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D2.2 · Identify a modelling question and required information

Learn to identify a modelling question and required information through clear examples and targeted practice.

Ontario Grade 9 Mathematics

Data

Ontario MTH1W — Data and Mathematical Modelling, Expectation D2.2

Have you ever wondered how a store decides how many staff members to schedule on a busy weekend? Or how a city planner figures out whether a new bus route will reduce crowding? These real-world decisions are not solved with a single formula. Instead, people ask a careful question, decide what information they need, and then build a mathematical model — a simplified mathematical description of a real situation — to help them think it through. This lesson focuses on the very first and most important step of that process: identifying a good modelling question and figuring out exactly what information you would need to answer it. Getting this step right shapes everything that comes after.

What you will learn

Explain what a mathematical modelling question is and how it differs from a routine calculation question.
Identify whether a given question can be explored through mathematical modelling.
List the specific information needed to begin building a model for a given situation.
Distinguish between information that is essential and information that is not needed for a particular modelling question.

What Is Mathematical Modelling?

A mathematical model is a simplified description of a real situation that uses numbers, operations, or relationships to help us understand or predict something. The key word is simplified — a model does not capture every detail of reality. It captures the details that matter most for the question being asked.

Think about predicting how long it takes to fill a bathtub. You do not need to know the colour of the tub or the brand of the faucet. You do need to know how fast water flows in and whether the drain is open. A model strips away distracting details and keeps only what is mathematically useful.

Modelling is a process, not a single calculation. The process has several stages: ask a question, gather information, build a model, use the model to get results, check whether the results make sense, and revise if needed. This lesson is entirely about the first two stages — asking the right question and identifying the right information.

A mathematical model is a simplified mathematical description of a real situation.
Models keep the details that matter and leave out details that do not affect the answer.
The modelling process always starts by identifying a clear question and the information required to answer it.

What Makes a Good Modelling Question?

Not every question is a modelling question. A routine calculation question has one correct answer found by applying a known procedure — for example, 'What is 15% of CAD 80?' A modelling question is open enough that you must make decisions about what to include, what to assume, and what information to collect.

A good modelling question has three features. First, it is connected to a real situation that actually matters to someone. Second, it cannot be answered just by looking up a single fact or doing one calculation — it requires you to use mathematical relationships. Third, it is specific enough that you could realistically gather information to answer it.

Compare these two questions: (A) 'How much does a bus ticket cost?' and (B) 'About how much money does an average Grade 9 student in Ontario spend on transportation in one school year?' Question A is a look-up fact. Question B requires you to define 'average student,' decide which types of transportation count, estimate how often a student travels, and find or estimate costs. Question B is a modelling question.

A helpful test: if someone could answer the question in five seconds by checking one source, it is probably not a modelling question. If a thoughtful person would say 'well, it depends on…', that is a strong sign you are looking at a modelling question.

A modelling question is rooted in a real situation and cannot be answered by a single look-up or one-step calculation.
Good modelling questions are specific enough to be investigated but open enough to require mathematical thinking.
If the answer to a question depends on several factors, it is likely a modelling question.

Identifying the Required Information

Once you have a modelling question, the next task is to list the information you would need to answer it. This step is called identifying required information. Required information is any data, measurement, rate, or fact that directly affects the answer to your modelling question.

Here is a structured way to identify required information. Start by reading your modelling question carefully and underlining the key quantities — the things that are being measured or compared. Then ask yourself: 'What would change the answer if it were different?' Every factor whose change would change your answer is required information.

For example, consider this modelling question: 'How many bags of popcorn should the school canteen prepare for a movie night?' The key quantities are the number of people attending, how many bags each person is likely to eat, and possibly how large each bag is. Each of these would change the answer if it changed, so all three are required information. The colour of the popcorn bags is not required information because it does not affect how many bags are needed.

It is also important to separate required information into two types: information you already know (given information) and information you would need to find out or estimate (information to gather). Being clear about what you already know versus what you still need prevents you from starting a model with missing pieces.

