. Imagine a jump discontinuity on the order of a few micrometers, like the width of a hair. If you are tracing ... View answer 6 votes The spin states of an electron form a two-dimensional vector space over the complex numbers. Designate "spin up" and "spin down" for a basis. The vector space structure is a ... View answer 6 votes For the expression$x^x$we could focus on finding occurrences of$x\ln(x)$. One direction is Stirling's approximation$\ln(N!)\sim N\ln(N)$so$N!$is like$N^N$. Another direction is that the prime ... View answer 6 votes I think this is an interesting question. In the US undergraduate mathematics curriculum, one often finds a sequence of courses "Abstract Algebra I" and "Abstract Algebra II." I ... View answer 6 votes You are not being pedantic. The name of the process that slices is$g$, and the result of slicing$x$is$g(x)$. On the other hand, the textbook presentation seems to be for students who are just ... View answer 6 votes When I introduce parametric curves in Calculus 3, I like to bring my Etch A Sketch to class. It is a drawing toy from about 1960. The knob on the left encodes$x(t)$, and the knob on the right$y(t)$. ... View answer 6 votes If you know what you are doing, then you are wasting your time. Anonymous View answer Accepted answer 6 votes The awkwardness of "guessing" in the division algorithm is an artifact of the base-ten representation of numbers. If you represent in binary, then your only possible "guess" is 1. In binary, your ... View answer 6 votes I really appreciate your question. It gets to something I think about often when teaching calculus. You should read about Ed Dubinsky's notion of reflective abstraction, rooted in Piaget. At one ... View answer 6 votes Although I have serious issues with "outcomes based" education, its philosophy does combat the pernicious notion that grades should have a normal distribution centered on "average." It is not ... View answer 6 votes I have always found it instructive and fun to address this by thinking graphically. Students in Calculus II will know the graphs of the rational functions$f(x)=\frac{ax+b}{(x-c)(x-d)}$and$g(x)=\...

Your quest for measurable learning outcomes suggests a framework rooted in Bloom’s Taxonomy and the higher-education assessment industry that has been built around it. As I think about your question, ...

Another example of a crystalized definition is Dedekind's approach to defining finite. First we define a set $S$ be infinite if it is equivalent to a proper subset, i.e., if there is an injection $f:S\... View answer Accepted answer 5 votes Let me build on the idea of Steven Gubkin in his comments. One way to visualize this scenario is to use Ford circles. The standard picture is to plot a circle tangent to the$x$-axis at$\frac{p}{q}$... View answer 5 votes Boltzmann's equation for entropy is$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 ... 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 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 5 votes Think of the pedagogical goal of the problem as being to generate discussion about assumptions being made, just as you are doing. The goal is not to come up with a pre-determined correct number. ... View answer 5 votes Gauss said "You have no idea how much poetry there is in a table of logarithms." The first paragraph of this paper might get you pointed in the right direction ON THE DISTRIBUTION OF PRIMES—GAUSS’ ... View answer 5 votes I have a colleague whose hobby is to always figure out how to a evaluate any given limit without L'Hospital's rule. Every week in the lunch room, there is yet another example. I think a very good ... View answer 5 votes Students in the class could google their birth dates before class and come to class knowing the day of the week on which they were born (MTWRF). In class you then test the hypothesis that days of ... View answer 5 votes The university math department where I work runs a math competition for high school students each year. One year there was a problem with answer$7/2$. May students put$3\frac{1}{2}$down as the ... View answer 5 votes Many places have Calculus I as a co-requisite for University Physics I. The physics instructors like students in thier class to be familiar with derivatives of exponential functions before the end of ... View answer 4 votes Straightedge and compass construction is an interesting way to thing about this as an educator. This perspective would not be something you actually take into an elementary school classroom. Rather it ... View answer 4 votes A triangle is born from three non-collinear points and the axiom that two points determine a line. In the context of neutral geometry, a triangle has no structure other than three lines and three ... View answer 4 votes A classical example of proof by exhaustion is to establish an integer$N$is prime by trial divisions. One uses trial division by candidates for divisors up to$\sqrt{N}\$. This can be made slightly ...