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.

Last night while binge drinking and reading through chains of articles on Wikipedia I had the best idea. I realized there are sections of the site I never explored. So I immediately closed List of animals with fraudulent diplomas and the n+1 articles on Permian fauna I had open, then went to the Wikipedia reference desk. Now the reference desk is pretty cool and it's one more reason why Wikipedia is an internet gem, anyone can ask a question about any topic, and anyone can answer.
Most of the questions in the mathematics section are what one would expect; people asking about things in homework assignments they don't understand, people asking for help deciphering arcane mathematics articles on the site, and people questioning their own understanding of things. Every now and then you'd find people asking why some proof of [insert famous conjecture here] published in [insert obscure journal here] wasn't cited on Wikipedia and why it wasn't accepted as a proof by the mathematical community. But I found something a little more spicy than that, I found a user that claims to have proven the Riemann hypothesis, the Collatz conjecture, the Goldbach conjecture, and created an elementary proof of Fermat's last theorem.
Is the following proof of Riemann Hypothesis correct?

Riemann Hypothesis states that the real part of all non-trivial zeros of the Riemann zeta function, or ζ(s) = Σ(k=1 to ∞) 1/k^s = 0, equals one-half. For the non-trivial zero, s, a complex number, we have s = a + bi where Re(s)= a = 1/2.

If I had $1 for every "proof" of the Riemann hypothesis I've seen where the writer starts by trying to find s such that 1+1/2^s+1/3^s+1/4^s+⋯=0 I'd probably be lounging on a beach in the Caribbean right now. The problem here is that the Dirichlet series for ζ(s) only converges when the real part of s is greater than 1, and this series is never 0 where it converges. So analyzing only this series will not be helpful.

Fact III: The sum of the complex conjugate pairs of non-trivial zeros, s = a + bi and s' = c + di where ζ(s) = Σ(k=1 to ∞) 1/k^s = 0 and ζ(s') = Σ(k=1 to ∞) 1/k^s' = 0, of the Riemann zeta function equals one according to the Fundamental Theorem of Arithmetic and the Harmonic Series (H):(Note: Euler and others have proven that there exists an infinite set of primes in H. And that the divergence of H is a key reason for that result.)

If s' is the complex conjugate of s=a+bi then why not just write s'=a-bi instead of s'=c+di? Or why not write s=σ+it instead, as this is a fairly standard way to write a non-trivial zero in literature on the topic? Sure, this isn't bad math per say, but it's pretty bad notation. Also, s+s'=1 always only if the Riemann hypothesis is true and this would have nothing to do with the fundamental theorem of arithmetic or the harmonic series! They have already assumed the Riemann hypothesis is true before they've done anything!
The bit where they talk about primes in the harmonic series is somewhat odd. It looks like they think the divergence of the harmonic series implies the divergence of the sum of reciprocal primes (which it doesn't, the implication is the other way around) and they seem to treat the harmonic series like a set.
After this our writer slaps his four facts together in some convoluted way that I can't decipher and declares victory.

Therefore, according to Facts I, II, III, and IV, we have:
k^(1/2) ≤ k^a ≤ k, k^(1/2) ≤ k^c ≤ k, and a + c = 1.
Hence, k^a = k^c = k^(1/2) which implies a = c = 1/2. Riemann Hypothesis is true! Riemann was right!

Then they make some final notes where they try to rewrite the harmonic series using some underexplained ideas about prime gaps and says

There are infinitely many more positive integers than there are prime numbers, or prime numbers have a zero density relative to the positive integers, and prime numbers generate the positive even integers efficiently so that gaps between two consecutive prime numbers increase without bound.

which is true in the sense of natural density for sure, so why not just say that? Using the phrase "infinitely many more" makes it sound like cardinality. Saying "so that gaps between two consecutive prime numbers increase without bound" makes it look like they're saying all prime gaps become larger as we increase through the sequence of primes, this isn't necessarily true although it's statistically something we should expect. The existence of arbitrarily large prime gaps is true though and isn't hard to prove, but they did not prove it in any of what was written and it's not the same as what they said.
Is the following proof of Goldbach Conjecture correct?

Keywords: π(*):= Odd Prime Counting Function and Fundamental Theorem of Arithmetic (FTA) Goldbach conjecture states every positive even integer is the sum of two prime numbers. (We count one as prime in the sense of additive number theory outside of the FTA.)

What? The parenthetical here is so strange. Additive number theorists don’t take 1 to be prime and they have no reason to do so.
The writer then tries to make a probabilistic argument from a system of linear equations defined over a set of odd primes less than an even number e>2,

Therefore, e ≠ p + q over S, (p,q є S) , implies the following system of equations over S, 1 = e - n1 * q1, 3 = e - n2 * q2, ..., pk = e - nk * qk, according to the Fundamental Theorem of Arithmetic where 1 < qj ≤ (nj * qj)^.5 ≤ nj for 1 ≤ j ≤ k where pj, qj є S and nj is a positive integer. Note: If qj = 1, then nj є S, or nj is an odd prime less than e.

and this last sentence is what they try to base their argument on. They attempt argue that for every even number e>2, the probability that an equation of the form p=e-1q doesn't show up goes to 0. Which would mean that it's likely that e=p+q.
Even if their probabilistic manipulations made sense this obviously still wouldn't prove the Goldbach conjecture. Showing that it's "probably true" isn't a proof that it's true. As if to attest to the writer's own doubt,

In addition, empirical evidence has confirmed the validity of the conjecture for all positive even integers up to at least an order of 10^18. Therefore, we conclude the conjecture is true.

If you proved it, why do you need to test it empirically?
Is the following elementary proof of Fermat's Last Theorem correct?

x^n+y^n=z^n for n > 2. I begin the proof by assuming there exists an integral (positive integer) solution to equation one for some n > 2. Equation one becomes with some algebraic manipulation, 2. x^n=z^n-y^n = (z^(n/2)+y^(n/2))*(z^(n/2)-y^(n/2)).

Okay.

Now that I have factored the right side of equation two, Fermat, the great French mathematician and respectable jurist, made I believe the next logical and crucial step.

Any evidence that Fermat did what you're about to do?

He factored the left side as well, x^n, with the help of an extra real variable, Ɛ, such that 0 < Ɛ < n . I have the following equation, x^n = x^(n/2+Ɛ/2)* x^(n/2-Ɛ/2) = (z^(n/2)+y^(n/2))*(z^(n/2)-y^(n/2) ). This equation implies x^(n/2+Ɛ/2)= z^(n/2)+y^(n/2) and x^(n/2-Ɛ/2) = z^(n/2)-y^(n/2).

