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D2.3 · Plan and collect data for a mathematical model

Learn to plan and collect data for a mathematical model through clear examples and targeted practice.

Ontario Grade 9 Mathematics

Data

Ontario MTH1W — Expectation D2.3

Have you ever noticed that the longer you leave a phone charging, the higher the battery percentage climbs? Or that the more students in a fundraiser, the more money the class raises? Both situations involve two quantities that change together. In mathematics, we describe those relationships with a mathematical model — a rule, equation, or graph that represents a real-world pattern. But before you can build a model, you need data. This lesson walks you through how to plan what data to collect and how to record it carefully so the pattern becomes clear. Good data collection is the foundation of every reliable model.

What you will learn

Explain what a mathematical model is and why data are needed to build one.
Identify the independent variable and the dependent variable in a real situation.
Write a clear data-collection plan that states what to measure, how to measure it, and how many data points to collect.
Collect or organize data in a table so the relationship between two variables can be seen.
Describe the trend shown in a data table and connect it to the type of model the data might suggest.

What Is a Mathematical Model?

A mathematical model is an equation, table, or graph that captures a pattern in real data. The model lets you describe what has already happened and make reasonable predictions about what might happen next.

Think about a candle burning. As time passes, the candle gets shorter. You could measure the candle's height every five minutes, record those numbers, and then find an equation that fits the pattern. That equation is your model.

A model is only as trustworthy as the data behind it. If you collect data carelessly — measuring at random times, using a bent ruler, or skipping values — the pattern will be blurry and your model will give poor predictions. Careful planning before you collect a single number is therefore essential.

A mathematical model represents a real-world relationship using an equation, table, or graph.
Models are built from data, so data quality directly affects how useful the model is.
A model lets you describe and predict — but only within the range of situations it was built to cover.

Independent and Dependent Variables

Every data-collection plan starts with identifying two variables. A variable is any quantity that can change. In a relationship between two variables, one variable causes or controls the change, and the other responds to it.

The independent variable is the one you choose or control. In the candle example, time is the independent variable because you decide when to measure. The dependent variable is the one that responds — candle height depends on how much time has passed.

A quick way to remember the difference: ask yourself, 'Does one value depend on the other?' The answer names the dependent variable. The thing it depends on is the independent variable.

When you set up a data table, always put the independent variable in the left column and the dependent variable in the right column. This makes the relationship easy to read and matches the convention used when plotting points, where the independent variable goes on the horizontal axis (

x

) and the dependent variable goes on the vertical axis (

y

y \text{ depends on } x

The independent variable is the one you control or choose; label it $x$ .
The dependent variable changes in response; label it $y$ .
Always list independent values in order (usually smallest to largest) in your table.
The relationship can be written as: $y$ depends on $x$ , or $y$ is a function of $x$ .

Writing a Data-Collection Plan

A data-collection plan is a short written description you create before you start measuring. It prevents confusion and makes sure someone else could repeat your experiment and get the same kind of data.

Your plan should answer four questions. First: What am I measuring? Name both variables clearly (for example, 'time in minutes' and 'candle height in centimetres'). Second: How will I measure? State the tool and unit (for example, 'a ruler marked in millimetres, reading to the nearest millimetre'). Third: How often or over what range? State the values of the independent variable you will use (for example, 'every 5 minutes from 0 minutes to 40 minutes'). Fourth: How many data points will I collect? More points give a clearer picture of the pattern — aim for at least six to eight points for a simple relationship.

Choosing your independent variable values evenly spaced is a smart habit. Even spacing makes it easier to spot whether the dependent variable rises by the same amount each time (a linear pattern) or by a growing or shrinking amount (a non-linear pattern).

You should also note any conditions you will keep the same throughout, called controlled conditions. For the candle, you would keep the room draft-free and use the same candle brand every time. Controlled conditions stop outside factors from hiding the true pattern.

State clearly: what you measure, the tool and unit, the range of the independent variable, and the number of data points.
Use evenly spaced values of the independent variable to make patterns easier to spot.
Record controlled conditions so the data reflect only the relationship you are studying.
Aim for at least 6–8 data points to see a reliable pattern.

Recording and Reading a Data Table

Once you have your plan, you record measurements in a two-column table. The left column holds the independent variable values in order. The right column holds each corresponding dependent variable value.

After filling in the table, look at how the dependent variable changes as the independent variable increases. If the dependent variable goes up or down by roughly the same amount each step, the pattern is linear — it can be modelled by a straight-line equation of the form

y = mx + b

, where

m

is the rate of change (how much

y

changes per unit of

x

) and

b

is the starting value when

x = 0

If the differences between consecutive

y

-values are not equal but are growing or shrinking steadily, the pattern may be non-linear. At the Grade 9 level, you do not need to name the exact type of non-linear model — you simply note that the pattern is not linear and describe the trend in words (for example, 'the height decreases more slowly as time increases').

