The worst ALGEBRA MISTAKES I see as a Math Prof **don't do these**
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- Опубліковано 12 лип 2024
- Please tell me you don't screw these up!!
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In this video I'm sharing the five biggest categories of algebraic mistakes that people make in calculus courses. Because often it isn't the calculus, it is the algebra that is the problem. What algebra mistakes do you make?
0:00 Intro
0:39 Cancellation
2:16 Maple Learn
3:48 Fractions
6:34 Everything is Linear
8:58 Brackets
10:12 Exponents
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The real victim in cancel culture is algebra.
🤣🤣🤣🤣🤣🤣🤣
I’m dead 💀 😂 😆
😅😅😅💀💀💀
Exactly, we live in an age where feelings matter more than the truth! It feels like I can divide by it, so we ought to be allowed to do it.
No the real victims are college student's cause of algebra
I once made this mistake on my Numerical Analysis exam: 3 + 3 = 9. The professor laughed-out-loud when he gave me back my test, in good faith, of course!
I once wrote - 3-6= 3 lol
lol I remember having such equations like (x-1)(x+1) + 1 = 0 and I'd simply say x-1=0 or x+1=0.
in short words. be happy whenever you see fractions or multiplication because addition/subtraction ruins almost everything
I did the same thing on a calc exam too, and at the last step too, missing out on the answer :(
A bunch of people on this sets functions and relations test recently put 1 + 1 = 1
I had 3x+3=27 and I put down 3 as an answer. As my brain interpreted it as 3^3 but x=8. Now I double check and take my time with medium sized problems. Taking your time when you can is very important it could be the difference between an A and a D
"there is a tendency to cancel everything that is in sight that looks vaguely related, you've got to pause and think: "hold on, is this the exact same thing?" It is only then when you are allowed to cancel". Damn, I wanted to be lectured on math, not in life
The maths mistake I made that I can't forget was : 2 + 3 = 6 (instead of 5). My mind processed multiplication in place of addition. And that was the only thing I missed in that paper and that maths course that year. I can't forget because my teacher told us it was impossible to hit an 100/100 but with just this mistake, it costed me 3 marks and ended up 97/100.
It taught me to always recheck my work.
that's a classic!
Another one...-3^2 vs (-3)^2, mainly a calculator error when students are trying to square -3 but they get -9 in the calculator based on how they enter the expression. Squaring a binomial and forgetting the middle term is probably the most common algebra mistake I see in my students. Second most common is "cancelpallooza". Great Video!
Oh ya that's a big one. Basically all versions of parentheses failures where parentheses are either implicitly or explicitly in the wrong spot.
@@DrTrefor that is weird because every calculator I program always uses Math.pow(x, y) method. Take -3^2 or (-3)^2, both seen as Math.pow(-3, 2), which then always returns 9, as it breaks it down into -3 * -3, which can only equal 9, regardless of parentheses.
Math.pow(-3, 3) = - 27
Math.pow(-3, 4) = 81
And so forth.
Your voice is exactly similiar to Zach Star. Wow
I just noticed that
Lol, yes they're very similar for real.
I just looked up algebra mistakes (I’m having my hs finals, and now I’m just polishing my stuff), and just watched his video before this :P
Is better
One algebra mistake that i often see and do sometimes is to not use parenthesis in functions like cos,sin or ln.
we often see sinx or lnx instead of sin(x) or ln(x) which is a problem when used in a longer formula. For example ln x+1. Unambiguous syntax is really bad.
You mean _ambiguous_ syntax is really bad. Unambiguous syntax is really _good._
*"smash the like button cause you know we're mathematicians we like algorithms, UA-cam likes algo-"*
**Me on tears**- enough man enough are you gonna make me cry?
haha:D
Such great points. One of the things I realized after a while in college was that it is these holes in my knowledge, things that I should have learned earlier, and mastered, but didn't, that were tripping me up in areas like Proofs and Calc (other subjects too) . When I was doing A level maths I found cancellation to be a nightmare, I really didn't know my stuff. Only after lots of practice did it come more naturally.
practice is really just so important for these types of things
omg I’m in the same position doing my A levels and it feels like i never truly got the full theory on just how the basics work and function because i still make stupid unintuitive mistakes when im doing new questions/concepts and not thinking and i feel like this is the kind of thing that should just be intuitive if i need to build on this knowledge
I like that theres no "have a fantastic day and see you next time" its just "in the next video lets do some more math" cause thats enough kindness
that's all math majors need
wow...i really like your T-shirt
Isn't it cool? Best birthday gift from my wife ever:)
@@DrTrefor where did she get it? the one on your link isn't the same one
I just clicked the link and scrolled through 'related items' and found it... unfortunately some of the graphs are wrong in both versions (such as the y=logx graphs)
In my first day of teaching Math back in September of 2008, a 12th grader did the same thing. I was astounded.
