"HONORS" ALGEBRA STUDENT FAILS A BASIC ALGEBRA TEST
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- Опубліковано 9 лют 2025
- A student who thought he knew all about Quadratics came to City Tutoring yesterday and realized he'd been misled about his Math level in public school. What's going on in our schools, America?
Attention ALL: There is a typo in PART A #6 -- the equation should read as: y = 2(x^2 - 16x). Even so, the student was not even able to identify that it was not a quadratic, and did not ask me about it. An honors student should be able to recognize a basic quadratic form, and the fact that I was not even asked about the typo is very telling.
What is your opinion on the Art of Problem Solving Academy and Beast Academy math curriculums? What do you think is the best K-12 math curriculum? I grew up using the University of Chicago School Mathematics Project books...have you heard of them? Have you thought about designing your own K-12 math curriculum?
For question 1 in Part B, I thought about this in terms of the product of order pairs. Which set of ordered pairs results in the largest product?
(10, 300), (11, 285), (12, 270), (13, 255), (14, 240), (15, 225), (16, 210), (17, 195), (18, 180), (19, 165), (20, 150).
The values are 3000, 3135, 3240, 3315, 3360, *3375*, 3360, 3315, 3240, 3135, 3000. There is a "symmetry" in this list of products such that the dollar amount increases until the fare is raised to $15, but thereafter it starts to fall; thus, the most profitable fare for him to charge is $15 resulting in a revenue of $3375. Is this the correct answer?
I had university of University Chicago school mathematics also. I don't remember if it was good or not, but I went on to do pretty well in math. I guess it did the job.
@@tommiest3769the equation for total revenue is (10+x)*(300-15*x) for fare increase x. This is a quadratic equation and you can solve for its maximum value (because it's a concave down parabola).
I do wonder about the honors student. Did you talk to them about how they came up with the answers and why they didn't solve the problems? 1 and 2 part A feels like a technical issue with not reading the question for standard quadratic equation, but 3 part A feels alarming. Frankly speaking, if that's the work of honors student, how's the performance of non-honors student?
@@tommiest3769 I am not really keen on AOPS because it does not take a set theory approach, and their books are mostly geared to train students who want to do well in Math Olympiads. Nothing wrong with that at all, and in fact their material is quite good and funny, BUT I would not use it in a regular structured math class. Yes, as far as the Chicago School....in fact, those are the materials we mostly use at City Tutoring, and take the same approach. Nice solution, by the way.
Why did I click on this video thinking it would be smth like someone doing finite fields failing a group theory test
LMAO
So real. Literally thought the same...
It’s because there is basic algebra, and then there is fundamental algebra
Yep, same haha
same LOL
I remember taking an intro to analysis test. When I got the test back the teacher had knocked off some points for not having, "Let epislon > 0" and "There exists a delta > 0." I went up to the teacher and told him I just forgot. He told me, "I bet you won't forget again." He was right.
An excellent response! Nowadays, I've had students tell me "I know this, but I just didn't put the decimal point like you wanted"....well, following directions is part of the learning process. Tough!
Yeah my professor got fired for grading that way. He taught analysis which was fine. We were mathematics majors, but he also taught complex analysis which engineering students took. They all complained to their engineering professors who were outraged at the mathematical rigor. They'd never seen anything so awful in their lives. They complained to the dean and nothing could be done. He got fired for demanding engineers be mathematically rigorous. Such is the state of education. I loved it. He taught me so much about math and proof writing. If you aren't channeling Rudin, then you lose points. Lol. And he got fired for it.
It's like forgetting the +C on an indefinite integral.
@@InfiniteQuest86 "Oh, I forgot the '+' on your A+ grade, so you just get an A"
I'm sorry but that's not a minor mistake, the whole proof doesn't work without that part.
Back in the 2000s, I was taking calc II. Due to scheduling during finals my teacher had a conflict so I had to take my final in an algebra class. One of the students asked the teacher something to the effect of, “How many points do I need to make on the final to receive a passing grade.” I could practically feel the disappointment emanating from the teacher. If only there were some mathematical tool to solve such a problem.
Dayyum! I used to walk into most math finals knowing what I needed to score to get an A.
@@briant7265: Why not with ALL finals? It's the same principle and process, whether the class is math or art history, etc.
@rogergeyer9851 OK. Really, all finals where I had a chance at an A. Which was most finals. I was trying to be a little bit humble, and you wrecked it. 😁
And this was in the 1980's, before they handed out A's like candy at Halloween.
As a retired college math instructor, I had some nightmarish flashbacks as I viewed this. And grade inflation is a real thing. Very very real.
Understandable because I am a student and after seeing this I'm speechless. Aren't these students screwed up?
He sounds genuinely pissed off.
I get that, especially when we should want the future of this country to be far more intelligent than we are.
That being said, this guy screwed up on two of the questions (6 with it being mistyped and the whole set notation question. To me, that puts me off because you are mad at a student and a system, when you are making the very same mistakes that are screwing with these students. I personally expect better.
good. someone should be. i feel sad for all the American students, I mean, how is the majority (students attending public schools) supposed to keep up with children from other parts of the world, and with people of more privilege, who can perhaps pay for a private tutor, or even a private school? having your education reduced to the point where an "honours" student can't solve a basic quadratic question? holy fucking shit.
@@troybaxter Test was pretty poorly written not going to lie. Nothing you wouldn't be able to overcome if you just asked to clarify them. It seems like this student at the very least seemed to be over confident and complacent with their understanding. It's evident of this from their lack of showing work or asking clarifying questions when met with ambiguous questions.
@@jjeastside the student is not blameless here, BUT the blame should be put upon the instructor and teachers first. It's their duty to lead and hold themselves to a higher standard, and if they can't do that, then either they own up to their mistakes or step down.
Real leaders do that! Not whine about it on UA-cam and deflect their own mistakes!
@troybax ter ye this video does come off as pretty pretentious. The student is there for tutoring, not for ridicule.
My personal opinion is that a hardworking student will always find a way to learn successfully regardless of the instructor. However, the same hardworking teacher cannot teach a student that doesn't want to learn.
This student in this case does want to learn and was just misled in their proficiency in math. This isn't directly the students fault but a fault on the educational system as a whole.
If you're not at the level you need to be in your classes, your grades should reflect that, so you have an understanding of what you are capable of. That will allow you to adjust what you need to work on so you can keep up with the curriculum. It's not the students 'job to interpret these fake 100% and A's as failing grades.
I remember halfway through Algebra II when we were going through warm-ups in the morning and our teacher was like... "Okay, here we have an exponential curve with positive slope, and here we have an exponential curve with negative slope..." and someone raised their hand and asked him how we could tell if a line had positive or negative slope... Halfway through Algebra II... The teacher was a great guy and just explained the concepts as politely as possible without shaming them but holy cow. This was in 2002 or so... Believe me, it's been this way for some time.
So it's reached a point where the current teachers are likely from the classes where people were so confused. And that makes it really tough to improve
Well Bush did start No Child Left Behind in 2002.
in all fairness #6 13:41 makes no sense because there is no square to complete, you just get y = -30x when you expand
I'm at a loss here as well.
You're right, though I think the creator's pinned comment addresses this well enough
He put 2(x - 16x) but it's a typo, he meant 2(x² - 16x)
@@sen8078thanks that makes so much more sense
I've completed my masters in maths and went onto teaching at secondary. I quit after two years, as I was being told off for teaching maths mastery, mathematical thinking and being too strict! With being too demanding with robust attitudes from students towards the discipline. School was interested in exam pass rates ratings more than the quality of learning, understanding and retention. Sad world.
I had to work so hard to earn my degree, whilst I seen some students on my course who cheated all their way through a degree and earned HONS as well.
I believe you. Unfortunately, due to "virtual" settings and the rise of AI, student cheating has become even more rampant. It is so bad, that all exams should be done in person, supervised, and homework should no longer be a part of the evaluation of grades, since all they do now is take a picture and try to use photomath or Chatgpt. It gets really complicated for cheaters though when they get to my class, since I actually send people to the board and make them prove their work, and they have no computer or phone to guide them. They truly have to "know", but that is because I manage my own academy. I would never ever subject myself to teaching in a public school today, given the problems you mention.
