Lesson 9: Introduction to Word Problems
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What are word problems?
A word problem is a math problem written out as a short story or scenario. Basically, it describes a realistic problem and asks you to imagine how you would solve it using math. If you’ve ever taken a math class, you’ve probably solved a word problem. For instance, does this sound familiar?
Johnny has 12 apples. If he gives four to Susie, how many will he have left?
You could solve this problem by looking at the numbers and figuring out what the problem is asking you to do. In this case, you’re supposed to find out how many apples Johnny has left at the end of the problem. By reading the problem, you know Johnny starts out with 12 apples. By the end, he has 4 less because he gave them away. You could write this as:
12 — 4
12 — 4 = 8, so you know Johnny has 8 apples left.
Word problems in algebra
If you were able to solve this problem, you should also be able to solve algebra word problems. Yes, they involve more complicated math, but they use the same basic problem-solving skills as simpler word problems.
You can tackle any word problem by following these five steps:
- Read through the problem carefully, and figure out what it’s about.
- Represent unknown numbers with variables.
- Translate the rest of the problem into a mathematical expression.
- Solve the problem.
- Check your work.
We’ll work through an algebra word problem using these steps. Here’s a typical problem:
The rate to rent a small moving van is $30 per day, plus $0.50 per mile. Jada rented a van to drive to her new home. It took two days, and the van cost $360. How many miles did she drive?
It might seem complicated at first glance, but we already have all of the information we need to solve it. Let’s go through it step by step.
Step 1: Read through the problem carefully.
With any problem, start by reading through the problem. As you’re reading, consider:
- What question is the problem asking?
- What information do you already have?
Let’s take a look at our problem again. What question is the problem asking? In other words, what are you trying to find out?
The rate to rent a small moving van is $30 per day, plus $0.50 per mile. Jada rented a van to drive to her new home. It took 2 days, and the van cost $360. How many miles did she drive?
There’s only one question here. We’re trying to find out how many miles Jada drove. Now we need to locate any information that will help us answer this question.
There are a few important things we know that will help us figure out the total mileage Jada drove:
- The van cost $30 per day.
- In addition to paying a daily charge, Jada paid $0.50 per mile.
- Jada had the van for 2 days.
- The total cost was $360.
Step 2: Represent unknown numbers with variables.
In algebra, you represent unknown numbers with letters called variables. (To learn more about variables, see our lesson on reading algebraic expressions.) You can use a variable in the place of any amount you don’t know. Looking at our problem, do you see a quantity we should represent with a variable? It’s often the number we’re trying to find out.
The rate to rent a small moving van is $30 per day, plus $0.50 per mile. Jada rented a van to drive to her new home. It took 2 days, and the van cost $360. How many miles did she drive?
Since we’re trying to find the total number of miles Jada drove, we’ll represent that amount with a variable—at least until we know it. We’ll use the variable m for miles. Of course, we could use any variable, but m should be easy to remember.
Step 3: Translate the rest of the problem.
Let’s take another look at the problem, with the facts we’ll use to solve it highlighted.
The rate to rent a small moving van is $30 per day, plus $0.50 per mile. Jada rented a van to drive to her new home. It took 2 days, and the van cost $360. How many miles did she drive?
We know the total cost of the van, and we know that it includes a fee for the number of days, plus another fee for the number of miles. It’s $30 per day, and $0.50 per mile. A simpler way to say this would be:
$30 per day plus $0.50 per mile is $360.
If you look at this sentence and the original problem, you can see that they basically say the same thing: It cost Jada $30 per day and $0.50 per mile, and her total cost was $360. The shorter version will be easier to translate into a mathematical expression.
Let’s start by translating $30 per day. To calculate the cost of something that costs a certain amount per day, you’d multiply the per-day cost by the number of days—in other words, 30 per day could be written as 30 ⋅days, or 30 times the number of days. (Not sure why you’d translate it this way? Check out our lesson on writing algebraic expressions.)
$30 per day and $.50 per mile is $360
$30 ⋅ day + $.50 ⋅ mile = $360
As you can see, there were a few other words we could translate into operators, so and $.50 became + $.50, $.50 per mile became $.50 ⋅ mile, and is became =.
Next, we’ll add in the numbers and variables we already know. We already know the number of days Jada drove, 2, so we can replace that. We’ve also already said we’ll use m to represent the number of miles, so we can replace that too. We should also take the dollar signs off of the money amounts to make them consistent with the other numbers.
$30 ⋅ day + $.50 ⋅ mile = $360
30 ⋅ 2 + .5 ⋅ m = 360
Now we have our expression. All that’s left to do is solve it.
Step 4: Solve the problem.
This problem will take a few steps to solve. (If you’re not sure how to do the math in this section, you might want to review our lesson on simplifying expressions.) First, let’s simplify the expression as much as possible. We can multiply 30 and 2, so let’s go ahead and do that. We can also write .5 ⋅ m as 0.5m.
30 ⋅ 2 + .5 ⋅ m = 360
60 + .5m = 360
Next, we need to do what we can to get the m alone on the left side of the equals sign. Once we do that, we’ll know what m is equal to—in other words, it will let us know the number of miles in our word problem.
We can start by getting rid of the 60 on the left side by subtracting it from both sides.
The only thing left to get rid of is .5. Since it’s being multiplied with m, we’ll do the reverse and divide both sides of the equation with it.
.5m / .5 is m and 300 / 0.50 is 600, so m = 600. In other words, the answer to our problem is 600—we now know Jada drove 600 miles.
Step 5: Check the problem.
To make sure we solved the problem correctly, we should check our work. To do this, we can use the answer we just got—600—and calculate backward to find another of the quantities in our problem. In other words, if our answer for Jada’s distance is correct, we should be able to use it to work backward and find another value, like the total cost. Let’s take another look at the problem.