I’m STILL surprised even after all these years
which thing?
@@webknowledge9989the same thing.
@@webknowledge9989 that thing
Great video and timing(just before my exam) ! This video should be shown before every mathematics course not only algebra.
good luck on your exam! And don't make any silly mistakes lol:D
Misconceptions about algebra don’t seem to be my biggest problem. It’s writing things incorrectly despite understanding how they work. Linear algebra is going to be death of me 😂.
Yes. My primary issue is transcription errors. Are there any tips for this? There are so many opportunities in Linear Algebra!
you are so down to earth and compassionate... considerate to those that do want to learn.... you do everything you can to not intimidate us .... we who know that math is important and how it opens to us the sciences that may interests us. Praise the lord !
What? Are you kidding? I am fucking scared to death now that this overly smiling guy will come kill me in my sleep next time I forget a parenthesis. D:
Brilliant videos prof :) Keep the quality up
Glad you like them!
Will definitely help once I get into Ap calculus.
7:34 "All those others, not linear!"
y=-x on the third row: Am I a joke to you?
Yet seriously, that was a very nice video! Keep it up! :)
hahah missed that one:D
I love that last one. I'm learning pre-algebra (and algebra in my free time) and I know a lot of these.
3:24 this is so mathematiciany
I once said that a piecewise function f(x) was continuous by calculating the limits from both sides and seeing they were equal, without even considering that f(a) was not defined at the piece break 'a' I was calculating continuity at, so it fails the very first step of the 3-part definition of continuity. Another time, I was asked for differentiability at a piece break, and I differentiated both equations of the function and once again took the limits on either side and saw them to be equal. I immediately said f(x) was differentiable at x=a without checking for continuity, and f(x) turned out not to be continuous so I got it wrong. Luckily these two mistakes weren't on the same test. They aren't really algebra mistakes or calculus mistakes, they're just intuitive errors. Obviously if you look at a graph and a point is not shown to exist, that's not continuous. And if you can't even say a function is continuous at a point, you can't take the slope of the tangent line at that point either.
Me after watching this video : I'm gonna pretend I didn't see that
Another error I saw in another video that is related to the ambiguous multiplication/division issue: Typing 1/2x into a calculator and calculator does (1/2)x when you probably mean 1/(2x).
That kind of thing is also very good for trolling on the Internet. Ask for 6/3x when x=2. Let people fight over whether the answer is 1 or 4.
Thanks Professor 👍🏻
Amazing video as always!
Thanks so much!!
Maple learn is dope! 😎 If I didn't watch this video then I would never have seen it! Thanks a lot man.
Glad you liked it!
tell me that after using it for a math class. the thing is sooo dumb man. it might legit give you the wrong answer if you write something similar to 4x instead of 4*x. lmao
first ten seconds is so true in any high school calculus class, so much could be a achieved in an ap calc ab or bc class if the calc teacher wasnt forced to go back and have to reteach what should have been knowledge known prior to the course. Of course we as the students wouldn't know this because we're just trying to get through each course. You can't blame the student for not knowing certain topics that are "basic agl II and pre-calc topics," any student taking an ap calulus class isn't in there by chance, and aren't incapable of learning, the teachers need to take responsibility for theses things, and it's so apparent in my ap calculus class. Sorry for rant, but this is brought up every time our teacher asks if we were ever taught a certain skill in a pervious course. Fortunately, I was blessed with an amazing teacher and she makes sure all of the skills we were supposed to learn before this class, are strong enough for her to actually use them while teaching us new content.
I just subbed to your channel, because I want to improve my Math grades
You can do it!
Hello Dr Trefor, greetings from Costa Rica.
Love to see your videos, and also in this case your t-shirt haha
Greetings! I’ve noticed a few viewers from Costa Rica actually!
I dont make that kind of mistakes, but sometimes when i dont have much time to finish a test and i have to think as fast as i can (which is really stressful) it is possible to do mistakes that i wouldnt do if i was calm😅.
THAT T-SHIRT IS AMAZING BY THE WAY.
7:06 ln(a+b)=/=ln(a)+ln(b)
Ah yes, another victim of Log(1+2+3)
As an old timer, I'm of the opinion is that not sufficient homework assignments in HS is at the root of this deficiency in algebra.
Saaaaaaaaaaaaaaaad
It definitely is true that these types of things are a lot about practice, practice, practice. But it also needs to be practice with some intentionality so you identify and correct bad habits.
@@DrTrefor There's nothing like putting pencil to paper to ingrain math problem solving skills in the mind and in the brain. Reliance on technology to teach math and engineering is short-circuiting the creation of enduring neural pathways which cement those skills.
