Conjugate - MATH

New Rules is my favorite on TDC: Magic and the current votes in the Survivor game sadden me so to give New Rules some justice here's my Korean Lesson on it. Previous TXT Korean Lessons: Maze in the Mirror, Crown, We Lost The Summer. I am an intermediate Korean learner. As usual, I will provide the original lyrics, a breakdown, my English translation and a Quizlet: New Rules
​

I think I’m feeling all right 아니면 아니었나 hey 반복의 수학시간 선들에 갇힌 내 삶

아니면: or maybe
아니었: past tense of 아니다 (to not be)
-나: questioning conjugation
반복: repetition, repeat
의: possession marker
수학시간: math class
선들: lines, boundaries
에: location particle
갇히다 (갇힌): to be locked away
내: my (informal)
삶: life, existence
​
I think I'm feeling all right
Or maybe not hey
Repeating math class
My life is trapped in the lines
​

왼발, 오른발이면 오른발, 왼발이 편해 왜 이런지는 나도 몰라 세상 모든 게 선악과 (yum yum)

완발: left foot
오른발: right foot
이다: (-이) to be (is)
-면: if X
편하다 (편해): to be comfortable (comfortable)
왜: why
이런: like this, such
이런지는: like this
나: I, me (informal)
-도: also, as well
모르다 (몰라): to not know
세상: the whole world
모든: everyone
게: things
선악과: good and evil, forbidden fruit
​
left foot, right foot
right foot, left foot is comfortable
why it's like this I don't know either
everything in this world is forbidden fruit (yum yum)
​

잠긴 옥상엔 더 괜히 가고 싶어 정해진 문자 말투 난 지겨워 지겨워

잠긴: locked
옥상: rooftop
-엔: 에 + 는 topic marker = 에는 shortened to 엔
더: more
괜히: in vain
가고 싶다: to want to go
정해진: fixed, definite, regular
문자: text
말투: one's way of speaking
난: 나 (I, me) + 는 topic marker = 나는 shortened to 난
지겹다 (지겨워): to be boring
​
Without reason I really want to go to the locked rooftop
A set message tone, it's so boring so boring

(왜) 페북, 인스타에선 (왜) 쌤이 왜 이렇게 많아 (왜, 왜) 수업이 왜 이렇게 많아 다 비켜 비켜줘

페북: Facebook
인스타에선: Instagram
쌤: teachers, word of address for teachers
이렇게: like this
많다 (많아): to be a lot
수업: lessons
다: all, everyone
비키다: to step aside
주다 (줘): to please do something for me
​
(why) Facebook, Instagram
(why) why are there so many teachers
(why why) are there so many classes
Everyone move out of the way
​

나 벌받을게 뻔해도 더 짜릿한걸 원해 갖다 줘 넘지 말라면 I want it 하지 말라면 I do it Stupid, but 난

벌받: punishment
뻔하다: to be clear, obvious
V-해도: even though
짜릿한걸: thrilling
원하다 (원해): to wish, hope for
갖다: to have, hold
넘다: to cross
말하다: to talk
-라면: if you say
난: 나 (I, me, informal)+ 는 topic marker= 나는 shortened to 난

펑크이고 싶어 Let me go, let me go, let me go go go Let me go, let me go, let me go go go
펑크이고 싶어 Where we go? Where we go? Where we go? Don’t know Let me go, let me go, let me go go go

펑크이: Punk
-고 싶다: to want to do/be X
​
I want to be punk
Let me go, let me go, let me go go go
Let me go, let me go, let me go go go
​
I want to be punk
Where we go? Where we go?
Where we go? Don't know
Let me go, let me go, let me go go go
​

에스컬레이터는 미끄럼틀이 되고 기대지 말라면 기대고 Trouble, trouble, let me go Trouble, trouble, let me go

에스컬레이터: escalators
는/은: topic markers
미끄럼틀: slide
이/가: subject markers
되다: to become
기대다: to lean on, against
-지: don't
​
The escalator becomes a slide and
If they tell me not to lean on it I'll lean and
Trouble, trouble, let me go
Trouble, trouble, let me go
​

나의 투명한 족쇄를 깨고 룰을 만들어 새로 Trouble, trouble, let me go I’m trouble maker, let me go

투명한: transparent
족쇄: shackles
를/을: object marker
깨다: to break
-고: and
룰: rule
만들다: to make
새로: new
​
Breaking my transparent shackles and
Making new rules
Trouble, trouble, let me go
I'm trouble maker, let me go
​

네가 좋아하는 상식 그건 누구의 상식 검증받고 왔니? 뭐 어디 석사님 박사님께?

네: your (informal, speech)
좋아하다 (좋아하는): to like (that one likes)
상식: common sense
그건: that thing (topic)
누구: who
검증받: verified
뭐: what
어디: where
선사님: master
박사님께: professors/doctors
​
That common sense that you like
Whose common sense is that
Has it been verified?
What from the MBA's and professors?
​

내가 만든 상식 Do it what I like it 핸들 내가 잡지 I’m trouble maker

만든: that which is made, made
핸들: handle
잡다: to hold
​
The common sense I've made
Do it what I like it
I take the handles
I'm trouble maker
​

나 벌받을게 뻔해도 더 짜릿한걸 원해 갖다 줘 넘지 말라면 I want it 하지 말라면 I do it Stupid, but 난
펑크이고 싶어 Let me go, let me go, let me go go go Let me go, let me go, let me go go go
펑크이고 싶어 Where we go? Where we go? Where we go? Don’t know Let me go, let me go, let me go go go
에스컬레이터는 미끄럼틀이 되고 기대지 말라면 기대고 Trouble, trouble, let me go Trouble, trouble, let me go
나의 투명한 족쇄를 깨고 룰을 만들어 새로 Trouble, trouble, let me go I’m trouble maker, let me go
I think I’m feeling all right 아니면 아니었나 반복의 수학시간 선들에 갇힌 내 삶
왼발, 오른발이면 오른발, 왼발이 편해 왜 이런지는 나도 몰라 세상 모든 게 선악과

~see above~

There’s a lot of math out there. Some things show up all the time on the ACT. Other things don’t. I need to know this information in order to make the questions and question-selection algorithms for Mathchops. So I went through every question from the last 10 ACTs and figured out which skills showed up. Then one of my partners helped me make a Python script and we did a bunch of data analysis. What follows is a list of the 75 most common skills, along with an estimate of how likely they are to appear on your actual test.
Guaranteed To Show Up: These have to be rock solid because A) they’ll definitely show up and B) they’ll often be combined with other skills.
Fractions – All four operations. Mixed numbers.
Average – Also called the arithmetic mean. There is always a basic version and usually an advanced one, like the average sum trick (see below).
Probability – Know the basic part:whole versions. There is usually a harder one also (like one with two events).
Percents – Know all basic variations. More advanced ones are common also.
Exponents – All operations. Fractional and negative exponents are very common too (see below).
Linear Equations/Slope – Find the slope when given two points. Be able to isolate y (to create y = mx + b). All the standard stuff from 8th grade Algebra.
Solving Equations – Be very comfortable with ax + b = cx + d. Distribute. Combine like terms. You also need to be able to create these equations based on word problems.
Picking Numbers – You never have to use this but it will be a useful option on every test.
Ratio – Part:part, part:whole.
Quadratic skills – Factor. FOIL. Set parenthesis equal to zero. Graph parabolas.
Area/Perimeter of basic shapes – Triangles, rectangles, circles.
Negatives – Be comfortable with all operations.
SOHCAHTOA – Every variation of right triangle trig, including word problems.
Plug in answers – Like picking numbers, it’s not required but it’s often helpful.
Extremely Likely (> 80% chance):
Function shifts – Horizontal shifts, vertical shifts. Stretches. You should recognize y = 2(x+1)^2 - 5 right away and know exactly what to do.
Average sum trick – 5 tests, average is 80. After the 6th test, the average is 82. What was 6th test score?
MPH – The concept of speed in miles per hour shows up every time.
Median – Middle when organized from low to high. Even number of numbers. What happens when you make the highest number higher or the lowest number lower?
Radicals – Basic operations. Translate to fractional exponents.
System of Equations – Elimination. Substitution. Word problems.
Angle chasing – 180 in a line. 180 in a triangle. Corresponding angles. Vertical angles.
Time – Hours to minutes, minutes to seconds
Pythagorean Theorem – Sometimes asked directly, other times required as part of something else (like SOHCAHTOA or finding the distance between two points).
Apply formula – they give you a formula (sometimes in the context of a word problem) and you have to plug stuff in.
Composite function – As in g(f(x)).
Factoring – Mostly the basics. Almost never involves a leading coefficient.
Matrices – Adding, subtracting, multiplying. Knowing when products are possible.
Very Likely (> 50% chance):
Absolute Value – Sometimes basic arithmetic, sometimes an algebraic equation or inequality.
Fractional Exponents – Rewrite radicals as fractional exponents and vice versa.
Multistep conversion – For example, they might give you a mph and a cost/gallon and then ask for the total cost.
Probability, two events – If there's a .4 probability of rain and a .6 probability of tacos, what is the probability of rain and tacos?
Remainders – Can be simple or pattern based, as in “If 1/7 is written as a repeating decimal, what is the 400th digit to the right of the decimal point?”
Midpoint – Given two ordered pairs, find the midpoint. Sometimes they’ll give you the midpoint and ask for one of the pairs.
Weird shape area – It’s an unusual shape but you can use rectangles and triangles to find the area.
Periodic function graph – The basics of sine and cosine graphs (shifts, amplitude, period).
Circle equations – (x-h)^2 + (y-k)^2 = r^2. Sometimes you have to complete the square.
Negative exponents – Know what they do and how to combine them with other exponents.
Shaded area – The classic one has a square with a circle inside.
Counting principle – License plate questions.
Logarithms – Rewrite in exponential form. Basic operations.
Imaginary numbers – Powers of i. What is i^2? The complex plane.
LCM – Straight up. In word problems. In algebraic fractions.
FOIL – This has to be automatic.
Worth Knowing (>25% chance):
Ellipses – Know how to graph basic versions.
Scientific notation – Go back and forth between standard and scientific notation. All four operations.
Vectors – Add, subtract, multiply (scalar), i and j notation.
Permutation – You have 5 plants and 3 spots. How many ways can you arrange them?
Volume of a prism – Know that the volume = area of something x height. Sometimes the base will be a weird shape.
c = product of roots, -b = sum of roots – Use when in x^2 + bx + c form. Usually not required but often helpful.
Difference of two squares – (x + y)(x - y) = x^2 - y^2
Arithmetic sequence – Usually asks you to find a specific term, sometimes asks you to find the formula.
Law of Cosines – They almost always give you the formula. Then you just have to plug things in.
Triangle opposite side rule – There is a relationship between an angle and the side across from that angle?
Change the base – If 9^x = 27^5, what is x?
Similar triangles – Relate the sides with a proportion.
Probability with “or” – 3 reds, 5 blue, 6 green. Probability of picking a red or blue?
Probability with “not” – 3 reds, 5 blue, 6 green. Probability of picking one that’s not red?
Factors – The basic concept and greatest common factor, with numbers and variables.
30:60:90 – Know the basic relationships. Sometimes required for advanced trig questions.
Volume of a cylinder – They’ll usually give it to you but not always.
Trapezoid – Usually basic area questions.
Domain – Usually you can think of it as “possible x values”.
Conjugates – Rationalize denominators that include radicals or imaginary numbers. Know that imaginary roots come in pairs.
Exponential Growth/Decay – Be comfortable with this: Final = Initial(1+/- rate)^time.
Weighted average – Class A has 8 kids and an average of 70. Class B has 12 kids and an average of 94. What is the combined average of the two classes?
Inverse trig – Use right triangle ratios to find angles.
Parallelogram – Know that adjacent angles add to 180. Area formula.
Use the radius – A circle will be combined with another shape and you have to use the radius to find the essential info about that other shape.
Value/frequency charts – They’ll tell you the value and frequency and then ask about mean or median.
3:4:5 – Recognize 3:4:5 right triangle relationships.
Algebra LCD – Find the lowest common denominator, then combine the numerators.
5:12:13 – Recognize 5:12:13 right triangle relationships.
System of equations with three equations – Usually a word problem. Involves substitution.
Compare numbers – Radicals, fractions, decimals, absolute value.
Translate points – Images, reflections.

