Math - Blog Posts
Pascal’s Triangle
I'm nervous
But I showed up and here I am
I'm choosing to go down swinging, hard
I have to know that I tried
Even if I know I'm gonna get my butt kicked,
I like to at least attempt to kick back
My life has trained me for the sport of butt kicking
But it sometimes leaves me feeling pooped out
Every time I yawn I roar like a lion
A silent roar of sleep deprivation
But a roar of determination
Call me stupid
Call me crazy
But I'm gonna get this right no matter how many times it takes
I'll get it eventually
You can annoy me and make me feel uncomfortable but you cannot get through my stubborn head
I'm nearly impossible to brainwash
Without other methods being used
Looking at the big picture
Can be daunting
But you can just use fractions
And break everything up
Shatter it thoroughly
Take a step back
Breathe
And look with new eyes of simplicity
One step
At
A
Time
Let the miracles happen, and have fun storming the castle
fuck you *turns your coordinates polar*
Looking at my son: oh you like math?
Cocks gun and looks at my wives: I know what we’re having for dinner tonight :)
Class 1-A nerds vs Trump & Shigaraki
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Female Aerospace Pioneers - Elizabeth “Bessie” Coleman In 1921 Elizabeth “Bessie” Coleman, a Texan, became the first civilian (non-military) licensed African-American pilot in the world. She went to France to learn to fly after her brothers, who served in World War II, told her French women were allowed to fly. When she returned to the United States she did air shows: barnstorming, parachute jumping, and giving demonstrations. Coleman would only perform if the audience was not segregated and all people got to enter the show through the same gate. This courageous and adventurous woman fell from the open cockpit of a plane during a test flight which ended her life on April 30, 1926. She was enshrined in the National Aviation Hall of Fame in 2006. Read more #OnTheBlog: http://www.groovylabinabox.com/female-aerospace-pioneers/ #WOAW16 #womenshistorymonth #flysafe #femalepilot #genav #pilotlife #femalepilot #femalepilots #avgeek #STEMists #GroovyLabInABox #STEMist #GroovyLab #WomenInSTEM #GirlsInSTEM #aviation #science #technology #engineering #math #STEM #instascience #homeschooling #homeschoolscience #aerospace #STEMEducation #STEMEd #aviators #AirAndSpace #SciGirls #WomenInTech
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When algorithms surprise us
Machine learning algorithms are not like other computer programs. In the usual sort of programming, a human programmer tells the computer exactly what to do. In machine learning, the human programmer merely gives the algorithm the problem to be solved, and through trial-and-error the algorithm has to figure out how to solve it.
This often works really well - machine learning algorithms are widely used for facial recognition, language translation, financial modeling, image recognition, and ad delivery. If you’ve been online today, you’ve probably interacted with a machine learning algorithm.
But it doesn’t always work well. Sometimes the programmer will think the algorithm is doing really well, only to look closer and discover it’s solved an entirely different problem from the one the programmer intended. For example, I looked earlier at an image recognition algorithm that was supposed to recognize sheep but learned to recognize grass instead, and kept labeling empty green fields as containing sheep.
When machine learning algorithms solve problems in unexpected ways, programmers find them, okay yes, annoying sometimes, but often purely delightful.
So delightful, in fact, that in 2018 a group of researchers wrote a fascinating paper that collected dozens of anecdotes that “elicited surprise and wonder from the researchers studying them”. The paper is well worth reading, as are the original references, but here are several of my favorite examples.
Bending the rules to win
First, there’s a long tradition of using simulated creatures to study how different forms of locomotion might have evolved, or to come up with new ways for robots to walk.
Why walk when you can flop? In one example, a simulated robot was supposed to evolve to travel as quickly as possible. But rather than evolve legs, it simply assembled itself into a tall tower, then fell over. Some of these robots even learned to turn their falling motion into a somersault, adding extra distance.
[Image: Robot is simply a tower that falls over.]
Why jump when you can can-can? Another set of simulated robots were supposed to evolve into a form that could jump. But the programmer had originally defined jumping height as the height of the tallest block so - once again - the robots evolved to be very tall. The programmer tried to solve this by defining jumping height as the height of the block that was originally the *lowest*. In response, the robot developed a long skinny leg that it could kick high into the air in a sort of robot can-can.