The rate to rent a small moving van is $30 per day, plus $0.50 per mile. Jada rented a van to drive to her new home. It took 2 days, and the van cost $360. How many miles did she drive?
According to the problem, the van costs $30 per day and $0.50 per mile. If Jada really did drive 600 miles in 2 days, she could calculate the cost like this:
$30 per day and $0.50 per mile
30 ⋅ day + .5 ⋅ mile
30 ⋅ 2 + .5 ⋅ 600
60 + 300
360
According to our math, the van would cost $360, which is exactly what the problem says. This means our solution was correct. We’re done!
While some word problems will be more complicated than others, you can use these basic steps to approach any word problem. On the next page, you can try it for yourself.
Practice!
Let’s practice with a couple more problems. You can solve these problems the same way we solved the first one—just follow the problem-solving steps we covered earlier. For your reference, these steps are:
- Read through the problem carefully, and figure out what it’s about.
- Represent unknown numbers with variables.
- Translate the rest of the problem into a mathematical expression.
- Solve the problem.
- Check your work.
If you get stuck, you might want to review the problem on page 1. You can also take a look at our lesson on writing algebraic expressions for some tips on translating written words into math.
Problem 1
Try completing this problem on your own. When you’re done, move on to the next page to check your answer and see an explanation of the steps.
A single ticket to the fair costs $8. A family pass costs $25 more than half of that. How much does a family pass cost?
Problem 2
Here’s another problem to do on your own. As with the last problem, you can find the answer and explanation to this one on the next page.
Flor and Mo both donated money to the same charity. Flor gave three times as much as Mo. Between the two of them, they donated $280. How much money did Mo give?
Problem 1 Answer
Here’s Problem 1:
A single ticket to the fair costs $8. A family pass costs $25 more than half that. How much does a family pass cost?
Answer: $29
Let’s solve this problem step by step. We’ll solve it the same way we solved the problem on page 1.
Step 1: Read through the problem carefully
The first in solving any word problem is to find out what question the problem is asking you to solve and identify the information that will help you solve it. Let’s look at the problem again. The question is right there in plain sight:
A single ticket to the fair costs $8. A family pass costs $25 more than half that. How much does a family pass cost?
So is the information we’ll need to answer the question:
- A single ticket costs $8.
- The family pass costs $25 more than half the price of the single ticket.
Step 2: Represent the unknown numbers with variables
The unknown number in this problem is the cost of the family pass. We’ll represent it with the variable f.
Step 3: Translate the rest of the problem
Let’s look at the problem again. This time, the important facts are highlighted.
A single ticket to the fair costs $8. A family pass costs $25 more than half that. How much does a family pass cost?
In other words, we could say that the cost of a family pass equals half of $8, plus $25. To turn this into a problem we can solve, we’ll have to translate it into math. Here’s how:
- First, replace the cost of a family pass with our variable f.
- Next, take out the dollar signs and replace words like plus and equals with operators.
- Finally, translate the rest of the problem. Half of can be written as 1/2 times, or 1/2 ⋅ :
f equals half of $8 plus $25
f = half of 8 + 25
f = 1/2 ⋅ 8 + 25
Step 4: Solve the problem
Now all we have to do is solve our problem. Like with any problem, we can solve this one by following the order of operations.
- f is already alone on the left side of the equation, so all we have to do is calculate the right side.
- First, multiply 1/2 by 8. 1/2 ⋅ 8 is 4.
- Next, add 4 and 25. 4 + 25 equals 29 .
f = 1/2 ⋅ 8 + 25
f = 4 + 25
f = 29
That’s it! f is equal to 29. In other words, the cost of a family pass is $29.
Step 5: Check your work
Finally, let’s check our work by working backward from our answer. In this case, we should be able to correctly calculate the cost of a single ticket by using the cost we calculated for the family pass. Let’s look at the original problem again.
A single ticket to the fair costs $8. A family pass costs $25 more than half that. How much does a family pass cost?
We calculated that a family pass costs $29. Our problem says the pass costs $25 more than half the cost of a single ticket. In other words, half the cost of a single ticket will be $25 less than $29.
- We could translate this into this equation, with s standing for the cost of a single ticket.
- Let’s work on the right side first. 29 — 25 is 4.
- To find the value of s, we have to get it alone on the left side of the equation. This means getting rid of 1/2. To do this, we’ll multiply each side by the inverse of 1/2: 2.
1/2s = 29 — 25
1/2s = 4
s = 8
According to our math, s = 8. In other words, if the family pass costs $29, the single ticket will cost $8. Looking at our original problem, that’s correct!
A single ticket to the fair costs $8. A family pass costs $25 more than half that. How much does a family pass cost?
So now we’re sure about the answer to our problem: The cost of a family pass is $29.
Problem 2 Answer
Here’s Problem 2:
Flor and Mo both donated money to the same charity. Flor gave three times as much as Mo. Between the two of them, they donated $280. How much money did Mo give?
Answer: $70
Let’s go through this problem one step at a time.
Step 1: Read through the problem carefully
Start by asking what question the problem is asking you to solve and identifying the information that will help you solve it. What’s the question here?
Flor and Mo both donated money to the same charity. Flor gave three times as much as Mo. Between the two of them, they donated $280. How much money did Mo give?
To solve the problem, you’ll have to find out how much money Mo gave to charity. All the important information you need is in the problem:
- The amount Flor donated is three times as much the amount Mo donated
- Flor and Mo’s donations add up to $280 total
Step 2: Represent the unknown numbers with variables
The unknown number we’re trying to identify in this problem is Mo’s donation. We’ll represent it with the variable m.
