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2

At my school (a community college in California) the curriculum is set up so that students first take an algebra course in which essentially every function is a linear function, and only later do they do anything at all with nonlinear functions. Nonlinear functions are just more complicated, so they require more thought about the logic. Thinking about logic ...


5

I am writing this based on pure observation (e.g., entering year four of teaching this topic to secondary school students, and having co-taught a minicourse for teachers on absolute value functions$^\star$). There are a lot of definitions/interpretations of absolute values: the (abstract) axiomatic one; the piecewise or "case-based" one (provided by Joel ...


7

Your last para was very reasonable. (I was going to give a mean sarcastic answer, but can't now.) We can crowdsource this: Frank Ayres, First Year College Math (algebra 1 to precalc; Schaum's Outline) 1958 but still in print: Only has geometric mean of 2 objects, but does spend quite a lot of time on geometric progressions. And also discusses getting ...


-1

Because it's intricate. Yes, even just one or two "if then"s still makes something intricate! We are meat, not silicon. Just explaining a rule or set of rules is not adequate for us, if we have to remember some intricacies. Instead of tacitly searching for some lock-key explanation idea (what is the hurdle and how do we adroitly remove it), I recommend ...


13

(My answer is just a guess and not based on any formal research.) I suspect the absolute value function may be difficult to understand because it involves "negative numbers that aren't negative." One way to define the absolute value of $x$ is: $$|x|=\left\{\begin{array}{rl}-x, & x<0\\x, & 0\le x\end{array}\right.$$ I think the $-x$ confuses ...


3

Although I admire the enthusiasm of the OP, I'd like to make a point similar to @guest's post: There are quite a few calculus textbooks out there. Each thumbnail below is the cover of a calc textbook. I would be wary of investing too much time in such a crowded environment without investigation of how it is already populated. Google images link Also, there ...


4

You mention the desire for a book with content and exercises, but less fluff. But also, seem not to have done a lot of comparison research (other commenters recommend this to you, also). Two specific texts to look at would be: Schaum's Outline and Granville, et al. Both have a short, simple presentation. And there is definitely a student market for books ...


2

A nice challenge for a calculus class with a little physics: If particles are thrown out from a common point in all directions at the same speed, then allowed to fall freely, the shape they will sweep out is a parabola. (Of course, the trajectory of each particle is also a parabola, that's a simpler fact.) The Fourth of July might suggest some examples: ...


3

I've found that students are not very clear on the image that is being invoked when I call $z = x^2-y^2$ by its traditional name of "saddle point", but they are all very clear on what a Pringles potato chip looks like.


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