what is the difference between word problems and problem solving

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Problem-Solving vs Word Problems

I remember preparing for an interview for my first teaching position in the 90’s. I was told that I would likely be asked to explain my approach to teaching problem-solving. I jumped on the Internet to research problem-solving and craft my response. What I found was that problem-solving in math basically meant teaching students to solve word problems. I ended up getting the job and, for a number of years, taught what I  thought was problem-solving. What I’ve come to find out, however, is that while we certainly need to teach students strategies for solving word problems, problem-solving is so much more than solving word problems.

Problem-Solving > Word Problems

Think for a minute about a problem you’ve solved recently. I’ll give you a personal example. My current car lease ends next month, and I have to decide what to do. Usually, I just turn in my old car and lease another one. This year, however, is different. We are in the midst of an unprecedented shortage of new cars, driving new car prices way up. Not the best time to buy or lease a new car. At the same time, used car prices are surging and many used cars are selling at close to their original MSRP. Once again I jumped on the Internet to research the situation. I found out that I might be able to purchase my car at lease-end and turn around and sell it at a higher price! But that would leave me without a car. So, I have decided to purchase my car at lease-end and hold onto it until new car prices start to come back down. I should still get a trade-in value on my current car higher than what it will cost me to purchase it at lease-end. Of course, all this sounds great in theory and seems to be the right decision based on the data, but I won’t really know if I made the best decision until sometime next year.

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I think you’d agree that what I described was some heavy-duty problem-solving with pretty significant consequences. Yet not a word problem in sight. You see, true problem-solving is messy and goes way beyond solving word problems.

George Polya is often called the Father of Problem-Solving. In 1945, he outlined a 4-step process for solving problems in his ground-breaking book  How to Solve It .  You can see the four steps pictured below.

Now think about the process I went through while solving my car problem. Don’t you see the four steps in what I did?

The problem is that well-intentioned teachers have tried to turn the problem-solving process, which is inherently messy, into an algorithm—if you do these steps, then you can easily solve problems. This is why we see students boxing numbers, underlining questions, and looking for “key” words, all shortcuts that basically give students permission to not read and understand word problems.

So how do we teach students to become problem solvers? Well, it might sound simplistic, but we give them rich problems to solve and get out of the way. Again, with the best of intentions, teachers often provide too much support and students come to depend on it.

I recently facilitated a book study on the book Productive Math Struggle: A 6-Point Action Plan for Fostering Perseverance , by John J. SanGiovanni, Susie Katt, and Kevin J. Dykema. It is a fabulously useful and easy-to-read book, chock full of implementable ideas. In other words, it’s a book you will  use and not just read. It’s no coincidence that the first three chapters all deal with creating a climate where productive struggle can thrive. Let’s face it, many math classrooms still run on the premise that math is about regurgitating a memorized procedure. Not much thinking involved. First, we as teachers need to embrace the idea of teaching through productive struggle. Then, we need to set students up for success as we introduce struggle into our lessons. 

One way to increase productive struggle and thinking in our classroom is to flip the sequence of our instruction. Rather than the traditional direct teaching approach of I do, We do, and You do , we flip the process so students are given a problem to solve before direct instruction.

Here’s an example. Say students have been using a part/whole diagram to represent join/result unknown word problems . So they have been practicing identifying if each number in a word problem represents one of the parts or the whole and creating part/whole diagrams, such as this one. By looking at the diagram, you can probably construct the word problem they were solving, right?

Now you would like to introduce a new structure—join/change unknown. It’s a more complicated type of problem. Here’s an example of this type of problem.

Mariana had $20. Her grandmother gave her some money for her birthday. Now Mariana has $28. How much money did Mariana’s grandmother give her for her birthday?

I could proceed to teach this new structure with a scripted lesson:  Boys and girls, you have been using a diagram to solve join/result unknown problems. Today, I’m going to show you how to use the diagram to solve a new structure—Join/change unknown.  I would model a problem or two, we would work a couple together, and then they could practice more on their own. A typical I do, We do, You do lesson.

But instead, what if we read the new problem out loud together, and then I commented, Huh. This problem sounds a little different. Work with a partner to see if you can solve it using your part/whole diagram.  In other words, the You do comes first! Would every pair of students be successful in solving the problem? Probably not. But after all students have the opportunity to struggle with it, think of the rich discussions we can have. It’s likely that at least some students will determine that it’s a new structure and then I can come along behind and put a name to it.

So I hope you will commit to thinking of problem-solving as something beyond just solving word problems and give students the opportunity to productively struggle in your classroom. I think you’ll see engagement soar!

If you want more information on addition/subtraction structures check out this post .

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Cy Fair does model drawing!! It works

Howdy, neighbor! Yes, model drawing is so powerful. I’m looking forward to next year when this whole group comes up knowing it!

I’ve been using model drawing with my first graders and it’s been powerful to see how they understand what’s happening in the problem, especially those tricky missing addend problems. This is the most confident I’ve felt with first graders and problems in a very long time!

I totally agree, Nilda! It really helps them to visualize the math that is happening in the story!

Looks like a great lesson!

Tara The Math Maniac

Thanks, Tara!

Hi Donna I am a math coach in Massachusetts and want institute bar models in all grade levels, K-5, for the 2017-2018 school year. Can you recommend a resource for me to use for research prior to professional development? Thanks Meg

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Word Problems and Problem-Solving

Today’s challenge is all about reflecting on problem-solving–how we teach it AND how we make it accessible to students.  I am hoping something in this post resonates with you–and if it does, please consider dropping into the Facebook group and sharing your thoughts!

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Problem-Solving or Solving Problems?

By Carolyn Marchetti,

In both math and science, problem-solving is a critical skill.  It is a process that students will use throughout their schooling, work life, and beyond.  By developing problem-solving skills, our students not only learn how to tackle a math or science problem but also how to logically work their way through any types of problems that they face.  Our textbooks include word problems after every lesson – so this incorporates problem-solving skills, right?  Not necessarily.

I was at a conference over 10 years ago and heard a presenter say, “Problem-solving is what you do when you don’t know how to solve a problem”.  Solving problems, like the typical word problems found in our texts, on the other hand, is applying a known method to a problem that has already been solved before.

Here’s an example of how the majority of textbooks phrase a lesson — “Today we are going to learn how to multiply fractions.  Here are the steps to multiplying 2 fractions.   Here are some non-contextual examples to hone your skill”.  Then most follow with ‘real-life’ word problems which, more times than not, involve fractions that require multiplication. This is a routine of practicing skills.  I’m not saying that this isn’t important, just that problem solving is much more than this.

As teachers, we need to know the differences between the 21st-century skill of problem-solving and the traditional way of solving problems, and we especially need to learn how to recognize and even create true problem-solving experiences for our students.

I would like to give some tips on creating a problem-solving classroom by using an example of a task that I used when doing professional development with math teachers. The task is called The McDonald’s Claim Problem.  There are several versions of this task on the internet, but basically, it goes like this:

• Wikipedia reports that 8% of all Americans eat at McDonald’s every day.
• There are 310 million Americans and 12,800 McDonald’s in the United States.
• Do you believe the Wikipedia report to be true? Create a mathematical argument to justify your position.

Tips on creating a problem-solving classroom:

• Engage students in real-world problems that students can relate to and have a prior understanding of.  For McDonald’s, it was an interesting problem because it engaged students in prior learning – they’ve all been to McDonald’s and have all used Wikipedia.  For other tasks, videos can be used to spark interest.  For example, Dan Meyer’s 3 Act Tasks are one way to spark interest.  Another suggestion is to use a career video like the Defined STEM videos that are included with each performance task.  These videos grab students interest by answering the question of “When will I ever use this?”.
• Use group work for problem-solving. Students can frequently help each other, and talking about a problem helps them think more critically about the steps needed to solve the problem.  By discussing the problem, students will start to realize that problems have multiple solution strategies, and some may be more effective than others.  For the McDonald’s problem, I would have teachers work in groups of 4-5. There would be many discussions among the members before even starting the task. Discussions around what does “eat at” mean?  Does the drive-through count?  Does the question mean the same 8% eat there every day?  These questions and discussions had teachers brainstorming ideas before deciding on a course of action to solve the problem.
• There should not be a direct path to the solution.  Even better if there is not one right answer, like the McDonald’s problem, but these are harder to find.  Monitor student progress and solutions.  When they get stuck, answer their questions with other probing questions.  When the math teachers would ask me questions regarding the McDonald’s problem, I would always come back with “What does your group think?”, to encourage them to collaborate and come to a consensus.
• Have students share their solutions. When everyone is finished, have groups present their solution to the others, especially the ones that went about the problem in different or unique ways. Having the groups share their solutions and justifications is very important for others to see various ways of problem-solving. For the McDonald’s problem, even though groups often used calculations to solve the problem and would get the same numbers, many had a different answer of “yes or no” depending on their reasoning. Hearing the different reasons from other groups can be very enlightening.  I heard a lot of “I never thought of it that way”, which is a powerful aspect of problem-solving.

There are many other tips I can give, which I will continue in a later post.  For now, I would like to leave you with a quote from a colleague: “It is better to answer 1 problem 5 different ways than to answer 5 different problems”.  In one short sentence, that is the difference between problem-solving and solving problems.

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Why Word Problems Are Such a Struggle for Students—And What Teachers Can Do

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Give Cindy Cliche a math word problem, and she can tell you exactly where most students are going to trip up.

Cliche, the district math coordinator in the Murfreesboro City school district in Tennessee, has spent decades teaching elementary schoolers how to tackle their first word problems and now coaches teachers in how to do the same. Kids’ struggles, for the most part, haven’t changed, she said.

Take this problem, which students might work on in 1st grade: There are some bunnies on the grass. Three bunnies hop over, and then there are five total. How many bunnies were there to begin with?

The problem is asking about a change: What’s the starting, unknown quantity of bunnies, if adding 3 to that quantity equals 5? In other words, x + 3 = 5 . But most 1st graders don’t make that connection right away, Cliche said. Instead, they see the numbers 3 and 5, and they add them.

“Nine times out of 10 they’re going to say, ‘eight,’” Cliche said. “They’re number pluckers. They take this number and this number and they add them together or they take them apart.”

This is one of the biggest challenges in word problem-solving, educators and researchers agree—getting students to understand that the written story on the page represents a math story, and that the math story can be translated into an equation.

Making this connection is a key part of early mathematical sense-making. It helps students begin to understand that math isn’t just about numbers on a page, but a way of representing relationships in the world. And it’s one of the ways that kids learn to unite conceptual understanding of problems with the procedures they will need to solve them.

“When students struggle [with word problems], it tends to be everything else they have to do to get to the calculation,” said Brian Bushart, a 4th grade teacher in the West Irondequoit schools in Rochester, N.Y.

There are evidence-backed strategies that teachers can use to help students make these connections, researchers say.

These approaches teach students how to understand “math language,” how to devise a plan of attack for a problem, and how to recognize different problem types. And though they provide students tools and explicit strategies, these techniques are designed to support kids’ sense-making, not circumvent it, said Lynn Fuchs, a research professor in the department of special education at Vanderbilt University.

The goal, she said, is “understanding the full narrative of what’s being presented.”

How word problems are used in early grades

Story problems serve a few different purposes in early grades, said Nicole McNeil, a professor of psychology at the University of Notre Dame who studies students’ cognitive development in math.

They can help connect children’s preexisting knowledge to the math they’re learning in class—"activating that knowledge kids have in their everyday life, and then showing, how do mathematicians represent that?” McNeil said.

Cliche likes to use word problems in this way to introduce the concept of dividing by fractions.

“We’ll tell the kids, ‘I have three sandwiches here and I need to divide them in half so that everyone will get a piece,’” she said. “‘How many people can I feed?’”

After students solve the problem, Cliche introduces the operation that students could use to divide by fractions—marrying this conceptual understanding with the procedure that students would use going forward.

But word problems can also be used in the opposite direction, to see if students can apply their understanding of equations they’ve learned to real-world situations, McNeil said.

And there’s another, practical reason that teachers practice word problems: They’re ubiquitous in curriculum and they’re frequently tested.

There are lots of different kinds of problems that kids could work on in math classes, said Tamisha Thompson, a STEAM (for science, technology, engineering, the arts, and math) instructional coach in the Millbury public schools in Massachusetts, and a doctoral student in learning sciences at Worcester Polytechnic Institute.

Many story problems have one right answer, but there are also problems that could have multiple answers—or ones that aren’t solvable. Spending more time with a broader diversity of problems could encourage more creative mathematical thinking, Thompson said. “But we’re really driven by standardized tests,” she said. “And standardized tests typically have one right answer.”

In general, between 30 percent and 50 percent of standardized-test items in math feature these kinds of story problems, said Sarah Powell, an associate professor in the department of special education at the University of Texas at Austin.

“Until things change, and until we write better and different tests, if you want students to show their math knowledge, they have to show that through word problem-solving,” Powell said.

Why students struggle with word problems

Sometimes, students struggle with word problems because they don’t know where to start.