Step 3: Translate the rest of the problem
Here’s the problem again. This time, the important facts are highlighted.
Flor and Mo both donated money to the same charity. Flor gave three times as much as Mo. Between the two of them, they donated $280. How much money did Mo give?
The important facts of the problem could also be expressed this way:
Mo’s donation plus Flor’s donation equals $280
Because we know that Flor’s donation is three times as much as Mo’s donation, we could go even further and say:
Mo’s donation plus three times Mo’s donation equals $280
We can translate this into a math problem in only a few steps. Here’s how:
- Because we’ve already said we’ll represent the amount of Mo’s donation with the variable m, let’s start by replacing Mo’s donation with m.
- Next, we can put in mathematical operators in place of certain words. We’ll also take out the dollar sign.
- Finally, let’s write three times mathematically. Three times m can also be written as 3 ⋅ m, or just 3m.
m plus three times m equals $280
m + three times m = 280
m + 3m = 280
Step 4: Solve the problem
It will only take a few steps to solve this problem.
- To get the correct answer, we’ll have to get m alone on one side of the equation.
- To start, let’s add m and 3m. That’s 4m.
- We can get rid of the 4 next to the m by dividing both sides by 4. 4m / 4 is m, and 280 / 4 is 70.
m + 3m = 280
4m = 280
m = 70.
We’ve got our answer: m = 70. In other words, Mo donated $70.
Step 5: Check your work
The answer to our problem is $70, but we should check just to be sure. Let’s look at our problem again.
Flor and Mo both donated money to the same charity. Flor gave three times as much as Mo. Between the two of them, they donated $280. How much money did Mo give?
If our answer is correct, $70 and three times $70 should add up to $280.
- We can write our new equation like this:
- The order of operations calls for us to multiply first. 3 ⋅ 70 is 210.
- The last step is to add 70 and 210. 70 plus 210 equals 280.
70 + 3 ⋅ 70 = 280
70 + 210 = 280
280 = 280
280 is the combined cost of the tickets in our original problem. Our answer is correct: Mo gave $70 to charity.
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You can solve many real world problems with the help of math. In order to familiarize students with these kinds of problems, teachers include word problems in their math curriculum. However, word problems can present a real challenge if you don’t know how to break them down and find the numbers underneath the story. Solving word problems is an art of transforming the words and sentences into mathematical expressions and then applying conventional algebraic techniques to solve the problem.
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Read the problem carefully.[1]
A common setback when trying to solve algebra word problems is assuming what the question is asking before you read the entire problem. In order to be successful in solving a word problem, you need to read the whole problem in order to assess what information is provided, and what information is missing.[2]
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Determine what you are asked to find. In many problems, what you are asked to find is presented in the last sentence. This is not always true, however, so you need to read the entire problem carefully.[3]
Write down what you need to find, or else underline it in the problem, so that you do not forget what your final answer means.[4]
In an algebra word problem, you will likely be asked to find a certain value, or you may be asked to find an equation that represents a value.- For example, you might have the following problem: Jane went to a book shop and bought a book. While at the store Jane found a second interesting book and bought it for $80. The price of the second book was $10 less than three times the price of he first book. What was the price of the first book?
- In this problem, you are asked to find the price of the first book Jane purchased.
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3
Summarize what you know, and what you need to know. Likely, the information you need to know is the same as what information you are asked to find. You also need to assess what information you already know. Again, underline or write out this information, so you can keep track of all the parts of the problem. For problems involving geometry, it is often helpful to draw a sketch at this point.[5]
- For example, you know that Jane bought two books. You know that the second book was $80. You also know that the second book cost $10 less than 3 times the price of the first book. You don’t know the price of the first book.
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Assign variables to the unknown quantities. If you are being asked to find a certain value, you will likely only have one variable. If, however, you are asked to find an equation, you will likely have multiple variables. No matter how many variables you have, you should list each one, and indicate what they are equal to.[6]
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Look for keywords.[7]
Word problems are full of keywords that give you clues about what operations to use. Locating and interpreting these keywords can help you translate the words into algebra.[8]
- Multiplication keywords include times, of, and factor.[9]
- Division keywords include per, out of, and percent.[10]
- Addition keywords include some, more, and together.[11]
- Subtraction keywords include difference, fewer, and decreased.[12]
- Multiplication keywords include times, of, and factor.[9]
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Write an equation. Use the information you learn from the problem, including keywords, to write an algebraic description of the story.[13]
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Solve an equation for one variable. If you have only one unknown in your word problem, isolate the variable in your equation and find which number it is equal to. Use the normal rules of algebra to isolate the variable. Remember that you need to keep the equation balanced. This means that whatever you do to one side of the equation, you must also do to the other side.[14]
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Solve an equation with multiple variables. If you have more than one unknown in your word problem, you need to make sure you combine like terms to simplify your equation.
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Interpret your answer. Look back to your list of variables and unknown information. This will remind you what you were trying to solve. Write a statement indicating what your answer means.[15]
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Solve the following problem. This problem has more than one unknown value, so its equation will have multiple variables. This means you cannot solve for a specific numerical value of a variable. Instead, you will solve to find an equation that describes a variable.
- Robyn and Billy run a lemonade stand. They are giving all the money that they make to a cat shelter. They will combine their profits from selling lemonade with their tips. They sell cups of lemonade for 75 cents. Their mom and dad have agreed to double whatever amount they receive in tips. Write an equation that describes the amount of money Robyn and Billy will give to the shelter.
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Read the problem carefully and determine what you are asked to find.[16]
You are asked to find how much money Robyn and Billy will give to the cat shelter. -
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Summarize what you know, and what you need to know. You know that Robyn and Billy will make money from selling cups of lemonade and from getting tips. You know that they will sell each cup for 75 cents. You also know that their mom and dad will double the amount they make in tips. You don’t know how many cups of lemonade they sell, or how much tip money they get.
