The Mathematical Modelling Process

A mathematical model uses math (an average, a rate, an equation, or a graph) to describe a real situation so you can answer a question about it. Models help people make decisions: how much food to order for a school dance, when to leave for the bus, or how long it will take to save for something. On this page you’ll learn the steps of the modelling cycle and practise them on real-life questions.

Key ideas

What a model is (and isn’t)

A model is a simplified version of reality. It keeps the parts that matter for your question and ignores the rest. For example, “my phone loses about $0.28$ percentage points of battery per minute of video” is a model. It ignores screen brightness, the age of the battery, and whether apps are running in the background.

That’s fine. A model doesn’t have to be perfect. It has to be good enough to answer the question and honest about what it leaves out.

Models are used everywhere to make decisions:

• City planners use models of traffic and population to decide where to build roads and schools.
• Weather forecasters use models to predict tomorrow’s temperature.
• A family can use a simple model of its water use to decide whether a low-flow shower head is worth buying.

The modelling cycle

Building a model is a process, and you often go around it more than once.

The modelling cycle. Step 6 often sends you back to the start with a better question or better data.

1. Ask a question of interest. Make it specific enough to answer with numbers. “Do we use a lot of water?” is vague. “How many litres of water does our household use for showers in a year?” is answerable.
2. Identify the information you need. What quantities go into the answer? For the shower question: how much water the shower uses per minute, how long showers last, and how many showers there are.
3. Make a plan. Decide where the data will come from: first-hand (you measure, time, count, or survey) or second-hand (a trusted source such as Statistics Canada or a product label). Write down your assumptions, and decide what varies and what stays the same.
4. Collect the data. Carry out your plan and record the results in a table.
5. Display and analyse the data to build a model. Choose a graph that fits the data, then find a model (see the table below).
6. Answer the question and judge the model. Use the model to answer the original question. Then ask: how well does it fit? What are its limitations? What predictions can it make?

Then revise if needed: collect more data, fix an assumption, or ask a sharper question.

Assumptions, and what varies

An assumption is something you decide to treat as true so the problem becomes manageable. Good models state their assumptions out loud, because if an assumption is wrong, the answer might be too.

It also helps to sort the quantities in your situation:

• What varies: quantities that change from one measurement to the next (the length of each shower).
• What stays the same: quantities you treat as constant (the flow rate of the shower head).

Choosing a model

The kind of data you collect suggests the kind of model to build.

Your data	A good display	A model to try
One quantity measured many times (shower lengths, walk times)	dot plot, histogram, or box plot	an average: mean or median
An amount per unit (litres per minute, dollars per week)	a table	a rate: total = rate $\times$ amount
Two quantities that change together	scatter plot	a line of best fit $y = mx + b$

For one-variable data, you can also use quartiles and box plots to see how spread out the values are, and different graphs to show them. The Grade 9 curriculum writes a line as $y = ax + b$; this site usually writes $y = mx + b$. They mean the same thing: the slope is the rate of change and the $y$-intercept is the initial value.

Judging a model

When you report your answer, include three things:

• Fit. Do the data points sit close to the line, or cluster tightly around the average? A model that fits well gives more trustworthy answers.
• Limitations. Which assumptions might be wrong? Was the sample small, or collected in an unusual week?
• Predictions. What can the model predict, and how far can you trust it? Predictions inside the range of your data (interpolation) are safer than predictions far outside it (extrapolation).

Data in society

Today, huge amounts of data (big data) are collected automatically, often without people noticing. Phones record locations. Fitness apps record heart rates and sleep. Transit cards, like Ontario’s PRESTO card, record trips. Streaming services record everything you watch.

This data can help: traffic apps reroute drivers around a crash, and health agencies can plan where clinics are needed. But it raises real questions:

• Privacy and storage. Who can see the data? How long is it kept? Could it be sold, or stolen in a data breach?
• Use. Is it used only for what people agreed to? Could it be used to treat people unfairly?
• Representation. Data can be shown in ways that mislead: a graph whose vertical axis doesn’t start at zero, a time range picked to hide a trend, a biased sample, or a claim that one thing causes another just because they’re correlated.