There’s a lot of math out there. Some things show up all the time on the ACT. Other things don’t. I need to know this information in order to make the questions and question-selection algorithms for Mathchops. So I went through every question from the last 10 ACTs and figured out which skills showed up. Then one of my partners helped me make a Python script and we did a bunch of data analysis. What follows is a list of the 75 most common skills, along with an estimate of how likely they are to appear on your actual test.
Guaranteed To Show Up: These have to be rock solid because A) they’ll definitely show up and B) they’ll often be combined with other skills.
Fractions – All four operations. Mixed numbers.
Average – Also called the arithmetic mean. There is always a basic version and usually an advanced one, like the average sum trick (see below).
Probability – Know the basic part:whole versions. There is usually a harder one also (like one with two events).
Percents – Know all basic variations. More advanced ones are common also.
Exponents – All operations. Fractional and negative exponents are very common too (see below).
Linear Equations/Slope – Find the slope when given two points. Be able to isolate y (to create y = mx + b). All the standard stuff from 8th grade Algebra.
Solving Equations – Be very comfortable with ax + b = cx + d. Distribute. Combine like terms. You also need to be able to create these equations based on word problems.
Picking Numbers – You never have to use this but it will be a useful option on every test.
Ratio – Part:part, part:whole.
Quadratic skills – Factor. FOIL. Set parenthesis equal to zero. Graph parabolas.
Area/Perimeter of basic shapes – Triangles, rectangles, circles.
Negatives – Be comfortable with all operations.
SOHCAHTOA – Every variation of right triangle trig, including word problems.
Plug in answers – Like picking numbers, it’s not required but it’s often helpful.
Extremely Likely (> 80% chance):
Function shifts – Horizontal shifts, vertical shifts. Stretches. You should recognize y = 2(x+1)^2 - 5 right away and know exactly what to do.
Average sum trick – 5 tests, average is 80. After the 6th test, the average is 82. What was 6th test score?
MPH – The concept of speed in miles per hour shows up every time.
Median – Middle when organized from low to high. Even number of numbers. What happens when you make the highest number higher or the lowest number lower?
Radicals – Basic operations. Translate to fractional exponents.
System of Equations – Elimination. Substitution. Word problems.
Angle chasing – 180 in a line. 180 in a triangle. Corresponding angles. Vertical angles.
Time – Hours to minutes, minutes to seconds
Pythagorean Theorem – Sometimes asked directly, other times required as part of something else (like SOHCAHTOA or finding the distance between two points).
Apply formula – they give you a formula (sometimes in the context of a word problem) and you have to plug stuff in.
Composite function – As in g(f(x)).
Factoring – Mostly the basics. Almost never involves a leading coefficient.
Matrices – Adding, subtracting, multiplying. Knowing when products are possible.
Very Likely (> 50% chance):
Absolute Value – Sometimes basic arithmetic, sometimes an algebraic equation or inequality.
Fractional Exponents – Rewrite radicals as fractional exponents and vice versa.
Multistep conversion – For example, they might give you a mph and a cost/gallon and then ask for the total cost.
Probability, two events – If there's a .4 probability of rain and a .6 probability of tacos, what is the probability of rain and tacos?
Remainders – Can be simple or pattern based, as in “If 1/7 is written as a repeating decimal, what is the 400th digit to the right of the decimal point?”
Midpoint – Given two ordered pairs, find the midpoint. Sometimes they’ll give you the midpoint and ask for one of the pairs.
Weird shape area – It’s an unusual shape but you can use rectangles and triangles to find the area.
Periodic function graph – The basics of sine and cosine graphs (shifts, amplitude, period).
Circle equations – (x-h)^2 + (y-k)^2 = r^2. Sometimes you have to complete the square.
Negative exponents – Know what they do and how to combine them with other exponents.
Shaded area – The classic one has a square with a circle inside.
Counting principle – License plate questions.
Logarithms – Rewrite in exponential form. Basic operations.
Imaginary numbers – Powers of i. What is i^2? The complex plane.
LCM – Straight up. In word problems. In algebraic fractions.
FOIL – This has to be automatic.
Worth Knowing (>25% chance):
Ellipses – Know how to graph basic versions.
Scientific notation – Go back and forth between standard and scientific notation. All four operations.
Vectors – Add, subtract, multiply (scalar), i and j notation.
Permutation – You have 5 plants and 3 spots. How many ways can you arrange them?
Volume of a prism – Know that the volume = area of something x height. Sometimes the base will be a weird shape.
c = product of roots, -b = sum of roots – Use when in x^2 + bx + c form. Usually not required but often helpful.
Difference of two squares – (x + y)(x - y) = x^2 - y^2
Arithmetic sequence – Usually asks you to find a specific term, sometimes asks you to find the formula.
Law of Cosines – They almost always give you the formula. Then you just have to plug things in.
Triangle opposite side rule – There is a relationship between an angle and the side across from that angle?
Change the base – If 9^x = 27^5, what is x?
Similar triangles – Relate the sides with a proportion.
Probability with “or” – 3 reds, 5 blue, 6 green. Probability of picking a red or blue?
Probability with “not” – 3 reds, 5 blue, 6 green. Probability of picking one that’s not red?
Factors – The basic concept and greatest common factor, with numbers and variables.
30:60:90 – Know the basic relationships. Sometimes required for advanced trig questions.
Volume of a cylinder – They’ll usually give it to you but not always.
Trapezoid – Usually basic area questions.
Domain – Usually you can think of it as “possible x values”.
Conjugates – Rationalize denominators that include radicals or imaginary numbers. Know that imaginary roots come in pairs.
Exponential Growth/Decay – Be comfortable with this: Final = Initial(1+/- rate)^time.
Weighted average – Class A has 8 kids and an average of 70. Class B has 12 kids and an average of 94. What is the combined average of the two classes?
Inverse trig – Use right triangle ratios to find angles.
Parallelogram – Know that adjacent angles add to 180. Area formula.
Use the radius – A circle will be combined with another shape and you have to use the radius to find the essential info about that other shape.
Value/frequency charts – They’ll tell you the value and frequency and then ask about mean or median.
3:4:5 – Recognize 3:4:5 right triangle relationships.
Algebra LCD – Find the lowest common denominator, then combine the numerators.
5:12:13 – Recognize 5:12:13 right triangle relationships.
System of equations with three equations – Usually a word problem. Involves substitution.
Compare numbers – Radicals, fractions, decimals, absolute value.
Translate points – Images, reflections.

First of all I hope you all are doing great!
It might sound kind of obsessive but I've been trying to find my type from a long time ago, and I've considered even the strangest possibilities (according to close friends) just because I overthink the functions a lot and I consider I become more and more biased with time, changing the meanings of things just to make them fit into different functions whenever I come up with a new theory. It sounds convincing to everyone, but not to me... I really have no one I can talk about mbti in this way because people close to me don't understand or are very into it haha, I believe I don't know myself enough (which I should work on, and that's why I really like mbti but I feel I worsen things instead of finding some light hahaha).
Anyways, so sorry for the long introduction and thanks a lot for reading! <3
• How old are you? What's your gender? Give us a general description of yourself.
I'm a 22 year old female and I'm from Colombia. I am interested in several things though I am not good at all of them haha. I played tennis almost during 9 years (or 10) and people don't know why since I am not very good for sports but I've always been very disciplined. I love languages and I think I have a knack for them, I love to paint, draw, sculpt or anything I can do with my hands since it helps me to focus only on what I'm doing and stops my thoughts or helps me clarifying them, I love to write and read (yeah yeah, infp haha) and love to learn new things. My childhood dream was to become astronomer but I am terrible at math and teachers never understood how I always got the right answers with the wrong (or not indicated) methods haha.
• Is there a medical diagnosis that may impact your mental stability somehow?
I don't think so... I've had problems like everyone else and hard times as well, but I'm ok I guess haha.
• Describe your upbringing. Did it have any kind of religious or structured influence? How did you respond to it?
I've always said that my dad taught me to love knowledge and my mom taught me to spread it. My dad always told me Jesus was a philosopher whose ideas were distorted since I was very little and up until now I fiercely believe in that haha, there was a very difficult time in which I used to get upset when my mom talked about God (she's very catholic) but now I like to listen to everyone and get ideas from their beliefs. I believe God is a misunderstood artist whose plans we don't fully understand but I like to compare him to those writers who always imagine tragic endings for their characters... (?)
• What do you do as a job or as a career (if you have one)? Do you like it? Why or why not?
I currently work in a call center, even thought I hated the idea 4 years ago I've learn a lot from it! Now I don't feel anxious with calls and I am very thankful that because of my situation I was forced to work in one hahaha, I can express myself better and explain my ideas to other without fear of rejection or making them feel upset with me. I don't plan to work in one of them my whole life, I am actually saving money to start my university studies in something related to languages and teaching! :)
• If you had to spend an entire weekend by yourself, how would you feel? Would you feel lonely or refreshed?
I would definitely love it! Even though I am pretty talkative I always have a hard time when spending a lot of time with my family or friends because I feel I am more productive when by myself. I feel very refreshed after having a long time by myself and I can learn a lot of things!
• What kinds of activities do you prefer? Do you like, and are you good at sports? Do you enjoy any other outdoor or indoor activities?
I love spending time in nature, places like parks or mountains are my favorite ones! As I mentioned I played tennis and through that experience I realized how important it is to take care of your body and your mind at the same time, but I am not naturally skilled for sports haha. It'd be redundant if I talked about my hobbies once again.
• How curious are you? Do you have more ideas then you can execute? What are your curiosities about? What are your ideas about - is it environmental or conceptual, and can you please elaborate?
I've always said my mind fluctuates in a lot of hypocognitions and I can never accomplish them entirely because there are just a lot of them! Most of the time I need to create something in order to understand a concept, an idea and/or felling I have that I don't understand in the very moment. For example I like to draw things that come to my mind when feeling something I don't understand and I'll understand that feeling three months later (or more)... Not a very practical technique, but getting better at it hahaha.
• Would you enjoy taking on a leadership position? Do you think you would be good at it? What would your leadership style be?
I love to help people in general, but being in charge is a different thing, a work I admire but wouldn't be able to execute. Still, if I ever had to be in a leadership position under stress I would be kind of bossy to be honest, when normal I'd be very active and funny as well but would burn out easily.
• Are you coordinated? Why do you feel as if you are or are not? Do you enjoy working with your hands in some form? Describe your activity?
I am very coordinated with specific tasks, not very multitasking, when investing a lot of time on certain skill I can be very coordinated, though but yes! I adore working with my hands! (Going back to the hobbies thing again).
• Are you artistic? If yes, describe your art? If you are not particular artistic but can appreciate art please likewise describe what forums of art you enjoy. Please explain your answer.
I am very artistic, I guess, specially because creating things is my way to understand myself, I always end up drawing like cute things that have a weird aspect with a lot of self-references I don't like to explain to people. Other thing I like to do is to ask people what do they thing a tale/drawing/sculpture means and try to figure out how I feel or what I meant based on that.
• What's your opinion about the past, present, and future? How do you deal with them?
Ugh, I am very nostalgic when thinking about the past, I wish I could get some sensations back but I understand that I cannot retrocede so that ideas makes me feel better at times.
Present moment? Whaaaat is thaaaaat? Haha, kidding; I've learnt to appreciate what I have in this very moment because it might change at any moment and I am afraid of that!
Future is basically the reason why I wake up and think I need to work harder to save money and be in the place I want to be... Not very good to think a it every single moment, but working on that.
• How do you act when others request your help to do something (anything)? If you would decide to help them, why would you do so?
Before, it was very hard for me to say no to my immediate family and it still is but I've become better at it * makes uncomfortable noises *. I feel I owe what I am to my parents and sisters so that's why I help them. I help people in general if I feel I am able to do it and it will really make their situation better. No regrets when helping, because if you regret why did you do it in the first place?
• Do you need logical consistency in your life?
I feel logic is the most importan thing in my life, not because I consider I am logical at all, but because it needs to be my main focus for being able to accomplish things in life and not losing track. Logic is as beautiful as feelings when wisely used.
• How important is efficiency and productivity to you?
I don't think I am naturally a productive or efficient person but life has pushed me to be so, currently I think those are amazing traits people should work on!
• Do you control others, even if indirectly? How and why do you do that?
I do it all the time, most of the time because I want to have a balance between personal and collective goals and I want things to occur in the most convenient way.
• What are your hobbies? Why do you like them?
Uhm, I've written a lot of stuff I know, I already talked about them haha, artistic thingies that might make people thing I am an ixfp haha. I love to watch youtube videos as well and memes hehe.
• What is your learning style? What kind of learning environments do you struggle with most? Why do you like/struggle with these learning styles? Do you prefer classes involving memorization, logic, creativity, or your physical senses?
I love learning by context and I have that very clear since I've noticed it in my languages classes, if I cannot get (pseudo)philosophical with it, I don't wanna learn it haha if it does not lead everyone to a conversation where they express their ideas about it it's not working. But maybe I need to thing more about it. For certain stuff I like repetition, conjugation and stuff is better when repeating it, that depends on what your learning, several techniques can be used for different results... (?)
• How good are you at strategizing? Do you easily break up projects into manageable tasks? Or do you have a tendency to wing projects and improvise as you go?
I have a talent for planning but not for executing haha, recently I have learnt that executing is important and maybe have gotten better at it, but definitely not natural for me.
• What are your aspirations in life, professionally and personally?
I want to have mental, health and economic stability to help my family and myself as well, if after that I can help others I'll be happy to! I want to know more every day and be excellent at the things I like to do and have to do.
• What are your fears? What makes you uncomfortable? What do you hate? Why?
I hate not knowing things, it always turns out to be something awful when I don't know how a person is going to react, what can I expect from a job interview, what is making me uncomfortable, and so on and so on... Not knowing what needs to be known is awful though ignorance saves me at times from getting depressed haha. Besides, who am I to decide what needs to be known and what does not?
• What do the "highs" in your life look like?
Peace, love, philanthropy, rainbows and tons of knowledge.
• What do the "lows" in your life look like?
Desire for absolute destruction, when helpless wanting to control everything, very bitter and cynical.
(I currently feel a bit of both since I am feeling fine and neutral, the examples on the highs is when I am very very high that it becomes something bad).
• How attached are you to reality? Do you daydream often, or do you pay attention to what's around you? If you do daydream, are you aware of your surroundings while you do so?
What is surroundings? Haha, I have a very hard time focusing in the real moment and when I finally can do it I forget to think haha, have to do one of those at the same time. Daydreaming 85% to be honest.
• Imagine you are alone in a blank, empty room. There is nothing for you to do and no one to talk to. What do you think about?
First thing I would think about would be the love of my life haha, cheesyyyy. After that I would start overthinking about everything else, when on the empty room for hours I'd start thinking about what others are going to do to me or my family or the reasons why I ended up in there, then ways to escape after years of being there haha... Not really sure. (?)
• How long do you take to make an important decision? And do you change your mind once you've made it?
Woah, that's related to the last thing I've said hahaha, it takes years or months for me to decide something, but when decided there is no coming back.
• How long do you take to process your emotions? How important are emotions in your life?
I guess I explained that as well before, understanding feelings and thoughts (which I like to combine) through tangible things like drawings, writings and others.
• Do you ever catch yourself agreeing with others just to appease them and keep the conversation going? How often? Why?
That happens to me often with strangers or people I am not very close to. When confident in the conversation or if I see the person is getting my point and is respectful I can give a little bit more of info, if my family, most of the time I will speak up my mind regardless of the outcome of it hahaha.
• Do you break rules often? Do you think authority should be challenged, or that they know better? If you do break rules, why?
I don't break rules often just in case I break them once for solid and valid reasons I will be heard and maybe that specific rule will be reviewed and changed hahaha. The fact that someone is in a position of authority does not make them being right every single time, it means a lot of stuff: they either worked very hard for it, or they bribed someone or whatever but one has to be very attentive and aware of the person that is leading, they are humans after all, haha.
​
Well, I think this was a lot and maybe no one is going to read it but thanks a lot for giving us this space, maybe after writing so much about myself I learnt something new about me!
​
Cheers!!! :)
​
Sorry for the typos and for getting more and more concise in the latest answers, I got tired hahahaha. :(

