Focus the object
Define the object, dataset, time period, variable or mathematical structure as precisely as possible.
IB Mathematics · Complete topic list
900 topic ideas – page 27 of 36.
Topics 651 to 675 with explanations, methods, course and equipment guidance.
Mathematics and applications
Compare 900 ideas clearly.
The list mixes calculus, statistics, modelling, geometry, number theory, computer science, sport, environmental topics and other areas. Each entry includes a short explanation and visible methods such as differential calculus, integral calculus, statistics or regression.
Planning guidance, not official topic approvalThe IA, EE, AA, Math AI, SL and HL classifications are editorial guidance. The current subject guide, assessment session, mathematical depth, focus and school approval remain decisive.
1
Select an idea. Titles and areas are starting points, not finished research questions.
2
Check A and C. These codes give an initial indication of assessment type, course and level.
3
Read P, M and S. They show possible independent direction, tools, and safety or data-protection needs.
| No. | Topic idea | A | C | P | M | S |
|---|---|---|---|---|---|---|
| 651 | Further water-flow and hydraulic experimentsNozzle shape and discharge coefficient Compare sharp-edged, rounded, and conical nozzles at equal outlet area and water head. |
3 | 6 | 1361112 | 35131620 | 0 |
| 652 | Further water-flow and hydraulic experimentsPipe diameter and volume flow rate Vary the internal pipe diameter while keeping length and pressure head constant. |
3 | 6 | 1361112 | 15131620 | 0 |
| 653 | Further water-flow and hydraulic experimentsPipe length and pressure loss Use pipes of equal diameter but different lengths and measure flow rate or pressure loss. |
3 | 6 | 161112 | 15131620 | 0 |
| 654 | Further water-flow and hydraulic experimentsNumber of pipe bends and flow loss Investigate pipes containing zero to several bends of different angles. |
3 | 6 | 161112 | 15131620 | 0 |
| 655 | Further water-flow and hydraulic experimentsInlet shape and flow rate Compare sharp-edged, funnel-shaped, and rounded pipe inlets. |
3 | 6 | 1361112 | 5131620 | 0 |
| 656 | Further water-flow and hydraulic experimentsUniformity of a perforated pipe Vary hole diameter or spacing along a pipe and measure discharge from each hole. |
3 | 9 | 1361112 | 1516 | 0 |
| 657 | Further water-flow and hydraulic experimentsOpening angle of a triangular weir Pass water through V-shaped weirs with different opening angles. |
3 | 6 | 161112 | 151316 | 0 |
| 658 | Further water-flow and hydraulic experimentsWidth of a rectangular weir Vary the width of a rectangular weir and measure discharge and water depth. |
3 | 9 | 1361112 | 151620 | 0 |
| 659 | Further water-flow and hydraulic experimentsVenturi constriction and pressure difference Build tubes with different constriction ratios and measure pressure before and inside the throat. |
3 | 6 | 12361112 | 5131620 | 0 |
| 660 | Further water-flow and hydraulic experimentsPump head and pump performance Measure the flow rate of a small water pump at different delivery heights. |
3 | 9 | 1361112 | 13516 | 0 |
| 661 | Buoyancy, floating bodies, and water resistanceSalt concentration and immersion depth Float the same object in solutions of different densities. |
3 | 6 | 1361112 | 1511131620 | 0 |
| 662 | Buoyancy, floating bodies, and water resistanceHull shape and maximum payload Build boats from equal-sized sheets using different cross-sections. |
3 | 9 | 12361112 | 251620 | 0 |
| 663 | Buoyancy, floating bodies, and water resistanceCentre-of-mass height and stability Change the height of a weight in a model boat and measure the heel angle. |
3 | 9 | 12361112 | 125151620 | 0 |
| 664 | Buoyancy, floating bodies, and water resistanceDraft and oscillation period of a floating body Displace a floating body vertically and record its oscillation. |
3 | 9 | 1361112 | 12561516 | 0 |
| 665 | Buoyancy, floating bodies, and water resistanceTowing speed and hydrodynamic drag Tow the same body through water at different constant speeds. |
3 | 9 | 1361112 | 5121620 | 0 |
| 666 | Buoyancy, floating bodies, and water resistanceObject shape and drag coefficient Compare spherical, cubic, cylindrical, and streamlined objects with equal frontal area. |
3 | 6 | 1361112 | 511131620 | 0 |
| 667 | Buoyancy, floating bodies, and water resistanceAir volume in a Cartesian diver Vary the trapped air volume and determine the pressure at neutral buoyancy. |
3 | 9 | 13681112 | 51620 | 0 |
| 668 | Buoyancy, floating bodies, and water resistanceBallast position and trim angle Move a weight along a model boat and measure its trim angle. |
3 | 9 | 12361112 | 1251620 | 0 |
| 669 | Buoyancy, floating bodies, and water resistanceBubble diameter and rise velocity Generate bubbles of different sizes and record their rise through water. |
3 | 9 | 136101112 | 135121620 | 0 |
| 670 | Buoyancy, floating bodies, and water resistanceHydrometer stem diameter and sensitivity Build hydrometers with different stem diameters and compare their sensitivity. |
3 | 9 | 12361112 | 151620 | 0 |
| 671 | Surface tension, capillarity, and dropletsCapillary radius and rise height Measure water rise in capillaries of different radii. |
3 | 6 | 161112 | 15131620 | 0 |
| 672 | Surface tension, capillarity, and dropletsSurfactant concentration and contact angle Add different concentrations of detergent and measure contact angle on the same surface. |
3 | 6 | 161112 | 135131620 | 0 |
| 673 | Surface tension, capillarity, and dropletsSurface coating and contact-angle hysteresis Compare untreated glass, wax, plastic, and other coatings. |
3 | 9 | 181112 | 12516 | 0 |
| 674 | Surface tension, capillarity, and dropletsNozzle diameter and droplet mass Determine droplet mass at detachment from nozzles of different diameters. |
3 | 9 | 11112 | 125151620 | 1 |
| 675 | Surface tension, capillarity, and dropletsDrop height and spreading factor Drop equal-sized droplets from different heights onto the same surface. |
3 | 2 | 16101112 | 1571316 | 0 |
No topic ideas match this combination.
