How Can I Overcome My Aversion to Manual Calculations?

[Mastermind01]

I've developed a sort of aversion to doing huge calculations most of the time (at home) I just want to plug it into the calculator and do it . This is affecting my exams, I tend to put off questions which have a lot of calculation to do and then ultimately do the calculation wrong (We're not allowed calculator use here at school in exams) . I mean, I just don't see the point . I've got the equations , the questions is solved. This tendency has been extended to simplifications as - well huge equations in variables which need to be simplified to get the answer. This is hurting me in exams . Can anyone suggest something , how to basically, stop being lazy about number crunching ?

P.S : We spend a lot of time in class proving trigonometry identities , are they at all useful anywhere? I find no connection at all.

 

[micromass]

P.S : We spend a lot of time in class proving trigonometry identities , are they at all useful anywhere? I find no connection at all.

Did you really just ask whether trigonometric identities are useful?

 

[Mastermind01]

Did you really just ask whether trigonometric identities are useful?

I mean yes.. Not the basic ones . Questions like

Prove that:

(1 - tanA) / (1 + tanA) = (cotA - 1) / (cotA + 1)

What's the point in proving the above?

P.S - I did ask two questions though , the other one is that I have developed an aversion to calculations , how to overcome that?

 

[micromass]

I mean yes.. Not the basic ones . Questions like

Prove that:

(1 - tanA) / (1 + tanA) = (cotA - 1) / (cotA + 1)

What's the point in proving the above?

The point is not so much that it is an important identity, but that you are able to prove identities like this when they show up. Trigonometric identities show up everywhere in math, engineering and physics. Being able to manipulate them well is an important skill.

 

[Mastermind01]

The point is not so much that it is an important identity, but that you are able to prove identities like this when they show up. Trigonometric identities show up everywhere in math, engineering and physics. Being able to manipulate them well is an important skill.

They do? Higher up I suppose, thanks for answering . What about my other question?

 

[Choppy]

I think it's easy to fall into a trap of thinking that if you've arrived at the correct formula, you undertand the problem and the rest is just a trviality, but I think that's a bad habit that a lot of young students tend to pick up from some more outspoken professors for whom the details of the calculation really are a triviality.

Consider for example the Drake Equation. Just because it's written down, doesn't mean that we now know the number of intelligent civilizations in the galaxy. The details lie in the values and uncertainties of the variables themselves.

Anyone who has ever taught undergraduate labs can also attest to the value of carrying calculations through to completion. Inevitably there are groups that will hand in results that are orders of magnitidue off, or that are in wrong units, or that otherwise just don't make sense.

In my own field (medical physics) we do a lot of very basic calculations. The formulas are well known. They're programmed into computers. But still we often have to perfom them ourselves to make sure that what the computers are telling us make sense. People's lives can depend on the results.

So the details are often important. And if you're making frequent errors in performing them, they haven't become a trivial matter for you yet. Hopefully this at least can give you some justification for working through them.

 

[Mastermind01]

So the details are often important. And if you're making frequent errors in performing them, they haven't become a trivial matter for you yet. Hopefully this at least can give you some justification for working through them.

Computers are programmed to be much better than me at calculations. Why do I need to master them? And that's another thing , how can I improve my calculation (speed maintaining accuracy) . I'm in grade 10 (second year of high school) I don't want to spend too much time behind improving calculation , but I understand when we have to check calculator results..

 

[CalcNerd]

This is only a suggestion and shouldn't be carried to extremes. Buy a $2-4 calculator with the basic functions and sq root key. Use it and the learn the trig identities for 30, 45, and 60 degrees for sin, cos and tan. Learn how to interpolate the trig values for 15 and 75 degrees and use those tools for the next 3-6 weeks. That will sharpen your trig skills and should enlighten you to the more advanced trig identities too.

The el cheapo calculator will allow you to blow through the truly repetitive number crunching while being too simple to do any advance math with.

