Vectors, Logs and Errors — Teaching Maths Through Real Experiments

Why hands-on science is the perfect classroom for applied mathematics

Maths becomes far more meaningful when students see it working inside a real experiment.
At Philip M Russell Ltd, whether we’re running physics demos, environmental measurements, PASCO sensor work, or chemistry investigations, we use experiments to teach the mathematics behind them — especially vectors, logarithms, and error analysis.

When formulas move off the page and into the lab, students finally understand why these ideas matter.

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Teaching Vectors Through Experiment

1. Forces in Equilibrium

A simple force table or three-spring setup shows that forces aren’t just numbers — they have direction.
Students learn to:

• break forces into components

• add vectors head-to-tail

• predict equilibrium points

• measure discrepancies between theoretical and experimental results

2. Motion and Velocity Vectors

PASCO SmartCarts or video analysis help demonstrate:

• velocity as a vector

• turning acceleration into vector changes

• how direction changes even at constant speed

Perfect for explaining why sailors talk about apparent wind, or why drones drift in crosswinds.

3. Field Vectors

Magnetic field mapping or electric field plate experiments teach:

• vector fields

• direction of force lines

• superposition

Students physically see vector addition happen in space.

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Teaching Logarithms Through Data

Students often struggle with logarithms until they realise how often nature behaves logarithmically.

1. Radioactive Decay & Half-Lives

A Lascells cloud chamber or decay simulation demonstrates that:

• exponential decay becomes a straight line on a log plot

• ln(N) vs time removes curvature

• the gradient gives decay constant

Logs suddenly become a tool, not a hurdle.

2. Sound Intensity

Decibels are defined logarithmically.
Using a sound sensor, students see how:

• doubling intensity ≠ doubling dB

• logs help us compress huge dynamic ranges

Great for linking physics to music and video production.

3. pH Calculations

Acid–base experiments reveal that:

• pH is a logarithmic scale

• small concentration changes create big pH shifts

• titration curves simplify when plotted using log concentration

Pairing this with your DIY Arduino pH meter makes the maths very real.

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Teaching Errors Through Measurement

Every experiment is an opportunity to teach uncertainty, precision, and propagation of error.

1. Random vs Systematic Errors

Using repeated measurements with SmartCarts or light gates helps students see:

• scatter around a mean

• bias from misaligned equipment

• the importance of calibration

2. Percent Uncertainty

A metre ruler, thermometer, or electronic balance lets students calculate:

• absolute error

• fractional error

• percentage error

They learn that no measurement is perfect — only quantified.

3. Error Bars on Graphs

When plotting Hooke’s Law or Ohm’s Law, students add:

• vertical error bars for measurement variation

• horizontal bars for instrument uncertainty

Seeing error bars helps them judge whether theoretical models match reality.

4. Propagation of Errors

When combining measurements (e.g., power = IV), students explore:

• how uncertainties combine

• why small errors can grow quickly

Excellent preparation for A-Level and university science.

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Why Experiments Make Maths Easier

Because they answer the question students always ask:

“When will I ever use this?”

In the lab, the answer is obvious:

• Vectors explain forces, fields and motion.

• Logs linearise curved data.

• Errors separate good science from guesswork.

Students don’t just learn maths — they experience it.

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The Takeaway

Vectors, logs and errors aren’t abstract topics.
They are the mathematical language of experiments.
By teaching them through real measurements, we give students deeper understanding, stronger skills, and the confidence to analyse the world around them.

Hands-on experiments turn maths into meaning.