IB Mathematics: Analysis and Approaches Complete Study Guide
IB Diploma Program Mathematics · 2027IB Mathematics: Analysis and Approaches Complete Study Guide
- SL and HL
- 5 syllabus topics
- Paper 1 without a calculator
- Formula booklet provided
IB Mathematics: Analysis and Approaches develops exact algebra, functions, geometry and trigonometry, statistics and probability, and calculus. This guide organizes those topics around the decisions students make on written papers: choose a model, preserve its conditions, show the calculation, and state a conclusion that answers the question.
International Baccalaureate does not publish a per-topic weighting for this exam. The subject guide gives assessment-component weights and warns students not to assume that the five topics receive equal emphasis. Keep every area active rather than treating an estimated planning percentage as an official score share.
- Choose before calculating. State the function, distribution, diagram, derivative, integral, or proof structure that governs the work.
- Keep conditions visible. Record domains, intervals, units, calculator rules, model assumptions, and required precision.
- Check a second way. Use substitution, a graph, a table, a derivative, an integral, or a limiting case to test the result.
What IB Mathematics: Analysis and Approaches covers
The five topics appear in syllabus order. Each study route connects its central decision to representations, common errors, formulas, and newly written worked examples. The names and links below match the five topic pages.
SL and HL written papers plus the mathematical exploration
At SL, Paper 1 and Paper 2 each last 90 minutes, carry 80 marks, and contribute 40% of the final grade; the internal mathematical exploration contributes 20%. Paper 1 does not allow a calculator, while Paper 2 requires a graphic display calculator.
At HL, Paper 1 and Paper 2 each last 120 minutes, carry 110 marks, and contribute 30%. Paper 3 lasts 60 minutes, carries 55 marks, contributes 20%, and requires a graphic display calculator. The internal mathematical exploration contributes the remaining 20%.
| Level and component | Time | Marks | Weight | Calculator |
|---|---|---|---|---|
| SL Paper 1 | 90 minutes | 80 | 40% | Not allowed |
| SL Paper 2 | 90 minutes | 80 | 40% | Required |
| SL exploration | Internal assessment | 20 | 20% | Individual mathematical exploration |
| HL Paper 1 | 120 minutes | 110 | 30% | Not allowed |
| HL Paper 2 | 120 minutes | 110 | 30% | Required |
| HL Paper 3 | 60 minutes | 55 | 20% | Required |
| HL exploration | Internal assessment | 20 | 20% | Individual mathematical exploration |
Students receive a clean mathematics formula booklet in the examination, but they must still select the right relation, state its conditions, and know how to derive or reconstruct results that the booklet does not provide. Check the official subject guide for later changes. These details are stated as of 2026-10-09.
Model a geometric saving plan
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
A student follows a monthly saving plan.
A student deposits $120 in month 1 and increases each later deposit by 5%. Find (a) the month-8 deposit and (b) the total deposited through month 8.
Show your working.
Give monetary answers to the nearest cent.
A. Use 120(1.05)⁷ for the eighth deposit and 120((1.05)⁸−1)/(1.05−1) for the total.
✅ Correct! 😊 Correct. Month 1 has exponent 0, so month 8 has exponent 7, while the total requires the finite geometric-sum formula for eight terms.
B. Use 120(1.05)⁸ for the eighth deposit and multiply that result by 8 for the total.
❌ Not quite 😭 This shifts the sequence index by one and treats unequal deposits as equal. Neither expression represents the requested quantities.
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C. Use 120+7(0.05) for the eighth deposit and add the first and eighth deposits for the total.
❌ Not quite 😭 This treats a 5% increase as a fixed arithmetic difference. It also omits the six deposits between the endpoints.
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D. Use 120(1.05)⁷ for both the eighth deposit and the total.
❌ Not quite 😭 The term formula correctly finds one deposit, but it does not accumulate all eight deposits. The second part needs a sum.
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Full explanation
- Step 1The deposits form a geometric sequence with first term 120 and common ratio 1.05.
- Step 2Because the first deposit has exponent 0, the eighth is 120(1.05)⁷=168.852… .
- Step 3The total is S₈=120((1.05)⁸−1)/(1.05−1)=1145.892… .
- Step 4Rounding and labeling give an eighth deposit of $168.85 and a total of $1,145.89.
Let P(n) be the statement 1+3+5+⋯+(2n−1)=n² for positive integers n.
Prove P(n) by mathematical induction.
Show each stage of the proof.
A. Assume the sum is k², then add the next odd number 2k+1 to get k²+2k+1=(k+1)².
✅ Correct! 😊 Correct. This explicitly uses the induction hypothesis and turns the next partial sum into the required expression for k+1.
B. Check that the formula works for n=2 and n=3, so it works for every positive integer.
❌ Not quite 😭 Checking examples supports the pattern but does not establish the general implication. No induction hypothesis or step is shown.
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C. Assume the formula is true for k+1 and subtract 2k+1 to obtain k².
