D. Data · Data, Statistics and Mathematical ModellingWhat Is Mathematical Modelling?
D2.1 A2About 60 minutes
Learning goals- I can explain what a mathematical model is and why models need assumptions.
- I can describe the steps of the mathematical modelling process.
- I can describe how mathematical models are used in real life to inform decisions.
- I can explain how different knowledge systems can contribute to a model.
Words to know
mathematical model
A mathematical description of a real situation, such as an equation, graph, table or simulation.
mathematical modelling
The process of creating, testing and using a mathematical model to answer a question about a real situation.
assumption
Something you accept as true to simplify a problem.
prediction
An estimate of a value or outcome, made using a model.
limitation
A way in which a model does not match reality, or a situation where it does not work well.
revise
To change and improve a model after testing it.
"Snow day tomorrow?"
The night before a big storm, school boards decide whether to cancel buses. They do not flip a coin. They look at weather forecasts, which come from computer models that use enormous amounts of data: temperature, pressure and wind measurements from weather stations, balloons and satellites. The forecast says something like "20 to 30 cm of snow". The board uses that prediction, plus what it knows about its roads and buses, to make a decision.
That is mathematical modelling: using mathematics to represent a real situation so we can understand it, predict what might happen and make better decisions.
What is a mathematical model?
A mathematical model is a mathematical description of a real situation. It can be:
- an equation or formula (C = 4 + 2d for a taxi fare),
- a graph or regression line from data,
- a table, a diagram or a map,
- a simulation run on a computer.
Every model is a simplification. To build one, you make assumptions: statements you accept as true to make the problem manageable (for example, "every snow day uses the same amount of salt"). The statistician George Box famously wrote that all models are wrong, but some are useful. A model is useful when it is close enough to reality to help answer the question.
The modelling process
Modelling is not a single calculation. Ontario's math curriculum describes four connected components that you often go through more than once: understand the problem, analyse the situation, create a mathematical model, and analyse and assess the model. In this course you finish by reporting what you found, so we add a fifth step:
- Understand the problem. What is the question? What would a useful answer look like?
- Analyse the situation. What information do you need? What varies and what stays the same? What assumptions will you make? Where will you get data?
- Create a mathematical model. Collect and organize data, choose a representation, and build an equation, graph or other model.
- Analyse and assess the model. Use it to answer the question. How well does it fit? What are its limitations? Does the answer make sense?
- Communicate the solution. Report the answer, the model, the assumptions and the limitations to the people who need it.
If the model does not fit well, you go back, change an assumption or collect more data, and revise.
Why modelling matters: real decisions
- Public health: models of how illnesses spread help decide where to send vaccines and staff.
- Weather and climate: forecasts guide snow-day decisions, flight schedules and farmers' planting.
- Electricity: utilities forecast how much power people will use on the coldest winter evenings and hottest summer afternoons, so they can supply enough.
- Transit and roads: models of traffic help plan bus routes, traffic lights and snow-clearing routes.
- Wildlife and fisheries: population models help set fishing limits so fish stocks can recover.
- Sports: teams model player performance to plan line-ups and strategy.
- Personal decisions: comparing phone plans, planning a budget or deciding how much paint to buy are small models too.
Good models combine data with knowledge of the situation. For example, some Inuit communities in the Arctic take part in sea-ice monitoring programs, such as SmartICE, that combine Inuit knowledge of ice and travel with sensor measurements to help people travel safely on the ice. Different knowledge systems can make a model better.
Worked exampleExample 1: Identify the parts of a model
Jordan is ordering pizza for a class party of 28 students. Jordan writes: "Assume each student eats 3 slices and each pizza has 8 slices. Number of pizzas P = 3n ÷ 8, rounded up." Identify the question, the variables, the assumptions and the model, and use it.
- Question: How many pizzas should be ordered?
- Variables: n = number of students (here 28), P = number of pizzas.
- Assumptions: every student eats exactly 3 slices; every pizza has 8 slices; nobody is absent.
- Model: P = 3n ÷ 8, rounded up (you cannot order part of a pizza, and running out is worse than leftovers).
- Use it: P = 3(28) ÷ 8 = 84 ÷ 8 = 10.5, so order 11 pizzas.
- Assess: Some students may eat 2 slices and some 4, and some may not eat pizza at all. Jordan could improve the model by asking the class first.
Order 11 pizzas. The model is useful, but its answer is only as good as its assumptions.
Worked exampleExample 2: How a model informs a decision
A town's public works department has used a model for several winters: salt needed (tonnes) = 2.5 × number of snow days. Last winter had 52 snow days. A long-range forecast suggests this winter could have between 50 and 70 snow days. How much salt should the town order, and what are the risks?
- Last winter: 2.5 × 52 = 130 t (useful to check the model against what was actually used).
- Low estimate: 2.5 × 50 = 125 t. High estimate: 2.5 × 70 = 175 t.
- Decision: ordering 175 t protects against running out, but costs more and needs storage. Ordering 125 t saves money but risks a shortage during a storm.
- Limitations: the model assumes every snow day needs the same salt. Freezing rain days may need more; light flurries less. Very cold days also matter because salt works less well at very low temperatures.
A reasonable decision is to order about 150 t now and arrange to buy more if the winter turns out snowier. The model does not make the decision, but it gives the town a clear, defensible starting point.
Just for funA modeller, a meteorologist and a cat walk out into a snowstorm. The modeller says "I predicted 25 cm." The meteorologist says "I forecast 20 to 30." The cat just goes back inside. (The cat's model, "snow is cold", has never been wrong.)
Hands-on activityHow long will the milk last?
