6th Grade · 6.EE.B.6

What is a variable in 6th grade math

What is a variable? In 6th grade math, a variable is a letter, such as $n$ or $x$ , that stands for a number. Your child learns to use variables to write expressions for real situations. Your child also learns that the letter can mean one unknown number or any number from a set, depending on the problem.

What the standard says

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

6.EE.B.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

What it means, in plain words

The short answer to "what is a variable" in 6th grade is a letter that stands for a number. Children met the idea in earlier grades as a box or a blank, as in $4 + \square = 10$ . In 6th grade, a letter takes the place of the box. Any letter works. Many teachers pick one that fits the problem, like $h$ for hours.

Variables can play two roles. In $n + 12 = 20$ , $n$ is one unknown number, and only 8 makes the equation true. In a pay rule such as $9h$ dollars for $h$ hours of work, $h$ can be any number of hours in a set, such as 1 to 40. The expression gives the pay for each of them.

Writing expressions with variables turns a situation into math. A pizza costs 9 dollars, plus 2 dollars for each topping. If $t$ is the number of toppings, the cost is $9 + 2t$ dollars. Each part of the expression means something: 9 is the starting price, 2 is the price of one topping and $t$ is how many toppings there are.

Your child should be able to:

Explain that a variable is a letter that stands for a number
Write an expression with a variable for a real situation
Tell whether a variable stands for one unknown number or any number in a set
Explain what each number and letter in an expression means in the situation

Example

A 60-inch ribbon is cut into $p$ equal pieces. Write an expression for the length of each piece, then find the length when there are 5 pieces.

Cutting into equal pieces means dividing, so each piece is $\frac{60}{p}$ inches long.
Here $p$ can be any number of pieces in a set, such as 2, 3, 4, 5 or 6.
For $p = 5$ : $\frac{60}{5} = 12$ .
The expression is $\frac{60}{p}$ , and with 5 pieces each piece is 12 inches long.

Example

A family drives 18 miles before lunch and $m$ miles after lunch. The whole trip is 45 miles, so $18 + m = 45$ . What does $m$ stand for?

The trip has a fixed total of 45 miles.
So $m$ is one unknown number: the miles driven after lunch.
Only one number works: $18 + 27 = 45$ .
Here $m$ stands for one unknown number, 27 miles.

Where children usually get stuck

Apples cost 2 dollars each. Your child writes $2a$ and reads it as "2 apples".
The letter feels like a label, a short way to write a word. In algebra it stands for a number, here the number of apples.
What helps. Have your child say the variable in full: " $a$ is the number of apples". Then $2a$ reads as "2 dollars times the number of apples".
Your child will not leave a letter in an answer like $9 + 2t$ and asks what the real number is.
Until now every answer was a number. An expression with a letter left in it feels unfinished.
What helps. Explain that the expression is a rule, not a final answer. Ask your child to use it for 1, 2 and 3 toppings. It gives 11, 13 and 15 dollars.
For a pizza that costs 2 dollars per topping plus a 9-dollar base, your child writes $2 + 9t$ .
The numbers go into the expression in the order they appear in the words. Nobody asked which amount repeats for each topping.
What helps. Ask which amount goes up with every topping. That number goes next to the letter: $9 + 2t$ .