Basic Math Symbols

SymbolSymbol NameMeaning / definitionExample=equals signequality5 = 2+35 is equal to 2+3≠not equal signinequality5 ≠ 45 is not equal to 4≈approximately equalapproximationsin(0.01) ≈ 0.01,x ≈ y means x is approximately equal to y>strict inequalitygreater than5 > 45 is greater than 4Download the printable chart here- Basic Math Symbols

2. Algebra Symbols

SymbolSymbol NameMeaning / definitionExamplexx variableunknown value to findwhen 2x = 4, then x = 2≡equivalenceidentical ton/a≜equal by definitionequal by definitionn/a:=equal by definitionequal by definitionn/a~approximately equalweak approximation11 ~ 10≈approximately equalapproximationsin(0.01) ≈ 0.01∝proportional toproportional toy ∝ x when y = kx, k constant∞lemniscateinfinity symboln/a≪much less thanmuch less than1 ≪ 1000000≫much greater thanmuch greater than1000000 ≫ 1( )parenthesescalculate expression inside first2 * (3+5) = 16[ ]bracketscalculate expression inside first[(1+2)*(1+5)] = 18{ }bracessetn/a⌊x⌋floor bracketsrounds number to lower integer⌊4.3⌋ = 4⌈x⌉ceiling bracketsrounds number to upper integer⌈4.3⌉ = 5x!exclamation markfactorial4! = 1*2*3*4 = 24| x |single vertical barabsolute value| -5 | = 5f (x)function of xmaps values of x to f(x)f (x) = 3x+5(f ∘ g)function composition(f ∘ g) (x) = f (g(x))f (x)=3x,g(x)=x-1 ⇒(f ∘ g)(x)=3(x-1)(a,b)open interval(a,b) = {x | a < x < b}x∈ (2,6)[a,b]closed interval[a,b] = {x | a ≤ x ≤ b}x ∈ [2,6]∆deltachange / difference∆t = t1 - t0∆discriminantΔ = b2 - 4acn/a∑sigmasummation - sum of all values in range of series∑ xi= x1+x2+...+xn∑∑sigmadouble summation📷∏capital piproduct - product of all values in range of series∏ xi=x1∙x2∙...∙xnee constant / Euler's numbere = 2.718281828...e = lim (1+1/x)x , x→∞γEuler-Mascheroni constantγ = 0.5772156649...n/aφgolden ratiogolden ratio constantn/aπpi constantπ = 3.141592654...is the ratio between the circumference and diameter of a circlec = π⋅d = 2⋅π⋅rDownload the printable chart here- Algebra Symbols

3. Geometry Symbols

SymbolSymbol NameMeaning / definitionExample∠angleformed by two rays∠ABC = 30°📷measured angle n/a📷ABC = 30°📷spherical angle n/a📷AOB = 30°∟right angle= 90°α = 90°°degree1 turn = 360°α = 60°degdegree1 turn = 360degα = 60deg′primearcminute, 1° = 60′α = 60°59′″double primearcsecond, 1′ = 60″α = 60°59′59″📷lineinfinite line n/aABline segmentline from point A to point B n/a📷rayline that start from point A n/a📷arcarc from point A to point B📷= 60°⊥perpendicularperpendicular lines (90° angle)AC ⊥ BC∥parallelparallel linesAB ∥ CD≅congruent toequivalence of geometric shapes and size∆ABC ≅ ∆XYZ~similaritysame shapes, not same size∆ABC ~ ∆XYZΔtriangletriangle shapeΔABC ≅ ΔBCD|x-y|distancedistance between points x and y| x-y | = 5πpi constantπ = 3.141592654...is the ratio between the circumference and diameter of a circlec = π⋅d = 2⋅π⋅rradradiansradians angle unit360° = 2π radcradiansradians angle unit360° = 2π cgradgradians / gonsgrads angle unit360° = 400 gradggradians / gonsgrads angle unit360° = 400 gDownload the printable chart here- Geometric Symbol

4. Set Theory Symbols

SymbolSymbol NameMeaning / definitionExample{ }seta collection of elementsA = {3,7,9,14}, B = {9,14,28}|such thatso thatA = {x | x∈📷, x<0}A⋂Bintersectionobjects that belong to set A and set BA ⋂ B = {9,14}A⋃Bunionobjects that belong to set A or set BA ⋃ B = {3,7,9,14,28}A⊆BsubsetA is a subset of B. set A is included in set B.{9,14,28} ⊆ {9,14,28}A⊂Bproper subset / strict subsetA is a subset of B, but A is not equal to B.{9,14} ⊂ {9,14,28}A⊄Bnot subsetset A is not a subset of set B{9,66} ⊄ {9,14,28}A⊇BsupersetA is a superset of B. set A includes set B{9,14,28} ⊇ {9,14,28}A⊃Bproper superset / strict supersetA is a superset of B, but B is not equal to A.{9,14,28} ⊃ {9,14}A⊅Bnot supersetset A is not a superset of set B{9,14,28} ⊅ {9,66}2Apower setall subsets of A n/a📷power setall subsets of A n/aA=Bequalityboth sets have the same membersA={3,9,14}, B={3,9,14}, A=BAccomplementall the objects that do not belong to set A n/aA'complementall the objects that do not belong to set A n/aA\Brelative complementobjects that belong to A and not to BA = {3,9,14}, B = {1,2,3}, A \ B = {9,14}A-Brelative complementobjects that belong to A and not to BA = {3,9,14}, B = {1,2,3}, A - B = {9,14}A∆Bsymmetric differenceobjects that belong to A or B but not to their intersectionA = {3,9,14}, B = {1,2,3}, A ∆ B = {1,2,9,14}A⊖Bsymmetric differenceobjects that belong to A or B but not to their intersectionA = {3,9,14}, B = {1,2,3}, A ⊖ B = {1,2,9,14}a∈Aelement of, belongs toset membershipA={3,9,14}, 3 ∈ Ax∉Anot element ofno set membershipA={3,9,14}, 1 ∉ A(a,b)ordered paircollection of 2 elements n/aA×Bcartesian productset of all ordered pairs from A and B n/a|A|cardinalitythe number of elements of set AA={3,9,14}, |A|=3#Acardinalitythe number of elements of set AA={3,9,14}, #A=3📷aleph-nullinfinite cardinality of natural numbers set n/a📷aleph-onecardinality of countable ordinal numbers set n/aØempty setØ = {}A = Ø📷universal setset of all possible values n/a📷0natural numbers / whole numbers set (with zero)📷0 = {0,1,2,3,4,...}0 ∈📷0📷1natural numbers / whole numbers set (without zero)📷1 = {1,2,3,4,5,...}6 ∈📷1📷integer numbers set📷= {...-3,-2,-1,0,1,2,3,...}-6 ∈📷📷rational numbers set📷= {x | x=a/b, a,b∈📷and b≠0}2/6 ∈📷📷real numbers set📷= {x | -∞ < x <∞}6.343434 ∈📷📷complex numbers set📷= {z | z=a+bi, -∞<a<∞, -∞<b<∞}6+2i ∈📷Download the printable chart here- Set Theory Symbols

5. Calculus & Analysis Symbols

SymbolSymbol NameMeaning / definitionExample📷limitlimit value of a function n/aεepsilonrepresents a very small number, near zeroε → 0ee constant / Euler's numbere = 2.718281828...e = lim (1+1/x)x , x→∞y 'derivativederivative - Lagrange's notation(3x3)' = 9x2y ''second derivativederivative of derivative(3x3)'' = 18xy(n)nth derivativen times derivation(3x3)(3) = 18📷derivativederivative - Leibniz's notationd(3x3)/dx = 9x2📷second derivativederivative of derivatived2(3x3)/dx2 = 18x📷nth derivativen times derivation n/a📷time derivativederivative by time - Newton's notation n/a📷time second derivativederivative of derivative n/aDx yderivativederivative - Euler's notation n/aDx2ysecond derivativederivative of derivative n/a📷partial derivative n/a∂(x2+y2)/∂x = 2x∫integralopposite to derivation ∫ f(x)dx∬double integralintegration of function of 2 variables ∫∫ f(x,y)dxdy∭triple integralintegration of function of 3 variables ∫∫∫ f(x,y,z)dxdydz∮closed contour / line integral n/a n/a∯closed surface integral n/a n/a∰closed volume integral n/a n/a[a,b]closed interval[a,b] = {x | a ≤ x ≤ b} n/a(a,b)open interval(a,b) = {x | a < x < b} n/aiimaginary uniti ≡ √-1z = 3 + 2iz*complex conjugatez = a+bi → z*=a-biz\* = 3 + 2izcomplex conjugatez = a+bi → z = a-biz = 3 + 2i∇nabla / delgradient / divergence operator∇f (x,y,z)📷vector n/a n/a📷unit vector n/a n/ax * yconvolutiony(t) = x(t) * h(t) n/a📷Laplace transformF(s) =📷{f (t)} n/a📷Fourier transformX(ω) =📷{f (t)} n/aδdelta function n/a n/a∞lemniscateinfinity symbol n/a

I am currently using my paid vacation this week so I have been spending 8-10 hours a day using Khan Academy and SL to figure out this calculus course.
Unfortunately I am struggling pretty bad. I have put roughly 9 hours a day in since Monday (bar Tuesday where I just enjoyed myself) and am only on derivatives(section 3). I finally got a grasp of it all and then these find the velocity derivatives & conjugate limits have really thrown me for a loop. Thank God for the the power rule but ones I can't solve with that rule are extremely difficult to me.
I also took the first quiz and got a 60% so I definitely plan on taking it again and at the very least I would like to finish & pass this course but if I am having this much of an issue this early on then I am super scared for what's to come.
Bluntly I wanna know if I should just swap to another major since I do know CS requires a lot of math and I am just struggling to get it at the calculus level. But on the other hand it has only been 3 days so maybe I am just rushing too much? I really need some outside input because I am having serious self doubts.

Close your eyes. Take a deep breath. Relax your body, and here we go: Once upon a time...
Many, if not all, languages have a very special expression with which traditional tales begin, and for anyone interested, this Wikipedia article lists some of them.
​
The Evra equivalent of this stock phrase is:

• I esìv un est
• IPA j‿eˈsiv(u)‿un‿ˈ(n)ɛst (mostly like [jeˈsivu ˈnɛst])

​
Though, I don't want to reveal its meaning yet. In fact, let's take a step back and start from the beginning.
​
Yesterday morning, I was organizing some entries in my Evra dictionary, when I came to the verb esìr. Now, when it comes to Evra, I really love to mix things up a bit, and words of Romance and Germanic origins merge with other words with similar meanings or similar sounds, so to make Evra kind of familiar and exotic at the same time. The verb esìr makes no exceptions. It is a mix of:

• Latin exeō, exīre (depart; avoid, evade; from which Italian uscire, Romanian a ieși, and English exit, as well as issue)
• Latin existō, existere (exist; appear, emerge; become; from which Italian esistere, Romanian a exista, and English exist)

​
So here's what the entry esìr looks like in my Evra dictionary:

1. (intr., formal) go out (gen., bi + gen.; in + dir.) (ex. La esìt (bi) kas. – lit., “She exited from house”, “She went out from home”; La esìt in gearte – lit., “She exited in garden.”, “She went out to the garden.”)
2. (intr., impersonal) exist (ex. Di onvren ge i esim pa. – lit., “The ghosts here exit not.”, “Ghosts do not exist.”)
3. (trans., generic, also impersonal) issue, release (for public use), put on sale, make available (ex. En i esìnt la kemil de mi-l logò sil. – lit., “It here exited the T-shirt with the logo theirs.”, “They issued the T-shirt with their logo.”)
4. (trans., hyperbolic use, colloquial) say (an offence), call (offensively), give (a kick, a slap, etc…); react suddenly and angrily (ex., Un el esìt ir en orko! – lit., “And he exited to-him a moron!”, “And he called him a moron!; La esìt ir e bel pamèt! – lit., “She exited to-him a very smack!”, “She gave him a good smack!”)
5. (trans., generic, catenative) turn out (to be), result (to be); become apparent or known (as) (ex. La esìt a sèr la rovan. – lit., “She exited to be the robber.”, “It turned out she was the robber.”)
6. (trans., generic, copulative, idiomatic, familiar) be, look, seem, appear, result (in), act (as); give an impression (ex., Mi se verti-la, la esìt e dima. - lit., "With these words-there, she exited an idiot.", "She was an idiot with (by saying) those words")
7. (trans., generic, idiomatic, rifl., also + is) get or come out, conclude, result, cope, deal with (bi + gen.) (ex. La pita dus s-esìnt a bel gut. – lit., “The pressed sweet exited very well.”, “The cake came out very well.”; El ni s-esìnt tan gut bi tuides el pàes. – lit., “He not exited much good from death the father.”, “He didn’t deal very well with his father’s death”)
8. (Publishing industry) issue, come out, publish (ex. Si esim di lihvis. – lit., “They exit books.”, “They publish books.”)
9. (Math) equal (ex. Ek mos ek esi vìe. – lit., “One more one exits two”, “One plus one equals two.”)
10. (Information technology, impersonal, dative construct) say, give, return (a value, an error, etc…) (ex: È i esi e fal. – lit., “To-me here exits an error.”, “It’s giving me an error.”)
11. (Journalism, formal) issue, make (statements), do (interviews), talk (to the press) (ex., El Vesehre esìt nìe. – lit., “The President exited nothing.”, “The president had no comment.”)
12. (Economy, Finance, formal) issue, release; put in circulation
13. (Law, formal) issue, enact, enter into force (ex., E ne sat-rik s-esìnt. - lit., "A new law self exited.", "A new law has been enacted.")