Required information is any value or fact that directly affects the answer to the modelling question.
Ask 'would the answer change if this detail were different?' to test whether something is required.
Separate required information into what is already known and what still needs to be found or estimated.
Details that do not affect the answer are not required information — leave them out.

A Framework: From Question to Information List

A clear, repeatable framework helps you move from a modelling question to a complete information list. Think of it as three linked questions you ask yourself in order.

Step 1 — What am I trying to find? Write the modelling question in your own words and identify the quantity the model must produce. This quantity is called the output. For example, if the modelling question is 'How long will it take to save enough money to buy a CAD 350 phone?', the output is the number of weeks (or months) needed to save.

Step 2 — What quantities affect the output? List every factor that influences the output. These are your required inputs. For the savings question, the inputs would be: the current amount of savings already set aside, the amount saved each week, and the target price. Notice that a simple relationship connects them: if you let

s

represent savings already set aside and

r

represent the amount saved each week, then after

n

weeks the total saved is

s + r \times n

. The model is useful only if you know

s

r

, and the target.

Step 3 — Which of these inputs do I already know, and which do I need to find? Sort your list. Mark each input as 'known' or 'to find.' This tells you exactly what research or estimation you must do before you can use the model.

\text{Total saved after } n \text{ weeks} = s + r \times n

Step 1: Identify the output — what quantity does your model need to produce?
Step 2: Identify the inputs — what quantities affect the output?
Step 3: Sort inputs into 'already known' and 'need to find or estimate.'
A simple relationship between inputs and output is the beginning of the model itself.

Putting It Together: Reading a Situation Critically

Real modelling situations often come with extra details mixed in with the essential ones. Part of your skill as a modeller is reading a situation critically — keeping what you need and setting aside what you do not.

Suppose you read this: 'Amara loves hiking. She hikes every Saturday on a trail near her home in Sudbury, Ontario. The trail is 8 kilometres long and mostly flat. She wants to know if she is improving her pace over the school year. She always brings a water bottle and wears a red jacket.' Your modelling question might be: 'Is Amara's hiking pace improving over the school year?' Required information: her hiking time (or pace) recorded each Saturday over the school year, and the trail distance (8 km) since pace = distance ÷ time. Not required: the location (Sudbury), the colour of her jacket, or what she brings to drink. These details are part of the story but have no effect on the mathematical model of pace over time.

Practising this critical reading skill — separating story details from mathematical inputs — is one of the most valuable habits you can build in Grade 9 mathematics.

\text{Pace} = \frac{\text{distance}}{\text{time}}

Real situations mix essential and non-essential details; a modeller must tell them apart.
Required information directly appears in the mathematical relationship you will use.
Non-required details are part of the context or story but do not change the answer.
Critical reading of a situation is a core modelling skill.

Required vs. Not Required Information — Granola Bar Fundraiser Example

Detail from the Situation	Does it affect the number of bars to sell?	Required Information?
Fundraising target: CAD 200	Yes — changing this changes how many bars are needed	Yes — already known
Ingredient budget: CAD 40	Yes — limits how much can be spent on making bars	Yes — already known
Ingredient cost per bar	Yes — affects profit per bar	Yes — need to find
Selling price per bar	Yes — affects profit per bar	Yes — need to find
Cafeteria is painted blue	No — has no place in the calculation	No — leave it out

Worked example

Identifying the Modelling Question and Required Information for a Fundraiser

Read the following situation carefully, then answer these two tasks: (A) Write a clear modelling question for this situation. (B) List all the required information and label each item as 'already known' or 'need to find.'

Situation: The Grade 9 class at Westview Secondary wants to raise money for a class trip by selling homemade granola bars at lunch. They have CAD 40 to spend on ingredients. They need to raise a total of CAD 200 for the trip. They are trying to figure out how many granola bars they need to sell. The school cafeteria is painted blue.