Ah yes, if ab=cd then a=c and b=d. Everyone knows that! Eventually, after a few more lines, the author concludes

However, (1/4)^(1/n) is not a rational number, a ratio of two whole numbers, for n > 2. This implies the right side of equation five is not a positive integer. This contradicts my assumption that y is a positive integer. Thus, Fermat’s Last Theorem is true, and Fermat was right!

It's so easy now, Fermat's last theorem obviously just reduces to knowing 1/4^1/n is irrational for n>2. How did nobody see this before?
Is the following proof of the Collatz Conjecture correct?

Proof of the Collatz Conjecture: Suppose there exists a sequence, S’={n0, n1, n2, …} that does not converge to one, or nk ≠ 1 or nsub(k-r) ≠ 2^µ over S’ for all kϵ ℕ where r

It's obvious that hailstone sequences don't converge, so the “does not converge to one” bit is irrelevant. Here the fundamental error is same error as in their attempted proof of the Goldbach conjecture; they think making a probabilistic argument in favor of the conjecture being true is the same thing as proving it. Lots of other basic little details are also wrong, but I'll just look at one:

From a given positive integer, n, we obtain the maximum positive odd integer, n0 > 7, by repeated division of n by 2.

What is n here, the starting number? What if n is odd? We'd have to 3n+1 it first, not divide by 2. Even if n is even the first odd number we hit once we finish dividing by 2 is not the maximum odd number in its hailstone sequence, this is easy to see starting with n=22.

[https://pubchem.ncbi.nlm.nih.gov/periodic-table/png/Periodic_Table_of_Elements_w_Chemical_Group_Block_PubChem.png ] or [https://ptable.com/#Properties ]
In the last 14 posts, I have attempted to present the main points/useful information from a whole academic year of general chemistry. A significant fraction of the material taught in general chemistry is obsolete, but I am also skipping over any of the information that is actually beneficial to have somewhat memorized, all of the math, etc. Generally speaking, people don’t seem to have much trouble retaining information that is useful to them, so unless you’re having to pass a series of exams I would not worry about any of the details if you don’t want to. Maintaining a degree of rigor and intellectual honesty is important, but at the same time knowing a theory should enhance your understanding of the real world instead of detracting from it.
In any case, we have atomic nuclei with positively charged protons and non-charged neutrons surrounded by somewhat amorphous clouds of negatively charged electron density generated by a discrete number of negatively charged electrons moving around at high speed. How nuclei, orbitals, and electrons interact is chemistry, and given the complexity in chemical reactions that is evident (particularly in biology) it should come as no surprise that the behavior of electrons, elements, and molecules is also extremely complex. We as a species have spent many centuries of unified time and uncountable person-millennia of effort grappling with aspects of the complexity of chemical behavior, before discovering relatively recently that everything is derived from quantum mechanics and none of the simple mathematical models are particularly valid. The discovery of quantum mechanics started in the early 1900s to the 1920s or so in the physics community and has led to a progressive series of major improvements in the way we think about the world that is still underway. The information gained has led to our disastrous exploration of nuclear fission in heavy elements but also to the development of much more potent instrumentation, semiconductors, computers, and a better, if not necessarily more comforting, understanding of the universe that we live in.
Looking at chemistry specifically, our goal as a species needs to be to do as little chemistry as possible while still ensuring our survival. Where chemical reactions are unavoidable, we need to take care to ensure that the resulting waste is as non-toxic, biodegradable, and/or easily denaturable as possible. Simple molecules such as carbon dioxide can cause problems when emitted in bulk, and more complex molecules tend to be nastier in much lower quantities and concentrations (eg polychlorinated biphenyls/PCBs). As creatures with cellular machinery that is mostly made of organic molecules, we are going to be most interested in organic reactions despite our historical inability to make much sense of the complicated electronics and molecular orbitals of organic reactions. Unfortunately, this means that we will not be able to skip as many of the details, and if I want to try for complete coverage I would expect to see a few tens of posts. The main difference between general and organic chemistry is that a significant fraction (possibly even most) of the general chemistry material is obsolete and/or irrelevant, while the majority of organic chemistry material is both important and relevant. So this may take a while, and I’m going to wish that I still had access to the ChemDoodle software that is set up for organic structures. On ubuntu linux, the GChemPaint program seems similar and is free, and I guess that I’m about to find out how well that it works.
I will do my best to relate concepts back to the mental picture of how chemical compounds interact that you are hopefully building up as I introduce them, but as always things are usually going to be messy. The list of high level topics in organic chemistry as defined by my undergraduate study guide is as follows: structure, bonding, intermolecular forces of organic molecules, acids and bases in organic reactions, nomenclature, isomers, principles of kinetics and energy in organic reactions, preparation and reactions of (alkenes, alkynes, aldehydes, ketones, alcohols, sulfides, carboxylic acids, amines, aromatic compounds), organic reaction mechanisms, principles of conjugation and aromaticity, and spectroscopy. I have not yet decided if this is the order in which I would like to present these concepts, but hopefully you can see that this is a large amount of material. As a final note, organic chemistry is mostly the chemistry of hydrogen, carbon, nitrogen, and oxygen with trace quantities of several other elements participating at times. Organic molecules are interesting both because of the wide range of properties and behaviors that they exhibit and also because of our desire to understand our biology, and we are studying mainly the chemistry of the 1s, 2s, and 2p valence orbitals in small atoms.

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

Data science is one of the fastest-growing technologies in the world. There are lots of job opportunities in the data science field. That is the reason the majority of students are getting enrolled in data science. Most of the students think that data science is all about computer science, but it is not true. It is a combination of statistics, math, and computer science.
Therefore, whenever students want to enroll in data science, they should have a basic knowledge of math, computer science, and statistics. But they still don’t know what math to learn for data science. Even some of the students have the question in their mind is how much math for data science and how important is math for data science. Apart from that, the students even ask what math is required for data science. Here in this blog, we will talk about math for data science. Likewise, statistics for data science, mathematics for data science is also crucial.
If you talk about basic math for data science, then you should know the basic function, variables, and equation of mathematics i.e., binomial theorem and many more. Apart from that, you should also have the basic knowledge of logarithm, exponential, polynomial function, rations numbers real numbers, complex numbers, series sums, and inequalities. Let’s have a look at the basic math required for data science:-

Essential math for data science

Table of Contents

Calculus

Calculus is one of the crucial topics of math needed for data science. Most of the students find it difficult for them to relearn calculus. Most of the data science elements depend on calculus. But as we know that data science is not pure mathematics. Therefore you need not learn everything about calculus. But it would be best if you learn the basic principles of calculus and how the principle can affect you, models.
Apart from calculus, you should also have good command over basic geometry, theorems, and trigonometric identities. Here are some calculus topics that you should know for data science, functions of a single variable, limit, continuity, differentiable, mean value theorem, indeterminate forms, maxima, minima, product and chain rule infinite series, integration concepts, beta and gamma functions, partial derivatives, limit, continuity, partial differentiation equation.

