Write and Check Expressions for Word Problems | Grade 6 Math | LevelKite

You can already turn a word phrase into an expression. Now you will do it for a whole story.

A fair charges 4 dollars to get in. You pay that one time. Each ride then costs 3 dollars. How much does a visit cost? It depends on how many rides you take. The table shows the arithmetic for three different visits.

The cost of a visit to the fair

Number of rides	Arithmetic	Cost in dollars
$1$	$3 \cdot 1 + 4 = 3 + 4$	$7$
$2$	$3 \cdot 2 + 4 = 6 + 4$	$10$
$5$	$3 \cdot 5 + 4 = 15 + 4$	$19$

Read down the middle column. The steps never change: multiply 3 by the number of rides, then add 4. Only the number of rides changes.

A variable is a letter that stands for a number that can change or is not known yet. Let $r$ be the number of rides you take. An algebraic expression is a math phrase made of numbers, variables and operation signs. One expression gives the cost in dollars for any number of rides: $3 \cdot r + 4$ , written $3r + 4$ for short.

Key concept

An expression is a model of the story

An expression models a situation when it gives the same value as the story for every number that makes sense for the variable. You build a model by making each part of the expression say what a part of the story says. In $3r + 4$ , the $3r$ says you pay 3 dollars for each of $r$ rides. The $4$ is the entry fee, which you pay once, so you add it one time. Always state what the variable stands for, and build from what the story means.

Use these steps to write an expression for a word problem.

Name the unknown. Choose a letter. Say exactly what it stands for and its unit, such as the number of rides or the price in dollars.
Decide what the story does. Does it put amounts together, take an amount away, repeat equal groups, or share equally? That tells you whether to add, subtract, multiply or divide.
Build the expression one part at a time. In a subtraction, the amount you start with goes first. In a division, the amount you share goes first.
Check it. Read each part back into the story. Then test a number that makes sense in the story. Find the answer from the story, and find it from your expression. The two answers must agree.

Worked example

Comparing two amounts

Sam has 5 fewer stickers than Tia. You do not know how many stickers Tia has. Write an expression for the number of stickers Sam has.

Name the unknown. The story compares Sam's amount to Tia's, but it does not give Tia's amount. Let t be the number of stickers Tia has.
$t = \text{number of stickers Tia has}$
Decide what the story does. Having 5 fewer means Sam's amount is Tia's amount with 5 taken away. Taking an amount away is subtraction.
Build the expression. You start with Tia's amount, so t goes first. Then subtract the 5.
$t - 5$
Check it. Read it back: start with Tia's stickers and take 5 away. That is what the story says. Now test a number. If Tia has 12 stickers, Sam has 5 fewer than 12, which is 7. Replace t with 12 in the expression. It gives 7 as well.
$12 - 5 = 7$

Answer

Sam has $t - 5$ stickers, where $t$ is the number of stickers Tia has.

Worked example

Sharing a price, then adding more

Four friends share the price of a pizza equally. Each friend also buys a drink for 2 dollars. You do not know the price of the pizza. Write an expression for the total amount one friend pays, in dollars.

Name the unknown. The story does not give the price of the pizza. Let p be the price of the pizza in dollars.
$p = \text{price of the pizza in dollars}$
Write one friend's share of the pizza. Sharing equally among 4 friends means dividing by 4. The price is the amount you share, so p goes first.
$p \div 4 = \frac{p}{4}$
Add the drink. Each friend pays 2 more dollars one time, so add 2 to the share.
$\frac{p}{4} + 2$
Check it. Read it back: one of 4 equal shares of the price, plus 2 dollars for a drink. Now test a 12-dollar pizza. From the story, each share is 12 divided by 4, which is 3 dollars, and the drink makes 3 + 2 = 5 dollars. Replace p with 12 in the expression. It gives 5 as well.
$\frac{12}{4} + 2 = 3 + 2 = 5$

Answer

One friend pays $\frac{p}{4} + 2$ dollars, where $p$ is the price of the pizza in dollars.

Now switch roles. Someone gives you an expression for a story, and you decide whether it is correct. Use the same two checks.

Read it back. Say what each part of the expression means in the story. Does the variable go with the right number? Does each operation match what the story does? In a subtraction or a division, are the parts in the right order?

Test it. Pick a number that makes sense for the variable. Find the answer from the story, then from the expression. If the two answers are different, the expression is wrong.

One caution: an expression that looks different is not always a mistake. For the fair, $3r + 4$ and $4 + 3r$ are both correct, because adding in either order gives the same sum. Subtraction is not like that. For Sam's stickers, $t - 5$ is correct and $5 - t$ is not.

Worked example

Find and fix a mistake

A swim club charges a sign-up fee of 15 dollars, paid one time, and 8 dollars for each month. Ana lets $m$ be the number of months. She writes $15m + 8$ for the total cost in dollars. Is Ana's expression correct? If not, fix it.

Read Ana's expression back. The first part multiplies 15 by the number of months, so it says you pay 15 dollars every month. The second part adds 8 one time.
$15m + 8$
Compare with the story. The story says the opposite: you pay 8 dollars every month and 15 dollars one time. Ana put the variable with the wrong number.
Test 2 months to confirm. From the story, two months cost 8 + 8 = 16 dollars, and the fee makes 16 + 15 = 31 dollars. Replace m with 2 in Ana's expression. It gives 38, not 31, so the expression is wrong.
$15 \cdot 2 + 8 = 30 + 8 = 38$
Fix it. The monthly price repeats once for each month, so multiply 8 by m. You pay the fee once, so add 15 one time.
$8m + 15$
Test the fixed expression with 2 months to confirm the fix. It gives 31, the same as the story's answer.
$8 \cdot 2 + 15 = 16 + 15 = 31$

Answer

Ana's expression is not correct, because it charges 15 dollars every month and 8 dollars once. The total cost for $m$ months is $8m + 15$ dollars.

Testing both expressions for the swim club, with costs in dollars

Months	Cost from the story	Ana's expression, 15m + 8	Fixed expression, 8m + 15
$1$	$8 + 15 = 23$	$15 \cdot 1 + 8 = 15 + 8 = 23$	$8 \cdot 1 + 15 = 8 + 15 = 23$
$2$	$8 + 8 + 15 = 31$	$15 \cdot 2 + 8 = 30 + 8 = 38$	$8 \cdot 2 + 15 = 16 + 15 = 31$
$3$	$8 + 8 + 8 + 15 = 39$	$15 \cdot 3 + 8 = 45 + 8 = 53$	$8 \cdot 3 + 15 = 24 + 15 = 39$

Takeaway. An expression is a model of a story. To write one, name the variable and its unit, then build each part from what the story means, not from the order of its words. To find a mistake in an expression, read each part back into the story and test numbers that make sense. Different answers prove a mistake. Matching answers are a good sign, but check the meaning of every part too.