[Image: Tall robot flinging a leg into the air instead of jumping]
Hacking the Matrix for superpowers
Potential energy is not the only energy source these simulated robots learned to exploit. It turns out that, like in real life, if an energy source is available, something will evolve to use it.
Floating-point rounding errors as an energy source: In one simulation, robots learned that small rounding errors in the math that calculated forces meant that they got a tiny bit of extra energy with motion. They learned to twitch rapidly, generating lots of free energy that they could harness. The programmer noticed the problem when the robots started swimming extraordinarily fast.
Harvesting energy from crashing into the floor: Another simulation had some problems with its collision detection math that robots learned to use. If they managed to glitch themselves into the floor (they first learned to manipulate time to make this possible), the collision detection would realize they weren’t supposed to be in the floor and would shoot them upward. The robots learned to vibrate rapidly against the floor, colliding repeatedly with it to generate extra energy.
[Image: robot moving by vibrating into the floor]
Clap to fly: In another simulation, jumping bots learned to harness a different collision-detection bug that would propel them high into the air every time they crashed two of their own body parts together. Commercial flight would look a lot different if this worked in real life.
Discovering secret moves: Computer game-playing algorithms are really good at discovering the kind of Matrix glitches that humans usually learn to exploit for speed-running. An algorithm playing the old Atari game Q*bert discovered a previously-unknown bug where it could perform a very specific series of moves at the end of one level and instead of moving to the next level, all the platforms would begin blinking rapidly and the player would start accumulating huge numbers of points.
A Doom-playing algorithm also figured out a special combination of movements that would stop enemies from firing fireballs - but it only works in the algorithm’s hallucinated dream-version of Doom. Delightfully, you can play the dream-version here
[Image: Q*bert player is accumulating a suspicious number of points, considering that it’s not doing much of anything]
Shooting the moon: In one of the more chilling examples, there was an algorithm that was supposed to figure out how to apply a minimum force to a plane landing on an aircraft carrier. Instead, it discovered that if it applied a *huge* force, it would overflow the program’s memory and would register instead as a very *small* force. The pilot would die but, hey, perfect score.
Destructive problem-solving
Something as apparently benign as a list-sorting algorithm could also solve problems in rather innocently sinister ways.
Well, it’s not unsorted: For example, there was an algorithm that was supposed to sort a list of numbers. Instead, it learned to delete the list, so that it was no longer technically unsorted.
Solving the Kobayashi Maru test: Another algorithm was supposed to minimize the difference between its own answers and the correct answers. It found where the answers were stored and deleted them, so it would get a perfect score.
How to win at tic-tac-toe: In another beautiful example, in 1997 some programmers built algorithms that could play tic-tac-toe remotely against each other on an infinitely large board. One programmer, rather than designing their algorithm’s strategy, let it evolve its own approach. Surprisingly, the algorithm suddenly began winning all its games. It turned out that the algorithm’s strategy was to place its move very, very far away, so that when its opponent’s computer tried to simulate the new greatly-expanded board, the huge gameboard would cause it to run out of memory and crash, forfeiting the game.
In conclusion
When machine learning solves problems, it can come up with solutions that range from clever to downright uncanny.
Biological evolution works this way, too - as any biologist will tell you, living organisms find the strangest solutions to problems, and the strangest energy sources to exploit. Sometimes I think the surest sign that we’re not living in a computer simulation is that if we were, some microbe would have learned to exploit its flaws.
So as programmers we have to be very very careful that our algorithms are solving the problems that we meant for them to solve, not exploiting shortcuts. If there’s another, easier route toward solving a given problem, machine learning will likely find it.
Fortunately for us, “kill all humans” is really really hard. If “bake an unbelievably delicious cake” also solves the problem and is easier than “kill all humans”, then machine learning will go with cake.
