Real problems don’t come with an equation. Make one — then find out where it breaks.
Two students are given the population of a town for the last ten years. One writes a rule that multiplies by the same number every year. The other writes a rule where the growth slowly eases off. Both rules match every figure they were given.
Now run both of them fifty years ahead. One says the town has outgrown the country. The other says it has almost stopped growing. Both fitted the data. Only one can be believed — and the question is why.
That is modelling. No real problem arrives with an equation attached. Somebody has to make one up, by deciding what matters and what can be ignored. Every model leaves something out. The work is finding what was left out, and when it starts to matter.
Every session is a new situation, begun and finished in ninety minutes: newspapers and forests, crickets and temperature, a car braking, a herd of deer, a ferry timetable. Each one has a place where the model stops telling the truth. Finding that place is the session.
What you need before starting: Arithmetic, fractions and percentages, and drawing a simple graph. Taking Out of Reach first helps, because the algebra used here is built there, but it is not required.
Nobody is handed a formula. The situation goes up, the group decides what matters, builds a model, and then spends the rest of the ninety minutes pushing it until it gives way.
The questionHow much forest does one newspaper cost?
CoversPaper weight, then trees, then how closely trees are planted: a chain of simple steps with one answer at the end. When one answer comes out a thousand times too big, the group hunts for the broken link.
LandsA model is a chain of decisions, and one weak link spoils the whole answer.
The questionDrive to work at 30 km/h and home at 60 km/h. What was your average speed?
CoversNearly everyone says 45. Work out how long each journey took, and it isn’t.
LandsAn answer can feel completely right and still be wrong — and a model shows you why.
The questionRecycling has been rising steadily for years. Draw a straight line through it and look ahead.
CoversThe line fits the figures well. Run it far enough and it says more than 100% of plastic is being recycled.
LandsA model can work today and say something impossible tomorrow.
The questionWater pours into two containers at a steady rate. How does the depth rise? Sketch it before anyone measures.
CoversOne container fills evenly and the other does not. Sketch first, then measure, and keep the wrong sketches on the board: they teach the most.
LandsThe shape of a graph carries meaning, even with no numbers on it.
The questionCrickets chirp faster when it is warm. Can you tell the temperature by counting chirps?
CoversDraw a line of best fit by eye. It works well across the temperatures measured, and says ridiculous things outside them.
LandsFitting the numbers you have tells you nothing about the numbers you don’t.
The questionHow far does a car travel before it stops?
CoversThe distance covered before the driver reacts, plus the distance spent braking. One grows steadily with speed, the other much faster — and the units have to be watched, or the answer is wrong.
LandsReal things are often two simple parts added together, and the parts take turns to matter.
The questionA goat is five years old. How old is that in human years?
CoversTry one rule for every age, then two rules: one for young goats and one for older ones.
LandsUsing different rules for different stages is not cheating. It is often the honest model.
The questionSound travels faster in warm air. Is temperature the whole story?
CoversBuild a rule from temperature, then discover that the dampness of the air matters too.
LandsSometimes a model fails because the question left something out.
The questionA herd grows each year from the size it was the year before. How many deer in fifty years?
CoversRun it year by year. Within a few decades there are more deer than the land could ever feed.
LandsNothing grows forever. The only question is when the model stops being true.
The questionEach bounce reaches a fixed fraction of the one before, so in the model the ball bounces forever. Does that mean it never stops?
CoversAdd up the distances and the times. Infinitely many bounces make a finite total, and the group argues about it, loudly.
LandsA model can be trusted where intuition fails, once you understand why.
The questionYou walk up a moving escalator. How long does the journey take?
CoversBuild the rule, then decide for which escalator speeds it makes any sense at all — and defend the answer.
LandsEvery model has a range where it means something. Finding that range is part of the work.
The questionA ‘hundred-year flood’ happened last year. Is the town safe for another ninety-nine?
CoversWork out what the phrase really means, and how likely it is that the next flood still has not come after ten, twenty or fifty years.
LandsA model read wrongly does real harm. Reading it rightly is a skill.
The questionTwo rules both claim to describe how long a satellite takes to go round. Which one should you trust?
CoversDecide from the shape of the data, not from a calculation, and argue for your choice.
LandsChoosing between models is a mathematical judgement, and it can be argued for.
The questionRaise the price of a pizza and you sell fewer. What price makes the most profit?
CoversPut a profit rule and a demand rule together. Find the best price — and then the more useful answer: the range of prices the business can live with.
LandsThe best answer is a single point. A business needs a range.
The questionTwo designs for a cat enclosure. Which one should be built?
CoversCompare space, floor area and headroom. The mathematics narrows the choice to two, and then a person has to decide.
LandsMathematics narrows a choice. People still have to make it.
The questionCan a ferry service keep its promised journey time?
CoversUse random numbers to imitate the wait for a ferry, model the crossing with speed and distance, put the two together, and answer: yes or no.
LandsModels do answer questions. They answer them with conditions attached.
Groups of four, Grades 6 to 8. Every session is a whole problem, started and finished in ninety minutes, so a student who misses a week misses one problem, not a chapter.
A group opens when four students are ready.