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.

Hi,
I'm currently enrolled in a M.Sc. course in Data Science & Engineering, and I have to decide which optional course to take among the following (I can choose only 1), please take a brief look at the syllabus.
1) OOP
syllabus:

Basic features (1 credit) • Object-oriented programming, java, eclipse • Classes, attributes, methods and constructors, objects • Packages and visibility rules • Strings, wrapper classes • Arrays
Inheritance and interfaces (2 credits) • Inheritance • Abstract classes, interfaces • Functional interfaces, lambda expressions • Exceptions • Generic types
Standard libraries (3 credits) • Collections: sets, lists, maps • Streams • Files • Dates • Threads • Graphical interfaces, Swing, JavaFX
Software Engineering principles (2 credits) - Software life cycle - Design using UML - Design Patterns - Configuration management - Testing

2) Numerical Optimization & Stochastic Optimization
syllabus:

Convex optimization: - gradient descent method; conjugate gradient method - Numerical differentiation - Newton and quasi-Newton methods - Globalization techniques - Alternating direction method of multipliers (ADMM)
Constrained optimization: - Interior point methods - Projected gradient method - Active set
Stochastic optimization - Static simulation-based optimization (parametric optimization) - Dynamic simulation-based optimization (control optimization)

3) Decision Making & Optimization
syllabus:

Linear programming: modeling techniques, basic concepts of the Simplex Method, and duality (10% of the course).

2. Computational complexity: problem classes P, NP, NP-complete, and CoNP-complete (10% of the course).
3. Exact optimization methods: Branch and Bound, Cutting Planes, and Dynamic Programming (20% of the course).
4. Heuristic optimization methods: greedy algorithms, GRASP, Beam Search, meta-heuristics (Tabu Search, Simulated Annealing, Genetic Algorithms, ACO, VNS, RBS), and math-heuristics (30% of the course).
5. Decision making under uncertainty: Stochastic Programming with recourse, Measures for Stochastic Programming, Progressive Hedging method (10% of the course).
6. Nonlinear Programming: theoretical conditions for unconstrained and constrained optimization, algorithms for unconstrained and constrained optimization (20%).

Of course, the best answer is "it depends on what you have already done, and what you would like to do", so I try to give a brief introduction. I come from a B.Sc. in Electronic Engineering, and this is the only reason why I'm considering taking OOP. I don't have much problem with programming, but I feel like I don't have some skills because my Bsc was not in CS.
Regrading the other 2 courses, they are not the only math courses in my degree, I have many others ( such as ML&DL, Math for ML, Statistics for Data Science, Network Dynamics & Learning, Computational Linear Algebra), but still, they might be interesting.
I don't want to work as a software developer, I'm more interested in research, but do you think I should take OOP anyway to fill some gaps? Can you give me some examples where Decision Making and Numerical/Stochastic Optimization could be useful? (As I said the most important topics of both courses are also covered partially in other courses)

Hi,
I'm currently enrolled in a M.Sc. course in Data Science & Engineering, and I have to decide which optional course to take among the following (I can choose only 1), please take a brief look at the syllabus.
1) OOP
syllabus:

Basic features (1 credit) • Object-oriented programming, java, eclipse • Classes, attributes, methods and constructors, objects • Packages and visibility rules • Strings, wrapper classes • Arrays
Inheritance and interfaces (2 credits) • Inheritance • Abstract classes, interfaces • Functional interfaces, lambda expressions • Exceptions • Generic types
Standard libraries (3 credits) • Collections: sets, lists, maps • Streams • Files • Dates • Threads • Graphical interfaces, Swing, JavaFX
Software Engineering principles (2 credits) - Software life cycle - Design using UML - Design Patterns - Configuration management - Testing

2) Numerical Optimization & Stochastic Optimization
syllabus:

Convex optimization: - gradient descent method; conjugate gradient method - Numerical differentiation - Newton and quasi-Newton methods - Globalization techniques - Alternating direction method of multipliers (ADMM)
Constrained optimization: - Interior point methods - Projected gradient method - Active set
Stochastic optimization - Static simulation-based optimization (parametric optimization) - Dynamic simulation-based optimization (control optimization)

3) Decision Making & Optimization
syllabus:

Linear programming: modeling techniques, basic concepts of the Simplex Method, and duality (10% of the course).

