Maths IA · Algebra and functions · topic 3 of 40
Which function best describes pitch against string tension, using a tuner app? · IB Maths IA idea: guide, sample graph and step-by-step plan
measurement quality decides the markTighten a guitar string and the note rises. But by how much? You're going to turn a tuner app into a data logger, and let the data pick the function.
- ✓ you play an instrument or have a guitar to hand
- ✓ you can measure carefully and repeat readings
- ✓ you like fitting models to your own data
The big idea
← the log-log trick!Physics suggests the frequency rises as the square root of the tension, which is a power function with exponent one half. Your job is to collect your own data with a tuner app, decide how to turn note and cents readings into frequencies, then fit models and test which one fits best: linear, power, exponential. A log-log plot turns a power law into a straight line. Measurement quality decides the mark. At HL, use HL-level maths for criterion E: derive the model's form, use logarithms and error propagation, and compare models with residuals, not R² alone.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Fit f = k T^p to your data and give the exponent p with its uncertainty.
→ Your question becomes whether p is close to one half, with an interval to judge it. Needs logarithms or non-linear regression.Plot ln f against ln T and read the gradient as the exponent.
→ A linear regression gives the exponent. At HL, show the algebra behind the linearisation and propagate the uncertainty.Work out how to turn a note name and cents offset into hertz, and say what the app is assuming.
→ A proper unit conversion with exponentials. Your frequency uncertainty becomes something you can actually state.Repeat for two or three string lengths or thicknesses.
→ Your model gets extra parameters. You test whether the exponent stays the same.Fit linear, power and exponential models and compare residual plots, not only R².
→ The evaluation asks which model is best for the data and why, with evidence.A violin string, a ukulele, a rubber band or a bass string.
→ A personal angle. The materials differ, so the constants and the data quality change.Your Maths IA research question: run the check
how will you convert cents?Write your own question first. Then make sure it ticks every box:
- Names the independent variable (tension, and how you will set it) and the dependent variable (frequency, and how the app gives it).
- Says how you will convert the app's output (note and cents) to hertz.
- Says which models you will compare and how you will judge them.
- Says what you will keep fixed (length, string, plucking spot, room).
- Is narrow enough for 12 to 20 pages, with at least seven tension values and repeats.
The set-up
One way to measure. Draw yours, with your real lengths and masses, and say what the app measures.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Your instrument or string, your question, and what you predict about the shape of the relationship.
Where a power law is expected, the idea of a log-log plot, and how note and cents turn into hertz.
The set-up, how tension is set, how many readings, a table with units and uncertainties.
Frequency conversions shown once with a worked example. Averages and uncertainties.
Linear, power and exponential models, the linearisation algebra, parameters to a sensible precision.
Residual plots, error bars, the exponent compared with the theory. R² alone is not a judgement.
Measurement limits of a phone app, plucking effects, hidden assumptions, and what you would change.
Answer your aim and say which model you believe and why.
Sources for the physics and the cents formula.
What your Maths IA graph could look like
error bars both ways ✓Made-up data, to show you the form of a strong graph, not the answer. Your readings will look different. example only
- Axis titles give the quantity, the unit and the uncertainty in what you measured.
- Error bars on both directions. The frequency from the app is the less certain.
- A straight line drawn too, so the reader sees what a simple model would do.
- The scale starts near the data, not at zero. Say so in your text.
- Another graph to consider: plot ln f against ln T. If a power law holds, the gradient is the exponent and the residuals are easy to read.
Your game plan
trial run on the app firstTick steps off as you go. We'll remember on this device.
0 of 10 done
- PredictWrite down what you expect: a straight line, a curve, how steeply. Why?
- Trial run (1 lesson)Try the app on a string at one tension. How steady is the reading? How does it change when you pluck harder or in a different place? How noisy is the room?
- Plan the tensionsChoose at least seven tensions across a wide range and a way to set each accurately (hanging masses). Say why.
- Convert unitsWork out how to turn note and cents into frequency. Check it on a known note.
- Collect dataTake repeated readings at each tension. Record the spread and note anything odd.
- ProcessConvert, average and state the uncertainty in frequency and tension.
- Fit and compareFit linear, power and exponential models, linearising where it helps. Plot the residuals for each.
- InterpretWhat does the exponent mean? How close is it to theory, in terms of its uncertainty?
- ReflectWhat limits the quality of the data? Which assumption is hardest to test?
- Write, cut, checkWrite in the layout order. Show the conversion and one fit by hand and put the rest in tables. Run the checklist before your teacher's draft.
Maths you'll need
Traps to avoid in this Maths IA
- Trusting the app's number without a repeatability test. A tuner can lock onto a harmonic.
- Treating cents as hertz. They are a log scale, so convert before you do any fitting.
- Choosing a model because software offers it, with R² as the only evidence.
- At HL, doing only a software regression. Criterion E needs derivation, logarithms and uncertainty propagation.
- A hanging mass over a bench edge with no safety thought: a falling mass and a snapping string.
- Too few tension values, so that the models cannot be told apart.
- Changing the plucking place between readings.
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This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.