Linear algebra

Linear algebra is a crucial part of computer science, and it also plays the same part in data science. In data science, the computer uses linear algebra to perform the given calculation easily. It is also helpful when you need to perform the Principal Component Analysis. That is used to reduce the dimensionality of the data. Apart from that, it is best for neural networks. Data scientist use it to perform the representation and processing of the neural networks. Most of the models in data science are implemented with the help of linear algebra.
If you know the basic principle of linear algebra, it can be quite easy for you to apply transformation to the matrices in the data set’s existing model. The linear algebra topic you should know for data science is scalar multiplication, linear transformation, transpose, conjugate, rank, determinant, inner and outer products, matrix multiplication rule, matrix inverse, square matrix, identity matrix, triangular matrix, unit vectors, symmetric matrix, unitary matrices, matrix factorization concepts, vector space, linear least square, eigenvalues, eigenvectors, diagonalization, singular value decomposition.

Probability and statistics

Probability and Statistics work as the backbone of data science. If you want to learn data science, then you should have the basic knowledge of probability and statistics. Most of the students find statistics the toughest subject for them. But for data science, you need not have a strong command over statistics—all you need to cover the basics of statistics and probability for data science. The statistics concepts of data science are not super hard for students. Even if you can solve the basic problems in statistics, you can easily learn statistics for data science.
You should clear your basic concepts of probability and statistics before starting your data science learning journey. It is also the best answer for how to learn math for data science. The probability and statistics concepts you should know are data summaries and descriptive statistics, central tendency, variance, correlation, basic probability, probability calculus, Bayes’ theorem, conditional probability, chi-square, uniform probability distributions, binomial probability distribution, t distributions, central limit theory, sampling, error, random number generator, Hypothesis testing, confidence intervals, t-test, ANOVA, linear regression and regularization.

More Math in Data Science

Discrete math

The discrete math needed for data science. Most of the students think that is why it is needed for data science. The major reason for the use of discrete math is dealing with continuous values. With the help of discrete math, we can deal with any possible set of data values and the necessary degree of precision. The math in computers is based on discrete mathematics. The reason is that computers work in machine language.
Therefore the bits are used to present every value on the computer. Data science uses a large number of discrete math concepts that are used to solve the problems. Some of the discrete math topic that you should know for data science sets, power sets, subsets, counting functions, combinatorics, countability, basic proof techniques, induction, inductive, deductive, propositional logic, stacks, queue, graphs, array, hash tables.

Graph theory

The graphs are crucial for data science. There are a large number of problems in graphs that can be solved by graph theory. The data scientist used graph theory to create a fraud detection system with the help of data science. Graph theory is also helpful in data visualization in data science. We use different types of graphs in data science to visualize the data. Every graph is used to represent different kinds of data. We can use the same graphs again and again to represent the different data sets. Therefore the proper graph theory will help you get a good command over data visualization in data science. In the graph theory, you should know about the graphing and plotting, Cartesian and polar coordinates, and conic sections.

Information theory

Information theory is also widely used in math for data science. You should have the basic knowledge of information theory for data science. It is quite helpful when you want to build a decision tree. And want to maximize the information that you have retained from the Principal Component Analysis. It is best for a large number of optimizations in data science modes.
The optimization in the data science model is quite helpful in saving plenty of data space in the data science warehouse. Because sometimes the data science model contains the unwanted values in the data warehouse that put the extra burden on the system. If you have the proper knowledge of information theory, then you can easily optimize the data, science models.

Conclusion

It might be clear in your mind what math to learn for data science. In this blog, we have discussed the essential math for data science. We have categorized of math concepts for you. So that it can be easy for you to know how much math is required for data science. If you want to learn math for data science, then clear your basic concepts in math. It will help you to master most of the data science concepts. You should practice each concept manually or with the help of your computer. In the end, I would like to say that, start practicing these math topics to start learning data science.

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.

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

Exponents:
a¹=a
a⁰=1
a^-m= 1/a^m
a^m/aⁿ= a^m-n
a^m * aⁿ = a^m+n
(a^m)ⁿ= a^m\n)
a^mb^m=(ab)^m
a^m/b^m= (a/b)^m
a^m/n= ⁿ√a^m
(a/b)^-n = (b/a)⁺ⁿ
√a+b= √ a + √ b
x^y=x*x^y-1
(8^4/3)= (8^1/3)⁴=16
√x= x^1/2
√4 * √6 = √24
√x = ± x
³√8 =2
9 = 3²
a^m-aⁿ doesn't equal a^m-n
When there is no number > √ 9¹⁰ then ²√
_______________________________________________________________________________________________________________________________
Algebra:
y=mx+b
m=slope
b=y-intercept
m= y2-y1/x2-x1
y-intercept=x=0
x-intercept=y=0
Slope:0:horizontal
Slope:vertical:undefined
Perpendicular opposite resicprocal (m1*m2)=-1, same y-intercept, slope ex: 2/3 would be -3/2
Parallel: No solution, (m1=m2), b1 doesn't equal b2
One solution: equal to each other, intersect, m1 doesn't equal m2.
Infinite solutions: m1=m2, b1=b2
Ax+By= C, m= -A/+B
By=-Ax+C
Y=-A/B*X+C/B
Don't forget PEMDAS
(x+y)² DOESN'T MEAN x²+y²: FOIL (x+y)(x+y)
Multiply by a number in Elimination to make them equal and then subtract
System of equations: put in slope-intercept form
^ No solution: A1/A2=B1/B2 doesn't equal C1/C2
Tip (only works with 1/x): Add denominator to get numerator and multiply denominator to get the denominator. Ex: 1/3+1/4 = 7/12
I=P*R*T/100
To save time: make equations and solve
Absolute Value= 2 solutions ± (if you divide by a negative: flip the sign)
Inequality: 0>: above line, <0: below line, =equation: line itself
If it asks for the sum of all values: -B/A
Use the Quad formula when: can't factor(get zeros), want to know the # of solutions
b²-4ac < 0 = no solution
b²-4ac > 0 = 2 solutions
b²-4ac = 0 = 1 solution
If it asks for: rational fractions, zeros, x-intercepts, roots: FACTOR
If you are stuck: Factor, divide, or multiply
Sqrt Property: ax²=c or (ax+b³)²= C, x²=c=x= ± C, c>0= 2 solutions, c=0: 1 solution, c<0: two solutions
linear equation: ax+b = cx+d, a doesn't equal c
Paraobla: y=a(x-m)(x-n), m&n= solutions, roots, zeros, x-intercepts
a= direction/stretch, big "a": skinner, smaller "a": fatter, a>0: U, a<0: upside down U, c= y-intercept
y=ax²+bx+c, y=a(h-h)²+k, (h,k)= max,min, vertex, h=-b/2a
ROUND RIGHT.
Multiply fractions to make it simpler.
D= R*T
y=ab^x or y=a(1+r)^x, a=intial value, b=growth/decay rate, 1+r= growth, 1-r: decay.
9-x= -1(x-9)
x+9/9+x=1
tangent=parallel
Outliers affect mean not median.
special mean= all known numbers/ total +unknown = desired mean
Data: The answer usually doesn't contain "All, never,etc"
Correlation: Pattern
Standard Deviation: How spread the data is
Cube: V= s3, SA = 6s2
Sum of angles in polygons: (n-2)= 180, n = # of sides
(x-h)2+(y-k)2 = r2, center: (h,k)
Complex numbers:
i, -1, -i, 1, i, then cycle, i*-i = 1, dividing: conjugate of a-bi = a+bi, You also FOIL
Length of Arc: n degree / 360 degree * 2 * pi * r
Area of sector: Ø/360 degrees * pi* r2
Right triangles ONLY:
a2+b2=hypotenuse (longest side) (c) 2
SOH = S= Opposite/Adjacent
CAH = C = Adjacent/Hypotenuse
TOA= T= Opposite/ Adjacent
Vertical/Aletrnate interioexteriorangles are EQUAL.
1 degree = pi/180, Deg to Rad: Pi*rad=180, Rad to Deg: *180/Pi
I couldn't write some Geometry rules, how to complete a square, and how to divide a polynomial.
Am I missing any other thing my August 24th squad, (because I am so stupid I still get 630 on the math section after TONS of practice and my goal is an 800)?? LET ME KNOW IF THIS HELPED!