2. Computational complexity: problem classes P, NP, NP-complete, and CoNP-complete (10% of the course).
3. Exact optimization methods: Branch and Bound, Cutting Planes, and Dynamic Programming (20% of the course).
4. Heuristic optimization methods: greedy algorithms, GRASP, Beam Search, meta-heuristics (Tabu Search, Simulated Annealing, Genetic Algorithms, ACO, VNS, RBS), and math-heuristics (30% of the course).
5. Decision making under uncertainty: Stochastic Programming with recourse, Measures for Stochastic Programming, Progressive Hedging method (10% of the course).
6. Nonlinear Programming: theoretical conditions for unconstrained and constrained optimization, algorithms for unconstrained and constrained optimization (20%).

Of course, the best answer is "it depends on what you have already done, and what you would like to do", so I try to give a brief introduction. I come from a B.Sc. in Electronic Engineering, and this is the only reason why I'm considering taking OOP. I don't have much problem with programming, but I feel like I don't have some skills because my Bsc was not in CS.
Regrading the other 2 courses, they are not the only math courses in my degree, I have many others ( such as ML&DL, Math for ML, Statistics for Data Science, Network Dynamics & Learning, Computational Linear Algebra), but still, they might be interesting.
I don't want to work as a software developer, I'm more interested in research, but do you think I should take OOP anyway to fill some gaps? Can you give me some examples where Decision Making and Numerical/Stochastic Optimization could be useful? (As I said the most important topics of both courses are also covered partially in other courses)

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

A language can be described as a skill set. A language learner acquires fundamental skills, at first performing them only with difficulty and concentration. Through practice, those skills become automated, leaving the mind and attention free to acquire more complex skills.
Practice, therefore, is key to language-learning. But not just any form of practice will suffice. In fact, one popular form of practice is extremely ineffective: massed practice. I will be offering a brief explanation here and applying it to learning Latin. For a deeper introduction to the topic, see Make It Stick: The Science of Successful Learning.
All practice involves repetition, but different learning strategies distribute that practice differently. The term "massed practice" refers to practice that is highly concentrated in at least one of two ways. First, let's consider massed vs. spaced practice. Here, massed practice is basically cramming, trying to concentrate a large volume of practice into one or few sessions. I won't talk much about this, because I think just about everyone knows by now that cramming is bad for you and that spaced repetition is the answer. But I will make one point to teachers or autodidacts. If you're quizzing or testing only new material, you are in fact cramming. Anything worth testing once needs to be tested on multiple, separate occasions.
You may not be so familiar with the second kind of massed practice, massed vs. interleaved. Here's an example. Let's say that med students are learning to perform a particular procedure that has ten steps. They're going to practice it ten times today. That's a practice volume of 100 total operations. There are two basic ways we could structure this practice session. The massed practice approach would be to perform step 1 ten times, then step 2 ten times, etc. The interleaved approach would be to practice steps 1-10 in order as a complete circuit, completing the circuit 10 times. Then, after all the practice, each med student would perform the complete procedure, for a grade.
Many studies like this have been carried out, and they show a similar trend. Learning by massed practice leads to better initial gains and more self-reported improvement, but less retention between practice sessions and slower long-term improvement. Why might that be the case? The answer has two parts. First, our mind is always seeking the most efficient mental process to complete a task. Second, our memory takes its cues from our attention and stores processes along with their associated goal and context.
So, under the massed practice approach, our med student approaches step 1. Her brain recognizes that the goal is to complete step 1, and it calls from memory whatever it can to help. On the second round, it does the same thing. On the third, maybe the same. But at some point, let's say the fourth round, the brain realizes that it's superfluous to keep asking what the goal is, and to verify the procedure for that goal. It would be more efficient to simply issue a command: "repeat what I just did last." This "repeat what I just did" can be quickly reissued as many times as necessary.
So, how many times did our med student practice step 1? An outside observer, or even the med student's conscious mind, would say 10. But the subconscious parts of the brain responsible for memory would say 3. Only 3 times did the brain actively decide on the goal and connect a specific procedure to it, resulting in long-term strengthening of that association. The other 7 times it just refined motor patters.
By contrast, under the interleaved approach, our med student didn't practice the exact same step multiple times in a row. For each operation, the brain had to reorient itself, recognize the goal, and call up the appropriate procedure from long(er)-term memory. This way, all 100 operations contributed to long-term learning.
Note: There are lots of examples of this, and not all of them require the tasks to be ordered, as the medical procedure does. There have been studies on shooting baskets from different positions, solving different types of math problems, etc.
--------
So, how can we incorporate interleaved practice into Latin? I'm going to give three example of practicing the present infinitive, from worst to best.
The worst (i.e., most massed) form of practice I can think of goes something like this. On the left side of the page is a column of verbs, all in the exact same form (amo, porto, etc.), all from the first conjugation. Next to each verb is a blank. The instructions are to fill in the black with the present active infinitive corresponding to each verb.
Why is this so bad? Because the procedure is exactly the same each time. At first the brain recognizes the goal as "write the present active infinitive corresponding to each verb" and either works out or calls up the procedure: "Take the verb, drop the ending, add the verb's theme vowel, add -re." But after a while, the brain will realize, "Hey, I can complete this exercise just by changing all the -o endings to -are." Now, goal and procedure become one: "drop -o, add -are." And the person doing this can speed through it incredibly fast because there's basically no thinking, no mental processing, to be done at this point. And because the person doing it did it so fast, they'll probably think, "Wow, I really know this!" Which is precisely the opposite of the truth.
Next, let's look at a form of practice with some interleaving. We'll do the same exercise, but the verbs in the left column now have different forms and are taken from different conjugations: amo, docemus, capiunt, etc. Now the brain has to work a little harder, both at removing the ending from the stem, and at deciding which new ending to add. But this form of practice is still somewhat massed, because the goal in every case is to make a finite verb into an infinitive, and the procedure for doing so is still pretty similar: remove an ending, add []re. Can we do even better?
The optimal form of practice for long-term retention would add a functional, contextual element. This exercise consists of sentences missing a verb and asks the student to write in either a present tense or infinitive verb, whichever allows the sentence to make good sense. For example:
(doceo) Iucundum est bonos discipulos ....
(scio) Volo omnem consilium tuum ....
(dico) Marcus .... me esse stultum.
(fero) Consules Romani nobis auxilium ....
This final exercise is especially useful for long-term retention because practicing form and function together reinforces both, and because the context of the practice task is so similar to the context of the real task for which we are practicing, comprehending and producing Latin.

