Math Teacher Shows TOP 10 MISTAKES students make

i've been teaching high school math for 10 years and every year students in every grade are making the same mistakes here are the top 10 mistakes that are costing you marks and frustrating your teacher oh this is an easy one i've got negative 3 squared i remember the trick my teacher told me if i've got an even exponent on a negative base i know my answer is going to be positive so this one uh negative 3 times negative three that's positive nine perfect the answer is actually not positive nine you've just misinterpreted what the base of the power is let me make some notes over here showing you how you can correct this so negative 3 squared that negative is not part of the base of the power it's not in brackets with the 3. so negative 3 squared you need to think of that as a negative 1 multiplied by 3 squared which means what we have is negative one times three times three which is of course negative nine so for that question the base of the power is just three there's a negative in front of the power what you thought it was you thought it was this if the question was written like this negative 3 in brackets squared that would mean the base of the power is negative 3 which means we have negative 3 times negative 3 and that is positive 9. so make sure you know this is not the same thing as this pay attention to what the base of the power actually is oh i love it when stuff cancels out in questions this one's easy i got a 2 in the numerator a 2 in the denominator anything divided by itself cancels out right so all i'm left with is x plus 8 easy [Music] no sorry that's not correct but this is actually a super common mistake to actually forget to divide both of the terms by the denominator you need to divide the 8 by 2 as well let me show you how to do that okay one way you could do this that you don't make that mistake would be to separate this into two fractions being added since the 2x and the eight are both being divided by two i could separate this into two x over two plus eight over two which is x plus 4. notice the 2x and the 8 are both being divided by 2. another way you could do this i could common factor a 2 from the numerator giving me 2 times x plus 4 all over 2 and now that the numerator is in factored form you can look to cancel or reduce factors with each other so i see a factor of 2 in the numerator a factor of 2 in the denominator something divided by itself is 1 so i can cancel those out and i'm left with just x plus four so two different ways to think about getting that same correct answer but x plus eight no okay i just need to simplify this i remember distributive property i know you just take whatever's in front of the brackets and multiply it by everything in the brackets and then the brackets are gone so i've got this nine i'll leave that there minus and then i have to do five times x that's five x and five times negative eight that's negative forty and then i can collect my like terms nine minus forty is negative thirty one so i've got negative five x minus thirty one all right to start this one i'm just going to start collecting some like terms i've got a 9 minus 5. that's 4. it's going to be easier if i do that first and now i'll distribute the 4 in i've got 4x and then minus 4 times 8 that's 32. nice okay both of you actually just made one small mistake having to do with orders of operations let's correct them student number one over here on the left you shouldn't have just distributed the five you should have distributed a negative five if you were wanting to get rid of the brackets all at once if you did it the way you did it this product would still have to be in brackets and you'd have to remember to subtract both of these making this constant term wrong we'd be subtracting negative 40 which means plus 40 so you'd have 9 plus 40 which is 49. so you'd have negative 5x plus 49 not negative 5x minus 31. but that's not even how i would do this question like i said i would distribute the negative 5 and then that would allow me to get rid of the brackets all in one step so you'd have 9 and then negative 5 times x is negative 5x and negative 5 times negative 8 is positive 40. so when you collect your like terms like i said you'd have negative 5x plus 49. all right what happened over here okay this so yeah this this student didn't follow the correct order of operations we can't do a subtraction first this negative 5 is being multiplied by something so we'd have to do the multiplication before the subtraction so uh no right away i know the answer is going to be wrong because they didn't follow the correct order of operations always remember bed mass exponential equations these ones are always tricky for me okay i think oh all right this one's actually pretty easy i can do this one right three times two that's just six so i've got six to the x equals 36 and i know my powers of six six squared is 36 and my answer has got to be two [Music] before i show you where the mistake in this question is let me remind you that when solving equations you can always check to see if your answer is right or wrong by plugging into the original equation so if i plug 2 into the original equation i'd have 3 times 2 squared does that equal 36 well 2 squared's 4 3 times 4 and 3 times 4 is 12. uh 12 is not 36. so that shows