It's no big secret that the Stormlight Archive has a lot more mathematical depth to it than most fantasy stories. On an explicit level, the planet has very precise physical parameters required to make the world "work". On a social level, symmetries are considered holy, mathematics is put on a pedestal for being integral to the work of Stormwardens, and even laypeople show appreciation for the abstract beauty of math. On a higher level, the highstorms are mathematically predictable to a far greater extent than real weather patterns are, the Dawncities are shaped after cymatic patterns, and the world map of Roshar itself seems to be based on a fractal structure called a Julia set!
Now, Brandon is a master worldbuilder. He undoubtedly knows math has limited mainstream appeal, yet he decided to incorporate some really quite heavy mathematical concepts into his story. How odd! Why would he do such a thing? Is he simply trying to pander to the nerdiest subset of his fanbase? Have the Evil Librarians joined with the Math Teachers to get you to read discreet math? Is the Stormlight Archive going to teach you math?
Well, probably not. But maybe I will! In this post, I aim to provide an overview of the math in the Stormlight Archive. I also consider what the math might mean, and why I think Brandon decided to write so much math into a fantasy story. If you have no interest in the mathematics, you can skip ahead to the last two sections where I try to answer these meta-questions.

Symmetries and invariants

Let's start with symmetries - they are considered holy by the Vorin church. One might brush this off as an insignificant and quirky part of the worldbuilding, but what if the reasoning goes deeper than that? Why do they consider symmetry to be holy?
To begin to answer this, let us first look at what symmetry actually means. When a mathematician speaks of symmetry, they mean something slightly more abstract than you might expect; A symmetry is a reversible operation that leaves something invariant - and "invariant" is just a fancy word for "unchanged".
We sometimes phrase it by saying that something is "invariant under symmetry" or "an invariant of the symmetry" to mean that the symmetry operation leaves the thing unchanged. For instance, some things look the same when viewed in a mirror, and we can call these "the invariants of mirroring". Each such thing can be described as being "mirror-symmetric", or as being "invariant under mirroring".
To really get a feel for the terminology, let us take a look at some more examples.
Examples:

A clock display is symmetric under the symmetry operation of "advancing/reversing time by 24 hours". This kind of symmetry is so important that it has its own name: "periodic". It means that the pattern repeats at regular intervals.
A circle is symmetric under rotation by any number of degrees, so we just say it's "rotationally symmetric".
A ketek is symmetric under "mirroring word order with conjugation and typesetting changes allowed". The added conditions can be generalized quite easily. For instance, each third of the First Ideal can be considered symmetric under "mirroring both word order and the meaning of nouns".
Einstein's theory of relativity is all about the laws of physics that are invariant under the change of reference frame. E = mc² is one such invariant: the mass-energy of something doesn't change just because you started moving relative to it.

Now, everything has some symmetry, but certain symmetries are considered more beautiful than others. Einstein's theory of relativity is often praised for its many symmetries, whereas the trivial "leaving-it-alone-symmetry" is not particularly interesting. To most humans, there is something aesthetically appealing about symmetry, and this likely ties to the evolutionary advantage of pattern recognition. Regular patterns feel safe, and we instinctively pay attention to any deviation from the pattern. We generally like wearing similar socks and shoes, for instance.
Could this be the reason why Brandon chose to make symmetry holy in the dominant religion of the Stormlight Archive? Simply aesthetics? Let us investigate by looking at some of the more esoteric symmetries he shows off in the books.