I remember several years ago, I was honestly shocked at a situation that happened. A friend of mine needed a favor for a substitution, so since I already had the credentials, he told his school about me and they let me substitute one of his classes for a week. I was used to doing things the way they were always done in my own school, and also what I do with students of my own, so when his 8th graders got loud, I first told them to quiet down, etc. When they would not, I shouted at them "old school" way and the class got real quiet. The next day, the school principal asked me about it and little did I know she was actually upset. She even questioned my credentials, asking me "are you really a teacher"? Yes lady, and I have been teaching probably longer than you have without taking a dime from government. Of course after that, they never invited me back. I'll be damned if I let students get out of control and do as they please.
But also some people aren’t cut out for teaching either.
Small distinction to make but important nonetheless: There is compelling evidence that the erosion of public education isn't a product of DEI; it's a problem of standardized testing incentives and the No Child Left Behind way of standardizing knowledge. (At least in the US) Teachers teach to produce results on things like MAP tests. Teaching to a test leaves gaps in knowledge. This problem is compounded by pressure from parents for "good" grades, this can appear to be DEI but is distinct from it. Overall the problems present erode teacher agency even further. But. That's like my opinion man. Almost every financial incentive is based on standardized testing, which we have slowly lowered the standards of over time to produce better "results". This is why fundamental gaps appear when the true measure of proficiency is given
As much as I admired him since he was my professor, I agree with you about DEI. Professor Cromwell means well, but he doesn't know what it's like to be non white and constantly having to justify your presence. That was my one area of disagreement with him. Great guy overall, but unaware of what minorities face.
I'm so glad someone mentioned this. He seems like he means well but to start off this video with political mobo jumbo is absurd.
@@deeplydispirit yeah that's always like that in his class too. Like hell be talking derivatives and suddenly and randomly we get like 3 minutes or more discussing inflation lol. Fun class, but wonky. It was definitely harder for minority students given the very conservative nature of him
I always get a kick out of people blaming DEI for everything. I mean there exist people who were educated before DEI was a thing, who are bad at math.
I don't know enough about this fellow to form any conclusion so I will not do so here. My main point was to direct attention to a specific mislabeling of a problem, I don't want this to turn into a "pile on" effect of bashing.
The American public education system is horrendously lacking and exists only to provide students with diplomas they may present to employers. This is not due to DEI, but to a lack of state funding, lax hiring requirements (caused by an instructor shortage), poorly organized boring curriculum, and much more. Mathematics is incredible; it demands respect and academic rigor, but engaging, proper instruction is key. I try to instill a love of math in my tutees, but I never sugarcoat things. I wish you and your students well, you are doing very good work.
If DEI didn’t block sorting students into classes based on ability, then there wouldn’t be much of a problem. Instead, teachers are forced to teach to lower 10%, dragging everyone down.
@@electricpaper269 but what you're describing is not DEI, that is "No Child Left Behind".
The shuttle question is just (10+n)(300-15n), the vertex of that is going to be at n = 5, so 15 is what maximises the profit.
I read it as $10 per fare for $300, which would make a capacity of 30 passengers per round trip then. It'd be (10+n)(30-15n). Vertex formula -b/2a gives us 120 / -30 = -4, which plugging in would give us n = $6 for a max revenue of $540.
@@musicboxboy6995 There is a typo in the problem. Alfred has a dollar sign infront of the number of people the bus carries in a day. if you look at 14:43, you can take another look at the problem. I think the typo influened how you set up the problem, which led you to the wrong answer.
@walter274 Not really great with math and its been a while since I have taken algebra, but why is it (300-15n) instead of (300-15(n+1)) since it says that the manager expects to lose 15 passengers for each INCREASE of $1 in the fare?
@@joshuajoab9313 No worries. These problems are a little tricky because there are four parts to them. The first thing to realise is that you need make two seperate equations, the first is for the amount of money that is charged. That equation is 10, which is his current rate + the amount he increases the ticket price, which we're going to call n, so we have (10 + n). Next we need a seperate equation that models how our number of customers changes as we raide the price. We have 300 passaengers a day, and we will loose 15 of them for every dollar we increase the ticket price. We have a variable for the increase in ticket price from our first equation, which is n. We start with 300 and lose 15 riders per dollar increase, so our second equation is (300-15n). For the third part we need to think about how we figure out how much total money the bus makes. For this we do the number of riders times how much the rider pays. As it stands now the bus makes 300*10. Since the amount charged and the number of riders changes, we need to replace the fixed number of riders, 300, with out equation for number of riders (300-15n), next we replace our fixed price, 10, with our equation for price (10+n). We now multiple (300-15n)*(10+n), which equals 3000 - 150n + 300n -15n^2. We then simplify which gives us 3000 + 150n - 15n^2. The last step is to find the maximum we can do that by graphing the equation (you do this online at Desmos) and looking at it to find the maximum, or knowing some fact about parabolas you can use -b/2a, which is -150/-30 which equals 5. A in the number in front of the n squared term, B is the number in front of the n term. There is a more general way of doing this with calculus, which is over kill for this problem. I think the mistake that you were making is that you were trying to do everything with one equation.
@@joshuajoab9313 because n=0 means 15*0, which is 0. And we begin counting from 0. However in your proposed change (the +1), n=0 would result in subtracting 15 on the initial case, which does not represent the question.
I feel like some of these questions are asked in a way that kids don’t see in school. That already puts them at a disadvantage.
Perhaps a fair point, although the student who took this test was very familiar with vocabulary such as "the discriminant", among other things. Could you point to which questions are asked in a way that students don't see in school? Thank you
@@citytutoring I don’t know if it’s the case that I just wasn’t super focused but number 4 sounded difficult at first glance because it said “over the real numbers” but once I read it again, I understood what it was asking. Also, number 5 writes the function in set notation which I am not sure that students see in school.
@@GreenMeansGOF Thank you for your feedback! Yes, you are correct. The student indeed confirmed that they do NOT get exposed to any formal notation at his school, sadly. Thankfully, he decided to ask to opt out of his class and do an independent study with us, for credit. My hope is to expose him to the real rigorous mathematics.
@@citytutoring Not just formal notation, it is incorrect notation. It makes no sense to define a function as a set of functions and then ask for the axis of symmetry of that function. What is this? Functional Analysis? Clearly, you meant to ask for a function f : R -> R SUCH THAT f(x) = 3x^2-4x+2. The whole point of a function is that you have real number outputs as you intended, and you have real numbered outputs.
The student most likely ignored your notation and focused on the equation inside your set, managing to derive the answer you wanted. However, this kind of over-complication of simple ideas is clearly confusing the student you're trying to "help".
although you're correct i think this type of question encourages them to think and attempt to solve the problem using their own logic instead of being fed formulas and ways to do problems, etc. I experience this issue a lot in, for example, my AP calculus class. Past mathematicians and scientists are able to prove formulas and formulate laws because they are capable of thinking with their own logic and in some classes that is slightly lacking.
grade inflation is also not just a US thing, its pretty much a global phenomenon. but the gaps between grades and actual ability are definitely much more obvious in the US
While I agree with your analysis of the student’s lack of understanding with some fairly standard concepts, I do think this test is very poorly written. There are a lot of details like the wording in question 7 of part A or the typos in question 6 of part A and question 1 of part B that may be obviously typos but can really throw students off. A test like this can’t be given to a full-sized class.
Not only can it not be given to a full class, but I think it actually is kind of a disservice. He is getting mad and rightfully so, while simultaneously contributing to the same problem with his own careless mistakes.
For a tutor, you sure did make too many mistakes in this short piece to be regarded as competent or a good motivator for your students. Is that the fault of DEI too?
Why bring unnecessary politics into what's clearly a universal degradation of standards in education?
Rather than blame DEI, which only request that talent search be opened to all qualified persons, and not only those with connections, the root cause of deficiencies in educational systems everywhere can be attributed the influence of entitled parents who harass and run off competent teachers, who by the way are poorly paid. The fact that very few dedicated teachers remain to suffer the threats and humiliation of condescension by parents and the powers that control our educational systems is a potent cause of the failure of standards.
The solution: Pay teachers adequately and give them the freedom and security to do their jobs, without fear; then perhaps, the educational system will begin to attract and retain competent and dedicated teachers once more!
"too many mistakes" is exaggerated. There were two typos (and the second typo was easy to figure out that even the student asked me about it). I would much rather go by the hundreds of student evaluations in my possession to know whether I am a good motivator or not. That data would contradict your view. I agree with maintaining the independence of teachers, of course. I also have often said similar things about a certain group of parents who do not respect teachers. As for DEI, no, I'm sorry but that is not correct. You do not fix unfairness by institutionalizing a new flavor of it. DEI is the very opposite of meritocracy, and it is unacceptable in any free society. You replace some perceived injustice, with new biases and identity-driven politics, which further destroys standards. No thank you.