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Assign variables to the unknown quantities. Since you have three unknowns, you will have three variables. Let
equal the amount of money they will give to the shelter. Let equal the number of cups they sell. Let equal the number of dollars they make in tips.
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Look for keywords. Since they will “combine” their profits and tips, you know addition will be involved. Since their mom and dad will “double” their tips, you know you need to multiply their tips by a factor of 2.
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Write an equation. Since you are writing an equation that describes the amount of money they will give to the shelter, the variable
will be alone on one side of the equation.
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Interpret your answer. The variable
equals the amount of money Robyn and Billy will donate to the cat shelter. So, the amount they donate can be found by multiplying the number of cups of lemonade they sell by .75, and adding this product to the product of their tip money and 2.
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Question
How do you solve an algebra word problem?
Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary’s College.
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Expert Answer
Carefully read the problem and figure out what information you’re given and what that information should be used for. Once you know what you need to do with the values they’ve given you, the problem should be a lot easier to solve.
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Question
If Deborah and Colin have $150 between them, and Deborah has $27 more than Colin, how much money does Deborah have?
Let x = Deborah’s money. Then (x — 27) = Colin’s money. That means that (x) + (x — 27) = 150. Combining terms: 2x — 27 = 150. Adding 27 to both sides: 2x = 177. So x = 88.50, and (x — 27) = 61.50. Deborah has $88.50, and Colin has $61.50, which together add up to $150.
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Question
Karl is twice as old Bob. Nine years ago, Karl was three times as old as Bob. How old is each now?
Let x be Bob’s current age. Then Karl’s current age is 2x. Nine years ago Bob’s age was x-9, and Karl’s age was 2x-9. We’re told that nine years ago Karl’s age (2x-9) was three times Bob’s age (x-9). Therefore, 2x-9 = 3(x-9) = 3x-27. Subtract 2x from both sides, and add 27 to both sides: 18 = x. So Bob’s current age is 18, and Karl’s current age is 36, twice Bob’s current age. (Nine years ago Bob would have been 9, and Karl would have been 27, or three times Bob’s age then.)
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Word problems can have more than one unknown and more the one variable.
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The number of variables is always equal to the number of unknowns.
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While solving word problems you should always read every sentence carefully and try to extract all the numerical information.
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Article SummaryX
To solve word problems in algebra, start by reading the problem carefully and determining what you’re being asked to find. Next, summarize what information you know and what you need to know. Then, assign variables to the unknown quantities. For example, if you know that Jane bought 2 books, and the second book cost $80, which was $10 less than 3 times the price of the first book, assign x to the price of the 1st book. Use this information to write your equation, which is 80 = 3x — 10. To learn how to solve an equation with multiple variables, keep reading!
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Solving Word Problems in Mathematics
What Is a Word Problem? (And How to Solve It!)
Learn what word problems are and how to solve them in 7 easy steps.
Real life math problems don’t usually look as simple as 3 + 5 = ?. Instead, things are a bit more complex. To show this, sometimes, math curriculum creators use word problems to help students see what happens in the real world. Word problems often show math happening in a more natural way in real life circumstances.
As a teacher, you can share some tips with your students to show that in everyday life they actually solve such problems all the time, and it’s not as scary as it may seem.
As you know, word problems can involve just about any operation: from addition to subtraction and division, or even multiple operations simultaneously.
If you’re a teacher, you may sometimes wonder how to teach students to solve word problems. It may be helpful to introduce some basic steps of working through a word problem in order to guide students’ experience. So, what steps do students need for solving a word problem in math?
Steps of Solving a Word Problem
To work through any word problem, students should do the following:
1. Read the problem: first, students should read through the problem once.
2. Highlight facts: then, students should read through the problem again and highlight or underline important facts such as numbers or words that indicate an operation.
3. Visualize the problem: drawing a picture or creating a diagram can be helpful.
Students can start visualizing simple or more complex problems by creating relevant images, from concrete (like drawings of putting away cookies from a jar) to more abstract (like tape diagrams). It can also help students clarify the operations they need to carry out. (next step!)
4. Determine the operation(s): next, students should determine the operation or operations they need to perform. Is it addition, subtraction, multiplication, division? What needs to be done?
Drawing the picture can be a big help in figuring this out. However, they can also look for the clues in the words such as:
– Addition: add, more, total, altogether, and, plus, combine, in all;
– Subtraction: fewer, than, take away, subtract, left, difference;
– Multiplication: times, twice, triple, in all, total, groups;
– Division: each, equal pieces, split, share, per, out of, average.
These key words may be very helpful when learning how to determine the operation students need to perform, but we should still pay attention to the fact that in the end it all depends on the context of the wording. The same word can have different meanings in different word problems.
Another way to determine the operation is to search for certain situations, Jennifer Findley suggests. She has a great resource that lists various situations you might find in the most common word problems and the explanation of which operation applies to each situation.
5. Make a math sentence: next, students should try to translate the word problem and drawings into a math or number sentence. This means students might write a sentence such as 3 + 8 =.
Here they should learn to identify the steps they need to perform first to solve the problem, whether it’s a simple or a complex sentence.
6. Solve the problem: then, students can solve the number sentence and determine the solution. For example, 3 + 8 = 11.
7. Check the answer: finally, students should check their work to make sure that the answer is correct.
These 7 steps will help students get closer to mastering the skill of solving word problems. Of course, they still need plenty of practice. So, make sure to create enough opportunities for that!
At Happy Numbers, we gradually include word problems throughout the curriculum to ensure math flexibility and application of skills. Check out how easy it is to learn how to solve word problems with our visual exercises!
