6th Grade · 6.EE.C.9

Independent and dependent variables in 6th grade math

Independent and dependent variables are the two amounts in a relationship where one changes because the other does. In 6th grade math, your child decides which is which, writes an equation that links them and shows the relationship in a table and on a graph. Children also meet these words in science experiments, where they mean much the same thing.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

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.

What it means, in plain words

A variable is a letter that stands for an amount that can change. When two amounts are linked, one usually depends on the other. The independent variable is the amount you choose or that changes on its own, such as the number of minutes you walk. The dependent variable is the amount that results, such as the distance you cover. A quick test is to say "the ___ depends on the ___". "The distance depends on the minutes" makes sense. The other way around does not.

Independent and dependent variables in 6th grade come with an equation. If you walk 80 meters each minute, the distance $d$ in meters after $m$ minutes is $d = 80m$ . The dependent variable stands alone on one side. The independent variable sits inside the rule on the other side.

The same relationship can be shown in a table and on a graph. In a table, the independent variable usually goes in the first column. On a graph, it goes along the horizontal axis (the x-axis), and the dependent variable goes up the vertical axis (the y-axis). Each row of the table becomes one point. Your child learns to read dependent and independent variables in all three forms and to see that they show the same relationship.

Your child should be able to:

Decide which of two amounts is the independent variable and which is the dependent variable
Write an equation such as $d = 80m$ that gives the dependent variable from the independent one
Fill in a table from an equation and plot the pairs on a graph
Explain how the equation, the table and the graph show the same relationship

Example

A pool fills with 15 gallons of water each minute. Write an equation for the water $w$ in gallons after $m$ minutes, and say which variable is which.

The amount of water depends on how long the pool has been filling.
So $m$ (minutes) is the independent variable and $w$ (gallons) is the dependent variable.
Each minute adds 15 gallons, so $w = 15m$ . After 4 minutes, $w = 15 \times 4 = 60$ gallons.
The equation is $w = 15m$ , with minutes as the independent variable and gallons as the dependent variable.

Example

A recipe uses 2 cups of flour for each batch of cookies. Make a table for 1, 2 and 3 batches. What does the graph look like?

The flour $f$ depends on the number of batches $b$ , so $f = 2b$ .
1 batch needs 2 cups, 2 batches need 4 cups and 3 batches need 6 cups.
On the graph, batches go along the horizontal axis and cups of flour go up the vertical axis.
The points $(1, 2)$ , $(2, 4)$ and $(3, 6)$ lie on a straight line that rises 2 cups for each batch.

Where children usually get stuck

Your child graphs the cups of flour along the bottom and the batches up the side.
Either axis seems fine until your child connects the horizontal axis with the amount that drives the change.
What helps. Have your child say the sentence "the ___ depends on the ___" first. The amount in the second blank goes along the bottom.
Tickets cost 7 dollars each. Your child writes $n = 7c$ , where $n$ is the number of tickets and $c$ is the total cost.
The letters went down in the order the words appeared, so the 7 is attached to the wrong amount.
What helps. Test a value. Two tickets should cost 14 dollars. Put $c = 14$ into $n = 7c$ and you get 98 tickets, so the letters are swapped. The right equation is $c = 7n$ .
In a table for $d = 80m$ , your child correctly writes 240 next to 3 minutes but cannot say what that row means.
The table becomes number work, and the two columns lose their names.
What helps. Ask for one sentence per row: "After 3 minutes, the distance is 240 meters." The sentence ties the numbers back to the situation.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
A bowling alley charges $3 for each game. The total amount paid depends on the number of games.
If we graph this relationship using points (x, y), what do x and y represent? Use the dropdowns below.
x represents the
, and y represents the
.
Show the answer and how to solve it
x represents the $\textbf{number of games}$ , and y represents the $\textbf{total amount paid}$ .
1. 1Find the two quantities in the story that can change. The number of games can change because a person can play 1 game, 2 games, 3 games, and so on. The total amount paid can change because it depends on how many games are played. The $3 per game is a fixed rate, and 3 dollars is just a number; neither can change, so they are not variables.
2. 2Decide which changing quantity is x. x names the quantity that changes first. The story says the total amount paid depends on the number of games, so the number of games changes first. So x represents the $\textbf{number of games}$ .
3. 3Decide which changing quantity is y. y names the quantity that changes because of x. The total amount paid changes because of the number of games, so y represents the $\textbf{total amount paid}$ .
Common mistake. A common mistake is swapping x and y so that x is the total amount paid. But the total amount paid changes because of the number of games, not the other way around, so the number of games is x. Another common mistake is picking dollars per game or 3 dollars; those are fixed values, not quantities that can change.
The idea underneath. Variables name the quantities that can change in a situation. The quantity that changes first is x, and the quantity that changes because of x is y. Rates and fixed numbers are not variables, even though they appear in the story.

