problem solving vocabulary words

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Word Problem Vocabulary Sorts {FREE Vocabulary Cards}

By thisreadingmama 10 Comments

Word problems. Just the sound of that term may bring back bad memories. I know it does for me. As an elementary student, I despised word problems with ever fiber of my body. Why? Because, as a struggling reader , I struggled to read and comprehend them. I also wasn’t sure what the problem wanted me to do. It would have helped tremendously if I had studied some word problem vocabulary among other things (like some steps to problem solving).

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Word Problem Vocabulary Sorts

I’m definitely not an expert in math, but I do know that there is reading involved with word problems. And I do know that lack of vocabulary  and lack of  figuring out unknown words  leads to a breakdown in comprehension, no matter what kids are reading.

Just recently, I created some word problem vocabulary cards {FREE download at the end of this post} for my 3rd grader, ALuv. As we’ve worked through the different operations this year, we’ve talked about these terms. Now that we’re at the end of our year, I thought it was time to do a little vocabulary review {before that fun standardized end-of-the-year test}.

One of the things we did with our word cards was sort them in our tabletop pocket chart . {Seeing that it’s LEGO week , I tried to make the background of each card look like LEGO bricks .}

We also incorporated our LEGO bricks into the word sorts, too. I wrote the vocabulary words on LEGO bricks and we read and sorted them together. He really liked doing this, as he is a wee bit obsessed with LEGO bricks {or building with anything, really}.

These terms are not meant to just be memorized and sorted. They are meant to be applied to real word problems. So, we have looked for these terms as one the steps to solving word problems.

Books for Teaching Problem Solving to Kids

Since my brain is wired more for literacy, here are a couple of math books I’ve purchased and used heavily as a classroom teacher and as a homeschooling mom.

About Teaching Mathematics by Marilyn Burns is one that I devoured. I’m kind of shocked at the price of the newer edition {the 3rd edition is a little less}, but it is well worth it, in my opinion. Burns’ focus is helping kids move beyond just computation {although that is thoroughly covered} to applying that knowledge to real life situations and problems.

If you teach in the elementary grades, Introduction to Problem Solving {3-5 grades} has been a life-saver for me. After purchasing the problem solving book for PreK-2nd grades a few years ago, which can longer be purchased, it seems}, I purchased the one for 3rd through 5th grades. The problem solving steps are taught as well as a chapter on each of the problem solving strategies, like making a table, choosing an operation, or finding a pattern. By the way, if you teach PreK-2nd grades, I’d also recommend this one for you!

I have one more recommendation that I have not used, but heard good things about from a Reading in Math teacher training I did this past year, Teaching Struggling Readers to Tackle Math Word Problems for Grades 3-5. The thing that struck me about this book was how it ties the basic reading comprehension strategies with comprehending word problems.

 Download these FREE Word Problem Vocabulary Cards HERE .

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May 8, 2015 at 10:11 am

Such a cool use of blocks, great idea!

May 10, 2015 at 10:57 pm

I love the idea of using legos to review the related vocabulary terms! That’s a fun and hands-on way to practice! I actually just wrote about the importance of teaching and understanding math vocabulary so I’m excited to see this! 🙂 As always, thanks for sharing!

May 11, 2015 at 3:20 pm

So cool. I sent you an email, Bethany. 🙂

December 6, 2017 at 7:59 am

I love both ideas of lego stacking vocab words and the pocket chart for different ways to represent + and -,etc. So creative!!!!!! Thanks for sharing!

August 20, 2021 at 3:52 pm

Do you have any ideas/suggestions for a great-grandchild with dyslexia? I do not understand dyslexia and the continuing problems with reading. One day the word is known by sight, the next day it is not. Sounding out every word is time consuming and leads to complete frustration. I am concerned that “I can’t read. I hate school. I hate reading.” thoughts have developed. Any help you can send is appreciated.

May 12, 2022 at 2:06 pm

Check out my dyslexia series! You can find them on the blog. I’m also going to be releasing a dyslexia handbook soon!

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Adapting Math Word Problems for ELLs

To make word problems less confusing, especially for English language learners, change the language, not the math. Here are some ideas.

All students have a right to rigorous and challenging math classes, and word problems are a ubiquitous part of elementary and middle school math. Complex language structures or overly challenging vocabulary, however, can sometimes create barriers for students that impede access to a rigorous and challenging math curriculum. This is particularly true for English language learners (ELLs).