When you see data in the news or an app, ask: who collected it, how, and what might they want me to think?

Worked examples

Example 1: Saving for a bike

Maya wants a bike that costs $480 before tax. She has $60 saved. Over the last six weeks she earned $40, $55, $35, $50, $45, and $45 babysitting, and she plans to save all of it. About how many weeks will it take her to afford the bike?

Solution. Work through the cycle.

Question: How many weeks until Maya can buy the bike?

Information needed: the price with tax, what she has now, and how much she saves per week.

Assumptions: her weekly earnings stay about the same as in the past six weeks, she spends none of it, and the price doesn’t change. What varies: her weekly earnings. What stays the same: the price.

Model. Her earnings vary, so use an average. The mean weekly earnings are

$$\frac{40 + 55 + 35 + 50 + 45 + 45}{6} = \frac{270}{6} = 45$$

so the model is “Maya saves about $45 per week”. The price with $13\%$ HST is

$$480 \times 1.13 = 542.40$$

She still needs $542.40 - 60 = 482.40$ dollars. At $45 per week:

$$482.40 \div 45 \approx 10.72 \text{ weeks}$$

Answer. After $10$ weeks she won’t quite have enough, so it takes about $11$ weeks.

Check: $60 + 11 \times 45 = 555$, which is more than $542.40. After $10$ weeks she would have $60 + 450 = 510$ dollars, which is not enough.

Example 2: Water for showers

A family of four wants to know: how many litres of water do we use for showers in a year? They plan to use the answer to decide whether to buy a low-flow shower head.

Solution.

Information needed: the shower’s flow rate (litres per minute) and the total shower time per year.

Plan and data. They hold a $4$ L bucket under the shower and time how long it takes to fill: $25$ seconds. Then everyone writes down the length of every shower for one week. The log shows $24$ showers with a total of $192$ minutes.

Assumptions: the flow rate stays the same, the week they recorded is a typical week, and they shower like this all $52$ weeks of the year.

Model. The flow rate is a rate model:

$$\frac{4 \text{ L}}{25 \text{ s}} = 0.16 \text{ L/s} \quad\Rightarrow\quad 0.16 \times 60 = 9.6 \text{ L/min}$$

Water per week:

$$192 \text{ min} \times 9.6 \text{ L/min} = 1843.2 \text{ L}$$

Water per year:

$$1843.2 \times 52 = 95\,846.4 \text{ L}$$

Answer. The family uses roughly $96\,000$ L of water for showers each year. That’s about $96$ m³, since $1$ m³ $= 1000$ L.

Using the model to decide. A low-flow shower head uses about $7.6$ L/min. With the same shower times:

$$192 \times 7.6 \times 52 = 75\,878.4 \text{ L}$$

That would save about $96\,000 - 76\,000 = 20\,000$ L per year.

Limitations. One week is a small sample. Exams, holidays, or a summer sports season could change shower times. A better plan would record two or three weeks at different times of year.

Example 3: When will my battery hit 20%?

Leo fully charges his phone and streams video, recording the battery level every $30$ minutes.

Time, $x$ (min)	$0$	$30$	$60$	$90$	$120$	$150$
Battery, $y$ (%)	$100$	$91$	$83$	$74$	$66$	$58$

How long can he stream before the battery reaches $20\%$?

Solution. Two quantities change together, so make a scatter plot. The points lie almost exactly on a straight line, so a linear model makes sense. Technology (a spreadsheet, graphing calculator, or Desmos) gives the line of best fit:

$$y = -0.28x + 99.7$$

The slope means the battery drops about $0.28$ percentage points per minute. The initial value, $99.7$, is close to the true starting level of $100\%$.