you it's not the right answer but what did you do wrong you can't multiply the base of a power by a constant that's not how it works so what we would have to do to solve this question is divide both sides by three to start off with so would have two to the power of x equals 36 over three which is 12 and really what we're looking for is what exponent goes on 2 to get 12 and logarithmic functions can find us missing exponents the exponent would equal a log base 2 of 12. and you'd get an approximate value for that if you wanted but that's the exponent that would satisfy this equation okay fractions fractions are sometimes tricky to remember all these rules but um okay i think the first okay so i know bad mass i have to do the multiplying first so i'm going to multiply okay i have to do three times a fifth okay three times one is three three times five is fifteen okay that looks good plus one over two and then adding fractions okay so three plus one is four fifteen plus two is seventeen four over seventeen i think that's good so you've made a couple mistakes with fraction rules here the first of which three times a fifth isn't three fifteenths let me show you how you can avoid making that mistake of thinking you multiply a constant into the numerator and denominator right it only it only multiplies by the numerator let me show you why so you can think of that three anytime you have a whole number and you're working with fractions i find it helps a lot if people rewrite that whole number as a fraction over one so that's three think of it as a three over one so i've got three over one times one over five and when multiplying fractions you multiply the numerators three times one is three and multiply the denominators one times five is five so the correct product of the first two is three over five the three just gets multiplied by the numerator and when adding fractions do not forget to get a common denominator one fraction has a denominator of five the other denominator of two a common denominator between those two would be the product of them ten so i'll make this one be ten by multiplying top and bottom by two and this one be ten by multiplying top and bottom by 5 and that gives me 6 over 10 plus 5 over 10 and when adding fractions you keep the common denominator don't add the denominators like you did over here you keep the common denominator and just add the numerator 6 plus 5 11. okay i remember my teacher saying i have to do the opposite operation to move a term between sides of the equation um okay on the left i see a three so when i move to the right i think i'm going to divide by three which means okay the three is gone from the left which means i just have the x left and on the right i've got six over three which is two oh this equation over here negative two times x equals eight the opposite of multiplying by negative two well the opposite of negative two would be positive two and the opposite of multiplying is dividing so i would divide by positive 2. so i get x equals 8 over 2 x equals 4. there we go both of those done so you're right you do have to do inverse operations to isolate a variable it's just you chose the wrong operation so you got incorrect answers and don't forget you can check to see if your answers are right by plugging into the original equation if i plug 2 in for x into the first equation 3 divided by 2 isn't 6. if i plug 4 in for x into the second equation negative 2 times 4 is not 8. all right let me show you how to correct the first one to avoid making this mistake i think you should try and approach it from more of the balance method when solving equations so whatever you do to one side do it to the other side so when your variable's in the denominator let's multiply both sides of the equation by that variable to get it out of the denominator so x times 3 over x equals 6 times x i just multiply both sides by x and the reason why i did that is because now we have x over x on the left which cancel out and we've got 3 equals 6x then divide both sides by 6 and i get a half equals x and if we check that in the original equation 3 divided by a half is 6. all right let's correct this one i understand you're thinking here but if we divided both sides by 2 that doesn't cancel out this negative two negative two divided by two is negative one so we'd have negative x equals four what we should do instead is to the original equation divide both sides by negative two the opposite of multiplying by negative two is dividing by negative two and you can see why because these factors of negative two cancel out leaving us with x equals negative four if we check that in the original equation negative two times negative four is positive eight i love it when these easy distributive property questions show up in higher level courses i know how to do distributive property i can just multiply these right into there sine times x plus y well that's just sine times x plus sine times y and same with log log times x plus 4 that's just log times x plus log times 4. [Music] so both of these are incorrect it's a super common mistake to think that sine and log can be treated like numbers and distribute them into their own arguments sign and log are functions though not numbers what we have with both of these aren't a product of things it's not sine times x plus y and log times