@citytutoring
Okay, maybe I exaggerated on the number of mistakes, but blaming typos is a little too much of a copout. Typographical errors result from carelessly written words, mostly occurring in a haste. To claim "typos" presupposes that the errors can be caught during edits or reviews; if not, or if the writer reads the errors without realizing the errors, then the fault is no longer strictly typographical. You read a faulty statement without bating an eye, and that was jarring to watch.
Like you said, you may have a ton of wonderful reviews under your belt; however, people go by what they can only observe in real time. If the aim is to raise the standards, then let's not create unnecessary room for doubt as to one's competence in that pursuit.
Still, I beg to differ on the attributions to DEI, which, by the way, only encourages that the workforce be diverse and be inclusive of all stakeholders who have requisite qualifications to do job. DEI encourages a true competition of talents, and discourages the age-old nepotistic ways of hiring for positions.
If an education board cannot afford to hire or retain competent teachers, that's not the fault of DEI. If parents insistently interfer with teachers' ability to meaningfully educate students, that's not the fault of DEI.
Yes, there's political interference in the education of students, but the interference comes in different hues that have little to do with attempts to make an educational workforce more representative of the catchment population, particularly when such attempts only serve to encourage a competitive and diverse workforce.
@@citytutoring hi! just wondering, do you disagree with the fundamental idea with DEI (people given less opportunities via unlucky circumstances should have more given to them to maintain equity) or just specific implementations in U.S. society that are unfair
@@SDon-wn8xb*batting
This sounds like a caricature of a teacher lmao
awful test, awful answers by the student
The worst part to me is how there was never any work shown. Getting the right answer in math means pretty much nothing if you got there the wrong way.
I’m 40 and just finished college algebra. My professor was absolutely incredible. I met him while taking a comp course last summer. I was passing in the hall and saw him engage with students so I spoke with him. I finished with a B. I sought out a tutor midway through the class just to reinforce the material. I think more people should do this.
Yes, people with money should spend extra on tutors because the education you already paid for is nit adequate to it. Of course, those without more money can just fail.
@@eh1702 I totally get where you’re coming from with this but now we have Internet resources like Khan Academy and a lot of schools do offer tutoring centers. I don’t really think you have to be rich to pass school but maybe in some places you do have to pay for a tutor. I do think there are good Internet resources out there. I’m self studying algebra right now so I don’t have to take remedial classes when go back to school.
You wouldn’t know what’s it’s like to be not white, not privileged, immigrant, rough city life, no mentors, first generation, learning disability. I don’t know much about DEI, but I find people who critique Diversity, Equity, Inclusion don’t offer solutions to what it is trying to solve. It’s very easy to turn a blind eye and say there’s no problem. Think of who benefits from that
yea he's kind of a hardass but he did give the solution that cannot be given in traditional schooling because parents will blow up the teachers. How has this kid been given A's in math when he can't even answer these questions. He supposedly has a mother who is well educated as he said the mother has a background in mathematics. Like even forget about DEI, think about Bush's No Child Left Behind. You cannot just keep passing the kid in hopes they'll eventually get it because soon they'll be 18 and out of school. Regardless of the grade, his teachers, and maybe even his parents should be helping him understand the logic instead of pumping up his ego by giving him good grades. It's ok to admit you don't know the material or are struggling with some learning disability, but it is not ok to pretend there is no problem. Just acknowledge the struggle and work at it. If you have the mentality that you'll understand eventually, you will.
This guy sounds like an average mathematician and a mediocre teacher tbh. His critiques towards the grading inflation are completely valid and true though.
@@pikenersatierf he flopped his last two questions in his own exam!
As a white, privileged bisexualI agree with that guy. Don’t let woke poison you.
@@BritishEngineer
Whats woke exactly
"Number 2 is correct, by the way" (11:26) Huh? #2 says "Write a quadratic equation," but the answer given is not an equation. I also take exception to #5 being considered correct: an axis of symmetry is a line (like x = 2/3), not a number.
You are absolutely correct - however, when I went over the test with the student, he put it in correct form as an equation, he just didn't write it out that way initially, but I am confident he knows it. Thus, I didn't have a problem with it later given that he showed he knows. And I also agree with you about x = 2/3, but he was being sloppy, so I wasn't concerned that he didn't actually "know" since he's tested with it before at his school. Great points, thank you!
@@citytutoring Thanks for the response! I've seen that, often, students need to be reminded to pay attention to what the question is asking for and make sure they've actually answered the question.
A very interesting thought came to mind when you mentioned how when you attended primary school, grades were announced out loud to the class. The contrast of today’s confidential grading system certainly has had an impact on students socially. At least from personal experience I can remember receiving a grade and not being satisfied with it and because no one else knew what I got, I never had to confront reality on its own terms. Instead I could dwell or come up with excuses or scapegoats. What I’m trying to say is that having your grades announced out loud is probably the best way to go about it. It forces one to accept the outcome of their actions and into realizing the only sustainable way forward IS ACCEPTANCE. For different people this might not make any sense
i agree that grade inflation is a bad thing, but why did you blame it on DEI? would love to hear your argument on that
That's what immediately struck out to me. Grade inflation has been a thing longer than I heard of DEI, and I've never heard DEI advocaing for grade inflaton nor refusing to grant F grades.
Ikr this video rubs me the wrong way
@@LumberLopper I know right! If he didn't rant about DEI, the only disagreement that I have would be his tone when talking about the student (saying "That is ridiculous!" to a mistake is not necessarily a great educational approach). In fact, I would even agree with the gist of what he is saying since I have observed grade inflation and relaxation of mathematical education standards during my time in high school too.
However, taking into account the DEI rant, this all just feels like a geriatric "young people these days"... Maybe I am predisposed to feel averse to this sort of traditionalism, completed with a fancy suit and tie, but my mind is still screaming "ok boomer lol"
@@capnbug ok funny tangent but he is EXACTLY like the "guy raised by grandparents" caricature from @slappablejerk
@nek5935 dude exactlyyy. This is almost like public shaming. I think showing the scores calmly would speak for themselves. Of course I see the necessity of sparking urgency, but you can do that in a way that isn't through shame.
Fundamentally the issue is based on the fact funding is test based rather than population based. This forces schools to both cut down on good teachers and to deliberate inflate grades to continue getting funding.
Im from ireland and here we have the leaving cert maths exam and genuinely this exam is incredibly easy for any leaving cert student cant believe the student failed this
You should become a theater teacher because wow! the dramatization is incredible! With that being said, Hi, high school calculus student here. I do agree with the assessment that grade inflation is a serious issue and I'm lucky enough to attend a private school who's curricula is founded on not having any grade inflation, but I will say that the test was worded not completely clearly. Where the issue comes from in my eyes is that higher level institutions and universities are priding themselves more and more on their exclusivity, so in order to attend those school's students must have incredible grades, and if we move back to an era pre-grade inflation, then those administrations would not be meeting their admissions quota. I could be completely off base here, and theres also a chance that this kid's teacher is both incompetent and completely done with their job, but that's just my input.
By the way, the cost of air shuddle problem is $15. For every $1, 15 people leaves. Therefore, we have (300-15x) as the number of people left with respect to the dollar added. Then, we have (x+10) to indicate the total money. We can then multiply these two expressions together, because this will give us the maximum profit. (300-15x)(x+10) = y. Expand gives -15x^2+150x+... The reason why we don't care about the third term is because we know that this function has a maximum. A formula that is often used to find the x-coordinate of a function is -b/2a. We know that b = 150 and a = -15. -150/-30=5. Then, we add 5 to 10 to get $15.