Code legend
What A, C, P, M and S mean.
The table stays narrow on a phone by replacing long descriptions with numeric codes. Entries may contain several P and M codes.
AAssessment type
- 1
- Internal Assessment (IA)
- 2
- Mathematics Extended Essay (EE)
- 3
- Potentially suitable for an IA or EE, depending on focus and mathematical depth
CCourse and level
- 1
- Mathematics AA SL
- 2
- Mathematics AA HL
- 3
- Mathematics AA at SL or HL
- 4
- Mathematics AI SL
- 5
- Mathematics AI HL
- 6
- Mathematics AI at SL or HL
- 7
- Mathematics AA or AI at SL
- 8
- Mathematics AA or AI at HL
- 9
- Mathematics AA or AI at SL or HL
PWays to demonstrate independent direction and personal engagement
- 1
- Own data, measurements, observations or experiment
- 2
- Own photographs, drawings, constructions or models
- 3
- Own sport, video, GPS or tracker context
- 4
- Own school or class survey or observation
- 5
- Own everyday, household, consumer or financial data
- 6
- Local context: environment, buildings, traffic, climate or nature
- 7
- Own programming, simulation or algorithm
- 8
- Personal interest: music, art, games, design or another hobby
- 9
- Public data selected, prepared and analysed independently
- 10
- Own conjecture, proof idea, generalisation or theoretical comparison
- 11
- Own modelling decision, construction, optimisation or adaptation
- 12
- Critical comparison of assumptions, errors, limitations or ethical issues
MMeasuring instruments, tools or data access
- 0
- No specialist physical instrument; a calculator, CAS, spreadsheet or open data may be sufficient
- 1
- Ruler, tape measure, calliper or protractor
- 2
- Balance or precision scale
- 3
- Contact thermometer or temperature data logger
- 4
- Infrared thermometer or thermal camera
- 5
- Multimeter or another electrical measuring instrument
- 6
- Stopwatch or timer
- 7
- Camera or smartphone for photographic and video analysis
- 8
- GPS device or fitness tracker
- 9
- Microphone, sound-level meter or audio-analysis software
- 10
- Light meter, light sensor or solar sensor
- 11
- Conductivity, pH or salinity meter
- 12
- Weather instruments, such as an anemometer or rain gauge
- 13
- Computer, spreadsheet, CAS, GeoGebra, Desmos or Python
- 14
- Survey form, data sheet or observation record
- 15
- Force sensor or spring balance
- 16
- Laboratory glassware, measuring cylinder or pipette
- 17
- Telescope, binoculars or a suitable camera
- 18
- Humidity or material-moisture sensor
- 19
- Non-invasive physiology sensor, such as heart-rate or reaction-time measurement
- 20
- Physical model, 3D printer or material samples
- 21
- Specialist school laboratory equipment
SSafety and data protection
- 0
- Likely to be low risk within normal school practice
- 1
- Supervision recommended, for example for heat, electricity, sport or traffic observation
- 2
- Carry out only in a school laboratory or with qualified supervision
- 3
- Sensitive personal or health data: consent, anonymisation and preferably secondary data; no medical self-intervention
From heading to investigation
A topic becomes workable only through independent decisions.
The table is designed to speed up the first step. The actual research question emerges through focus, mathematical choice and critical checking.
Select the mathematics
Decide which models, proofs, statistical procedures or optimisation steps can genuinely answer the question.
Plan independent direction
Use your own data, comparisons, modelling choices, extensions or proof ideas rather than reproducing a standard procedure.
Reflect on limitations
Examine assumptions, sources of error, data quality, model limitations, safety and possible improvements.
Personal engagement and independent direction
A personal connection is more than one sentence in the introduction.
The P codes indicate possible ways to shape an investigation independently. Independent thinking becomes visible through justified decisions, appropriate data selection, personal model variants, meaningful comparisons and critical reflection. A code does not guarantee a particular mark.
Review the IA requirementsFrequently asked questions
Use the topic list correctly.
Are the titles finished research questions?
No. They name a possible direction. A question for assessed work must be focused more narrowly, matched to the course and level, and connected to a clear mathematical method.
What does A = 3 mean?
The broad direction could be developed as an IA or Mathematics EE, depending on focus and depth. An EE will normally require a substantially deeper mathematical argument and an appropriate research scope.
Is C an official IB classification?
No. C is editorial guidance for Mathematics AA or Math AI and SL or HL. Final suitability depends on the specific research question and the current requirements.
Do I need to own the listed instruments?
No. M indicates typical or possible tools. Many topics can use open data, a spreadsheet, CAS, GeoGebra, Desmos or Python. Adapt the topic to resources that are genuinely available.
How should topics involving health or personal data be handled?
Prefer anonymised or publicly available secondary data. Original data collection needs consent, data protection, school approval and a low-risk method. Diagnosis, medication changes and invasive self-experimentation do not belong in a Mathematics project.
The complete list is also machine-readable.
The same 900 entries are available as plain text and bilingual JSON for search, accessibility and AI systems.
Authoritative foundations
Check the current curriculum version before starting.
The catalogue complements the PreLearning explanations. Current official IB documents and the school's instructions remain authoritative.