Word problems can be introduced in Kindergarten and be used through all grades as an important part of an educational process connecting mathematics to real life experience.
Happy Numbers introduces young students to the first math symbols by first building conceptual understanding of the operation through simple yet engaging visuals and key words. Once they understand the connection between these keywords and the actions they represent, they begin to substitute them with math symbols and translate word problems into number sentences. In this way, students gradually advance to the more abstract representations of these concepts.
For example, during the first steps, simple wording and animation help students realize what action the problem represents and find the connection between these actions and key words like “take away” and “left” that may signal them.
From the beginning, visualization helps the youngest students to understand the concepts of addition, subtraction, and even more complex operations. Even if they don’t draw the representations by themselves yet, students learn the connection between operations they need to perform in the problem and the real-world process this problem describes.
Next, students organize data from the word problem and pictures into a number sentence. To diversify the activity, you can ask students to match a word problem with the number sentence it represents.
Solving measurement problems is also a good way of mastering practical math skills. This is an example where students can see that math problems are closely related to real-world situations. Happy Numbers applies this by introducing more complicated forms of word problems as we help students advance to the next skill. By solving measurement word problems, students upgrade their vocabulary, learning such new terms as “difference” and “sum,” and continue mastering the connection between math operations and their word problem representations.
Later, students move to the next step, in which they learn how to create drawings and diagrams by themselves. They start by distributing light bulbs equally into boxes, which helps them to understand basic properties of division and multiplication. Eventually, with the help of Dino, they master tape diagrams!
To see the full exercise, follow this link.
The importance of working with diagrams and models becomes even more apparent when students move to more complex word problems. Pictorial representations help students master conceptual understanding by representing a challenging multi-step word problem in a visually simple and logical form. The ability to interact with a model helps students better understand logical patterns and motivates them to complete the task.
Having mentioned complex word problems, we have to show some of the examples that Happy Numbers uses in its curriculum. As the last step of mastering word problems, it is not the least important part of the journey. It’s crucial for students to learn how to solve the most challenging math problems without being intimidated by them. This only happens when their logical and algorithmic thinking skills are mastered perfectly, so they easily start talking in “math” language.
These are the common steps that may help students overcome initial feelings of anxiety and fear of difficulty of the task they are given. Together with a teacher, they can master these foundational skills and build their confidence toward solving word problems. And Happy Numbers can facilitate this growth, providing varieties of engaging exercises and challenging word problems!
My students had been struggling with how to solve addition and subtraction word problems for what seemed like forever. They could underline the question and they could find the numbers. Most of the time, my students just added the two numbers together without making sense of the problem.
Ugh.
Can you relate?
Below are five math problem-solving strategies to use when teaching word problems on addition and subtraction using any resource.
So, how do I teach word problems? It’s quite complex, but so much fun, once you get into it.
How to teach addition and subtraction word problems
The main components of teaching addition and subtraction word problems include:
- Teaching the Relationship of the Numbers – As a teacher, know the problem type and help students solve for the action in the problem
- Differentiate the Numbers – Give students just the right numbers so that they can read the problem without getting bogged down with the computation
- Use Academic Vocabulary – And be consistent in what you use.
- Stop Searching for the “Answer” – it’s not about the answer; it’s about the process
- Differentiate between the Models and the Strategies – one has to do with the relationship between the numbers and the other has to do with how students “solve” or compute the problem.
I am a big proponent of NOT teaching keyword lists. It just doesn’t work consistently across all problems. It’s a shortcut that leads to breakdowns in mathematical thinking. Nor should you just give students word problem worksheets and have them look for word problem keywords. I talk more in-depth about why it doesn’t work in The Problem with Using Keywords to Solve Word Problems.
Teach the Relationship of the Numbers in Math Word Problems
One way to help your students solve word problems is to teach them the relationship of the numbers. In other words, help them understand that the numbers in the problem are related to each other in some way.
I teach word problems by removing the numbers. Sounds strange right?
Removing the distraction of the numbers helps students focus on the situation of the problem and understand the action or relationship of the numbers. It also keeps students from solving the problem before we talk about the relationship of the numbers.
When I teach word problems, I give students problems with blank spaces and no numbers. We first talk about the action in the problem. We identify whether something is being added to or taken from something else. That becomes our equation. We identify what we have to solve and set up the equation with blank spaces and a square for the unknown number
___ + ___ = unknown
Do you want a free sample of the word problems I use in my classroom? Click the link or the image below. FREE Sample of Word Problems by Problem Type
Differentiate the numbers in the Word Problems
Only after we have discussed the problem do I give students numbers. I differentiate numbers based on student needs. At the beginning of the year, we all do the same numbers, so that I can make sure students understand the process.
After students are familiar with the process, I start to give different students different numbers, based on their level of mathematical thinking.
I also change numbers throughout the year, from one-digit to two-digit numbers. The beauty of the blank spaces is that I can put any numbers I want into the problem, to practice the strategies we have been working on in class.
At some point, we do create a list of words, but not a keyword list. We create a list of actions or verbs and determine whether those actions are joining or separating something. How many can you think of?
Here are a few ideas:
Join: put, got, picked up, bought, made
Separate: ate, lost, put down, dropped, used
Don’t be afraid to use academic vocabulary when teaching word problems
I teach my students to identify the start of the problem, the change in the problem, and the result of the problem. I teach them to look for the unknown.
These are all words we use when solving problems and we learn the structure of a word problem through the vocabulary and relationship of the numbers.
In fact, using the same vocabulary across problem types helps students see the relationship of the numbers at a deeper level.
Take these examples, can you identify the start, change and result in each problem? Hint: Look at the code used for the problem type in the lower right corner.
