# Indirect Measurement Methods for IB Mathematics IA & EE

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Updated: 2026-09-11

Core principle: raw measurement -> model -> derived quantity -> mathematical analysis.

## Conductivity example
Measured: Uref, Usolution, temperature, electrode spacing L and electrode area A.
I = Uref / Rref
Rsolution = Rref * Usolution / Uref
σ = L / (Rsolution * A)

## Measurement chains
### 1. Calliper + scale
- Measured directly: Diameter, length, mass
- Derived indirectly: Density ρ = m/V
- Research question: Which geometric model gives the most realistic density of an object?
- Mathematics: Geometry, model comparison, relative error

### 2. Scale + timer
- Measured directly: Mass over time
- Derived indirectly: Evaporation rate dm/dt
- Research question: Which model describes evaporation under different airflow conditions?
- Mathematics: Difference quotients, regression

### 3. Scale + fan + temperature
- Measured directly: Mass, temperature, time
- Derived indirectly: Airflow-dependent evaporation
- Research question: How do airflow and temperature change the evaporation constant?
- Mathematics: Multiple regression, piecewise models

### 4. Measuring cylinder + timer
- Measured directly: Volume, time
- Derived indirectly: Flow rate Q = ΔV/Δt
- Research question: How does the flow rate change during drainage?
- Mathematics: Rates of change, regression

### 5. Temperature data logger
- Measured directly: Temperature over time
- Derived indirectly: Cooling constant
- Research question: Which container insulation minimises the cooling constant?
- Mathematics: Exponential regression, residuals

### 6. DS18B20 at several positions
- Measured directly: Temperature, position, time
- Derived indirectly: Temperature gradient ∂T/∂x
- Research question: How does a temperature gradient develop along a metal rod?
- Mathematics: Spatial regression, rates of change

### 7. Thermal imager
- Measured directly: Temperature field
- Derived indirectly: Area mean, gradient
- Research question: How does the area above a temperature threshold change?
- Mathematics: Image scaling, area analysis

### 8. CO₂ sensor
- Measured directly: CO₂ concentration, time
- Derived indirectly: Air-exchange rate
- Research question: Which ventilation strategy maximises the air-exchange rate?
- Mathematics: Exponential model, parameter comparison

### 9. EMF meter + tape measure
- Measured directly: Field strength, distance
- Derived indirectly: Exponent of a power law
- Research question: Which exponent n describes B(d) = a·d⁻ⁿ + c?
- Mathematics: Power regression, log-linearisation

### 10. HX711 load cell
- Measured directly: Digital raw readings
- Derived indirectly: Force or mass after calibration
- Research question: Which calibration model minimises sensor error?
- Mathematics: Calibration, residuals, uncertainty

### 11. Microphone + tape measure
- Measured directly: Frequency, tube length
- Derived indirectly: Speed of sound
- Research question: How accurately can the speed of sound be obtained from resonances?
- Mathematics: Regression of f against 1/L

### 12. Phone video + scale reference
- Measured directly: Pixel position, frame number
- Derived indirectly: Velocity, acceleration
- Research question: Which model describes a projectile path?
- Mathematics: Quadratic regression, derivatives

### 13. Phone gyroscope
- Measured directly: Angular velocity
- Derived indirectly: Angular position, damping
- Research question: How does the damping rate change with friction?
- Mathematics: Integration, exponential models

### 14. GPS + time
- Measured directly: Position, time
- Derived indirectly: Speed, path length
- Research question: How do GPS and mapped path lengths differ?
- Mathematics: Coordinates, numerical summation

### 15. GPS at a fixed point
- Measured directly: Coordinate sequence
- Derived indirectly: Spread, error ellipse
- Research question: How does the built environment change GPS spread?
- Mathematics: Covariance matrices, statistics

### 16. Compass + GPS/map
- Measured directly: Bearing, position
- Derived indirectly: Triangulated position
- Research question: How does position uncertainty grow for small intersection angles?
- Mathematics: Trigonometry, error propagation

### 17. ESP32-CAM + image analysis
- Measured directly: Pixel values, image time
- Derived indirectly: Area, spreading rate
- Research question: How can dye spreading be modelled?
- Mathematics: Scaling, power models, regression

### 18. Raspberry Pi + sensor logger
- Measured directly: Several time series
- Derived indirectly: Correlation, lag
- Research question: How are temperature, CO₂ and ventilation related over time?
- Mathematics: Time series, correlation, lag