The sanity of sanity checks

Once upon a time, I was an engineer. And in that environment, engineers would think it was insane to depend on a calculation, a computer program, or a sketch - without checking the result, and if possible, checking it in the simplest way possible.
Fast forward to now. You have calculating power in your hand that was beyond what I could do with a long computer program when I first entered an engineering career. Your problem is, you depend on it. The teacher gives you a problem and what do you do? You reach for the calculator.
So look: the human brain is infinitely more powerful than the best calculator you can put in your hand. Learn to reach for it first instead. Use it to set up your work, to help you understand why you're doing it, to help you recall how you did what you did, and to find out what it takes to do things right. THEN grab the calculator. Otherwise, the thing the calculator does best is give you the wrong answer fast.

Nearly all the problems you face in school are "given/find" types: you're given all the inputs and were to find a single output based directly on the inputs. In the workplace, in "real life," problems won't be like that. You'll have to find inputs on your own, and you may not even know what outputs are the ones that really matter. No calculator can help you there, bubba.

If the problem you're solving isn't to be repeated, if you just get to do it once or twice and then walk away and forget it, that's one thing. Then knowing how you did what you did doesn't matter so much. And yeah, most students who are working on, say, solving simultaneous equations, may wonder how it matters. If I had a buck for everyone who's asked me "how do you use this in real life?" I'd be rich. 
But let's just say what you're doing now might have parts that will be of use to you later. Then knowing how you did what you did matters a LOT. Then it might help to write a few things down. As an engineer, I used "the back of the envelope" or "on a cocktail napkin" to keep from losing things that are important. It's a time-honored tradition!
No matter what you will do in your life, it is VERY useful to develop habits that help you define problems, recognize right and wrong answers when you see them, and remember how you did what you did. That's what I mean by a sanity check: if you think you can get a good job someday without being able to do these things, you're insane.
Here are a few things engineers check, to make sure they are sane:
- Do I know everything that can affect the answer? Structural loads, for instance, can come from shear, compression/tension, and moments. Forget one and something's going to break.
- Have I considered the units? Engineers have lost a 12 or a 57.296 before, and that makes a big difference between right and wrong!
- Is there anything in the way of the right answer? Like not enough power, not enough room, not enough time, not enough money.

I can come up with more, but you get the idea. Brain before calculator, unless you want to be insane.
Here's an example of a satellite that was lost, for many of the same reasons someone you know struggles with Algebra 2. Cheers!
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