But as as I'm and old timer ...
@@lgl_137noname6
There is no skill needed here. There's conceptual misunderstanding based on this word "cancel".
The splash of this video actually made me scream from horror.
I am happy that my ap calc teacher taught me these rules when I first took calc because if I didn't know them than I think I would of done pretty terribly compared to what I did
I really think the cancelling errors is because it makes the problems look cleaner in the end. I think people forget that when you change the form of an expression then you are merely finding an equivalent form so (3e^x - 6)(3cosx) = (3)(e^x - 2)(3)(cos(x)). Dividing so that you get rid of a number is saying 3/3 = 1 so it doesn't change the value of that expression and you can only do that once when dividing. As a general rule in algabra If you can't show that your process gets the same number as the correct process then it is wrong.
Im a math tutor and the illegal cancelling of terms shows up a lot.
6:22 I always say brackets are the clearest thing to use because it's its own symbol, it isn't just a longer version of something else, or the same symbol again.
I also use different brackets ([7{2π}^2]/3)and different colors to help.
To see that people actually make these mistakes me a little happy :))
I see the exact same mistakes in my Ap Calculus class in high school.
0:39 Cancellation. 1 is the identity operator of multiplication and division. (0 is the identity operator of addition and subtraction e.g. 0=5-5.) I call it "removing the explicit multiplication by 1". n/n =1. (Don't divide by 0!) So, if you can change the expression such that it is multiplied by n/n, then that is multiplying by 1, which doesn't to be explicitly written. You can still cancel your cable service.
Oh! My favorite is just plain forgetting to carry over some symbol from one line to the next, like just forgetting about a minus sign somewhere. Whoops!
We need more like these videos
That last one blew my mind
I don't know why I'm here, I hate maths more than myself but this information is sacred
maths loves you though, ha:D
I still remember having a drink before a mid term (more than a quarter century ago...) I finished up a proof with the phrase that "since 2+2=5 the proof is true." The TA had a large quantity of red exclamation marks. My friend swore by the idea of having a drink to relax before an exam, I never did it again.
lol a classic
Well, most people don't become incapable of arithmetics after one drink.
My question is that how do you pull yourself together in this kinda situation 😂😂😂
At 8:50 I would also mention the correct sum of squares identity: (a+b)^2=a^2+2ab+b^2.
On my Calc 2 exam I made a small algebra error. Took a Maclaurin series, factored out a 4, but forgot to change the coefficient of x and got it wrong.
Because mathematicians do have feelings ;)
haha it's true!!
What mathematian have feelings?? 🤯 I thought they were gods so they don't have feelings 😂😅🤯
Thank you! Now I can easily refer others to a nice recap. Appreciate the welcoming tone in this whole constructive criticism. "Yes, it's tempting and I get it. Let's do it wrong all the way first and see where we'd get, then check the proper way."
I think you are gonna hit 100k mark soon...😊😊
All the best Sir 👍👍👍
Haha excited for that silver play button:)
sqrt(3)/3 = 3^.5/3^1 = 3^(.5 - 1) = 3^-.5 = 1/sqrt(3)
so you can cancel out the 3 in the bottom if you leave your exponent of the 3 to -.5.
Commenting for the algorithm.🙂
I laughed out loud 🤣 at some of these errors.
I had done all these mistakes 😂
And then there's that whole forgetting to multiply by the Jacobian!!
I barely passed my cal II not because I found it hard or I did not understand the material, but because of my sloppy Algebra that I didn't care about in high school. 🤦♂
Coordinates lol always mix up Len + sin(angle) for x and Len - cos(angle) for the y
1:03 we also must mention that
X 0 ( is not equal)
0:40 truly described twitter.
the digital pen u used was buyed or made in home? (yes you can make it in home*)
what a intresting video i love it waaiting for more
Here is a number for you. 27.. this is exactly how many seconds were needed for me to start liking you.
3:07 another mistake: forgetting that the expression is not defined for x=0
Sir please make video on olyampaid preparation...
Oopsie! Thanks for the video Dr. great as always!
Glad you enjoyed!
"All the others are not linear"
y=-x crying in the bottom
To be picky we don’t really cancel....if the numerator and denominator are equal, the answer is one (1)
My biggest problem is going too fast thus writing the wrong operator between plus/minus.
I NEED THAT SHIRT RIGHT NOW
My maths teacher made the first mistake. I pointed it out, she told me I was wrong. She wasn't a nice person anyway, the fact her maths teaching was bad didn't help
as a Teacher of Maths I'm truly fed up of my students doing these mistakes.
that thumbnail had me choke on my food lol
haha me too:D
I have a problem with these, what are some resources to learn it!
