9th June strip

[Shyal_malkes]

i honestly didn't think anyone was really crazy enough to actually solve it.

and you tore it apart!

btw
not that it proves you correct but i got the same answer...

[CasVeg]

i honestly didn't think anyone was really crazy enough to actually solve it.

and you tore it apart!

That's me; I'm a psycomath. And you haven't seen half of what I did. I also analysed the expression as a function of x:

Horizontal asymptote: y = 0
Vertical asymptotes: x = 1, x = 2
Removable discontinuity: (-4/3, -36/35)
Root: 4/3
No maxima of minima
Inflection point: ((4 - 2^(1/3) + 4^(1/3))/3, -1 + 2^(1/3) - 4^(1/3))
Domain: x != -4/3, 1, 2
Range: all real numbers
Positive on (1, 4/3), (2, infinity)
Negative on (-infinity, -4/3), (-4/3, 1), (4/3, 2)
Decreasing over entire domain.
Downward curvature on (-infinity, -4/3), (-4/3, 1), ((4 - 2^(1/3) + 4^(1/3))/3, 2)
Upward curvature on (1, (4 - 2^(1/3) + 4^(1/3))/3), (2, infinity)

Did I mention I did all of this on paper (or in my head)? I only used a calculator(TI-81) to check my work.

[Shyal_malkes]

that's the cool thing about math, the more you know what you're doing the less you need the calculater to do it.

[StrangeWulf13]

Isn't that a good argument for taking calculators out of the classroom?

I personally would like to improve my math skills in my head; a calculator is useful, but requires batteries. How do you figure stuff out if the battery is dead and there's no way to replace it?

Hmm... I wonder if I could buy an abacus and practice with that...

[Shyal_malkes]

my algebra II teacher had us on the final show a lot of our work and some of the graphing problems almost required a calculator (the instructions even told us to use one, i was just glad i had not forgotten it at home at the time)

being able to do math in your head is fine, but the head isn't a computer, it can calculate things as complex and meaningfull as morals and values, but it has a hard time doing the exact same meaningless mathematical motions over and over and over again and again and again all the time, calculators are made to do that.

besides how else are you gonna sneak a portable videogame into a classroom?

[Persephone_Kore]

Whether you want calculators in the classroom depends on what you're doing, honestly. If you're teaching arithmetic, no. If you're teaching how to analyze an equation algebraically or do proofs, no.

The answer may be yes if you're teaching something where you need the numerical value of a square root, or anything where you want the roots of an equation beyond... well, quadric, really, because while I think they've found formulas for cubic, quartic, and possibly quintic, everything beyond quadric is inordinately clunky and beyond quintic (if not starting with it), no universal formula exists.

Basically, there are things where a calculator is irrelevant, things where it's useful, and things where stopping to reach for one will actually slow you down if you know what you're doing and are in practice.

I need to work on my mental arithmetic myself, though. I'm not bad on paper (unless I get careless), but I can't safely skip as many steps (by doing them in my head) as I'd like.

[CasVeg]

How do you figure stuff out if the battery is dead and there's no way to replace it?

Do it with pencil and paper. Another thing that comes in handy are. . .handbooks.¹ Memorize important formulae and special values. Practice a lot. Practice more than you think is necessary and sufficient.

Another thing to watch out for are exact values. An ordinary scientific or graphing calculator can't do these, only the decimal approximations. (Note that all of the values that I gave are exact.) Examples of basic values that cannot be expressed exactly using most calculators are logarithms, nth degree roots, and fractions². It's not uncommon for a needed value to combine two or more of these. If an instructor asks for an exact value, you will almost inevitably need to do the work on paper. If your calculations involve a value that is represented by a symbol, an exact value must use that symbol.³

Calculators come in handy for approximations, but they aren't necessary for this.


1 Once you get to calculus, these are effectively required.
2 Many calculators can derive the exact value of a fraction if the denominator is sufficiently small.
3 There are some computer algebra systems that can do this, but they cost a lot and can't be used without a computer.