 

[Mastermind01]

This is only a suggestion and shouldn't be carried to extremes. Buy a $2-4 calculator with the basic functions and sq root key. Use it and the learn the trig identities for 30, 45, and 60 degrees for sin, cos and tan. Learn how to interpolate the trig values for 15 and 75 degrees and use those tools for the next 3-6 weeks. That will sharpen your trig skills and should enlighten you to the more advanced trig identities too.

The el cheapo calculator will allow you to blow through the truly repetitive number crunching while being too simple to do any advance math with.

The problem is not with trig values , it's with huge three-four digit multiplication and addition and simplification. We're not allowed to use a calculator here so have to do repititive number-crunching.

 

[symbolipoint]

The problem is not with trig values , it's with huge three-four digit multiplication and addition and simplification. We're not allowed to use a calculator here so have to do repititive number-crunching.

Finally you present the real trouble. You must become good with symbolic manipulations for numeric statements and proving identities, and managing a set of formulas. The reason you some level of learning that you really SHOULD be EXPECTED to use a small handheld calculator is to do the computations more efficiently. Still you absolutely MUST understand and know how to do the computations without a calculator. Are you at least permitted to estimate?

 

[Mastermind01]

Finally you present the real trouble. You must become good with symbolic manipulations for numeric statements and proving identities, and managing a set of formulas. The reason you some level of learning that you really SHOULD be EXPECTED to use a small handheld calculator is to do the computations more efficiently. Still you absolutely MUST understand and know how to do the computations without a calculator. Are you at least permitted to estimate?

Sorry , if I beat about the bush . Yes we are allowed to estimate . I understand how to do computations without a calculator I just find it pretty boring and make a lot of mistakes. How can I get better now?

 

[symbolipoint]

Just struggle with it as you are, and use estimation as much as necessary. I have the same feeling about doing the computational part as you do, but I just understand also that knowing how is necessary. Conditions in the classrooms will become better. At the level of "Algebra 1", you should generally not need a calculator. As you learn in Intermediate Algebra, you have more detailed computations to finish and for time efficiency, you SHOULD BE allowed to use a calculator, since by this stage, you are already expected to know how to do numeric computations, and you will have too many to do without rechecking for mistakes, and more concepts and skills must be taught and tested. You should always be required to setup your equations and derive necessary formulas, and to show all of these steps. Once that is done, you should need to show all of the arithmetic computation steps, but still a calculator should be allowed, maybe depending on what the goal for any test or assignment is. You could still use the calculator to go from step to step in the computation.

You're still in high school. Most of your college-level math courses in college or university will expect you already understand and are skilled with basic computation and will expect you to use a scientific or graphing calculator, frequently, including in the classroom.

 

[Intrastellar]

Are you talking about physics calculations or maths calculations ? Are you carrying the algebra all the way to the last step or are you plugging numbers into the equations at the first possible step ?
Can you show us an example of a question with these needless calculations or symbolic manipulations ? (Preferably at the homework section I suppose)
It is very possible that there are some very useful algebraic tricks which you are not proficient with, but which will speed the computations considerably.

I mean yes.. Not the basic ones . Questions like

Prove that:

(1 - tanA) / (1 + tanA) = (cotA - 1) / (cotA + 1)

This one is actually not so difficult, multiply the top and the bottom of the left hand side by cotA.
This is one of those very useful algebraic tricks which can speed computations considerably, I guess you approached this question by decomposing tanA into sinA and cosA ? If you can master these tricks by doing trigonometric proofs, then one can say that the trigonometric proofs were very useful.

 

[Mastermind01]

Are you talking about physics calculations or maths calculations ? Are you carrying the algebra all the way to the last step or are you plugging numbers into the equations at the first possible step ?

I'm talking about maths calculations. Carry the algebra all the way to the last step and then plug in numbers.

This one is actually not so difficult, multiply the top and the bottom of the left hand side by cotA.
This is one of those very useful algebraic tricks which can speed computations considerably, I guess you approached this question by decomposing tanA into sinA and cosA ? If you can master these tricks by doing trigonometric proofs, then one can say that the trigonometric proofs were very useful.

