IB Math AA SL & HL Prior Learning Formulae: Complete Guide

Reviewed against the current IB Mathematics: Analysis and Approaches guide. This page explains prerequisite knowledge, not a replacement for the five assessed syllabus topics.

Prior learning is mathematics the IB expects students to know before sitting a Diploma Programme mathematics examination. Schools may review these ideas early, especially when students arrive from different curricula, but an exam question can use them without reteaching them. Weakness in a prerequisite can therefore make a new AA topic feel harder than it really is. A calculus error may come from fractions; a vector error may come from Pythagoras; a probability error may come from set notation.

The official list covers five broad areas: Number and Algebra, Functions, Geometry and Trigonometry, Statistics and Probability, and one foundational rate relationship linked to Calculus. Most prerequisites apply to AA SL and AA HL. A smaller set is marked HL only, including greatest common factors and least common multiples, rational exponents, rationalising denominators, quadratic equations and inequalities with rational coefficients, and addition or subtraction of algebraic fractions.

What “prior learning” means in IB Mathematics AA

The IB guide says students enter Diploma Programme mathematics with varied previous experience, yet examination questions assume familiarity with the listed prerequisites. Teachers are expected to identify missing skills and address them early. That wording has two consequences for students. First, “we did this before IB” does not mean it is unimportant. Second, the prior-learning list is not an additional sixth syllabus topic with a separate formula table or guaranteed stand-alone question.

Prior learning supplies the language and operations used everywhere else. You need percentage fluency for compound change, algebraic fractions for calculus, interval notation for domains, function graphs for transformations, and basic probability diagrams before formal probability models. AA emphasizes analysis and reasoning, so slow arithmetic or unreliable algebra consumes time that should be spent interpreting the problem.

A ten-minute entry-skills diagnostic

Try these prompts without notes. They are not an IB practice paper; they sample the underlying actions that later questions assume. Mark each response as secure, slow or not yet understood.

1. Evaluate 3/4+2/5 and give the answer in simplest form.
2. Increase 240 by 17.5%.
3. Write 0.000052 in scientific notation.
4. Expand and simplify 3(2x−5)−2(x+1).
5. Rearrange v=u+at to make t the subject.
6. Solve 5−2x≤11 and show the solution on a number line.
7. Find the distance between (−1,2) and (5,10).
8. A right triangle has hypotenuse 13 and one side 5. Find the third side.
9. Find the mean of 4, 7, 7, 9 and 13.
10. A journey covers 156 km in 2.4 hours. Find the average speed.

Number systems, SI units and accuracy

Natural, integer, rational, irrational and real numbers

You should recognize common number sets and their nesting. Natural numbers ℕ are counting numbers; integers ℤ include negatives and zero; rational numbers ℚ can be written as p/q for integers p and non-zero q; irrational numbers cannot be written as such a fraction; real numbers ℝ include rational and irrational values. Conventions about whether 0 belongs to ℕ can vary, so follow the definition in context.

Classifying a number is not just vocabulary. It controls whether a result can be exact, whether a decimal terminates or recurs, and which operations preserve membership. For example, the sum of two rational numbers is rational, but the sum of a rational and an irrational number is irrational.

SI units and derived units

Be familiar with SI units for mass, time and length and with derived units such as m/s, m² and m³. Convert before calculating. One metre equals 100 centimetres, but one square metre equals 10,000 square centimetres and one cubic metre equals 1,000,000 cubic centimetres. The conversion factor must be squared or cubed with the dimension.

Rounding, significant figures and errors

Rounding should match the data and the requested accuracy. Decimal places count digits after the decimal point; significant figures begin at the first non-zero digit. Keep guard digits through working and round once at the end. Appreciate that a rounded measurement represents an interval rather than an exact point.

An estimate is a reasonableness check, not careless arithmetic. If a calculator reports 18,700 m for a classroom width, units or input are wrong. Order of magnitude, sign and plausible range should be checked before accepting a screen value.

Absolute value

The absolute value |a| is the distance from a to zero, so it is non-negative. Thus |−7|=7 and |x|=5 has two solutions, x=5 or x=−5. Geometrically, |x−a| is the distance between x and a on the real line. Absolute-value language later supports moduli, distances, inequalities and complex numbers.