Mailing list plug
If you enter your email, there will be cake!
math aid
OK SO in the days leading up to the biggest maths exam I’ve ever written (also my 4th last one ever ) i’ve found this website. now, symbolab is different to mathaway and wolfram alpha (which are both great!) in that it shows you all the steps and it doesn’t do that thing where it’s “free” but if you want the explanation you have to sign up and pay. it’s AMAZING. know why?
it does everything. not baby everything, but everything. Calculus?
what’re you looking for?
also, the interface is really easy to use, and it’s set out so well.
the website is https://www.symbolab.com and you will not be disappointed.
hello! precalculus is a pretty interesting class if you put your mind to it, and i found some great resources for it so i thought i would share! also, since trigonometry is part of this subject, that’s included too!
khanacademy
purplemath
precalc dictionary!
math forum
mathbff
course notes
notes by topic
amazing formula sheet
sparknotes: math
edX course!!
math study tips
video lessons
ucl instructional videos
quadratic formula song
math professor quotes
+ trig specific
khanacademy!
course notes
how to learn trig
great formula sheet
interactive unit circle
trig identities hexagon
basic trig + graphing
identities and equations
unit circle song!
more masterposts!!
algebra
AP chemistry
AP world history
studyblr-ing
the Everything Book
the pomodoro method
how to use flashcards
how to use sticky notes
welcome to high school
tiny study spaces
what’s in a pencil case
i hope this was of help to you! keep shining like the star you are and remember to be awesome today!!
- Aza
Don’t Let Calculus D(e)rive You Mad
I was always one of those people who thought some people were naturally good at math and if I wasn’t one of those people then there was nothing I could do about it. I thought I wasn’t “a math person” and would use that description as an excuse. Is math one of my weaker subjects? Sure but that’s mostly because I let years of bad habits get in the way of my current work. This caught up to me in my first semester of calculus (calc I) at university, where calculus was my worst class. Here’s the thing: if you’re not “a math person” make yourself one. In my second semester of calculus (calc II) I improved my mark by an entire letter grade (something I never thought possible). How? Through hard work and by understanding that I would have to work harder than some people because of my past study habits.
Know your pre-calculus well! You will struggle so much if you forget the basics. My prof said not having a good grasp of the basics is the number one reason why students will struggle with calculus. Invest time before/at the beginning of the semester to really review the stuff you learned in high school. (Khan Academy is the best way to review, in my opinion. They have challenge questions you can do for each section. Try a couple of questions for each section. If you can’t answer the question easily, watch the accompanying videos for that section first. Do this for sections you forget or know you struggle with.) Be confident in your basic mental math too, especially under pressure. I wasn’t allowed a calculator on any of my midterms or finals for calc and you don’t want to waste time on easy math that you should know lightning fast anyway.
Attend every lecture, especially if you’re even slightly confused. If you’re behind, try not to get even more behind by skipping class (obviously use your own judgement, but don’t skip unless it’s totally necessary). Don’t sit near the back of the class if you know you won’t pay attention.
Don’t just sit there and copy down notes. Be attentive in class and follow along with examples the best you can. If you get lost at a certain step in a problem put a star beside it. After class, study and attempt the problem on your own. If you still don’t understand, go to a TA or prof for help. They will be able to provide better help if they can see exactly where you got lost.
Keep your notes simple. I would use either blue or black pen for the majority of my notes and use one other colour to emphasize parts of my notes (indicate where I got lost, circle important follows, highlight which section of the textbook the class was at, etc.) Keep your notes neat and leave a gap, if you fall behind during a lecture (just remember to get the notes from someone else later). I also recommend using a grid paper notebook, for when you need to draw graphs.
Get a mini notebook! I bought a tiny notebook for cheap and filled it with a (very) condensed version of my notes, throughout the semester. I wrote down common derivatives and integrals, shapes of common graphs, important theorems and formulas, etc. This is especially helpful for calc II, because you’ll have all the necessities from calc I handy.
Advice for using Maple for math labs (if this applies to you): Pay attention to tutorials and ask questions. Complete as many assignment questions as you can in the lab/when a TA is present. If you have any other assignment questions to finish up make sure you work on them at least a few days before they’re due, so you have time to ask for help if you need it. Also, Maple can be a stupid program. You could be missing just one number, letter, or symbol and it won’t work. Or you could have it exactly right and it still won’t work (retyping your input in a new worksheet usually helps). To remedy these issues, I would work on assignments with friends and compare what our worksheets looked like. Oh and TAs love if you give your variables funny names or change the colours of your graph, because they’re all nerds (and so are you, so embrace it).
