Chapter 5 · Numerical expressions
Write Expressions for Word Problems and Fix Mistakes
Turn a word problem into one expression and say what each part means. Then find and fix mistakes such as numbers in the wrong order or missing parentheses.
A word problem is a short story with steps. First one thing happens, then another. You can record all of those steps in one line of math.
That line is a numerical expression: numbers joined by operation signs such as , , and , with no equals sign. It can also have parentheses, the curved marks ( ) that hold a group of numbers together. You already write expressions with two operations. Now you will write them for word problems, say what each part stands for, and find and fix mistakes in them.
Here is a story. An art room has tables. Each table has red markers and blue markers. How many markers are in the room?
Think about the steps in order. First, find the markers on one table: and together. Then take that amount times, once for each table. One expression records both steps.
| Part of the expression | What it stands for in the story |
|---|---|
the red markers on one table | |
the blue markers on one table | |
all the markers on one table | |
the number of tables | |
all the markers in the room: groups of |
The parentheses keep together as one amount: the markers on one table. They tell you to work out that part first.
You can say what the expression means before you work anything out. It is times as large as . When you do work it out, and , so the room has markers.
Key concept
One expression tells the whole story
A numerical expression can record every step of a word problem in one line. Each number stands for an amount in the story. Each operation sign stands for something that happens to those amounts. Parentheses hold together a group that the story treats as one amount, and you work out that group first. To check an expression, read it back in words. It should tell the same story.
Worked example
Write an expression: the kite shop
A shop has kites. It sells of them. Then it hangs the rest on walls, with the same number of kites on each wall. Write one expression for the number of kites on each wall. Say what each part stands for.
Find the first thing that happens: the shop sells kites. Start with the 32 kites in the shop and take away the 8 it sells. Write the starting amount first, because subtraction takes away from the amount you start with.
Find the next thing that happens: the shop splits the kites that are left equally among 4 walls. Write the amount being split first and the number of walls after the division sign. Put parentheses around the kites that are left, so that you divide the whole amount by 4.
Match every part to the story, so you know nothing is missing. 32 is the number of kites at the start. 8 is the number of kites sold. The part in parentheses is the number of kites left. 4 is the number of walls.
Check by working it out. Inside the parentheses, 32 minus 8 is 24. Then 24 divided by 4 is 6. That fits the story: 24 kites are left, and 6 go on each of the 4 walls.
Answer
The expression is . It stands for the kites left after the sale, , split equally among walls. Each wall has kites.
Now switch jobs. Someone else writes the expression, and you check it. Read the expression back in words and see whether it tells the same story. Check two things:
- Order. In each subtraction and division, does the amount you start with come first?
- Parentheses. If the story treats some numbers as one amount, do parentheses hold them together?
Order matters because subtraction and division are not like addition. and are the same total. But starts with and takes away , while starts with and takes away . The same two numbers tell two different stories.
| Words | What the words mean | Expression |
|---|---|---|
subtract from | start with and take away | |
less than | start with and take away | |
divide by | split into equal groups | |
share equally among | split into equal groups |
Worked example
Find the mistake: sharing pencils
A teacher has red pencils and blue pencils. She shares all of them equally among teams. Dev writes for the number of pencils each team gets. Find his mistake and fix it.
Read Dev's expression back in words. It says: start with 4 and divide it by the number of pencils. The story says the opposite: start with all the pencils and share them among 4 teams.
Check the parentheses. The story puts the 20 red pencils and the 16 blue pencils together as one amount, and Dev's parentheses hold them together. That part is right, so keep it.
Fix the order. Write the amount being shared first, and write the number of teams after the division sign.
Check the fixed expression. 20 plus 16 is 36, and 36 divided by 4 is 9. That fits the story: 36 pencils, with 9 for each of the 4 teams.
Answer
Dev wrote the numbers of the division in the wrong order. The correct expression is , and each team gets pencils.
Worked example
Find the mistake: games on sale
A game costs dollars. On sale day, each game costs dollars less. Tom buys games on sale day. He writes for the total cost in dollars. Find his mistake and fix it.
Read Tom's expression back in words. With no parentheses, you work out 4 times 15 first and then take away 3. That says: buy 4 games at the full price, then take 3 dollars off just one time.
Compare with the story. Every game costs 3 dollars less, so the 3 comes off the price of each game before you count the 4 games. The sale price of one game is one amount: 15 minus 3.
Fix the expression. Put parentheses around the sale price, so that you multiply the whole sale price by 4.
Check the fixed expression. 15 minus 3 is 12, and 4 times 12 is 48. That fits the story: 4 games at 12 dollars each.
Answer
Tom left out the parentheses. The correct expression is , so the games cost dollars, not dollars.
Takeaway: an expression is a word problem written in one line. Every number stands for an amount in the story, and every sign stands for something that happens. Whether you write an expression or check one, read it back in words. Make sure the amount you start with comes first in each subtraction and division, and make sure parentheses hold together each group that the story treats as one amount.