Quadratics was really popping when I've investigated the effects of varying coefficients on the graph, studied transformations and derived the 'k' of (h,k) from the standard form of the quadratic equations.
If a is negative , e.g. , -2x^2 + 3x + 5 = 0 , and you're not too comfortable with it (or even if you are, because eventually, overwhelming equations will be tackled effortlessly), just rearrange it...def one of the good practices.
When you verify your solution(s) and graph it, you'll realize you've done nothing wrong but rather something right. Gradually exercise this so that you don't get 'stuck' ; instead, work around the situation.
Imagine you get a 3x^2 - 7x + 8 = 10 ; 4x^2 + 10x = - 9
No, you're not going to factorize them by any method just yet. First, observe the right-hand sides (RHSs). Rearrange, simplify, and ensure that they're in the standard form of a quadratic equation, where the RHS equals 0. This is because you're finding the roots/zeroes/x-intercepts, which occur where the parabola expressed by ax^2+bx+c intersects the x-axis (y=0), hence y = f(x) = 0
y = (x+h)^2 + k, the axis of symmetry (basically, a vertical line) x = - h , because the actual form is always gonna be y = (x - h)^2 + k = (x- (-h) )^2 + k
Graph, observe and modify...you'll be fine.
The standard form " y = ax^2+bx+c = 0, a ≠ 0 " is genuinely the standard here. For example, derive the quadratic formula (and in doing so, you'll see how completing the square method brewed). Divide LHS & RHS by a (coefficient of x^2). Take the constant term to the RHS . There's another term, the linear term, apart from the x^2 on the LHS. Divide it by 2 and square it up, because that turns the left side into a perfect square. It will show up on both sides because you balance the sides of the equation by adding and subtracting or multiplying and dividing. Now, you'll see the RHS can have a common denominator and the LHS can be compressed into the (a+b)^2 form. Finally, make x the subject of the equation to call it a wrap.
Dad asked me a cool question [and I gave in hehe] about which graph has a crazy bunch of tangents, and I was like, it's got to be a straight line. It made me think about how there's so much in math I haven't touched yet. Like, I haven't even looked at angle of banking because it's in book 5, and who even knows what's in there without any notes or guides? He talked about his own undergrad school days. He self-learned his way through life, picked upon many books when he was in college (but for me, it’s middle school instead 😊) and even bigger ones when he was majoring in Aircraft Production Engineering. Even back in the 80s, it wasn't a surprise that almost many weren’t truly great, rather flunked well, because actually, no interest and just a filler space sums up much of his cohort, let alone putting in the time, effort and continuity [ always passable anytime, anyday ]. A fun fact: Dad’s scent in his former office room where I spent 95% of my time was on that big book for very long while! [it still was the case for a little over 2 years].
Right after 10th Grade finished, and I had months [could say the same before but post-Grade 9 is ….something] before going to do something and somewhere I hadn’t precisely decided yet, I gave in [cause I didn't have a reason not to…I desired clarity and was like “What’s up with the façade? I’ll tackle it” ] and was vividly quite serious because It was a subset of what the future had for me […a prelude to the intoxicating Mathematics]; collective seriousness I must say. One day, I kept going with my math chapters, just like always. I've got tons of math books in my folder and a dozen tabs opened with pages bookmarked for recording my progress. Dad cracked jokes about how I scroll through them, saying it's like watching a movie but with math pictures going up and down instead. He felt sleepy [so rare] and the next day morning, he said that “I was weeping in my heart seeing what you were upto”. At first, I was pissed when he wondered why I kept doing nothing much and clinging onto functions, sequences, and trig. The same sec, csc, and Pi rad, whole by 2?…. But I’ve been really “getting” into Math, not just floating on the surface but really understanding it [cause I had the mental time to give my mind the chance to absorb and assimilate]. It’s time for me to start new stuff. Logs at least and counting principles at best. Now, hearing that turned me off. I wish we’d actually done it more in the future, but the years before uni always had to be cumbersome, and mine were particularly overbearing + us parallel lines haven’t been about all that much. Funny how I say we’re parallel lines cause in all honesty, I can’t think it is not. But our styles and psyches are so congruent (way more than similar). Am I as good of a bamboo tree as his? Eh, I’m pretty sure I have a long way to go.
What I instinctively did was to prepare all things Trig in an all-encompassing cocoon and it made me happier than I thought it would.
I went ahead of unit circle, trigonometric ratios, the sine and cosine rules, bearings, calculating the area of a triangle using angles, trig functions, several trig formulae, and just all the geometry + coordinate geometry + Heron's formula (honorable mention! Fun in deriving it) + quadratics + logs and exponential functions + polynomials + arithmetic, quadratic and geometric sequences + graphical transformations + stats + prob [Props to Schaum]. It was like opening a new book that I had heard would be important but didn't quite understand how yet.