Just reading the problem can be the first hurdle. If early-elementary schoolers don’t have the reading skills to decode the words, or if they don’t know some of the vocabulary, they’ll struggle, said McNeil.

That can result in students scoring low on these portions of standardized tests, even if they understand the underlying math concepts—something McNeil considers to be a design flaw. “You’re trying to assess math, not reading twice,” she said.

Then, there’s math-specific vocabulary. What do words like “fewer than,” or “the rest,” mean in math language, and how do they prompt different actions depending on their placement in a problem?

Even if students can read the problem, they may struggle to figure out what it’s asking them to do, said Powell. They need to identify relevant information and ignore irrelevant information—including data that may be presented in charts or graphs. Then, they have to choose an operation to use to solve the problem.

Only once students have gone through all these steps do they actually perform a calculation.

Teaching kids how to work through all these setup steps takes time. But it’s time that a lot of schools don’t take, said Cliche, who has also worked previously as a state math trainer for Tennessee. Word problems aren’t often the focus of instruction—rather, they’re seen as a final exercise in transfer after a lot of practice with algorithms, she said.

A second problem: Many schools teach shortcut strategies for deciphering word problems that aren’t effective, Powell said.

Word problem “key words” charts abound on lesson-sharing sites like Teachers Pay Teachers . These graphic organizers are designed to remind students which math words signal different operations. When you see the word “more,” for example, that means add the numbers in the problem.

Talking with students about the meaning of math vocabulary is useful, said Powell. But using specific words as cues to add or subtract is a flawed strategy, Powell said, because “there is no single word that means an operation.” The word “more” might mean that the numbers need to be added together—or it might mean something else in context. Some problems have no key words at all.

In a 2022 paper , Powell and her colleagues analyzed more than 200 word problems from Partnership for Assessment of Readiness for College and Careers (PARCC) and Smarter Balanced math tests in elementary and middle school grades. Those tests are given by states for federal accountability purposes.

They found that using the key words strategy would lead students to choose the right operation to solve the problem less than half the time for single-step problems and less than 10 percent of the time for multistep problems.

Evidence-based strategies for helping struggling students

So if key words aren’t an effective strategy to support students who struggle, what is?

One evidence-based approach is called schema-based instruction . This approach categorizes problems into different types, depending on the math event portrayed, said Fuchs, who has studied schema-based instruction for more than two decades.

But unlike key words, schemas don’t tell students what operations to use. Instead, they help students form a mental model of a math event. They still need to read the problem, understand how that story maps onto their mental model, and figure out what information is missing, Fuchs said.

One type of schema, for example, is a “total” or “combine” problem, in which two quantities together make a total: “Jose has five apples. Carlos has two apples. How many apples do they have together?” In this case, students would need to add to get the answer.

But this is also a total problem: “Together, Jose and Carlos have seven apples. If Jose has five apples, how many apples does Carlos have?”

Here, adding the two numbers in the problem would bring students to the wrong answer. They need to understand that seven is the total, five is one part of the total, and there is another, unknown part—and then solve from there.

To introduce schemas, Vanderbilt’s Fuchs said, “we start with a child and the teacher representing the mathematical event in a concrete way.”

Take a “difference” problem, which compares a larger quantity and a smaller quantity for a difference. To demonstrate this, an early-elementary teacher might show the difference in height between two students or the difference in length of two posters in the room.

Eventually, the teacher would introduce other ways of representing this “difference” event, like drawing one smaller and one larger rectangle on a piece of paper. Then, Fuchs said, the teacher would explain the “difference” event with a number sentence—the formula for calculating difference—to connect the conceptual understanding with the procedure. Students would then learn a solution strategy for the schema.

Children can then use their understanding of these different problem types to solve new problems, Fuchs said.

There are other strategies for word-problem-solving, too.

• Attack strategies . Several studies have found that giving students a consistent set of steps they can use to approach every problem has positive effects. These attack strategies are different from schemas because they can be used with any problem type, offering more general guidance like reminders to read the problem and pull out relevant information.
• Embedded vocabulary. A 2021 study from Fuchs and her colleagues found that math-specific vocabulary instruction helped students get better at word problem-solving. These vocabulary lessons were embedded into schema instruction, and they focused on words that had a specific meaning in a math context—teaching kids the difference between “more than” and “then there were more,” for example.
• ‘Numberless’ problems . Some educators have also developed their own strategies. One of these is what’s called “numberless” word problems. A numberless problem has the same structure as a regular story problem but with the quantities strategically removed. An initial statement might say, for example, “Kevin found some bird feathers in the park. On his way home, he lost some of the feathers.”

With numberless problems, instead of jumping to the calculation, “the conversation is the goal,” said Bushart, the 4th grade teacher from New York, who has created a website bank of numberless problems that teachers can use .

The teacher talks with students about the change the story shows and what numbers might be reasonable—and not reasonable. The process is a form of scaffolding, Bushart said: a way to get students thinking conceptually about problems from the start.

Balancing structure and challenge

These approaches all rely on explicit teaching to give students tools that can help them succeed with problems they’re likely to see often in class or on tests.

But many math educators also use word problems that move beyond these common structures, in an attempt to engage students in creative problem-solving. Figuring out how much structure to provide—and how much challenge—can be a delicate balance.

These kinds of problems often require that students integrate real-life knowledge, and challenge them to “think beyond straightforward applications of mathematical situations,” said McNeil of Notre Dame.

There may be an extra number in the problem that kids don’t have to use. Or the problem might pose a question that would lead students to a nonsensical answer if they just used their procedural knowledge. For example: 65 students are going on a field trip. If each bus can hold 10 students, how many buses are needed?

Students might do the calculation and answer this question with 6.5, but that number doesn’t make sense, said McNeil—you can’t have half a bus.

In a 2021 study , McNeil and her colleague Patrick Kirkland rewrote some of these challenging questions in a way that encouraged students to think more deeply about the problems. They found that middle school students who worked on these experimental problems were more likely than their peers to engage in deep mathematical thinking. But, they were also less likely to get the problems correct than their peers who did standard word problems.

Other research, with young children, has found that teaching students how to transfer their knowledge can help them work through novel problems.

When students are given only problems that are all structured the same way, even minor changes to that format can prevent them from recognizing problem schemas, said Fuchs.

“What we found in our line of work is that if you change the way the word problem reads, in only very minor ways, they no longer recognize that, this is a ‘change’ problem, or a ‘difference’ problem,” she said, referencing different problem schemas.

In the early 2000s, she and her colleagues tested interventions to help students transfer their knowledge to more complex, at times open-ended problems. They found that when children were taught about the notion of transfer, shown examples of different forms of the same problem type, and encouraged to find examples in their own lives, they performed better on novel, multistep problems than their peers who had only received schema instruction.

The results are an example of how explicit instruction can lay the groundwork for students to be successful with more open-ended problem-solving, Fuchs said.

Exactly how to sequence this learning—when to lean into structure and when to release students into challenge—is an open question, McNeil said.

“We need more researchers focused on what are the best structures? What order should things go in? What is the appropriate scope and sequence for word problems?” she said. “We don’t have that information yet.”

A version of this article appeared in the May 10, 2023 edition of Education Week as Why Word Problems Are Such a Struggle for Students—And What Teachers Can Do

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What is a word problem?

A word problem is a few sentences describing a 'real-life' scenario where a problem needs to be solved by way of a mathematical calculation.

Word problems are seen as a crucial part of learning in the primary curriculum, because they require children to apply their knowledge of various different concepts to 'real-life' scenarios. 

Word problems also help children to familiarise themselves with mathematical language (vocabulary like fewer, altogether, difference, more, share, multiply, subtract, equal, reduced, etc.).

Teachers tend to try and include word problems in their maths lessons at least twice a week.  

What is RUCSAC?

In the classroom children might be taught the acronym RUCSAC (Read, Understand, Choose, Solve, Answer, Check) to help them complete word problems.

By following the acronym step by step children learn to apply a structured, analytical strategy to their calculations. They will need to understand what the problem is asking them to find out by reading the question carefully, choosing the correct mathematical operation to help them solve the query and finally checking their answer by using the inverse operation .

Word problem examples for Years 1 to 6

The following are example word problems that apply to each primary year group.

In Year 1 a child would usually been given apparatus to help them with a problem (counters, plastic coins, number cards, number lines or picture cards).

Sarah wants to buy a teddy bear costing 30p. How many 10p coins will she need?

Brian has 3 sweets. Tom has double this number of sweets. How many sweets does Tom have?

In Year 2, children continue to use apparatus to help them with problem-solving.

Faye has 12 marbles. Her friend Louise has 9 marbles. How many marbles do they both have altogether?

Three children are each given 5 teddy bears. How many teddy bears do they have altogether?

In Year 3, some children may use apparatus, but on the whole children will tend to work out word problems without physical aids. Teachers will usually demonstrate written methods for the four operations (addition, subtraction, multiplication and division) to support children in their working out of the problems.

A jumper costs £23. How much will 4 jumpers cost?

Sarah has 24 balloons. She gives a quarter of them away to her friend. How many balloons does she give away?

Children will also start to do two-step problems in Year 3. This is a problem where finding the answer requires two separate calculations, for example:

I have £34. I am given another £26. I divide this money equally into four different bank accounts. How much money do I put in each bank account?

• In this case, the first step would be to add £34 and £26 to make £60.
• The second step would be to divide £60 by 4 to make £15.

Children should feel confident in an efficient written method for each operation at this stage. They will continue to be given a variety of problems and have to work out which operation and method is appropriate for each. They will also be given two-step problems. 

I have 98 marbles. I share them equally between 6 friends. How many marbles does each friend get? How many marbles are left over?

Children will continue to do one-step and two-step problems. They will start to carry out problem-solving involving decimals . 

My chest of drawers is 80cm wide and my table is 1.3m wide. How much wall space do they take up when put side by side?

There are 24 floors of a car park. Each floor has room for 45 cars. How many cars can the car park fit altogether?

In Year 6 children solve 'multi-step problems' and problems involving fractions , decimals and percentages . 

Sarah sees the same jumper in two different sales: In the first sale, the original price of the jumper is £36.15, but has been reduced by a third. In the second sale, the jumper was priced at £45, but now has 40% off. How much does each jumper cost and which one is the cheapest?

In the past, calculators were sometimes used for solving two-step problems like the one above, but the new curriculum does not include the use of calculators at any time during primary school.

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What are the Differences in Meaning Between "Problem Solving" and "Solving Problems"

Problem Solving tells us what you do.

Solving Problems tells us what you do it to.

Is there any other difference in meaning in the context of math word problems?

• "He loves nothing better than problem solving; or, "He loves nothing better than solving problems." Each sentence says pretty much the same thing. –  rhetorician Commented Dec 25, 2014 at 2:45

2 Answers 2

In principle a connection of gerund and object can have three forms

1 the solving of problems (in Latin Grammar "problems" is called genetivus objectivus, i.e. "problems" corresponds to an object in a normal sentence with a finite verb.)

2 solving problems

3 problem solving/problem-solving

In 1 we have the full form with the and of. 2 is derived from 1 by dropping the and of. 3 is derived from 2 by putting "problems" in front position.

All three forms have the same meaning. Which form will be used is a matter of style. Form 2 is the common form.

Even "I'm solving problems" might be derived from form 1: I'm at/in the act of (the) solving (of) problems.

Well, "problem solving" is a noun (or, when hyphenated, an adjective); but "solving problems" is a present-progressive tense verb with an object.

Thus, "He has good problem-solving skills." But: "I am solving problems", rather than "I am problem solving".

But these are only matters of syntax. In answer to your question, no, there is no difference in meaning.

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Types of Word Problems

Types of word problems are always a challenge for students. There are so many processes involved in solving word problems that it can be challenging to pinpoint concerns. Is it reading comprehension? Is it computation? Are students overly relying on key words? And the list goes on.

One of my favorite strategies is to make sure students are familiar with all types of word problems. This Word Problems resource is very helpful for teaching this challenging skill!

Addition and Subtraction

There are three main types of addition and subtraction problems:

• joining problems
• separating problems
• comparing problems

Students should solve problems where the result is unknown, change is unknown, or start is unknown. When I begin teaching types of word problems, I use small numbers so students can focus on the meaning of the problem, rather than computation. This also gives students the opportunity to use counters to illustrate the problem, role playing and writing their own word problems.

Multiplication and Division

There are also three main types of multiplication and division word problems:

• equal groups

Within each type of problem, the product may be unknown, the group size may be unknown, or the number of groups may be unknown. Once again, I use counters to have students determine which equation to use to determine how to solve the different types of problems. Students should also notice the difference between addition/subtraction problems and multiplication/division problems.

In the resource, there are posters for each type of word problem. You can display these after you introduce that type of word problem and then refer to the posters throughout the year. You can even print this in a smaller size so students can add these to their math notebooks.

Types of Word Problems Booklet

A great way to help students develop an understanding of word problems is to have them write word problems. In the resource, there is a booklet is for addition and subtraction, a booklet for multiplication and division, and a booklet that combines both addition and subtraction. In the addition and subtraction booklet, I replaced all numbers with letters, because I wanted my students to not focus on a numerical answer, but the equation used to solve the problem. Over the years, I’ve found that my students’ primary focus was the answer, not the process.