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

I have studied some linguistics, but dropped out of the major because 90% of it was just literature. Coming just from dropping out of a maths major, I wasn't really interested in any literature, and much more interested in how the system of language works.
Anyway, in my effort to undo all my linguistic prejudice, I took to believing that every language is equally complex, whatever well-defined version of complexity (if there even is such a thing) is to be taken.
One of my Linguistics 101 professors said that, for example, while Portuguese has many different verb conjugations for different tenses, singular or plural, first, second or third persons, English has a plethora of verbs that in Portuguese are all compressed into a single verb, but in English can be further split into several differing but connected meanings. Shining, Gleaming, Glowing, Glistening, Glittering, all can only be translated to "Brilhando".
That was enough to convince me that in whatever area some language lacked complexity, it made up in another.
Is this true? Does it even make sense to ask? Can it be measured, or even defined, how complex a language is?

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.

He just posted this video, imo very nicely done with a step-by-step lab and math example. If you've ever wondered about complex conjugate matching in the Extra syllabus, this should help explain it.
https://www.youtube.com/watch?v=XPLqSDXZuF4

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

tl;dr I wanted to try something different. This is an introduction to Schanuel's conjecture, explaining its significance and hinting at related topics. There are some exercises, and theorems are stated, but not proved. Prerequisites: first-year university math, linear algebra, and preferably introductory abstract algebra. Those with Galois Theory under their belt should find everything simple and intuitive.
I. Transcendental Numbers
A complex number is called algebraic if it is the solution to some polynomial equation with rational coefficients. For example, √2 is an algebraic number, since it's a complex number and it solves the polynomial equation x² - 2 = 0. The number 2×3^1/3 is also algebraic: it is a solution of the polynomial equation x³ - 24 = 0.

Exercise: Prove that every rational number is algebraic.
Exercise: Explain why the golden ratio is algebraic.