x plus four it's sine of x plus y and log of x plus four you can't distribute a function into its own argument so don't try and treat sine and log as if they were numbers and try and do distributive property sine of x plus y it's actually fairly complicated there's an identity for that if you wanted to rewrite sine of the angle x plus y it's equal to sine x cos y plus cos x sine y and log of x plus four there's there's there's nothing we could do to simplify that you can't expand a log into its own argument so don't try and do that i just have some expressions here to simplify i remember my exponent laws um okay i've got things being multiplied i know how to multiply oh i think i remember okay you leave the base yeah and they're being multiplied 5 times 3 15. this next one okay this time the powers are being divided so i keep the base and eight divided by four two and my last one oh and this one here they're being multiplied oh four times four that's 16 okay so my base is 16 and then i think i remember something about when multiplying powers you um i think you add the exponents three plus two five sixteen to the five good okay so all three of these are incorrect let's start by looking at the first one so we do have powers of x being multiplied together and when you're multiplying powers with the same base you keep that base of x but you add the exponents 5 plus 3 is 8 so we'd have x to the 8. and if you want to see why you can take the time to write it out in factored form right x to the 5 means x times x times x times x times x and we have to multiply that by x cubed which means multiply by three more factors of x and if you look at how many x's we have they're being multiplied together there are eight of them x to the eighth if we look at the second one dividing powers with the same base yes you keep the same base you did that well but we have to subtract the exponents not divide them 8 minus 4 4. and if you want to see why write it out so we have 8 factors of x divided by 4 factors of x and you notice that 4 pairs will cancel out one pair two pairs three four so what do we have left in the numerator we have four factors of x hence x to the four okay the last one was so close um multiplying powers the same base yes you do add the exponents the exponent is 5 but you forgot what you did up here you keep the base the same it should be 4 to the power of 5. you don't multiply the bases together you keep the base the same and add the exponents oh solving an equation easy inverse of squaring is square rooting so on the other side if i move the squared it becomes square root i get the square root of 9 which is 3. and let me check and see if that's right if i plug it back in is 3 squared equal to 9. yup 9 equals 9. good i know i got the right answer this time okay you did get one correct answer for this question 3 is the correct answer 3 squared does equal 9 but it's not the only correct answer the inverse of squaring is actually plus or minus square rooting so you should have got plus or minus 3 as your answer right if you square negative 3 you also get 9 so that's also a solution to the equation so this is just an example how some equations can have more than one answer so make sure you find all answers that satisfy an equation this one okay the base of my power is being squared i can just put this exponent 2 on the x and the three so i've got x squared plus three squared oh and i'll make sure i simplify that three squared is nine perfect okay so this is incorrect you can't take the exponent and put it on both terms in the base if in the base the terms are being added or subtracted this is the most common mistake made in all the high school math courses so try never to make this mistake look at the base of your power if it's terms being added or subtracted you can't put the just put the exponent on both so how can you not make this mistake rewrite that x plus 3 squared as x plus 3 times x plus 3. that way you can see what you need to do is you need to find the product of these two by foiling it i need to do the product of the x's plus the product of x times 3 plus the product of 3 times x plus the product of 3 times 3 which gives me x times x is x squared plus x times 3 is 3x plus 3 times x is another 3x plus 3 times 3 is 9. so yes there is an x squared and a nine but there's also a middle term it's x squared plus six x plus nine i think maybe what you were thinking of when you tried to do this like let's say it was three times x squared yes there is an exponent rule for that if the base of the power is a product of two numbers we can put the exponent on both of the factors of the base so yes this would equal 3 squared times x squared which is 9x squared so in that case you can but if the base is a sum or difference of things you cannot do that you have to take the time to actually expand it out all right now that i've gone over these common mistakes with you i hope you never make the common mistakes that were made in this video why not test yourself now here's a eight question true or false test try and figure out if each of these statements are true or false post your answers down in the comments let's see how you do

Channel: JensenMath

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Id: _E6VEU491eA

Length: 18min 43sec (1123 seconds)

Published: Sun Dec 06 2020