Word problem solving process

Word problems often confuse students simply because the question does not present itself in a ready-to-solve mathematical equation. You can answer even the most complex word problems, provided you understand the mathematical concepts addressed. While the degree of difficulty may change, the way to solve word problems involves a planned approach that requires identifying the problem, gathering the relevant information, creating the equation, solving and checking your work.

Identify the Problem

Begin by determining the scenario the problem wants you to solve. This might come as a question or a statement. Either way, the word problem provides you with all the information you need to solve it. Once you identify the problem, you can determine the unit of measurement for the final answer. In the following example, the question asks you to determine the total number of socks between the two sisters. The unit of measurement for this problem is pairs of socks.

«Suzy has eight pairs of red socks and six pairs of blue socks. Suzy’s brother Mark owns eight socks. If her little sister owns nine pairs of purple socks and loses two of Suzy’s pairs, how many pairs of socks do the sisters have left?»

Gather Information

Create a table, list, graph or chart that outlines the information you know, and leave blanks for any information you don’t yet know. Each word problem may require a different format, but a visual representation of the necessary information makes it easier to work with.

In the example, the question asks how many socks the sisters own together, so you can disregard the information about Mark. Also, the color of the socks doesn’t matter. This eliminates much of the information and leaves you with only the total number of socks that the sisters started with and how many the little sister lost.

Create an Equation

Translate any of the math terms into math symbols. For example, the words and phrases «sum,» «more than,» «increased» and «in addition to» all mean to add, so write in the «+» symbol over these words. Use a letter for the unknown variable, and create an algebraic equation that represents the problem.

In the example, take the total number of pairs of socks Suzy owns — eight plus six. Take the total number of pairs that her sister owns — nine. The total pairs of socks owned by both sisters is 8 + 6 + 9. Subtract the two missing pairs for a final equation of (8 + 6 + 9) — 2 = n, where n is the number of pairs of socks the sisters have left.

Solve the Problem

Using the equation, solve the problem by plugging in the values and solving for the unknown variable. Double-check your calculations along the way to prevent any mistakes. Multiply, divide and subtract in the correct order using the order of operations. Exponents and roots come first, then multiplication and division, and finally addition and subtraction.

In the example, after adding the numbers together and subtracting, you get an answer of n = 21 pairs of socks.

Verify the Answer

Check if your answer makes sense with what you know. Using common sense, estimate an answer and see if you come close to what you expected. If the answer seems absurdly large or too small, search through the problem to find where you went wrong.

In the example, you know by adding up all the numbers for the sisters that you have a maximum of 23 socks. Since the problem mentions that the little sister lost two pairs, the final answer must be less than 23. If you get a higher number, you did something wrong. Apply this logic to any word problem, regardless of the difficulty.

After you finish this lesson, view all of our Pre-Algebra lessons and practice problems. After you finish this lesson, view all of our Algebra 1 lessons and practice problems.

Solving Word Problems

To solve a word problem using a system of equations, it is important to;
– Identify what we don’t know
– Declare variables
– Use sentences to create equations

An example on how to do this:

Mary and Jose each bought plants from the same store. Mary spent $188 on 7 cherry trees and 11 rose bushes. Jose spent $236 on 13 cherry trees and 11 rose bushes. Find the cost of one cherry tree and the cost of one rose bush.

Cost of a cherry tree:
Cost of a rose bush:
7 cherry trees and 11 rose bushes = $188

The y-values cancel each other out, so now you are left with only x-values and real numbers.

Then, you plug in your x-value into an original equation in order to find the y-value.

Cost of a cherry tree: $8
Cost of a rose bush: $12

Example 1

Three coffees and a cupcake cost a total of dollars. Two coffees and four cupcake cost a total of dollars. What is the individual price for a single coffee and a single cupcake?

Let’s solve this by following steps.

1. What we don’t know:

Cost of a single coffee

Cost of single cupcake

2. Declare variables:

Cost of a single coffee=

Cost of single cupcake=

3. Use sentences to create equations.

Three coffees and a cupcake cost a total of dollars.

Two coffees and four cupcake cost a total of dollars.

Now, we have a system of equations:

Let’s solve for one of the variables in one of the equations and then use that to substitute into the other.

Now, solve for the value of using the first equation.

Let’s solve the value of by substituting the value of to the bottom equation.

Distribute to each terms inside the parenthesis

Combine like terms

Now, let’s isolate the by subtracting on both sides.