When students have to write a word problem to reflect a certain style of word problem, it raises their level of thinking and problem solving significantly. This is a much more complex skill, so we typically practice writing word problems through guided practice first.

For additional practice solving different types of word problems, there are 22 addition and subtraction task cards and 18 multiplication and division task cards, for a total of 40 task cards. There are two task cards for each type of word problem, so all problem types are equally represented.

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4 thoughts on “types of word problems”.

Are your rounding riddles still available? They look very interesting. Thank you Julie

Yes, there’s a link in the post right above the picture.

I was looking for the blue topped poster that has a grid showing ALL the types of problems. I bought the file, but it was not in it. Where is that available?

Oh! That’s one of my math anchor charts.

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Modeling Word Problems

Using models is a critical step in helping students transition from concrete manipulative work with word problems to the abstract step of generating an equation to solve contextual problems. By learning to use simple models to represent key mathematical relationships in a word problem, students can more easily make sense of word problems, recognize both the number relationships in a given problem and connections among types of problems, and successfully solve problems with the assurance that their solutions are reasonable.

Why is modeling word problems important?

Mr. Alexander and teachers from his grade level team were talking during their Professional Learning Community (PLC) meeting about how students struggle with word problems. Everyone felt only a few of their students seem to be able to quickly generate the correct equation to solve the problem. Many students just seem to look for some numbers and do something with them, hoping they solve the problem.

Mr. Alexander had recently learned about using modeling for word problems in a workshop he had attended.  He began to share the model diagrams with his teammates and they were excited to see how students might respond to this approach. They even practiced several model diagrams among themselves as no one had ever learned to use models with word problems. Since part of their PLC work freed them up to observe lessons in each others' rooms, they decided they would watch Mr. Alexander introduce modeling to his students. 

So, two days later they gathered in Mr. Alexander's room for the math lesson. Mr. Alexander presented the following problem:

Lily and her brother, Scotty, were collecting cans for the recycling drive. One weekend they collected 59 cans and the next weekend they collected 85 cans.  How many cans were collected in all?

Mr. Alexander went over the problem and drew a rectangular bar divided into two parts on the board, explaining that each part of the rectangle was for the cans collected on one of the weekends and the bracket indicated how many cans were collected in all. Reviewing the problem, Mr. Alexander asked students what was not known, and where the given numbers would go and why. This resulted in the following bar model:

The class then discussed what equations made sense given the relationship of the numbers in the bar model. This time many students wrote the equation, 59 + 85 = ?, and solved the problem. In their discussion after the lesson, Mr. Alexander's teammates mentioned that they noticed a much higher degree of interest and confidence in problem solving when Mr. Alexander introduced the bar model. Everyone noticed that many more students were successful in solving problems once modeling was introduced and encouraged. As the class continued to do more word problems, the diagrams appeared to be a helpful step in scaffolding success with word problems.

Knowledge can be represented both linguistically and nonlinguistically. An increase in nonlinguistic representations allows students to better recall knowledge and has a strong impact on student achievement (Marzano, et. al., 2001, Section 5). In classic education research, Bruner (1961) identified three modes of learning:  enactive (manipulating concrete objects), iconic (pictures or diagrams), and symbolic (formal equation).  The iconic stage, using pictures and diagrams, is an important bridge to abstracting mathematical ideas using the symbols of an equation. Research has also validated that students need to see an idea in multiple representations in order to identify and represent the common core (Dienes, undated). For word problems involving the operation of addition, students need to experience several types of problems to generalize that when two parts are joined they result in a total or a quantity that represents the whole. Whether the items are bears, balloons, or cookies no longer matters as the students see the core idea of two subsets becoming one set. Dienes discovered that this abstraction is only an idea; therefore it is hard to represent. Diagrams can capture the similarity students notice in addition/joining problems where both addends are known and the total or whole is the unknown.  Diagrams will also be useful for missing addend situations. Like Bruner, Dienes saw diagrams as an important bridge to abstracting and formalizing mathematical ideas.  

Along with Bruner and Dienes, Skemp (1993) identified the critical middle step in moving from a real-life situation to the abstractness of an equation. While students need to experience many real-life situations they will get bogged down with the "noise" of the problem such as names, locations, kinds of objects, and other details. It is the teacher's role to help students sort through the noise to capture what matters most for solving the problem. A diagram can help students capture the numerical information in a problem, and as importantly, the relationship between the numbers, e.g. Do we know both the parts, or just one of the parts and the whole? Are the parts similar in size, or is one larger than the other? Once students are comfortable with one kind of diagram, they can think about how to relate it to a new situation. A student who has become proficient with using a part-part-whole bar model diagram when the total or whole is unknown, (as in the collecting cans problem in Mr. Alexander's class), cannot only use the model in other part-part-whole situations, but can use it in new situations, for example, a missing addend situation. Given several missing addend situations, students may eventually generalize that these will be subtractive situations, solvable by either a subtraction or adding on equation.

The work of Bruner, Dienes and Skemp informed the development of computation diagrams in some elementary mathematics curriculum materials in the United States. Interestingly, it also informed the development of curriculum in Singapore, as they developed the "Thinking Schools, Thinking Nation" era of reforming their educational model and instructional strategies (Singapore Ministry of Education, 1997). The bar model is a critical part of "Singapore Math."  It is used and extended across multiple grades to capture the relationships within mathematical problems. Singapore has typically scored near the top of the world on international assessments, a possible indicator of the strong impact of including the visual diagram step to represent and solve mathematical problems.

What is modeling word problems?

Models at any level can vary from simple to complex, realistic to representational. Young students often solve beginning word problems, acting them out, and modeling them with the real objects of the problem situation, e.g. teddy bears or toy cars. Over time they expand to using representational drawings, initially drawing pictures that realistically portray the items in a problem, and progressing to multi-purpose representations such as circles or tally marks. After many concrete experiences with real-life word problems involving joining and separating, or multiplying and dividing objects, teachers can transition students to inverted-V and bar model drawings which are multi-purpose graphic organizers tied to particular types of word problems.

Modeling Basic Number Relationships

Simple diagrams, sometimes known as fact triangles, math mountains, situation diagrams, or representational diagrams have appeared sporadically in some curriculum materials. But students' problem solving and relational thinking abilities would benefit by making more routine use of these diagrams and models.

Young children can begin to see number relationships that exist within a fact family through the use of a model from which they derive equations. An inverted-V is one simple model that helps students see the addition/subtraction relationships in a fact family, and can be used with word problems requiring simple joining and separating. The inverted-V model can be adapted for multiplication and division fact families. For addition, students might think about the relationships among the numbers in the inverted-V in formal terms, addend and sum , or in simpler terms, part and total , as indicated in the diagrams below.

A specific example for a given sum of 10 would be the following, depending on which element of the problem is unknown.

      6 + 4 = ?                   6 + ? = 10                       ? + 4 = 1

         4 + 6 = ?                 10 - 6 = ?                       10 - 4 = ?

While often used with fact families, and the learning of basic facts, inverted-V diagrams can also work well with solving word problems. Students need to think about what they know and don't know in a word problem - are both the parts known, or just one of them?  By placing the known quantities correctly into the inverted-V diagram, students are more likely to determine a useful equation for solving the problem, and see the result as reasonable for the situation. For example, consider the following problem:

Zachary had 10 train cars. Zachary gave 3 train cars to his brother. How many train cars does Zachary have now? 

Students should determine they know how many Zachary started with ( total or whole ), and how many he gave away ( part of the total ). So, they need to find out how many are left ( other part of the total ). The following inverted-V diagram represents the relationships among the numbers of this problem:

3 + ? = 10 or 10 - 3 = ?, so Zachary had 7 train cars left.

As students move on to multiplication and division, the inverted-V model can still be utilized in either the repeated addition or multiplicative mode. Division situations do not require a new model; division is approached as the inverse of multiplication or a situation when one of the factors is unknown.

Again, the inverted-V diagram can be useful in solving multiplication and division word problems. For example, consider the following problem:

Phong planted 18 tomato plants in 3 rows. If each row had the same number of plants, how many plants were in each row?   

Students can see that they know the product and the number OF rows. The number IN A row is unknown. Either diagram below may help solve this problem, convincing students that 6 in a row is a reasonable answer.

While the inverted-V diagram can be extended to multi-digit numbers, it has typically been used with problems involving basic fact families. Increasing the use of the inverted-V model diagram should heighten the relationship among numbers in a fact family making it a useful, quick visual for solving simple word problems with the added benefit of using and increasing the retention of basic facts.

Models and Problem Types for Computation

As children move to multi-digit work, teachers can transition students to bar model drawings, quick sketches that help students see the relationships among the important numbers in a word problem and identify what is known and unknown in a situation. 

With bar models the relationships among numbers in all these types of problems becomes more transparent, and helps bridge student thinking from work with manipulatives and drawing pictures to the symbolic stage of writing an equation for a situation. With routine use of diagrams and well-facilitated discussions by teachers, student will begin to make sense of the parts of a word problem and how the parts relate to each other.

Part-Part-Whole Problems. Part-Part-Whole problems are useful with word problems that are about sets of things, e.g. collections. They are typically more static situations involving two or more subsets of a whole set. Consider the problem,

Cole has 11 red blocks and 16 blue blocks. How many blocks does Cole have in all?

Students may construct a simple rectangle with two parts to indicate the two sets of blocks that are known (parts/addends). It is not important to have the parts of the rectangle precisely proportional to the numbers in the problem, but some attention to their relative size can aid in solving the problem. The unknown in this problem is how many there are altogether (whole/total/sum), indicated by a bracket (or an inverted-V) above the bar, indicating the total of the 2 sets of blocks. The first bar model below reflects the information in the problem about Cole's blocks. 

11 + 16 = ?  so Cole has 27 blocks in all.

A similar model would work for a problem where the whole amount is known, but one of the parts (a missing addend) is the unknown. For example:

Cole had 238 blocks. 100 of them were yellow. If all Cole's blocks are either blue or yellow, how many were blue? 

The following bar model would be useful in solving this problem.

100 + ? = 238 or 238 - 100 = ? so Cole has 138 blue blocks.

The answer has to be a bit more than 100 because 100 + 100 is 200 but the total here is 238 so the blue blocks have to be a bit more than 100.

The part-part-whole bar model can easily be expanded to large numbers, and other number types such as fractions and decimals. Consider the problem:

Leticia read 7 ½ books for the read-a-thon.She wants to read 12 books in all. How many more books does she have to read?   

The first diagram below reflects this problem. Any word problem that can be thought of as parts and wholes is responsive to bar modeling diagrams. If a problem has multiple addends, students just draw enough parts in the bar to reflect the number of addends or parts, and indicate whether one of the parts, or the whole/sum, is the unknown, as shown in the second figure below.

     12 - 7 ½ = ? or  7 ½ + ? = 12 so Leticia needs to read 4 ½ more books.

Join (Addition) and Separate (Subtraction) Problems.

Consider this joining problem:

Maria had $20.  She got $11 more dollars for babysitting.  How much money does she have now?

Students can identify that the starting amount of $20 is one of the parts, $11 is another part (the additive amount), and the unknown is the sum/whole amount, or how much money she has now. The first diagram below helps represent this problem.   

Consider the related subtractive situation:

Maria had $31.  She spent some of her money on a new CD.  Maria now has $16 left.

The second diagram above represents this situation. Students could use the model to help them identify that the total or sum is now $31, one of the parts (the subtractive change) is unknown, so the other part is the $16 she has left. 

Comparison Problems. Comparison problems have typically been seen as difficult for children. This may partially be due to an emphasis on subtraction as developed in word problems that involve "take away" situations rather than finding the "difference" between two numbers. Interestingly, studies in countries that frequently use bar models have determined that students do not find comparison problems to be much more difficult than part-part-whole problems (Yeap, 2010, pp. 88-89). 

A double bar model can help make comparison problems less mysterious. Basically, comparison problems involve two quantities (either one quantity is greater than the other one, or they are equal), and a difference between the quantities. Two bars, one representing each quantity, can be drawn with the difference being represented by the dotted area added onto the lesser amount. For example, given the problem:

Tameka rode on 26 county fair rides. Her friend, Jackson, rode on 19 rides. How many more rides did Tameka ride on than Jackson?

Students might generate the comparison bars diagram shown below, where the greater quantity, 26, is the longer bar. The dotted section indicates the difference between Jackson's and Tameka's quantities, or how much more Tameka had than Jackson, or how many more rides Jackson would have had to have ridden to have the same number of rides as Tameka.

26 - 19 = ?  or  19 + ? = 26; the difference is 7 so Tameka rode 7 more rides.