I would've been very lucky to have a tutor such as you, Sir. I attended high school in the mid-to-late 90's and absolutely struggled with Algebra 1. I realize how simple that subject is for most individuals, but I simply couldn't absorb what was happening outside of "just perform these steps". Looking back now, I was doomed before I even started because I first began slipping at fractions, and only kind-of started understanding how to manipulate negative numbers before we moved on. As soon as my brain could no longer associate things like these with visual references (like I always could with basic arithmetic) I very suddenly started requiring more time to interpret these new ideas than the curriculum could afford. They weren't yet afraid to fail students back then, I somehow scratched out a C by the end of Algebra 1. The following year was basic Geometry, but even to my surprise I bumped up to an A- simply because things like area, volume, how angles interacted with each other, etc. had all been things that my visually inclined brain had spent time pondering but until then, had no "language" to express the thoughts with. Everything there just made perfect sense, and I actually caught on quicker than most in the room. The next year was Algebra 2, and it was back to instant struggle. I was so ill-equipped from my lack of proficiency that now stretched several years back, that I was simply drowning. We were very rural; school was 30 miles from my home and there weren't any real tutors - be it volunteer or for hire. The Algebra 2 teacher "assigned" me a tutor at the school, but all it amounted to was me sitting in the Special Education room with the students who couldn't read (very discouraging for an otherwise "A" student in every other subject) with one of the aides reading off the answers to the questions and asking, "Does that make sense now?" because she didn't know Algebra either. That's as far as I went with math, and now at 45 years of age I couldn't have answered ANY of the questions on that test. I would've been very lucky to have a tutor such as you, Sir 👍
Thank you very much for your kind comment and perspective. That is indeed the challenge (but also the glory) of Math - Math is built upon foundations. If you miss the previous brick in your foundation, the foundation will fall and start to crack. I firmly believe in early intervention because of that reason. If I see, for example, a student start to slip up with fractions, or have no knowledge of fractions, they are immediately intervened and taken either to a lower level to start all over, OR, if it's just an issue of one topic, they have to brush up real fast. I always tell my students that you simply cannot, and will not, move forward in Math unless you absolutely master every previous concept, since every subsequent concept builds upon the previous.
I remember those "special education" rooms back in the 1990s, although in some ways I think they were a better alternative than the mainstreaming public schools have today, where basically all classes have become "special ed" in some ways. Not that I personally have anything against "special ed" students, but they need a very focused assistance that a "regular" classroom simply cannot (and should not be in the business of) providing.
Rural schools still face similar challenges. In my area, the challenges are similar as we have many districts that are struggling. If you ever need Math assistance, please contact me.
Thank you!
Algebra 2 student here. This is absolutely horrific. It's such a shame how bad the US education system has become. The way in which education boards demand mathematics to be taught emphasizes rote memorization and a complete ignorance of what lies behind the actual concepts. It's truly a shame and inexplicably disrespectful to the art and beauty of mathematics. I, too, used to be ignorant of mathematics, just as the student being discussed in this video. Sure, I was an "A" student, but that was only a reflection of how well I could memorize the formulas being thrown at me. Fortunately, in my own free time, I came to the realization of just how ignorant I truly was in mathematics. And this, I believe, will be a huge advantage above other students. On the topic of decreasing student attention spans and how that correlates to an unwillingness to read large amounts of text, as discussed by you, I have completely changed the method in which I learn these concepts: textbooks. Not Khan Academy, or UA-cam videos with other humans explaining the concepts, and certainly not the poor excuse of a learning method which is being taught in school. Textbooks are a gamechanger in learning mathematics. At this point, there's not even much value in attending school in terms of mathematical education: I learn it all myself with textbooks. Hopefully, university-level teaching of the concepts will be better.
Apologies for my long and passionate rant, ha. I'm glad people like you exist to discuss these problems. Thanks.
No, thank YOU for such a detailed analysis on the problems in our schools from your own experience suffering it! I will begin sort of "backwards" from your message -- I, too, am a passionate defender of textbooks. I know a lot of people think not having a textbook is "cool" (whatever that means!) or they think they are "smarter" and don't need one. I absolutely agree with you - the key is to have the RIGHT kind of textbook. At the risk of me sounding "arrogant", the reason I became such a rigorous sort of person in Math was precisely due to self-studying. I was NOT satisfied with the whole "that's just the rule, memorize it or else!" silliness at my secondary school. I learned a LOT of other positive things at my secondary school, such as being disciplined and tough, but Math was not one of them.
I also wanted to do a video on Khan Academy at some point. If I told you that Khan Academy didn't bother me, both as a concept and in presentation, I'd be telling you a lie. I am very annoyed at the thing, particularly when now a lot of Math education has become "just watch Khan Academy" and teachers are actually recommending it. I have not done a video yet on Khan Academy because since I was always annoyed by them, I very much ignore them and the only time I hear about Khan is when someone brings it up casually, much to my annoyance. I want to collect all the facts about Khan Academy before giving a full analysis of it. I wish we didn't have to deal with them and we could go back to real and pure math. Thank you again for your analysis! Always feel free to reach out if you need anything related to Mathematics from a rigorous perspective. Stay tuned, because you will absolutely love the book I am about to present to my audience very soon. It is indeed one of the best classics out there for Math!
It's not clear to me what you mean in #7 saying "8 and 0 are solutions to the equation y=bx-x^2. The equation has an infinite number of x and y pairs that would solve it. Unless you meant 8 and 0 are roots or zeros of the equation.
Exactly, the question all most insinuates that there is only one such unique function from the infinite family of functions: y=bx-x^2, for which (maybe some) solutions of this equation are x=0 and x=8. But (8,-64) and (0,0) are solutions when b=0, so are (8,0) and (0,0) when b=8. So we already have that x=0 and x=8 are solutions to at least 2 different functions from the family of functions even when x is constrained to be two fixed values (but there are actually an infinite number of them, I just picked two different b-values). The problem here is that y is not held fixed so that the number of function solutions converges to a unique one. So now already the second part of the question makes zero sense, the maximum value in the interval (-6,0) of which function? there is an infinite number of them to choose from. So do they mean x=0,8 are root solutions to the function? Because then that would make sense and this question would be fine.
"Solutions" are roots. (x,y) pairs that lie on the equation are not solutions.
@ Incorrect. Roots are just x values that satisfy the equation when y=0. Certain quadratic equations don't even have real roots. There are infinite (x,y) pair solutions to the equation, just as there are with a linear equation, or any other type of equation with x and y variables in it.
@@gtg309vIt's obvious in this scenario we're referring to solutions as roots; plus, this is common terminology in an Algebra class
@absceptual To be fair, that was the only conclusion that could of been made with this question. It's just that he is little pedantic with the use of mathematical language in all other questions (which is a good habit, i'm not denying that), but why not be as pedantic with the use of language in his question? Seems a little hypocritical, but everyone has a different way of doing things, so maybe this really is actually a common phrase to say roots == solutions for functions, so could just be me.
My child is in Honors Algebra 2 and his situation is nearly identical to this, with the difference being that he's currently earning a C. I fear this grade would be lower if it accurately reflected his actual understanding of the syllabus. I've observed the issues you describe, and some others as well. The district has weighted grades -- similar to grade inflation, it effectively elevates the grade level one point for Honors and AP classes. I believe the central issue is that the school(s) won't practice _mastery learning,_ largely because it's impractical to implement at classroom scale. It would require the teachers to individuate instruction -- each student has different strengths and weaknesses while the classrooms have to move along at the same pace.
I'm convinced that one reason students are unprepared for Algebra and Precalculus is that their minds have been clouded by the problem-based learning (PBL) so loved by policy makers and influential pedagogues. They fail to generalize and abstract-ify concepts because their minds are bogged down by low-level cognitive tasks that they practiced in elementary and middle school. How are they expected to decompose polynomials or factorize quadratic equations while they have no automaticity of arithmetic facts? Tape diagrams be damned! What happened to flash cards and multiplication tables?
Sal Khan said something like 'if a student gets an 80% grade and receives a B in a semester, they move on. Would you build a house atop a foundation made of 20% holes?'
Who is going to step in a backfill those holes? The schools? They have no incentive to do so. Parents? Most are unaware. Remedial maths in 1st year College? Not effective, generally. Tutors, coaches, and student grit seems the only way forward.
Thank you for sharing your experience! Yes, I am a proponent of mastery learning. At the same time, I also agree with you that given the constraints of your typical classroom, it is hard to implement seamlessly. However, schools used to do a much better job at this decades ago with proper tracking, entrance tests/level tests, and other performance metrics.
You also make an excellent point about PBL. I started noticing this trend even back in "my day", where they started obsessing over "graphing" and "word problems", without teaching the abstract generalities. All you have to do is for example compare an old 1980s SAT Math exam, to the newer exams to immediately notice the change.
Mr. Khan's comment is one with which I agree - and to top it off, a "B" is not universally the same across America. In some districts, a "B" student would be a C or D student in worse districts.
I believe your conclusion is the right one - we can only march on with what we have, with grit and determination. That is what I plan to do, on my end, to help our students.
I did college in the 1980s, and the teachers would review foundation material as it was needed. Physics 2 (e-m fields), we got a quick review of integration by parts. Queueing theory (graduate statistics), we got a sidebar on Laplace transforms. Good review plus application can fill in some gaps later, but it requires enough understanding in the foundation.
