Maths IA · Modelling · topic 76 of 97
The Keeling curve: modelling carbon dioxide · IB Maths IA idea: guide, sample graph and step-by-step plan
trend + wiggle: model each oneSince 1958 one mountain in Hawaii has recorded the carbon dioxide in the air, and the graph has a zigzag on top of a climb. Which function describes the climb, and what year does each model say we hit 450 ppm?
- ✓ you want a model that is checked against a famous real data set
- ✓ you like asking which of several fits deserves belief
- ✓ you care about a question where the prediction matters
The big idea
The Keeling curve combines two patterns: a steady rise over decades, and a regular wiggle that repeats every year as plants take up and release carbon dioxide. You can model them separately. Fit the annual means with a line, a quadratic and an exponential, and compare them by residuals as well as R². Then fit the seasonal swing with a sinusoid. All three trend models can fit the past well and still disagree about the future, so the prediction of when 450 ppm is reached is where the thinking happens. Ask what each model assumes about the rate of growth, then decide which assumption the data supports.
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 all three to the annual means and compare residual plots, not just R².
→ Models that look alike on one graph separate when you plot what is left over. Say which one the residuals support.Solve each model for the year it reaches 450 ppm and compare the answers.
→ An equation per model: linear, quadratic formula and logarithms. Explain why the three years differ.Fit a sinusoid to monthly data after removing the trend, and find its amplitude, period and phase.
→ A sinusoidal model with real parameters. Say when the peak occurs and why that month makes physical sense.Model the excess above a pre-industrial level with an exponential, so the baseline is a parameter you choose and justify.
→ A shifted exponential. Test how the doubling time of the excess depends on the baseline you pick.Find the growth in ppm per year from your models and compare it with the growth you measure between years.
→ Differentiation of each model against the data's own year-to-year changes. Which model has the right growth?Fit to data up to a chosen year, then predict the later years you held back and measure the error.
→ A real test of prediction. A model's fit to old data says little about its forecast skill.Your Maths IA research question: run the check
which data? which models?Write your own question first. Then make sure it ticks every box:
- Names the data (station, years, annual or monthly) and where it comes from.
- Names the models you compare and the number you will report: a prediction year, a growth rate or an amplitude.
- Says how models will be judged: residuals and a test on held-back years, not only R².
- Includes a decision to make, such as which model is more believable for the year 2040.
- States the limits: one station, a short record, no physics in the model.
The model
Measured carbon dioxide is a slow trend plus a repeating yearly swing. Draw yours: which function is the trend, and how big is the swing?
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.
What the Keeling curve is, why you model it, and what you expect the best trend model to be.
Station, years, units and source. What an annual mean is and how it removes the seasonal swing.
Every symbol defined. A line, a quadratic and an exponential fitted, with the parameters and what they mean.
Residual plots, R² and a held-back test, read in words. Which model is better and why.
A sinusoid on the monthly data: amplitude, period, phase and a check against the data.
The equation solved for each model, the answers compared, and why they differ.
Back to your aim: the best model, the prediction, and how much trust it deserves.
Short record, one station, what a model without physics cannot know. Say which assumption matters most.
The source of the data and every scientific claim.
What your Maths IA graph could look like
plot the residuals tooMade-up annual values to show the form of a strong graph, not the answer: data against two fitted models, with units. Yours will look different. example only
- Data and models on one set of axes, with both axes labelled in units.
- Fit more than one model, and give each a different line style.
- A residual plot beside it shows what the picture hides.
- Mark what you predict and say how far past the data it goes.
- Another graph to consider: year-to-year growth in ppm against the year.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 9 done
- Read and predictFind out what the Keeling curve shows and why the zigzag appears. Before fitting, sketch the trend you expect.
- Get the dataDownload NOAA's Mauna Loa annual means, and monthly means if you will model the seasons. Record the source and the date.
- Fit three trendsFit a line, a quadratic and an exponential to the annual means. Define every parameter and give its units.
- Trial run (1 lesson)Fit the three models to the first 20 years only and predict the next 10. Fix any mismatch now, such as the time origin.
- Read the residualsPlot the residuals of each model against the year and describe what you see in words.
- Model the seasonsRemove the trend from the monthly data and fit a sinusoid. Find the amplitude, period and phase, and check them against the data.
- Go further, with a reasonSolve each model for 450 ppm, or fit the excess above a baseline. HL: derive the least squares fit or the growth rate with calculus.
- Reflect as you goAfter every result, write one line on what it means for the real atmosphere and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes and units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Linear and quadratic regression – Fit and to annual means and compare coefficients, and residuals.
- Exponential models – Fit with a stated baseline , and solve it for the year using logarithms.
- Sinusoidal models – Fit to detrended monthly data and read the amplitude and peak month.
- Residual analysis – Plot residuals against year to detect curvature that hides, and test models on held-back years.
- Rates of change – Differentiate each trend to get ppm per year, and compare with the year-to-year growth in the data.
Key equations
Syllabus links
- 2.6Quadratic functions
- 2.9Exponential and logarithmic functions
- 3.7Circular functions and their graphs
- 4.4Correlation and the regression line of y on x
- 2.5Linear, quadratic, exponential, cubic and sinusoidal models
- 2.6The modelling process
- 4.4Correlation and linear regression
- 4.13Non-linear regression, residuals and R-squaredHL
- 5.9Further derivatives; chain, product, quotient rules; related ratesHL
Read up first
Data and sources to start from
Download the annual and monthly mean CO2 at Mauna Loa from NOAA's Global Monitoring Laboratory, and note the download date.
- NOAA Global Monitoring Laboratory: trends in atmospheric carbon dioxide
- Scripps CO2 Program, UC San Diego (Keeling curve)
References to cite
- Keeling, C. D. (1960). The concentration and isotopic abundances of carbon dioxide in the atmosphere. Tellus, 12(2), 200-203.
- NOAA Global Monitoring Laboratory. (n.d.). Trends in atmospheric carbon dioxide. https://gml.noaa.gov/ccgg/trends/
- Scripps CO2 Program. (n.d.). The Keeling curve. https://scrippsco2.ucsd.edu/
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
good fit ≠ good forecast- Choosing the model with the highest R² and stopping. All three fit the past well.
- Predicting far beyond the data and presenting the year as a forecast rather than a model output.
- Measuring time from year 0 in an exponential fit, which makes the numbers unstable and the parameters meaningless.
- Fitting a sinusoid to data that still has the trend in it.
- Mixing annual and monthly data without saying which one each result uses.
- Using radians or degrees inconsistently in the seasonal model.
- Calling a model 'right' because it fits, without saying what physics it leaves out.
More Maths IA ideas like this
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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.