​
And yeah, without any doubt, esìr has many meanings, but we can just reduce all of them to a lowest common denominator, that is roughly 'give out, put out, or stand out'.
​
Now that we know everything about esìr, let's take a look at the phrase I esìv un est.
​
The first word, i, is a weird clitic that can take a lot of roles in Evra. Here, in its preverbal position, it generally works as a dummy pronoun (just like it in "It rains"), and I call it impersonal i, in this case. Being an impersona pronoun, its presence obviously forces us to interpret the following verb as an impersonal one, and thus triggering the sense 2 of esìr (i.e., 'exist', and not 'exit').
​
The second word, esìv, is the 3rd person singular of esìr in its Narrative tense form, which is roughly equivalent to the Imperfect tense of Romance languages. We can translate it with the English expressions "It was existing" or "It used to exist".
​
The third word, un, is simply the conjugation 'and', but here comes the trick. Now, this little word is a bit special, because it can share 'some features' between the two phrases it links, just as in English. Let me explain better with some example. In English, we can say "a smart girl and a handsome boy". Here we have two 'entities': a "smart girl" together with a "handsome boy". But we can also say "a smart and handsome boy". Here, we have only one 'entity', the 'boy', which is a "smart boy" as well as a "handsome boy". Basically, what can 'and' do here is that of sharing the word 'boy' between the two phrases: "a smart boy" + "a handsome boy" = "a smart boy and a handsome boy". This is a key step in understanding the desired ambiguity of the whole "I esìv un est" expression, more in a moment.
​
The fourth and last word, est, is the 3rd person singular of esìr in its Past Attributive form, which is a weird past tense used in Evra, but which I'll not explain here in detail now. Just think to it as the -ed form of verbs in English. Now, the interpretation of this verb depends on the reading of un.
​
If un tells us to see i esìv and est as two entities, we'll translate I esìv un est as "It was existed and it turned out / became known". This gives us a simple, literal translation of the whole Evra expression.
​
However, if un tells us to see i esìv and est as one entity, we'll translate I esìv un est as "It was existing and existed", because the impersonal i gets shared between the two verbs, i.e. i esìv (= it was existing) + un + i est (= it existed). Obviously we have a contradiction here. On the one hand, we have the Narrative Tense (Imperfect) that tells us the facts in the tale continue and progress over time. On the other, the Past Attributive tells us the facts happened once only. We entered indeed the realm of Fantasy, where the scientific definition of Time succumbs to the emotional perception of it.
​
I esìv un est...
It was existing and existed, or became known,...