Do you mean on #7 0 and 8 are zeros of the quadratic function? If so, this really would throw anyone off. A solution of an equation in x and y is an ordered pair (x,y) that makes the equation true. However, if you mean zeros of the function, they are symmetric about the axis of symmetry, so the max would be (4,y(4)) and the function is increasing on (-inf,4]. If the interval of interest is (-6,0) there would not be a maximum because of the open endpoint at x=0. If the interval is (-6,0] then the maximum value would be at x=0.
The use of language in this test is so imprecise I would never have given this test. I would have expected a lot of WTF is this Dr. Mac?
The college board would have raked you over the coals.
Thank you for making this comment. My thoughts, too. It concerns me when educators are blind to their own faults.
A friend of mine was doing quadratics, and lost a mark every time he wrote that solutions were, for example, x= 2 or x= -1. Was told you need to write x=2 and x=-1. Now, I wouldn't deduct a mark for either answer, but when you stop to think about it, using "and" is incorrect.
i sighed physically when he mentioned equity
yea he's kind of a hardass but he did give the solution that cannot be given in traditional schooling because parents will blow up the teachers. How has this kid been given A's in math when he can't even answer these questions. He supposedly has a mother who is well educated as he said the mother has a background in mathematics. Like even forget about DEI, think about Bush's No Child Left Behind. You cannot just keep passing the kid in hopes they'll eventually get it because soon they'll be 18 and out of school. Regardless of the grade, his teachers, and maybe even his parents should be helping him understand the logic instead of pumping up his ego by giving him good grades. It's ok to admit you don't know the material or are struggling with some learning disability, but it is not ok to pretend there is no problem. Just acknowledge the struggle and work at it. If you have the mentality that you'll understand eventually, you will.
@@The-Entelechy stop yapping noone is reading allat
@@MasonMNKyou are so cooked bro it’s 1 paragraph
@@MasonMNKYou could watch a man talk for 18 minutes but can't read 1 paragraph? Wild
how tf else are you gonna sigh 😭
Im not sure if this is me being out of practice or if its me being unfamiliar with american terminologies but to be honest i could not answer a single question on that paper
I am doing a phd in green chemical technology so its been a while dince i have had to deal with much complicatex mass beyond just mass balance stuff so that could be it, i would appreciate you going through , explaining and answering these questions as a follow up
Same here!
In all fairness, I would have had a WTF expression on number 6 because you did not give a quadratic expression. You gave a linear expression. There is no square to complete.
I think the squared was just missing on the document; I imagine that the actual test had the correct expression
It is the right of the student to not have to read the teacher’s mind. Tell this to the reader in an AP exam when the student makes a linkage error. If the reader CAN tell that the student recovered from something like a transcription error, that’s one thing. But a straight up linkage error will get you destroyed because what you wrote is just literally not true.
Here, Cromwell is building his business by denigrating professionals who will not embarrass themselves the exact same way he has in this video. Cromwell needs to check himself before criticizing folks like me. He better be bulletproof when he makes a video like this one, or folks like me will expose him.
mpmcd: I already indicated to others who have asked that it was indeed a typo. Yes, of course it is the right of the student to not have to read minds, and I have no problem giving credit for "dud" questions when they occur, but it is also no less true that a student in "honors" should be able to identify a basic quadratic equation and realize the equation was not in quadratic form. The student did not even bat an eyelid, so even if he were to receive credit because of a typo, my critique still stands that an honors student should be able to identify a quadratic.
Also, how can I be denigrating professionals who do the same thing I do? That would be very silly wouldn't it? I am sorry if this channel is not to your liking. I will continue to produce content for my students and audience, because at the end day my number one priority is our students.
However, for the sake of clarity, I will pin a comment explaining the typo.
May you have a blessed day, sir.
@@citytutoring I do think it's unrealistic to expect a 15 year old coming to you to take an assessment to see what he or she knows to be able to push back about the problem in number 6 having a typo. As far as the test taker is concerned it's just another problem they don't know how to do. You can't hold their failure to do that against them. In the mindset of someone grading an exam that's going to be used to determine somones grade in a course, i don't like that you marked it wrong. From the point of view of someone just trying to see what the kid knows. It doesn't matter what you marked the question as, they clearly have some skill deficits.
Did you proof read any of this before giving it as an assessment? This is one of the worst tests I have ever seen
I'm glad y'all voted trump to remove the DEI BS from everything, good job Americans. Hope my country select similar leader.
16:46 The distributive property isn't necessary here. The statement would be true even in a non-distributive algebra.
-(a + b) is, by definition, the additive inverese of (a + b). We want to show that (-a) + (-b) is also the additive inverse of (a + b).
(a + b) + (-a) + (-b)
= a + b + (-a) + (-b) by associativity
= a + (-a) + b + (-b) by commutativity
= 0 + 0 by definition of additive inverse
= 0 by definition of additive identity
QED
Edit: changed "additive inverse" to "additive identity"
Good! Last line should've been something like "by 0 being the additive identity" rather than "definition of additive inverse."
@@bartholomewhalliburton9854 Yup, fixed
Tutored my nephew in AP Calculus. Total disaster. All “A” student in math courses through precalculus, but wasn’t even proficient in pre-algebra level maths. Bombed the course and the AP exam. Now attending a regional university and essentially repeating (through remedial courses) all the math (and English!) he should have learned prior to attending university. Wants to major in Electrical Engineering. If all goes well, this should take 6 years for a 4 year degree.
As a teacher in NY, I can tell you why:
1) We do rote learning where students get similar questions to answer
2) We don't have the time to have students do critical thinking. Shockingly, by Algebra 2, some students HAVE NOT even gotten a taste of critical thinking. And I know some people are saying, "What about Geometry?" but you should see our pass/fail rates for that course, its incredulous.
3) We teach towards Regents
It is rare that we get students who can do higher-level problems without a prompt for it. I did have my cases of students who can perform this task though. I can say that with these students, I can see how they can manipulate information they know and apply it to questions they have never seen before and they don't give up immediately. But I think this shows us that most students have Fixed Mindset rather than a Growth Mindset.
That discriminant problem is quite sad though since the student is in Algebra 2 Honors. This is a topic that they should have already seen in Algebra I. This shows that they do not understand the Quadratic Formula and its components.
To sum it up, we do not teach Math in the way that Math is truly supposed to be learned. There supposed to be discoveries, connections, and conjectures to be made but due to circumstances, this is rarely done.
I can't tell if this is satire.
But, to be in your 30s (presumably), unironically wearing a bow-tie, and talking in an old-fashioned and condescending voice to sound indignant about the current state of American education isn't the way to connect with your intended audience, unless your intended audience is non stakeholders who just want to watch some struggling kid get his test ripped apart. If you want to get students engaged in learning algebra, it's your job as a tutor to instill a passion in them. So much of what you say sounds intentionally verbose and stuffy that it's no wonder these kids don't have an attention span.
I'm a Mathematics and Computer Science major, and without question, the most effective way I've learned any math (including algebra) is from professors who love what they do and want us to love it as well. It's not from professors who ban phones, belittle incorrect work, and condescend to struggling students.
Sure, standards should be higher. But I can say with absolute confidence that this type of content is not helping anything.
I also have some perspective from the other side- I've tutored seven times at my university, help write exams, I've lectured in the statistics department, and generally engage in the pedagogy and curricula in the math and data science departments. I can say the same things from a teacher's perspective. It's more fun when everyone in the room is engaged and passionate, and saying "no phones! You're a bad student!"
I love what I do as both a student and instructor, and it makes a huge difference. Find that love, and you'll see vast improvements in the attitudes of your students.
And lose the bow-tie 😊
As an adult close to sixty years of age, you have encouraged me to relearn algebra. I don't think I can trust young tertiary educated people to help me out if I should have memory loss or help me with a refresher with mathematics..
That is so wonderful to read! I very much appreciate your comment and wish you all the best in your journey!
@@citytutoring Thank you Sir, I corrected the grammar so my comment might be understood by the younger generation reading it.
Damn those young people!
Kind of a weird ageist expression X)
The haughtiness that pervades this thread is the reason so many good mathematicians make terrible teachers...Arrogance does not fit well in learning...I do, however, believe in strict/ high standards,but that can be achieved with less hubris....
That is so neat that you kept a gradebook as a student.
I could always gauge how well I was doing learning math by doing the exercises in the the textbook and checking the answers in the answer key. The student doesn't understand the definition of real numbers. He has them confused with rational numbers.