Arithmetic, primes, factors, ratios and percentages

Fractions, decimals and order of operations

You should add, subtract, multiply and divide integers, decimals and fractions without losing signs or order. For addition, use a common denominator; for multiplication, cancel common factors before multiplying; dividing by a fraction means multiplying by its reciprocal. Follow brackets, powers, multiplication/division and addition/subtraction, resolving operations of equal rank from left to right.

Prime numbers, factors and multiples

A prime number has exactly two positive factors: 1 and itself. The number 1 is not prime. Prime factorization breaks a positive integer into prime powers and supports simplifying fractions and radicals. Factors divide a number exactly; multiples are created by multiplying it by integers.

The official guide marks greatest common factor/divisor and least common multiple as HL-only prior learning. Using prime powers, the GCF takes the smaller shared exponent and the LCM takes the larger exponent across all primes.

Ratio, proportion and percentage

A ratio compares quantities in the same units. Direct proportion has the form y=kx, while inverse proportion has y=k/x. Percentage change compares the difference with the original value, not the new value.

Use a multiplier for repeated change: an increase of r% uses 1+r/100 and a decrease uses 1−r/100. Reverse percentages require division by the multiplier. If a price after a 20% discount is 80, the original is 80/0.8=100, not 80×1.2.

Algebraic manipulation, substitution and exponents

Expand, collect and factorize

Expanding removes brackets by the distributive law; collecting combines like terms; factorizing reverses expansion. Terms are like only when their variable parts and exponents match. For example, 3x² and −5x² combine, but 3x² and 3x do not.

Rearranging formulas

To change the subject, perform inverse operations while preserving equality. Treat grouped expressions as units and identify restrictions. Rearranging v=u+at for t gives t=(v−u)/a, provided a≠0. When a variable appears more than once, collect its terms before factorizing it out.

Substitution and exponent expressions

Substitute values using parentheses, especially negatives. If f(x)=2x²−3x and x=−2, then f(−2)=2(−2)²−3(−2)=14. The square acts on the entire negative value.

SL and HL students should evaluate simple positive exponent expressions. Rational exponents are listed as an HL-only prerequisite. Use a^m/n=ⁿ√(a^m) where the real expression is defined. The HL list also includes simplifying surds and rationalising a denominator.

Scientific notation a×10^k, with 1≤|a|<10 and integer k, also appears in prior learning and then explicitly in assessed Topic 1. Use written ×10 notation rather than calculator E-format. Continue the assessed material in the AA SL/HL Number and Algebra guide.

Equations, inequalities, systems and sets

Linear equations and inequalities

Solve a linear equation by preserving equality and isolating the variable. Clear fractions using a common non-zero denominator when helpful. For inequalities, the same operations apply except that multiplying or dividing by a negative quantity reverses the inequality sign.

Understand <, ≤, > and ≥ and connect inequalities to intervals on the real number line. The solution −3≤x<4 includes −3 but excludes 4. Domain restrictions often use the same notation.

Quadratics and simultaneous equations

Solving quadratic equations and inequalities with rational coefficients is marked HL-only prior learning. Methods include factorization, completing the square, the quadratic formula and graph interpretation. A quadratic inequality requires roots plus a sign analysis; finding the roots alone does not finish the question.

All students should solve systems of two linear equations in two variables. Elimination combines equations to remove one variable; substitution expresses one variable in terms of the other. Check the ordered pair in both original equations.

Sets and algebraic fractions

Know basic set notation, including union A∪B and intersection A∩B. The union contains elements in either set; the intersection contains elements common to both. Venn diagrams show these relationships visually and later support probability.

Addition and subtraction of algebraic fractions is listed as HL-only prior learning. Factor denominators, identify a common denominator, combine numerators and state excluded values from the original expressions.

Function and graph foundations

Prior learning includes graphing linear and quadratic functions using technology and understanding mappings between sets. A function assigns each permitted input exactly one output. You should recognize the same relationship as ordered pairs, a table, a mapping diagram, an equation or a graph.

Linear graphs

The gradient m measures vertical change per unit horizontal change and c is the y-intercept. Parallel non-vertical lines have equal gradients. A horizontal line has gradient zero; a vertical line has undefined gradient. Units matter: on a distance-time graph, gradient can represent speed.

Quadratic graphs

A quadratic y=ax²+bx+c forms a parabola. Be able to use technology to display it and identify intercepts, roots, the axis of symmetry and vertex. Factorized form shows roots; completed-square form shows the vertex; expanded form shows the y-intercept c. Switching form is a mathematical choice, not cosmetic rewriting.

