D. Data · Two-Variable Data, Scatter Plots and Probability

One-Variable and Two-Variable Data

D1.1 A1About 40 minutes

Learning goals

I can tell the difference between one-variable data and two-variable data.
I can explain when a question needs two-variable data.
I can name the independent and the dependent variable in a two-variable question.
I can turn a one-variable question into a two-variable question.

Words to know

variable
A characteristic that can change from one person or thing to another, like height, age or favourite sport.

one-variable data
Data about one characteristic of each person or thing, like the heights of the students in a class.

two-variable data
Data about two characteristics of each person or thing, recorded as a pair, like each student's height and arm span.

relationship
A connection between two variables: when one changes, the other tends to change in a certain way.

independent variable
In a two-variable question, the variable you change, choose or think might affect the other one. It goes on the horizontal axis.

dependent variable
In a two-variable question, the variable you think might respond to the other one. It goes on the vertical axis.

Remember from Grade 7

In Grade 7 you collected qualitative, discrete and continuous data, organized it with percentages, drew circle graphs, and saw how adding or removing data changes the mean, median and mode. Almost all of that data was about one characteristic at a time. In Grade 8 you start asking how two characteristics are connected.

At the doctor's office

At a check-up, a nurse measures your height. If the clinic only wants to know how tall Grade 8 students are, one measurement per person is enough. That is one-variable data.

Now suppose the clinic asks a different question: "Do taller students tend to have longer arms?" One measurement is not enough any more. For each student, they need two measurements, height and arm span, kept together as a pair. That is two-variable data.

One variable or two?

One-variable data describes one characteristic of each person or thing.

Example: the number of minutes each student in your class reads on Saturday.
We use it to describe a distribution: the mean, median, mode, range and shape.
Graphs you know: bar graphs, histograms, stem-and-leaf plots, line plots and circle graphs.

Two-variable data describes two characteristics of each person or thing, recorded together.

Example: each student's reading minutes and their score on a vocabulary quiz.
We use it to look for a relationship: when one variable goes up, does the other tend to go up, go down, or not change in any pattern?
Each person gives an ordered pair, like (30 min, 18 points). In this unit you will graph these pairs on a scatter plot.

When do you need two variables? Whenever the question asks how one thing is connected to, related to or affected by another thing. Words like "Does ... affect ...?", "Is there a relationship between ...?" and "Do people who ... also ...?" are clues.

One-variable data fits on one number line: each person is one dot. Two-variable data needs two axes: each person is one point made from an ordered pair, like Person D (162, 165).

Question	Type of data	Why
How long does it take Grade 8 students to get to school?	One-variable	Only travel time is measured.
Does the distance from school affect how long the trip takes?	Two-variable	Distance and time are measured for each student.
What is the favourite winter sport in our class?	One-variable	One answer per student.
Do students who sleep more have faster reaction times?	Two-variable	Sleep and reaction time for each student.
How much snow fell each day in January?	Two-variable	Date and snowfall for each day (change over time).

Look for questions that connect two measurements of the same person, object or day.

Independent and dependent variables

In many two-variable questions, we think one variable might affect the other.

The independent variable is the one you change, choose, or that happens first.
The dependent variable is the one that might respond to it.

A quick test: put the variables into the sentence "Does ___ affect ___?" The first blank is the independent variable. For "Does the height you drop a ball from affect how high it bounces?", drop height is independent and bounce height is dependent.

Sometimes neither variable really causes the other, like height and arm span. Then choose the one you want to predict from as the independent variable.

Put the variables into "Does ___ affect ___?" The first blank is the independent variable (x-axis, across) and the second is the dependent variable (y-axis, up).

Worked example

Example 1: Sort the questions

Which questions need two-variable data?

What is the mass of each backpack in our class?
Is there a relationship between the mass of a backpack and the number of books in it?
How many hours did each student volunteer this year?
Do students who volunteer more hours also spend more time on clubs?

Question 1 measures one thing, backpack mass: one-variable.
Question 2 connects backpack mass and number of books for each backpack: two-variable.
Question 3 measures one thing, volunteer hours: one-variable.
Question 4 connects volunteer hours and club time for each student: two-variable.

Questions 2 and 4 need two-variable data. The clue: they ask about a relationship between two things measured on the same backpack or student.

Worked example

Example 2: Make it a two-variable question

Change this one-variable question into a two-variable question: "How many hours do Grade 8 students sleep on a school night?"

Keep the first variable: hours of sleep.
Think of something that might be connected to sleep: bedtime, screen time before bed, minutes of exercise, or the time you wake up.
Write a question that connects the two: "Is there a relationship between the time students go to bed and how many hours they sleep?"