The answer to question 1 in part B, the helicopter problem, is that he should charge $15. If x is the additional amount he wants to charge, he will lose 15x passengers (1 for each $15). So he will have 300 - 15x passengers each day. Each of them will pay 10 + x dollars, so the 300 - 15x passengers will produce (300 - 15x)(10 + x) dollars. He needs to find the x that makes that expression a maximum. One approach is to find the roots, which are -10 and 20, and then find the average. The average is 5. That's the additional amount he needs to charge, for a total of $15 fare to maximize profit.
A symphony of errors! A concerto of confusion! 😂❤
Isn't a square missing in question 6? Otherwise it is not a quadratic equation, so completing the square makes no sense.
Yes - please see my pinned comment addressing that. Thank you.
yeah and bro thiks he is galois or some shit while roasting high schoolers xd
Answer to B1.
Total fare revenue (300-15x)(10+x)=3000-5x+15x². Derivative zero gives 30x=5, so x=6.
Therefore this is highest when fare price is $16.
Because the net profit will also benefit from having to spend less fuel when transporting less passengers, so the actual profit optimum will lie at a higher fare price.
For set A
For qustion 1,2,3,4,
I think they are fine and good
I would expect from a high-school student to read them and solve them
For #5
I think the wording is a bit too harsh
writing a function in that way is really wired and also requriment (a) could've been written in a better way
(to note: im not an english speaker perhaps that way of writing is pretty normal but i dont think so and my knowledge in mathematical terms in english shouldn't be worse than an english speaker
But something to note)
for #6
Other than the typo there isnt much to say
but since you didnt fix the typo midtest i think you should either redistribute it marks to other qustion or give the mark for free (i know generous but you still made a mistake even if that little)
YES i do think a student should be able to tell that there is a mistake in fact its kinda of a crazy thing for me to hear no student noticed that
#7
its written horribly
I assume you mean x=8 when y=0
or that both x=0 and x=8 (roots) are solutions for y=0
(i saw other comments and those are the only thought they got its not a "me" think)
But
If it were truely that then the seconed part of the qustion make no sense
you will get b=8
and the whole function thats fine
but there isnt a greatest value between (-6,0)
the greatest value would be a number that get arbitrarily close to 0 but thats not really an answer
If it was less or equal to 0 then it would have been fine
Also even if 8 and 0 mean another thing
that qustion would require a bit of calculus which (i assume) student shouldn't be using it in that test
IF you thought them that the vertex would be the maximum value and you also thought ghem how to calculate it and 8 and 0 somehow should be interpreted in a way that make the vertex of a parabola between-6 and 0
Then that qustion would be fine (other than "8 and 0 are solutions" part its written in a bad way)
Overall for part A only #7 have a major problem in my eyes
Part B
pretty generous to give 8 options and only pick 3
but ok
#1
Beautiful one but i dont really think it fits well in an exam for high-school students
They already struggle to make equations out of heavy worded qustions and to ask them for maximum value is a bit too much
also similar to #7 it kinda off require calculus (unless you thought them that the vertex is the maximum value and how how to find it)
in genral i would like if this was put as a challenge qustion or change the requirements from finding maximum value to (find values where gain is 0$ which would be $0 and 30$ trips)
I know it would be more intuitive to ask for maximum gain cause that what generally people want but i dont really expect ordinary high-school students to solve it (if you made it for high level students then its a really well qustion which is probably the case sinsce you literally give them the ability to skip 5 qustions out of 8)
#2,#3,#4
Are good qustions and well done
#5 could be written in a better way i think you should write it as
Intercepts with the x axis at 3 and 6
And with the y axis at -2
(not sure thats what the qustion try to tell but thats the only way to get 3 points that i see and you cannot make a parabola equation where only x,y are arbitrary values with 2 points only)
#6 i really love those qustions nonetheless its unfair to expect from a high-school student to be able to prove something with a good amount of rigour
I dont think you give them an exact set of axiom to prove such thing nor do i think they really understand such thing well
How hard this qustion is completely depends on how much rigorous the prove have you want (which appear to be next to none since you give him full mark on his answer)
but you ended with giving them a full mark for an incomplete answer sad that such fundamental stuff is not thought well but we gotta deal with it
#7,#8 are totally fine
Overall
Only #7 in part A i can call as a major
The rest are either fine or minor
i dont know how school exactly work in the US so i will put base on age on what i expect them to be at
15y the qustions should be a bit easier
16y,17y they are fine other than what i mentioned before i have no problem
I recently took Algebra 2
Didn't do the best despite trying but did decent enough to progress to Precalculus.
In schools we are simply given a type of problem, and taught to work through the steps to solve.
I often struggle with definitions and the phrasing. We are trained to really look in our textbook, look at the type of problem
Complete said problem, and practice each 'set' of problems if you will.
I dont remember once were we took a dedicated time to go over Real, Rational, Imaginary numbers - etc.
And really went in and understood what they meant beyond how to solve the problems like a robot.
I remember we took 30 minutes in all of Algebra 2 to explain Imaginary Numbers, We took a small amount of time
In precalc (maybe a day or two) looking over differences between real Rational, Irrational.
The issue is we, like robots simply learn to mindlessly go through the steps - but not to look any deeper.
Some people just have not been told the truth. Or some people are downright naive. Still doesn’t make it right for you to outright bash these kids. You’re a grown man, act like it.
Furthermore, the student should receive no credit for the axis of symmetry since they didn’t provide an equation of any vertical line which bisects this parabola
I wanted to note that y=by-x^2 is a function, not an equation, also 0 and 8 would be not solutions but roots, or 0-points, however you like to call em
Thought this was going to be ASMR roleplay 😭😭
Frankly, education has devolved into what you can get the most while putting in the least effort in, evident in the irony of AP and Honors classes having the highest levels of cheating compared to normal classes among High Schools
Please make more videos! I love longer videos... I open talkative and conversational videos while studying... 😊
1:07 yeah just so you know when I was in highschool they were forced to give you a 50% on an assignment so long as you turned it in (including blank)
and everybody was allowed to retake everything lol. Good times....
Edit: I'm obviously saying forced to give you 50% as lowest score, they werent allowed to give grades below that.
is being allowed to retake everything a bad thing?
15:23 Kind of depends on whether you only read what the question actually asks, or if you know what you really want instead. With how it's phrased, there are only 30 customers ($300 with $10 per customer) and it sounds like you can only increase the price. A reasonable restriction if you consider the number of helicopter flights might be limited. With these conditions, $10 is the right answer.
Otherwise, the effective number of customers is
C(x) = 300 - 15x,
and the price is
P(x)=10 + x.
The full profit is
f(x) = (300 - 15x)(10 + x)
And you get the optimum by setting the derivate to 0:
f'(x) = 300 - 15x - 150 - 15x
= 15×10 - 15×2x
= 0
2x = 10
x = 5
And f''(x)=-30, so x=5 is a maximum.
So it should be made $5 more expensive, reaching a total profit of
(300-5×15)×$15 = $3,375.
As for the situation as a whole, I can see how grade inflation would hurt the progress of the best students, even if it prevents the students with worse grades from being discouraged. You could tweak the grading formula to try to change this, but I don't see why teachers couldn't just talk to their students. When I was in school, the teachers would have a special session every quarter year where they'd call one student after the other to discuss their performance. These discussions were way more informative than any grade, and it honestly made the grading system pretty redundant for most situations. It's a pretty obvious to me that learning about a subject because it's important to know is much more motivating than learning about a subject because I need good grades.
DEI has nothing to do with grade inflation
As a teacher, it's not that #1 is hard. It's that it's not the niche thing that is taught by teachers to suit the test. No concepts are ever taught. He wrote that because that is the collection of symbols he's used to seeing
EQUATIONS DON'T HAVE ROOTS
EQUATIONS HAVE SOLUTIONS
*POLYNOMIALS* HAVE ROOTS
Perhaps the student was so flabbergasted by that absurdity in the first question that he simply forgot to think throughout 💀
Maybe wait until you can write a quiz without all those errors before criticizing a student. 😂
Just to include my story, I was in honors Algebra in a state, then I moved, and was put into Algebra in that state. That state had superior algebra teachings. The teacher would've been willing to work with me stay in the course, however, I choose to drop down to pre-algebra.
As a result, in high school, I didn't not take honors geometry my first year, but honors algebra, except this time I actually was able to gain a solid foundation. Mathematics is something where it is sometimes better to move backwards and then forwards, in the pursuit of a solid foundation, as mathematics builds upon itself.