Then divide both sides by ,

And we’ll have

Then, let’s plug the value of into one equation to get the value of .

Cost of a single coffee=

Cost of single cupcake=

Example 2

The senior class at High School A rented and filled vans and buses with students. High School B rented and filled vans and bus with students. Every van had the same number of students in it as did the buses. Find the number of students in each van and in each bus.

Let’s solve this by following steps.

1. What we don’t know:

Students in each van

Students in each bus

2. Declare variables:

Students in each van=

Students in each bus=

3. Use sentences to create equations.

High School A rented and filled 8 vans and 8 buses with 240 students.

High School B rented and filled 4 vans and 1 bus with 54 students.

Now, we have a system of equations:

Let’s solve for one of the variables in one of the equations and then use that to substitute into the other.

Now, solve for the value of using the second equation.

Let’s solve the value of by substituting the value of to the bottom equation.

Distribute to each terms inside the parenthesis

Combine like terms

Now, let’s isolate the by subtracting on both sides.

Then divide both sides by ,

And we’ll have

Then, let’s plug the value of into one equation to get the value of .

Students in each van=

Students in each bus=

Video-Lesson Transcript

To solve a word problem using system of equations, it is important to:
1. Identify what we don’t know
2. Declare variables.
3. Use sentences to create equations.

Let’s have an example:

Mary and Jose each bought plants from the same store. Mary spent on cherry trees and rose bushes. Jose spent on cherry trees and rose bushes. Find the cost of one cherry tree and the cost of one rose bush.

Let’s solve this by following steps above.

1. What we don’t know:
cost of a cherry tree
cost of a rose bush

2. Declare variables:
cost of a cherry tree :
cost of a rose bush :

3. Use sentences to create equations.

For Mary:
cherry trees and rose bushes

For Jose:
cherry trees and rose bushes

Now, we have a system of equations

We can solve this by process of substitution, elimination or fraction.

Since the value of is the same for both equations, let’s do the process of elimination.

First, let’s multiply the first equation by

Here we’ll have a negated equation

Let’s do the process of elimination now

We’ll have

Then, let’s isolate by dividing both sides by

Now, we have

Remember, our declared variable?

cost of a cherry tree :

Since

Now we can say that

cost of a cherry tree :

Now, let’s solve for the value of by getting one equation and plugging the value of .

Let’s use the first equation to plug in

Let’s isolate by subtracting on both sides of the equation

Then divide by

And we get

Now, we know that cost of a rose bush is .

Ah, word problems. They are SO important for our students to understand, yet one of the most challenging parts of math instruction. Word problems are unique in that they are not JUST about math – they require reading comprehension, as well. Therefore, even students who excel at math can often struggle with word problems. As teachers, it can be difficult to know which word problem strategies to teach.

In this post, I am going to share some of the common strategies for teaching word problems and why they are or aren’t effective. I would also highly encourage you to check out this post, where I share ideas for reaching all your students in your word problem instruction.

Math Word Problem Solving Strategies

There are many word problem strategies out there, but some are more effective than others. Let’s take a look at some of the different strategies and whether or not they are beneficial.

Teaching Keywords in Word Problem Instruction

Growing up, many of us were probably told that certain words in story problems indicated a specific operation. For example, “more” means addition, “fewer” means subtraction, “each” means multiplication. And while often this is the case, sometimes it is not. For example, take the following problem:

Katie picked 4 apples fewer than Marvin. If Katie picked 12 apples, how many did Marvin pick?

In this problem, the word “fewer” is used. Normally this would indicate subtraction, but if you really look at what the question is asking, you will find that we actually need to add to solve. Because of this, the keyword strategy is not effective.

That being said, I personally believe the words and language used in a problem are still worth noting. The word “fewer” is still significant; the problem lies in assuming it automatically means subtraction. As a teacher, I always taught my students to pay attention to important words, but never taught them that certain words always mean a certain operation.

Instead of using the term “keywords,” I refer to them as “important words” or “context clues”. Because reading comprehension is a major part of solving word problems, we cannot entirely ignore the language used. I do think it’s worth taking the time to encourage students to look for certain words in problems, such as the words listed above. The important thing is that we do not tell them what to do with those words – only that they are important. More on this later, but I thought it was worth noting here.