As teachers, we strive to cultivate mathematical reasoning and help students apply math to real-world contexts. When designing instruction for our English language learners, we have to ensure that they are afforded access to rich math tasks but also attend to the unique challenges of students working to acquire an understanding of the language. Fortunately, by attending to our vocabulary choices and sentence structure, we can adapt word problems and ensure that all students have access to rich mathematical content.

Certain linguistic features commonly found in middle school math classes are especially  problematic. Passive voice, complex sentences, and long noun phrases or clauses can be very difficult for all learners, but especially multilingual students developing English proficiency. Unfamiliar vocabulary, novel context, and poorly worded or vague questions can also create barriers to understanding. Small changes that simplify language, however, can significantly improve accessibility and ensure that more students can tackle rich math tasks.

Adapting the Math Language

Use the active voice: The passive voice can obscure what is actually happening in a word problem. Use the active voice to show people engaging with the world. For instance, rather than “The ball was thrown by the girl,” revise the sentence structure to “The girl threw the ball.”

Separate complex sentences: Break up long, convoluted, and meandering sentences to express key ideas. Consider the difference between “A hot dog costs $3.75 and a side salad costs $1.65. If a group of 5 students ordered 6 hot dogs and 4 side salads, and they left an 18% tip, how much did they pay in total, including the tip?” and the revised problem, “A group of friends ordered 6 cheeseburgers at $6.50 each and 4 side salads at $1.65 each. They left an 18% tip on the total bill. How much did they pay in total, including the tip?”

Both versions require the same mathematical understanding, but the language of the second is clearer and more accessible.

Simplify verb tense: Lean toward simple present tense. “The maintenance crew repairs the AC unit” rather than “has been repairing.”

Center people in the problem: Humanize problems with people rather than impersonal subjects. “85% of parents supported the schedule,” not “85% of the votes supported....”

Use familiar vocabulary: Swap challenging terminology for more recognizable vocabulary. “The school is hosting a fundraiser by selling concessions during the basketball tournament. If they sold 322 hamburgers at $3 each and 211 hot dogs at $2 each, what was the total revenue from the concession stand sales?”

Here’s a suggested alternative: “The school wants to raise money by selling food at a basketball game. They sold 322 hamburgers for $3 each and 211 hot dogs for $2 each. How much did the school make from selling the food?” Of course, some students will require additional supports, such as pictures and labels for key vocabulary found in word problems.

Shorten clauses: Trim unnecessary clauses. Instead of “The math tutor, who has taught for 10 years, helps students,” use “The math tutor helps students. She taught for 10 years.”

Replace obscure questions: Be sure to look for vague questions that distract from the math and substitute clear, direct questions. Change “What was the resulting amount after the chef used 16½ cups of milk?” to “The chef used 16½ cups of milk to make ice cream. Calculate how much ice cream the chef made yesterday.” Is something missing here?

Consider the big idea: Notice that in the previous example, students do not have enough information to solve the problem. When adapting math word problems for English language learners, revise the construction of your questions to clarify the task at hand, but also be mindful to simultaneously help students to think like mathematicians. To paraphrase what math education innovator Dan Meyer notes in his TED Talk on math instruction , real-world problems do not contain a simple list of all the required information.

As you adapt math instruction for English language learners, be sure to design rich experiences and help them to develop a mathematical mindset. What additional information do I need to solve this problem? What can I do to find the missing information? English language learners need accessible English, but they also need experiences that help them develop habits of inquiry, problem-solving, and self-efficacy.

The key is to adapt language without watering down rich mathematical thinking and problem-solving. Be sure to maintain high expectations while providing appropriate linguistic support. With slight modifications to ensure comprehensible and accessible language, your English language learners can tackle the same meaningful math as their peers.

Equity in math education means meeting each student where they are and helping them reach meaningful goals. Adjusting language is one path toward creating a math community that works for everyone.

Remember, context matters: Real-world contexts allow students to see math as a meaningful tool, rather than an abstract set of rules. However, take care not to introduce obscure, unfamiliar contexts that overwhelm ELLs with new vocabulary. Similarly, jumping between many different contexts in short succession can impede understanding.