Okay, so, I am going to break down this guide into the subjects which I took. Use Control F to read about the subjects you want because this guide is quite long.
SL: English A Language & Literature, Spanish Ab Initio, Mathematics
HL: Biology, Chemistry, Economics
First of all, a huge shoutout to everyone on this sub for all of the help they gave me during the IB, specifically all of those resources and all of the memes to keep me going. A special thanks to the mods who keep the place in control too :).
~~~
English A Language & Literature SL
Paper 1:
With this paper, I cannot stress enough how much you need to PRACTICE. Practice is the absolute key to being successful on this paper. You could get literally any type of text on this paper, and for this reason you need to practice as much as possible on all of the possible text types (these can be found in the subject guide). Before the exam, try to memorise some of the conventions of each text type to show off to the examiner your text knowledge. I was a teacher who made each person in the class do a list of conventions for each text then send it to the class, but if not you may want to try and do this. I get that practice can take a ton of time, so for this reason just annotate the texts that come up in Paper 1's, you do not need to write the full essay. You also may want to make a list of all of the stylistic devices which come up, and their relevance (I have a sheet of these which I can upload if anyone wants it).
Specifically when actually writing this paper, you want to link all of your analysis to one main idea, which our teacher taught us to be the PURPOSE of the text. So, if in doubt during the exam, link things to the purpose of the text, and make sure you actually believe in the purpose that you are writing about, because if not you will struggle to avoid going on a tangent. In each of your analysis paragraphs start off with a topic sentence i.e. "X text uses Y feature to convey the purpose", then do your analysis then finish off with a link back to the purpose. If you are struggling to think of points to make in your essay, just think of the BIG 5 (Purpose, Themes, Stylistic Devices, Mood and Structure). Also, remember 1 thing, every single thing on the text is there for a reason, so you can analyse everything i.e. Pictures (I have a note sheet on how to analyse pictures as well, if anyone wants it let me know and I can upload it), Slogans, Titles, Captions, etc.
Paper 2:
First thing that I will say for this is please read the books, like there is no way around it. My teacher gave us a booklet of quotes for both texts that we studied for the exam (Miss Julie and Never Let Me Go), and it was still useless until I actually read both books. To be honest, there is nothing more valuable for Paper 2 then listening in class. When you read the books and listen to class discussion on them, you begin to understand the themes, moods, characters and plots further, and you begin to articulate your own opinions on the texts which is KEY for the exam. What you want to do ahead of the exam is make notes through specific quotes, and you want to link all of them to context. No matter which question you choose to answer, you must include context to score highly. During the exam you need to make a judgement call on which quotes that you have memorized fit the question best, and if the quotes do not fit the questions perfectly, don't worry. A big part to scoring highly on Paper 2 is your close analysis (i.e. talking about denotations and connotations of words and phrases), so if you do have to choose quotes which don't perfectly fit, you inbed analysis perfectly.
Also, ANALYSE your quotes before the exam, and memorize some of that analysis, because if you can memorize links to context and some of the more complex literary devices, it will help you when writing your essay. With your quotes, you want to be able to link all of them to at least one character, symbol and one piece of context. LitCharts can do this for you luckily, and it is really good at doing it, and I used them so much when revising for exams. Two final things before I finish the Paper 2 section: Have faith in yourself because it can screw you over when you change your strategy on the actual exam day (I learned about this from my mocks), and you do not need too many quotes to be successful, I think I had 7-8 for each book and I was fine. You want to PRACTICE as much as possible before this paper, and you do not have to write full essays, you can simply plan them and use your quotes for them.
IOC, FOA and Written Task:
Before I took this class, I absolutely hated English, and it was a huge relief to learn that you can have 50% of your final grade decided prior to even writing an exam, so take advantage of this! This means that your FOA, IOC and Written Task are incredibly important. If you nail these, you can afford to have a bad day on Paper 1 if your texts aren't too good, and it can be a source of relief if you don't think your exams went well. In your IOC, you want to prepare by looking at the extracts which your teacher has given you (if they give any), or read your book constantly and try to analyze any quote that you think is gold when reading (A good exersize for this is opening a random page of your texts, and just analysing everything). When it comes to the actual thing, I would recommend bringing 4 or 5 different highlighters into the exam, and highlighting the quotes with the theme you think that they link to, so that you have some structure set for your IOC, and then you can weave between these and make some creative points. You want to learn about your stylistic devices, links to the rest of the text and links to context as these are what can help you to score highly.
In your FOA, I'm not sure if your teacher will give you prompt on what you should do it on but if they do not, I would reccomend doing it on comparing two famous speeches. I did this with one of my best mates who I had a lot of trust in, and we compared a Winston Churchill speech to the Barack Obama Inaugural Speech. We both found this okay because the speeches have a TON of techniques inside them which you can show off in your FOA. So, if anyone were to ask me what to do an FOA on, I would say that. Just search up some of the world's most famous speeches, and choose one which interests you. No matter what topic you choose, analyse specific extracts on them for stylistic devices, aristotelian appeals (i.e. Ethos, Pathos, Logos (Which you can include in Papers 1 and 2 as well)), mood, themes and effects of what they do. Do video recorded practices before you do it and ask yourself questions on what is uncertain and what more you could include and you should be good.
Your written task on it's own is worth 20%, so try as hard as you can on making sure that you nail this completely. Our class was made to do 3 of these, and then we had to submit one, and I think doing 3 was the perfect amount. Even if you think that your first one is great, try as hard as possible on all 3, because naturally your analysis skills will get better over your time in the course so a similar amount of effort can produce better work. Plus, it gives you a choice on what you actually want to submit at the end of the course. Since you have a lot of independence on this, and it is technically not mean't to be an "essay", I would choose something that I enjoy, as you will put more effort into it. The written task I ended up submitting was on my IOC texts, as I surprisingly enjoyed writing that the most, but you have many options on what you can write it on (all the way from writing to an editor criticizing their recent article to writing as a person from your text to your family member (which is what I did)).
~~~
Spanish Ab Initio
Paper 1:
I got a 5 in Spanish Ab Initio (1 mark off of a 6), so I do not think that I can give you the best advice ever. But basically, in my opinion, the bottom line with this is that you need to do two things: Learn a ton of vocab ahead of the exam and do practice papers (add any words which you don't understand into something like a quizlet set so that you can learn it). I just want to give some fair warning before anyone takes this class, IT IS NOT EASY and effort needs to be made to do well in the exam (After exams I realized I probably should've revised a lot more for this, so don't be like me and do small amounts of revision over the two years). The grade boundaries are really high because fluent people take the exams, so you need to have a good understanding of Spanish to get a 7. Process of elimination can be really helpful for the Paper 1 exams if you are in doubt, and during reading time you want to skim through the texts and FOCUS ON WHAT YOU KNOW rather than dwelling on what you do not understand, because that will not get you anywhere.
Paper 2:
One thing that you should probably know before you do this exam is that 12% (3/25) of the marks are just FORMATTING, so please learn how to format all of the different text types. For this exam what you want to know is your conjugations for about 6/7 tenses which you can use (Present, present continuous, future, near future, conditional, imperfect and preterite were the ones I learned), but I would say to learn tenses continuously over the 2 years so that it becomes second nature to you after a while. I didn't do this and on the exam day I wanted to conjugate some irregular verbs, and struggled to as it does not stick to memory too well. The people who got level 7's in my class also knew some more of the complex tenses such as Pluperfect and subjunctive, but you don't need to know the full tense necessarily, just memorize some general phrases in these two tenses which you can use in your writing. Doing practice papers for both paper 1 and 2 will help you to get a grasp of common types of questions and topics which also come up, so practice!
Speaking Exam and Written Assignment:
A large chunk of your final Spanish Ab Initio exam grade is, similarly to English Lang Lit, decided before you actually take the exam. So, once again, I will say take advantage of this. When it comes to the speaking exam, a lot of it does come down to your luck on the day, especially when it comes to preparing for the picture which you may recieve. What I did to prepare for this initial part of the exam was think of all of the possible kinds of photos I could get (i.e. A market, street, beach, campsite, factory, etc.) and would think of what I would say for each picture in English, then simply translate those words to Spanish and make Quizlet sets with it. Following this, for the questions part of the exam, I thought of questions in specific topic areas (Family, individuals, holidays, environment, the area you live, sports, health, etc.) which could come up (Paper 2 writing prompts can actually help you to come up with these), and write model answers to these. I may have some sheets of possible questions, if you guys would like me to upload them. Oh, 1 more thing, during your prep time for the Speaking exam, when thinking about how to descirbe the picture, divide the picture into 9 equally sized squares, and describe them one by one. This enables you to actually describe the photo but also show to the examiner that you know your words for location, so memorize location words (i.e. On the right, next to, behind, etc.).
Regarding the written assignment, it took me a long time to think of a topic which actually interested me, and that I knew that I could score highly on. I initially wanted to do one on comparing a typical football matchday in England to that in Spain, but someone in my class had taken it, so mine was on public transport. And, if you are stuck on which topic to choose, I would say do one on public transport. I scored 19/20 on my written assignment, and doing a written assignment on public transport allowed me to show off a lot of knowledge. In order to make it incredibly clear to the examiner that you are formatting your assignment correctly, I would have seperately bolded sections which say: Description, Comparison and Reflection. You must remember that the reflection is worth the most marks, so you should use most of your words there, since your word limit is so low. In your description, you only need 3 facts about your topic in the Spanish speaking country and in your comparison I would recommend doing 2 similarities and 2 differences in the cultures as your writing is more balanced then. When writing your reflection, I would use the same facts as the ones in your comparison so that your writing flows and is easier to understand. In the reflection, try to give some opinion phrases, which are both negative and positive, and try to link it to wider topic areas (so for me, that was talking about the environment).
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Mathematics SL:
Paper 1 and 2:
Following learning everything on the syllabus (be sure to read the actual subject guide), past papers are your best friend. In my opinion, all of the textbooks that I came across for Mathematics SL were okay at teaching the topics, but when it came to the practice questions, they were average at best. The textbook questions just are never like the exam questions, and I feel like if I had spent more time doing past papers (starting from the very beginning), I could have finished with a level 7. The IB Questionbank is fantastic for this as it breaks down questions by topic and paper, so you know exactly what you are practising. If you can afford it, Revision Village is fantastic as well, because it does what the Questionbank does, but also breaks them down by difficulty and works you through problems. During the actual exam, check your work as you go, because it sucks to have done so much hard work on a section B question, only to find out that you made a small error in the first part.
The IB has started to like asking more obscure and application based questions in Mathematics SL now, so practice these as much as you possibly can. Also, when doing the actual exam, look at how many marks each question is worth, this can save you big time. I ended up missing out on a level 7 by one mark, and I was so annoyed to see that because I remember spending 5 minutes just staring at a 2 mark trigonometry question which was just asking about SOHCAHTOA. Wasting time on that question prevented me from answering a probability question (about 6-8 marks total) at the end of the paper, so MOVE ON if you do not understand what a question is asking. In Paper 2, you have got a calculator for a reason, so use it for all of the questions, and for questions where you do not have to actually write too much, write "used GDC" on the paper, and quickly sketch graphs as necessary, to make it clear to the examiner. On some questions which require more work, I would recommend checking and working backwards with a different method i.e. On a quadratic question which asks you to solve by completing the square, check with your graph or simple factorizing.
Internal Assessment / The Exploration:
The first thing I will say, and I believe this applies to all of the IA's is: Choose a topic which interests you. I ended up doing one on a topic which related to my HL Economics class to show some personal engagement, but I feel as though I would have done a bit better if I had chosen something which interested me more. In Maths, you really want to map out what your start point is and what you want to have learned by the end, then you can actually plan the logistics of what happens in between. It will also help you to stay motivated and avoid getting confused and stressed when writing it, which can mean that you put more effort into writing it as well.
In addition, I would say the IA does not have to be too complex, I ended up including topics which were a bit above SL level, but some people in my class scored higher than me even with just including SL material. Furthermore, I would say that once you have chosen a certain area of maths that you want to focus on, stick to it, and do not integrate more topics into it because you can really show off your use of mathematics if you have a strong focus in one area. Majority of the points in the IA are not actually specifically maths related, so make sure that you format your IA correctly, and make sure that is easy to both read and understand.
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Biology HL
Paper 1:
Okay, unfortunately it must be said, you kinda need to know everything for all 3 Biology HL papers because the topics which come up, especially in Paper 1's, vary year on year so you need to be prepared for anything. Paper 1 tests the most random areas of the syllabus, and requires you to know many small details in topic areas. To remember these specifics for this paper, I would recommend learning via quizlet sets and mnemonics (i.e. King Phillip Came Over For Gay Sex (Kingdom, Phylum, Class, Order, Family, Genus, Species) for the heirarchy of taxa (Yeah, its weird. I had the same reaction when our teacher told us it, but you remember it.)). On each of the 40 questions they test different areas of the syllabus, and now they love to test people on application points on the syllabus, so learn all of these. There are 2 general things which you can keep your eye out for: The first one being that whenever an image is shown, read the link to see if it gives any hints on the answer, you would be surprised how often it gives it away. The second being, if you know the order of the topics in the syllabus, this is typically the order in which they ask questions in Paper 1, so you usually know the first questions are on cells and the last ones are on human physiology (so if one of the options seems far fetched based on where it is found in the syllabus you know it is not true).
Paper 2:
First thing that I want to say for Paper 2 is practice data based questions, as you are doing revision for the actual exams and are memorizing content, take half an hour out of your Biology revision to just do data based questions. You need practice for those to be able to read graphs quickly, and be able to interpret many of them at once, so print them out of the past papers and just do them as you revise, because they are worth a lot of marks. SL data based questions are good to start off with because they are a bit shorter, but then you can ease yourself into the HL ones. Next, for those 3 mark questions which come at the end of the data based questions every year, learn some generic marking points which you can write if you have no clue what is going on because they are pretty similar every year (i.e. Effects in different animals aren't the same, you need more repeats, you need to test in more climates/places, etc.). For the rest of the paper, similarly to Paper 1, you just need to learn all of the material. I would personally use the Oxford Textbook to revise, complemented with The Science Codex and IB Dead websites because the Oxford textbook has a lot of extra info which you do not need to know. If you prefer to revise by watching, I would recommend Stephanie Castle, Crash Course and Alex Lee.
Although I did finish with a level 6, I was 1 mark off of a level 7, despite working at a high 5 and low 6 level throughout the course, and the one thing which made a big difference was taking all of the extended answer questions, seperating them topic by topic and compiling all of the markschemes together per specific syllabus point. The IB can only ask so many extended response questions, and by doing this and memorizing these markschemes, you get a good idea on the key words which the IB love to see, and implementing them becomes second nature to you. So, if you were to revise very last minute for your course, I would recommend doing this markscheme technique, as the people who score very highly usually do very well on their Paper 2 extended response questions. I would not recommend the Oxford Study Guide, the textbook is much better because the study guide is too condensed, and lacks details in some of the topics, for example in Chapter 5: Evolution. One more thing, make sure that you know ALL of the application points, the IB asks about them so much and when memorized they aren't hard marks to get.
Paper 3:
The one part to this paper which confused me the whole time was Section A, an area in which you could be asked about anything on the course, including your practicals. Pay attention when you do complusory practicals in class, you save a lot of time, as many people learn by doing things. Once you have done all of these practicals, what I did to revise was make a diagram of every practical and annotate it in as much detail as I could, and then on the side of it evaluate the pros and cons of the practical, and jot down its possible applications. That pretty much covers anything which could be asked about your practicals, and use the questionbank to find previous practical questions. And you know how I mentioned those application points before, well the IB has started to ask about them in Section A questions on Paper 3, so know them inside out before.
Section B for me was actually okay, I did Option D: Human Phys which our teacher had recommended and I found it very interesting. Similar 6 mark questions come up in this Option every year, and there is not too much to memorize at all. If you are confused on which option to learn, I would say learn Human Physiology. Again, here, the markscheme technique works fine to compile a bank of knowledge, and doing that with the resources that I have shown should be okay. They usually like to ask about similar things from each topic area, so when you practice past papers you get the gist of what these topic areas actually are. But, as I said with Papers 1 and 2, you just have to memorise the material here again. Make sure that you learn all of your diagrams here, as you need to in Paper 2, as well as definitions, as questions on labelling diagrams are common, and if you are completely stuck on one question, giving a few definitions can usually help you to pick up some marks.
Internal Assessment:
One bit of warning our teacher gave us before we did our IA's was don't worry if your experiment doesn't work completely, nobody's does. So, it's okay to have some errors in your experiment, and have to change your methodology a bit as long as you reflect on your changes and preliminary work in your IA. Online there are a bunch of what to include checklists, so use these as in my opinion they are pretty good and help to give your IA some sort of focus. Personal engagement marks are important, so imbed small bits of personal engagement into your IA as you are writing it, and as I had mentionned before, if you can reflect on your errors and preliminary work it shows personal engagement and reflection. The personal engagement doesn't have to be completely true, as there is only so much interest you can have in one experiment, and you want to save some pages for all of your reflection and analysis.
You want to make sure that you are plotting accurate graphs, and that the calculations associated with those data points are accurate, because those are marks that you can avoid. The page limit is quite low for the Biology IA, so do not make a title page or contents page, just number your sections as you go. I personally would recommend including statistical testing into your IA in order to do some numerical analysis of your data. You can do standard deviation on your graph's data points, and if you have space, and deem it appropriate, you could include another statistical test such as an ANOVA, which tests the relationship between variables. Just remember that the IA is worth 20%, so it is nice to have it as a safety net in case of a difficult exam.
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Chemistry HL
Paper 1:
For chem, as with all 3 papers, past papers are your friend because there are some common topics which come up in multiple choice exams and if you nail down those chapters you can score highly. The chapters which you need to nail down in order to be successful are: Stoichiometry, Kinetics, Energetics & Thermochemistry and Organic Chemistry. Oh, and one more chapter, BONDING. Bonding is the chapter which the whole course is built on, and if you understand this chapter understanding everything else will become a hell of a lot easier, especially in the tougher chapters such as organic chemistry and acids and bases. But, again, you can never predict an IB exam, so revise all of the chapters, but the chapters that I named before, especially Bonding, are very common topics on Paper 1 and Paper 2, so you want to make sure that you understand them inside out. Like in Biology HL, mnemonics and quizlet sets are good to remember things, such as equations and definitions. Mnemonics are especially useful to learn periodicity, where the IB likes to ask about the most random trends in the periodic table, so you should simple memorise those as they are marks that you don't want to be losing. Make sure that you know error calculations for this paper, as the final couple of questions are usually on this area, and nail balancing equations as the first few questions are usually related to this.
Paper 2:
Like in Biology HL, you literally need to know everything for this paper because there are too many areas which have been asked about before. But, luckily for us, we have good resources that are availale, such as Richard Thornley's Youtube channel and the Pearson textbook, which are both absolute gold. Richard Thornley goes through all of the topic areas in insane detail, but explains them in a simple way, so I would recommend watching his videos for the very specific areas such as magnetism, dimers, walden inversion, etc. Memorize all of the formulae that you need to know, particularly for Acids and Bases, because the calculation questions are quite similar every year (i.e. Gibbs free energy, pH calculations with pKa values, molar calculations, empirical formula and equilibrium constants). Paper 1 and Paper 2, like in Biology HL, were back-to-back for me so learning everything for this paper does help for Paper 1 as well. There is a very large amount of material in Chemistry HL course too, so review the subject guide closer to exam time to make sure you know everything.
Make sure that you know ALL of your organic mechanisms, because you just have to memorize them, and drawing them isn't too hard once memorized. The IB also really likes asking about ligands and coloured transition metals, so learning the markscheme for those classic 3-4 mark questions isn't a bad idea as they do not change too much whatsoever. Past papers are again very helpful here, because you see the topics which come up very often in papers and what the exam board likes to ask about. Learn your periodic trends, because they will always come up and they are marks which you really do not need to lose if you have memorized the material, so just be safe and memorize all of the trends (Although the data book can give some trends away, so keep your eye out for that if you forget them). Another shoutout to the IB Dead website, which has some good quality notes for Chemistry too. VSEPR Theory is your friend as well, it comes up way to often, so make sure that you memorize what the theory comprises of, and memorize all of your bond angles as well.
Paper 3:
I did the Biochemistry option, and if you do Biology HL, do Biochemistry because it overlaps with Biology quite a bit, and a lot of that memorization that you do for Biology is really helpful for Chemistry too. For section A, similarly to Biology, you can be asked about any of your complusory practicals, so check the subject guide for which practicals these are. Like I said for Biology as well, draw annotated diagrams of each experiment, then write the method used to obtain the data as well as the equiptment, then you can critique it by listing pros and cons of the experiment itself. If you practice past papers, many of them give away these pros and cons via previous questions on experiments, so you should try and do some as you are going through the course because then its one thing less that you have to worry about revising closer to exam time.
Regarding section B, for the most part, at least of Biochemistry, it's simply just memorisation. So you kinda need to learn everything for this unfortunately. Past papers will help you with this because there are common areas which are always asked about in most papers (i.e. Hydrolysis, condensation, peptides, DNA, etc.). The markschemes for these topic areas are similar so myou can learn these for some of the longer questions, and the markscheme definitions are the ones which you need to know so do not memorise other definitions for key terms. There are some data based questions here so again doing past papers will help you to practice these kinds of questions. For both biology and chemistry, you don't need to do full past papers at once, use the Questionbank to your advantage and practice questions in specific areas you need to practice.
Internal Assessment:
Similarly to Biology HL, find checklists online on what to include as they are quite detailed and usually cover all bases. The Science Codex website has fantastic IA examples for both Biology and Chemistry, so if you are stuck on how to structure each of your IAs, or what kind of information to include, use the model IAs there as an example as they scored very highly. Just like in Biology HL, you want to make sure that you nail your calculations and polish your graphs to make sure that there are no errors in them (Be sure to include error calculations, which you then discuss in your reflection and evaluation section).
Personal engagement again is just something that you can make up a bit and try to imbed it into the IA as you are writing it, but it helps if you are doing a topic which actually interests you. The big advantage for the Chemistry HL IA is that you don't have to do statistical testing like you can in the Biology HL IA, so it saves you space which you can use instead on calculating error. Make sure that you try quite hard on the IA, because with Chemistry HL exams they can be so unpredictable and difficult sometimes that it's nice for something to be there to help you in case the exam day isnt the best.
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Economics HL
Paper 1:
This paper is worth 30%, and with practice and past papers, is an exam which you can do very well on. Before I begin talking about anything else, for everything in Economics, even the IAs, use the Cambridge Revision Guide (Economics In A Nutshell), it's possibly one of the best revision guides I have ever used! So this paper is Micro and Macroeconomics, and to do well on the 10 and 15 mark questions, you need to memorise content from the revision guide. For anything that you do not understand in this book, or for extra detail, use EconPlusDal. Both of those resources together are insanely detailed but explained concisely enough that it is easy to follow and understand. The only hard work for this paper is finding real world examples (yes, they are kinda important, though you can make them up a bit if they sound realistic), so as you learn topics I would just search up that respective topic on Google, find some statistics and data to do with it and compile it in a document which is extensive before you sit the actual exam paper. All of the diagrams that you need to know are in the revision guide, and use a few diagrams in each of your responses, in order to visualise the theories which you are referring to.
In your body paragraphs to your responses, I used an acronym called DEED (Define, Explain, Example, Diagram), and that really helped to structure my answers to make sure I was hitting all of the points on the generic markscheme. However, in your 15 mark questions, where economic synthesis is also required, I used the acronym CLASPP (Conclusion, Long term + Short term, Assumptions, Stakeholders, Priorities, Pros + Cons) as that would cover all of the aspects of the synthesis for me. In Paper 1s every year, there is usually one Theory of the firm question in Microeconomics and one which is not Theory of the firm, so if you can nail down your knowledge on Theory of the firm, you typically have a nice question which you can answer most years (as there is only so much that they can ask on both aspects of Theory of the firm, although they do prefer to ask about market structures).
Paper 2:
This paper is also worth 30%, and I found it harder to revise for, because I absolutely despised Development Economics. Nonetheless, as I said with Paper 1, and as I will say with Paper 3, the Cambridge Study Guide is amazing to revise for this paper. In addition, since you do not need real world examples to complement your responses here, everything that you need to know is in that book. In this paper you dont have to worry as much about sticking to DEED and CLASPP, although you could use DEED on your 4 and 8 mark questions if you deem it to be an appropriate place to use it, but make sure ALL examples are from the text, as most of the marks come from there. Seriously, have a look at the markscheme to one of those 8 mark questions, you would be very surprised to see how 80% of those marking points are simply copying what is actually written inside that text booklet, so use it as much as possible!
Regarding those random definitions at the start, I would recommend just learning all of the terms in the glossary of the Cambridge Study Guide, as those definitions are very similar to the ones which usually appear in the markschemes, and aren't too long to learn (Use Quizlet if you want some more active revision!). For the 4 mark questions, do not forget Micro and Macroeconomics for Paper 2, as they can still be asked about, especially the Macroeconomics diagrams. Including some of the information from the passage in your 4 mark questions can add some more detail, and despite the question not explicitly saying to do it, it often helps to secure 4 points instead of just 3.
Paper 3:
I actually really liked this paper, and I believe that it is possible to score 100% on this paper, or at least close to it, if you just practice. Unfortunately, there is no formula booklet or anything in Economics HL to help you when writing this exam, but all of the equations you need to know are in the Cambridge Revision Guide, so learn your material from there. Regarding the 4 mark questions which you will get, they do repeat over time as there is only so much which can be assessed in this paper, so doing past papers will teach you which kinds of phrases to include in these 4 mark questions and which of these 4 mark questions usually comes up. Refresh reading points off of graphs and using those values to plug into equations to get answers, and using multiple equations to find your answers. For a lot of the small bits which have been asked before such as drawing MR curves or explaining why a profit maximisation would attract firms into a market is explained by EconPlusDal very well, so use his videos once again if you do not understand anything. If you don't think that your Paper 1 or Paper 2 went very well, Paper 3 is the paper which is there to help you out, and if you practice papers and learn all of your equations for this paper you should be good.
Internal Assessment/ Portfolio:
In Economics HL, you have to write 3 different mini-IAs, each 750 words max, which all combine to form a portfolio worth 20%. To start, I would recommend that you should do your third Economics HL IA in International Economics above Development Economics, because your International Economics article options are usually quite good compared to Development, and you can include more diagrams in International Economics. Generally speaking, focus most of your words in each of your IAs on your synthesis, because about 7 of the 15 marks on each of the IAs has something to do with the synthesis, and 2 extra marks for application, so you want to make sure that you nail that analysis really well.
Economic diagrams are key, so use them to talk about the theory related to the article as well, because then you hit two birds with one stone. In addition, I would recommend that you choose an article which talks about a problematic situation, compared to one which talks about a positive economic situation, because you can suggest more solutions and have more analysis when there are problems which need to be ammended. Other than that I would say that define your key terms well (The resources I have said do this for you), and bold key terms as you use them to make it very clear that you are using them.
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Well that's my guide done, hope you guys found it helpful :) If you have any questions just reply in the comments or drop me a PM and I'll respond as best as I can to you. Once again, thanks so much to this legendary sub for all of the help they gave during the IB exam period.
EDIT: Reddit didn't let me do a post with everything in it, so I will post a part two later with my advice on TOK, EE, CAS and some extra sections for people who want to apply for Medicine in the UK