Using Attack Strategies

Another common strategy for teaching word problems is what’s known as an attack strategy. Attack strategies involve a series of steps (or “plan of attack”) to follow when solving word problems. Common attack strategies include:

• RDW (Read the problem, Draw a model, Write the equation)
• CUBES (Circle the numbers, Underline the question, Box the keywords/context clues, Evaluate the problem, Solve & check)
• RUN (Read the problem, Underline the question, Name the problem type)
• FOPS (Find the problem, Organize information with a diagram, Plan to solve the problem, Solve the problem)

There are many more – these are just a few examples. Attack strategies can certainly be helpful when used correctly. They are easy to remember and give students a clear plan for solving. Many attack strategies use fun acronyms like the ones listed above; however even strategies that do not spell out words can still be effective.

These strategies are effective largely because they focus on reading and understanding the problem first, and then solving. As we all know, many students like to simply pull out the numbers and start doing math instead of actually taking the time to read the problem. Attack strategies help solve that issue.

Numberless Word Problems

In 2014, Brian Bushart popularized the idea of removing the numbers from word problems. This is to help students understand what is actually happening in the problem. He details the process in this blog post, which is a GREAT read and I highly recommend checking it out (once you’re done reading this one!).

In short, numberless word problems are effective because they cause students to take a step back and really look at what the problem is asking. Eventually, of course, you’ll want to add the numbers back in. But starting out with the numbers removed and engaging in a discussion of what is actually happening in the problem is an effective first step in gaining comprehension.

Schema-Based Instruction

Of all the word problem strategies out there, schema-based instruction is the one with the most research backing it. Schema-based instruction (or SBI) involves categorizing word problems into particular types, or schemas, which will help you determine how to solve the problem.

Schemas can be additive or multiplicative. There are 3 main additive schemas: combine, compare, and change. Combine problems involve putting together two or more numbers to find a total. Compare involves finding the greater set, lesser set, or the difference. In change problems, an amount either increases or decreases over time.

Likewise, there are also 3 main multiplicative schemas: equal groups, comparison, and proportions. Equal groups involves a unit or group being multiplied by a specific number to find a product. In comparison problems, a set is multiplied a number of times for a product. The proportions schema focuses on the relationships among quantities.

The idea behind schema-based instruction is that all word problems fit into one of these schemas. Each schema has a unique graphic organizer and process for solving a problem that fits that schema. SBI is often accompanied with an attack strategy – such as RUN, for example. Students will first Read the problem, then Underline the questions, and finally Name the problem type (schema) before solving. This last step of identifying the schema will help students understand how to solve the problem.

Part-Part-Whole Instruction

This isn’t really a strategy, per se. But, it is an important concept for students to understand in order to be successful with word problems. Understanding part-part-whole relationships is a critical aspect of number sense. My approach to teaching word problems involves a major emphasis on part-part-whole. Granted – I taught second grade, where word problems were mostly solved by addition and subtraction. Part-part-whole works with multiplication and division, too, but looks a little bit different.

My process for walking students through word problems always included having students identify whether each number represented a part or the whole, and which (a part or the whole) was missing. This was helpful because my students knew that a missing part means subtraction, and a missing whole means addition.

Identifying what each number represented is where the “keywords” I mentioned above come in. Words like “more,” “fewer,” and “each” – while they do NOT tell us how to solve – are important context clues to help us decide whether each number given is a part or the whole.

I used the CUBES attack strategy to go with this. However, instead of boxing “keywords,” we boxed “context clues”. Again, those words are still important – they just don’t tell us how to solve. The words are important because they tell us which numbers are greater or less than other numbers, or if a certain number represents a group. All are important things to know in determining the parts and the whole.

Once students identify what each number represents, they are able to solve. I’ve found that students who were correctly able to determine the parts and the whole in a word problem were VERY successful in finding the answer.

Free Training for Word Problem Strategies

Want to learn more about how to effectively teach word problems? I’m hosting a FREE video training series that you won’t want to miss! It’s launching THIS MONDAY, 9/21, so make sure you join the waitlist so you don’t miss out. Upon signing up, you will receive my free 30-page e-book for how to differentiate word problem instruction in your classroom. CLICK HERE TO JOIN THE WAITLIST, or sign up using the form below.

How do you teach word problems?

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Solving Word Problems in Mathematics

     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. 

     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;

     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. 

     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.

     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. 

Solving Word Problems Reminder

First, I reviewed with my class how to solve regular word problems. I’ve taught them the strategy, “Circle, Circle, Underline” (circle information and underline what they want them to do figure out)  to help zero in on the important stuff. We solved the problem and we were ready to move on. Then, I added a second question to our previous. I showed the students how to use information gathered from our first step to solve the second question. Once we solved, we headed back to our seats to take notes on the three-step method.