Thank you for patiently explaining everything starting from the absolute basics, answering all my questions no matter how many times I asked them differently. Months following it in primetime of high school (when trigonometry began alongside quadratics [these were quite the stars], stats, and graph transformations in general. The circular functions, trig ratios, their governing rules/laws, the area of a scalene triangle, ambiguous case, and angle-sum identities...), I was devoid of the pressure to rush through these concepts, allowing me to approach them with a happy-go-lucky attitude, yet with a depth of understanding that was gradually cultivated. And neither did I have to look at it much until mid ’20. I'm not sure why, but I felt this knack for sequences and series (also log applications) + further topics like MVT, global and local maxima and minima [and yes, the asymptote analogy falls under a personable ‘About me’ speech!]. I loved them. I plunged into them, not without facing opportunity costs in exploring other realms on the waitlist, though I pursued that love until I was subtly developing a future health problem because up until that point, all that I had ever done counted on me and stuck by my side.
I used to ask the same questions in different ways – excessively so. Okay, the information is readily available, and it's always a cute, helpful dump yard anyone will always try to declutter. And I'm confident you won't ask overly obvious questions. If "intention" is not condescending or mockery, I will always advocate for focusing on having conversations, by setting aside any "awkwardness”. My point is to let such a space evolve organically and patiently, always staying involved and sticking around. The real journey starts the moment your interest is piqued, or disinterest doesn’t takeover. We can’t always be on the same, precise pages. If that's a lot atm and you find yourself searching for an out, know that I understand. Well, just being on a nearby page is a gift in itself if it’s the same book. This naturally gets me thinking about the art of fitting in.
Quickly grab a rough paper and start reading the test paper rapidly (more like, skimming) but carefully in the reading time (so I want you to "ponder"). Just because you're really good at quadratics, don't quickly start substituting formulas with immense ecstasy (makes sense that I identified this as an alarming lesson in the future). Write the given info and what to find. Always extract the data available and the context of the problem you're working on.
Even if you think you can use the axis of symmetry because you know the other parts, pay attention to the details – like making sure you're using the right signs (plus instead of minus, and so on). Scrutinize the graph thoroughly (just know where you really are), and let that be your guiding principle. It's not as much a mystery as geometry and trigonometry (lowkey) are.
As I glance at the question, I first distinguish between the information provided (knowns) and what needs to be found (unknowns). This helps in formulating a plan by connecting the dots between what is known and what is to be discovered. Then, I recall the relevant tools to approach it.
If you hit a roadblock, you should be able to revisit earlier steps to understand what's actually going on. I've never valued space management this much. It can greatly influence time management, especially if you don't have endless time at hand. Moreover, reviewing is one thing and verifying is another. Both are significant to ensure that the solution logically follows from the given information.
Endless practice of the same topics over and over again is cool, but wait, it's not quickly clear if it involves a rewarding choice, i.e., not limited to one type, modifying a diagram or adjusting an equation with an additional variable - exploring different scenarios and their effects on the problem-solving process. This is a deeper way of thinking and learning.
About 9 out of 15 textbooks were similar to the other 6. From this, the final gist for math slowly came to form [Most of the textbooks share similar content. Working through them simultaneously is great]. The moment this is touted to be long, it's not an enticing topic anymore. Now this is when I let my mind know it needs to stay in contact with tasks when they’re longest.
When dealing with problems that involve substituting values when you know some of the variables in an equation, is it enticing because you "get it," or does it feel dull because it seems too obvious? Just work through around 30-35 challenging [you don't need to be picky much. Your mind will know better when it's all smokes. Your mind will sort it out once you're in the thick of it] problems to get used to the storyline.
I’ve personally loved studying Schaum’s differential equations, 4th edition. Shoutout to it (and all of my notes from the past) while I freshly explored Newtonian mechanics at school.
Today, I've been thinking about how I should just let my feelings about math flow like a simple wave in my mind, not making it too tangled. Even with it, untangle it or put it aside. We have enough of tangents. I’m rather sure I’m going to be done with binomial theorem (for example, governing equations to find the middle term and 6th term all…not just nCr), counting principles, limits and derivatives because writing a final gist for them don't need me to do a million problems like when I work on sequences and series, functions, logarithms, circular functions, and trigonometry.
Like, take figuring out the common ratio of a geometric sequence when you know the 4th and 6th terms. You see, if you divide the 6th term by the 4th and then halve it (because 6 minus 4 equals 2), you're on your way. And if you know the sum of 8 terms, shouldn't you be able to find the first term? And if the sum can't be more than 600, you just use inequalities to limit your answer and solve it. Then there's the snake-looking integration, where you really have to be sharp about adding and subtracting to find areas on a graph. That stuff needs you to be super clear about what you think you know.
To figure out the common ratio when you know the 4th and 6th terms of a geometric sequence, you use the relationship between terms. The ratio of the 6th term to the 4th term, squared (because the difference between 6 and 4 is 2), gives you the common ratio.
If you know the sum of the first 8 terms, you can use that information to find the first term. You use inequalities if there's a condition like the sum can't exceed 600. This helps narrow down the possible values for the first term.
When you're working out areas on a graph, remember to carefully add and subtract as needed; this step in integration really tests how well you understand the concept. Honestly, I'm not surprised to note that "graphs" is not an area where you hear many saying a happy YES.