I have no problem with proving trig identities, I just asked the use.. and got the answer

Thanks for the answers , @symbolipoint I'll try my best to carry on..

 

[symbolipoint]

Are you talking about physics calculations or maths calculations ? Are you carrying the algebra all the way to the last step or are you plugging numbers into the equations at the first possible step ?
Can you show us an example of a question with these needless calculations or symbolic manipulations ? (Preferably at the homework section I suppose)
It is very possible that there are some very useful algebraic tricks which you are not proficient with, but which will speed the computations considerably.This one is actually not so difficult, multiply the top and the bottom of the left hand side by cotA.
This is one of those very useful algebraic tricks which can speed computations considerably, I guess you approached this question by decomposing tanA into sinA and cosA ? If you can master these tricks by doing trigonometric proofs, then one can say that the trigonometric proofs were very useful.

montadhar is describing an example of using any known or common trigonometric identities, since some of them as here, can be applied in clever ways.

 

[Nidum]

P.S : We spend a lot of time in class proving trigonometry identities , are they at all useful anywhere? I find no connection at all.

In engineering quite simple problems can generate very complicated expressions with many trig terms . Being able to simplify these formulas rapidly is an essential practical tool .

Not a direct answer to your questions but in real world maths it is often possible to arrive at a sufficiently accurate answer to a numerical problem by making sweeping simplifications almost intuitively .

A branch of maths which is little explored but which engineers use all the time is backwards calculation - guess a plausible answer and see if it fits the maths . One good guess based on experience and one or two refining iterations often yield correct answers very quickly indeed .

 

[Dembadon]

I've developed a sort of aversion to doing huge calculations most of the time (at home) I just want to plug it into the calculator and do it . This is affecting my exams, I tend to put off questions which have a lot of calculation to do and then ultimately do the calculation wrong (We're not allowed calculator use here at school in exams) . I mean, I just don't see the point . I've got the equations , the questions is solved. This tendency has been extended to simplifications as - well huge equations in variables which need to be simplified to get the answer. This is hurting me in exams . Can anyone suggest something , how to basically, stop being lazy about number crunching ?

P.S : We spend a lot of time in class proving trigonometry identities , are they at all useful anywhere? I find no connection at all.

Symbolic solutions are infinitely more useful that numerical solutions because they can be manipulated to solve many different types of problems. Numerical solutions are only valid for the one problem on which you're working.

Proving different identities will help you derive expressions that are needed to solve a given problem, but aren't explicitly given in the problem statement. This will be required of you quite often as you progress through your program. There will be many occasions for which you need an expression of a certain form, and you will not be able to use a textbook or Google to look it up. You will have to perform some derivations to arrive at the expression(s) needed to solve the problem. Your calculator will not be able to do this for you, and as you've mentioned, your grades will suffer. Hence, being able to prove and derive things is not a pointless exercise.

 

[mathwonk]

when someone says he/she has an aversion to doing calculations, my best response is :"get over it". i.e. do as many as possible, until you get the idea.

 

[Nidum]

Hi Mastermind01 ,

Perhaps I can cheer you up with a bit of history :

From the earliest days of science and engineering and right into the late 1950's (and 1970's in a few cases) the only way to solve many problems was numerically and by essentially manual calculation .

One numerical calculation type that came up very often was solving FInite Difference equations . Finite difference methods were used for aerodynamic , stress and thermal analysis and usually resulted in huge equation sets .

These equation sets were worked through systematically by small teams of people , sometimes for days , until a verifiable answer was obtained .Special error checking methods were employed throughout the procedure . Calculations were often duplicated as well and interim results compared every hour or so .

Some help was available in form of logs , slide rules and (very) basic calculators but for many problems high precision was needed and that meant manual calculation for each and every equation term .

There is a brief description of teams of people doing manual stress calculations for balloon frames in Neville Shute's book .

Happy days .