Reading the table carefully is the bridge between raw data and a model. You are looking for a consistent rule that connects every

x

-value to its

y

-value.

\text{First difference} = y_{\text{next}} - y_{\text{current}}

Put independent variable values in order in the left column.
Calculate the change in $y$ between each pair of rows — called the first difference — to check for a linear pattern.
Equal first differences suggest a linear model: $y = mx + b$ .
Unequal first differences suggest a non-linear pattern; describe the trend in words.

Connecting Your Data to a Simple Linear Model

When first differences are equal, you can use two pieces of information from your table to write a linear equation. The starting value — the

y

-value when

x = 0

— becomes

b

. The constant first difference becomes

m

, the rate of change.

For example, suppose a candle is 24 cm tall at the start and shrinks by 3 cm every 10 minutes. The starting value is

b = 24

and the rate of change is

m = -3

per 10 minutes, or

-0.3

cm per minute. The model is

y = -0.3x + 24

, where

x

is time in minutes and

y

is height in centimetres.

This equation is the mathematical model. You can use it to predict the candle height at any time within the range of your data — for example, at

x = 25

minutes:

y = -0.3(25) + 24 = -7.5 + 24 = 16.5

cm.

Remember: the model is only reliable within or very close to the range of data you collected. Predicting too far beyond your data range is risky because you do not know whether the same pattern continues.

y = mx + b

A linear model has the form $y = mx + b$ , where $m$ is the rate of change and $b$ is the starting value.
Read $b$ directly from the table as the $y$ -value when $x = 0$ .
Read $m$ as the constant first difference divided by the step size of $x$ .
Use the model to predict $y$ for new $x$ -values, but stay within or close to your data range.

Dripping Tap Data Table with First Differences

Time, x (min)	Volume, y (mL)	First Difference (mL)
0	0	—
4	50	50
8	100	50
12	150	50
16	200	50
20	250	50
24	300	50

Worked example

Planning and Using a Data Table for a Dripping Tap

A student notices water dripping from a tap into an empty bucket. She plans to model the relationship between time and the volume of water collected. She measures the volume every 4 minutes for 24 minutes and records the values below.

Time (min): 0, 4, 8, 12, 16, 20, 24
Volume (mL): 0, 50, 100, 150, 200, 250, 300

Use her data to: (a) identify the independent and dependent variables, (b) calculate the first differences to confirm the pattern is linear, and (c) write the equation of the linear model.

Step 1 — Identify the variables
Ask which quantity the student controls: she chooses when to measure, so time is the independent variable ( $x$ , in minutes). Volume responds to time, so volume is the dependent variable ( $y$ , in millilitres).
$x = \text{time (min)}, \quad y = \text{volume (mL)}$
Step 2 — Calculate first differences
Subtract each volume value from the next one in the table. If all differences are equal, the pattern is linear.
$50 - 0 = 50, \quad 100 - 50 = 50, \quad 150 - 100 = 50, \quad 200 - 150 = 50, \quad 250 - 200 = 50, \quad 300 - 250 = 50$
Step 3 — Confirm the pattern
All six first differences equal 50 mL. Because the first differences are equal, the relationship is linear.
$\text{All first differences} = 50 \text{ mL}$
Step 4 — Find the starting value b
The starting value is the volume when time equals zero. Reading directly from the table: when $x = 0$ , $y = 0$ .
$b = 0$
Step 5 — Find the rate of change m
The volume increases by 50 mL every 4 minutes. Divide the change in volume by the change in time to find how many millilitres per minute.
$m = \frac{50 \text{ mL}}{4 \text{ min}} = 12.5 \text{ mL/min}$
Step 6 — Write the model
Substitute $m = 12.5$ and $b = 0$ into the linear model form $y = mx + b$ .
$y = 12.5x + 0 \quad \Rightarrow \quad y = 12.5x$
Step 7 — Use the model to predict
Predict the volume at $x = 10$ minutes, which is within the data range, as a check.
$y = 12.5(10) = 125 \text{ mL}$

Answer: The linear model is

y = 12.5x

. It predicts 125 mL of water collected after 10 minutes.

Check: At

x = 8

min the table shows 100 mL. The model gives

y = 12.5(8) = 100

mL. ✓ At

x = 20

min the table shows 250 mL. The model gives

y = 12.5(20) = 250

mL. ✓ Both match, confirming the model is correct.

Common mistakes and how to avoid them

Swapping the independent and dependent variables — for example, putting volume in the left column and time in the right column.