Cymatics

In the Way of Kings, the ardent Kabsal demonstrates cymatics by playing musical notes that make the sand on a plate be reshaped into symmetric patterns. One might think this is a magical fantasy process, but no - it's 100% real! I recommend taking a look at this music video by Nigel Stanford, which shows off even more cymatic patterns than Kabsal did.
What's happening here is that for certain frequencies, the soundwave through the plate has resonance nodes where the plate doesn't move much. Where the plate doesn't move, the sand can lie undisturbed. On the rest of the plate, the sand will instead get tossed about by the vibration of the soundwave. Thus, the distribution of sand will gradually adapt towards the stable configuration, meaning more and more sand will end up near the resonance nodes. After all, once a grain of sand reaches the node, it's probably going to stay there. Ironically for Kabsal, who was trying to use this to prove the existence of the Almighty, this is often used to highlight how evolution and natural selection works; if one state endures its environment better than other states, then the population will tend towards that state as the other states "die out". Similarly, the pattern of the sand gradually ends up looking like the pattern of resonance nodes.
In this dynamic environment, the symmetry operation of "progressing time" leaves the nodes invariant. That the cymatic pattern of sand also displays mirror symmetry is simply due to the physical system being a symmetric metal plate; the sand isn't seeking the symmetry itself, but rather the resonance nodes that are symmetrically distributed due to being on a symmetric plate. The Dawncities being shaped like cymatic patterns indicates that a similar process based on frequencies is likely to blame for the natural rock formations on Roshar, but one cannot from this conclude that it's "intentional" - at least not from an in-world scientific perspective. We, as readers, happen to know that Brandon made this world, and so the assumption of intent is far more valid. But why would the Almighty Brandon decide to make the Dawncities look like cymatic patterns?
At any rate, this ties symmetry to the worldbuilding - it's somehow related to the nature of Roshar, and Kabsal appeals to the aesthetics of symmetry as an argument for the Almighty's intentional design. It would be a major digression to go into the many fan theories regarding this in-world connection, but for this post we're mostly interested in looking at Brandon's thematic intent. He emphasizes the aesthetics through Kabsal, but contextualizes it with Jasnah's doubts. To paraphrase Wit, Brandon does not tell us what to think regarding this coincidental symmetry, but instead provides us with questions to think upon by providing different interpretations. To me, this suggests that Brandon's message is "symmetry is worthy of philosophical consideration." Some people will appreciate the symmetry as a supernatural and magical thing, whereas others will want to look behind the veil for a natural explanation.
However, these views are not mutually exclusive. Understanding does not have to dispel the magic and wonder, it may instead lead to deeper forms of magic. Understanding cymatics does not make it less amazing, and quantum teleportation is no less magical than fantasy teleportation simply due to being real. In fact, there are some deep mathematical patterns present in the Stormlight archive that the characters are unlikely to discover until they get access to graphing computers - it's time to look at fractals!

Fractals, approximations, and ideals.

A fractal structure has a very strange symmetry - it somehow contains itself in a suitable technical sense. Usually this amounts to containing copies of itself when you zoom in, which is called "unfolding symmetry". Let us begin with a moderately simple example of a fractal.
Imagine that you start out with the Triforce symbol from the Legend of Zelda. We define an operation where we replace each of the solid gold triangles with a smaller triforce. Then we do it again and again. Each time, we end up with a slightly different figure with more and more intricacies, so the triforce is clearly not invariant under this operation. Nor are any of the shapes we end up with after using the operation on the triforce a finite number of times, though it changes less and less each time. It's almost as though it's slowly approximating something. Is there a shape that this operation is symmetric for?
Yes, there is! It's called the Sierpinski Triangle, and is often nicknamed the "Triforce fractal". You can think of this as the shape you end up with "after" having done the above operation infinitely many times to the triforce - the thing that is approximated by the procedure described above! This one has somehow replaced all of its solid gold triangles with copies of itself, so it doesn't change when we apply the operation, nor does it change when we reverse the operation by zooming in on a corner.
Other famous fractals include the Koch curve, Peano curve, and the Mandelbrot set. Some are harder to visualize than others, because the notion of fractal can be mathematically extended into dimensions we can't really visualize.
It's the Mandelbrot set that is closest related to the Julia set of Roshar. Their symmetries are vaguer and their invariance is of a much more technical nature, one determined by certain multi-dimensional mathematical functions. These are often animated in terms of zooming in on a colored picture, with the color representing a measure of how much the symmetry is broken there. It's pretty, but unless you knew the mathematical foundations it can be difficult spot any obvious symmetries. It doesn't come as a big surprise that the Rosharans haven't spotted the pattern in their map - this particular Julia set is not something you can easily draw by hand, at least not without help from the spren.
What I want to emphasize is that this technical form of symmetry is not emphasized by Brandon, and was left as an exercise for the particularly nerdy readers to spot. That he would pick such an obscure piece of mathematics as the foundation for the world map suggests that aesthetics is not his primary reason for emphasizing symmetry in general - it should be a deeper theme, one that is fundamental to the physical nature of Roshar. In fact, this connection is made tighter by the Cryptics, who have fractal-like symbol-heads and are described the following way in Oathbringer:

Syl: “We honorspren mimic Honor himself. You Cryptics mimic … weird stuff?” Pattern: “The fundamental underlying mathematics by which natural phenomena occur. Mmm. Truths that explain the fabric of existence.”

This suggests, in my eyes, that the fractals are fundamental to the natural phenomena of Roshar, and the Julia set was almost definitely not just chosen for its aesthetics. While the Honorspren mimic the moral ideals of Honor, the Cryptics mimic the part of Honor that's concerned with rules and the platonic ideals of existence.
Interestingly, some of the theory of fractals can be phrased in the language of mathematical ideals. I won't go into too much technicality regarding these ideals (there are multiple notions), but in essence they are constructions that are invariant, but also inside of a larger invariant - much like the Sierpinski triangle is an invariant inside of... well, a filled-in triangle, which is also invariant if you replace its filled-in part with itself. To make a somewhat imprecise analogy, the Sierpinski triangle is like an ideal of certain symmetries on the full triangle.
The name "ideal" is a historical artifact in mathematics, and is not explicitly connected to moral ideals. Then again... what if Brandon, in his artistic vision, decided that the different notions of "ideal" should be thematically connected? Would the use of mathematical ideals mirror the philosophical ideals presented in the book? Is there such a thematic symmetry of ideals? If so, what could it mean when a symmetry or ideal is broken?

Symmetry breaking

On a fundamental level, physics is mostly about symmetries and the things that are invariant under them - or in layman's terms, patterns. As mentioned previously, the mass-energy E = mc² is an example of such an invariant, which doesn't change under the symmetry operation of changing reference frames. Most fields of physics focus on such symmetries, and incomplete models are often emphasized by some situation where the symmetry no longer holds. The process by which the symmetry of a simple model can be broken by taking into account the bigger picture is simply called "symmetry breaking", and it's very important in physics. However, it's also mathematically very complicated, so while the following example isn't exactly wrong, it's going to be slightly mathematically imprecise.
Example:
Consider a pen perfectly balanced by its tip on a table. If we model it as a perfect cone balanced in the center of an infinite plane, the system is symmetric with respect to rotation. A neat and aesthetically pleasing model of reality!
However, we know that this configuration is unstable, so realistically it's going to fall over. But our mathematical model should be able to predict not only that it will fall over, but also which direction it's going to fall. Without taking into account the bigger picture - the movement of the air, the vibrations in the table, and so on - we can't make such a prediction. In order to fix the symmetry being broken, physicists introduce an extra interaction or particle that makes it so that the system can be predicted by knowing - say - how the wind blows.
To draw an analogy: It's like solving a sudoku by using that you know there should only be one unique solution, thereby inferring how to eliminate ambiguity!
The famous Higgs boson - the one that got nicknamed "the God particle" for political reasons - was actually theorized this way. Experimentalists at CERN only found the particle several decades later.
In the Stormlight Archive, there are many symmetries that are imperfect, or slightly broken. A ketek has some leeway with conjugation and typesetting. A name shouldn't be perfectly symmetric, and the letter "h" lets you break the symmetry more. The Oathgate at the Shattered Plains violates the otherwise symmetric pattern. There are a lot of almost-Herald-analogues such as the Unmade, the Ten Fools, the Alethi Highprinces, and the naming conventions of the Dawncities.
If we invoke the assumption that Brandon considers mathematical ideals and philosophical ideals to be analogous, how can we interpret these failed symmetries? Are they coincidental, or are they extensions of the themes of the book? Could it be that it's not symmetry that is divine, but rather symmetry breaking? Are the splinters of Honor the Rosharan analogues of the "God particle"?
Few of these questions can be answered in any definitive manner. However, I personally find these to be very appealing ideas - after all, I think it's an aesthetically pleasing thematic pattern.