Step 1 — Read and identify the output
Ask: what quantity does the class need to find? They want to know how many granola bars to sell. This is the output of the model. Write it clearly so the modelling question is specific.
Output: number of granola bars to sell = n
Step 2 — Write the modelling question
A good modelling question names the real situation, identifies who it matters to, and asks for the output in a way that requires mathematical reasoning. Avoid vague language like 'how do we raise money?' — be precise about what the model must calculate.
Modelling question: How many granola bars must the class sell to raise CAD 200, given a CAD 40 ingredient budget?
Step 3 — List quantities that affect the output
Ask: what facts would change the number of bars needed if they were different? The total amount needed for the trip changes the target. The ingredient cost per bar changes how much profit each bar generates. The selling price per bar also changes the profit per bar. The starting budget of CAD 40 tells us the maximum they can spend on ingredients.
Inputs: total fundraising target, ingredient budget, cost per bar, selling price per bar
Step 4 — Identify a simple relationship linking the inputs to the output
Profit on each bar equals the selling price minus the ingredient cost for that bar. Total profit equals profit per bar multiplied by the number of bars sold. The class needs total profit to equal the fundraising target. Writing this out shows exactly which inputs are required and how they connect.
Total profit = (selling price - cost per bar) × n
Step 5 — Sort inputs into 'already known' and 'need to find'
Go through each input. The fundraising target of CAD 200 is stated in the problem — already known. The ingredient budget of CAD 40 is stated — already known. The cost per bar and the selling price per bar are not given in the situation — they need to be found or estimated before the model can be used.
Known: target = \200,\ budget = \40 To find: cost per bar, selling price per bar
Step 6 — Identify and discard non-required information
The colour of the cafeteria (blue) appears in the situation. Ask: would changing this detail change the number of bars needed? No — it has no place in the mathematical relationship. It is not required information and should not appear on the list.
$\text{Cafeteria colour} \notin \text{required information}$

Answer: (A) Modelling question: 'How many granola bars must the Grade 9 class sell to raise CAD 200 for their class trip, given a CAD 40 ingredient budget?' (B) Required information — Already known: fundraising target (CAD 200), ingredient budget (CAD 40). Need to find: ingredient cost per granola bar, planned selling price per granola bar. Not required: colour of the cafeteria.

Check: Check that every item on the required information list actually appears in the relationship — total profit equals (selling price minus cost per bar) times number of bars. Target: yes, it sets the total profit needed. Budget: yes, it limits spending on ingredients. Cost per bar: yes, it is in the formula. Selling price per bar: yes, it is in the formula. Cafeteria colour: no, it does not appear anywhere in the relationship, so correctly excluded.

Common mistakes and how to avoid them

Writing a modelling question that is too vague, such as 'How do we make money?' This gives no clear output and cannot guide the information-gathering step.

Correction: Be specific about who, what, and how much. A good modelling question names the situation and the quantity to be found, for example: 'How many granola bars must the class sell to raise CAD 200?'

Listing every detail from the situation as required information, including story details like names, colours, or locations that have no mathematical role.

Correction: For each detail, ask: 'Would a different value for this detail change my answer?' Only include details that pass this test. If changing the detail has no effect on the calculation, leave it out.

Forgetting to separate 'already known' information from 'need to find' information, then being surprised midway through the model that key values are missing.

Correction: Always sort your required information list into two columns: what you already know from the problem, and what you still need to measure, look up, or estimate.

Confusing a routine calculation question with a modelling question. For example, treating 'What is 15% of CAD 80?' as a modelling question.

Correction: A modelling question requires decisions about what to include and cannot be answered in one step from one source. If the question has a single, immediate answer, it is a calculation question, not a modelling question.

Skipping the step of identifying the output before listing required inputs, which leads to collecting information without knowing what it is supposed to produce.

Correction: Always identify the output first — what quantity must the model calculate? Once the output is clear, every input is chosen because it affects that specific output.

