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5

The problem underlying the discussion in the question can be summarized as that it is necessary to choose a branch cut to define a complex logarithm (or arctangent). It is a mistake and pedagogically a bad practice to allow negative values of $r$. It is a mistake because pairs $(r, \theta)$ with $r$ possibly negative have no right to be called coordinates. ...


2

(Second answer for follow-on question from Sue about contacting math historians.) Sorry, to be so simplistic, Sue, but I would just reach out* to some general math historians of any stripe (at a university that has those) and just correspond. Either with whoever you get or by trying to work your way to one with the specific subject matter expertise you ...


1

Here's another computer-graphics example, which may or may not count as "nowadays". When I was in college 25 years ago I started a project to write a ray-trace renderer, which I then continued to expand after college. Ray-tracing is an elegant model that just recently has gained the hardware support to make it feasible in real time, e.g.: https://developer....


5

I teach in the U.S. at a community college. Although I prefer distributing without drawing an area box when I'm doing math myself, I often show the box in class to help students see how things work. When we use an area model to help students visualize the workings of the distributive property, it makes much more sense for many students. In fact, the box ...


7

I run into this issue frequently. As a high school in-house math tutor, students visit to show me their quiz/test scores and ask about their work. The FOIL method is fine, if it works for the student. For those who are prone to making mistakes, I show them the Box method (call it what you will, that just my name for it). The benefit, if any, to this method ...


4

To demonstrate the distributive law, we often use an area model. Then $(2+4)\times(5+7+9)$ can be visually decomposed into the sum of 6 subproducts. Rather than actually drawing a grid, you might eventually start just labeling the edges and the products, without really caring about the relative sizes of the pieces. Finally, this ``table method'' could ...


0

I suggest looking at some intermediate general science texts. Often there will be some simple computational problems involving very minor* algebra. You might be able to find some that appeal to you although you will have to see what part of the algebra sequence they correspond to. Green stuff has been of interest going back at least to the seventies. And ...


1

A new car has a carbon footprint of $15.25$ tons of CO2. Each year you drive a car, you contribute $4.6$ tons of CO2. If you want to buy a new car, and emit less than $100$ tons of CO2 total over your lifetime, how many years could you drive the car? Would it be realistic to drive your car for that many years? What other assumptions in this model might ...


2

To supplement Sue's great suggestion: Another advantage of the artofproblemsolving.com is that it provides a whole lot of support for students like you, whether or not you purchase or use the textbooks available there. You'll find, e.g., see AoPS Community, which provide a number of online forums in which all students can participate, ask questions, e.g.,...


6

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 do so. In my thinking, the modern perspective of Cardano's formula is an algebraic perspective, not analytic, after Galois. Polynomials of degree 2, 3, 4 are ...


1

Sue: (Tiny help, but maybe something.) This came up in a Google search. http://archives.math.utk.edu/ICTCM/VOL10/C003/paper.pdf The first page has a couple small motivating stories/quotes about why people are fascinated by cubics. You could trace back the two quotes to the original sources and see if there's anything else that's helpful above the quotes....


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If the students are weak, lots and lots of tangibles: get them into the habit of drawing pictures and moving stuff around ("paper is cheap"). For example, integer arithmetic can either be a bunch of rules, or you can use the "chip model" to really get across certain ideas of how it works: https://youtu.be/EA8j7V677Z4?list=...


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