Correction: Always ask: which variable do I control or choose? That is the independent variable and belongs in the left column. The responding variable (dependent) goes in the right column.

Using unevenly spaced independent variable values, then treating unequal first differences as proof of a non-linear pattern.

Correction: First differences only reveal a linear pattern when the independent variable values are evenly spaced. Check your spacing before drawing any conclusion from the differences.

Collecting too few data points (only 2 or 3) and assuming a line fits perfectly.

Correction: Two points always form a straight line, but that does not mean the real relationship is linear. Collect at least 6–8 data points to see whether the pattern is genuinely consistent.

Using the model to predict far outside the data range — for example, using a candle model built from 0–40 minutes to predict 5 hours later.

Correction: A model is only reliable within or very close to the range of data collected. State the valid range when you report your model.

Forgetting units when reading or writing the rate of change

m

Correction: The rate of change always has units: dependent-variable units per independent-variable unit (e.g., mL per minute). Including units prevents errors when interpreting the model.

Lesson summary

A mathematical model is an equation, table, or graph that represents a real-world pattern and can be used to make predictions.
The independent variable ( $x$ ) is the one you control; the dependent variable ( $y$ ) is the one that responds.
A data-collection plan states what to measure, the tool and unit, the range and spacing of the independent variable, and the number of data points to collect.
Recording data in a table with the independent variable in order in the left column makes patterns easier to see.
Equal first differences in a data table indicate a linear relationship that can be modelled by $y = mx + b$ , where $m$ is the rate of change and $b$ is the starting value.
A model should only be used to predict within or very close to the range of data it was built from.

Check your understanding

Question 1

A student records how far a toy car travels (in cm) after being pushed for different durations (in seconds). Which is the independent variable?

Distance travelled in centimetres
Duration of the push in seconds
The mass of the toy car
The speed of the car at the end

Show answer and explanation

Duration of the push in seconds

The student controls how long the car is pushed, so duration is the independent variable. Distance is the response — it depends on how long the push lasts.

Question 2

A data table shows these volume readings (in mL) at equal time steps: 10, 18, 26, 34, 42. What are the first differences, and what do they tell you?

First differences are 8, 8, 8, 8 — the pattern is linear.
First differences are 10, 8, 8, 8 — the pattern is non-linear.
First differences are 8, 8, 8, 8 — the pattern is non-linear.
First differences are 2, 8, 8, 8 — the pattern is linear.

Show answer and explanation

First differences are 8, 8, 8, 8 — the pattern is linear.

18 − 10 = 8, 26 − 18 = 8, 34 − 26 = 8, 42 − 34 = 8. All first differences equal 8, so the pattern is linear.

Question 3

A linear model for a plant's height (cm) over days is

h = 0.6d + 5

. What does the value 5 represent?

The height increases by 5 cm every day.
The plant's height on day 5.
The plant's starting height on day 0, in centimetres.
The number of days needed to reach 5 cm.

Show answer and explanation

The plant's starting height on day 0, in centimetres.

h = 0.6d + 5

, the value added at the end (5) is

b

, the starting value — the height when

d = 0

, which is day zero.

Question 4

A student collects data from day 1 to day 20 and builds a model. A classmate uses the model to predict the value on day 500. What is the main problem with this prediction?

The prediction is fine because the model is an equation and equations always work.
Day 500 is far outside the data range, so the model may not be reliable there.
The model can only predict for even-numbered days.
The student should have used a graph instead of a table.

Show answer and explanation

Day 500 is far outside the data range, so the model may not be reliable there.

A model is only trustworthy within or near the range of data it was built from (days 1–20). Predicting at day 500 goes far beyond that range, so the prediction may be very inaccurate.

Key terms

Mathematical model: An equation, table, or graph that represents a pattern in real-world data and can be used to describe or predict values.
Independent variable: The variable you choose or control in an experiment; it goes in the left column of a data table and on the horizontal axis of a graph. Often labelled $x$ .
Dependent variable: The variable that responds to or depends on the independent variable; it goes in the right column of a data table and on the vertical axis. Often labelled $y$ .
Data-collection plan: A written description made before measuring that states what to measure, the tool and unit, the range of the independent variable, and the number of data points.
First difference: The result of subtracting one dependent variable value from the next in a table where the independent variable values are evenly spaced. Equal first differences indicate a linear pattern.
Rate of change: How much the dependent variable changes for each one-unit increase in the independent variable; represented by $m$ in $y = mx + b$ .
Starting value: The value of the dependent variable when the independent variable equals zero; represented by $b$ in $y = mx + b$ .
Linear model: A model of the form $y = mx + b$ where the first differences in the data table are all equal, meaning the relationship forms a straight line when graphed.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation D2.3. 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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