For compare problems, we use the terms, larger, smaller, more and less. Try out these problems and see if you can identify the components of the word problems.
Stop searching for “the answer” when solving word problems
This is the most difficult misconception to break.
Students are not solving a word problem to find “the answer”. Although the answer helps me, the teacher, understand whether or not the student understood the relationship of the numbers, I want students to be able to explain their process and understand the depth of word problems.
Okay, they’re first and second-graders. I know.
My students can still explain, after instruction, that they started with one number. The problem resulted in other another number. Students then know that they are searching for the change between those two numbers.
It’s all about the relationship.
Differentiate between the models and the strategies
A couple of years ago, I came across this article about the need to help students develop adequate models to understand the relationship of the numbers within the problem.
A light bulb went off in my head. I needed to make a distinction between the models students use to understand the relationship of the numbers in the problem and the strategies to solve the computation in the problem. Models and strategies work in tandem but are very different.
Models are the visual ways problems are represented. Strategies are the ways a student solves a problem, putting together and taking apart the numbers.
The most important thing about models is to move away from them. I know that sounds odd.
You spend so long teaching students how to use models and then you don’t want them to use a model. Well, actually, you want students to move toward efficiency.
Younger students will act out problems, draw out problems with representations, and draw out problems with circles or lines. Move students toward efficiency. As the numbers get larger, the model needs to represent the relationship of the numbers
This is a prime example of moving from an inverted-v model to a bar model.
Here is a student moving from drawing circles to using an inverted-v.
Students should be solidly using one model before transitioning to another. They may even use two at the same time while they work out the similarities between the models.
Students should also be able to create their own models. You’ll see how I sometimes gave students copies of the model that they could glue into their notebooks and sometimes students drew their own model. They need to be responsible for choosing what works best for them. Start your instruction with specific models and then allow students to choose one to use. Always move students toward more efficient models.
The same goes for strategies for computation. Teach the strategies first through the use of math fact practice, before applying it to word problems so that students understand the strategies and can quickly choose one to use. When teaching, focus on one or two strategies. Once students have some fluency in a few strategies, have them choose strategies that work for different problems.
Which numbers do you put in the blank spaces?
Be purposeful in the numbers that you choose for your word problems. Different number sets will lend themselves to different strategies and different models. Use number sets that students have already practiced computationally.
If you’ve been taught to make 10, use numbers that make 10. If you’re working on addition without regrouping, use those number sets. The more connections you can make between the computation and the problem-solving the better.
The examples above are mainly for join and separate problems. It’s no wonder our students have so much difficulty with compare problems since we don’t teach them to the same degree as join and separate problems.
Our students need even more practice with those types of problems because the relationship of the numbers is more abstract. I’m going to leave that for another blog post, though.
Do you want a FREE sample of the resource that I use to teach Addition & Subtraction Word Problems by Problem Type? Click this link or the image below.
How to Purchase the Addition & Subtraction Word Problems
The full resource is also available in my store for purchase and on Teachers Pay Teachers.
More Ideas for Teaching Word Problems
44 Responses
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This is great! I teach high school math, and always ask them to “Tell me the story” before we start looking at the numbers. If, in telling the story, they tell me a number, I stop them, and remind them that we’re just looking at what is happening, and ignore the numbers. They look at me like I am crazy, “Ignore the numbers?” Yes, I tell them. The numbers are not important until you understand the story, and even then, meh. I am thinking about giving them word problems without numbers, and use some of your suggestions. Maybe even let them put in numbers and solve their own problems. I’ve seen the word lists like you mentioned, and they’re ok, but they are not always true. Like, “how many all together?” usually means add, but in higher math, it could be addition in the form of repeated addition, aka, multiplication. Those little phrases are usually true for the early word problem problems, but as the students get older, they will need to be able to think about what the problem means, rather than just hunting for words and numbers. LOVE this approach!
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I love the perspective of a high school math teacher! This is why I want to emphasize teaching about the situation and action of a word problem. I know it can be so simple when students are young, but once they hit third grade and are doing both multiplication and addition within the same problem, boy, does it get complicated! Students really need to understand the problem. Using blank spaces has helped most of my students focus on what is happening in the problem. If you try it, I’d love to hear how it goes!
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Thank you for presenting your work in such an organized fashion. Your thought process is so clear a beginning teacher will be able to instruct children brilliantly! I appreciated the work samples you included. Hope you continue this blog, you’re very talented.
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Wow! Thank you for posting such an in-depth, organized lesson! My students, as well, struggle with the concept of word problems. This is wonderful!
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You did a great job presenting this information. I absolutely love your way of teaching students how to think about word problems. Superior work!
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Thank you so much! I have a lot of fun teaching word problems in the classroom, too.
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Hi Jessica,
I work with Deaf and Hard of Hearing students at the elementary level. The overall and profound struggle of the deaf child is that of access to language(written). For those children not born into Deaf, ASL, 1st language household, we, in many cases, consider these children to be language deprived. Math is typically the stronger subject for my students as it has been, up until recently, the most visual subject, one which requires less reading and more computation and visual or spacial awareness. When the Common Core rolled out, I looked at the Math, more specifically, the word problems with the addition of explaining ones answer, I thought…”if it isn’t already so difficult for my students to navigate the written language presented to them but to now need to explain themselves mathematically” I figured I would go on just blocking out the story and focus on numbers and key words/indicators…. After reading your blog on the topic of word problems and looking at your products I have decided to start a new!!! Knowing the story, for some of my students, might better help them visualize the WHY and the reality of the numbers and their relationships. Knowing the story will also provide context to real life scenarios, which will translate to them being able to better explain their result, outcome or answer. An ah-ha moment for me! Cheers!!!-
I am in my senior year @ UNCG for Deaf Education k-12 and we JUST discussed this today! Things like ‘CUBES’ and other key word memorization methods take away from the importance of understanding the story/situation. Being able to use these real life situations to make connections to the concept helps tremendously, even with large gaps in background knowledge/language. ASL provides the ability to SHOW the story problem, so I hope to take advantage of that when I teach math lessons. I love finding deaf educators!