When selecting contexts for word problems and examples, opt for familiar situations from students’ everyday lives that clearly illuminate the mathematical concepts. Additionally, aim to consistently revisit and reinforce the same contexts when teaching specific concepts, math models, or problem types. Repeated exposure across similar situations allows ELLs to digest both the linguistic and mathematical nuances. As comfort builds, you can broaden into new contexts, always taking care to explain unfamiliar vocabulary or scenarios that are essential to the problem.

The goal is to have students see math as meaningful while preventing contexts from distracting from the essential mathematical reasoning. Familiar, consistent contextualization keeps the focus on math concepts and problem-solving strategies.

The Power of Mathematical Models and Manipulatives

In addition to thoughtful verbal and written language adaptations, mathematical models and manipulatives provide critical visual and tactile scaffolds that support deeper understanding and reasoning for English language learners. Charts, ratio tables, coordinate planes, fraction models, graphs, algebra tiles , base-ten blocks, and more make concepts concrete while mitigating vocabulary barriers.

Leveraging models and manipulatives moves learning toward mathematical action. Students demonstrate conceptual connections nonverbally, allowing alternative pathways to develop understandings. All students access deeper thinking as teachers elevate mathematical visualization alongside precision in academic language.

When planning for math instruction and adapting for our multilingual learners, I’ve found these resources to be particularly helpful:

• Teaching Math to Multilingual Students, Grades K–8: Positioning English Learners for Success ,
• “ Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching ,” and
• Math Workshop: Five Steps to Implementing Guided Math, Learning Stations, Reflection, and More .

Ultimately, we want students to develop a deep conceptual understanding of mathematics and to grow their English language proficiency. Slight adaptations to language, the use of familiar real-world contexts, and deliberately incorporating mathematical models and manipulatives can help students to access the math curriculum and to acquire English.

We’d like to know—what strategies have you successfully used to help improve math accessibility for ELL students? Please comment and share.

Words for General IELTS - Lesson 24

Here you will learn some English words about problems and solutions, such as "damage", "tackle", "panacea", etc. that are needed for the IELTS exam.

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06.09.15 Active Practice: The Key to Vocabulary

Explicit Vocabulary instruction includes five steps:

• Careful and intentional selection of high-value words
• Framing of a definition that students can use and apply
• Guidance on ‘parameters of use,’ when and how a word occurs in language
• Active practice—time spent using the word in different settings and applications
• Maintenance and reinforcement

Of these, we believe the fourth step, active practice, is the most important. This may come as a bit of a surprise: its benefits are rarely plumbed but, as Isabel Beck et al. describe, practicing using words adeptly and accurately is the key to mastery.  We learn words by using them and seeing them over and over in different settings that are rich and challenging.  So a key part of vocabulary instruction is to put students in situations where they apply their nascent knowledge of a word. If we could make one small change to super-charge vocabulary instruction across American classrooms it would be to take the time teachers spend having students guess at word meanings and replace it with time spent using those words after having been given a definition.

This would mean replacing something like, “Who can tell us what ‘destitute’ means?” with first giving students a definition for destitute—“completely without the things needed to survive”—and asking questions in which students have to apply the word and its definition: “Who can tell me a situation that might cause someone who is rich to become destitute?”  “Good, could a person ever be destitute even if they still had money?”  “How is being destitute different from being poor?” “What’s a good way to help someone who is destitute?”

In each of these cases, the teacher would insist that the students use the word “destitute” in their answer, as in, “If you were shipwrecked on an island all by yourself with a trunk of money, you’d still be destitute, even though you’d have lots of cash.”

This process of active practice makes problem-solving a key part of vocabulary instruction. It merely places the problem-solving after a basic definition has been learned instead of asking students to make tenuous guesses of a word’s meaning, which often results in a subpar definition and surface understanding of a word. This switch results in making the problem-solving more rich and rigorous than in many typical vocabulary lessons (represented in diagram 1) where teachers spend time asking students to guess at a word’s definition, sometimes from context, sometimes from previous experience: “What does destitute mean here? Has anyone heard that word before?”

The idea is that this process is more rigorous than “just” telling students what a word means because they are problem solving. In fact, however, what students are doing is not always problem-solving. More often, they are guessing and that involves relatively low rigor, potentially only a few members of class, and can result in the perpetuation of mistaken ideas which can be just as easily be the ones students remember. In the second diagram, the definition comes early, but instead of just writing it down, students are asked to use it over and over and to problem-solve with it by applying it in various new settings. Because students start with a base of knowledge, this really is problem-solving and is likely to be both more rigorous and to result in greater mastery of the word because it results in students using and saying the word multiple times.