Do as many practice problems as you can. Calculus is a class where you learn by doing. Do questions till you understand the concept. If problems are recommended, treat them as if they’re actually due (otherwise you’ll just tell yourself you didn’t have enough time to do any practice problems). My number one mistake was not doing enough practice problems and just assuming I knew how to answer the problem (if you can’t answer the entire question from start to finish, then you don’t actually understand the concept).
Please don’t fall behind. Stay on top of things and prioritize what needs to be done (i.e. treat practice problems from the chapter you just learned on equal footing with the lab report you have due – if you treat it as a priority, you will get it done). But, if you do fall really behind, don’t wait until it’s too late to ask for help. Just remember, there’s always something you can do (even if you feel like you don’t know anything and there’s not enough time for any practice problems before your midterm). Identify what you need to learn before you can do anything else (i.e. work on understanding basic integration before you try to do something more complicated like trigonometric substitution) and fit in as many practice questions as you can.
Don’t give up! If you don’t understand a concept right away you just have to keep trying! For practice problems, try to find an answer without looking at your notes. If you can’t figure it out from there, look in your lecture notes and textbook for any relevant formulas, examples, or similar questions. Try to answer the problem again. If you get it, be sure to fully complete another practice problem without any outside references. If you can’t figure out an answer then you should seek help from another person!
Don’t forget everything you learned at the beginning of the semester – review, review, review! Check out this explanation on the curve of forgetting. If you continually review what you learned, for only short periods of time, you will remember so much more and save yourself time in the end!
Utilize the resources available to you. I have a list of online resources at the end of this post, but don’t overlook what’s right in front of you. Go to your prof’s office hours, ask a TA for help, and take advantage of any tutoring or study groups. My uni has a math and science centre where upper year students are always available to help other students with practice problems. If you join a course union, they sometimes offer free tutoring.
Study in a productive environment. This varies by person but personally I need a quiet environment, with ideally no noise or only instrumental music, bright/natural lighting, and nothing to distract me (I hide my phone and only have one pen or pencil out). If you like to listen to music when you study, math is one of those subjects where you can listen to music with words.
Improve your test-taking skills. (1) On an exam, understanding a concept is no use if it takes you forever answer the question. Do lots of practice problems till you immediately know how to answer any kind of question. Speed can be key on exams. (2) My strategy is to flip through the exam booklet as I get it. I answer the questions I can do easily, first, and leave the really difficult ones till the end. (3) Show all of your work! Don’t lose marks because you didn’t show all of your work. (4) Expect your exams to be challenging and prepare accordingly. Overlearn the material. Prepare specifically for the exam by completing past exams/practice exams in an environment that mimics the test-taking environment.
Get every mark you can, because the little marks make a big difference. If you don’t know how to answer a question on an exam, write down any formula or theorem that could relevant. If you try to figure out a solution and know that it’s most likely incorrect, but don’t have enough time/knowledge to find the correct answer, just leave your work there (don’t erase it). There’s always a chance you could be on the right track or nice markers will give you a point or two for trying. Something is always better than nothing.
Focus on the applications of calculus (it’ll make the semester a whole lot more interesting)! A physics major won’t necessarily use calculus the same way a bio or chem major might, but that doesn’t mean some calculus isn’t useful for all of those majors to know. I’ve always planned to major in biology and looking ahead at classes I will need calculus for biostatistics and genetics classes. Never tell yourself something isn’t useful because then you’ll never treat it like it’s useful. Also, my prof taught a whole lecture about how calculus could be used to account for all the variables that could affect population if a zombie apocalypse ever happened, so obviously calculus has at least one really important use :)
Resources
A bit of advice: These are called resources for a reason. It’s okay once in a while to use some of the resources to find a full solution for a practice problem, but don’t abuse it. It is so so easy to just look up the answer but you’re only hurting yourself in the end.
Desmos (Online graphing calculator - I’ve made it through so far without actually buying a graphing calculator)
Khan Academy (Step by step videos and practice questions! You can go your own speed with the videos! My top recommendation!!!)
Paul’s Online Math Notes (If your prof doesn’t provide you with decent lecture notes, these ones are great!)
Symbolab (They have a calculator for derivatives, integrals, series, etc. and I like the way they split up the steps to solve.)