At this point, you might wonder: are there any numbers that are not algebraic?! Euler wondered about the same thing in the 1700s. He called non-algebraic numbers transcendental, and suspected that e and π were both transcendental numbers. It was a good fifty years after Euler's death that Liouville managed to prove the existence of transcendental numbers for the first time. Liouville used a technique called Diophantine Approximation to prove that the number L=∑k∈ℕ 10^-k! is, in a certain technical sense, easier to approximate using rationals than any irrational algebraic number, and so L cannot be algebraic.
Liouville's proof required a fairly deep understanding of diophantine approximation of algebraic numbers. These days, we have a much quicker way of constructing transcendental numbers, using Cantor's diagonal argument. Since the algebraic numbers form a countable set, we can make a list of all the real algebraic numbers, and diagonalization will produce a real number absent from our list. That number cannot be algebraic, so it will have to be transcendental.

Exercise: Prove that the real algebraic numbers are countable.

Thanks to Cantor, first year undergraduates can answer questions that baffled the great Euler! Thus advances mathematics.
However, it's very hard to prove the transcendence of numbers that were not specifically constructed for the purpose of proving them transcendental. Euler's suspicions about e and π being transcendental were confirmed only in the late 19th century. The transcendence of the following numbers is still unknown: π+e, π−e, πe, π/e, π^π, e^e, π^e.

Exercise: To understand how difficult it is to prove a number transcendental, try and fail to prove that log(2)log(3) transcendental. Can you even prove it irrational?
Exercise (difficult): Prove that the sum of two algebraic numbers is algebraic. Conclude that at least one of e+π and e-π is transcendental.

II. Lindemann-Weierstrass Theorem
In 1882, Lindemann managed to show that e^z is transcendental for any algebraic z (except 0). Thanks to Euler's identity, this implies that π is irrational. Since e^iπ is −1, and -1 is not transcendental, iπ must be transcendental. Therefore, π itself must be transcendental.

Exercise: Using the transcendence of π, show that one cannot construct a square with the same area as a given arbitrary circle by using only a finite number of compass-and-straightedge operations.

Lindemann's result has a very powerful variant, known as the Lindemann-Weierstrass theorem. This theorem is best phrased in the language of algebraic independence, a common generalization of both algebraicity and linear independence.
Let z1,z2,...,zn be complex numbers. We say that these numbers are algebraically independent if there is no non-trivial polynomial identity between them. More formally, if f(z1,z2,...,zn) = 0 for some polynomial f(x1,...,xn) with rational coefficients, then f is the zero polynomial.
Now, we can state the Lindemann-Weierstrass and Gelfond-Schneider theorems:

Theorem (Lindemann-Weierstrass). Let z1,z2,...,zn be algebraic numbers which are linearly independent over ℚ. Then e^z1,e^z2,...,e^zn are algebraically independent.
Theorem (Gelfond-Schneider). Let x,y be algebraic irrationals. Then x^y is transcendental.

These two theorems are the real workhorses of transcendental number theory: they encompass essentially everything that a non-specialist would be able to prove.
III. Schanuel's Conjecture and Implications
There is a conjecture due to Schanuel which generalizes the Lindemann-Weierstrass theorem and would resolve all the questions that we encountered before.

Conjecture (Schanuel). If the numbers z1,z2,...,zn are linearly independent over ℚ, then at least n of the numbers z1,z2,...,zn,e^z1,e^z2,...,e^zn are algebraically independent.

Notice that we now allow arbitrary complex numbers: in contrast with the Lindemann-Weierstrass case, z1,z2,...,zn need not be algebraic!

Exercise: Assume Schanuel's conjecture. Prove that log(2) and log(3) are algebraically independent. What does that say about log(2)log(3)?

It remains an open problem whether there exist two algebraic numbers whose logarithms are algebraically independent. Proving Schanuel's conjecture would clearly revolutionize transcendental number theory. The conjecture would have applications in other fields of mathematics too. In constructive mathematics, one often wants to work with sets of real numbers that have decidable equality. The real algebraic numbers have this property: Cyril Cohen showed that these form a discrete Archimedean real closed field. But we would like a number field that admits more real functions, at the very least elementary functions such as exponentials and sines. Unfortunately, there are serious formal limitations on how far we can go. Miklos Laczkovich showed that any extension of the algebraic numbers that has the sine function must have both the equality relation and the positivity predicate undecidable.
What if we add the exponential function instead of the sine function? Then, assuming Schanuel's conjecture, Wilkie gave a decision procedure for checking equality (in fact, he proved a much stronger, model-theoretic result about the theory of real closed fields with an exponential function, from which this result follows). Without Schanuel's conjecture, this is still an open problem.
IV. Zilber's field
Boris Zilber and others started to slowly build up towards Schanuel's conjecture. Using Shelah's work on infinitary logics, Zilber was able to axiomatize a field B with the following properties (among others):