Please don't hate me for posting this Mods lol.
So, basically I have 5 months until I start college. Recently, I have made it a challenge to myself and my abilities to see how much math I can really handle.
To be honest, I've always been mediocre at math, but math has never ceased to amaze me with all its equations and levels of complexity. And I've also realized that it was mainly by lack of focus, dedication, diligence, patience, and commitment which has deterred my progress in math, not so much my intelligence (so I believe haha).
However, now that I've been reinvigorated to confront the new possibilities that await through the process of learning, I'm ready for a disciplined approach to not just understanding maths, but absorbing and utilizing the concepts of math fundamentally. In doing so, I've laid out a path to follow: from basic Algebra to college level stuff like Calc 3 and Differential Equations. This is the exact order and listing from calcworkshop.com, the resource I'll be making tremendous use of on this adventure. For anyone interested in an intuitive and simple to comprehend way of teaching, I highly suggest you check out the website (albeit it is a paid subscriber platform).
I obviously do not intend to finish this entire monstrosity of a course/program/ultimate math killer bootcamp or even half of it, but the crucial thing is that we try our best everyday in pursuit of some arbitrary (or specific) ,personal (or public) goal. And I don't think I'm naive either in thinking I'll just be able to breeze past difficult topics in just 5 months. On the contrary, I am more so inclined to master the basic fundamentals like algebra, trig, precalc than to half-heartedly tackle obscure problems in Linear Algebra just for the sake of doing "hard shit." I think most of you can agree that in order to do well in math or anything, it is paramount for one to build a strong foundation on which more complexity can be built.
So that is indeed my plan. I'm guessing I'll spend roughly 4 hours a day adhering to this. Currently, I'm on 1 (Algebra) D (Polynomials) and I'm excited to see where I'll end up in the coming weeks and months. I'm also trying to get better at programming as well during all this. So I'm doubly excited for all the challenges that lie at my feet, just waiting for me to snuggle up and devour them.
But why am I posting this to all you mathematical folks on this subreddit? Well, I have to admit, sometimes Reddit really fuels me to take fruitful actions. And by that I mean you guys motivate me a lot. Especially on those self-improvement subs, I see countless of people getting back on their feet after months or years of depressive, suicidal, and chaotic times. In a way, I'm doing this to make myself feel in tune with the potentiality of my existence (lol Jordan Peterson fans where you at?) by confronting something hard, that could possibly be useful, disguised as fear and illusion. It's 3am here, and I'm rather tired after spending the last few hours typing this gargantuan list up. But even if the following guide can help some of you folks who are struggling to find a direction, whether it's in math or in life, then I'm happy.
It's really just fun and games at the end. To be able to sit here and do math in order to educate myself with amazing resources at my disposal, as the world just keeps innovating and progressing (cough singularity is near cough), is a real dream for some folks. And if I could elaborate on this further, I'd say that there is a meaning or purpose to be found implicit in the act of doing something worthwhile, as challenging and as exhausting as it may be. Right between the lines of aptitude and stress is where we are able to flourish and grow.
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• 1. Algebra
  • A. Intro Algebra
    • Order of Operations
    • Real Numbers
    • Translating Algebraic Expressions
    • Algebraic Properties
    • Multiplying and Dividing
    • Combining Like Terms
    • Consecutive Integers
  • B. Solving Equations
    • One Step Equations Addition
    • One Step Equations Multiplications
    • Multi Step Equations
    • Variables on Both Sides
    • Linear Word Problems
    • Literal Equations
    • Absolute Value
    • Linear Inequalities
    • Absolute Value Inequalities
    • Inequality Word Problems
  • C. Exponents
    • Simplifying Exponents
    • Adding Exponents
    • Multiplying Exponents
    • Negative Exponents
    • Scientific Notation
  • D. Polynomials
    • Adding and Subtracting Polynomials
    • Multiplying Polynomials
    • FOIL Method
    • DRT Word Problems
  • E. Factoring
    • Integer Factorization
    • Greatest Common Factor
    • Difference of Squares
    • Trinomial Coefficient One
    • Trinomial Coefficient Not One
    • Factor by Grouping
    • Factoring Cubes
    • Solve by Factoring
    • Factoring Word Problems
  • F. Rationals
    • Rational Expressions
    • Multiplying Fractions
    • Dividing Fractions
    • Adding Fractions
    • Long Division
    • Proportion
    • Solving Equations
    • Percents
    • Graphing Linear Equations
    • Coordinate Plane
    • Finding Intercepts
    • Slope Formula
    • Slope Intercept Form
    • Parallel Perpendicular Lines
    • Point Slope Form
    • Graphing Linear Inequalities
  • G. Systems of Equations
    • Graphing Method
    • Substitution Method
    • Elimination Method
    • System of Inequalities
    • Linear Programming
    • Nonlinear Systems
  • H. Radicals
    • Square Root
    • Rational Exponents
    • Rationalizing
    • Multiplying Radicals
    • Conjugate Math
    • Solving Radical Equations
    • Completing the Square
    • Quadratic Formula
    • Pythagorean Theorem
  • I. Functions & Statistics
    • Relations and Functions
    • Set Notation
    • Linear Function
    • Central Tendency
    • Empirical Rule
    • Z Score
    • Line of Best Fit
    • Permutations and Combinations
    • Probability
• 2. Trigonometry (Pre-calc Part 1)
  • A. Trigonometric Functions
    • Interval Notation
    • Coterminal Angles
    • Reference Triangles
    • Reference Angles
    • Cofunctions
    • Degrees Minutes Seconds
    • Angles of Elevation and Depression
    • Applications of Right Triangles
  • B. Radian Measure
    • Degrees to Radians
    • Arc Length Formula
    • Unit Circle
    • Angular and Linear Velocity
  • C. Graphing Trig Functions
    • Graphing Sine and Cosine
    • Graphing Sine and Cosine with Period Change
    • Graphing Sine and Cosine with Phase Shift
    • Graphing Reciprocal Trig Functions
  • D. Trig Identities
    • Fundamental Trigonometric Identities
    • Steps for Verifying Trig Identities
    • Proving Trig Identities
    • Sum and Difference Identities
    • Double Angle Identities
    • Half Angle Identities
    • Product-Sum Identities
  • E. Trig Equations
    • Inverse Functions
    • Inverse Trig Functions
    • Solving Trigonometric Equations
    • Solving Trig Equations Using Inverses
  • F. Law of Sines and Cosines
    • Law of Sines
    • Law of Sine - Ambiguous Case
    • Law of Cosines
    • Heron’s Formula
  • G. Vector Applications
    • What is a Vector?
    • Displacement Vector
    • Force Vector
    • Velocity Vector
  • H. Polar Equations and Graphs
    • Polar Coordinates
    • Graphing Polar Equations
  • I. Complex Numbers
    • Complex Numbers in Standard Form
    • Complex Numbers in Polar Form
    • De Moivre’s Theorem
• 3. Math Analysis (Pre-calc Part 2)
  • A. Intro Math Analysis
    • Solving Equations
    • Solving Inequalities
    • Absolute Value Equations
  • B. Functions and Graphs
    • Transformations of Functions
    • Piecewise Functions
    • Composite Functions
    • Domain of Composite Functions
  • C. Exponentials and Logarithms
    • Exponential Equations
    • Graphing Exponential Functions
    • Logarithmic Functions
    • Solving Logarithmic Equations
    • Graphing Logarithmic Functions
    • Compound Interest Formula
    • Exponential Growth
  • D. Polynomial Function
    • Quadratic Polynomial
    • Synthetic Division
    • Zeros
    • Finding Zeros
    • Graphing Polynomial Functions
    • Polynomials Functions in Calculus
  • E. Rational Functions
    • Identifying Asymptotes
    • Graphing Rational Functions
    • Variation Equations
    • Rational Functions in Calculus
    • Partial Fractions
  • F. Conic Sections
    • Circle Conics
    • Ellipse Conics
    • Parabola Conics
    • Hyperbola Conics
    • Conics Review
    • Parametric Equations
  • G. Series and Sequences
    • Sequences
    • Geometric Sequences
    • Summation Notation
    • Geometric Series
    • Mathematical Induction
    • Binomial Theorem
• 4. Calculus 1
  • A. Limits
    • Finding Limits Graphically
    • Limit Rules
    • Squeeze Theorem
    • Limits at Infinity
    • Limits Review
    • Epsilon Delta Definition
    • Limits and Continuity
    • Limits Definition of Derivative
  • B. Derivatives
    • Definition of Derivative
    • Power Rule
    • Product Rule
    • Quotient Rule
    • Chain Rule
    • Derivative Rules
    • Derivative of Exponential Function
    • Derivatives of Logarithmic Functions
    • Trig Derivatives
    • Inverse Trig Derivatives
    • Hyperbolic Trig Derivatives
    • Derivative of Inverse Functions
    • Implicit Differentiation
    • Higher Order Derivatives
    • Logarithmic Differentiation
    • L’Hopital’s Rule
    • Particle Motion
    • Rate of Change
    • Equation of Tangent Line
    • Linear Approximation
    • Continuity and Differentiability
    • Derivatives Using Charts
    • Limit Definition of Derivative
    • Newton’s Method
    • Related Rates
  • C. Application of Derivatives
    • Application of Derivatives Lesson 1
    • Application of Derivatives Lesson 2
    • Application of Derivatives Lesson 3
    • Demand Function
    • Elasticity of Demand
  • D. Integrals (Same as Calc 2, A)
    • Riemann Sum
    • Sigma Notation
    • Integration Rules
    • Trig Integrals
    • Fundamental Theorem of Calculus
    • U Substitution
    • Arc Length Formula
    • Mean Value Theorem for Integrals
    • Particle Motion
    • Simpson’s Rule
    • Partial Fraction Decomposition
    • Integration by Parts
    • Advanced Trigonometric Integration
    • Trig Substitution
    • Improper Integrals
• 5. Calculus 2
  • A. Integrals (Same as in Calc 1)
  • B. Application of Integrals
    • Area Between Two Curves
    • Volumes with Known Cross Sections
    • Solids of Revolution - Disk and Washer Method
    • Solids of Revolution - Shell Method
    • Work and Hooke’s Law
    • Moments and Center of Mass
  • C. Differential Equations
    • Separable Differential Equations
    • Slope Fields
    • Exponential Growth and Decay
    • Euler’s Method
    • Logistic Differential Equations
  • D. Polar Functions
    • Polar Coordinates
    • Polar Graphs
    • Tangent Line in Polar Coordinates
    • Area in Polar Coordinates
  • E. Parametric and Vector Functions
    • Vectors Lesson
    • Parametric Lesson
  • F. Sequences and Series
    • Sequences and Series Intro
    • Nth Term Test
    • P Series Test
    • Geometric Series
    • Limit Comparison Test
    • Integral Test
    • Telescoping Series
    • Alternating Series
    • Ratio Test
    • Root Test
    • Sequences and Series Review
    • Radius and Interval of Convergence
    • Power Series
    • Maclaurin and Taylor Series Intro
    • Taylor Series
    • Binomial Series
    • Lagrange Error Bound
• 6. Calculus 3
  • A. Vectors and The Geometry of Space
    • Three-Dimensional Coordinate Systems
    • Vectors in 3D
    • Dot Product in 3D
    • Cross Product in 3D
    • Equations of Lines and Planes
    • Cylinders and Quadric Surfaces
    • Cylindrical and Spherical Coordinates
  • B. Vector Functions
    • Vector Functions and Space Curves
    • Derivatives and Integrals of Vector Functions
    • Arc Length and Reparameterization
    • Curvature
    • Unit Tangent Vectors
    • Motion in Space: Velocity and Acceleration
  • C. Partial Derivatives
    • Functions of Several Variables
    • Limits and Continuity
    • Partial Derivatives
    • Tangent Planes and Linear Approximations
    • Multivariable Chain Rule
    • Directional Derivatives and Gradient Vectors
    • Relative Extrema
    • Absolute Extrema
    • Lagrange Multipliers
  • D. Multiples Integrals
    • Double Integrals over Rectangles
    • Average Value and Double Integral Properties
    • Iterated Integrals
    • Double Integrals over General Regions
    • Double Integrals in Polar Coordinates
    • Applications of Double Integrals: Density, Mass and Moments of Inertia
    • Surface Area
    • Triple Integrals
    • Triple Integrals in Cylindrical Coordinates
    • Triple Integrals in Spherical Coordinates
    • Change of Variables in Multiple Integrals
  • E. Vector Calculus
    • Vector Fields
    • Line Integrals
    • Review of Line Integrals
    • The Fundamental Theorem of Line Integrals
    • Green’s Theorem
    • Curl and Divergence
    • Parametric Surfaces and Their Areas
    • Surface Integrals
    • Stoke’s Theorem
    • The Divergence Theorem
    • Vector Calculus Review
• 7. Linear Algebra
  • A. Linear Equations
    • System of Linear Equations
    • Reduced Row Echelon Form
    • Vector Equations for Matrix Algebra
    • The Matrix Equation Ax=b
    • Solution Sets of Linear Systems
    • Linear Independence
    • Linear Transformations
    • Applications of Linear Systems
  • B. Matrix Algebra
    • Matrix Operations and Determinants
    • Inverse Matrix
    • Invertible Matrix Theorem
    • Partitioned Matrices
    • Matrix Factorization
    • Applications to Computer Graphics
  • C. Determinants
    • Determinant of a Matrix
    • Cramer’s Rule
  • D. Vector Spaces
    • Vector Spaces and Subspaces
    • Null, Column, and Row Spaces
    • Linearly Independent Sets and Bases
    • Coordinate Systems and The Dimensions of a Vector Space
    • Rank
    • Change of Basis
    • Markov Chain Applications
  • E. Eigenvalues and Eigenvectors
    • Eigenvalues and Eigenvectors
    • The Characteristic Equation
    • Diagonalization
    • Eigenvectors and Linear Transformations
    • Complex Eigenvalues
  • F. Orthogonality and Least Squares
    • Inner Product, Length and Orthogonality
    • Orthogonal Sets
    • Orthogonal Projections and Decomposition
    • The Gram-Schmidt Process and QR Factorization
    • Least-Squares Problems
    • Applications to Linear Models: Least-Squares Lines
    • Inner Product Spaces
  • G. Symmetric Matrices and Quadratic Forms
    • Diagonalization of Symmetric Matrices
    • Matrix of the Quadratic Form
• 8. Differential Equations
  • A. Intro to DiffEqs
    • Ordinary Differential Equations
    • Initial Value Problems
  • B. First Order Differential Equations
    • Directional Fields
    • Separable Equations
    • Euler’s Method
    • First Order Linear Differential Equation
    • Solving Exact ODEs
    • Homogeneous Differential Equation
    • Bernoulli Differential Equation
    • Linear and Nonlinear Models
  • C. Second Order Differential Equations
    • Distinct Real Roots
    • Complex Roots
    • Repeated Roots
    • Wronskian
    • Undetermined Coefficients
    • Differential Operators
    • Variation of Parameters
    • Cauchy Euler Equation
    • Predator Prey Models and Electrical Networks
    • Mechanical Vibrations
  • D. Series Solutions
    • Power Series
    • Series Solutions
  • E. Laplace Transform
    • Laplace Transform Overview
    • Inverse Laplace Transforms
    • Initial Value Problems with Laplace Transforms
    • Translation Theorems of Laplace Transforms
  • F. Systems of Differential Equations
    • Homogeneous Linear Systems
    • Phase Plane Portraits