Slader (find your textbook on here and they’ll give you all the solutions to questions!)
Textbooks: I used the Single Variable Calculus: Early Transcendentals (8th edition, by James Stewart) and it was awesome. The way it was set up and all the examples really helped me (I just wish I had used it more)
This post by @quantumheels is seriously fantastic (and she has lots of good advice for other topics too, one of my favourite blogs)
My Other Posts:
AP lit tips, high school biology, how to ace intro psych, organization tips, physics doesn’t have to suck: how to enjoy and do well in your required physics classes, recommended reads, reminders for myself, using your time wisely on public transport, what i learned from university (first year), what i learned from high school
General Resources
Figure Out What You’re Missing
Tennessee Tech Placement Test
Berkeley Placement Test
To Learn Concepts
Videos/Playlists:
Khan Academy - videos and exercises
Khan Academy’s YouTube - videos
ProfRobBob - short filmed lectures
brightstorm - has problems, explanation, and transcript
Websites
TheMathPage - a list of topics and basic explanation, includes problems and step-by-step answers
Coolmath - a list of topics, colorful explanations, and examples
WolframMathWorld - topics, vocab, and quick summaries (probably more useful as a review resource/recap resource)
CK-12 - detailed curriculum, has practice and resources
Open Textbook Store - textbook, homework sets, teacher notes, lecture examples, sample quizzes and exams…
Free courses:
University of California, Irvine at Coursera
University of Texas, Austin at edx
Textbook PDFs
Precalculus by Carl stitz and Jeff Zeager
Precalculus by David H. Collingwood, K. david Prince, and Matthew M. Conroy
Other Resources:
A formula sheet
Practice Problems
Non-interactive:
University of California, Davis Resource - list of topics and example problems and answers
NYU - list of topics and example problems and answers, like lecture note format
Mathematics Vision Project - has a mix of algebra, geometry, and trig. These are the “workbooks” that my school uses.
Interactive
IXL - list of many topics and questions that increase in difficulty when you answer correctly.
List of Topics to Be Solid With Before Calculus
I took this list of topics from various internet sources and my own teacher. Resources to learn each of these are pretty easy to find. I haven’t taken Calculus yet (just have been freaking out about it), so if anyone has insight, please share!
Essential Alg/Precalc
*in italics are those my teacher stressed for my class in particular
find the equation of a line
trig (properties and values, unit circle, graphs, trig ratios (solve triangles using trig ratios & define), trig relations/identites)
different kinds of equations and graphing (polynomial, absolute value, exponential, polar, parametric, inequalities [esp those with absolute value], logs)
logs
limits
vectors
algebraic manipulation (factoring, completing the square, simplying expressions, solving equations)
–manipulate polynomials
formula manipulation
manipulation of abstract function expressions
function transformations (differences between y = f(x) + c and y = f(x+c) & the like, inverse functions)
using your calculator
analytical geometry
Memorize the following graphs visually - will help with derivatives (x^2, x^3, sqrt(x), 1/x, 1/(x^2), logx)
![Hey!! :] A Lot Of You Guys Seem Confused About Math Like I Was When I Used To Study It. However, Now](https://64.media.tumblr.com/aa4997bfece8a9aea8157c94baf4cae9/tumblr_nv2qm71eyA1uu5l5yo1_500.png)
hey!! :] a lot of you guys seem confused about math like i was when i used to study it. however, now that i don’t anymore i would really like to help people out with their math studies by making a masterpost. i was always mediocre at math so bear with me please, here goes:
how to study math
a guide
solving problems
check the math section here
how to take perfect math notes
websites for you
the best math site!!
khan academy
MATH CHEAT SHEETS
calculus cheat sheets
mathblrs
mathway
symbolab
iformulas
a facebook page
google does math for you
sparknotes [i use this mainly for english, but there are a lot of resources on it]
calculators
algebra
a complete list of online math resources
a website to help you stay interested in math
+ another
videos + audio
TEDed math
khan academy on youtube [fav!!]
