D. Data · Data, Statistics and Mathematical Modelling

Asking a Question of Interest

D2.2 A1About 60 minutes

Learning goals

I can pose a question of interest that is specific, measurable and answerable with data.
I can improve a vague question.
I can identify the variables, the factors to keep the same and the information needed to answer a question.

Words to know

question of interest
A specific, measurable question about a real situation that can be answered by collecting and analysing data.

variable
A quantity or characteristic that can change or be measured.

controlled variable
A factor kept the same so it does not affect the results.

primary data
Data you collect yourself.

secondary data
Data collected by someone else, such as Statistics Canada.

It all starts with a good question

You have probably heard someone ask a question that can't really be answered: "Is winter the best season?" That is an opinion. Compare it with: "In our town, how does the daily low temperature affect the number of people at the outdoor rink?" That question can be answered with data and a model.

The first step of mathematical modelling is to understand the problem: pose a clear question of interest and figure out what information you need to answer it.

What makes a good question of interest?

A good modelling question is:

Specific: it names exactly what you will study (which people, place, time).
Measurable: the answer depends on quantities you can count or measure, not opinions.
Answerable with data you can actually collect or find.
Useful or interesting: someone cares about the answer, or it helps a decision.

Many good questions are one of these kinds:

Relationship: How does x affect y?
Prediction: How long / how many / how much will...?
Comparison: Which is better, A or B, and by how much?
Decision (optimization): What is the best amount / time / choice?

Narrowing a topic into a question of interest. Each step makes the question more specific and measurable until it can be answered with data.

Weak question	Problem	Improved question of interest
Is tobogganing fun?	Opinion; not measurable	How does the angle of a hill affect the time for a sled to travel 20 m?
Do people like the cafeteria?	Vague; "like" is not defined	What percent of Grade 9 students at our school buy cafeteria lunch at least 3 days a week?
How much does stuff cost?	Too broad	How has the price of a 4 L bag of milk at one grocery store changed over the past 12 months?
Is my phone slow?	No measurable variable	How does the number of open apps affect the time my phone takes to open the camera?

Turning weak questions into questions of interest.

Identifying the information you need

Once you have a question, list:

The variables. Which quantity might explain or change the other (independent), and which quantity responds (dependent)? Include units.
What else could affect the result. These are things that should stay the same (controlled) or that you need to record.
How much data you need and over what range.
Where the data will come from: your own measurements, a survey, or an existing source such as Statistics Canada or Environment and Climate Change Canada.

A planning map for a question of interest. Naming the variables, what must stay the same and what data you need comes before collecting any data.

Worked example

Example 1: Improve a question

A student writes the question: "Does homework matter?" Improve it so it can be answered with data, and identify the information needed.

"Matter" is vague. Decide what you want to measure: maybe test marks.
Make it specific: "For Grade 9 students in my math class, how is the average number of minutes of math homework per night related to the mark on the unit test?"
Independent variable: minutes of math homework per night (min). Dependent variable: unit test mark (%).
Information needed: each student's homework minutes (tracked for the unit) and their test mark. Ethics: students must agree to share, and marks must be kept anonymous.
Other factors: prior marks, attendance, tutoring. Record them or mention them as limitations.

Improved question: How is the average number of minutes of math homework per night related to the unit test mark for students in my class? It is specific, measurable and answerable.

Worked example

Example 2: A decision question

A community skating rink asks: "How many litres of hot chocolate should we make for each public skate?" Identify the information needed.

This is a prediction/decision question. The dependent variable is litres of hot chocolate sold.
Possible independent variables: the number of skaters and the outdoor temperature (colder days may sell more).
Information needed: for many past public skates, the number of skaters, the temperature and litres sold. Also the size of a serving (for example, 250 mL).
Other factors: weekends vs weekdays, holidays (such as the winter break), price changes.
Data source: the rink's own sales records, plus temperatures from Environment and Climate Change Canada's historical weather data.