The artistic interpretation

After all is said and done, why did Brandon bother to incorporate all of this math in the Stormlight Archive?
It seems unlikely to be for marketing reasons, and he is neither mathematician nor physicist - he takes advice from the professionals in his writing group on topics like these. This leads me to believe that this isn't a case of the mathematics being a physical necessity for the story to work, but rather that the world is built in a way to make the artistic themes agree with an educated layman's understanding of physics and math. So, if the world really had the kinds of intent that we often describe when we anthropomorphize the forces of nature (in other words, spren), what intents could be read out of this form of physics?
Of course, I cannot tell you for sure what Brandon's intent really was when he decided to emphasize symmetries. I can only tell you what artistic themes I get out of it as a mathematician with a physics background, with the caveat that I am biased towards making the interpretation that appeals the most to my own preferences. Perhaps you will agree that my interpretation enriches the story, or perhaps you will disagree. Truthfully, I doubt I got everything "right" regarding Brandon's intent, but I do believe at least some of his intent is fuzzily reflected in my interpretation.

Symmetric names are holy, but the perfect name should slightly break the symmetry.

In this, I see the theme that the perfect and unchanging ideals of Honor are to be idealized, but our imperfections are what make us human. For example, Shallan explains how humans can pretend that a word is symmetric even when it is not, much to Pattern's chagrin. The honorspren Syl acts as though there is only one universal and perpetual notion of Honor, until Kaladin challenges her with very difficult and human questions about perspectives, thereby inducing change and improvement to Syl's world view. Even the Stormfather, the shadow of Honor himself, grows a conscience about the innocents killed when he does his duty.
On a meta-level, I also see the imperfection-induced-improvement as the influence of Cultivation on Honor's perfect ideals, much like a physical theory can be improved by symmetry breaking.

The moons of Roshar are in a slightly unstable orbit.

A stable orbit would be too perfect for the themes of this book. Instead of destabilizing the moon's orbit by way of tidal forces as is happening to the Earth's moon, these moons are in orbits that are unstable on an astronomical scale, but stable on a practical scale. Since the three moons very likely connect to the three Shards of the Rosharan system, this would likely be part of a theme. At first glance, one might think that this is something Ruin and Preservation would be up to, but I suspect it's again an interplay between Honor and Cultivation. Things must be allowed to change, but do so close to the ideal.

The predictive model for highstorms is very good, but not perfect.

The periodic symmetry is slightly broken, like Honor was broken. The Stormfather may be the Cognitive Shadow of Honor, but if Honor represents symmetry then it's thematically appropriate that his shadow is only almost-symmetric.

Odium's forces have fuzzy analogues to Honor's: An Everstorm instead of the Highstorm. 9 Unmade instead of 10 Heralds, Voidbinding instead of Surgebinding. Different ways to form spren bonds. Carapace instead of Shardplate. Moash instead of Kaladin.

Due to Honor's influence, a symmetry could be expected to form, some kind of equilibrium. However, in each case, it's more of a fuzzy analogy than a perfect match. I see this as reinforcing the theme of reality being too complicated for neat and symmetric models to capture all of the nuance. Especially Kaladin's arc emphasizes that the split between "us" and "them" isn't a clean cut, despite that being an appealing world view.

The cycle of desolations is broken.

The model of previous times cannot predict what will happen. This is revolutionary to Rosharan historians like Jasnah, who seem to assume that history repeating itself is a universal pattern. She might be a heretic, but I doubt she avoided a cultural appreciation for symmetry that would make the cyclic world-view appealing to her. No, Jasnah. Symmetry has been broken, the old model no longer fits. Not only have the old powers returned - new forces also stir.

The Oathgate in the Shattered Plains is discovered by a scout pointing out that Shallan's map is wrong.

Shallan's map was drawn based on assumptions of symmetry, and it was the symmetry breaking that let her intuit the location of the Oathgate. Much like the discovery of the Higgs boson, a theoretical model with unrealistic symmetries was vindicated by symmetry breaking.

Roshar is shaped as an approximate Julia Set.

Reality can only approximate a true fractal, but I interpret this as an analogy saying that it's still worth approximating an ideal even if the goal cannot ever be attained. I personally suspect that the disagreements between the old maps and the new aren't due to the "modern cartography techniques" as Shallan assumes (they had Windrunners back in the days, their maps should be way better back then), but instead due to Roshar being in the process of changing shape to gradually approximate a fractal structure. It'll never reach perfection in finite time, but that's okay, because the world is being Cultivated to approximate Honor's ideals. Journey before destination.

The Cryptic spren that represent mathematical physics have fractal-like symbol heads.

The heads of the Cryptics emphasize that complex mathematical rules and patterns are fundamental to the nature of Roshar. As any simplified mathematical model of reality will contain both truths and lies-by-oversimplification, I believe the Cryptics grant access to the Surge of Transformation to make the physical world adapt to match a platonic ideal. This emulates the way Honor moulded Roshar by rigid rules - it worked, even though his rules were too flawless for reality.

Concluding remarks

In conclusion, my interpretation is that Honor provided a supposedly-perfect mould: the ideal and the symmetric. Mathematically consistent rules that hypothetically could bring Honor to as many as possible. Roshar's innate investiture adapted the physical reality to approximate spiritual ideals, but the world didn't quite fit them because reality is more complex than any simple rule can account for. Importantly, mortals can break oaths and rules, which Honor - like a mathematician trying to model sociology - could not see until it was too late. Now, Cultivation's influence is letting the symmetries be broken to facilitate growth, as mold spreading from the decaying mould of Honor. What was once considered perfect is changed to fit an evolving world.
Characters break and mend in weird ways, ranging from Shallan's personalities to Dalinar's scars. Perfection is an imperfect concept, but everyone strives for unreachable goals because the journey is more important than the destination, and remaining invariant is no journey at all.