Again, time spent guessing a word’s meaning is far less productive than time spent using and applying the word’s meaning in increasingly complex situations, but the former tends to dominate what teachers do with new words. For example, consider whether it is more rigorous to have students answer the question, “What do you think clandestine might mean?” or something like, “How is clandestine similar to and different from  surreptitious ?” or “How could camouflage help you do something in a clandestine manner?  Could you use blaring noise to help you do something in a clandestine manner?  How?”

Active Practice to Master Meaning

Mastering meaning involves students using a word to illustrate its degree of meaning with fluidity, expertise, and a depth of understanding.  There are four different ways you could ask students to practice mastering meaning and they should be done both verbally and in writing:

• When a word would (and would not) apply. “Would it be accurate to say that Aunt Alexandra is acting like a tyrant in this scene? Explain.”
• Combining multiple new words: “Could a tyrant ever be humble? Tell me why or why not.”   
• Narrate the story:  “Can a group ever exert tyranny over another group? How? Explain how a group of people in To Kill a Mockingbird make decisions that are tyrannical?”
• Define a change: “How would it sound different if Scout said that Aunt Alexandra’s opinion was “indirectly stated” rather than “obliquely observed”? How is it different to state that Aunt Alexandra is being tyrannical as opposed to say ‘bossy”?

Active Practice to Master Usage

Asking students to practice correctly avoids students misapplying words incorrectly. It’s much harder to unlearn bad habits than it is to build new ones, so it’s a good idea to have students practicing a variety of uses. There are two ways to do this:

• Change the form: “In its adjective form we would say ‘tyrannical.’ Would Atticus agree that Aunt Alexandra is tyrannical?
• Create a sentence with the word and/or other parameters: “Write a sentence in which you describe Aunt Alexandra looking ‘obliquely’ at Scout. Be sure to describe what Scout has done to earn such a glance.”

It’s quite a common practice to ask students to create sentences with new words, but you’ll notice in the example above, that adding additional parameters adds rigor and helps to ensure that students are using a word correctly.

 Three Keys to Active Practice

Practicing incorrectly can sometimes do more harm than not practicing at all. When practicing, be sure that students are accurately applying the word. It is not uncommon for an eager student to lose the meaning of a word in his/her earnest attempts to apply the word wherever the definition seems to fit. Whether it’s active practice to master meaning or usage, there are three important keys to getting the most out of it.  The first is that you want to ensure that students actually use the new word in their answers (you would be surprised by how often we forget this!). What happens instead of using a new word is that student will use a sentence that describes a word instead.  For example, if a teacher asks a student to describe a time when they are sentimental, the student might start the sentence without using the word, “when I look at photo albums of my baby brother when he was little” – while that may demonstrate the meaning and accurate use of the word, unless the teacher pauses the student to revise their answer and use the word in their sentence, then the student doesn’t actually get practice with the word.

The second key to successful active practice is that you want students’ answers to illustrate their understanding of the word. We often hear students say something like “I detest broccoli.” Without further explanation, it’s not clear if they truly know what it means to “detest.” In this case, we want to push students a bit further to expand their sentences to illustrate the meaning of the word, as in “I detest broccoli because it’s bitter and doesn’t taste good.” Avoid questions that are mundane or obvious in order to ensure that students have to rigorously apply a word and its definition.  The result of simple fill in the blank questions, or only asking questions that ask students to use the word to describe their own experiences can have two negative outcomes.  The degree of the word may not match the context (e.g. “I was irate when my sister got ice cream before I did.”) because they match a simple fill-in-the blank formula or they may generate a surface level of understanding about a word (e.g., “I adore my teddy bear.”) without understanding its deeper meanings or connotations. Provide prompts that support students precise use of the word (e.g., “Write a sentence about why a mouse would likely despise a snake.” versus “Write a sentence using despise.”) or in which they combine similar words with slightly different meanings (e.g., “Write a sentence in which you describe something you dislike and something you detest.  Use ‘despise’ and ‘detest’ in your answer and illustrate how their meanings are slightly different.”) Asking these rigorous application questions increases the quality of their practice as well as deepens their understanding of the word and its definition.