So whatever things might be like, knowing yourself (including your work ethic, the manageable scope) and therefore ‘at some point, enough of this chapter’ is important, especially as the magnitudes of resources can change a lot from high school to college, for example. [I totally feel this advice coming for me ha.]
So, I'm trying to figure out how to make sense of all these jumbled thoughts and turn them into something you can easily understand. This might not cover everything I've talked about in my past videos, but here goes. I've been trying to juggle learning a bunch of stuff - like a lot of chemistry, a bit of physics, some economics, and then there's math with all its differentiation, not to forget English [where I have to analyze books for the next paper] and Telugu [ooh yes, reading/writing with ease has never been a problem while not having to academically engage with it that much...the beauty of being deeply analytical with a poetic edge, love for anything artful and this very great language] . And guess what? Classes begin in just three days. It’s been about a month. Why couldn’t I get done with these way earlier?
With all this in mind, at the nucleus of it all, the realization of what I’m about to discuss with you requires a “certain” amount of time to fully take shape. All sides pulling, the system striving for equilibrium? Sounds about right.
I've noticed my journey is pivoted by what I've learned from reading, ridiculously reading off of texts. Because I've been slow, I might end up (terribly) repeating stuff in my final notes that I want to share with you. At a time when I was focusing on certain subjects, I started a routine at least a month before, waking up every day at 3 am to start my work. Now, this is a problem because if I say I didn’t work out 3 out of 4 chapters I learned in my high school junior year, it might seem like I don’t understand the real thing. However, my lens says It’s actually about the gaps: the spaces between what’s known and what’s unknown, the knowledge that isn’t captured in textbooks or typically taught in classrooms. Make the connections! Pretty paranoia. Questioning the sanctity of waking early : think about the solitary hours when the world sleeps, and our thoughts have the space to breathe.
As the final high school years kickstarted, the challenging of all were unamusing [leaving me hungry for some actual adrenaline rush]. Thus, I floated (after learning to swim..ofc!) in an array of resources and a bittersweet part of the corollary was acing that rather-feel-good test, missing out on just 2 points out of what was answered (that would've had a much better effect overall). Yet again, I'll have to admit it depends on where we stand that makes all the difference. Less time, more time. Less effort, more effort.
I haven’t read the 2nd, 3rd, and 4th chapters of my chemistry book yet, but I’ve got the gist of it [well, except for most of the 4th chapter]. Usually, it’s not that hard to go through textbooks chapter by chapter and piece everything together, especially once you get into a flow. But here’s the thing: I haven’t finished the 2nd chapter, the 3rd chapter isn’t pulling me in yet, and that means I’ve got to read carefully and quickly [really making sure I get each part before I can sum it all up but also satisfying the adrenaline rush].
On one side, I feel like I've only done 10% of the work and need to push harder. On the other side, I feel like I've done a lot more than it sounds and that I'm on the right track. This conflicting situation makes me feel like it ain't really enough [like oil that should be getting thicker but is just evaporating away.] It seems possible to finish the 2nd to 6th chapters of chemistry in one day, and the 5th to 7th chapters of economics in the same amount of time. But skipping straight to the end or not engaging deeply with the material just makes things more unsatisfying. It's easier to handle the chapters and understand the content when I focus and take it seriously, which I've been loving lately. [It's much healthier to note that being a snail pays off somewhere, sometime.] I think for these 'zones' in my mind to activate and then the classroom setting being more of
‘The fear of missing something’ can drive us nutso. Valid or not, read, read, and read! And hopefully, make a good gist. We strive to distill the essence of our findings. It’s a time-consuming process, but we persist, preparing summaries and snapping photos of notes across four subjects. But it's necessary to make sure I don't miss anything, especially since the syllabus can soon change.
I intentionally didn't want to go through any other chapters after the first one in Chemistry. Was that ignorance? Self-destruction? Neither. The truth is, I don’t fully, exactly know everything in chapters five or six (or most of the first five, for that matter) but the core concepts, the ones that matter, actually make sense to me. Except for a few specific areas, like stoichiometry and the likes of it, the rest is fine. Most of it, after all, was covered back in middle school, and I’ve already been through those parts before.
There's nothing brain-teasing about the 2nd, 3rd, 5th, and 6th [excluding the 4th because it's a lot abstract in this head than it already was at a first glance] chapters in chemistry. Similarly, [The thing is, you'll have to learn to love your thought process, nourish it and cherish it. Just getting by on luck or piecemeal efforts could work for now, but it won't help in the long run, especially for those who really want to learn. If you're just cobbling together code snippets and have “yeah, whatever” kinda idea without the what, why, when, how, and where, you're not really learning anything. Have them at the back of your mind, so that you can gradually let the new stuff assimilate and marinate! This half baked trip doesn’t cut it for me, not when the loss of interest and not much relevance to personal, academic and professional goals prevail. What’s amusing to me is that when it should support me, it instead brings reverse benefits, complicating things and creating "defining moments", reasons to toss me out. Smh.] the whole journey isn't so unbearable or something [it can get that way though!], yet lateral thinking makes it dreaded, for a strong foundation of concepts is a must for firm achievements [Still love investigative pattern tasks. That part’s stuck with me ever since].
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