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 :).
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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).
~~~
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.
~~~
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

Going to start off with some factors, and then just link to some webpages, some of them are news articles, yes, but they show concepts really well.

Chinese, Japanese, and even Turkish have simplified numbers, eleven is ten-one. Very logical. Chinese children, at the age of 4 usually can count to 40, while English children usually reach that at age 5. A simplified counting system allows for easier learning of the numbers and logically finding patterns to count higher. Also, some people claim our memory runs on a two second memory loop, so having quicker numbers to say can cram in more numbers in two seconds, so you can remember sequences better. Personally, I have my doubts, I don't think Èr is quicker to say than 'Two', but overall, it could be much easier and simpler overall, I guess.
In Chinese you can say three-ten-four plus four-ten-two makes what? And the equation is all there, 34+42, while to say the same thing to an English speaker they need to translate it quickly (not a big thing, but still a thing nonetheless) from thirty four and forty two to 34+42. Even thought they didn't say this sentence on any news article, I feel it plays a factor, in Chinese they have a character for each word, so each number is it's own word, you don't need to go one is 1, you just think yĩ is yĩ (一 is 一), you don't need to go from your writing system to your number system (I understand the Chinese use the 12345678910 number system often, but just in their base language, the rest of this is the case), and I feel conlanging can take this as having a single letter meaning a letter. So maybe A is 1, and T is 2, or whatever, work with it how you want. Or maybe you'll make a logographic system.
One of the links also talks about the hard work of rice farming, and how hard work and focus is a large factor in mathematics, so if you want to make a conculture good at math, make something in their past or current history require lots of hard work every day to survive.
The same link also talks about how fractions are more simple. In English, they say, we say 'three fifths', but in Chinese they say 'Out of 5 parts, take 3'. I can see how that would be easier, however, for purposes of conlanging, I personally feel that it is a little longer than necessary, but that is just my opinion.
Children who have a tonal language are much better at identifying musical pitches and such, in simple words when the language has pitches to make different words, their brain becomes better at it, and it bleeds over to other areas, AKA music.
Measure words. Chinese requires a measure word for everything. In English I don't say 'I have a water', you say 'I have a cup of water' or something like that. Chinese is like that, but for everything. You need a measure word, you don't say 'one person', you need a word like 'cup, box, handful', etc etc (not for person, it's a different measure word), but studies show that it helps with mathematical skills, and although I cannot find where I heard that, I believe the speaker was talking about how it forces them to think about the relationship of the word, whether or not it's 5 bushels of grapes, or 5 actual grapes.
Some of you probably know of the Pormpuraaw, the culture who's language doesn't have left and right, but north, east, south, west, etc. And you don't have a left foot, you have a south foot, or a west foot, or whatever direction you are facing. The people from these culture are able to tell very VERY quickly, even in a new environment, assuming they didn't lose direction via some means, exactly where north is at any given moment. For conlanging, don't assume 'My speakers will never be able to keep track of what time of day it is accurately, how hot it is, etc etc!', just try, make your conjugations for whatever you want, push the boundaries of human limits and they will move for you. Also, side note, their way of saying hello requires them to say their heading, so saying hello in your language can force your speakers to remember whatever your cool idea is
English speakers write left to right, however, Arabic writes right to left, and as such this dictates their thought, their perception of time. When organizing a series of picture by what time they happened in, English Speakers will most likely arrange them earliest on the left and working right, while Arabic Speakers will do it opposite. Same can work for any language, in whatever way they write. Interesting note from the Youtube video: The Pormpuraaw consider it from East to West, and they will organize it according to that on how they sit, they may organize it coming to them, going away from them, left to right, right to left, etc, depending on which way they are sitting
Russian Colours. Russian has two different words for one word in English. Blue. In Russian they differentiate between light blue and light blue. In the Youtube video she talked about how when the shade of blue was lowered Russian Speakers gave a response when it changed from light to dark, because to them it changed colours, while English speakers did nothing. Russian speakers are also faster to differentiate between the colours, and have better colour recognition of those colours, because to them, it's one of their primary colours. Yes, we have maroon, cyan, amber, crimson, ivory, etc, but when was the last time you saw major children's books, or even classroom settings teach this colours to the students. It's mostly Red, Orange, Yellow, Blue, Purple, sometimes Pink, Black, White, Grey, Brown, sometimes Violet (Personally I wasn't taught violet that much, and the only place I really learned it is from the colours of the rainbow, but I don't know what it looks without looking it up, and even then it looks like a shade of blue to me). For conlanging you can use this to classify more base colours (don't go crazy, because then you end up with Cyan and such, words that exist but rarely used, limit yourself to somewhere under 20, maybe, or if you go over, maybe include a conculture note or something on how you expect the education culture to teach them), or even sounds, classify things such as harsh, soft, or maybe other stuff, smells, temperatures, tastes, or anything you can find on here: https://en.wikipedia.org/wiki/Sense the key is just being creative on how you classify them and describe them.
Language Genders- The other language I speak decently (not enough to really converse in, I'll rather be humble than be called out), is French, and all romance languages like it are famous for genders, even German has genders. Throughout my French class I've learned a few things about how French speakers (or even the class I took in spanish), think of objects. To take an example from the Youtube video 'bridge' is feminine in German, but masculine in Spanish. A Spanish speaker might describe it as long, strong, etc, very masculine terms, while a German speaker might describe it as beautiful, elegant, etc. Genders can influence how a person thinks about something. In French television is feminine, while fridge is masculine. A French speaker, when asked to imagine a conversation between them in their head (or maybe a kids show or something), will have the television take on a feminine personality, and the fridge a masculine one. German also has a third, neutral, gender, but I am unsure of how that impacts the psych. However, think about this when creating your language. You don't have to copy Romance languages, or even German. You can have a Strong Masculine, Weak Masc, Weak Fem, Strong Fem, etc (maybe a neutral, maybe a God ones, who knows). Each one having a different aspect, and that can impact them by thinking of Strong Masculine items as very very masculine, and weak masc ones as less masculine, and more closer to a neutral terms (I'm not talking about 'Alpha males' or anything like that, just how strongly one might consider a word masculine or feminine). However, the only note of caution I can say is personally, I don't see a lot of use for genders, unless you want to differentiate between words (French does that for at least one word, the masc and fem articles makes it a different word. I can't remember it though), or unless you want to add culture or naturalism to your conlang.
Clause of the Environment/Actions/Blame. I don't feel I have the skills to explain this well enough without copy and paste, but basically some languages are more biased on the completion or final product of a verb, so a speaker of that language may say about a picture 'The man is walking towards the store', while in one where the verb is more important and the final product isn't stressed, they might say 'The man is walking/strolling' or something like that. When shown a man accidentally knocking something over, as shown in the Youtube video, some speakers might say the vase broke or the vase broke itself, while a different language speaker might say 'He broke the vase' or 'He knocked it over', and in some languages you don't say 'I broke my arm' unless you meant to broke it, as they don't consider accidents at true causes or something like that. For conlanging keep this in mind when you think up your grammar, what will be emphasized, how will this manifest in your speaker's world view?
Tenses- This should be something, if nothing else is, known to just about everyone well versed in cool facts about linguistics. A study has shown that speakers of languages who have weak future tense tend to have strong future-orientated actions, such as saving for the future, while those with strong future tense don't as much. This is because those with strong future tense separate the future from the present, AKA 'That's a problem for future Homer, GLAD I'M NOT THAT GUY!'. So 'I am going to each tomorrow' is strong future tense, while a weak future tense language would say 'I eat/am eating tomorrow' and while that doesn't sound as weird to an English speaker, there are better examples of that (that I can't for the love of me think of right now). For conlanging, I know it's difficult to swallow for some conlangers, but having past present future tenses isn't always the best, or even having more, 5, 6, 10, etc tenses, while very clarifying, it separates times a lot, and in cultures like that you can write in some bad habits, or lack of good habits, because to them the present is all that really matters, because everything else is so separated from them.
Also, the Youtube video mentions that some languages (natural languages that exist in our world,) don't have an exact number system, she says how they have trouble keeping exact quantities. You can take from this and consider nothing in language as necessary. Personally, in one of my languages (I consider it the best one I have so far, and it works perfectly in all ways), everything is a noun. I don't want to explain it more until I finish it, but it works well. You can even take out pronouns. Maybe take relationship as pronouns. If a big brother is talking to his little sister, maybe he won't say 'I, me, etc', but instead say 'older brother', and instead of saying 'you, your, etc' he says 'little sister'. And he changes 'big brother' to 'son' when talking to parents. Maybe for meeting strangers he says 'humble friend' or something. Who knows. But also keep in mind on how these may impact it, and try to think to yourself how this can impact the speaker's mind. Removing the pronouns may make the speaker be more 'down to earth', make him less 'american' and more 'Japanese' so to speak, what I mean is he thinks of himself less as himself, the best person, whatever, but part of a team, part of something, with every sentence he reminds himself he isn't special, and has to think of other people. This might breed empathy, and that may happen to cause him to work harder even because he considers himself a small part of a large society, and it can't work if everyone doesn't give it his/her all. I am not saying that is what happens, I am just saying it would make sense for that to happen in my mind.
Personal note, experiment with sign language as well, create a whistle language, like Silbo, to pair along with your culture, simply, explore. People make conlangs all the time because it's relatively easy (yes, it is hard, but it's not too hard to learn and create, just timely and dedication-requiring), but few create sign languages. Explore, learn. The more we create together the more we learn together.
EDIT: I forgot to mention E-Prime. Taking the verb 'to be' out of a language removes the ability to play god in your language, so you can't make the apple be red because you said it is, instead, it looks red to you. However, this example becomes a lot better with feelings. You are not depressed, you feel depressed. Cool read, find it here: https://en.wikipedia.org/wiki/E-Prime

