Maths IA · Algebra and functions · topic 1 of 40
Bridge arch: can a cubic or quartic fit a real shape? · IB Maths IA idea: guide, sample graph and step-by-step plan
gentle start, but the evaluation is the workSoftware will fit a cubic, a quartic, a degree-9 monster to your arch and give you a lovely R². The real question is which one deserves belief.
- ✓ you like real shapes (bridges, arches, hills)
- ✓ you can take a good photo and measure from it
- ✓ you enjoy asking whether a model deserves to be believed
The big idea
← R² is not the answerChoose a real arch, measure points from a scaled photo, and fit polynomials of different degree. A higher degree always hugs the points more closely, so the fit alone cannot decide. You must judge: do the residuals have a pattern, does the shape make sense for the arch, and what happens outside your data? Symmetry, the span and the height give you facts to test each model against. The evaluation is the real work. At HL, software fitting is not enough for criterion E: derive the least squares method, or use calculus on the model (slope, arc length, area), with the algebra shown.
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 →
Work out the least squares quadratic with the normal equations, then compare with software.
→ You show real understanding of the method. It needs partial derivatives or matrices, which are beyond AA (matrices are AI HL), so introduce them carefully. The strongest route for criterion E.Fit a quadratic with the symmetry built in, then fit a cubic and look at the odd term.
→ Your evaluation uses a physical fact to judge the model, instead of R² alone.Compare a parabola, a catenary (cosh) and a circular arc for the same arch.
→ You compare model types, not only degrees. The cosh function is in neither course, so introduce it, and find out which shape suits the way your arch is loaded.Find the area under the arch, the slope at the base, and the arc length.
→ Your fitted function does something useful, and you can compare the numbers with the published ones.Fit using only part of the points and predict the rest, including the ends.
→ Residuals on unseen points show which degree is trustworthy and which is overfitted.A suspension cable, a roller-coaster track, or a hill profile instead of an arch.
→ A new context with a different expected shape, so the choice of model changes with it.Your Maths IA research question: run the check
what would change your mind?Write your own question first. Then make sure it ticks every box:
- Names the structure and the quantities you will measure (horizontal position, height), with how you will get them.
- Says which models you will compare and what the comparison criteria are, not only R².
- Includes a fact about the real structure that you can test the model against (symmetry, span, height).
- Is clear about what you will do beyond the software: derivation, calculus, or an independent check.
- Is narrow enough for 12 to 20 pages: one structure, a few models, done properly.
The picture to draw
A scaled photo with axes on it, and measured points. Draw yours with your own origin, your own scale and the points you chose.
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 chosen structure, why it interests you, and what you will decide: which model, if any, deserves belief.
The photo or map, your scale, how you chose the origin and points, and the measurement uncertainty in each direction.
The least squares idea explained, the equations, and the fitted models with parameters to a sensible number of significant figures.
Residual plots, a test with symmetry or known dimensions, and the problem of overfitting. R² on its own is not a judgement.
A catenary or a circular arc, compared fairly with your best polynomial.
Calculus on the model (area, slope or length), compared with a known value if you can find one.
Limits of the photo (distortion, perspective), the choice of points, and what you would measure next.
Which model you believe, with evidence, and what that says about a real arch.
Sources for the structure's dimensions and any method you used.
What your Maths IA graph could look like
Made-up measurements, to show you the form of a strong graph, not the answer. Yours will come from your own photo. example only
- Axis labels give the quantity, the unit and the measurement uncertainty.
- Error bars on both directions, because both come from your scaled photo.
- Both fits drawn so the reader sees where they agree and where they part.
- The scale shows the whole span, so the ends of the fits, where they differ most, are visible.
- Another graph to consider: a residual plot for each model, with the residual against x. A pattern in the residuals is the sign that the model is wrong.
Does a higher degree help?
rank them: why stop at 2?The root mean square residual always falls as the degree rises, so it cannot choose the model alone. Chart it with your numbers and ask how much each step really gains.
Your game plan
use YOUR photo, YOUR pointsTick steps off as you go. We'll remember on this device.
0 of 10 done
- Choose and predictPick a real arch. Before measuring, write down what degree you expect to be needed and what facts (symmetry, span, height) the right model must respect.
- Trial run (1 lesson)Take a photo square-on, with a known length in view. Measure a few points and check how large your uncertainty is. Fix the photo problems now.
- Plan your pointsDecide how many points and where, including near the ends. Say why. This is evidence of your own decisions.
- Collect and tabulateMeasure the points, convert with your scale, and record the uncertainty. Check the table against the photo.
- Fit the modelsFit several degrees. Show how the method works for one of them by hand before you rely on software.
- Judge themPlot residuals, check symmetry, and test each model on points that you left out. Decide what you will accept as evidence of a good model.
- Try a different typeFit another kind of curve and compare it with your best polynomial fairly.
- Use calculusUse the model to find something (area, slope at the base, length) and compare with a known value or a second method.
- ReflectWhere did software give you a result that the real arch contradicts? What would you change in your measurements?
- Write, cut, checkWrite in the layout order. Keep one worked fit 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
- Choosing the degree by R²: it climbs with every extra term, so it proves nothing.
- Using perspective-distorted photos: points far from the centre are in the wrong place.
- No uncertainty on the measured points, so the residuals cannot be judged.
- At HL, only running a regression in software. Criterion E asks for a derivation or sophisticated calculus too.
- Never testing the model outside the measured points, so overfitting goes unnoticed.
- A model that is never compared with a fact about the real arch.
- Copying the dimensions from a website instead of measuring: the personal data is lost.
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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.