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
Notebooks cost $3 each, and a total of $15 was spent. The equation below represents the situation, where n is the number of notebooks.
$\displaystyle 3n = 15$
Mia's claim: "n represents one unknown number because this equation finds a specific number of notebooks." Leo's claim: "n represents many possible numbers because a letter can stand for any number."
Select who is correct about what n represents.
A
Only Mia is correct.
B
Only Leo is correct.
C
Both students are correct.
D
Neither student is correct.
Show the answer and how to solve it
Only Mia is correct because in 3n = 15, n is being used to find one unknown number of notebooks.
1. 1Look at the equation and the notebook story. The total is fixed at 15, so this equation is about one specific situation.
2. 2The equation means 3 times the number of notebooks equals 15. Because the total is already fixed, there is one specific missing count to find.
3. 3Mia is correct: n represents one unknown number. Leo is wrong because a letter can mean many values in some settings, but the fixed total here makes n one specific missing count.
Common mistake. A common mistake is agreeing with Leo that a letter always can stand for any number. Even though variables can represent many values in some contexts, 3n = 15 fixes the total at 15, so n here is one specific missing number.
The idea underneath. A variable's role comes from what the math statement is doing: a rule can allow many values, but an equation with a fixed total is asking for one missing value.
2On level
Each situation uses n. Drag each situation to the expression that matches it.
Connect each item on the left to its match on the right.
Situation
Expression
A library charges 5 dollars to join and 2 dollars for each book borrowed. Total cost for n books.
A coin jar has 24 coins and a player uses 6 coins each round. Coins left after n rounds.
A savings account starts with 10 dollars and earns 3 dollars each week. Total after n weeks.
$\displaystyle 24 - 6n$
$\displaystyle 10 + 3n$
$\displaystyle 5 + 2n$
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
The correct matches are library cost → $5 + 2n$ , coin jar → $24 - 6n$ , and savings account → $10 + 3n$ .
1. 1The library charges a flat 5 dollars to join, which is the constant. Then it charges 2 dollars for each book, so 2 is multiplied by n. The matching expression is $5 + 2n$ .
2. 2The coin jar starts with 24 coins, which is the constant. The player uses 6 coins each round, so 6 is subtracted for each n. The matching expression is $24 - 6n$ .
3. 3The savings account begins with 10 dollars, the constant, and gains 3 dollars every week, so 3 is multiplied by n. The matching expression is $10 + 3n$ .
Common mistake. A common mistake is swapping the starting amount and the per-unit rate. For the library, a student might write $2 + 5n$ instead of $5 + 2n$ , mixing up which number is the one-time fee and which is the per-book charge.
The idea underneath. In a context, the constant tells the amount you start with, and the coefficient of n tells how much is added or taken away each time.
3On level
In a sticker collection, n is the number of sticker packs opened.
$\displaystyle 15 + 4n$
Select the description that matches the whole expression.
A
A sticker collection starts at 4 stickers and gains 15 stickers each pack.
B
A sticker collection starts at 15 stickers and gains 4 stickers each pack.
C
A sticker collection starts at 15 stickers and loses 4 stickers each pack.
D
A sticker collection gains 4 stickers each pack.
Show the answer and how to solve it
The expression $15 + 4n$ means the collection starts at 15 stickers and gains 4 stickers each pack, so the description with a starting count of 15 and a 4-sticker gain each pack is the correct match.
1. 1Look at the expression and remember that n stands for the number of sticker packs opened.
2. 2The 15 is the starting number of stickers. The 4n means 4 stickers are added for each pack opened.
3. 3The plus sign shows the collection is growing, so the matching story must start at 15 stickers and gain 4 stickers each pack.
4. 4The description that says a sticker collection starts at 15 stickers and gains 4 stickers each pack is the correct match for $15 + 4n$ .
Common mistake. A common mistake is swapping the 15 and 4 and choosing the story that starts at 4 stickers and gains 15 per pack. In $15 + 4n$ , 15 is the starting amount and 4 is the amount added each pack.
The idea underneath. In an expression like a + bn, the constant is the starting amount and the term with n shows the repeated change.
4Stretch
Notebooks cost 6 dollars each, and n is the number of notebooks in one purchase that totals 48 dollars, so 6n = 48.
Select the statement that best describes what n means.
A
One unknown number — one fixed value
B
One unknown number — many possible values
C
Any number in a given set — one fixed value
D
Any number in a given set — many possible values
Show the answer and how to solve it
The correct choice is One unknown number — one fixed value. The total is fixed at 48 dollars, so n stands for one specific notebook count.
1. 1The sentence says this one purchase totals 48 dollars — that total is already fixed, not left open for any purchase.
2. 2Dividing both sides by 6 gives n = 8. Only that one value makes the equation true; no other count of notebooks adds up to exactly 48 dollars at 6 dollars each.
3. 3Because one specific total locks in one answer, n is used as one unknown number — one fixed value.
Common mistake. A common mistake is choosing any number in a given set because notebook counts can vary from purchase to purchase. In this sentence, the total is already fixed at 48 dollars, so n is used to find one specific count.
The idea underneath. A variable represents one unknown number when the context locks in one case, but it represents any number in a given set when the rule can work for many cases.