maths common error

7/26/2019 Maths Common Error

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COMMON MATHS ERRORS

We students commit a lot of silly mistakes while solvingmathematical problems and lose many marks because of it.

Therefore in this project I have tried to compile common errors

committed by the students. Moreover for many people math is a

very dicult subject and they struggle with it. Hence the intent

of this section is also to address certain attitudes and

preconceptions many students have that can make maths very

dicult.

rrors in algebra

The topics covered here are errors that students often make in

doing algebra. Mistakes are made by students in all level of

classes and many of these mistakes given here are caused by

people getting la!y or getting in a hurry and not paying attention

to what they"re doing. #y slowing down$ paying attention to what

you"re doing and paying attention to proper notation you can

avoid the vast majority of these mistakes%

Division by zero

Everyone knows that the problem is that far too many people also say that

2/0=0

or 2/0=2 Remember that division by zero is undefined! You simply annot divide

by

zero

"ere is a very #ood e$ample of the kinds of havo that an arise when you divide

by zero


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%o& we've mana#ed to prove that ( = 2! )ow& we know that's not true so learly we made amistake somewhere *he mistake was in step + Reall that we started out with the assumption

a = b "owever& if this is true then we have a b = 0 %o& in step + we are really dividin# by

zero *hat simple mistake led us to somethin# that we knew wasn't true& however& in most

ases your answer will not obviously be wron# ,t will not always be lear that you aredividin# by zero& as was the ase in this e$ample You need to be on the lookout for this

kind of thin#

Remember that you -.)'* divide by zero!

Bad/lost/Assumed Parenthesis

*he first error is that people #et lazy and deide that parenthesis aren't needed at ertainsteps or that they an remember that the parenthesis are supposed to be there f ourse

the problem here is that they often tend to for#et about them in the very ne$t step "ere are some

e$amples

E$ample ( s1uare $


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%ine we are subtratin# a polynomial we need to make sure we subtrat the 3"4Epolynomial! *he only way to make sure we do that orretly is to put parenthesis around it

)ote the use of the parenthesis *he problem states that it is 56 times the 3"4Einte#ral not 7ust the first term of the inte#ral 8as is done in the inorret e$ample9

Additive Assumptions

%ine 2(x +y) = 2x + 2y students assume that everythin# works like this "owever& here isa whole list in whih this doesn't work


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Canceling Errors*hese errors fall into two ate#ories %implifyin# rational e$pressions and solvin#

e1uations 4et's look at simplifyin# rational e$pressions first

So note that whenever dividing anything ( in this case x ), the term has to

be divided from all the terms of the numerator (in this case )

)ow& let's take a 1uik look at anelin# errors involved in solvin# e1uations


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"ere's the orret way to solve this e1uation :irst #et everythin# on one side thenfator!

Therefore, while some simplification is a good and necessary thing, you shouldNEVE divide out a term as we did in the first attempt when solving! "f you do this you#"$$ loose solutions!

Proper Use of Square Root*here seems to be a very lar#e misoneption about the use of s1uare roots out there %tudents

seem to be under the misoneption that

*his is not orret however %1uare roots are .43.Y% positive or zero! %o the

orret value is *his is the )4Y value of the s1uare root!

*his misoneption arises beause they are also asked to solve thin#s like

-learly the answer to this isx = and often they will solve by ;takin# the

s1uare root< of both sides *here is a missin# step however "ere is the proper solution

tehni1ue for this problem

)ote that the shows up in the seond step before we atually find the value of

the s1uare root! ,t doesn't show up as part of takin# the s1uare root

Ambiguous Fractions*his is more a notational issue than an al#ebra issue sin# a ;/< to denote a fration& for instane2/6 isn't really a problem but what about 2/6x> *his an be either of the two followin#


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frations

,t is not lear from 2/6x whih of these two it should be! You& as the student& may know

whih one of the two that you intended it to be& but a #rader won't .nother e$ample of this kind

whih also inludes the use of parenthesis is for the frations of suh kind ften students

who use ;/< to denote frations will write this is fration as a + b/c + d *hese students know

that they are writin# down the ori#inal fration "owever& almost anyone else will see the

followin# *his is definitely )* the ori#inal fration %o& if you ?%* use ;/< to

denote frations use parenthesis to make it lear what is the numerator and what is the

denominator %o& you should write it as

Trig Errors

Proof for the first e$ample& let's 7ust pik a ouple of values forx andy and plu# into the first

e$ample

where 8in radians9 = (@0 8in de#rees9

%o& it's lear that the first isn't true and we ould do a similar test for the seond e$ample

it may be possible for some values of $ suh as as both the sideswould beome zero but these are not appliable for any other random values .lso these rules #o

same for the other tri#onometri funtions as well

Poers of trig functions

Remember that if n is a positive inte#er then

but sin $An 8sin $9An beause ,n 4"% we takin# the sine then s1uarin# result and in theseond we are s1uarin# thex then takin# the sine

!nverse trig notation


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*he 5( in is )* an e$ponent& it is there to denote

the fat that we are dealin# with an inverse tri# funtion

COMMON ERRORSPa" attention to restrictions on formulas

%any a times the students forget about the restrictions on any formula or they may thin&that the restriction is so obvious that they don't need to understand it! Then eventuallythey forget all about it as it is very rare that those restrictions would be used!:or instane& in an al#ebra lass you should have run aross the followin# formula

so let's take an e$ample to understand the restrition behind it

%o learly we've #ot a problem here as we are well aware that *he problem arose

in step 6 *he property that , used has the restrition that a and b an't both be ne#ative,t is okay if one or the other is ne#ative& but they an't B*" be ne#ative!,#norin# this kind of

restrition an ause some real problems as the above e$ample

shows

.nother e$ample is in alulus ne of the most basi formula in alulus is

,n order to use this formula n ?%* be a fi$ed onstant! ,n other words you an't use

the formula to find the derivative ofx^x sine the e$ponent is not a fi$ed onstant ,f you

tried to use the rule to find the derivative ofx^x you would arrive at

and the orret derivative is&


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%o& you an see that what we #ot be inorretly usin# the formula is not even lose to the

orret answer

*oo often students make the followin# lo#i errors %ine the followin# formula is true

where a and b anCt both be ne#ative there must be a similar formula for

,n other words& if the formula works for one al#ebrai operation 8ie addition&

subtration& division& and/or multipliation9 it must work for all *he problem is that this usuallyisn't true! ,n this ase

%o& don't try to e$tend formulas that work for ertain al#ebrai operations

to all al#ebrai operations

Calculus errors

Proper use of the formula for ?any students for#et that there is a restrition onthis inte#ration formula& so for the reord here is the formula alon# with the restrition

*hat restrition is inredibly important beause if

we allowed n = ( we would #et division by zero in the formula

THIS ISNT TRUE!!!!!! *here are all sorts of problems

with this :irst there's the improper use of the formula& then there is the division by zeroproblem

*he orret inte#ral of

!mproper derivative notation

3hen asked to differentiate f (x) =x (x6 2) , sometimes answers are li&e

*his is a#ain a situation where you may know whatyou're intendin# to say here& but anyone else who reads this will ome away with the idea that

and that islearly )* what you are tryin# to say "owever& it ,% what youare sayin# when you write it this way

*he proper notation is


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Some more errors that should be avoided

Do not assume that you will do everythin# orretly and 7ust write the answer .lways

hek what you do

.lways hek whether your answers make sense or not %uppose you are alulatin# interest then the amount of money should #row& so if you end up with less than you started you've made a mistake somewhere

)ever assume that if somethin# is easy in lass it will be easy on the e$am as well

maths common error

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