❌ Not quite 😭 This assumes the case that must be proved. The logical direction is reversed, so the induction argument does not advance.
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D. Assume the sum is k² and add 2k−1 to obtain k²+2k−1.
❌ Not quite 😭 The next term after 2k−1 is 2k+1. Adding the previous final term does not form the sum through k+1 terms.
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Full explanation
- Step 1Base case: for n=1, the left side is 1 and the right side is 1², so P(1) is true.
- Step 2Assume P(k): 1+3+⋯+(2k−1)=k² for some positive integer k.
- Step 3Add the next odd number: k²+(2k+1)=k²+2k+1=(k+1)².
- Step 4Therefore P(k+1) follows from P(k), so the statement holds for every positive integer n.
Fifteen terms to use precisely
These definitions come from the guide's appendix. Connect each term to a formula, graph, table, interval, or written justification so it does mathematical work in a response.
- Arithmetic sequence
- A sequence with a constant difference between consecutive terms.
- Convergence
- Approach to a finite limiting value as the number of terms grows.
- Modulus
- For a complex number, its distance from the origin.
- Induction
- A proof method using a base case and an implication from one integer to the next.
- Range
- The set of outputs produced on the stated domain.
- Vertical asymptote
- A vertical line approached when function magnitude grows without bound.
- Modulus inequality
- An inequality describing a restriction on distance from a point.
- Cosine rule
- A triangle relation connecting three sides and one included or opposite angle.
- Vector
- A quantity with magnitude and direction represented by components.
- Parameter
- A variable that traces positions in a line, curve, or model.
- Random variable
- A numerical value assigned to each outcome of a chance process.
- Normal model
- A continuous symmetric bell-shaped model defined by mean and standard deviation.
- Density
- A nonnegative function whose interval areas give continuous probabilities.
- Stationary point
- A point where the first derivative is zero.
- Velocity
- The derivative of position with sign indicating direction.
What to expect and how to use this guide
Are the practice problems copied from IB examinations?
No. Every practice context, value, dataset, function, and solution in this guide is newly written for learning.
Can students use a calculator on every written paper?
No. A calculator is not allowed on Paper 1. A graphic display calculator is required for Paper 2 and for HL Paper 3.
Is a formula booklet available in the examination?
Yes. Students must have access to a clean mathematics formula booklet, but they still need to choose relations, state conditions, and show reasoning.
Does each syllabus topic have a published examination percentage?
No topic-by-topic percentage is published in the subject guide used here. The guide publishes assessment-component weights and cautions against assuming equal emphasis among topics.
Is this guide produced or endorsed by International Baccalaureate?
No. This is an independent AskSia study guide. AskSia is not affiliated with or endorsed by the International Baccalaureate Organization. IB and International Baccalaureate are registered trademarks of the International Baccalaureate Organization.
What is the official IB Mathematics: Analysis and Approaches exam structure?
SL has Paper 1 and Paper 2, each lasting 90 minutes and carrying 80 marks, plus an Internal Assessment worth 20%. HL also has Paper 3; the official guide states 60 minutes and 55 marks. If a timetable adjustment is not supported by an official source, this guide does not state it.
Does the official formula booklet or data sheet give me everything I need?
No. A formula booklet is provided in the examination, but students must know derivations that are not in it and justify their working.
Build a repeatable route from model choice to justified answer
Pass 1 — identify the mathematical job. Decide whether the prompt asks for an exact algebraic form, function feature, geometric relation, probability model, derivative, integral, optimization, or proof. Mark the domain, interval, support, units, and calculator rule before choosing a procedure.
Pass 2 — reproduce complete examples. Work from Problem to Calculation to Answer without hiding sequence indices, inverse restrictions, angle branches, distribution conditions, derivative tests, integral bounds, or units. Use the four-column check to name the model element, likely error, and evidence needed for credit.
Pass 3 — connect representations. Turn a solved expression into a labeled graph, diagram, or table, and turn a visual representation back into a defensible expression. Check signs, scale, intercepts, asymptotes, areas, parameter values, support, and limiting behavior in a second representation.
Pass 4 — practice under both tool settings. Rework exact algebra, proof, identities, and calculus without a calculator for Paper 1. For Paper 2 and HL Paper 3, record the equation, viewing region, distribution inputs, or numerical method used by the graphic display calculator; a display value alone does not show the reasoning.
Use the formula booklet actively. Practice finding the relevant relation, naming its variables and conditions, and connecting it to the current representation. The booklet is available in the examination, but students still need to know derivations and methods that are not printed there.
Mix all five topics. International Baccalaureate does not publish per-topic exam percentages, so do not drop a topic because of an estimated planning weight. Alternate focused repair with mixed-paper work, and keep an error log that records the decision that should have been made first.
AskSia is not affiliated with or endorsed by the International Baccalaureate Organization. IB and International Baccalaureate are registered trademarks of the International Baccalaureate Organization. Exam structure is based on official International Baccalaureate information as of 2026-10-09; every practice setting, value, dataset, function, and solution in this guide is newly written for learning.