IB Mathematics · Complete topic list

900 topic ideas – page 23 of 36.

Topics 551 to 575 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.

Work
Course
Level

Showing 25 of 25 topic ideas

Mixed topic list for the Mathematics IA and Mathematics Extended Essay
No. Topic idea A C P M S
551 Water and flowWater temperature and flow velocity

Measure the flow rate through a narrow tube at constant pressure head and different temperatures. Compare linear, polynomial, and viscosity-based models.

Functions & modellingGeometryAlgebra
3 9 161112 35121620 0
552 Water and flowStreambed roughness and velocity

Use smooth plastic, sandpaper, sand, and gravel as interchangeable beds. Investigate how a defined roughness index relates to velocity and discharge.

Functions & modelling
3 9 1361112 25121620 0
553 Water and flowChannel slope and discharge

Vary the slope of a model channel while keeping other conditions constant. Test whether a power model describes the relationship between slope and discharge.

Functions & modellingAlgebra
3 9 161112 51620 0
554 Water and flowBend radius and velocity distribution

Build channels with different bend radii and film floating tracer particles. Compare velocities along the inner and outer sides of each bend.

Statistics & probabilityGeometry
3 6 1236811 15121620 0
555 Water and flowChannel sinuosity and travel time

Compare straight and meandering channels with the same elevation drop. Model travel time as a function of path length and number of bends.

Functions & modellingStatistics & probabilityGeometryAlgebra
3 6 1361112 1561620 0
556 Water and flowCross-sectional shape and discharge

Compare rectangular, triangular, and semicircular channels with similar cross-sectional areas. Include wetted perimeter and hydraulic radius in the model.

Functions & modellingGeometry
3 9 161112 1351620 1
557 Water and flowWater depth and wave speed

Generate small waves in a long shallow tray and determine their speed using video analysis. Compare a linear model with a square-root model.

Functions & modellingTrigonometryGeometry
3 9 1261112 157121620 0
558 Water and flowObstacle density and flow loss

Place different numbers of equal-sized stones or rods in the channel. Measure travel time, upstream water level, and discharge.

Functions & modelling
3 9 1361112 5616 0
559 Water and flowArtificial aquatic plants and flow

Simulate aquatic plants using flexible strips or stems. Separately investigate the effects of plant density, height, or flexibility.

Functions & modelling
3 9 1681112 1516 0
560 Water and flowGrain size and onset of sediment transport

Gradually increase the flow until sand or gravel grains of different sizes begin to move. Model critical velocity as a function of grain diameter.

Functions & modellingGeometryAlgebra
3 9 1361112 135121620 1
561 Water and flowParticle size and settling velocity

Allow spherical particles or sorted sand grains to settle through a water column. Compare linear, quadratic, and other power models.

Functions & modellingNumber theoryGeometryAlgebra
3 9 168101112 511121620 0
562 Water and flowTemperature and dye dispersion

Place an equal-sized dye drop into still water at different temperatures. Use image analysis to measure coloured area or colour variance over time.

Statistics & probabilityGeometryTime-series analysis
3 6 161112 35671316 0
563 Water and flowSolution concentration and flow rate

Compare water containing different proportions of sugar or glycerol in a narrow tube. Investigate how concentration changes the flow rate.

Geometry
3 9 161112 516 0
564 Water and flowWater head and jet range

Allow water to leave through a side opening in a container. Vary the water height above the opening and model the horizontal range.

Functions & modelling
3 9 161112 151620 0
565 Water and flowOutlet area and drainage time

Drain identical containers through openings of different sizes. Also record water level over time and develop a differential-equation model.

Functions & modellingDifferential equationsGeometryProofTime-series analysisAlgebra
3 2 136101112 5612131620 0
566 Water and flowHeight difference and siphon performance

Vary the height difference between two water containers while using the same tube. Measure flow rate and drainage time.

Geometry
3 9 1361112 13561112 0
567 Water and flowSoil composition and infiltration

Compare defined mixtures of sand, soil, and clay. Record cumulative water infiltration as a function of time.

Algebra
3 9 13681112 561618 0
568 Water and flowFilter grain size and filtration performance

Investigate how the grain size and layer depth of a sand filter affect flow time and turbidity.

Geometry
3 9 161112 13561216 0
569 Water and flowDecay of a water vortex

Create a reproducible vortex and film the angular velocity of a marker. Test exponential and power-decay models.

Functions & modellingTrigonometryAlgebra
3 9 1361112 15121620 0
570 Water and flowWater-wheel blade angle

Build water wheels with different blade angles. Measure rotational speed, torque, or electrical output and determine an optimal angle.

Functions & modellingDifferential calculusTrigonometryOptimisation
3 9 1261112 135151620 1
571 Ice, freezing, and strengthAir-bubble fraction and flexural strength

Photograph equal-sized ice samples before loading and calculate the visible bubble-area fraction. Compare it with maximum breaking load.

Functions & modellingDifferential calculusGeometryOptimisation
3 9 121112 2371115 1
572 Ice, freezing, and strengthBubble size and fracture strength

Determine mean bubble diameter, variation, and bubble count using image analysis. Use multiple regression to predict strength.

Functions & modellingStatistics & probabilityRegressionNumber theoryGeometry
3 6 1101112 13713 0
573 Ice, freezing, and strengthFreezing rate and trapped air

Freeze identical amounts of water at different rates, for example using different insulation layers. Measure bubble content, transparency, and strength.

Functions & modelling
3 9 1681112 31216 0
574 Ice, freezing, and strengthIce temperature and breaking stress

Test standardized ice beams at several temperatures below 0 °C. Calculate a comparable breaking stress from failure load and sample geometry.

Geometry
3 9 11112 2351520 1
575 Ice, freezing, and strengthSalinity and ice strength

Produce ice samples with different accurately measured salt concentrations. Investigate strength, density, and fracture behaviour.

Functions & modelling
3 9 11112 31116 0
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.

Focus the object

Define the object, dataset, time period, variable or mathematical structure as precisely as possible.

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 requirements
Frequently 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.

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