Comparison problems express several differently worded relationships. If Tameka rode 7 more rides than Jackson, Jackson rode 7 fewer rides than Tameka.  Variations of the double bar model diagram can make differently worded relationships more visual for students. It is often helpful for students to recognize that at some point both quantities have the same amount, as shown in the model below by the dotted line draw up from the end of the rectangle representing the lesser quantity. But one of the quantities has more than that, as indicated by the area to the right of the dotted line in the longer bar. The difference between the quantities can be determined by subtracting 19 from 26, or adding up from 19 to 26 and getting 7, meaning 26 is 7 more than 19 or 19 is 7 less than 26.

Comparison word problems are especially problematic for English Learners as the question can be asked several ways. Modifying the comparison bars may make the questions more transparent. Some variations in asking questions about the two quantities of rides that Tameka and Jackson rode might be:  

• How many more rides did Tameka ride than Jackson?
• How many fewer rides did Jackson go on than Tameka?
• How many more rides would Jackson have had to ride to have ridden the same number of rides as Tameka?
• How many fewer rides would Tameka have had to ride to have ridden the same number of rides as Jackson?

Comparisons may also be multiplicative. Consider the problem:

Juan has 36 CDs in his collection. This is 3 times the amount of CDs that his brother, Marcos, has. How many CDs does Marcos have?  

In this situation, students would construct a bar model, shown below on the left, with 3 parts. Students could divide the 36 into 3 equal groups to show the amount that is to be taken 3 times to create 3 times as many CDs for Juan.

  36 ¸ 3 = ? or  3 x ? = 36             12 + 12 + 12 = ? (or 3 x 12 = ?)

so Marcos has 12 CDs.                    so Juan has 36 CDs.

A similar model can be used if the greater quantity is unknown, but the lesser quantity, and the multiplicative relationship are both known. If the problem was:

Juan has some CDs. He has 3 times as many CDs as Marcos who has 12 CDs. How many CDs does Juan have?

As seen in the diagram above on the right, students could put 12 in a box to show the number of CDs Marcos has; then duplicate that 3 times to sow that Juan has 3 times as many CDs. Then the total number that Juan has would be the sum of those 3 parts. 

Multiplication and Division Problems. The same model used for multiplicative comparisons will also work for basic multiplication word problems, beginning with single digit multipliers. Consider the problem:

Alana had 6 packages of gum. Each package holds 12 pieces of gum. How many pieces of gum does Alana have in all?

The following bar model uses a repeated addition view of multiplication to visualize the problem.

12 + 12 + 12 + 12 + 12 + 12 = 72 (or 6 x 12 = 72)

so Alana has 72 pieces of gum.

As students move into multi-digit multipliers, they can use a model that incorporates an ellipsis to streamline the bar model. For example:

Sam runs 32 km a day during April to get ready for a race. If Sam runs every day of the month, how many total kilometers did he run in April?

30 x 32 km = 30 x 30 km + 30 x 2 km = 960 km

Sam ran 960 km during the 30 days of April.

Since division is the inverse of multiplication, division word problems will utilize the multiplicative bar model where the product (dividend) is known, but one of the factors (divisor or quotient) is the unknown. 

Problems Involving Rates, Fractions, Percent & Multiple Steps. As students progress through the upper grades, they can apply new concepts and multi-step word problems to bar model drawings. Skemp (1993) identified the usefulness of relational thinking as critical to mathematical development. A student should be able to extend their thinking based on models they used earlier, by relating and adapting what they know to new situations. 

Consider this rate and distance problem:

Phong traveled 261 miles to see her grandmother. She averaged 58 mph. How long did it take her to get to her grandmother's house?

The following model builds off of the part-part-whole model using a repeated addition format for multiplication and division. It assumes that students have experience with using the model for division problems whose quotients are not just whole numbers. As they build up to (or divide) the total of 261 miles, they calculate that five 58's will represent 5 hours of travel, and the remaining 29 miles would be represented by a half box, so the solution is it would take Phong 5½ hours of driving time to get to her grandmother's house.

Even a more complex rate problem can be captured with a combination of similar models.  Consider this problem: 

Sue and her friend Anne took a trip together.  Sue drove the first 2/5 of the trip and Anne drove 210 miles for the last 3/5 of the trip.  Sue averaged 60 mph and Anne averaged 70 mph.  How long did the trip take them?

There are several ways students might combine or modify a basic bar model. One solution might be the following, where the first unknown is how many miles Sue drove. A bar divided into fifths represents how to calculate the miles Sue drove. Since we know that the 210 miles Anne drove is 3/5 of the total trip, each one of Anne's boxes, each representing 1/5 of the trip, is 70 miles. Therefore, Sue drove two 70 mile parts, or 140 miles, to equal 2/5 of the total trip.

The diagram now needs to be extended to show how to calculate the number of hours. Anne's 210 mile segment, divided by her 70 mph rate will take 3 hours, as recorded on the following extension of the diagram. Sue's distance of 140 miles now needs to be divided into 60 mph segments to determine her driving time of 2 1/3 hours.  So, the total trip of 350 miles would take 5 1/3 hours of driving time, considering the two driving rates. 

Certainly, a foundation of using simple bar model drawings needs to be well developed in early grades for students to extend diagrams with understanding in later grades. The Sue and Anne rate-time-distance problem would not be the place to begin using bar models!  But, by building on work in earlier grades with models, this extended model makes the mathematics of this complex problem more transparent, and helps students think through the steps. 

Consider a simpler multi-step problem:

Roberto purchased 5 sports drinks at $1.25 each. Roberto gave the cashier $20. How much change did he get back?

Again, there may be student variations when they begin to extend the use of diagrams in multi-step or more complex problems.  Some students might use two diagrams at once, as show below on the left.  Others may indicate computation within one diagram, as shown in the diagram on the right.

With routine experience with bar modeling, students can extend the use of the models to problems involving relationships that can be expressed with variables.  Consider this simple problem that could be represented algebraically:

Callan and Avrielle collected a total of 190 bugs for a science project.  Callan collected 10 more bugs than Avrielle.  How many bugs did Callan collect?

Let n equal the number of bugs Avrielle collected, and n + 10 equal the number of bugs Callan collected.  The following model might be created by students:

Since n + n = 180 (or 2 n = 180), n = 90.  Therefore, Callan collected 90 + 10 or 100 bugs and Avrielle collected 90 bugs for a total of 190 bugs collected together.

Planning and Instruction

How do I intentionally plan for and use modeling?

If modeling is not a way you learned to identify the important information and numerical relationships in word problems, you may want to review some of the resources on problem types (see Carpenter's book in References and Resources section below), or bar modeling (see books by Forsten, Walker, or Yeap in the References and Resources section below).  You may also want to practice the different types of models.  Decide which are most accessible for your students, and start with introducing one model at a time, helping students determine what is unknown in the problem, and where that unknown and the other numerical information should be placed in the bar model.  A question mark, box, or a variable can be used for the unknown.  As students become comfortable with that model, introduce, and compare and contrast a second model with the known model. 

You might introduce bar model drawings, or inverted-V diagrams, when there is a unit in your curriculum that contains several word problems.  If word problems are sporadic in your curriculum, you might introduce a "Word-Problem-of-the-Day" format where students solve a problem, or cluster of related problems, each day. 

To emphasize model drawings, you might have students take a set of problems, and classify them as to which model would help them solve the problem, or do a matching activity between word problems and model drawings.  Ask students to explain why a particular equation matches a model and would be useful in finding the solution.  Another activity is to present a bar model with some numerical information and an unknown.  Then ask students to write a word problem that could logically be solved using that model.  Ask students to explain why the word problem created matches a diagram well.  As students use models for solving word problems, they may generate different equations to solve a problem even though their models are the same.  Plan for class discussions where students may discuss why there can be different equations from the same bar model. 

Several studies have shown that students who can visualize a word problem through modeling increase their problem solving ability and accuracy.  This has been particularly documented in Singapore and other high performing countries where bar modeling is used extensively across grades.  Students are more likely to solve problems correctly when they incorporate bar model drawings.  On difficult problems, students who have been able to easily generate equations with simple problems often find that bar model drawings are especially helpful in increasing accuracy as problems increase in difficulty or involve new concepts (Yeap, 2010, pp. 87-89).

TALK:  Reflection and Discussion

• Are there particular types of word problems that your students solve more easily than others?  What characterizes these problems?
• Identify some basic facts with which your students struggle.  How could you incorporate those facts into word problems, and how might the use of the inverted-V model help?
• How do bar model drawings help extract and represent the mathematical components and numerical relationships of a word problem?
• With which type of word problems would you begin to show your students the use of bar model drawings?

DO:  Action Plans

• Select several story problems from your curriculum, MCA sample test items, or the Forsten, Walker, or Yeap resources on bar model drawing. Practice creating a bar model for several problems.  Compare your models with others in your grade level, team, or PLC group.  Practice until you feel comfortable with various model drawings.
• Investigate the types of multiplication and division problems, and how bar models can be used with different types such as measurement and partitive division, arrays, equal groups, rates.  The Carpenter resource may be helpful.
• Select some problems from your curriculum that are of a similar type.  Which bar model would be helpful in solving this type of problem?  Practice using the model yourself with several problems of this type.  How will you introduce the model to your students?
• Identify some basic facts with which your students struggle.  Craft some rich word problems utilizing these fact families.  Introduce the inverted-V diagrams with the word problems to make sense of the information in the word problem, and discuss strategies for solving the problems.
• Initiate a "Word-Problem-of-the-Day".  Students might want to keep problem solving notebooks.  Begin with problems of a particular type, and show students how to use a bar model to represent the information in a problem.  Cluster several problems of a given type during the week.  What improvements do you see in student selection of appropriate equations, accuracy of solutions, and ability to estimate or justify their answers as they increase the use of bar models to solve the word problems?  A quick way to disseminate the "Word-Problem-of-the-Day" is to duplicate the problem on each label on a sheet of address labels.  Students can just peel off the daily problem, add it to their problem solving notebook or a sheet of paper and solve away.
• When your district is doing a curriculum materials review, advocate to include a criteria that requires the use of visual models in helping students make sense of mathematical problems. 
• Watch some of the videos of students using models on the Powerful Practices CD (see Carpenter and Romberg in References and Resources Section). 

References and Resources

Bruner, J. S. (1961). The act of discovery.  Harvard Educational Review, 31, pp. 21-32, in Yeap, Ban Har. (2010). Bar modeling:  A problem solving tool.  Singapore: Marshall Cavendish Education. 

Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L. & Empson, S. B. (1999). Children's mathematics: Cognitively guided instruction.  Portsmouth, NH:  Heinemann.  (Book and CD)

Carpenter, T. P. & Romberg. T. A. (2004). Modeling, generalization, and justification in mathematics cases, in Powerful practices in mathematics & science.  Madison, WI:  National Center for Improving Student Learning and Achievement in Mathematics and Science.  www.wcer.wisc.edu/ncisla   (Booklet and CD)

Dienes, Z. (undated). Zoltan Dienes' six-state theory of learning mathematics. Retrieved from http://www.zoltandienes.com

Forsten, C. (2009). Step-by-step model drawing:  Solving math problems the Singapore way.  Peterborough, NH: SDE:  Crystal Spring Books.  http:// www.crystalspringsbooks.com

Hoven, J. & Garelick, B. (2007). Singapore math: Simple or complex? Educational Leadership, 65 (3), 28-31.

Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement.  Portsmouth, NH:  Heinemann.

Marzano, R. J., Norford, J. S., Paynter, D. E., Pickering, D. J., & Gaddy, B. B. (2001).   A handbook for classroom instruction that works.  Alexandria, VA:  Association for Supervision and Curriculum Development.

Singapore Ministry of Education. (1997). Retrieved http://moe.gov.sg

Skemp, R. R. (January, 1993). "Theoretical foundations of problem solving: A position paper."  University of Warwick. Retrieved from http://www.grahamtall.co.uk/skemp/sail/theops.html

Walker, L. (2010). Model drawing for challenging word problems:  Finding solutions the Singapore way.   Peterborough, NH: SDE:  Crystal Spring Books.  http:// www.crystalspringsbooks.com

Yeap, B. H. (2010). Bar modeling:  A problem solving tool.  Singapore: Marshall Cavendish Education.  http:// www.singaporemath.com

Understanding Multiplication and Division in Word Problems

So, the fun and games of learning how to make arrays, skip counting on number lines and using models to solve multiplication problems has lead to the equally exciting task of solving division problems. Everyone seems to be making great progress and making meaning of multiplication and division in real world problems, UNTIL … wait for it … wait for it … We mix the two together!

And here the real “fun” begins … A few students usually have an intuition about the structure of the problems and just “get it” with out much help, but the majority of students need direct, systematic instruction paired with hands on or pictorial examples to really, truly, deeply, understand the difference between multiplication and division in word problems.  But where can we start?

Start with The Same Story!

I like to take the same story and write 3 word problems that use the same story, but ask 3 different questions.  This really helps my students to see how multiplication and division can be the same and different in a word problem and help them to see to importance of analyzing the question.   How many times have you had a student choose multiplication to solve a division problem because they “saw the word each”?  Solving similar problems and then discussing the problems in relation to each other can really get students thinking deeper about problem solving.