Hello EthDev,
After several years of searching, I recently found the basics of a game theoretic framework for describing both the real (objective) aspects of a game and the players (subjective) thoughts. One immediate economic application of this theory would be towards describing a social credit score system and its affects on the thoughts of people as well as how to combat a social credit score system. However, I suspect people can find other uses for this this in crypto-economics.
Lexicographic Ordering of the Complex Numbers and their Interpretation
Before we can construct complex game theory, we need to (lexicographically) order the complex numbers in a line. To compare complex numbers x and y: First, compare their real parts and then use their imaginary parts to break a tie.
With this ordering on a line, the interpretation of the complex numbers is now different than the ordering that places the complex numbers on a plane. Complex numbers no longer correspond to a scaling and rotation of the plane. Instead, the number i (sqrt(-1)) is now a positive number and the number -i (-sqrt(-1)) is a negative number. Contradictions arise if we assume that the number i can represent a positive, real quantity. (Construct a square of side length i. It's area is -1, which is a negative number, despite that all sides are positive in length. This contradicts that all real areas constructed from real, positive lengths are positive) Since anything that exists but is not real is, by definition, imaginary, the imaginary numbers represent imaginary quantities. In other words, any time an imaginary number shows up (and the lexicographic ordering is used), it must represent a mental quantity subjective to some observer.
Complex Game Theory (A game theory where both the thoughts and actions of the players are scored)
Now that we have the lexicographic ordering of the complex numbers, we can take minima and maxima over the complex numbers. We can construct the complex game theory simply as a complex generalization of the real game theory. Now, the moves of each agent and each agent's utility function are complex.
Complex Differential Game Theory and the Equation for Minimal Rationality Strategies
In the complex differential game theory, each agent has a start location (complex) and a goal location (complex) as well as a utility function (complex) which scores each path (complex) from the start location to the goal.
It helps to consider the worst possible strategies because, in the event that our game has unique worst strategy for going from a start point to an end goal, considering the worst strategies gives us a deterministic physics. Since any agent wishes to maximize its own utility function, the only agents who will attain a minimum of the utility function (i.e. who will take a worst strategy) are those who do not know any better because they are minimally rational.
For any strategy of agent-k that minimizes S_k, agent-k's utility function, we know that (S_k).real is also at a minimum. By real-valued Calculus of Variations, we therefore know that δ_k(S_k.real) = 0, where δ_k is the infinitesimal variation due to an infinitesimal change in agent-k's path. I conjecture that (δ_k S_k).real = δ_k(S_k.real) and (δ_k S_k).imaginary = δ_k(S_k.imaginary), so that we know (δ_k S_k).real = 0. Furthermore, δ_k S_k is a functional, which means that δ_k S_k.imaginary is also a functional. The only natural functionals in the calculus of variations are δ_j for some agent-j, its complex conjugate, its adjoint, and its transpose. By definition of δ_k, δ_k S_k.imaginary is linear in δ_k. Therefore, δ_k S_k.imaginary = a_k δ_k + b_k δ_k-adjoint, where a_k and b_k are real numbers.
δ_k S_k = i (a_k δ_k + b_k δ_k-adjoint) for some a, b in the reals.
That is as far as I have gotten with the equation, however, I conjecture that "a_k = b_k < 0". This would give the family of equations
δ_k S_k = -i hbar_k (δ_k + δ_k-adjoint)
where I have defined hbar_k = -a_k = -b_k. This family of equations (one for each agent-k), if correct, generalizes Schwinger's Quantum Action to a game theoretic setting. In other words, if this equation is correct, then the minimally rational, but thinking, agents are the particles of quantum mechanics, and the math of how to handle minimally rational, thinking, agents has largely already been worked out.
Edit: I realize that my argument above is very hand-wavy. Unfortunately, I am just beginning my study of Stochastic Calculus of Variations, so I cannot yet do better.

The site talkie is one of the biggest names out there for practicing your language skills online with native speakers. Designed for language learners to find native speaking teachers and exchange partners via Skype, the free version of their service includes some great benefits, including access to their learner community. This access allows you to connect with other learners and arrange informal language exchanges via Skype or email.

Websites for Finding Free Language Courses

When most of us think about taking a language course, we’re picturing a classroom full of wobbly desks and chalkboard dust, plus a three- or four-digit tuition fee. For those who are looking for all the structure and formality of a traditional course but aren’t sure about stepping back into the classroom or forking over those fees, check out some of these free online language courses, available in nearly as many languages as there are learners.

1. Live Lingua

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Heading off the list is Live Lingua, the Internet’s largest collection of free public domain language learning materials, according to the site itself. It’s not hard to believe. There are thousands of free e-books, audio recordings and foreign language video materials available in over 130 different languages.
It’s also one of the most user-friendly websites hosting the US Foreign Service Institute’s public domain language courses, which were developed by the US Department of State for expert language learning and are now freely available on various sites across the Web.

2.Learn language .com

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Learn language.com is a website with links to extensive resources on and in 19 different languages. You can learn over 1400 words in your target language for free with their vocabulary lists and verb conjugation charts, and there’s plenty more free learning to be done in the nine languages in which Learn language.com maintains its own Web-based courses.

3. Open Culture

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Open Culture is an e-learning website that hosts “the best free cultural and educational media on the web,” and when it comes to languages, they’re not bluffing. Open Culture maintains a list of free courses in 48 languages across the Web, from Amharic to Yiddish. With these collected resources from governments, universities and respected private institutions across the world, you’re sure to find free, high-quality lessons in 48 languages that are learned far and wide.

4. Surface Languages

Surface Languages maintains an extensive database of free online learning resources, as well as its own beginner audio courses in Italian, Portuguese, French and Polish. Additionally, there’s a handy section with audio and flashcards in Afrikaans and Romanian, and a “recently added” sidebar that shows you the latest additions to their ever-growing language learning library.

5. Internet Polyglot

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Internet Polyglot offers many of the same kinds of resources as the sites listed above, with the awesome added advantage of its “quick start menu,” which allows you to choose not only what language you’re learning but also what language you’re learning it in. Do you already speak Spanish and want to get started on Portuguese? Just select “Spanish” for the language you speak, and get started with a composite mental exercise to strengthen one language while building another.

6. Headstart2 Defense Language Institute

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The Defense Language Institute (DELI), like the Foreign Service Institute, is a government service that makes high-quality language learning programs available for free. Headstart2 is one of the best and most easily navigable services hosting complete DELI courses. After quickly registering for an account, you’ll be launched straight into interactive lessons with maps, images, sound, cultural notes and more.

Websites with Video-based Language Learning Content

The science shows that switching on the TV is great for language learning. If you want your brain to soak up the sights and sounds of what actual everyday speech sounds like and how it’s used in your target language, tune in and kick back for some quality input-based learning with these websites.

7. Streetcar

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For the tube-loving language learners out there, Streetlamps hard to beat. Although it’s not necessarily designed as a language learning tool, it offers free TV streaming from over 100 countries around the world in nearly as many languages.
This is an especially exciting resource if you’re learning a less commonly studied language with less widely available video material. Tuning into your favorite Albanian or Nepali TV station can easily make up for the lack of other learning materials in your language.

8. Fluent U

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Fluent U is a totally different world of language learning.
If you’re into learning with video, TV and movies (which you definitely should be), you’ve got to at least grab your free 15-day trial and spend a couple weeks binge-learning here with our video and audio libraries, which feature many of the same things native speakers of your target language are watching in their sweatpants while eating potato chips at home.
Two main features set Fluent U apart from the others on this list. The first is the sheer range of available content: movies, news, documentaries, cartoons, music videos, funny YouTube videos or whatever else you like to watch, it’s there.
The second is that it uses real-world video. Instead of contriving some slowly-spoken and articulately-pronounced (and usually terribly boring) videos for learners, Fluent U directs you to authentic video content appropriate to your level with built-in learning tools that ensure you can keep up.
Starting out with the free trial will give you a priceless learning boost, and if you subscribe after the trial period you’ll be paying less monthly for unlimited video content than you’d be shoveling out hourly for lessons with even the most affordable tutors.
Here’s just a brief taste of the content you’ll find on Fluent U:
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Fluent U App Browse Screen.
Fluent U has interactive captions that let you tap on any word to see an image, definition, audio and useful examples. Now native language content is within reach with interactive transcripts.
Didn’t catch something? Go back and listen again. Missed a word? Hover your mouse over the subtitles to instantly view definitions.
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Interactive transcript for Carlos Beaut song.
You can learn all the vocabulary in any video with Effluents “learn mode.” Swipe left or right to see more examples for the word you’re learning.
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Fluent U Has Quizzes for Every Video
And Fluent U always keeps track of vocabulary that you’re learning. It uses that vocab to give you a 100% personalized experience by recommending videos and examples.
Start using Fluent U on the website with your computer or tablet or, better yet, download the Fluent U app from the iTunes store or Google Play store.

