๐ Purpose: Learn the modelling workflow โ choose an appropriate theoretical model, determine a reasonable domain, find parameters by substitution or solving systems, test the model (residuals, fit, reasonableness), and use the model while being aware of limitations (especially extrapolation). Includes step-by-step worked examples and practical GDC/exam tips.
| Term | Definition / Practical note |
|---|---|
| Model | A mathematical formula chosen to represent a real phenomenon (linear, quadratic, exponential, etc.). Always state assumptions and units. |
| Parameters | Constants in the model (e.g., slope m, intercept c, coefficients a, b). They are estimated from data or given conditions. |
| Domain | Set of x-values where the model is valid. Often limited by context (e.g., time โฅ 0). |
| Fit | How well the model reproduces observed data. Measured by residuals or summary statistics (e.g., R2). |
| Extrapolation | Prediction outside the data range โ risky because model assumptions may fail outside observed domain. |
๐ The modelling process โ concise steps
- Understand context: identify variables, units, and what is being modelled.
- Choose candidate models: linear, quadratic, exponential, direct/inverse, cubic, sinusoidal โ choose based on shape & context.
- Determine reasonable domain: physical or contextual limits (e.g., time โฅ 0, concentration โฅ 0).
- Fit parameters: use substitution (exact points) or set up simultaneous equations; use technology if needed.
- Test model: check residuals, plot fit, compute simple summary (e.g., R2 when available), check sensitivity.
- Interpret & use: explain numerical results in context; avoid unjustified extrapolation.
๐ Finding parameters โ worked examples (full working shown)
Example A โ Find a linear model through two data points
Data: (x1, y1) = (2, 5) and (x2, y2) = (5, 11). Seek f(x) = m x + c.
Step 1 โ compute slope m:
m = (y2 โ y1) รท (x2 โ x1) = (11 โ 5) รท (5 โ 2) = 6 รท 3 = 2.
Step 2 โ find intercept c:
Use one point, e.g. (2,5): 5 = 2ร2 + c โ 5 = 4 + c โ c = 1.
Model: f(x) = 2x + 1.
Check: For x = 5, f(5) = 2ร5 + 1 = 11 (matches data). Always substitute at least one data point to verify.
Example B โ Find a quadratic passing through three points
Data: (0,1), (1,4), (2,11). Seek f(x) = a x2 + b x + c.
Step 1 โ set up equations by substitution:
- x = 0: aยท02 + bยท0 + c = 1 โ c = 1.
- x = 1: aยท12 + bยท1 + c = 4 โ a + b + c = 4 โ with c = 1 โ a + b = 3.
- x = 2: aยท22 + bยท2 + c = 11 โ 4a + 2b + c = 11 โ with c = 1 โ 4a + 2b = 10.
Step 2 โ solve the 2ร2 linear system:
From a + b = 3 โ b = 3 โ a. Substitute into 4a + 2b = 10:
4a + 2(3 โ a) = 10 โ 4a + 6 โ 2a = 10 โ 2a = 4 โ a = 2.
Then b = 3 โ a = 3 โ 2 = 1. We already have c = 1.
Model: f(x) = 2x2 + 1x + 1.
Check quickly: f(1) = 2(1)2 + 1 + 1 = 4; f(2) = 8 + 2 + 1 = 11 โ matches data.
๐ Solving systems & using technology
โข For models with three unknown parameters (e.g., quadratic) you can solve 3 linear equations in 3 variables by substitution, elimination, matrix methods or by using your GDC to solve the linear system (recommended for speed and accuracy).
โข At SL you are expected to set up and solve up to three linear equations in three variables โ show the algebraic setup even if you use a calculator to obtain the numeric solution.
๐ Test & reflect โ measures of fit and reasonableness
- Residuals: For each data point (xi, yi) compute residual ri = yi โ f(xi). Small residuals โ good fit (but consider pattern of residuals).
- R2 (if available): A summary measure of proportion of variance explained โ useful but not the only criterion.
- Plot residuals: Look for systematic patterns; random-looking residuals indicate model captures main structure.
- Sensitivity: Test how small changes in parameters (ยฑฮด) change predictions โ indicates model robustness.
- Parsimony: Prefer simpler models that explain data well (avoid overfitting with unnecessarily complex models).
Illustration โ Quick residual check (linear example):
Using f(x) = 2x + 1 from Example A and an additional observed point (3, 7): predicted f(3) = 2ร3 + 1 = 7, residual r = 7 โ 7 = 0. Residuals of 0 indicate perfect match for those points (but dataset may be exact by construction).
๐ Interpreting results & being careful with extrapolation
โข Always interpret numeric answers in context (include units). E.g., “x = 7 km” rather than just “7”.
โข State domain: predictions outside the observed x-range are extrapolations and may be unreliable. Give a short justification when extrapolating (physical constraints, reason for steady growth, etc.).
โข Discuss model limitations: measurement error, omitted variables, or non-stationarity (if data-generating process changes over time).
๐งฎ GDC Tips
- Use regression tools to fit models (linear, quadratic) โ but show algebraic derivation/verification in the exam.
- Use matrix solve or system solver for simultaneous equations (3ร3). Save the calculator output or note the method used in your working.
- Use Table/Trace to compute residuals quickly and inspect patterns visually.
- When using technology, round final numerical results to appropriate significant figures and state that values were obtained with a GDC.
๐ง Examiner Tip
In modelling questions always: (1) state the model with variable definitions and units, (2) show step-by-step parameter calculations, (3) verify by substitution, (4) test residuals or a simple goodness-of-fit check, and (5) interpret the result (including domain and limitations).
๐ TOK Perspective
When is a model a useful simplification and when is it misleading? Consider the trade-off between simplicity and explanatory power: a simple model may be robust but miss subtleties; a complex model may fit the data well but overfit and generalize poorly.
๐ Quick modelling checklist
- Define variables and units; write the chosen model form clearly.
- Give reasons for model choice (shape of data, context, physical law).
- Show full algebra when finding parameters (substitution/elimination or matrix set-up).
- Check model: substitute back, compute residuals, and comment on fit.
- State domain and caution explicitly about extrapolation.