The final key is that we also want to ask students to use different forms of the word so that they are able to practice using a word flexibly and deeply, and to consistently correct inaccurate part of speech or tense in their answers so that they are practicing using the word in its correct forms.  Do not accept for example, “The pond was scarce of water.”  If we accept incorrect forms of new vocabulary words during practice then students will most assuredly make those mistakes beyond the classroom.  Following these three keys of active practice can support teachers in asking students to rigorously apply their burgeoning vocabulary, and they are useful in giving feedback to students on their practice as they use new words and apply their meanings in a variety of contexts.

3 Responses to “Active Practice: The Key to Vocabulary”

Obviously, beginning a vocabulary lesson by asking “Who knows what ___ means?” is not a good use of time. But I would argue that giving an example of the word in context and having the students infer the meaning offers some advantages over simply giving a definition. First, it’s more rigorous that simply listening to a given definition. I understand the idea that you want teachers to follow the definition with more rigorous activities, but the entire vocabulary lesson should be rigorous. Also, it’s more typical of what happens in real vocabulary acquisition. We learn most of our vocabulary from context. When students learn how to use context to help them understand a word’s meaning, this is a skill that will help them learn words beyond the relatively few that are explicitly taught in the classroom. There is a strong research base suggesting that teaching students to infer from context is more effective than simply giving definitions.

This brings me to a concern that I have with your books. I love them. I read them and recommend them to others. But you never make use of the research that’s out there. Everything is anecdotes. Your books are great, but they could be even better if you would back up your opinions with research.

I do appreciate and benefit when supporting research is cited.

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Teaching Math Word Problem Key Words (Free Cheat Sheet)

Published: November 20, 2023

Contributor: Jeannette Tuionetoa

Disclosure: This post may contain affiliate links, meaning if you decide to make a purchase via my links, I may earn a commission at no additional cost to you. See my disclosure for more info.

Tackling word problems in math can be challenging for kids to learn. We called them story problems when I was in school. If your kids are learning math key words so they can solve word problems, they you’ll find these free cheat sheets and worksheets for word problem key words helpful. 

Math Word Problem Key Words

There is no doubt that mathematical operations using words are difficult for kids. They go from counting numbers to doing math equations with numbers.

Then all of a sudden… there are words, just words . All of a sudden algebraic expressions and mathematical operations are POOF – words.

The lack of numbers and shift in mindset can completely throw off a lot of students. If kids have difficulty with reading, then that is yet another struggle for kids as they try to learn basic problems in math.

Teaching students about challenging math keywords just got easier! Be sure to download our free math key words cheat sheet at the end of this post. 

Why do some kids struggle with word problems?

A key proponent in different operations in math is learning the key words that prompt kids to understand which operation skill they need to use to solve the problem.

This means that they should master regular math problems first and be able to read with comprehension. You will shortly find that if these two skills aren’t somewhat mastered first, then word problems will become an issue.

Many times math is a subject best taught in sequential order. If one step is missed, then the future steps falter. This is much like how it is when teaching word problems.

The best thing for your children is for them to first:

• Be able to read well.
• Understand math concepts and phrases.
• Know to not rush, but focus on math key words, identify relevant information, and understand the text.
• Get to know the keywords for math word problems

What are keywords for math word problems?

Key words in mathematical word operations are the words or phrases that will signal or show a student which type of math operation to choose in order to solve the math word problem.

The keywords for math word problems used in operations are a strategy that helps the math problem make sense and draw connections to how it can be answered.

Basically, when using key words, students must decipher whether they need to solve the math equation via addition, subtraction, multiplication, or division.

What are the common keywords for math word problems?

Thankfully, there are math key words that our children can learn that help them work through their word problems. They are prompts that point them in the right direction.

Just like a different language needs words translated for comprehension, students translate the words… into math .

Keywords for Math Word Problems

Learning these math keywords will help with problem solving:

Addition Math Key Words:

• increased by
• larger than
• in addition to
• how much in all

Subtraction Math Key Words:

• how many more
• how many less
• shorter than
• smaller than

Key Words for Multiplication Word Problems:

• multiplied by
• double/twice

Key Words used for Division Word Problems:

• equal group
• how many in each

You can print off a free math key words cheat sheet that has the above math key words for word problems and add it to your homeschool binder . Find the download link at the bottom of this post. 