For example, my 3 word problems might be:

• Jeff had 4 plates of cookies to take to his neighbors.  He put 6 cookies on each plate.  How many cookies was Jeff giving his neighbors?
• Jeff had 4 plates of cookies to take to his neighbors. He had 24 cookies and put an equal number of cookies on each plate.  How many cookies did he put on each plate?
• Jeff had 24 cookies to take to his neighbors.  He put the cookies on plates. Each plate had 6 cookies.  How many plates did he use?

The three problems have the same story, but different clues are given and different questions are asked for each problem.  ( Get this FREEBIE here)

As we work through the problems, I ask my students these questions:

• What is the question asking me to find?
• Does the story tell me the total?
• Does the story tell me the number in each group?
• Does the story tell me the number of groups?

Draw pictures or make diagrams!

We write down the question and clues. Then draw a picture …

After discussing how we made our picture we make our number sentence.

Write a number sentence and label the answers.

Compare and Reflect!

Continue to have meaningful discussion as you practice!

As you continue working guided problems with students ask them to explain why they are choosing to multiplication or division. Encourage students to question each other when working in small groups or with partners.  Have them write their own examples of word problems using the same story, but different questions.

And, of course, practice, practice, practice!    Download this FREE page to practice!

Share this:, 3 thoughts on “ understanding multiplication and division in word problems ”.

These are great ideas for the interchangeable nature of multiplication and division. I know for me, as a kid, word problems were the bane of my existence. My brain seemed to go “offline” once I realized I was attempting to solve a math problem. It’s neat to see you offer your students those questions as ways to assist them in figuring the problem out for themselves.

Great ideas for teaching a hard concept!

Thanks Linda!

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Breadcrumbs

How to Know which Operations to Use in Word Problems

When working out what operation to use in a word problem, there are some key phrases students should look out for. Let’s take you through the key phrases and use them in examples.

Key phrases for addition word problems

Find the total. How many altogether? How many are there?

If asked the above, the word problem involves addition.

For example:

There are 14 goldfish in a water tank. Kerri put in 23 more goldfish in the tank. How many goldfish are there?

Answer: 14+ 23= 37. There are 37 goldfish in the tank.

Addition word problem worksheets

Starting in kindergarten, we’ve created addition word problems for students to practice. By grade 3, we’ve compiled mixed number word problems with addition as well.

Here’s an example of word problems of addition with sums of 50 or less from our grade 1 word problem section:

Key phrases for subtraction word problems

What is the difference? How many more? How many less?

If asked the above, the word problem involves subtraction.

For example: Mrs. Sheridan has 11cats. Mrs. Garrett has 24 cats. How many more cats does Mrs. Garrett have than Mrs. Sheridan?

Answer: 24 –11 = 13. She has 13 more cats.

Subtraction word problem worksheets

Starting in kindergarten, we’ve created subtraction word problem for every grade. By grade 3, you’ll find subtraction included in the mixed word problems as well.

In grade 2, students work on 1-3 digit subtraction word problems .

Key phrases for multiplication word problems

How many of the same thing repeated?

If asked the above, the word problem involves multiplication.

For example: Tyler, an animal rights advocate, decided to build his own animal sanctuary to protect different animals.

500 yards away there was the aquatic reserve for freshwater organisms. If there are 6 lakes in that region, each having 175 different fish, how many fish does he have in total?

(How many groups of 175 fish are repeated across 6 lakes.)

Answer: 6 x 175 = 1,050. He has a total of 1,050 fish.

Multiplication word problem worksheets

Starting in grade 3, students work on multiplication word problems .

Key phrases for division word problems

How many equal groups? If share evenly, how many…? The same number.

If asked the above, the word problem involves division.

For example: Sheila has started writing a list of the gifts that she plans to give to her family and friends this Christmas.

For her classmates, she made colorful paper stars which will be placed in small clear bottles. She was able to prepare 45 paper stars. How many stars will be placed in each bottle if Sheila has 9classmates?

Answer: 45 ÷ 9 = 5. There will be 5 stars placed in each bottle.

Division word problem worksheets

Starting in grade 3, we have division word problems for students to practice.

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Fluency, Reasoning and Problem Solving: What This Looks Like In Every Maths Lesson

Neil Almond

Fluency reasoning and problem solving have been central to the new maths national curriculum for primary schools introduced in 2014. Here we look at how these three approaches or elements of maths can be interwoven in a child’s maths education through KS1 and KS2. We look at what fluency, reasoning and problem solving are, how to teach them, and how to know how a child is progressing in each – as well as what to do when they’re not, and what to avoid.

The hope is that this blog will help primary school teachers think carefully about their practice and the pedagogical choices they make around the teaching of reasoning and problem solving in particular.

Before we can think about what this would look like in practice however, we need to understand the background tothese terms.

What is fluency in maths?

Fluency in maths is a fairly broad concept. The basics of mathematical fluency – as defined by the KS1 / KS2 National Curriculum for maths – involve knowing key mathematical facts and being able to recall them quickly and accurately.

But true fluency in maths (at least up to Key Stage 2) means being able to apply the same skill to multiple contexts, and being able to choose the most appropriate method for a particular task.

Fluency in maths lessons means we teach the content using a range of representations, to ensure that all pupils understand and have sufficient time to practise what is taught.

Read more: How the best schools develop maths fluency at KS2 .

What is reasoning in maths?

Reasoning in maths is the process of applying logical thinking to a situation to derive the correct problem solving strategy for a given question, and using this method to develop and describe a solution.

Put more simply, mathematical reasoning is the bridge between fluency and problem solving. It allows pupils to use the former to accurately carry out the latter.

Read more: Developing maths reasoning at KS2: the mathematical skills required and how to teach them .

What is problem solving in maths?

It’s sometimes easier to start off with what problem solving is not. Problem solving is not necessarily just about answering word problems in maths. If a child already has a readily available method to solve this sort of problem, problem solving has not occurred. Problem solving in maths is finding a way to apply knowledge and skills you have to answer unfamiliar types of problems.

Read more: Maths problem solving: strategies and resources for primary school teachers .

We are all problem solvers

First off, problem solving should not be seen as something that some pupils can do and some cannot. Every single person is born with an innate level of problem-solving ability.

Early on as a species on this planet, we solved problems like recognising faces we know, protecting ourselves against other species, and as babies the problem of getting food (by crying relentlessly until we were fed).

All these scenarios are a form of what the evolutionary psychologist David Geary (1995) calls biologically primary knowledge. We have been solving these problems for millennia and they are so ingrained in our DNA that we learn them without any specific instruction.

Why then, if we have this innate ability, does actually teaching problem solving seem so hard?

Mathematical problem solving is a  learned skill

As you might have guessed, the domain of mathematics is far from innate. Maths doesn’t just happen to us; we need to learn it. It needs to be passed down from experts that have the knowledge to novices who do not.

This is what Geary calls biologically secondary knowledge. Solving problems (within the domain of maths) is a mixture of both primary and secondary knowledge.

The issue is that problem solving in domains that are classified as biologically secondary knowledge (like maths) can only be improved by practising elements of that domain.

So there is no generic problem-solving skill that can be taught in isolation and transferred to other areas.

This will have important ramifications for pedagogical choices, which I will go into more detail about later on in this blog.

The educationalist Dylan Wiliam had this to say on the matter: ‘for…problem solving, the idea that pupils can learn these skills in one context and apply them in another is essentially wrong.’ (Wiliam, 2018)So what is the best method of teaching problem solving to primary maths pupils?

The answer is that we teach them plenty of domain specific biological secondary knowledge – in this case maths. Our ability to successfully problem solve requires us to have a deep understanding of content and fluency of facts and mathematical procedures.

Here is what cognitive psychologist Daniel Willingham (2010) has to say:

‘Data from the last thirty years lead to a conclusion that is not scientifically challengeable: thinking well requires knowing facts, and that’s true not simply because you need something to think about.

The very processes that teachers care about most—critical thinking processes such as reasoning and problem solving—are intimately intertwined with factual knowledge that is stored in long-term memory (not just found in the environment).’

Colin Foster (2019), a reader in Mathematics Education in the Mathematics Education Centre at Loughborough University, says, ‘I think of fluency and mathematical reasoning, not as ends in themselves, but as means to support pupils in the most important goal of all: solving problems.’

In that paper he produces this pyramid:

This is important for two reasons:

1)    It splits up reasoning skills and problem solving into two different entities

2)    It demonstrates that fluency is not something to be rushed through to get to the ‘problem solving’ stage but is rather the foundation of problem solving.

In my own work I adapt this model and turn it into a cone shape, as education seems to have a problem with pyramids and gross misinterpretation of them (think Bloom’s taxonomy).

Notice how we need plenty of fluency of facts, concepts, procedures and mathematical language.

Having this fluency will help with improving logical reasoning skills, which will then lend themselves to solving mathematical problems – but only if it is truly learnt and there is systematic retrieval of this information carefully planned across the curriculum.

Performance vs learning: what to avoid when teaching fluency, reasoning, and problem solving

I mean to make no sweeping generalisation here; this was my experience both at university when training and from working in schools.

At some point schools become obsessed with the ridiculous notion of ‘accelerated progress’. I have heard it used in all manner of educational contexts while training and being a teacher. ‘You will need to show ‘ accelerated progress in maths ’ in this lesson,’ ‘Ofsted will be looking for ‘accelerated progress’ etc.

I have no doubt that all of this came from a good place and from those wanting the best possible outcomes – but it is misguided.

I remember being told that we needed to get pupils onto the problem solving questions as soon as possible to demonstrate this mystical ‘accelerated progress’.

This makes sense; you have a group of pupils and you have taken them from not knowing something to working out pretty sophisticated 2-step or multi-step word problems within an hour. How is that not ‘accelerated progress?’

This was a frequent feature of my lessons up until last academic year: teach a mathematical procedure; get the pupils to do about 10 of them in their books; mark these and if the majority were correct, model some reasoning/problem solving questions from the same content as the fluency content; set the pupils some reasoning and word problem questions and that was it.

I wondered if I was the only one who had been taught this while at university so I did a quick poll on Twitter and found that was not the case.

I know these numbers won’t be big enough for a representative sample but it still shows that others are familiar with this approach.

The issue with the lesson framework I mentioned above is that it does not take into account ‘performance vs learning.’

What IS performance vs learning’?

The premise is that performance in a lesson is not a good proxy for learning.

Yes, those pupils were performing well after I had modeled a mathematical procedure for them, and managed to get questions correct.

But if problem solving depends on a deep knowledge of mathematics, this approach to lesson structure is going to be very ineffective.

As mentioned earlier, the reasoning and problem solving questions were based on the same maths content as the fluency exercises, making it more likely that pupils would solve problems correctly whether they fully understood them or not.

Chances are that all they’d need to do is find the numbers in the questions and use the same method they used in the fluency section to get their answers – not exactly high level problem solving skills.

Teaching to “cover the curriculum” hinders development of strong problem solving skills.

This is one of my worries with ‘maths mastery schemes’ that block content so that, in some circumstances, it is not looked at again until the following year (and with new objectives).

The pressure for teachers to ‘get through the curriculum’ results in many opportunities to revisit content just not happening in the classroom.

Pupils are unintentionally forced to skip ahead in the fluency, reasoning, problem solving chain without proper consolidation of the earlier processes.

As David Didau (2019) puts it, ‘When novices face a problem for which they do not have a conveniently stored solution, they have to rely on the costlier means-end analysis.

This is likely to lead to cognitive overload because it involves trying to work through and hold in mind multiple possible solutions.

It’s a bit like trying to juggle five objects at once without previous practice. Solving problems is an inefficient way to get better at problem solving.’

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Fluency and reasoning – Best practice in a lesson, a unit, and a term

By now I hope you have realised that when it comes to problem solving, fluency is king. As such we should look to mastery maths based teaching to ensure that the fluency that pupils need is there.

The answer to what fluency looks like will obviously depend on many factors, including the content being taught and the year group you find yourself teaching.

But we should not consider rushing them on to problem solving or logical reasoning in the early stages of this new content as it has not been learnt, only performed.

I would say that in the early stages of learning, content that requires the end goal of being fluent should take up the majority of lesson time – approximately 60%. The rest of the time should be spent rehearsing and retrieving other knowledge that is at risk of being forgotten about.

This blog on mental maths strategies pupils should learn in each year group is a good place to start when thinking about the core aspects of fluency that pupils should achieve.

Little and often is a good mantra when we think about fluency, particularly when revisiting the key mathematical skills of number bond fluency or multiplication fluency. So when it comes to what fluency could look like throughout the day, consider all the opportunities to get pupils practicing.

They could chant multiplications when transitioning. If a lesson in another subject has finished earlier than expected, use that time to quiz pupils on number bonds. Have fluency exercises as part of the morning work.

Read more: How to teach times tables KS1 and KS2 for total recall .

What about best practice over a longer period?

Thinking about what fluency could look like across a unit of work would again depend on the unit itself.

Look at this unit below from a popular scheme of work.