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Hello Jessica,
I think your strategy is interesting.
I already subsribe, but how to get your free sample of addition word problems.
Thank you-
Hi, Kadek,
It looks like you’ve already downloaded the free sample. Let me know if you’re not able to access it.
Jessica
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Hi, Jessica,
I already got it yesterday.
Thank you so much for your free sample.
Kadek
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I purchased your word problems pack and LOVE it! My 2nd graders are forced to slow down and analyze the story. We’ve had some GREAT discussions in math lately. Another strategy I like to do in problem solving is show the word problem but leave the question out. Kids brainstorm what questions could we ask to go with the problem. Fun stuff happening in math!
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— so do you wait on teaching compare until they are a little good at joining and separating?
“The examples above are mainly for join and separate problems. It’s no wonder out students have so much difficulty with compare problems, since we don’t teaching them to the same degree as join and separate problems. Our students need even more practice with those types of problems because the relationship of the numbers is more abstract. I’m going to leave that for another blog post, though”-
I totally forgot that I was going to do a follow-up post on compare problems! Thank you for reminding me!
I do introduce join and separate problems first, but I don’t necessarily wait on teaching compare problems until students are proficient solving join/separate problems. Students will progress at different rates and I don’t want to wait to teach something that others’ might be ready to learn. I teach compare problems with a lot of physical modeling first and then we move into using a bar-model as the written model. The other thing I do with these types of problems is use concrete sentence frames. Sometimes, especially my English learners, need some of the vocabulary and sentence structure to better understand the relationship of the numbers.
I vary when I teach them every year. I often do it around Halloween, when we talk about pumpkins and who had a larger pumpkin or more seeds. I also do it when we measure our feet and we discuss the size of feet. It’s a great problem type for measurement, although you can compare any two quantities. Although I have taught a problem type, we continue to use it all year long as we relate to the math around us.
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Hi. Just wondering if you did have a follow up post on compare problems. Thank you!
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Not yet, but it’s on the plan for this month. I took a (long) break from doing FB lives and am starting back up again. That is one that I’ll do this month. I don’t have an exact date yet – kinda depends on when I can get my kids out of the house! 🙂
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I cannot wait to try this with my students! We are getting their baseline today and then we are going to start on Thursday. I wish I could pick your brain about this and how you teach this beginning to end. Do you start by teaching them the vocabulary and just labeling the parts (start, change, result)?
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I love how you teach student to label parts of the word problem while trying to solve it (S for Start, C for Change, etc.) You seem to have easily clarified the steps of solving problems in very clear (and cute) kid friendly language. Nice job. Thank you for sharing.
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I love this idea of having the students organize the information. My question is how do you teach them when to add or multiply or subtract/divide? At that point do they look for works like equal groups?
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I’m sorry…I have one more question. Can you apply this method to multi-step word problems?
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Yes! Each “step” in the word problem would have its own equation, which may be dependent on the first equation. You’re using the same process, reading the problem for a context, setting up an equation, then giving students the numbers. With second graders, I do a lot of acting out for multistep problems, as it’s generally a new concept for them.
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We don’t look for keywords but set up an equation based on the situation or context of the word problem. The situation in the word problem will illustrate the operation, like someone dropping papers, adding items to their cart, sharing something with friends, etc. The situation will tell the operation.
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I love , love , love this concept my year 1 pupils easily grasps the lesson. Thanks a bunch! Do you have strategies like this for multiplication and division?
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Hi Jessica, this is simply great. My 7 year old struggles with worded problems and I’ll try to method with him and hopefully it’ll help him grasp the methodology better. On w different note, I’ve been trying teach him how to solve simple addition and subtraction in the form of an equation. For example 15+—= 43 or 113- = 34. But despite multiple attempts of explaining the logic using beans and smaller numbers, he is struggling to understand. Would you have any tips on those.
Many thanks,
Varsha
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Thank you so much! This is super helpful for me. I’m currently student teaching in a 2nd grade class. My cooperating teach is EXTREMELY uncooperative and hasn’t/won’t help me in planning lessons. She told me to teach word problems and despite my follow up questions I don’t know what exactly they’ve done already this year or where to start. This post gave me lots of ideas and helped me prepare for last minute shifts as I teach without a plan (unfortunately). If I wasn’t a poor college student I would definitely buy the pack, especially after getting the free samples! These samples are so helpful!
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Thanks! It is very interesting! Good!
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Hey! You’re amazing! I’ve heard that this really help kids comprehend better & I want to try it! I sent my info but haven’t received the freebie.
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Hi, Gisel,
You need to confirm your email address before I can send you any emails. The confirmation may have gone to your spam folder. I also have a different email address than the one for this comment. Feel free to fill out the contact form if you need me to switch the email address. For now, I’ll assume that this comment is a confirmation and manually approve it.
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This worked amazingly well! My second graders were having such a tough time understanding how to do word problems. This strategy helped most of them with the ability to understand how to do word problems and demonstrate their knowledge on testing. Most importantly, after learning this strategy, the students kept asking for more problems to solve.
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What pacing do you suggest for introducing the different types of problems? Should students master one type before moving on to another?
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Great question! I would consider your students, grade level, and curriculum. I generally spend more time at the beginning of the year, with easier problem types to establish routines. Some problem types are complementary and easier to teach and practice after students learn one. I also cycle back through problem types as we learn new computation strategies. For instance, in second grade, we do single-digit addition at the beginning of the year, mid-year we move onto two-digit addition and mid- to end-of-the-year we do three-digit addition. We will cycle through problem types we’re already learned but increase the complexity of the numbers.
