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Spectrum Math Tutor: Dependent and Independent Variables

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Independence & Dependence

You're probably familiar with what it means to be independent or dependent on something. A person's income is dependent upon their job because without that job they wouldn't get paid. A car's speed is independent of the amount of windshield wiper fluid it has because that has no effect on the speed whatsoever. In general, if you're independent of something, it means it cannot affect you in any way, and if you're dependent on something, changes in whatever it is you're dependent on will affect you.

These concepts of independence and dependence also have meaning in math, specifically with variables. In this lesson, we will learn what independent and dependent variables are and how to identify them.

Deciphering Variable Types

In a math equation, variables are the symbols or letters that represent numbers whose values can change. Variables can either be dependent or independent on other variables. Dependent variables rely on other variables to find their value, and independent variables do not rely on other variables to find their value.

So, to sum it up, an independent variable will change the value of a dependent variable, but a dependent variable cannot change the value of an independent variable, and the value of the dependent variable is determined by the value of the independent variable.

We can best see how the two types of variables differ in word problems, so let's look at one.

Example

Sarah makes $10 an hour and works between 6 to 8 hours a day. How much money does she make in a single day?

Let's start by breaking down the components of our word problem. We have one constant, a value that does not change, and two variables. The constant is her wage, $10 per hour, and the variables are how much money she makes in a day ( m), and how many hours she works ( h).

So, now which variable is dependent and which is independent? Since we know what the variables physically represent, we can figure this out by determining which variable depends upon the other to find its value. This will be our dependent variable.

Does Sarah need to know how many hours she works to find out how much money she makes in a day

- or -

Does Sarah need to know how much money she makes in a day to find out how many hours she works?

In this problem, it's the first option. Since Sarah's wage is hourly, how many hours she works determines how much money she makes.

So m, money made in a day, is our dependent variable, and h, hours worked, is our independent variable.

To see how the independent variable changes the value of the dependent variable, we can set up an equation for this word problem and solve it. Here, the amount of money Sarah makes is equal to her wage multiplied by the hours she works.

Sarah can work for 6, 7, or 8 hours in a day. We can see how this changing value for the independent variable affects the value of the dependent variable.

m = $10 * 6 = $60

m = $10 * 7 = $70

m = $10 * 8 = $80

When you know what each variable represents in the real world, you can use logic to figure out whether each variable is independent or dependent. There is no one size fits all formula that will automatically tell you what type of variable each is; you must figure it out yourself.

Equations Without Context

So, we've seen that you can figure out which variables are independent and dependent when you know what they represent, but what if you don't know that? Often in a math class you're given nothing more than an equation like the following:

When you're given an equation with a single stand-alone variable on one side of the equals sign, and a combination of constants and variables on others, the stand-alone variable is often the dependent variable. The variables on the other side of the equals sign are then independent.

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independent & dependent variables in math - video & lesson transcript | study.com

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Independent & Dependent Variables in Math - Video & Lesson Transcript | Study.com

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You're probably familiar with what it means to be independent or dependent on something. A person's income is dependent upon their job because without that job they wouldn't get paid. A car's speed is independent of the amount of windshield wiper fluid it has because that has no effect on the speed whatsoever.

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Although children may be too young to be familiar with the methods for experimentation, they are not too young to understand the meaning of the concepts. If you can explain and demonstrate the concepts underlying the scientific method in an age-appropriate way, kids can grasp how it works. With your help and a few basic rules in scientific experimentation, a child can run her own simple experiments.

A "variable" is a word for a quantity or condition that can change. Variables can be continuous or they can be discrete. Continuous variables can have many values. For example, time is continuous and can have many values. Plant growth, amount of sunlight or amount of water flowing are all continuous variables. Discrete variables are ones that have a few, distinct values. Something can be either on or off, present or absent, or have only several countable possibilities. A kitchen light can be on or off or a person may have blue, brown, green or hazel eyes.