B has cardinality continuum (the same cardinality as the set of real numbers), is algebraically closed, has characteristic 0, and admits a surjective exponentiation morphism.
The kernel of this exponentiation morphism is an infinite cyclic group generated by a transcendental element (think Euler's identity).
Schanuel's conjecture holds on B.

There is exactly one field that obeys all of these properties and Zilber's other axioms. If this field is isomorphic to the complex numbers, then Schanuel's conjecture holds. If this field is not isomorphic to the complex numbers, then either Schanuel's conjecture or another related property (strong exponential-algebraic closure) fails for the complex numbers. Originally, model-theorists thought that B will turn out to be different from ℂ, but public opinion has shifted, and now they expect the isomorphism to hold - but the proof is, as usual, completely out of reach.
Apart from Schanuel's conjecture, proving that Zilber's field is in fact the complex numbers would settle various interesting open questions. For example: it is well-known that the only two field automorphisms of ℂ that fix the real numbers are the identity map and complex conjugation. However, if we only need to fix the rationals (which every field automorphism fixes), we can construct other, "wild" automorphisms.

Exercise: Construct uncountably many wild automorphisms of the field ℂ. (Hint: Hamel bases)

But what about automorphisms of ℂ that send the exponential map to an exponential map (i.e. exponential field automorphisms)? How many do we have? Remarkably, we don't know the answer for ℂ: nobody ever managed to construct one apart from the trivial ones. However, we know that B has many "wild" exponential field automorphisms, so proving that Zilber's field is the complex numbers would immediately settle this question. On the model-theoretic side, it would settle yet another open question - that membership in the reals is not definable in (ℂ,exp).
There have been attempts to distinguish B from ℂ using topology: the usual topology of the complex numbers is not available in Zilber's field, but we could try to prove analogues of theorems of complex analysis for the definable functions in B. Remarkably, Picard's theorem holds for these functions on B!
V. Conclusion
I hope that this effort-post managed to inspire some people to look into either Transcendental Number Theory or Model Theory, or at least convince them that Schanuel's conjecture might have implications to their interests. I am not an expert on this particular topic, so there will be mistakes. Feedback and corrections are welcome. However, answering questions will have to be a community effort. edit: Thanks for the gold!

I can't seem to solve new problems on my own without having seen a similar problem before. Let me explain through an example.
I just finished my final for my undergraduate ODE/LA course. One of the questions on the exam was on complex numbers. It asked me to solve for z^4 + z^2 + 1 = 0 (z is complex). I thought about the problem and tried to go back to what I learned from the lectures and homework problems, but I couldn't come up with the solution.
Our professor uses his own lecture notes and there is no textbook for the course. My prof. dedicated one class to the complex numbers and we only had one homework set. The lecture notes included the following: z = x + iy, where z is a complex number, the definition of the modulus and argument, what a complex conjugate is, what ln(z) means, and Euler's formula. It was a brief lecture and the rest of class was spent solving for roots of unity (z⁵ = -1, stuff like that) and complex eigen values.
I tried to apply the techniques from the homework and lecture (replace z with re^iƟ, x + iy, and ln(z)), but I still couldn't come up with the answer. I looked at the solution after the exam and the "trick" to the problem is to create a new variable and set that equal to z^2. You also have to use the quadratic formula.
I did everything else on the exam correctly, but that's only because I've seen similar problems before. In the time I was given, I would never have thought to create that new variable. I also didn't even know that I could use the quadratic formula. I later asked my friend if he got the answer and he said he did after some thought.
I've had this problem ever since I started taking math classes: I have to have seen a similar problem before or else I can't solve the problem at all. My question is, what exactly is preventing me from solving new problems and how can I fix the issue?
Is it because I lack mathematical "insight" or "intuition"? Is that something I can gain? A lot of people say it's just common sense, but I honestly wouldn't have been able to solve that problem even if I was given more time (maybe I'm just dumb). I've started researching "mathematical thinking" books like the one written by G. Polya, but I'm not a math major and I'm not sure that I would be able to successfully go through a book like that and be able to solve all the problems.