You can read more about our simple chant to help with 1 step word problems here! 

I used the gradual release process as teaching structure for this method and it worked really well.

Modeling Two-Step Word Problems

First, the kiddos wrote out the 3 steps to take to solve a multi-step word problem.

Practice With a Buddy

Then, it was their turn to practice with a buddy. I have my class in Kagan style groupings, so I had them turn to their shoulder partner for this part.

They talked it through with their buddy and then glued it in.

Independent Practice

Finally, they were ready to try the process on their own. They worked through the word problems just like they did with their buddy. And because their buddy was still sitting right by them, they could also double check if they had an issue.

I spot-checked each student’s notebook as they finished. This was a perfect chance for me to see who still needed help with multi-step problems.

My orange soda smelly marker check was their ticket to move on to math centers. If students were still struggling, I pulled them to my small group while the other students worked through their tubs for the week.  The following day, we continued to use the 3 step process to solve multi-step problems. I wanted students to try the process more on their own again. So I passed out another half sheet and glue it into their notebooks.

I gave them 10 minutes to work through the problem. This was WAAYY more time then they would really actually need, but I wanted them to really work through the problem and it gave me a chance to spot check with the students who struggled from the previous day.

These differentiated cards are PERFECT for all learners in your 2nd or 3rd-grade classroom as they have word problems for 2-3 digit addition, subtraction, simple multiplication, and division. They also include multi-step problems for every section!

So do you have a little trick for solving multi-step problems? This way worked great with my students! You can grab the sheets I used here for FREE! Just confirm your email so I know you’re not a robot, and then you can download the freebie!

A Picture is Worth a Thousand Words

“Today, we are going to solve math word problems.” When students hear this from their math teacher, their faces drop, sweat starts to form on their foreheads and they refuse to make eye contact. As a math teacher, I understand. I understand the anxiety and want to make the learning process with word problems more enjoyable. This is where reading, writing and, math collide and the real world of math begins. Let’s talk about how to solve word problems with pictures.

When students struggle with word problems in school, they also have difficulty tackling them for homework. Most students want the “one-size-fits-all” formula for word problems but unfortunately, that does not exist. However, drawing a picture will help students visualize the problem and will start advancing their learning stage from the concrete to the abstract. Let’s take a look at how to solve a word problem using pictures:

Steps for Solving Word Problems using Pictures

1. Read the entire problem: Get all the facts – Underline key word
2. Answer the question: What am I looking for?
3. Draw a picture or diagram: Visualize as a real world situation
4. Solve the problem: Set up the equation and solve
5. Check your solution: Is this answer reasonable?

Word Problem Examples – Example A

Cody has 6 pencils on his desk, Jonah has 4 more than Cody and Vinny has three less pencils than Jonah. How many pencils are there in all?

• Read the entire problem  √
• What am I looking for? How many pencils do Cody, Jonah and Vinny have altogether?  √
• Draw a picture or diagram √

• Solve the problem    √

      6 + 10 + 7 = 23 pencils

• Check your solution  √

      This answer is reasonable

Example B

There are 3 fish tanks labeled   X, Y, and Z. Y weighs 6 times as much as X and twice as much as Z. If Z is 36 lbs. heavier than X, find the total weight of X, Y and Z.

• Read the entire problem  √
• What am I looking for?  What is thetotal weight of fish tanks X, Y and Z?  √
• Draw a picture or diagram √

• Solve the problem    √

18 + 108 + 54 = 180 pounds

• Check your solution  √

      This answer is reasonable

Visualizing a word problem with pictures is a strategy that will help motivate many students to begin the process of solving them.  This works especially for students that may become bored by the excess amount of words instead of numbers or for those students who become overwhelmed by the information and want to break it down into a simpler form.   Another benefit of using pictures when solving word problems involves communicating the results.  The pictures act as justification for answers, make the problems easier to understand and therefore, secure the learning process.

Meet our Guest Blogger: Jan Rowe

Jan is one of Educational Connections’ top tutors. She has twelve years of classroom teaching experience and holds a Virginia and Florida teaching license in middle school mathematics and elementary education. Jan has been with Educational Connections for over a year, working with over twenty students. Her tutoring goal is to help each student understand their learning style so they can improve the speed and quality of that learning. When she is not tutoring or teaching, Jan loves to play scrabble, go hiking and play Frisbee with her new puppy.