How can we help kids learn keywords for solving numberless word problems?

Teach kids steps for solving word problems until it becomes a habit or they get comfortable with the steps. First, they can look for the important information and write those down. (Read the problem carefully). Next, kids need to define or find the variables in the math equation.

From the keywords, kids can now determine what math operation to use. Translate the words to math. Then, kids can solve the math equation. This is where the skills of solving numbered equations are important.

Finally, students have to put their answers in the form of a word sentence. NOTE: Many times kids think after solving the equation they are done. However, the key to making sure they understand that word problems need word answers.

Different Strategies to Familiarize Keywords in Word Operations

You can use some of these keywords for math word problems as vocabulary words in your homeschool.

Students can display subtraction, addition, multiplication, and multiplication handy reference posters on a bulletin board in your homeschool area. Students can also just list them on dry erase boards . These are perfect visual reminders for what keywords go with what math word problems.

Your students can also keep their keywords for math word problems with them as they study. They can place the list of keywords in a math folder or in an anchor chart – and then in their math folder.

Kids can keep the keyword poster sets in their math notebooks or keep them in a word problem journal .

Their strategy for learning word problem keywords all depends on how they best absorb information.

Students may do well using a combination of these methods. Either way, all of these different strategies can be used to get them comfortable in identifying the route to solve math word equations.

Math Word Problem Keywords Cheat Sheets & Teaching Aids:

We created a free pdf download Word Problem Key Words Cheat Sheet that you can find at the bottom of this post. It’s great to use as a reference for math word problems.

Word Problem Clue Words

Get a Clue Free Download – Check out these word problem clue word handouts and posters to help your students with word problems. There 5 pages in all that will be handy for your kids in trying to find the correct answer while using the correct operation.

Addition and Subtraction Word Problem Keywords

Subtraction Keywords/Addition Keywords – Until your kids memorize keywords and what they mean, this freebie can help. Grab these simple black and white printable signs. They will help kids look for keywords like larger numbers for subtraction word problems or addition keywords like in addition to . 

Story Problem Key Words

Words to Math – Keywords in math problems are essentially turning words into math. This graphic organizer printable is a quick reference for your students to use with numberless math word problems. Place them in a notebook chart or your homeschool classroom wall as a visual reminder.

Word Problem Key Words Poster

Key Word Posters for Math Problems – Grab these word problem keyword handy reference posters for subtraction, addition, division, and multiplication. Each poster has its specific theme and specific words to solve all problem types. Kids will enjoy having practiced with these math key words posters.

Word Problem Key Words Worksheets

Fun Key Word Sorting Activity – Your kids have now studied some keywords for math word problems helpful for problem solving in mathematical operations. Use this word problem sorting activity to test their knowledge in a fun engaging way. Add this fun activity to your test prep materials.

World Problems Worksheets with Key Words – These word problems worksheets use key phrases to help your students identify the phrases that will help them determine which math operation to use.

Word Problem Key Words for Math

Math word problems are probably the first opportunity students get to understand how math relates to real world situations. The applications can be relevant in their real life experiences like going to the market.

However, the benefit to word problems doesn’t stop there…

With word problems, students develop their higher-order thinking and critical thinking skills.

Different types of word problems guide your students to applying math various math concepts at the same time. They have to know basic number sense, basic algebra skills, and even geometry when they attempt multiplication word problems.

If we do it the right way, kids won’t see word problems as a dreadful experience in math. Understanding word problems is a learning curve and doesn’t come easily to kids.

Identify Learning Gaps

Another important aspect of word problems is that they tell a parent/teacher if a child needs help in areas like reading comprehension or math number operations skills. This type of word math is a great evaluation of your student’s thinking processes.

We can, however, help make it a better experience for them by teaching it the right way.

Free Math Key Words Cheat Sheet Instant Download

You won’t want to miss our free Word Problem Key Words Cheat Sheet PDF download for different ways kids see keywords in various types of problems in mathematics. This math tool is everything your student needs and the perfect resource to reference keywords in math operations.

Includes the keywords that will help your children solve and recognize word problems for:

• Subtraction
• Multiplication

Instant Download: Math Word Problems Keywords Cheat Sheet

Jeannette tuionetoa.