An independent variable is a variable that you can control. One way to explain it to a child is that it is the variable that the child can change during the experiment. For example, in an experiment on the effect of light on plant growth, the child can control how much light a plant receives. He can put one plant near a window and another plant in a dark closet.

A dependent variable is the variable that you observe and measure. You have no control over the dependent variable; you want to observe what happens to the dependent variable when you change the independent variable. For example, if the child is testing the effects of light on plant growth, how much the plant grows after some period of time would be the dependent variable. The child can measure how much the plant in the closet grows relative to how much the plant near the window grows.

Besides independent and dependent variables, every good experiment needs to control certain variables so that they do not influence the outcome of the experiment systematically. A controlled variable is one that you keep the same for all the conditions of your experiment. A good scientist needs to think through all the variables that might need to be controlled so that they do not interfere with the experiment. For example, to get reliable results for the plant experiment, both the plant near the window and the plant in the closet need to receive the same amount of water so that testers would know that it was the differences in light and not the differences in water that made one plant grow more than the other.

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What Are Independent & Dependent Variables in Science for Kids?

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Although children may be too young to be familiar with the methods for experimentation, they are not too young to understand the meaning of the concepts. If you can explain and demonstrate the concepts underlying the scientific method in an age-appropriate way, kids can grasp how it works.

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Personally, I'm a little dyslexic. I still mix up right and left, and I find the words "dependent" and "independent" a little confusing. So I think "cause" (the x variable) and "effect" (the y variable). This is not exactly right and will make some scientists apoplectic, but as a sort of memory aid, it might help you too.

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Complete your note sheet and raise your hand to discuss with teacher.

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There are two types of experiments when it comes to any experiment. The variable varies depending on which is being tested in the experiment and which one is controlled in the experiment. Test out how well you understood the two in class by taking the quiz below. All the best!

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I can create a table of two variables that represents a real-world situation in which one quantity will change in relation to the other (S .

I can explain the difference between the independent variable and the dependent variable and give examples of both (R).

I can determine the independent and dependent variable in a relationship (S).

I can write an algebraic equation that represents the relationship between the two variables (S).

I can create a graph by plotting the dependent variable on the x-axis and the independent variable on the y-axis of a coordinate plane (S).

I can analyze the relationship between the dependent and independent variables by comparing the table, graph, and equation (R).

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Paper and Pencil

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Name:_____________________________________ Date:______

Introduction: In this lab, you will use a simulation to further your understanding of independent and dependent variables. You will be studying the level of European Corn Borer infestation in various environments.

Open the Simulator at: biol.co/sci-ecb
(alternately, use google to search for "mhhe virtual labs")

1. Read the introduction. You will change the independent variable, then you will observe and measure the dependent variable. In this experiment, what will always be the dependent variable?

2. Explain what is meant by the term "yield" with regard to this experiment. What factors can affect a corn's yield?

3. Open the Greenhouse Handbook and scan to find information about Bt corn. What makes Bt corn different from the other variety? What is the purpose of creating Bt corn?

4. What are some questions or worries that people might have about transgenic plants?

5. Read the instructions for setting up your chambers. Here you can make some decisions about what level of ECB infestation you will start with. Record your measurements in the data table.

6. Describe the effects of the ECB infestations you used. Were all corn varieties equally effective at controlling the ECB? How do you know?

7. Did the level of ECB infestation change your results?

8. What is the purpose of taking an average?

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Virtual Lab - Dependent and Independent Variables

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Name:_____________________________________ Date:______ Introduction: In this lab, you will use a simulation to further your understanding of independent and dependent variables. You will be studying the level of European Corn Borer infestation in various environments. Open the Simulator at: biol.co/sci-ecb (alternately, use google to search for "mhhe virtual labs") 1.

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Do only if you are way ahead!

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Both the independent variable and dependent variable are examined in an experiment using the scientific method, so it's important to know what they are and how to use them. Here are the definitions for independent and dependent variables, examples of each variable, and the explanation for how to graph them.