So overall, if you want a language that kind of primes children to excel, a tonal language with a good number, not just two (it would work, but I imagine it wouldn't as well, but on the same hand, don't go over board, draw the number at triple digits (I'm joking, try not to go over 20, I would draw the line at more like 12 max, personally)), have measures words needed for everything, you don't need a separate word for everything, but maybe a measure words for all vehicles, maybe groups, maybe plant foods, etc, etc. Think about how you want to work that. Have a weak future tense or no future tense language, Have the most simple and shortest words you can, making the symbols for the numbers very similar to the letters that make it up, if not the same (Like how 'me' in Japanese is eye, maybe have a syllabary where the word for one is a single syllabary, like 'me' is 'め'). Find a short, easy, VERY self explaining, simple, and whatever form of describing various mathematical functions, such as decimals, fractions, scientific notation, etc. I found a nice post where someone talks about numbers and it is an excellent post to think about and guide your conlang. Look specifically at the comment by ethuank1. Remove the verb TO BE from your language and require users to find other ways around it. ETC ETC. From there you can add in some base colours, add in absolute directions instead of relative, make the hello say what direction and time they are facing/it is, if you want, etc etc.
https://www.reddit.com/conlangs/comments/78kllc/does_your_conlang_have_its_own_numeral_system/doxcayi/?context=3

I hope this post was interesting, and yes, a lot of this stuff you might have known, but I decided to put a lot of what I've found on here, kind of to help me remember, and kind of to help others. Sorry for anything I got wrong, if anything, and I would love it if you can list some other cool functions you can think of. As for these, what do you think, and how do you incorporate this into your conlangs? If I inspire you to add anything or make anything in your conlangs or concultures, please tell me, it would make me very happy to know that my post helped another. Anyways, thank you very much, and I hope everyone has a good day!

Side note: Before posting this I thought of Toki Pona, and just want to mention to those that don't know it that it is a language with very few words and very VERY simple, and not meant to express complex ideas, but meant to force the user to think in simple terms, and from there induce a zen like and happy state to the speaker.

http://www.messagetoeagle.com/kuuk-thaayorre-language-uses-cardinal-direction-to-define-space/
https://www.google.com/imgres?imgurl=http://masterrussian.com/learnwords/img/russian-colors-dictionary.png&imgrefurl=http://masterrussian.com/learnwords/russian-colors.htm&h=403&w=403&tbnid=I6Iw4d-7icpEMM:&q=russian+colours&tbnh=160&tbnw=160&usg=AI4_-kS8D5rtn0U4qESkEBzgkwMuITZeFw&vet=12ahUKEwjiwpmZ7fzfAhUU8YMKHVMxAWoQ9QEwAHoECAYQBg..i&docid=WY_4lk3TE6_-ZM&sa=X&ved=2ahUKEwjiwpmZ7fzfAhUU8YMKHVMxAWoQ9QEwAHoECAYQBg
https://en.wikipedia.org/wiki/Sense
https://medium.com/@fesja/6-reasons-asians-are-better-at-math-ab88fcb3db0
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4341514/
https://www.wsj.com/articles/the-best-language-for-math-1410304008
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0181074
https://www.theatlantic.com/business/archive/2013/09/can-your-language-influence-your-spending-eating-and-smoking-habits/279484/
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3627212/
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0185047
https://www.sciencedaily.com/releases/2017/01/170118163756.htm

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