These foundations lead directly into the assessed AA SL/HL Functions formulae, including domain, range, inverse functions and transformations.

Geometry, circles, perimeter, area and volume

The official prior-learning list begins with geometric concepts such as point, line, plane and angle. Know angle measurement in degrees, the triangle angle-sum theorem and properties of common quadrilaterals: parallelogram, rhombus, rectangle, square, kite and trapezoid. A diagram should be read from stated information, not from appearance alone.

Perimeter and plane area

Perimeter measures boundary length; area measures surface. Compound-shape questions become manageable when you divide the shape into familiar pieces or subtract gaps. Use perpendicular height, not a sloping side, in triangle, parallelogram and trapezoid area formulas.

Know circle terms: centre, radius, diameter, arc, sector, chord, tangent and segment. A radius meets a tangent at a right angle. Distinguish a sector—the region between two radii and an arc—from a segment—the region between a chord and its arc.

Three-dimensional shapes

Be familiar with prisms, pyramids, spheres, cylinders and cones. The formal prior-learning list requires volumes and surface areas of cuboids, prisms, cylinders and compound three-dimensional shapes. A prism has a constant cross-section, so volume equals cross-sectional area times perpendicular length.

Coordinate geometry, Pythagoras and right-triangle trigonometry

Distance and midpoint in the Cartesian plane

The distance formula is Pythagoras applied to horizontal and vertical changes. Keep each coordinate difference inside parentheses. For points (−1,2) and (5,10), the changes are 6 and 8, so distance is √(36+64)=10 and midpoint is (2,6).

Pythagoras and its converse

The hypotenuse is opposite the right angle and is the longest side. The converse states that if side lengths satisfy a²+b²=c² with c the longest, the triangle is right-angled. Pythagoras finds lengths, not angles.

Right-angle trigonometry

Label sides relative to the chosen angle. The adjacent side is not the hypotenuse. To find an angle, use an inverse trigonometric function and ensure the calculator is in degree mode when the question uses degrees. To solve a triangle, combine trigonometry, Pythagoras and the 180° angle sum as appropriate.

Bearings, compass directions and transformations

Bearings are measured clockwise from north and written with three figures, such as 065° or 240°. Draw north lines at relevant points and mark the clockwise angle. Back bearings differ by 180°, adjusted into the range 000° to 360°.

Know translations, reflections, rotations and enlargements. A complete description includes the necessary parameters: translation vector; mirror line; rotation centre, angle and direction; or enlargement centre and scale factor. Negative scale factors place the image on the opposite side of the centre and reverse orientation.

Statistics and probability foundations

Data displays and simple statistics

Prior learning includes collecting data and representing it in bar charts, pie charts, pictograms and line graphs. Choose a display that matches the data and purpose. Bar charts compare categories; line graphs show ordered change, often over time; pie charts show proportions of a whole.

Know mean, median, mode and range for discrete data. The mean uses every value but is influenced by extremes. The median is the middle after ordering and is more resistant. The mode is most frequent and can be absent or non-unique. Range measures spread using only the extremes.

Simple probability, Venn and tree diagrams

Tree diagrams organize sequential events: multiply probabilities along a path and add mutually exclusive paths that produce the requested outcome. Venn diagrams organize set relationships. Fill intersections before outer-only regions to avoid double counting.

Do not assume independence or equal likelihood without evidence. A probability of 1 means certain and 0 means impossible. If calculated probabilities exceed 1 or sum incorrectly across a complete set of outcomes, review the model.

Calculus foundation: speed, distance and time

The listed Calculus prior learning is the elementary rate relationship speed=distance/time. It prepares students to interpret gradients and later derivatives as rates of change. Align units before substitution: kilometres and hours produce km/h; metres and seconds produce m/s.

AA SL versus AA HL prior-learning differences

Most official prerequisites apply to both levels. The following Number and Algebra skills are explicitly marked HL only in the current guide.

HL readiness does not mean never making an arithmetic error. It means these operations are sufficiently fluent that you can use them inside a multi-step argument. If several HL-only prerequisites are unfamiliar, discuss preparation with your teacher rather than trying to memorize advanced formulas without the supporting algebra.