Leo’s data with the line of best fit. The dashed part of the line is a prediction beyond the data.

Set $y = 20$ and solve:

$$\begin{aligned} -0.28x + 99.7 &= 20 \\ -0.28x &= -79.7 \\ x &= \frac{-79.7}{-0.28} \approx 284.6 \end{aligned}$$

Answer. About $285$ minutes, or roughly $4$ hours and $45$ minutes.

Judging the model. The fit is excellent: the points are very close to the line. But $285$ minutes is far beyond the last measurement at $150$ minutes, so this is extrapolation. Many phones also drain at a different rate when the battery is low or warm. Leo should treat “about $4\tfrac{3}{4}$ hours” as an estimate, and test it by streaming longer.

Example 4: Revising a model

Five weeks after making her model in Example 1, Maya has $260 saved, not the $285 her model predicted. Revise the model and update the answer.

Solution. The model predicted $60 + 5 \times 45 = 285$ dollars. She has less, so her real saving rate was lower (maybe she spent some money, or had fewer jobs).

Revised rate, using what actually happened:

$$\frac{260 - 60}{5} = \frac{200}{5} = 40 \text{ dollars per week}$$

She still needs $542.40 - 260 = 282.40$ dollars:

$$282.40 \div 40 = 7.06 \text{ weeks}$$

After $7$ more weeks she would have $260 + 280 = 540$ dollars, just short. So she needs $8$ more weeks, for $13$ weeks in total instead of $11$.

This is the cycle in action: real data showed the first model was a bit too hopeful, so Maya changed the rate and made a new prediction.

Common mistakes

Asking a question that can’t be measured. “Is our phone use bad?” has no numerical answer. Rewrite it as something you can collect data on, like “How many hours per day does each person in our home use a phone?”

Hiding the assumptions. Every model makes assumptions. If you don’t write them down, nobody (including you) can tell why the answer might be off. List them in step 3 and look back at them in step 6.

Trusting a prediction far outside the data. A line that fits $0$ to $150$ minutes may not hold at $300$ minutes. Say clearly when you’re extrapolating, and treat the answer as a rough estimate.

Rounding the wrong way in context. If $10.72$ weeks of saving are needed, $10$ weeks is not enough. When you need to reach a goal, round up to the next whole week.

Using too little data. One week of showers or one test run of a battery might be unusual. More data, collected at different times, gives a more reliable model.

Treating the model as the truth. A model is a tool, not a fact. If new data disagrees with it (like Maya’s savings), revise the model instead of ignoring the data.

Practice

1. (Warm-up) A student asks: “How much does my family spend on groceries in a month?” List the information needed and one way to collect it.

Solution

Information needed: the cost of each grocery trip and how many trips happen in a month.

One way to collect it: keep every grocery receipt for a month (or check the family’s bank or card records, with permission) and add the totals. This is first-hand data.

2. (Warm-up) You want to know how long it takes to fill a backyard pool with a garden hose. Name one thing that varies, one thing you would treat as staying the same, and one assumption.

Solution

Sample answer. Varies: the depth of water in the pool as it fills. Stays the same: the flow rate of the hose. Assumption: the water pressure doesn’t drop while the pool fills (for example, nobody else in the house is running water).

3. (Warm-up) Which kind of model (an average, a rate, or a line of best fit) fits each question best?

• (a) How tall will my bean plant be on day $20$, using its height each day for two weeks?
• (b) What is a typical number of text messages I send per day?
• (c) How much will $35$ L of gas cost if gas is $1.55$ dollars per litre?

Solution

(a) A line of best fit: two quantities (day and height) change together.

(b) An average (mean or median) of the daily counts.

(c) A rate: cost $= 1.55 \times 35 = 54.25$, or $54.25.

4. (Core) A garden hose fills a $10$ L bucket in $40$ seconds. About how many hours will it take to fill a pool that holds $18\,000$ L? State an assumption.

