Independent and dependent variables

Here you will learn about independent and dependent variables, including what they are, how to use them to represent real world problems and how to solve those problems.

Students will first learn about independent and dependent variables as part of expressions and equations in 6 th grade.

What are independent and dependent variables?

Independent and dependent variables are types of variables that change in relation to each other.

The independent variable change causes a measurable change in the dependent variable. In equations, it is typically represented by the variable x. When graphed, it is represented on the x -axis. It can also be called the input.

The dependent variable is determined by the independent variable. In equations, it is typically represented by the variable y. When graphed, it is represented on the y -axis. It can also be called the output.

For example,

Each pair of socks costs \$ 8. The total cost in dollars, y, depends on the number of pairs of socks, x.

This can be shown with an equation, in a table or on a graph:

Table:

Each pair of socks adds \$ 8 to the total cost. Always define the units of each variable in a table.

Equation: 8x=y; \, x= pairs of socks; y= total cost (\$)

The total cost will always be 8 times the number of pairs of socks, since each pair of socks costs \$ 8. After writing an equation, always define the units of each variable.

Graph:

As the pairs of socks increase, so does the total cost, by \$ 8 for each pair. Always label the units of the axes and give the graph a title.

What are independent and dependent variables?

Common Core State Standards

• Grade 6 – Expressions and Equations (6.EE.C.9)
  Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
  
  For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d=65t to represent the relationship between distance and time.

How to represent a real world problem with independent and dependent variables:

In order to represent a real world problem with independent and dependent variables:

1. Define the independent and dependent variable.
2. Define the relationship between the variables.
3. Create or use a representation to solve.

Independent and dependent variables examples

Example 1: find the value of the dependent variable

A train is travelling 55 miles per hour. How many miles will the train travel after 6 hours? Show the relationship in a table, then solve.

1. Define the independent and dependent variable.

The time the train travels (in hours) and the distance it travels (in miles) are the two variables. Since the number of miles traveled depends on how many hours the train has been traveling, it is the dependent variable y. This makes the hours the train has been traveling the independent variable x.

2Define the relationship between the variables.

The rate given, 55 miles per hour, shows that for each hour, the train will travel a distance of 55 miles.

You can model this in a table:

Relationship:

The total miles are always 55 times the hours

OR

For every additional hour, 55 more miles are traveled.

3Create or use a representation to solve.

To solve, you can extend the table:

OR

The total miles are always 55 times the hours, so 55\times{6}=~\ldots

Either way you solve, you see that 6 hours is 330 miles.

Example 2: find the value of the independent variable

On his birthday, Michael’s grandmother always gives him money. She gives him \$ 10 more than his age. How old was he when she gave him \$ 25?

Show the relationship in a table, then solve.

Define the relationship between the variables.

Michael always gets \$ 10 more than his age.

You can model this in a table:

Relationship:

The birthday money is always 10 more than the age

OR

For every year added, \$ 1 more is given.

Create or use a representation to solve.

You can extend the table…

OR

The birthday money is always 10 more than the age, so x+10=25.

To find x, think about what number added to 10 is equal to 25.

Either way you solve, you see that Michael was 15 years old.

Example 3: creating the equation

Daniel practices the piano 5 hours each week. Write an equation to calculate Daniel’s total practice hours after some weeks.

Define the relationship between the variables.

The rate given is 5 hours for every week.

You can model this in a table:

Relationship:

The total practice hours are always 5 times the weeks of practice.

Example 4: creating the equation

A store is having a sale and all the clothes are half off the original price. Write an equation to calculate the sale price.

Define the relationship between the variables.

The sale price is half the original price.

You can model this in a table:

Relationship:

The sale price is always 0.5 \, (or \cfrac{1}{2}) times the original price.

Example 5: comparing the equation and the graph

A marathon runner aims to run 20 miles. The equation and graph show the distance left to run after running some miles.

Equation: 20-x = y, where x=~\text{miles ran} and y=~\text{miles left to run}

Graph:

Explain why the graph and equation show the same relationship.

Define the relationship between the variables.

The equation shows that the total number of miles is always 20.

• When x=1, \, y=19 since 20-1=19.
• When x=2, \, y=18 since 20-2=18.
• When x=3, \, y=17 since 20-3=17.

If you look closely at the graph, you see this same relationship:

• The point (1, \, 19) shows that after 1 mile ran, 19 miles are left.
• The point (2, \, 18) shows that after 2 miles ran, 18 miles are left.
• The point (3, \, 17) shows that after 3 miles ran, 17 miles are left.

Example 6: comparing the table and the graph

A recipe book has two versions of the same recipe, including a “light recipe” for people that want to eat less salt. The equation and the table show the relationship between the salt in the two recipes.

Equation: \cfrac{1}{3} \, x=y, where x=~\text{regular recipe (grams)} and y=~\text{light recipe (grams)}

Table:

Explain why the table and equation show the same relationship.

Define the relationship between the variables.

The equation shows that the grams of salt in the light recipe are always \cfrac{1}{3} times the grams of salt in the regular recipe.

• When x=10, \, y=3\cfrac{1}{3} since \cfrac{1}{3}\times{10}=\cfrac{10}{3}=3\cfrac{1}{3}

• When x=20, \, y=6\cfrac{2}{3} since \cfrac{1}{3}\times{20}=\cfrac{20}{3}=6\cfrac{2}{3}

• When x=30, \, y=10 since \cfrac{1}{3}\times{30}=\cfrac{30}{3}=10

If you look closely at the table, you see this same relationship:

Teaching tips for independent and dependent variables

• Use a real word classroom example as an intro into this topic. For example, ask students to write down an item from the school cafeteria and the price. Then ask them to create a table of how many 1-5 items will cost. Use this as a jumping off point to introduce the variables and create tables, equations and graphs.

• Use the analogy of cause and effect to help students determine which variable is the independent and which is the dependent.

• Tie this into the grade level statistical analyses topics, helping students understand how independent and dependent variables connect to research design/research methods. This helps them better understand quantitative research and its connection to variables.

Easy mistakes to make

• Confusing the operation that connects the independent and dependent variable
  Pay close attention to the real world context and the variable being described. Then describe it in your own words to make sense of it. Then try creating a table or graph with possible values. Use this to help you decide whether the relationship between the variables is addition, subtraction, multiplication or division.

• Confusing the independent vs dependent variable in real world situations
  It can be tricky to immediately distinguish between independent and dependent variables. Consider the context: which variable affects the other? For some contexts, patterns will develop (such as price almost always being the independent variable). However, it can be hard to tell the difference for more subtle examples. Sometimes, either variable could be independent, depending on the specific context.
  
  For example,
  A recipe calls for 4 cups of blueberries for every 1 cup of cherries. This relationship could be written with either variable as independent.
  
  Cherries: 4x=y, where x=~\text{cups of cherries} and y=~\text{cups of blueberries}
  Blueberries: \cfrac{1}{4} \, x=y, where x=~\text{cups of blueberries} and y=~\text{cups of cherries}
  
  Note: Even though the equations are different, they represent the same relationship and both could be used to solve.

• Confusing qualitative and quantitative data
  Independent and dependent variables can only be defined for a relationship with quantitative data.

Related types of graphs lessons

• x and y axis
• Plot points on a graph
• Interpreting graphs
• Graph transformations

The next lessons are

• Types of graphs
• Graphing linear equations
• Rate of change

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