Lesson summary

A mathematical model is a simplified mathematical description of a real situation that uses numbers and relationships to help understand or predict something.
A modelling question is rooted in a real situation, cannot be answered by a single look-up or one-step calculation, and asks for a quantity that depends on several factors.
The three-step framework is: (1) identify the output, (2) list all inputs that affect the output, (3) sort inputs into 'already known' and 'need to find.'
Required information is any fact or value that directly appears in the mathematical relationship connecting inputs to the output.
Non-required information appears in the story of the situation but does not change the answer — leave it off the list.
Getting the modelling question and required information right is the foundation of the entire modelling process; a vague question or a missing input makes the later steps impossible.

Check your understanding

Question 1

Which of the following is the best example of a modelling question?

What is the population of Toronto according to the 2021 census?
About how many litres of water does a typical Ontario household use in one month?
What is 20% of CAD 150?
How many days are in the month of March?

Show answer and explanation

About how many litres of water does a typical Ontario household use in one month?

Option B is a modelling question because it cannot be answered with a single fact or one-step calculation — it requires you to define 'typical household,' identify what types of water use to include, and estimate or research usage rates. The other options all have single, direct answers that need no mathematical model.

Question 2

A student is building a model to answer: 'How long will it take a candle to burn down completely?' Which piece of information is NOT required?

The height of the candle in centimetres
The rate at which the candle burns, in centimetres per hour
The colour of the candle
Whether the candle has already been burning for some time

Show answer and explanation

The colour of the candle

The colour of the candle does not appear in the mathematical relationship — time equals remaining height divided by burn rate. Changing the candle's colour would not change how long it takes to burn. All other options directly affect the calculation.

Question 3

A student writes this modelling question: 'How many people are there?' What is the main problem with this question?

It uses the word 'how many,' which is not allowed in a modelling question.
It is too vague — it does not name a real situation, a specific group, or a reason for modelling.
It is a perfectly good modelling question because the answer requires counting.
It is a routine calculation question because counting is a one-step process.

Show answer and explanation

It is too vague — it does not name a real situation, a specific group, or a reason for modelling.

A good modelling question must name the specific real situation and the quantity to be found. 'How many people are there?' gives no context, no defined group, and no reason for modelling — so it cannot guide any information-gathering step. It is too vague, not a routine calculation.

Question 4

You are building a model for the question: 'How many weeks will it take to save CAD 180 for a concert ticket if you already have CAD 30 saved and save CAD 15 each week?' You write the relationship as: total saved =

30 + 15 \times n

, where

n

is the number of weeks. Which item below is correctly identified as required information that is already known?

The name of the performer at the concert
The number of weeks needed — this is what the model calculates
The weekly savings amount of CAD 15
The distance to the concert venue

Show answer and explanation

The weekly savings amount of CAD 15

The weekly savings amount of CAD 15 appears directly in the relationship and is stated in the problem, so it is required information that is already known. The performer's name and the distance to the venue have no role in the savings relationship. The number of weeks is the output — the quantity the model calculates — not a required input.

Key terms

Mathematical model: A simplified mathematical description of a real situation, using numbers, operations, or relationships to understand or predict something.
Modelling question: A question rooted in a real situation that cannot be answered with a single look-up or one-step calculation and requires mathematical reasoning about several connected factors.
Output: The quantity that the mathematical model is designed to calculate or predict — the answer the model produces.
Input: A quantity or piece of information that is fed into the model and directly affects the output.
Required information: Any fact, measurement, or value that directly appears in the mathematical relationship and whose change would change the model's output.
Given information: Required information that is already stated or known within the problem situation.
Information to gather: Required information that is not yet known and must be measured, researched, or estimated before the model can be used.
Modelling process: A cycle of steps used to apply mathematics to real situations: identify a question, gather information, build a model, use the model, check results, and revise as needed.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation D2.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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