Independent Variable

The independent variable is the condition that you change in an experiment. It is the variable you control. It is called independent because its value does not depend on and is not affected by the state of any other variable in the experiment. Sometimes you may hear this variable called the "controlled variable" because it is the one that is changed. Do not confuse it with a "control variable," which is a variable that is purposely held constant so that it can't affect the outcome of the experiment.

Dependent Variable

The dependent variable is the condition that you measure in an experiment. You are assessing how it responds to a change in the independent variable, so you can think of it as depending on the independent variable. Sometimes the dependent variable is called the "responding variable."

Independent and Dependent Variable Examples

In a study to determine whether how long a student sleeps affects test scores, the independent variable is the length of time spent sleeping while the dependent variable is the test score.

You want to compare brands of paper towels, to see which holds the most liquid. The independent variable in your experiment would be the brand of paper towel. The dependent variable would be the amount of liquid absorbed by the paper towel.

In an experiment to determine how far people can see into the infrared part of the spectrum, the wavelength of light is the independent variable and whether the light is observed (the response) is the dependent variable.

If you want to know whether caffeine affects your appetite, the presence/absence of a given amount of caffeine would be the independent variable. How hungry you are would be the dependent variable.

You want to determine whether a chemical is essential for rat nutrition, so you design an experiment. The presence/absence of the chemical is the independent variable. The health of the rat (whether it lives and can reproduce) is the dependent variable. If you determine the substance is necessary for proper nutrition, a follow-up experiment might determine how much of the chemical is needed. Here, the amount of chemical would be the independent variable and the rat health would be the dependent variable.

How to Tell the Independent and Dependent Variable Apart

If you are having a hard time identifying which variable is the independent variable and which is the dependent variable, remember the dependent variable is the one affected by a change in the independent variable. If you write out the variables in a sentence that shows cause and effect, the independent variable causes the effect on the dependent variable. If you have the variables in the wrong order, the sentence won't make sense.

Independent variable causes an effect on the dependent variable.

Example: How long you sleep (independent variable) affects your test score (dependent variable).

This makes sense, but:

Example: Your test score affects how long you sleep.

This doesn't really make sense (unless you can't sleep because you are worried you failed a test, but that would be a different experiment).

How to Plot Variables on a Graph

There is a standard method for graphing the independent and dependent variable. The x-axis is the independent variable, while the y-axis is the dependent variable. You can use the DRY MIX acronym to help remember how to graph variables:

DRY MIX

D = dependent variable
R = responding variable
Y = graph on the vertical or y-axis

M = manipulated variable
I = independent variable
X = graph on the horizontal or x-axis

Test your understanding with the scientific method quiz.

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Definition and Examples of Independent and Dependent Variables

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Get the definitions for independent and dependent variables, examples of each type of variable, and the explanation for how to graph them.

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Before beginning this section, I draw the table of values for the Bridge weights on the whiteboard for all to see. I like to ask students to think of the different ways in which they can name both columns. Some possible answers are "input/ output", "x/y", "domain/range".

I indicate that they will learn one more way of naming these columns. I say, "Comparing data in tables is really important so there are a lot of different ways of talking about the tables." Then, I ask:
Which value is dependent on the other? Does the number of wooden boards depend on the maximum weight they can hold, or does the bridge maximum weight depend on the number of wooden boards?
Most of my students will likely say the latter. Then I write on the board:

I tell the class that another way of naming the columns is using the terms independent and dependent variable and proceed to ask which is which in our bridge case. If students seem confused I will ask, "Which part of the bridge task can you manipulate or change?"

Most of the time, my students will find it logical that these measurements (number of boards, maximum weight) represent the independent variable and that the weight the bridge can hold is the dependent variable.

I annotate my notes on the board by writing the Function Statement and stating the dependent variable and the independent variable.

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Before beginning this section, I draw the table of values for the Bridge weights on the whiteboard for all to see. I like to ask students to think of the different ways in which they can name both columns. Some possible answers are "input/ output", "x/y", "domain/range".