9. Innovative Language

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Since its inception in 2005, Innovative Language has striven to become one of the leading language lesson providers.
Innovative Language is a massive system of free video and audio lessons in 34 languages. It’s updated with new lessons every week, with material geared towards absolute beginners all the way up to advanced learners nearing fluency. You may recognize it from its podcast sites (for example, Portuguesepod101) These are the language-specific sites you’ll be directed to once you enter your email and pick a language. Here’s just a sampling of the many popular foreign languages they currently offer:

• Spanishpod101
• Frenchpod101
• Germanpod101
• Chineseclass101
• Russianpod101

It focuses on featuring audio and video material made by professional teachers, and it’s one of the most prolific and consistent language sites in the game. They teach grammar, vocabulary, everyday conversations, real-life situations and culture. So, basically, you’ve got all your bases covered.
Although they have a paid version of the service, thousands of professionally produced audios and videos are given absolutely free.
You can benefit from this flood of language content regardless of your state of fluency—they’ve got something for absolute beginners as well as advanced learners.

10. YouTube TV Channels

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Did you know that YouTube has an insane number of TV shows available from around the world? Just go to the International TV section and scroll through the shows sorted by language.
This one is particularly helpful for learners of Indian languages, as it includes Hindi, Tamil, Marathi and others widely spoken through the Indian subcontinent, but other options like Korean, Japanese, Spanish, Greek, Danish and more are available as well.

Massively Open Online Courses (Moo Cs) for Free Language Learning

Did we mention that we think technology is awesome? A lot of the world’s best universities agree with us. That’s why so many of them are making lesson materials and entire university courses available online for free. Here are some of the best massively open online courses for learning a language.

11. MIT Open Courseware

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MIT was one of the university hipsters making their course content available online before it was cool. Under the amazing Global Studies and Languages section of their Open Course ware website, you’ll find courses ranging from Chinese II to Contemporary French Politics, all designed to help you engage not only with the language you’re studying but also with its greater cultural context.

12. Ed X

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Ed X is one of the biggest MOOC (massive open online course) websites out there. You can find a lot of college-level courses here, created and taught by actual professors from top universities. As a language learner, you can consider taking a dedicated language class, or if you’re more advanced, taking a course in your target language. For example, right now you can sign up to take an algebra class entirely in French!
You can actually earn college credits on Ed X, through Arizona State University. Some of the courses have certificates you can get for completing them. In general, it costs money to get a verified certificate for completing a course (so make sure you check the details before you sign up), but many of the courses themselves are still free.

13. Coursers Language Learning

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The courses offered on Coursers are run by professors from world-renowned institutions, and it’s even possible to earn a certificate from many of the courses.
Coursers, another big name in online education, also maintains a Language Learning section under its online catalogue. Many of the courses will be geared toward beginners, like First Step Korean, and others will engage more generally with language learning, like the Miracles of Human Language course from Leiden University. Still other classes are taught on other topics in foreign languages, so you can learn about writing, math, engineering or literature in the foreign language you’re learning—though this approach is recommended for higher-level learners.
On Coursers, there are currently 122 classes taught in Chinese, the second most common language after English, and a long list of other languages trail after. Spanish has 79 classes. French has 41 classes. You get the idea. There are a ton of classes out there for you to try. You can even learn about entrepreneurship in Khmer!
Whatever you find here, it’s coming from a respected university or educational institute, and if it’s not on offer now, sign up to get an alert when it is!

14. Carnegie Mellon Open Learning Initiative

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The OIL at Carnegie Mellon isn’t specifically made for language courses, but there are several excellent ones on offer. From elementary Spanish and French to “Arabic for Global Exchange,” you’ll find frequent courses available in the world’s biggest and most commonly learned languages.

15. Alison

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Each of the eight languages offered on Alison’s language learning platform include at least an introductory course, and others include more advanced and detailed courses in popular languages like French. Best of all, if you complete all modules and score at least 80% on all your course assignments, you’ll be rewarded with a fancy (and free) certificate!

16. The Mega List of MOOC from Web Techno and Translation

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The blog Web Techno and Translation smartly and helpfully recommends taking an MOOC given in the foreign language you’re studying, and gives you a decent list to get started on doing so. Once you’ve broken through the intermediate level and are ready to start doing more with your language skills, why not try taking a course conducted in that language? Whether you want to study Croatian history in Croatian or systems administration in Spanish, the goal is the same: by focusing on learning about a topic instead of the language itself, you’ll learn more naturally.

Free Websites for Connecting with Native Speakers of Your Language

The biggest secret to effective language learning isn’t really a secret at all: You’ve gotta talk! And who better to talk with than a native speaker?
It’s okay if you can’t hop on a plane this weekend—instead, just click over to one of these websites, where you can connect with native speakers of your target language for free.

17. talkie

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The site italki is one of the biggest names out there for practicing your language skills online with native speakers. Designed for language learners to find native speaking teachers and exchange partners via Skype, the free version of their service includes some great benefits, including access to their learner community. This access allows you to connect with other learners and arrange informal language exchanges via Skype or email. If you're looking to invest a little more to have a private tutor or teacher at some point, you can do that on talkie too.

18. The Polyglot Club

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The Polyglot Club is an awesome all-purpose language learning website with tons of free features for connecting you with native speakers of your target language. You can find language exchange partners, submit written texts for correction, hang out in chat rooms, browse through target language videos and even attend their language events and meetups for connecting with even more learners!

19. Hi Native

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Hi Native is an innovative app that takes a nontraditional approach. Rather than giving you flashcards and courses or even facilitating Skype sessions, Hi Native allows you to ask questions to native speakers of the language you’re learning. Whether it’s about the difference between two confusing words, getting a pronunciation check or figuring out how and when to use the subjunctive, just send a text and wait for a response.

20. Word2Word

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Word2Word is all about connecting the world, and they aim to do so by providing you with an insane amount of free online language learning resources. Not only are there chat rooms and meetups, but the site hosts resources on everything from typing characters in other alphabets and scripts to a Dictionary of Period Russian Names. If you’re looking for language exchange and don’t mind falling down the rabbit hole of linguistic Gerry, this is definitely the place to get started.

21. Lang-8

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For many of us, speaking comes easily, but when it comes time to put pen to paper and communicate in writing, all of a sudden we feel like we don’t speak a word of the language. Lang-8 is a community of native speakers of various world languages who work together to improve each other’s writing in languages they’re learning by offering revisions and tips on the posts made in their languages—just don’t forget to pay it forward by helping others out with your native language too!

Free Vocabulary Games and Flashcards

There are boring flashcards, and then there are the kind that turn language learning into a game and wake up the competitor in all of us. To build your vocabulary while having some fun, these sites offer some fantastic flashcard exercises and other language learning games.

22. Duo lingo

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How could we even make a list without mentioning everyone’s favorite pushy little green bird? Duo lingo user-friendly vocabulary games made language learning cool again by unlocking the power of gaming for language learning.
They use the gaming strategy to make sure you retain more of what you learn than you would from your standard, flipping-through-printed-flashcards approach. The truly dedicated will be rewarded by being asked to translate sentences of increasingly comical ridiculousness as they work up through the levels of their language.

23. Mesmerism

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Mesmerism gives you a more hands-on option for flashcard learning with its “memes.” You can use the cards provided by Mesmerism and other users, or you can create your own mnemonic devices by composing and finding images for your own cards. Somehow, repetition doesn’t seem so repetitive when you can turn all your flashcards into irreverent memes and personal in-jokes tailored to you and your own brain.

24. Digital Dialects

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For beginners and those suffering from 90s nostalgia, Digital Dialects is a techno-retro online game site with animated and interactive lessons in a couple dozen languages. Listen to the voice as it names fruits, then click them and drag them into a basket, or match English phrases to the ones being spoken in your target language. It’s a simple platform free of distraction, and it offers that j NE said quo for the Nintendo kids who still have a knack for learning in 16-bit.

General Language and Language Learning Resources

For those of us with a deep and passionate love for language and languages, we want to know everything about them. Not just how to get from the airport to your hotel or how to make small talk during your coffee break, but the details.
Who speaks the language we’re learning, and where do they speak it? What’s its history, and what can it teach us about the cultures that created it and that are created by it? These websites include the kind of encyclopedic information necessary for thorough, holistic learning, as well as free courses and learning resources.

25. Omnipotent

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If you’re learning one language or find yourself in a poly amorous love affair with all languages, Omnipotent should live on your browser’s bookmarks bar. This online encyclopedia of writing systems and languages is like the Wikipedia of all things languages and linguistics.
Not only does it provide linguistic and cultural information on hundreds of world languages, but it also offers some of its own free video lessons. Under its “courses” page, Omnipotent maintains an impressive list of links to learning resources for specific languages, most of which are free.

26. BBC Languages

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Another goldmine for the language and linguistics geeks in the room, BBC Languages offers free information and learner resources for 40 world languages. While some are more thorough than others, the seven languages listed on their main Languages homepage include extensive links to target language media (like TV streaming services, online newspapers and magazines in French, Chinese and Greek). The kinds of resources offered vary from language to language, but most include overviews and phrasebooks, as well as BBC’s own introductory lessons, on-site courses and links to help you find classes in your area.

27. Thought Co.

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Thought Co. provides a long list of language learning resources like many others on this list, but also has its own sites dedicated to several of the languages on its list. About French, for example, includes educational videos, blog posts on French learning topics and guides to language essentials. About German offers much of the same. Explore multiple languages and find links to extensive resources across the Web.

28. Wiki travel Phrasebooks

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You don’t have to be planning a vacation to take advantage of Wiki travel! The user-maintained wiki has a compilation of travel phrasebooks, ranging from well-known world languages like Arabic and Portuguese to less popular choices like Kannada and Parliament.
Even some of the most skeleton-like articles on languages like Zulu include guides to pronunciation, basic phrases, numbers, time, transportation, eating, shopping and common problem-solving vocabulary, making it an incredible resource for practically-inclined learners who just want to get out and use their language.

Free Language Learning Websites for Specific Languages

Some of the best free sites out there are the ones that focus on a single language, building an entire website packed with information, lessons and more for learners who know what language they want and know they don’t want to break the bank learning it. Here are some of our favorite websites for learning that one language.
French
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29. The French Experiment — Free French lessons, stories, reviews and cats!
30. London Speaks French — Vocabulary and grammar lessons with a helpful pronunciation tool to compare your voice to natives’ voices.
For dedicated Francophile looking to max out their online learning time, browse through some of these best websites to learn French.
German
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31. Deutschmark.com — With 10 beginner lessons and 24 advanced grammar lessons, this free resource is a great starting point for beginning learners and those intimidated by the infamously frustrating German grammar.
32. Deutschmark — This greeter Online-Deutschmarks has helpful audio clips and interactive lessons, as well as links to in-person courses in Berlin and other parts of Central Europe.
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33. Deutschmark Welles — The German international broadcast’s online Deutschmark offers free courses by level, from beginner to advanced.
For video-loving learners, don’t forget about all the great YouTube videos for German learners out there!
Spanish
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34. Study Spanish.com — Don’t be fooled by the simple name. With exhaustive information and lessons on all aspects of the language, it’s a Spanish linguistic powerhouse and a goldmine for learners.
35. Spanish Games — Learn Spanish with fun, interactive vocabulary games.
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36. 123 Teach Me — A mega-resource for all things Spanish, including free online games, courses, word of the day and a handy verb conjugation.
If you like thinking outside the box when it comes to language learning, you can also try out some nontraditional websites for learning Spanish.
Chinese
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37. CCTV — Chinese public television, as well as a BBC-like array of written media and resources for Chinese language learners.
38. Chinese-Tools.com — A 40-lesson online course with a learner forum and Chinese dictionary.
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39. Zhengzhou Red — In-context vocabulary and culture lessons in a conversation-like format.
More of a bookworm? Try getting started with some free and cheap Chinese e-books!
Japanese
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40. Japanese Online — A simple beginner’s resource with four units on the basics, topic modifiers, sentence structures and Japanese traditions.
41. Easy Japanese — Flashcards and games geared toward writing and speaking the language.
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42. NH World — This Japanese public broadcaster maintains its own online magazine, including language lessons, audio clips and video resources for learners.
Is it grammar that’s got you down? Check out these websites for mastering your Japanese grammar.
Arabic
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43. Arabic Online — This website from the European Union aims to help beginners familiarize themselves with the Arabic language, and offers several paid courses beyond the beginner level.
44. Arabic Keyboard — Aside from helping you with tricky Arabic script and the grammar that goes along with it, this site includes super helpful, practical information about the Arabic language, its dialects and where and how to use them.
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45. Arabic Reading Course — For absolute beginners, this letter-by-letter course is the perfect jump-start to get you reading and writing in Arabic.
Portuguese
46. Learn Portuguese with Rafa — Beyond introductory lessons on things like counting and ordering food, Rafa maintains a sidebar full of links on everything from traditional recipes to doing business in Portuguese.
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47. Tod Munro Pod — An exciting So Paulo-based podcast with basic tips in English and the rest in beautiful Brazilian Portuguese.
48. Practice Portuguese — For those looking to learn Iberian (European) Portuguese, get started with the free Practice Portuguese Podcast.
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49. Really Learn Portuguese — Podcasts and flashcards for beginner, intermediate and advanced learners, all brought to you by two friendly Brazilians.
These 49 websites prove it: Money is no object for determined language learners.
The first step is just clicking one of the links above.
Haven’t started learning any one language yet? Choose your language and head over to Omnipotent or BBC Languages to read up on it. Build up your basic vocab with a flashcard app or your favorite language game, immerse your brain in quality video content and connect with natives around the world via Skype or your social network of choice.
With money concerns firmly out of the picture, now all you need to do is make the time for language learning, and you’re all set to start heading towards fluency!