They recommend 20 days to cover 9 objectives. One of these specifically mentions problem solving so I will forget about that one at the moment – so that gives 8 objectives.

I would recommend that the fluency of this unit look something like this:

LY = Last Year

This type of structure is heavily borrowed from Mark McCourt’s phased learning idea from his book ‘Teaching for Mastery.’

This should not be seen as something set in stone; it would greatly depend on the needs of the class in front of you. But it gives an idea of what fluency could look like across a unit of lessons – though not necessarily all maths lessons.

When we think about a term, we can draw on similar ideas to the one above except that your lessons could also pull on content from previous units from that term.

So lesson one may focus 60% on the new unit and 40% on what was learnt in the previous unit.

The structure could then follow a similar pattern to the one above.

Best practice for problem solving in a lesson, a unit, and a term 

When an adult first learns something new, we cannot solve a problem with it straight away. We need to become familiar with the idea and practise before we can make connections, reason and problem solve with it.

The same is true for pupils. Indeed, it could take up to two years ‘between the mathematics a student can use in imitative exercises and that they have sufficiently absorbed and connected to use autonomously in non-routine problem solving.’ (Burkhardt, 2017).

Practise with facts that are secure

So when we plan for reasoning and problem solving, we need to be looking at content from 2 years ago to base these questions on.

Now given that much of the content of the KS2 SATs will come from years 5 and 6 it can be hard to stick to this two-year idea as pupils will need to solve problems with content that can be only weeks old to them.

But certainly in other year groups, the argument could be made that content should come from previous years.

You could get pupils in Year 4 to solve complicated place value problems with the numbers they should know from Year 2 or 3. This would lessen the cognitive load, freeing up valuable working memory so they can actually focus on solving the problems using content they are familiar with.

Read more: Cognitive load theory in the classroom

Increase complexity gradually.

Once they practise solving these types of problems, they can draw on this knowledge later when solving problems with more difficult numbers.

This is what Mark McCourt calls the ‘Behave’ phase. In his book he writes:

‘Many teachers find it an uncomfortable – perhaps even illogical – process to plan the ‘Behave’ phase as one that relates to much earlier learning rather than the new idea, but it is crucial to do so if we want to bring about optimal gains in learning, understanding and long term recall.’  (Mark McCourt, 2019)

This just shows the fallacy of ‘accelerated progress’; in the space of 20 minutes some teachers are taught to move pupils from fluency through to non-routine problem solving, or we are somehow not catering to the needs of the child.

When considering what problem solving lessons could look like, here’s an example structure based on the objectives above.

Fluency, Reasoning and Problem Solving should NOT be taught by rote 

It is important to reiterate that this is not something that should be set in stone. Key to getting the most out of this teaching for mastery approach is ensuring your pupils (across abilities) are interested and engaged in their work.

Depending on the previous attainment and abilities of the children in your class, you may find that a few have come across some of the mathematical ideas you have been teaching, and so they are able to problem solve effectively with these ideas.

Equally likely is encountering pupils on the opposite side of the spectrum, who may not have fully grasped the concept of place value and will need to go further back than 2 years and solve even simpler problems.

In order to have the greatest impact on class performance, you will have to account for these varying experiences in your lessons.

Read more: 

• Maths Mastery Toolkit : A Practical Guide To Mastery Teaching And Learning
• Year 6 Maths Reasoning Questions and Answers
• Get to Grips with Maths Problem Solving KS2
• Mixed Ability Teaching for Mastery: Classroom How To
• 21 Maths Challenges To Really Stretch Your More Able Pupils
• Maths Reasoning and Problem Solving CPD Powerpoint
• Why You Should Be Incorporating Stem Sentences Into Your Primary Maths Teaching

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What makes mathematical word problem solving challenging? Exploring the roles of word problem characteristics, text comprehension, and arithmetic skills

• Original Article
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• Published: 17 December 2019
• Volume 52 , pages 33–44, ( 2020 )

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• Nonmanut Pongsakdi   ORCID: orcid.org/0000-0003-2937-6610 1 ,
• Anu Kajamies 1 , 2 ,
• Koen Veermans 1 ,
• Kalle Lertola 3 ,
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• Erno Lehtinen 1  

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In this study we investigated word-problem (WP) item characteristics, individual differences in text comprehension and arithmetic skills, and their relations to mathematical WP-solving. The participants were 891 fourth-grade students from elementary schools in Finland. Analyses were conducted in two phases. In the first phase, WP characteristics concerning linguistic and numerical factors and their difficulty level were investigated. In contrast to our expectations, the results did not show a clear connection between WP difficulty level and their other characteristics regarding linguistic and numerical factors. In the second phase, text comprehension and arithmetic skills were used to classify participants into four groups: skilful in text comprehension but poor in arithmetic; poor in text comprehension but skilful in arithmetic; very poor in both skills; very skilful in both skills. The results indicated that WP-solving performance on both easy and difficult items was strongly related to text comprehension and arithmetic skills. In easy items, the students who were poor in text comprehension but skilful in arithmetic performed better than those who were skilful in text comprehension but poor in arithmetic. However, there were no differences between these two groups in WP-solving performance on difficult items, showing that more challenging WPs require both skills from students.

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1 Introduction

Word problems (WPs) serve many purposes in mathematics education. They bring variation to the practising of basic mathematical operations and prepare students to use mathematical skills in everyday situations outside the classroom. WPs differ from other mathematical tasks that are presented in mathematical notation because the problem is laid out through text that describes a situation and a question(s) that must be answered by performing a mathematical operation(s) derived from the descriptions in the text (Verschaffel et al. 2000 ). The text comprehension theory of Kintsch and his collaborators (Kintsch 1998 ; van Dijk and Kintsch 1983 ) has been widely applied to describe processes required to solve mathematical WPs (Kintsch and Greeno 1985 ; Reusser et al. 1990 ). This theory distinguishes between textbases (networks of propositions within the text) and situation models (model of the situation described in the text) as two aspects necessary for an adequate mental representation of text (Kintsch 1998 , p. 107).

Solving WPs is complex, as the (complete) process involves a number of phases (Montague et al. 2014 ; Niss 2015 ; Verschaffel et al. 2000 ). Depaepe and colleagues ( 2015 ) reviewed descriptions of different WP-solving processes (e.g., Blum and Niss 1991 ; Burkhardt 1994 ; Mason 2001 ; Verschaffel et al. 2000 ) and concluded that, essentially, the process comprises six phases that are not necessarily performed sequentially: (1) understanding and defining the problem situation and developing a situation model; (2) developing a mathematical model based on a proper situation model; (3) working through the mathematical model to acquire mathematical results; (4) interpreting the results with respect to the original problem situation; (5) examining whether the interpreted mathematical result is appropriate and reasonable for its goal; and (6) communicating the acquired solution of the original WP. According to this description, WP-solving requires students not only to apply mathematical concepts and procedures (e.g., arithmetic relations) but also to construct a mental representation (Verschaffel et al. 2015 ) that demands different levels of text comprehension (van Dijk and Kintsch 1983 ; Reusser et al. 1990 ).

In this study, we focused on the more demanding WPs that cannot be solved without going through important phases of the complex problem-solving process summarized by Depaepe and colleagues ( 2015 ). Instead of dealing with the difficulty from the experts’ predefined point of view, we focused on the difficulty as it appears in students’ performance (see actor-oriented theory by Lobato 2012 ). This study aims to (a) explore WP-item characteristics regarding linguistic and numerical factors and their difficulty levels, using item response theory (IRT) modelling; (b) examine an association among WP-solving, text comprehension and arithmetic skills; and (c) investigate whether students with different levels of text comprehension and arithmetic skills differ in their performance on WPs at different levels of difficulty.

1.1 Linguistic and mathematical factors contributing to WP difficulty

Previous studies have investigated different factors contributing to WP difficulty. For example, in the 1980s, researchers began investigating the difficulties that students encounter when solving various WPs, starting from simple arithmetic WPs (e.g., change, combine, compare: Carpenter 1985 ; Cummins et al. 1988 ; De Corte and Verschaffel 1987 ; Greer 1987 ; Riley and Greeno 1988 ) and progressing to more complex WPs requiring non-routine thinking (Lee et al. 2014; Verschaffel and De Corte 1997 ). Based on a recent literature review by Daroczy and colleagues ( 2015 ), the factors influencing WP difficulty could be distilled into three components: linguistic factors, numerical factors, and interaction between linguistic and numerical factors (e.g., reading direction and numerical process, order of number word system).

Prior studies reported that linguistic factors, such as the number of words in the WP statement, influence the difficulty of WPs (Jerman and Rees 1972 ). However, later research has shown that superficial textual features such as the number of words hardly explains the difficulty of WPs. For example, Lepik ( 1990 ) reported different findings after investigating students’ performance on WPs. Linguistic factors, including the length of a WP statement, were not significant predictors of the proportion of correctly solved WPs but were strong predictors of the WP-solving time. In addition to this basic quantitative property of WPs, empirical evidence has convincingly shown that the semantic structure of WPs strongly impacts WP difficulty and the strategies that young children apply when solving arithmetic WPs (Cummins et al. 1988 ; De Corte and Verschaffel 1987 ; LeBlanc and Weber-Russell 1996 ; Riley and Greeno 1988 ). Semantic structure refers to meaningful relations between the known and unknown sets involved in the WP (e.g., whether the text of a simple additive WP involves a combination of two sets, a dynamic change in a start set, or a comparison between the magnitudes of two sets: see Riley and Greeno 1988 for WPs with different semantic structures). There are evidently significant differences in the probability of solution for problems both within a specific semantic type and between these semantic structure types (Cummins et al. 1988 ; De Corte and Verschaffel 1987 ). According to the literature, many young children have difficulty at the stage of comprehending sentences (Cummins 1991 ), while difficulties for older students may be more closely connected to the overall demands of arriving at an integrated representation of the situation described in text (LeBlanc and Weber-Russell 1996 ). In WPs, an additional linguistic factor is related to the role of a situation model in comprehending the meaning of the problem text. In routine WPs, it is enough to understand the propositions presented in the text, whereas in WPs requiring non-routine thinking, the construction of an adequate situation model is necessary in order to understand the problem (Kintsch 1998 ; Reusser 1992 ). Linguistic factors, such as irrelevant information and implicit information, have been found to influence students’ comprehension of WP-situations, which is essential to the construction of situation models. According to Englert and colleagues ( 1987 ), irrelevant numerical information negatively impacts students’ WP-solving, while irrelevant linguistic information did not affect their performance. Concerning implicit information, researchers suggested that many unsuccessful problem solvers often rely on the direct translation strategy (looking for numbers and keywords) and fail to provide correct answers, especially when problems include important implicit information, which they should infer on the basis of the situation described in the text (Hegarty et al. 1995 ). Another factor influencing WP difficulty, which can be seen as an extended aspect of a situation model, is the necessity of using realistic considerations requiring a non-direct translation of the situational model into a mathematical one. WPs that demand the use of realistic considerations were reported to be very difficult for many students (e.g., Verschaffel and De Corte 1997 ; Verschaffel et al. 2000 ).

Several studies have shown that numerical factors, such as number properties, required operations and number of solving steps, influence difficulty too. For example, Koedinger and Nathan ( 2004 ) investigated the effect of decimal numbers on students’ WP-solving performance. Their results indicated that whole-number problems are significantly easier than decimal-number problems. Apart from the effect of number type, the type of operation required (e.g., addition and subtraction; multiplication and division) appears to have an impact on children’s solution strategies and varies widely in difficulty (De Corte and Verschaffel 1987 ; De Corte et al. 1988 ). Various kinds of arithmetic calculation errors can result from the type of operation required (Kingsdorf and Krawec 2014 ). In addition to the required operations, the number of solving steps was reported to have an impact on WP difficulty. For instance, Quintero ( 1983 ) examined students’ problem-solving performance on WPs with a ratio and revealed that two-step ratio WPs are more difficult than single-step ones. Problems can also require mathematical thinking that goes beyond the basic arithmetic, such as combinatorial reasoning, which has proved to be difficult for children (English 2005 ).

1.2 Associations between WP-solving, text comprehension, and arithmetic skills

A substantial number of studies have examined an association between WP-solving performance, arithmetic and text comprehension skills. For example, Fuchs and colleagues ( 2006 , 2018 ) reported that arithmetic skills are related to WP-solving performance and can be seen as an essential foundation for WP-solving. However, their studies indicated that, although arithmetic skills are a necessary foundation, they are not sufficient for WP-solving given that WPs also require text processing when constructing a mental representation. Furthermore, this is evident in several studies that have found associations between WP-solving performance and text comprehension even after controlling for general cognitive abilities (e.g., working memory) or other factors that may be involved (e.g., technical reading, calculation skill, gender) (Boonen, de Koning et al. 2016 ; Boonen, van der Schoot et al. 2013 ; Swanson et al. 1993 ; Vilenius-Tuohimaa et al. 2008 ). Although the association between WP-solving performance, text comprehension, and arithmetic skills has received much attention in previous research (e.g., Boonen et al. 2013 , 2016 ; Fuchs et al. 2006 , 2018 ; Swanson et al. 1993 ; Vilenius-Tuohimaa et al. 2008 ), these studies typically investigated WP-solving performance on WPs that have simple semantic structures and did not pay any attention to the differences in the difficulty levels of WPs. There are well-established findings in the literature on how different semantic problem types have different difficulty levels (LeBlanc and Weber-Russell 1996 ) and result in different errors (Carpenter 1985 ; Cummins et al. 1988 ; Greer 1987 ; Riley and Greeno 1988 ). However, there is still a lack of studies on more demanding WPs that require non-routine thinking (e.g., including more complex structures, involving different factors contributing to their difficulty).