2On level

The graph below shows the relationship between Hours studying and Quiz score.

The table below shows the two students' statements.

Student	Statement
Owen	Claim: "Quiz score is independent; Hours studying is dependent because changing Quiz score changes Hours studying."
Ava	Claim: "Quiz score is independent; Hours studying is dependent because Quiz score has the larger values on the graph."

Which answer is correct?

Owen

Ava

Both students

Neither student

Show the answer and how to solve it

The answer is Neither student because both wrongly call Quiz score the independent variable; Hours studying is independent and Quiz score depends on it.

1Look at the graph. Hours studying is on the x-axis and Quiz score is on the y-axis.
2Hours studying is the independent variable because it is the quantity you choose or track first. Quiz score changes as a result, so it is the dependent variable.
3Owen reverses the quantities, claiming Quiz score is independent. Ava also says Quiz score is independent, this time because it has the larger values, but value size does not decide variable roles. Both students are wrong, so the answer is Neither student.

Common mistake. A common mistake is calling Quiz score the independent variable, either by swapping the quantities or by assuming the larger values must be independent. Quiz score depends on Hours studying.

The idea underneath. The independent variable is the quantity you choose or track first, and the dependent variable changes because of that quantity.

3On level
Use the equation below to complete the y-column in the table.
$\displaystyle y = 2x + 3$
Enter each missing y-value.
x y
1
3
5
Show the answer and how to solve it
The completed y-column is 5, 9, 13 because 2(1) + 3 = 5, 2(3) + 3 = 9, and 2(5) + 3 = 13.
1. 1The equation gives the rule for every row: replace x with the row's x-value, multiply by 2, then add 3.
2. 2For the first row, substitute x = 1.
3. 3Now do the same for x = 3 and x = 5.
4. 4So the missing y-values are 5, 9, and 13.
Common mistake. A common mistake is multiplying each x-value by 2 but forgetting to add 3. That would give 2, 6, and 10 instead of 5, 9, and 13.
The idea underneath. Substituting means replacing x with the number from that row and then using the same equation each time.
4Stretch
Equation A, Equation B, and Equation C are shown below.
$\displaystyle \begin{array}{l} \text{Equation A: } y = 2x + 1 \\ \text{Equation B: } y = 4x + 1 \\ \text{Equation C: } y = x + 3 \end{array}$
Drag each equation label to the graph that matches it.
Connect each item on the left to its match on the right.
Equation label
Graph
Equation A
Equation B
Equation C
Use Tab to navigate between items. Click or tap a card to select it, then click or tap a card on the opposite side to create a match. You can also drag a card or its connector dot to a card on the opposite side.
Show the answer and how to solve it
Equation A matches the graph through (0,1) and (2,5), Equation B matches the graph through (0,1) and (2,9), and Equation C matches the graph through (0,3) and (2,5).
1. 1Start by looking at the three equations. Substituting x = 0 into each one gives the y-axis starting point.
2. 2Substitute 0 for x in each equation to find the y-axis starting point.
3. 3Now substitute x = 2 to find the second highlighted point on each graph.
4. 4Equations A and B both start at (0,1), so their second points are needed to tell them apart. Equations A and C both pass through (2,5), so their y-axis points are needed to tell them apart.
5. 5The extra graph shows (0,1) and (2,7). Equations A and B both give 1 when x = 0. At x = 2, they give 5 and 9, not 7. Equation C gives 3 when x = 0, not 1. No equation matches both points of the extra graph.
Common mistake. A common mistake is matching by only one highlighted point. Equations A and B both start at (0,1), and Equations A and C both pass through (2,5), so both points must be checked for every equation.
The idea underneath. A linear equation gives points on its graph, and two matching points are enough to identify the correct graph.