You need: your family's milk (or juice, or cereal), a measuring cup, paper and a pencil.
- Understand: your question is "How many days will a full container last in our home?"
- Analyse: list what you need to know (the amount in a full container, the amount used per day). List your assumptions (for example, "we use about the same amount every day"). Many Ontario families buy milk in a 4 L bag of three smaller bags: check yours.
- Create: for 2 or 3 days, record how much is used (measure what you pour, or measure what is left each evening). Find the mean daily use, d. Write a model: days = total ÷ d.
- Assess: use your model to predict when the next container runs out. Then check whether it really does. Was your prediction close? Why or why not (weekends, guests, baking)?
- Communicate: write a two-sentence note to your family with the prediction and one limitation.
What to look for: a clear question, at least two assumptions, a model written as an equation, and a check against what actually happened.
Your turn
1In your own words, what is a mathematical model?
A mathematical description of a real situation (an equation, graph, table, diagram or simulation) that is used to understand it, make predictions or make decisions.
2List the five steps of the modelling process.
- Understand the problem. 2. Analyse the situation. 3. Create a mathematical model. 4. Analyse and assess the model. 5. Communicate the solution. (Then revise if needed.)
3Why do models need assumptions?
Real situations are too complicated to include every detail. Assumptions simplify the situation so it can be described with mathematics.
4Give two real examples of models that help make decisions in your community.
Examples: weather forecasts used to cancel school buses; electricity demand forecasts; traffic models for timing lights; transit ridership models for scheduling buses; snow-clearing plans; budgets.
5A model predicts 11 pizzas for 28 students. Name one assumption that, if wrong, would make the prediction too low.
"Each student eats 3 slices": if many students eat 4 slices, 11 pizzas (88 slices) would not be enough for 28 × 4 = 112 slices.
6Explain the quote "All models are wrong, but some are useful."
Every model simplifies reality, so its answers are never perfectly exact. But a model can still be close enough to help you understand, predict and decide, which makes it useful.
More practice: warm-up
1Name three forms a model can take.
Any three of: equation, graph, table, diagram, map, simulation.
Something you accept as true to simplify a problem.
3True or false: a model is finished after the first try.
False. Modelling is a cycle; you revise when needed.
4Use the salt model S = 2.5d to find the salt for 40 snow days.
5Which step of the modelling process includes deciding what data you need?
Step 2, analyse the situation.
More practice: on your own
1A school wants to know how many water bottle filling stations to install. List three pieces of information it would need.
Examples: the number of students and staff; how often people refill bottles per day; how long one fill takes; the busiest times (breaks and lunch); the cost of each station.
2A model of a bus route assumes the bus always travels at 30 km/h. Describe a situation where this assumption fails, and how it affects predictions.
During a snowstorm or rush hour, the bus moves more slowly, so the model would predict arrival times that are too early.
3Use the pizza model to find the number of pizzas for 45 students. Then find it if each pizza had 12 slices.
8 slices: 3(45) ÷ 8 = 135 ÷ 8 = 16.875, so 17 pizzas. 12 slices: 135 ÷ 12 = 11.25, so 12 pizzas.
4Explain how combining Indigenous knowledge with sensor data can improve a model of safe sea-ice travel.
Sensors give precise measurements at specific places and times. Inuit knowledge, built over generations of travel, includes where ice is usually dangerous, how currents and weather affect it, and how it changes through the season. Together they give a more complete and reliable picture than either alone.
5A store models weekly ice-cream sales using only the temperature. Suggest two other variables that might improve the model.
Examples: whether it is a holiday weekend (such as Canada Day), whether there is a sale, school vacation, rainy days, local events.
Challenge
1A hockey arena wants to model how much hot chocolate to stock for each game. Write a simple model (with variables and assumptions), use it for a game with 1 800 fans, and name one limitation.
Sample: assume 20% of fans buy one cup each. Cups C = 0.2 × F, where F is the number of fans. For 1 800 fans: C = 0.2 × 1 800 = 360 cups. Limitation: on a very cold night more fans might buy, and some buy two; past sales data would make the percentage more accurate.
2The salt model predicted 130 t for last winter's 52 snow days, but the town actually used 160 t. Suggest two reasons and one way to revise the model.
Reasons: more freezing-rain days, more very cold days, new roads added, or heavier storms. Revision: use actual data to change the rate to 160 ÷ 52 ≈ 3.1 t per snow day, or build a model that treats different kinds of storm days differently.
3Explain why a model used to make a public decision (like closing schools for a storm) should be communicated with its assumptions and limitations.
People affected by the decision deserve to know why it was made and how certain it is. Sharing assumptions and limitations helps people judge the decision, trust the process, prepare for uncertainty, and suggest improvements.
Quiz
1. Which is the best description of a mathematical model?
2. Why do modellers make assumptions?
3. Which step comes right after "Create a mathematical model"?
4. A pizza model gives P = 3n ÷ 8. For n = 30, how many pizzas should be ordered?
5. Which is an example of a model informing a decision?
6. A model's prediction is very different from what actually happened. What should a modeller do?
Big idea: D2.1 asks students to describe the value of mathematical modelling and how it is used in real life to inform decisions. The next four lessons (D2.2 to D2.5) walk through the modelling process step by step, ending with a full modelling project.
How to help: Point out models in daily life: weather forecasts, a GPS arrival time, a recipe scaled for more people, a budget. Ask "What assumptions does it make? When would it be wrong?"
Common difficulties:
- Thinking a model must be perfect to be useful.
- Forgetting to check whether an answer makes sense (for example, 10.5 pizzas).
- Seeing modelling as one calculation rather than a process that gets revised.