Various students may experience WP difficulty differently. Moreover, the effects of students’ skills and WP characteristics may be interrelated. For example, linguistically rather weak students (poor in text comprehension) may face challenges with linguistically complex WPs (e.g., including semantically complex features, long WP statements) and arithmetically rather weak students with arithmetically complex WPs (Daroczy et al. 2015 ). This assumption seems reasonable since empirical evidence has clearly shown associations between WP-solving performance, text comprehension, and arithmetic skills (Boonen et al. 2013 , 2016 ; Fuchs et al. 2006 , 2018 ; Swanson et al. 1993 ; Vilenius-Tuohimaa et al. 2008 ). However, it raises important questions as to whether it is possible to identify which WPs are linguistically or arithmetically complex and whether both features can explain the difficulty of more demanding WPs.

Our aim in this study is to deepen our understanding of the associations of linguistic and mathematical WP characteristics and WP-solving skills. Previous investigations about these associations were mostly conducted with simple arithmetic WPs (e.g., Boonen et al. 2013 ; Fuchs et al. 2006 , 2018 ; Swanson et al. 1993 ; Vilenius-Tuohimaa et al. 2008 ). In this study, the investigation was carried out with more demanding WPs and the item-level difficulty of WPs was scrutinized. This study had a focus on a joint investigation of linguistic and numerical WP characteristics and took into account students’ skills in text comprehension and arithmetic, while previous studies often focused on one or the other aspect or skill (Daroczy et al. 2015 ). In identifying the difficulty level of WPs, previous research typically relied on classical test theory (CTT) in which the proportion of individuals answering the item correctly is used as the index for the item difficulty (Finch and French 2015 ; Parkash and Kumar 2016 ; Stage 2003 ). However, item difficulty index derived from CTT is often criticized because it is dependent on the sample (Chalmers 2012 ; Stage 2003 ). A widely recommended alternative to CTT is the item response theory (IRT) modelling (De Ayala 2009 ; Finch and French 2015 ; Reckase 2009 ), in which the difficulty level estimated with IRT refers to a probability of a correct response at a given level of participant ability (Finch and French 2015 ). With IRT, it is possible to obtain item characteristics (e.g., item difficulty level) that are not dependent on the examinee group (Parkash and Kumar 2016 ; Stage 2003 ).

The present study has the aim of answering the following research questions:

Are there linguistic or mathematical features that explain the level of difficulty of various WPs?

Are students’ text comprehension, arithmetic, and WP-solving skills correlated with each other?

How do different patterns of students’ text comprehension and arithmetic skills predict their performance in WPs of different levels of difficulty?

2.1 Participants and overall design

Participants comprised 891 fourth-grade students, including 446 boys and 445 girls, from different elementary schools situated in cities, small towns, and rural communities in southern Finland. All of them had Finnish as their mother tongue. All participants were asked to complete text comprehension, arithmetic, and WP-solving tests in a classroom situation as a part of the Quest for Meaning project. The data were partly used in a previous study (Kajamies et al. 2010 ). The University of Turku’s ethical guidelines were followed. Permissions were obtained from both the schools and the students’ guardians.

2.2 Measures

2.2.1 mathematical word problems.

Students’ WP-solving performance was measured with a WP test containing 15 items (Kajamies et al. 2003 ; see Appendix 1 ). These WPs were created in such a way that they could not be solved using straightforward strategies (e.g., by requiring students to develop a proper situation model, avoiding keywords in the WPs, and giving numerical information in a written format). Two WPs (WP6 and WP13) were constructed based on original items used in earlier studies (Verschaffel et al. 2000 ). The students had no time limit to complete the WP test. All WPs were assessed by giving one point for each correct answer and zero points for an incorrect answer or no response. Cronbach’s alpha for the whole test was 0.76. The number of words, irrelevant information, implicit information, the use of realistic considerations, the required problem-solving steps and arithmetic operations, and the use of decimal numbers were all noted in order to investigate WP characteristics that may influence WP difficulty level (see Table  2 ).

2.2.2 Text comprehension

Text comprehension skills were assessed with the Finnish Standardized Reading Test (Lindeman 1998 ). The students were given 48 multiple choice questions about the four different texts they had to read. One point was awarded for each correct answer, making a maximum score of 48 for text comprehension. The Kuder–Richardson coefficient of internal consistency (CR20) was 0.87 (Lindeman 1998 ). Text comprehension was seen as an important measure of the linguistic skills of 4th graders.

2.2.3 Arithmetic skills

Arithmetic skills were measured with a time-limited (10 min) RMAT test (Räsänen 2004 ). The test begins with simple computations and ends with algebraic tasks. According to Räsänen ( 1993 ), the RMAT is comparable to the WRAT-R (Jastak and Wilkinson 1984 ). Both of them contain similar instructions, but the RMAT closely follows the Finnish mathematics curriculum (e.g., the role of fractions is not emphasized) and includes more computational tasks. Therefore, it can assess more basic arithmetic than the WRAT-R (correlations were 0.547–0.659, n  = 2673, Räsänen 2004 ). The total number of correct solutions in the RMAT is here used as an indication of the students’ arithmetic skills. The maximum score was 56. The alpha coefficient was 0.92–0.95 (Räsänen 2004 ).

2.3 Analysis

Analyses used in the present study are separated into two phases. The first phase investigated item characteristics and employed item response theory (IRT) modelling to classify WPs based on their difficulty level. The second phase used k-means clustering to categorize students into groups based on their text comprehension and arithmetic skills. In addition, the one-way analysis of variance (ANOVA) was used to determine whether students with different text comprehension and arithmetic skills differ in their performance in mathematical WP-solving.

2.3.1 Item response theory (IRT)

IRT is widely employed in assessment and evaluation research in the fields of education and psychology. IRT is an approach of testing, which is based on the relationship between participants’ performance on a particular test item and the level of his or her performance in general in all items measuring the skill in question. In technical terms, IRT attempts to model individual response patterns by determining how underlying latent trait levels (i.e., ability) interact with the item’s characteristics (e.g., item difficulty, discrimination ability) to form an expected probability of the response pattern (Chalmers 2012 ; Embretson and Reise 2000 ). In this study, IRT analyses were conducted using the R 3.2.3 with ltm (latent trait models) package, which was developed to analyze multivariate dichotomous data using latent variable models (Rizopoulos 2006 ).

One type of IRT models called 2PL model (two-parameter logistic model) was applied to investigate the difficulty level of WPs. It expresses the relationship between individuals’ level of the latent trait (his or her WP-solving ability) and the probability of endorsing a given item (answering the WP correctly) in the form of a logistic model (Finch and French 2015 ). Relative fit indices (AIC, BIC, Item fit) were examined to see whether the model fits the individual items well.

\(\theta = \text{students' ability};\,a_{j} = {\text{discrimination value of item}}\,j;\, b_{j} = {\text{difficulty level of item}}\,j.\)

2.3.2 Unidimensionality test

For selecting a suitable IRT model, the dimensionality of a set of test items has to be tested. A primary assumption underlying the 2PL model is that there is only one latent trait being measured by the set of items (unidimensionality). There are many ways to test the unidimensionality assumption (see Finch and French 2015 ; Verhelst 2002 ). One approach is to use the bootstrap modified parallel analysis test (BMPAT, Finch and Monahan 2008 ), which was developed based on Horn’s ( 1965 ) parallel analysis method for indicating the number of factors. The BMPAT works by checking the second eigenvalue of the observed data to see whether it is larger than the second eigenvalue of the data under the assumed IRT model. If the BMPAT test results are statistically significant for the second eigenvalue (p < 0.05), it means that the data are not unidimensional (Finch and French 2015 ).

2.3.3 2PL model and unidimensionality test

Table  1 shows the fit indices for the 2PL model. The results indicate that the 2PL fits all items well, and based on the BMPAT result, the observed data are unidimensional (p > 0.05), and the results support the primary assumption underlying the 2PL model.

3.1 Difficulty level and WP characteristics concerning linguistic and numerical factors

Item difficulty values estimated by the IRT-analysis and linguistic and numerical factors of WPs were examined, and the results are presented in Table  2 and Fig.  1 . According to Finch and French ( 2015 ), the item difficulty estimates are centred at 0. Therefore, the negative values indicate relatively easy items, while the positive values represent somewhat difficult items. The order of items was arranged based on their difficulty level, from the easiest item (WP1) to the most difficult one (WP13).

Item characteristic curves (ICC) of 2PL model

Overall, the results showed that the association between WP characteristics concerning linguistic and numerical factors and their difficulty level is not simple and straightforward. There was no significant correlation between the number of words and the difficulty value (r(13) = 0.21, p = 0.490). Within these WPs, the need for realistic considerations did not explain the difficulty, because the two WPs requiring realistic consideration (WP6 and WP13) were located at the different ends of the difficulty dimension. Also the existence of irrelevant linguistic or mathematical information was distributed equally with the difficulty dimension of the WPs and did not distinguish between easy and difficult WPs. However, implicit information seemed to explain WP difficulty: the eight WPs with the lowest difficulty value had no implicit information, whereas four of the five WPs with highest difficulty values (WP12, WP5, WP4, and WP10) had implicit information.

Neither the solving steps (r(13) = 0.22, p = 0.468) nor the number of possible operations (r(13) = 0.42, p = 0.157) correlated significantly with the difficulty value, but both of the WPs including decimal numbers appeared to be relatively difficult.

The success rates of WP9 (6.6%) and WP13 (7%) were extremely low, and the curves (see Fig.  1 ) suggest that the two items were very difficult. Therefore, within the model, these items’ difficulty could not be properly estimated (extremely high standard errors). None of the aspects used in the analysis (Table  2 ) explained the extreme difficulty of WP9. One explanation might be that it was the only WP that required combinatorial reasoning. WP13 required deep understanding of the real-life situation described in the problem. Because only a few students could solve these two extremely difficult WPs, they were excluded from further analyses.

For further analyses, WPs were classified as easy (WP1, WP6, WP3, WP14, and WP15) and difficult items (WP2, WP8, WP7, WP12, WP5, WP4, WP11, and WP10) based on their difficulty values in the IRT analyses. Cronbach’s alphas for each subgroup were 0.61 and 0.64, respectively.

3.2 Associations between text comprehension, arithmetic, and WP-solving skills

To investigate the interrelation among text comprehension, arithmetic, and WP-solving skills on easy and difficult items, Pearson correlations were calculated. The students who did not complete all the three tests were excluded from the analyses (N = 55). The correlation matrix is shown in Table  3 , and it revealed a significant correlation between text comprehension skills with both easy (r(836) = 0.41, p < 0.01) and difficult items (r(836) = 0.43, p < 0.01). Arithmetic skills also showed a significant correlation with both easy (r(836) = 0.52, p < 0.01) and difficult items (r(836) = 0.53, p < 0.01).

3.3 Individual differences and how those differences relate to WP-solving performance

To investigate individual differences in text comprehension and arithmetic skills and how those differences relate to WP-solving performance, first, students were categorised based on their text comprehension and arithmetic skills. K-means clustering was conducted on the z-scores of the variables. As a result, students were classified into four different groups: poor in text comprehension and arithmetic skills (N = 154); poor in text comprehension but skilful in arithmetic skills (N = 197); skilful in text comprehension but poor in arithmetic skills (N = 288); skilful in both skills (N = 197) (see Fig.  2 ). Then, ANOVAs were conducted to compare students’ text comprehension and arithmetic skills between groups. The results of the ANOVAs indicate that there was a significant difference between groups in their text comprehension (F(3, 832) = 787.666, p < 0.001) and arithmetic skills (F(3, 832) = 519.959, p < 0.001). Post-hoc tests (Bonferroni) revealed that the differences in text comprehension and arithmetic skills were significant in all group comparisons (all ps < 0.001). Descriptive information concerning text comprehension and the arithmetic skills of students in each group is presented in Table  4 .

Four clusters of students based on their text comprehension and arithmetic skills. M−L− very poor in both skills, M + L− skilful in arithmetic but poor in text comprehension, M−L + poor in arithmetic but skilful in text comprehension, M ++L ++ very skilful in both skills

Later, ANOVAs were conducted to investigate whether students in each group differ in their performance in mathematical WP-solving on easy and difficult items. The results of these ANOVAs show that there was a significant difference between groups in mathematical WP-solving performance on both easy items (F(3, 832) = 102.636, p < 0.001) and difficult items (F(3, 832) = 116.554, p < 0.001).