Edit:
For the student, he doesn't seem to understand the difference between an expression and an equation. Doesn't understand what "simple" form is (I think for me that'd be standard form? y = ax^2 + bx +c ?). Doesn't know what a real number is, considering he claims pi isn't one (which is actually algebraic vs. transcendental). Doesn't know the rational root theorem. For the "x < 0" one, I think he didn't know how to approach it, and he only know that x cannot be zero lest 1/x be undefined (though, if that was his thoughts, I'd expect x > 0 and x < 0). I think for this one it's primarily not knowing how to approach the problem, perhaps not understanding that, because r is fixed, this is all just with "x" and he can multiply through and solve for solutions, of which there is none.
The 6th problem made no sense to me, as there is no square (at least not rendered). y = 2(x - 16x) = 2 * (-15x) = -30x. There is no completing the square of a linear equation.
For the 7th, he doesn't seem to understand the relationship between x & y of input and output, able to use the roots given to solve for b, thus gaining the equation, and then answer your second question.
For the word problem, I think it's called linear optimization if I recall, but anyways. If cost is c, and people is p, our profit function is Profit = p * c.
The relationship between people and cost is 300 when cost is 10, so let us normalize, and, then 15 people are lost per additional, hence p = 300 - 15 * (c - 10). Thus you get a quadratic for the profit function to be solved for, cost of $15.
Thank goodness they got the 6th written problem right, otherwise, that would be pretty bad as it's just distributivity. Will say, if we want to be pedantic it can be said that you don't "distribute" the negative. It would be better to write it as -1 * (a + b) and then distribute the number, as that is the symbol the property acts on.
I also recommend to any math students that are young looking at AoPS or other math competition stuff, a good refinement tool.
Yes, so just to clarify, #6 has a typo, unfortunately. However, appallingly, the student did not even bat an eyelid to recognize that there was something "off" about it.
You put it beautifully: "Mathematics is something where it is sometimes better to move backwards and then forwards, in the pursuit of a solid foundation, as mathematics builds upon itself."
This is PRECISELY the reason we do what we do at City Tutoring. I absolutely agree with you and based on that thinking, I have been able to help a lot of students. You simply cannot be in an "advanced" course if you cannot do simple basic foundational concepts in Algebra. Math is built upon a foundation brick by brick. You miss one brick in the beginning, and your entire foundation will collapse.
Thank you for sharing your perspectives.
Answer to question 5 is also wrong.
5a) among others, asks for an equation, not an expression. So correct answer would be x=2/3
5b) The student's answer does not specify the value of the minimum which is f(2/3) = 0.
You are the Platonic ideal of a math tutor.
Peak physiognomy.
justified glaze
@@olivernorth7418 Excellent glaze.
encouraging content this is not the mainstream but is clearly important insights thanks brotha
I have been tutoring math and science for over 20 years, and I agree that the schools don’t teach. Students know less and less each year. This is includes honors and regular classes. I find that many schools let students retake tests for higher scores. They rely too heavily upon calculators and cannot do simple math. Many students cannot graph without a calculator doing it for them. I also think it is appalling that calculators are allowed on the SAT and AP tests.
This is why the US is behind other countries!
My brother has taught some of the college-level "math for teachers" classes. It is abysmal. They are taught essentially no math and they still struggle to even get a C or passable grade on such basic concepts. The only responsible thing to do would be to tell the administration that they don't understand math and are incapable of teaching it. Yet, they don't and then end up incorrectly teaching wrong things to thousands of unsuspecting students. It should be criminal. This ruins lives.
Absolutely! People look at me like I have three heads when I suggest penalizing these sorts of teachers for mathematical malpractice. I usually get a look of "you can't be serious"! and I am dead serious. I cannot think of any other profession that allows this to such an extent. Like, for example, can you imagine all the lawsuits if a doctor was shown to be incompetent at what he was doing and knowingly so? Now of course, education might not be the same as health, but it is actually your FUTURE and thousands of our students across America are being misled, mistreated, and taught wrong. I get hundreds of students a year who literally say to my face that they never knew these things. Just this morning, for example, half of my new Saturday class did NOT even know the SYMBOL for "element"! These are TENTH graders who are also planning, so they say, to go to places like MIT and Harvard.
@@citytutoring Exactly!
I just found your channel and it's really quite interesting! There definitely is a big problem with education in our country, we gotta improve it.
FRIES. BAG. NOW.
Im 23, these questions felt like i was at highschool again, it was really fun solving them, its incredible how hard they seemed at the time ahahahaha
I remember I did "great" in high school math, but had to take additional math classes once I got into my university as I did not feel confident in my ability. I ended up having to take 2 semesters of Calculus, which I passed, but it was definitely a struggle as I was not set up for success from the start.
Sadly, many students in our schools today find themselves in a similar situation. It seems that the only thing "honors" means these days is the fact that students are expected to be "good" even when they have not mastered the basics!
@@citytutoring In my anecdotal experience I had a good teacher and was in algebra II honors around 3 years ago. She gave out long homework assignments that were meant to challenge us, and they were fun- it seemed significantly better than what my friend was doing in the normal algebra class when she decided to stay rather than move to honors (the class was much slower paced and didn't cover as much content (we covered the majority of pre-calc by the end of the year))
I "qualified" for calculus in college, but didn't feel confident about my algebra and trig from high school. I signed up for algebra and trig the same semester (I got a funny look from the registration person. This was before it was online/computerized.), breezed through them and then went on to calculus.
Hi, I'm a Malaysian studying an actuarial degree. While I may not know the math syllabus in America or even the state your student is in, I can understand and relate to the mindset this student has when doing the test.
Luckily, I am fortunate to realise the gap in knowledge between high-school math and "proper Math". Though, I think it is reasonable enough to understand why many high-school graduates are like this is because of the lack of nuance to these things they learn.
To a normal teen, a determinant may be an expression or even formula you use to solve a root for x. But they do not have this exposure to the correct reasoning and thought process to accept this 'fact'.
Though I do share the same frustration on this issue, I do feel your dumbfoundedness towards this student's answers is slightly exaggerated. While it may be not acceptable, I am sure (or hope) you may already know why they think or give these answers.
Additionally, on the matter on electronic devices in the classroom, it may be disrespectful but I think many students may find it helpful or maybe meaningful to interact with mathematics using their smartphone.
I just wanted to share my 2 cents and I appreciate your effort in highlighting this concern. I think you're a great math teacher 👌
15 year olds can solve whatever paper that was
That helicopter question has a typo
$300 passengers makes it seem like 30 passengers, so $10 would be optimal because only carrying 15 passengers would make $165
However, if the question meant 300 passengers, the optimal fare would be $15 (225 passengers but $3,375 profit)
Nevermind I realize this was addressed as a typo already
They removed some of the probability curriculum from Pre-Calculus classes at my school and now kids are just losing all their money to bad bets. 😂
As a pre-calc 1 student, I would fail your test. Maybe I don't recognize the format of the questions(?) Although I took algebra 2, 10 years ago. But before I started Intermediate Algebra at the beginning of this school year, I took a "crash course" 1 credit math class that used the Aleks app, which required me to take an assessment. Then it determined that I needed to learn various parts of high school and college math over again, and it got me up to speed to where I was able to jump into Intermediate Algebra. I did get a 4.0 in the Intermediate Algebra this past fall term, perhaps an inflated or curved grade. Pre-calc is making sense so far. All that said, I probably would get a zero on your test. Perhaps I forgot everything I'd learned in Algebra II
When I was in school in the late 80's they changed the grading scale so that an A included the scores of what was previously a B. The new B included all of scores of the previous C and D. And by time you got down to F you had to score less than 50% to actually fail.
I believe it - back in the day when I was in 7th/8th grade, anything below a 70 was considered an F. Then, years later, that went down to anything below 65. Now, anything below 60. For me, it's still anything below a 70 should be an F. We also had an 80 as C, not a B like most schools today. If anything, we had grade deflation!
@@citytutoring The old scale for my school was A 95-100, B90-94, C 85-89, D 80-84, F anything below 80. The new scale was A 90-100, B80-89, C70-79, D50-69, F below 50. I remember the one time I got an A on the new scale and now ashamed I was as it felt like I was cheating because I knew I really only got a B on the test.
I know of cases in public high schools where a student is enrolled in Calculus despite struggling with basic factoring.
Not a teacher but a college student (engineering not math, but declared math minor).