Formula booklet, calculator and communication

The official mathematics formula booklet supports the assessed course, but prior learning is assumed. Do not expect every elementary area, fraction, equation or trigonometric fact to be highlighted at the moment you need it. Familiarity and recognition still matter.

Technology can graph functions, evaluate expressions and check solutions. Use it as verification, not as a substitute for understanding. On a technology paper, write the model and interpret the screen result. On a no-technology paper, exact arithmetic and algebra are essential. In both cases, use accepted notation, show key steps, label units and state restrictions.

Fourteen-day prior-learning repair plan

For each day, review one concept, complete several single-skill questions and finish with one mixed problem. Reattempt incorrect work from a blank page after a delay. Once the foundations are secure, move into the shared Number and Algebra guide and, for HL, the AA HL-only extension.

How to prove that your prior learning is exam-ready

Finishing a worksheet is not the same as being ready to use a skill inside an IB question. A stronger test is transfer: can you identify the hidden prerequisite when the question does not name it? For example, a calculus question may quietly require factorisation before you solve for stationary points; a probability question may depend on exact fraction arithmetic; and a geometry model may fail if centimetres and metres are mixed. The foundation is ready when you can select the method, execute it accurately and explain why the answer is reasonable.

Use four checks. First, complete a short mixed set without headings such as “percentages” or “quadratics.” Second, mark it and classify each error as a knowledge gap, method-choice error, algebra slip, calculator entry error or communication issue. Third, correct the work from a blank page without copying the solution. Fourth, attempt a parallel question two or three days later. A delayed correct solution is better evidence of learning than an immediate correction made while the worked answer is still visible.

Accuracy is only one part of readiness. Time matters because prerequisite steps should not consume the whole question. Exact manipulation should also remain visible: avoid replacing fractions or radicals with rounded decimals too early. Finally, communicate with mathematical discipline. Define variables, preserve equality signs, state units, label diagrams and reject impossible roots when the context requires it. These habits turn elementary knowledge into dependable examination technique.

Mixed prior-learning examples with complete reasoning

Example 1: percentage, algebra and units in one model

A cylindrical container has radius 4 cm and height h cm. Its height increases by 15%, while its radius stays fixed. The original volume is V = π(4)²h = 16πh cm³. A 15% increase means multiplying the height by 1.15, not adding 0.15 cm. The new volume is π(4)²(1.15h) = 18.4πh cm³. Therefore the volume also increases by 15% because radius is unchanged and volume is directly proportional to height. This solution combines percentage multipliers, substitution, algebraic simplification and cubic units. A reasonableness check confirms that the new volume is larger but nowhere near double.

Example 2: coordinates, Pythagoras and exact form

Let A = (−2, 3) and B = (4, 11). The horizontal change is 4 − (−2) = 6 and the vertical change is 11 − 3 = 8. The distance is √(6² + 8²) = √100 = 10 units. The midpoint is ((−2 + 4)/2, (3 + 11)/2) = (1, 7). Notice the different purposes: distance uses squared changes and a square root, whereas midpoint averages the coordinates. Sketching the two points gives a quick check that the midpoint lies between them and that a 6–8–10 right triangle is plausible.

Example 3: tree-diagram probability and exact arithmetic

A bag contains three red counters and two blue counters. Two counters are taken without replacement. The probability of red then blue is (3/5)(2/4) = 3/10. The probability of blue then red is (2/5)(3/4) = 3/10. Since the two routes are mutually exclusive, the probability of obtaining one counter of each colour is 3/10 + 3/10 = 3/5. “Without replacement” changes the second denominator and numerator, so writing both branches prevents a common error. The result 3/5 is between zero and one, and symmetry shows why the two successful orders have equal probability.

Example 4: rearranging a formula before substitution

Suppose speed v = d/t, and you need the time for a 150 km journey at 60 km h⁻¹. Multiply both sides by t and divide by v to obtain t = d/v. Substitution gives t = 150/60 = 2.5 hours. Converting 0.5 hour to 30 minutes produces 2 hours 30 minutes. Keeping units in the calculation helps distinguish 2.5 hours from 2 hours 5 minutes, a frequent decimal-time mistake.

These examples show how prior learning appears in authentic questions: several elementary ideas cooperate inside one solution. When revising, do not stop after memorising isolated formulas. Practise deciding what each quantity represents, which relationships apply, whether the units agree and how to present an exact or appropriately rounded final answer.

Frequently asked questions