math tv
mathview
terry v
mathdude podcasts
games
mathsframe
absurd math
how to pass math [my method]
work out a lot of previous exam papers if ur allowed to buy them + stuff
if not, take examples from ur text book and work them out
you have to know the methods + get used to using them
this helps u become faster in the long run especially during the exam
time yourself when working out something [especially a past paper]
work as hard as u can but remember to take breaks. this is really important!!! i used to cry a lot bc of math idk why…
stay calm + focused, math needs hard work + determination
don’t worry if you fail, you can always take the exam another time. as long as you did your best, you should be proud of yourself! <3
hope this is of some help to you guys. if you want to ask me anything or request a masterpost, you can do that by messaging me!! xx
yesterday I realised that I barely know anything in maths so I’m having to sort myself out - trying to go over 2+ topics a day, making what I like to call “emergency notes”, so far so good but #prayforzoë
1/100 ➵ 190216
My love for perfection always ends up in procrastination, so I’m really excited to finally start with this challenge, yay! These are my Algebra notes for my big exam in July…and I’m quite happy with this writing system now.
Scans of the inside covers of Strang’s Calculus, which you can legally-download for free here from the MIT website. This is my all-time favorite math or physics textbook. Scanned it so I could cut and paste it into my new sketchbook, wanna try and make a ~cool artistic~ reference poster out of it, ‘cuz I’ve been real into that idea since I took notes about rings for the algebra midterm on a big piece of watercolor paper.
I’m taking the AB Calc AP this year (yikes), so here are some of the resources I’ve found so far! I’ll add more as I find them.
Free Practice Tests & Questions
1969-1988 Multiple Choice Questions
2006 Practice Exams (AB & BC, with answers)
Varsity Tutors
College Board Released FRQs
Peterson’s Practice Test
GetAFive Practice Questions
4Tests Practice
Booooooks
The Princeton Review (3 practice exams)
REA Crash Course (online practice exams)
Barrons (AB & BC, 5 practice exams each)
Kaplan (6 practice exams & 2 diagnostics)
5 Steps to a 5 (3 practice exams)
COW Math (online calculus books)
Peterson’s (online, AB & BC)
Multiple Choice Workbook
Videos
HippoCampus
Khan Academy (so many worked answers)
WOWmath (free response questions)
Other Resources
PDF Reference Sheets (from EE, but here in a handy folder)
Interactive Mathematics Lessons
Visual Calculus (tutorials & drills)
College Board FRQ Index
MIT OpenCourseware Exam Prep
Brightstorm
Mr. Calculus
GetAFive
Paul’s Online Math Notes
Study Guides
Elaine Cheong’s Study Guide
University of Houston Study Guide
Final Review Sheet
Calculus Cheat Sheet
I hope this helps you out! There are more useful posts from my study series here.
Took around 2 hours to finish my A4 sheet of handwritten notes for my calculus test! The topic is confusing so I need to work hard in order to get good results!! 😊 Studygram: acadehmic
TRIG REVIEW #2: IDENTITIES, EQUATIONS, & POLAR GRAPHS
So this trig review is going to be about trig identities, how to solve trig equations & oblique triangle problems, and last but not least, polar graphs.
Trig Identities: You have to know these identities (except half-angle and sum/difference). Also the bottom two are necessary for solving trig integrals, so memorize those!
Trig Equations: Here are various example problems showing how to solve trig equations.
Oblique Triangle Equations: You only have to know the equilateral equation for area of cross sections in Calc BC. The rest is just extra info. that is nonetheless helpful to know. (:
Polar Coordinate:
Polar Graphs: Memorize these trig graphs for finding polar area! (All graphs come from Wolfram Alpha)
And that’s basically all the trig you need to know for Calc BC. Good luck! If you haven’t seen my part 1 post, you can find it here.
calculus?
more like calculost. please don’t ask me what the derivative of a log base is because i might (i will) combust into ashes of weeb culture & self deprication
A list of math books
…to read in lieu of an actual math class. So far I have
- Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter
- I am Strange Loop by Douglas Hofstadter
- Chaos by James Gleick
- Category Theory by Steve Awodey
Heres the thing you gotta understand about statistics.
“Increases your chances by 80%” does not mean “there is now an 80% chance”.
If your chances were previously 10%, your chances are now 18%, not 90%.
if your chances were roughly 1%, they’re now just slightly less than 2%.
thats how that works.