Also for exponents, when we got something like this
ₓ²
b
this is considered as b^(x²) rather than (bˣ)^2, when we have more than one power, we always start from the top right and then go down one by one finally down to the bottom left(unless there is a parenthesis).
I'd say that's another similar mistake for the complex fraction 16/4/2, but we have the rule "top right to bottom left" for the exponents.
Where did you get your shirt?
Twitter has been silent since this dropped.
7:33 actually, y = -x is also linear ;)
Haha true!
Sometimes however : ln(1+2+3)=ln(1)+ln(2)+ln(3)
Yup that's a common one, thinking ln is just a linear function
But I like this one because it is correct. I wouldn’t discount points, maybe even add one :-)
Hi. In 10:12 you wrote x^-3/2 is same as x^3/2. How is that possible?
Thanks, could you tell about the tab model you're using? thanks
It is an ipad air
@@DrTrefor thanks , what is the best option for teaching? (university level) (using projection with data show) thanks
@@SkanderTALEBHACINE honestly they are all getting pretty good, so for me it is just about gettting one with a good stylus for writing that's all i care about
@@DrTrefor Thanks
Love the shirt 😆
hey!
can i ask what tablet and pen you're using!
Either iPad Air or iPad Pro
No link to the T-shirt?
SMASHED that like button. Destroyed that like button
That's what I'm talking about lol
I have a question:
for x + 3 = 5 we see the solution is 2.
but isn't the solution also -8?
To get both solutions we can square both sides and solve the quadratic equation: i.e.
(x+3)^2 = 25
...
...
x = 2, x = -8
But clearly if we substitute back into x+3 = 5 where x=-8, we don't get 5.
So my question is, why is squaring both sides of an equation considered a valid mathematical operation?
Use change a linear eqn into quadratic. So quadratic get 2 answers, -8 for eqn (x+3)²=25. Don't make quadratic answer to linear answer.
Quadratics have at MOST 2 solutions, sometimes you only have 1 solution which is valid and satisfies the equation. Also you shouldn't be squaring linears in the first place
The problem seems to be the word "cancel". You should try avoiding that when teaching students and just speak of equivalence tranformations and implications.
You need a firm Algebraic foundation to do calculus. Doing excessive cancellations is plain carelessness. My worst mistake is accidently changing a negative number to a positive one. You can do pages of calculations correctly. That one simple mistake will screw up the answer entirely. Watch those tiny details! PS: PEMDAS does not always work in basic math, and should never be taught in schools.
What I see is that people makes these mistakes when they don't know the correct method of solution. If they get an integral which they remember how to compute, they can do it correctly without algebra mistakes, but on the next exercise they can make the most absurd algebra mistakes in an effort to get anywhere because they don't remember how to solve that particular type of problem.
I think that’s right, all sorts of “creative” solutions happen when you don’t know what to do
When i see two things that look alike i a soon they cancel regardless of their signs
Maple appears to ignore domain restrictions.
The core Maple engine definitely can. The Maple Learn web app which is based on the Maple engine is quite a bit more limited right now in its early phases, this is something I can bring up with the developers. What type of functionality specifically were you thinking about?
Заранее спасибо
Your t shirt is so cool
I know the concept and how to do
But i haven't got full score becz of these mistakes😭😂
My common mistake is dropping the negative sign when solving. 😁😁
oh I HATE negative signs lol, bane of my existence:D
@@DrTrefor ohhhhh yes i didnt know a math proffessor shares my opinion
4:00 Actually, I think I would prefer the 16/4/2/1 interpretation, if not indicated otherwise by brackets like 16/(4/2), because this is the normal order of operations that we all learn in school. Also, I think it is mentally easier to imagine a "silent" one as the last denominator than it is to insert it into the middle of a fraction.
If you see that written down, you just have to slap the face of whoever made it! No need to interpret anything
@@davidp.7620 -- Wrong! You need to get educated.
*is equivalent to*
@@robertveith6383 the whole point of Math notation is to talk about numbers, sets, functions and so on in an unambiguous way. If you make things deliberately confusing it's not the reader's fault for being "uneducated".
i don't think uni students could make this kind of mistake!
You wouldn't think so, but they all happen surprisingly often!
It might come as a surprise that depending on how maths is taught prior to uni, students have wildly different misconceptions or aversions. Teachers can instill some PTSD variations in students for each subject available. E.g. I had to learn integrals by heart in high-school because even our teacher had no idea what was going on and that person left a stain on what maths was for me. Only until years later in engineering it started to make more sense and I still have work to do in order to overcome that bitter taste left behind. That's why I'm glad channels like this one exist: they maintain a cosy space to come back and try again.
Nice shirt with the shape of graphs.
that's a really cool shirt! 😂
In an alternate universe, (a ± b)ⁿ = aⁿ ± bⁿ
Not even algebra is safe nowadays