TL;DR Somewhat experienced practitioner looking for graduate level treatment of topics in math/stats used in ML.
Hi,
I have a CS background, so sometimes I find myself lacking the math/stats knowledge to understand certain topics.
I'm looking for a book to learn things at the graduate/PhD level in these topics:

Statistics
Calculus
Optimization
Linear algebra

Is there ONE book that is along the lines of "necessary mathematical background for ML" kind of thing?
I build complex NN's in the ML research department of one of the big-4 tech companies. I give my background to clarify that I am not looking for a book to teach me what Gaussian is, how to apply Bayes rule, or how backpropagation works.
To give a few examples, my hope is to gain enough background knowledge to comfortably understand topics such as:

Statistical learning theory
VC dimensions and SVMs
Proof of expressiveness of neural networks.
Optimization procedures such as conjugate gradient methods
Eigenvalues/vectors deeply and intuitively.

Thanks!

Hello, I am learning Quantum mechanics, and I have learned that there are functions that represent a particle in a state in which a certain quantity doesn't change, this are eigenfunctions of a certain operator, and there are two kinds of eigenfunctions here, those which have discrete eigenvalues, and those who have continuos eigenvalues. The first one lie on a Hilbert space and the second one do not. The problem are the second ones, because their inner product is not guaranteed to exist. The book (Griffiths, Quantum mechanics, second edition) then say that sometimes we can find that inner product, and makes an example of the momentum operator. First we get the eigenfuntions, which are eipx where p is actually the eigenvalue, it ended up in the exponent because the momentum operator is a derivative multiplied by a complex number. Anyway, when you try to get the magnitude of one of this functions you get infinite, because the inner product is the integral in all space of the product of the function and it complex conjugate. Then the autor says: "What if we make the inner product of two different eigenfunctions?" and then it turns out to simplify to a Fourier transform whose result is a delta function. That bit is here. So we have the Dirac delta function of (p1-p2), d(p1-p2), and why couldn't p1=p2? (actually this is the part that confuses me, see at the end), so all the inner products of these eigenfunctions are Dirac deltas, using this property the author shows that any function in Hilbert space can be represented by a combinations of these functions, like a linear combination but instead of summations we use integrals. The bit that bugs me here is that doing the inner product of a function with itself if infinite if you do it one way, and a Dirac delta if you do it some other way. Also, this whole thing, I understand it mathematically but I don't get it, it is not like the linear combinations, those make sense to me, but using this functions to represent other functions doesn't seem right to me. I know this works but I feel like I don't know why, even though I can explain it (maybe this is the nature of studying quantum mechanics?).
Also I'll post it in some math subreddit too.

So I've been taught that if, for example, 1 + i is a root of a polynomial, then 1 - i is also definitely a root. I was never taught WHY this is true, and wherever I search it up shows some overly complicated explanation with math symbols I've never seen before. So, why is this always true?
(I probably could've found a better sub for this but since complex conjugates come up occasionally in the SAT I came here)

My goal is to integrate the first Chern class of the tautological line bundle. To this end I calculated the connection matrix for the connection obtained by projecting the canonical flat connection of C² onto the line bundle and using the frame e([1:z]) = e_1 + z e_2, where {e_1, e_2} is the canonical basis of C² . This yields for the only element of the connection matrix the 1-form w = z*/(1+|z|² ) dz, with z* the complex conjugate of z. This is still in accordance with exercise 18, page 24 of http://www.staff.science.uu.nl/~crain101/DG-2015/main10.pdf
Now to compute the curvature matrix I wanted to use k = dw + w ^ w, using ^ to indicate the wedge product. The second summand is 0 (right?) so we get k = dw = d/dz* (z*/(1+zz*)) dz* ^ dz = -1/(1+zz*)² dz* ^ dz. However, this disagrees with both example 9.1 page 3 of http://www.maths.manchester.ac.uk/~tv/Teaching/Differential%20Geometry/2008-2009/lecture9.pdf and https://mathoverflow.net/questions/227233/first-chern-class-of-the-tautological-line-bundle-over-mathbbcpn
What am I doing wrong? Also, if anyone could show how one actually goes about calculating the integral I would appreciate it

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