Jeannette is a wife, mother and homeschooling mom. She has been mightily, saved by grace and is grateful for God’s sovereignty throughout her life’s journey. She has a Bachelor in English Education and her MBA. Jeannette is bi-lingual and currently lives in the Tongan Islands of the South Pacific. She posts daily freebies for homeschoolers!

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Module 10: Linear Equations

Apply a problem-solving strategy to word problems, learning outcomes.

• Approach word problems with a positive attitude
• Use a problem solving strategy for word problems
• Translate more complex word problems into algebraic expressions and equations

 Approach Word Problems with a Positive Attitude

The world is full of word problems. How much money do I need to fill the car with gas? How much should I tip the server at a restaurant? How many socks should I pack for vacation? How big a turkey do I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? If my sister and I buy our mother a present, how much will each of us pay?

Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student in the cartoon below?

Negative thoughts about word problems can be barriers to success.

When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.

Start with a fresh slate and begin to think positive thoughts, like the student in the cartoon below. Read the positive thoughts and say them out loud.

When it comes to word problems, a positive attitude is a big step toward success.

If we take control and believe we can be successful, we will be able to master word problems.

Think of something that you can do now but couldn’t do three years ago. Whether it’s driving a car, snowboarding, cooking a gourmet meal, or speaking a new language, you have been able to learn and master a new skill. Word problems are no different. Even if you have struggled with word problems in the past, you have acquired many new math skills that will help you succeed now!

Use a Problem-Solving Strategy for Word Problems

In earlier chapters, you translated word phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. Since then, you’ve increased your math vocabulary as you learned about more algebraic procedures, and you’ve had more practice translating from words into algebra.

You have also translated word sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. You had to restate the situation in one sentence, assign a variable, and then write an equation to solve. This method works as long as the situation is familiar to you and the math is not too complicated.

Now we’ll develop a strategy you can use to solve any word problem. This strategy will help you become successful with word problems. We’ll demonstrate the strategy as we solve the following problem.

Pete bought a shirt on sale for $[latex]18[/latex], which is one-half the original price. What was the original price of the shirt?

Solution: Step 1. Read the problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don’t understand, look them up in a dictionary or on the Internet.

• In this problem, do you understand what is being discussed? Do you understand every word?

Step 2. Identify what you are looking for. It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!

• In this problem, the words “what was the original price of the shirt” tell you what you are looking for: the original price of the shirt.

Step 3. Name what you are looking for. Choose a variable to represent that quantity. You can use any letter for the variable, but it may help to choose one that helps you remember what it represents.

• Let [latex]p=[/latex] the original price of the shirt

Step 4. Translate into an equation. It may help to first restate the problem in one sentence, with all the important information. Then translate the sentence into an equation.

Step 6. Check the answer in the problem and make sure it makes sense.

• We found that [latex]p=36[/latex], which means the original price was [latex]\text{\$36}[/latex]. Does [latex]\text{\$36}[/latex] make sense in the problem? Yes, because [latex]18[/latex] is one-half of [latex]36[/latex], and the shirt was on sale at half the original price.

Step 7. Answer the question with a complete sentence.

• The problem asked “What was the original price of the shirt?” The answer to the question is: “The original price of the shirt was [latex]\text{\$36}[/latex].”

If this were a homework exercise, our work might look like this:

https://ohm.lumenlearning.com/multiembedq.php?id=142694&amp;amp;theme=oea&amp;amp;iframe_resize_id=mom1

We list the steps we took to solve the previous example.

Problem-Solving Strategy

• Read the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don’t understand, look them up in a dictionary or on the internet.
• Identify what you are looking for.  Determine the constants and variables in the problem.  A constant is a number in the problem that is not going to change.  A variable is a number that you don’t yet know its value.
• Name what you are looking for. Choose a letter to represent that quantity.
• Translate words into algebraic expressions and equations.  Write an equation to represent the problem. It may be helpful to first restate the problem in one sentence before translating.
• Solve the equation using good algebra techniques.
• Check the answer in the problem. Make sure it makes sense.
• Answer the question with a complete sentence.

Translate word problems into expressions

One of the first steps to solving word problems is converting an English sentence into a mathematical sentence. In the table below, words or phrases commonly associated with mathematical operators are categorized. Word problems often contain these or similar words, so it’s good to see what mathematical operators are associated with them.