To be fair asking a true or false for a very simple square root problem and expecting students not to skip showing their work is a slight oversight. If this was timed and it's that far down the paper, not everyone is going to remember to show their work that far down or have time for it if they were stumped earlier on (plus they may skip around on problems and forget). I've seen many teachers start re-emphasizing it by just putting "(show work)" in the true or false questions. I know it's repetitive but it is very helpful. Also, side note, as a students myself rarely are we taught useful critical thinking or knowledge retention practices; it's a terrible problem in the U.S. It just comes with the territory of quantity over quality.
I've probably used a sloppy method to find #1, but I believe it to be $6, because 6x90=540, and 5x105=525, and 7x75 also equals 525. Meaning that since the profit margin of both values decreases, it is indicative that $6 is the most profitable solution to the equation.
By the way, those questions are slightly more difficult, and for some of them we are not taught to the depth that those questions necessitate in our current school system. I definitely believe that our education has been degraded if these are considered basic algebra.
Writing from Italy, we have the same issues you're talking about at the start of the video
I am currently an undergraduate majoring in physics and chemistry with a minor in mathematics at the University of North Texas in Denton. My math background from the US education system is extremely weird. In middle school, I qualified for advanced courses via testing. The school I attended proceeded to lose my test and refuse to allow me to take accelerated math courses or to retest. I therefore took math in high school on a normal track with all the other kids. It was horrible. Nothing but memorization, no concepts, no appreciation. I hated every day of it.
Fast forward two ish years and I fail my first semester of algebra at the University of Texas in Rio Grande Valley. I similarly hated that course. Our instructor was a grad student (seeking a masters degree) who came into the room with a single sheet of paper, would lecture for maybe half an hour, then dismiss us. I could see the disappointing common core watermark from my seat.
Fast forward a year and I had just CLEP'd out of algebra after a summer of teaching myself. I somehow stumbled into a general chemistry course at Temple College with my now research mentor. I fall in love with chemistry, and that drive motivates me to reteach myself mathematics.
Now after just over a year of essentially learning math from the ground up, I've completed Calculus 2 and am studying Linear Algebra. Hell, I even tutor now and have two semesters of experience under my belt. I couldn't have done it if I chose to let the US math education be my one and only. It failed me, and I saved myself. Plain and simple.
So to students out there, know that your education/knowledge is always something you can expand, and I can guarantee that everything you learn in school is far more beautiful than you will ever know if you rely on your education singly. But also know that with modern resources, anything you could possibly want to learn is out there for you.
Respect
Im 1995 and thats how it was the whole time
Examples:
I thought spanish in high school was so interesting. And it was also very chill. But we learned everyday
I remember being shocked that my classmates had no idea how to conjugated known words. We had spent months, everyday, where all we did was look at conjugations on the board!
In university, i cheated my way through nearly every class. Between leniency, outright giving you As, etc., it wasnt always easy mind you, but it very very doable
Everything was just very fast, and required so much deep development for every little topic coming at you week by week, either you slow down, you fail students, or students actually dedicate their time and effort on getting these topics down during their days every week (most students have split priorities with friends, fun, work, etc.)
Just a couple experiences from me
for the shuttle fare, I got $15. equation I set up was y=(-15x+300)(x+10), found the mid-point which is the maximum, that being when x=5, so $10 plus 5 increases of $1 is $15.
bro fr getting ur question wrong is kinda wild
For the question about optimal fares, realize that there is zero revenue when the fare is zero though anyone who desires to ride will ride. There will be zero revenue at $30 because when he increases the fare by $20 he will have driven off 20*15 or all 300 passengers. Because our model implies that the number of passengers is a linear function of price, and the revenue is this linear function of price multiplied by price, revenue is a concave down quadratic (because revenue is positive at a fare of $10) and the end behavior of a quadratic necessitates that a quadratic that is positive between its real zeroes is concave down. Therefore the vertex is located halfway between the zeroes of 0 and 30, and the optimal price is $15. The optimal revenue will be 15*(300-15*5) or 15*225 or 3375 dollars.
This is not a hard question whatsoever if you have mastered both quadratic and linear functions and can think in multiple representations and know how to make sense of fact patterns.
The distribution of negatives question illustrates a major problem. Our middle grades through our tenth grades have failed to teach the basics of abstract algebra (ring properties of integers and field properties of rational, real, and complex numbers). The student should have recognized that this is a case of applying the distributive property of multiplication over addition. This is the basis of factoring by GCF as well as the number one error in the BC calculus exam.
How does question 4 even make sense? 0 is a rational number, therefore the question implies that by making r=0 the equation makes sense.
if you are a teacher the worst thing that you possibly do is to make a youtube video of a student that failed a basic algebra test just try to be a good person for once and like confront the student in a nice way reather than roating him publicly its such a bad thing to do
He is outing the student in any way . . . . He is explaining a phenomenon that is happening throughout the country using a personal anecdote
@@rohanghoshdastidar1047 it doesent matter because by explaning in such a way really makes the situation worse. by roasting a student publicly shows that you are teaching bcs you value the money/social statious more than actualy enjoying the subject thus it makes harder students to like maths
@@rohanghoshdastidar1047 and quadratic equations and like graphing a parabola is such a easy thing. every body who studys for like a month can do it there is no need to flex on that or take it that seriously
@@rohanghoshdastidar1047 and like every ppl needs a lvl of math in their life like most of ppl solve DEs in mathamatica when you can sole it analytically nobody comes and roasts them. there is no point doing this if he does what ever work he ll be doing we should bw ok with that
Respected sir, this is the state of majority of students throughout the world. This is s fact.
One thing to consider (I disagree with a lot of your channel but I made inappropriate comments earlier for which I apologize) is that because there usually aren’t final exams administered in many math classes students often learn a concept before a test but are unable to maintain their understanding and knowledge long after. So this is a likely explanation why a student with an A average in a honors Algebra class failed this exam. Realistically I can see this student earning a B- back in the day we assume my theory about not retaking universal knowledge is accurate.
theres nothing wrong with this video, why are people commenting hateful things hes just sharing his opinion and you gotta respect it even if u dislike it
I think the DEI comment was just questionable even I was confused on that part. The problem is the no child left behind act that encourages even low performing students to still pass, this includes anyone not just those under DEI which are minorities.
Baltimore's smartest student
In the first answer, they confused 1+sqrt(3) with 1*sqrt(3) - they probably aren't used to working with numbers with multiple terms as roots of an equation.
There is no answer to 7 - the function is increasing over the inerval, so the close x is to 0, the greatee the value of the function, but x cannot be 0, so x = -epsilom for arbitrarily small epsilon
4 is false because when t is negative, the right side is positive while the lett side is positive
We call it "maths" not "math" in the UK, what a crazy world. How do you find teaching students with lower grades (even if their grades are inflated, the weaker students in general I mean) or is City Tutoring more for the higher attainers to prepare for University?
@Mrooooole: We do both, but speaking personally, I very much enjoy the "lower grade" students. For me, it is much more enriching to watch a student grow who had initially started out as a failure. I also enjoy the higher attainers, but the atmosphere in the classroom with that group is completely different. Both are challenging, in different ways, but yes, I still love a good group of what we call "remedial" students and watch them grow in Math.
Luckily students have this site as a resource. Tutors and teachers can determine what the best remedial material to study is, and refer the students there.
For question 8, since to of the points on the x axis equal distance from the origin then we can say that the points (--5,0) and (5,0) are the radii of a circle. Calculating the distance of (-3,4) from (0,0) is 5 which is equal to the radius of the same circle. Therefore. since the points form a triangle that is inscribed in a semicircle then the triangle is a right triangle, Euclid book III prop 31
There are two key points for the solution to Part B question 1 .(pls kindly point out if I make mistakes)
first is that (Cost).(People)=Income
so, if x=required increase
10+x = Cost
300-15x = People [since we are losing 15 people per increase and total people(initially) are 300 so decrease becomes 300-15x]
Now,
(10+x)(300-15x)
= 10(300-15x)+x(300-15x)
=3000-150x+300x-15(x^2)
= -15(x^2)+150x+3000
let y=(-15x^2 + 150x + 3000)
Now, notice that coefficent of x^2(or leading coefficient) is negative. Remember in such cases the parabola(which is the graph of a quadratic) opens downwards.
In such cases y has maximum value at vertex. This is because as the parabola proceeds downwards , the value of y gradually decreases.
So the second key point is --> The maximum value of a downward parabola is at the vertex
It is interesting to note that in case of an upward parabola , vertex actually represents minimum value.
Kindly let me know of any errors or how I could have wriiten a more concise and overall better answer