The rink needs past records of skaters, temperature and litres sold (plus serving size) for many sessions. Those will let it build a model like litres = (rate) × skaters.

Worked example

Example 3: The bounce project

For the next three lessons we will follow one project: "How high will a tennis ball bounce when dropped from different heights, and from what height must it be dropped to bounce back up 1 m?" Identify the information needed.

Independent variable: drop height (cm), measured from the floor to the bottom of the ball.
Dependent variable: bounce height (cm), the highest point of the first bounce.
Keep the same: the same ball, the same floor surface, dropping (not throwing), measuring to the bottom of the ball.
Data needed: at least 5 or 6 drop heights over a wide range (for example, 50 cm to 200 cm), with 3 trials at each to reduce measurement error.

We need pairs of (drop height, bounce height) over a wide range of heights, collected the same way every time. In the next lesson we plan how to collect them.

Just for fun

Funny but true: some of the most useful questions in science sounded silly at first. "Why does a ball bounce lower each time?" leads straight to how energy is lost as heat and sound, which matters for designing safer helmets and better sports surfaces.

Hands-on activity

Question workshop

You need: paper and a pencil, and a family member to interview.

Choose a topic you care about: sports, gaming, music, pets, cooking, weather, transit, your phone.
Write three questions about it. Then improve each one so it is specific, measurable and answerable with data.
For your best question, make a planning map like the one in this lesson: independent variable, dependent variable (with units), what to keep the same, and what data you need.
Read your question to a family member. Ask them: "Could you tell exactly what I will measure?" If not, revise.
Keep this page: you will plan and collect data for it in the next lesson.

What to look for: a final question that names the variables and units and could be answered with collectable data.

Your turn

Is "Which pizza is the best?" a good question of interest? Explain and improve it.

Identify the independent and dependent variables: "How does the number of hours of sunlight per day affect the growth (cm per week) of a bean plant?"

Classify the question type: "Should our class sell hot chocolate or cookies at the fundraiser to earn more money?"

For "How does outdoor temperature affect the number of skaters at the rink?", list three pieces of information needed.

Name one thing that should stay the same in the bounce project, and explain why.

Write a question of interest about your own phone or computer use that could be answered with data.

More practice: warm-up

Name the four qualities of a good question of interest.

Which variable goes on the x-axis?

Is "What is the best colour?" measurable?

Give one existing Canadian source of data.

Why should you include units when you name a variable?

More practice: on your own

Improve: "Is the bus late a lot?"

For "How does the mass of a snowball affect how far it rolls down a hill?", list two things to keep the same.

A student wants to know how screen time affects sleep. Write a question of interest and identify the variables.

Why might a question need data from an existing source instead of your own measurements? Give an example.

Explain why "How does studying affect marks?" is too broad even though it is about measurable things.

Challenge

Write a question of interest that would need two independent variables, and say why both matter.

A city asks: "Where should we put a new bike lane?" Rewrite this as one or more questions of interest that data could answer, and list the data needed.

Explain how the question of interest affects every later step of the modelling process.

Check your understanding

Quiz

Which is the best question of interest?

In "How does the temperature of water affect how long sugar takes to dissolve?", what is the dependent variable?

Why should the same ball be used for every drop in the bounce project?

Which question is an opinion, not a question of interest?

A question asks how the date of the first snowfall in Ottawa has changed over the last 50 years. Where is the data most likely to come from?

For parents and teachers

Big idea: D2.2 asks students to identify a question of interest requiring the collection and analysis of data, and the information needed to answer it. This is step 1 of the modelling project that continues through the next three lessons.

How to help: Help your teen pick a topic they genuinely care about; motivation matters for a project. Play "Could you measure that?" when they write a question.

Common difficulties:

Questions that are opinions or too broad.
Mixing up independent and dependent variables. Ask "Which one do you change or choose? Which one do you measure in response?"
Choosing a question whose data cannot realistically be collected in a few days.