Some examples follow:

• “[latex]x\text{ is }5[/latex]” becomes [latex]x=5[/latex]
• “Three more than a number” becomes [latex]x+3[/latex]
• “Four less than a number” becomes [latex]x-4[/latex]
• “Double the cost” becomes [latex]2\cdot\text{ cost }[/latex]
• “Groceries and gas together for the week cost $250” means [latex]\text{ groceries }+\text{ gas }=250[/latex]
• “The difference of [latex]9[/latex] and a number” becomes [latex]9-x[/latex]. Notice how [latex]9[/latex] is first in the sentence and the expression.

Let’s practice translating a few more English phrases into algebraic expressions.

Translate the table into algebraic expressions:

In this example video, we show how to translate more words into mathematical expressions.

For another review of how to translate algebraic statements into words, watch the following video.

The power of algebra is how it can help you model real situations in order to answer questions about them.  Let’s use this approach with another example.

Yash brought apples and bananas to a picnic. The number of apples was three more than twice the number of bananas. Yash brought [latex]11[/latex] apples to the picnic. How many bananas did he bring?

https://ohm.lumenlearning.com/multiembedq.php?id=142722&theme=oea&iframe_resize_id=mom2

Twenty-eight less than five times a certain number is [latex]232[/latex]. What is the number?

Following the steps provided:

• Read and understand: we are looking for a number.
• Constants and variables:  [latex]28[/latex] and [latex]232[/latex] are constants, “a certain number” is our variable, because we don’t know its value, and we are asked to find it. We will call it [latex]x[/latex].
• Translate:  five times a certain number translates to [latex]5x[/latex] Twenty-eight less than five times a certain number translates to [latex]5x-28[/latex], because subtraction is built backward. “is 232” translates to “[latex]=232″[/latex] since “is” is associated with equals.
• Write an equation:  [latex]5x-28=232[/latex]

[latex]\begin{array}{r}5x-28=232\\5x=260\\x=52\,\,\,\end{array}[/latex]

[latex]\begin{array}{r}5\left(52\right)-28=232\\5\left(52\right)=260\\260=260\end{array}[/latex]

In the video that follows, we show another example of how to translate a sentence into a mathematical expression using a problem solving method.

In the next example, we will apply our Problem-Solving Strategy to applications of percent.

Nga’s car insurance premium increased by [latex]\text{\$60}[/latex], which was [latex]\text{8%}[/latex] of the original cost. What was the original cost of the premium?

https://ohm.lumenlearning.com/multiembedq.php?id=142735&theme=oea&iframe_resize_id=mom3

https://ohm.lumenlearning.com/multiembedq.php?id=142761&theme=oea&iframe_resize_id=mom4

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• Write Algebraic Expressions from Statements: Form ax+b and a(x+b). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/Hub7ku7UHT4 . License : CC BY: Attribution
• Question ID 142694, 142722, 142735, 142761. Authored by : Lumen Learning. License : CC BY: Attribution . License Terms : IMathAS Community License, CC-BY + GPL
• Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

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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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Word of the Day

What it means.

An exodus is a situation in which many people leave a place at the same time—in other words a mass departure or emigration .

// The resort town eagerly anticipated the mass exodus from the cities to its beaches as summer approached.

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exodus in Context

“Experts link lower rents to a possible drop in demand after population losses during a recent exodus from parts of Southern California. As the state’s population has stagnated, some believe demand may cool and dampen rent growth.” — Anthony de Leon, The Los Angeles Times , 14 Mar. 2024

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The Biblical book of Exodus describes the departure of the Israelites from Egypt, so it's no surprise that the word exodus , uncapitalized, has come to refer more generally to any mass departure. Exodus was adopted into English (via Latin) from the Greek word Exodos , which literally means “the road out.” Exodos was formed by combining the prefix ex- , meaning “out of,” and hodos , meaning “road” or “way.” Indeed, many roads led out of hodos into English; other hodos descendants include episode , method , odometer , and period . While exodus is occasionally encountered in reference to an individual’s leaving (e.g., “his/her/their exodus”), such usage is likely to raise the eyebrows of editors who feel it should only refer to the departure en masse of a large group of people, as when novelist Nnedi Okorafor writes in her science fiction novel Lagoon (2015): “Everyone was trying to get somewhere, be it a church, a bar, home or out of Lagos. Then there was the exodus of people … to the parts of the city that had the least chance of flooding if the water rose too high.”

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