As shown in Table  4 , the results of post hoc comparisons using the Bonferroni test reveal that students who were very poor in both skills had the lowest mathematical WP-solving performance on both easy (M = 0.23, SD = 0.25) and difficult items (M = 0.12, SD = 0.14) (all ps < 0.001), while students who were very skilful in both skills had the highest mathematical WP-solving performance on both easy (M = 0.71, SD = 0.23) and difficult items (M = 0.53, SD = 0.22) (all ps < 0.001). In addition, students who were poor in text comprehension but skilful in arithmetic skills (M = 0.53, SD = 0.26) performed significantly better than those who were skilful in text comprehension but poor in arithmetic skills (M = 0.46, SD = 0.27) on easy WPs (p< 0.05). However, there were no differences in these groups’ performance on difficult WPs (p = 0.14), showing that more challenging WPs require students to also be competent in text comprehension.

4 General discussion

Previous empirical evidence has convincingly shown that WP-solving performance is related to both text comprehension (Boonen et al. 2013 , 2016 ; Swanson et al. 1993 ; Vilenius-Tuohimaa et al. 2008 ) and arithmetic skills (Fuchs et al. 2006 , 2018 ). However, these studies mainly examined WP-solving performance on arithmetic WPs with simple semantic structures without paying any attention to the differences in the difficulty level of WPs. This study focused on WP-solving performance when dealing with demanding WPs in which a solution to these WPs requires that students develop a proper situation model and cannot rely solely on superficial coping strategies, such as the keyword approach. The first aim of the study was to investigate WP-item characteristics regarding linguistic and numerical factors and their difficulty level, using item response theory (IRT) modelling. We wanted to find out whether the selected linguistic factors (the length of WP statement), those influencing difficulty in developing situation model (irrelevant information, implicit information, and the use of realistic considerations), and numerical factors (number properties, required operations, and number of solving steps) could explain the difficulty level of these demanding WPs. Overall, the results revealed that superficial linguistic factors did not clearly explain WP difficulty. However, WPs which required inference of implicit information belonged all to the difficult WPs. Both WPs including decimal numbers (WP7 and WP11) were difficult but other numerical factors did not predict the difficulty of WPs. These results are not surprising given that the structure and context of these demanding WPs that require non-routine thinking are fairly different, and there is no common strategy to solve these problems. Individual items seem to have unique factors in their deeper structure that contribute to the item’s difficulty level. For example, the main factor that contributes to the difficulty level of the extreme item WP13 seems to be the demand to use real-world knowledge in the modelling process (see Verschaffel and De Corte 1997 ), while the factor influencing WP difficulty for WP10 could be the difficulty in developing a situation model from the WP statement. One possible explanation for the finding that some of the superficially linguistically similar items appeared to be more difficult is that, in these difficult items, the deep structure is different. For example, the numbers needed in the calculations were not directly given and the students had to infer them from textual expressions.

The second aim of this study was to examine an association between text comprehension, arithmetic, and WP-solving skills on easy and difficult items. In line with prior studies (Boonen et al. 2013 , 2016 ; Fuchs et al. 2006 , 2018 ; Swanson et al. 1993 ; Vilenius-Tuohimaa et al. 2008 ), the results showed that text comprehension correlated with WP performance. In this study, we showed that the connection was equally strong with both easy and difficult items. Similar connections also occurred with arithmetic skills; the results indicated that there is an association between arithmetic skills and performance in both easy and difficult items.

The last aim of this study was to investigate individual differences in text comprehension and arithmetic skills and their relationships with WP-solving performance. Students were categorised, based on their text comprehension and arithmetic skills, into four different groups: very poor in both skills; poor in text comprehension but high in arithmetic skills; skilful in text comprehension but poor in arithmetic skills; very skilful in both skills. As expected, the students who were weak in both skills had the lowest mathematical WP-solving performance in both easy and difficult items, while students who were strong in both skills had the highest WP-solving performance in both easy and difficult items. In addition, students who were poor in text comprehension but strong in arithmetic skills performed better than those who were skilful in text comprehension but poor in arithmetic skills in easy WPs. However, there were no differences in the performance of these two groups on difficult WPs. This shows that arithmetic skills are needed in all WPs, but in more challenging WPs, the role of text comprehension skills becomes important as well. These results are in concordance with previous studies reporting that text comprehension skills have a prominent role in overcoming textual complexities (De Corte et al. 1985 ; De Corte et al. 1990 ), for example, when students deal with WPs containing semantically complex features (Boonen et al. 2016 ).

The main conclusion of the study was that the difficulty of WPs is not based on the surface linguistic features, but there seem to be features in the structure of the WP texts requiring deeper comprehension (Kintsch 1998 ), which can explain the differences in the levels of difficulty. However, the evidence for this conclusion is not convincing because the WPs used in this study were not systematically planned for the comparison of these features. Future studies are needed with WPs in which surface and deep structure features of the WPs are systematically varied. Further, in investigating individual differences in WP-solving performance, the study focused mainly on text comprehension and arithmetic skills, but other general cognitive abilities (e.g., working memory, motivation) were not included.

Lastly, to examine WP-solving performance and WP difficulty, the study relied merely on WP-test achievement. In future studies, more detailed observations of students’ WP-solving processes, for example, through stimulated recall interviews, are needed to understand students’ challenges in solving different demanding WPs.

5 Limitations of the study

A major limitation of the study is that in the formulation of WPs, different mathematical and linguistic features were not systematically varied. One consequence of the missing systematic design of the WPs was that it was impossible to make clear theory based sub-categories of the WPs. In addition some widely studied linguistic aspects such as different sematic structures (e.g. LeBlanc and Weber-Russell 1996 ) were not clearly represented in the WPs. Another limitation is that WP-solving takes much time and it was not possible to collect the data with a larger number of WPs. This aspect also limited the opportunity to find reliable sub-categories representing different mathematical and linguistic aspects of problems.

6 Educational implications

Mathematical WPs can be valuable content for mathematics education, and the use of WPs requiring non-routine thinking and real-world knowledge of the modelling process, instead of mere routine problems, has been recommended by many researchers (CTGV 1992 ; Mason and Scrivani 2004 ; Pongsakdi et al. 2016 ; Verschaffel and De Corte 1997 ). It is, however, important to be aware of the remarkable differences in the levels of difficulty of the WPs, which are not always self-evident for the teacher or textbook authors. Moving from routine to realistic non-routine tasks also requires novel teaching strategies and teacher scaffolding (Pongsakdi et al. 2016 ; Pongsakdi et al. 2019 ). The results of this study show that demanding non-routine WPs require high levels of text comprehension skills. Thus, practising with more demanding WPs is not only beneficial for mathematics learning but can also be an effective way to improve advanced text comprehension skills.

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Open access funding provided by University of Turku (UTU) including Turku University Central Hospital. This research is supported by the Grant 312528 of the Strategic Research Council (Academy of Finland) for the last author in affiliation with University of Turku.

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Pongsakdi, N., Kajamies, A., Veermans, K. et al. What makes mathematical word problem solving challenging? Exploring the roles of word problem characteristics, text comprehension, and arithmetic skills. ZDM Mathematics Education 52 , 33–44 (2020). https://doi.org/10.1007/s11858-019-01118-9

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Comparison Word Problems

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In these lessons, we will learn how to solve comparison word problems using either bar models or comparison bars.

Printable Word Problem Worksheets for 1st Grade: Addition Word Problems Addition/Subtraction Word Problems Comparison Word Problems

The following diagrams show the three types of comparison word problems: Difference Unknown, Unknown Big Quantity, Unknown Small Quantity. Scroll down the page for examples and solutions.

Types of Comparison Word Problems

There are three main types of comparison word problems.

Difference Unknown Connie has 15 red marbles and 28 blue marbles. How many more blue marbles than red marbles does Connie have? This is a subtraction problem. 28 - 15 = 13

Unknown Big Quantity Connie has 15 red marbles and some blue marbles. She has 13 more blue marbles than red ones. How many blue marbles does Connie have? This is an addition problem. 15 + 13 = 28

Unknown Small Quantity Connie has 28 blue marbles. She has 13 more blue marbles than red ones. How many red marbles does Connie have? This is a subtraction problem. 28 -13 = 15

How to solve comparison word problems using Bar Models or Tape Diagrams?

This video explains how to use bar modeling in Singapore math to solve word problems that deal with comparing. This technique of using model drawings to solve word problems is recommended by the Common Core mathematics standards.

Example: Adam has 11 fewer lollipops than Hope. If Adam has 16 lollipops, how may lollipops does Hope have?

Bar Model (Comparison) This video employs a visual way to solve world problems using bar modeling. This type of word problem uses the comparison model. Because the part is missing, this is a subtraction problem.

Example: Cayla did 88 sit-ups in the morning. Nekira did 32 sit-ups at night. How many more sit-ups did Cayla do than Nekira?

How to solve comparison word problems using Comparison Bars?

This is another strategy that we can use for story problems that involves comparison.

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• Ellie has 9 goldfish. Laney has 5 more goldfish than Ellie. How many goldfish does Laney have?
• Mark earned $428 doing yard work. Troy earned $186. How much less did Troy earn?
• Billy has 679 gumballs. He has 278 more gumballs than Lee. How many gumballs does Lee have?
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Comparison Bars

• Claire has 8 marbles. Sasha has 15 marbles. How many more marbles does Sasha have than Claire?
• Bill read 5 books. Beth read 2 more books than Bill. How many books did Beth read?
• Beth read 8 books. Bill read 3 fewer books than Beth. How many books did Bill read?

Example: The Nature Center has a collection of snakes. The redbelly snake in the collection is 9 inches long. The eastern ribbon snake is 21 inches long. How much longer is the eastern ribbon snake than the redbelly snake?

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Solving Math Word Problems: A Step-by-Step Guide

When tackling math word problems, it’s essential to break down the problem into manageable parts and solve it systematically. Here’s a structured approach to help you solve these problems effectively:

Two Key Steps

• Convert the words into mathematical expressions.
• Combine these expressions into a single equation.
• Follow mathematical rules to find the solution.

Tips for Success

• Start by reading the entire problem to understand the situation.
• List all the known quantities and the variables (unknowns) you need to solve for.
• Attach units of measurement to your variables (e.g., miles, gallons, inches). This helps keep track of what each variable represents.
• Clearly state what you’re trying to find, including its units.
• Write out your steps clearly. This helps you stay organized and reduces errors.
• Explain each step as you go, which can clarify your thinking and make it easier to track your progress.
• Visual aids like graphs or pictures can be very helpful. Label them clearly.
• Certain words in the problem indicate specific mathematical operations. Recognizing these can guide you in forming the correct equation.

Key Words and Operations

Different words in a problem suggest different mathematical operations. Here’s a guide:

Addition (+)

• Key Words : increased by, more than, combined together, total of, sum, added to
• What is the sum of 8 and y? → 8+y8 + y8+y
• Express the number of apples (x) increased by two. → x+2x + 2x+2
• What is the total weight of Alphie the dog (x) and Cyrus the cat (y)? → x+yx + yx+y

Subtraction (−)

• Key Words : less than, fewer than, reduced by, decreased by, difference of
• What is four less than y? → y−4y – 4y−4
• What is nine less than a number (y)? → y−9y – 9y−9
• What if the number of pizzas (x) was reduced by 6? → x−6x – 6x−6
• What is the difference between my weight (x) and your weight (y)? → x−yx – yx−y

*Multiplication (× or )

• Key Words : of, times, multiplied by
• What is y multiplied by 13? → 13y13y13y or 13×y13 \times y13×y
• Three runners averaged “y” minutes. What was their total running time? → 3y3y3y
• I drive my car at 55 miles per hour. How far will I go in “x” hours? → 55x55x55x

Division (÷ or /)

• Key Words : per, a, out of, ratio of, quotient of, percent (divide by 100)
• What is the quotient of y and 3? → y/3y/3y/3 or y÷3y ÷ 3y÷3
• Three students rent an apartment for $x per month. What will each pay? → x/3x/3x/3 or x÷3x ÷ 3x÷3
• “y” items cost a total of $25.00. What is their average cost? → 25/y25/y25/y or 25÷y25 ÷ y25÷y

Common Phrases and Their Translations

• Example: “30 miles per gallon” → 30 miles/gallon\text{30 miles}/\text{gallon}30 miles/gallon
• For example, “1.5 less than x” is x−1.5x – 1.5x−1.5, not 1.5−x1.5 – x1.5−x.
• Example: “The ratio of x and y” is x/yx/yx/y.
• Example: “The difference of x and y” is x−yx – yx−y.

Practice Examples

• Solution: 20x−6\frac{20}{x – 6}x−620​
• Solution: y+9y + 9y+9
• Solution: y+9y\frac{y + 9}{y}yy+9​
• Solution: (y+2)−9(y + 2) – 9(y+2)−9 or y−7y – 7y−7
• Solution: y+30y + 30y+30

By following these steps and recognizing key words, you’ll be better equipped to translate word problems into equations and solve them accurately.

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