I would make sure a majority of your class has mastered the process of reading a word problem and identifying the parts. Also, be sure you’re separating student mistakes between computational or mathematical errors and problem-solving errors. As I said in the beginning, I’d take the cues based on your students, grade level, and curriculum. Some years I have spent more time because my students needed more time. Other years I was able to move quicker.
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Makes sense! Thanks so much!
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I absolutely LOVE this post. Thank you for sharing it! I teach third grade and my babies are struggling with what to actually DO in a word problem. I’m going to be trying this with them immediately. Do you have any suggestions for how to incorporate it with multiplication and division problems?
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Thank you for sharing this wonderful resource! Could you explain how you teach your students to use the inverted V model? I noticed the 3 points are labelled as start, change, and result differently for each problem. I am very interested in teaching my students this model!
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Can you go over for me about “start, change and results”? Thanks.
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My 9yr old struggles with word problems to. He’s good in performing the calculations but struggles with tracking and comprehension of word problems. I look forward to giving your tips a try.
Thank you for sharing!!
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Really its fantastic strategy. Great ideas!
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Thank you for sharing this great resource. Teaching math word problems to students with disabilities is never easy. I have to come up with a variety of different ways to teach my students on how to make word problem with connections to the real world.
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First of all ,thanks for sharing this article. you explained it very well and my children learn so many things from this article. i wish you will post more article just like this one
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We recommend reading any word problem at least three times. By the conclusion of the third reading, you should be magically whisked away to your home in Kansas.
Read it once…
…to get a general idea of what’s going on. Is the problem about money? Height and width? Distances? Money? Yeah, we already said money, but it’s important. Can you draw a picture in the margin that helps you visualize the problem?
Twice…
…to translate from English into math. Read the problem carefully this time, figuring out which pieces of information are important and which ones aren’t. Then ignore the unnecessary bits. This may sound a bit harsh, but they’re big boys. They’ll get over it.
Three times…
…a lady. Wait, that’s not it.
…to make sure you answered the right question. To be sure, read the question one last time before drawing a box around your final answer. Even if this method of double-checking saves you only once out of every 100 times, it’ll help you in the long run. One out of 100 is better than zero out of 100, but now we’re moving into some advanced mathematics.
Sample Problem
The local department store was having a sale. Holla! Gabe bought a pair of shoes for $21, although they would’ve been cheaper if he’d bought penny loafers. He also bought some shirts that were on sale for 25% off their normal retail price of $18 each. Gabe, always the bargain hunter, spent $75 total. How many shirts did he buy? Also, does he really think any of them will go with those shoes?
Read the problem:
Once…
…for a general idea of what’s going on: Gabe went shopping and spent money. Sounds like an old familiar story. Time for an intervention, friends of Gabe. Preferably before he maxes out his Diner’s Club card.
Now we need to figure out how many shirts he bought, so we read the problem:
Twice…
…to translate from English into math. Remember, we can translate a bit at a time, sort of like how Gabe pays for some of his major purchases when he has them on layaway. Sheesh, Gabe. Get a hold of yourself.
(total amount Gabe spent) = (amount he spent on shoes) + (amount he spent on shirts)
We know he spent $75 total, of which $21 was spent on shoes. This gives us the equation:
$75 = 21 + (amount he spent on shirts)
The problem has gotten smaller. It’s depressing when that happens with cake, but great when it happens to a word problem. Now all we need to do is come up with a symbolic expression for how much Gabe spent on shirts, or the cost per shirt times the number of shirts:
(amount he spent on shirts) = (cost per shirt)(number of shirts)
Since the shirts are 25% off their normal price of $18, they cost $18 – 0.25(18) = $13.50 each. What a deal, and real polyester, too!
We need to introduce a variable for the number of shirts; s will do the trick nicely.
(amount he spent on shirts) = 13.5s
When we plug this into the earlier equation, we find that:
75 = 21 + 13.5s
Finally, we’ve reduced this thing to a super-simple-looking equation! Good riddance, vestiges of language! Begone, nouns and verbs!
Things look much nicer now, right? No worrying about shirts or shoes or prices or Gabe’s uncontrollable shopping addiction. For the moment, we can forget about the word problem and solve the equation. The answer is s = 4, by the way. In case you were interested.
Three times…
…to make sure we’re answering the right question. We want to know how many shirts Gabe bought. Is that the answer we arrived at? We found that s = 4, and s was the number of shirts Gabe bought, so we’re all done. Four new shirts for Gabe, and three of them feature a Hawaiian pattern. Gabe, if you insist on buying far more clothes than you need, can’t you at least have a decent fashion sense?
When translating from English into math, some information can be ignored. We don’t care that «the local department store was having a sale.» Gabe might, but we certainly don’t. We care about statements that tell us numbers, and statements that tell us what the question is. Any extraneous information has been placed there simply as a decoy. We’re not going to fall for that. Wait a second…two for one? We’ll grab our jacket and meet you there.
Some people find it helpful to underline the important pieces of information in a word problem. You’re like an actor highlighting in a script the lines that are important for him to remember. Unlike an actor, however, you can always look back at the original problem if you draw a blank. Also, you don’t need to wear any stage makeup.
In the problem we just did, the important bits might look something like this:
The local department store was having a sale. Gabe bought a pair of shoes for $21 and some shirts that were on sale for 25% off their normal retail price of $18 each. If Gabe spent $75 total, how many shirts did he buy?
As you practice, you’ll become better at figuring out which parts of the word problem you can ignore and which parts are important.