"Does the time a student goes to bed affect how many hours they sleep on a school night?" Each student would give two numbers: bedtime and hours of sleep. (Many other good answers are possible.)

Worked example

Example 3: Which variable is which?

A class wonders: "Does the temperature outside affect how long it takes the snow on the school steps to melt?" Name the independent and dependent variables.

Use the sentence: "Does temperature affect melting time?"
The temperature comes first and might cause a change.
The melting time is what might respond.

Independent: outside temperature (°C). Dependent: time for the snow to melt (minutes or hours).

Think like a mathematician

A good two-variable question names both variables clearly, says who or what is measured, and gives units. "Does sleep affect school?" is too fuzzy. "Is there a relationship between hours of sleep and score on a 20-question mental-math quiz for Grade 8 students?" is clear and can be answered with data. Asking clear questions is a key part of thinking critically (Strand A).

Just for fun

Why does the dependent variable never pick the restaurant?

It always waits to see what the independent variable does first.

Hands-on activity

Variable hunt at home

You need: paper, a pencil, and a few things around the house (a grocery flyer, a nutrition label, the weather forecast).

Look at a nutrition label on a box of cereal or crackers. List the variables it gives (serving size, calories, sugar, fibre and so on).
Write two one-variable questions using the label or the flyer. Example: "How much sugar is in one serving of different cereals?"
Write two two-variable questions. Example: "Do cereals with more sugar per serving also have more calories per serving?"
For each two-variable question, circle the independent variable and underline the dependent variable.
Talk with a family member: which of your questions could you actually answer with the boxes in your kitchen?

What to look for: each two-variable question names two things measured on the same item and asks how they are connected.

Your turn

Is this one-variable or two-variable data? The number of goals each player on a hockey team scored this season.

Is this one-variable or two-variable data? Each player's minutes of ice time and number of goals.

Name the independent and dependent variables: "Does the number of hours a plant gets light affect how tall it grows in 3 weeks?"

Which question needs two-variable data? (a) What is the most popular pizza topping? (b) Does the price of a pizza depend on its diameter?

Turn this into a two-variable question: "How many minutes do students spend on homework each night?"

A survey records each student's favourite subject and favourite season. Is this two-variable data?

More practice: warm-up

One or two variables? Each student's mass.

One or two variables? Each student's mass and backpack mass.

One or two variables? The daily high temperature and the number of people at a splash pad each day.

In "Does ___ affect ___?", which blank holds the independent variable?

What is a pair of data values called when it is written like (160, 158)?

More practice: on your own

Name the independent and dependent variables: "Does the number of hours a car is plugged in affect how many kilometres it can drive?"

Liam asks, "Do Grade 8 students like winter?" Explain why this question gives one-variable data, and write a two-variable question about winter.

Why does a two-variable question need the two values from the same person kept together?

A town records the year and its population for the last 20 years. Is this one-variable or two-variable data? What kind of graph might show it?

Mila wants to know whether older students have bigger shoe sizes. Which variable is independent? Explain.

Challenge

A coach records each runner's time for 100 m and the average time of the whole team. Mo says this is two-variable data. Is he right? Explain.

For the question "Is there a relationship between the time students wake up and the time they arrive at school?", explain why the variables could be chosen either way, and say which you would put as independent and why.

Write a two-variable question that a scatter plot cannot show, and explain why.

Check your understanding

Quiz

Which question needs two-variable data?

In "Does the amount of fertilizer affect the height of a plant?", what is the dependent variable?

Two-variable data is used mostly to look for...

Which is an example of one-variable data?

Each student's data is written as (bedtime, hours of sleep). What is this pair called?

Which graph is made for showing the relationship between two numerical variables?

For parents and teachers

Big idea: The Ontario curriculum (D1.1) asks Grade 8 students to identify situations involving one-variable data and situations involving two-variable data, and explain when each type is needed. Until now your child has mostly worked with one variable at a time. Two-variable data lets them ask how things are connected, which leads to scatter plots later in this unit.

Common difficulties:

Thinking that "two groups" means two variables (boys' heights and girls' heights is still one variable, height). Ask: "How many measurements did each person give?"
Mixing up independent and dependent. Use the sentence "Does ___ affect ___?"
Writing fuzzy questions like "Does sleep affect school?" Push for measurable variables with units.

How to help: At dinner, play "one or two?": name something you could measure ("the price of gas", "gas price and the day of the week") and have your child decide. The lesson also builds A1 skills: asking clear questions and thinking critically about what data is needed.