Solution

Rate: $\dfrac{10 \text{ L}}{40 \text{ s}} = 0.25$ L/s, which is $0.25 \times 60 = 15$ L/min.

$$18\,000 \div 15 = 1200 \text{ min} = \frac{1200}{60} \text{ h} = 20 \text{ h}$$

About $20$ hours. Assumption: the hose’s flow rate stays the same the whole time.

5. (Core) Ana timed her walk to school (in minutes) on seven days: $14$, $15$, $13$, $16$, $21$, $14$, $12$.

• (a) Find the mean and the median.
• (b) She walks to school and back on about $190$ school days a year. Use the mean to estimate her total walking time for the year, in hours.
• (c) Why is the mean (not the median) the better choice for a total?

Solution

(a) Sum: $14 + 15 + 13 + 16 + 21 + 14 + 12 = 105$, so the mean is $105 \div 7 = 15$ min. In order: $12, 13, 14, 14, 15, 16, 21$, so the median is $14$ min.

(b) Two trips a day:

$$15 \times 2 \times 190 = 5700 \text{ min} = \frac{5700}{60} \text{ h} = 95 \text{ h}$$

(c) The mean is the total divided by the number of trips, so mean $\times$ number of trips gives back the total. The median ignores how long the slow days (like $21$ min) were, so it would underestimate the total.

6. (Core) A class models the height of a sunflower with $h = 0.8d + 3$, where $h$ is the height in centimetres and $d$ is the number of days since it sprouted. They measured it for $30$ days.

• (a) Predict the height on day $20$.
• (b) The model predicts a height of $163$ cm on day $200$. Is that prediction trustworthy? Explain.

Solution

(a) $h = 0.8(20) + 3 = 16 + 3 = 19$, so about $19$ cm.

(b) Check: $0.8(200) + 3 = 163$. But day $200$ is far beyond the $30$ days of data (extrapolation). Plants don’t grow at a steady rate forever: they slow down, stop, or die at the end of the season. The prediction is not trustworthy.

7. (Core) A free fitness app records your location, heart rate, and sleep. Describe one benefit, one risk, and one question you should ask before agreeing to share your data.

Solution

Sample answer. Benefit: it can show patterns in your sleep and exercise that help you make healthier choices. Risk: your location history could reveal where you live and go to school, and it could be shared with advertisers or exposed in a data breach. Question to ask: “Who can see my data, how long is it stored, and can I delete it?”

8. (Challenge) A store’s ad shows a bar graph of monthly sales: $960 in March and $990 in April. The vertical axis starts at $950, so the April bar looks four times as tall as the March bar.

• (a) Explain why the bars look four times different.
• (b) What was the actual percent increase?

Solution

(a) Starting at $950, the March bar shows $960 - 950 = 10$ and the April bar shows $990 - 950 = 40$. Since $40 \div 10 = 4$, April’s bar is four times as tall, even though sales barely changed.

(b)

$$\frac{990 - 960}{960} \times 100\% = \frac{30}{960} \times 100\% = 3.125\%$$

Sales rose by only about $3.1\%$. A fair graph would start the axis at $0.

9. (Challenge) A family plans a $450$ km drive (about the distance from Toronto to Ottawa). Their car uses $7.5$ L of gas per $100$ km, and gas costs $1.55 per litre.

• (a) Build a model and estimate the cost of gas for the round trip.
• (b) List two assumptions, and say how the family could make the model better.

Solution

(a) Gas for one way:

$$450 \times \frac{7.5}{100} = 33.75 \text{ L}$$

Cost one way: $33.75 \times 1.55 = 52.3125$, about $52.31. Round trip: $2 \times 52.3125 = 104.625$, about $104.63.

(b) Sample assumptions: the car really uses $7.5$ L per $100$ km on this trip (highway driving, air conditioning, and a full car all change it), and the gas price stays at $1.55. To improve the model, they could check the car’s actual fuel use on a recent highway trip and look up current gas prices along the route.