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Visual Models for Problem Solving in 1st Grade

May 10, 2020

As students enter 1st grade, they continue to work on math comprehension using early structures, like the Kindergarten journal we introduced last week, but now we begin to add visual models to the mix!

Let’s recap a child’s developmental journey through problem solving:

• In the early childhood years, a child needs lots of developmentally appropriate experience interacting with real objects in a physical world . 
• The physical world is captured in a quantitative picture , which young children observe and use as a springboard for mathematical conversations.
• We transition into a more structured math work mat to help young students be able to connect numbers to words and words to numbers, still using familiar situations from real life.
• The math work mat gives way to a formal math journal in Kindergarten that makes use of math comprehension skills. It provides a structure for students to explain their understanding of numbers within real world situations that will carry on throughout elementary school. 

Each of the stages of development builds on the skills developed in the previous step, so it is important that students aren’t rushed through these stages. The goal is to teach students the why behind the how so they aren’t just memorizing procedures but truly understand what is happening as they solve problems.

This 1st grade year is the last stage in the Math4Littles progression , in my opinion. After this, there isn’t much scaffolding, so we really want to carefully implement all the previous stages of problem solving before we turn the students loose, because we don’t want them to start guessing and checking. In taking students on this developmental journey, we are trying to build them a solid foundation for visual models to help them to understand problem solving. 

How We Used to Teach Problem Solving

When I was teaching 1st grade, I remember a strategy that we used for problem solving called the C.U.B.S. method. Each of those letters stood for a step in the problem solving process so students could remember what to do: C – circle the numbers, U – underline the word, B – box the operation, S – solve the problem. Seems like a simple process that gives kids a really great structure to start to understand what words problems are asking, right? But what I realized is that this strategy doesn’t hold up long term.

“Shannon has 5 lollipop and Scott has 4 more lollipops than Shannon. How many do they have all together?” 

I watched students follow that procedure with this type of problem. They circled the 4 and 5, underlined important information and put a box around the words all together , which means add because we’ve all seen the T-charts of addition/subtraction vocabulary – it says difference , it means we’re going to subtract, if you see all together , we’re going to add. But that strategy gives me 4, 5, and all together . If you go back to the question, you’ll realize the answer isn’t 9.

As I often do, I asked myself why ? Why isn’t it 9? A little more reading comprehension is required to decode that answer. The problem says I had 5 lollipops. Scott had 4 more than me, which means he also had 5. Adding that up, he had 9 and I had 5, so there were 14 all together.  

Why are we teaching kids procedures with concepts they don’t understand? Sometimes the strategies that we teach in math are conditional, meaning they only work for a certain amount of kids or a certain length of time. Then you have to worry about teaching them when to apply it and the rules for applying it, and what was meant to make things easier for students ends up being more complicated.

When we start working with strategies, I want to be able to find that vertical zip, meaning if I show you how this strategy might work in first grade, it has to work as the child gets older too so that they don’t have to learn a whole new set of strategies every year because every teacher teaches it differently. Honestly, the CUBS method would probably work for 75% of the problems in first grade. Students are doing more advanced part-whole addition problems, part-whole subtraction, part-whole missing addends, and they’ll start doing a few multi-step problems, all of which fit in the part-whole family, for which the CUBS method works well. But when you move out of that genre of problems, it falls apart. 

In the Kindergarten journal, we featured part-whole addition, part-whole subtraction, part-whole missing addend, a few problems with teen numbers, and a mixed review. The journal is very structured because it is intended to start students thinking about what they’re reading in the story problem: We have a story, a sentence form, a quick draw area, a number bond, a 10-frame, and a computation area. As they transition to 1st grade, how do we remove some of that scaffolding while still keeping it developmentally appropriate?

We have to be really careful with the way we make this transition, because very quickly, students can jump to the “circle the numbers, box the word” strategy and many times they just appeal to us because they don’t know what to do. It’s a word problem and it’s confusing, so they just add because we’re talking about adding that week. 

Additive Comparison Problems 

Additive comparison problems, where I have an amount and you have the same amount but you may have more or less than I do, are introduced after students have spent some time working on multi-step part-whole problems.

This type of problem is really a play on language, in my opinion, which makes it really confusing for kids to understand exactly what it is asking. So, we really want kids to take a step back to understand the additive comparison problems, which are coded AC in our journals. I find that building these problems with unfix cubes is a good way to start.

Let’s take this problem: Shannon has 10 pet rocks and Sherry has 4 pet rocks. How many more rocks does Shannon have than Sherry?

In some ways it seems like this might be a missing addend problem, but in fact we’re really comparing my pet rocks to Sherry’s pet rocks and we’re asking how many more does one have than the other. This really requires students to take it to the concrete level and make a bar model with unifix cubes.

I put 10 cubes to represent Shannon’s pet rocks, and then I’ll use different color cubes to show Sherry’s 4. Then, I want to compare the lengths of those two bars and figure out what the problem is really asking, which is the gap between where Sherry’s bar stops and Shannon’s bar stops. The question mark is asking for how many more does Shannon have? 

Sometimes, the language of an additive comparison problem might be reversed and say how many less does Sherry have? Since it is a play on words, which sometimes becomes confusing for students, we really need to put thought into how we go about teaching kids to do a problem like this.

Visual Models for Additive Comparison Problems

If I were to line up all the programs we work with, every one of them has bit of a different name for visual models: model drawings, tape diagrams, bar models, unit bars. We’re going to universally call them visual models for word problems. 

These aren’t the little quick draws we’ve been doing in Kindergarten because, as students get older and the problems get more complex, I’m not going to be able to draw 13 ducks and then 9 more because it will take too long! Instead, I want to put it into a visual model that has these units.

This first grade year is a transitional time where kids are going from the quick draw to what I’m going to call proportional bars, which have a length of individual cubes that are representative of the quantities we’re talking about in the problem.

I just was working with a first grade teacher last week on a Zoom call, and this teacher had not been able to attend our workshop on their campus about visual models. She, like most teachers I work with, didn’t understand why visual models were so important. She thought her students should be able to do quick draws and didn’t understand why they had to do boxes. She told me she was a big proponent of encouraging students to solve problems in different ways, so why would she possibly want to teach students a procedure like this and make them solve word problems in this way. 

After I took her through the same progression of problem solving we’ve been going through in our blog the past few weeks, she was sold! I took her up through fifth grade to help her see why it is that, in 1st grade, we’re asking students to stop doing quick draws and start to use a visual model that has a unit bar with different pieces. This proportional model is also a great transition into using a non-proportional bar.

Let’s say I had 92 pet rocks and Sherry has 45 pet rocks. A quick draw clearly won’t work for this problem, and I don’t have enough room on my paper to draw a proportional model for those numbers. But I can draw a longer bar that represents Shannon’s rocks, write in 92 rocks, and draw a shorter bar to show Sherry’s 45 rocks so I could see the proportionality. 

The hardest thing to remember when we do visual models for word problems is that it actually has nothing to do with math! We’re not actually solving the problem on the model; we are solely using a reading comprehension strategy.  

One of the biggest misconceptions we addressed when we started rolling out the 1st grade journal samples that I’ll be using in this video, was that the total doesn’t go on the line. If the problem asks for a total, we represent that in the visual model with a question mark. 

We also want to make sure that we label the visual model. For example, putting a B above the books that Erin had and an L above the books she got at the library. 

The whole point of this process is to provide a systematic way for students to work through problems that doesn’t stop working after 1st grade or when you start working on a different type of story problem. In fact, this strategy carries through multiplicative comparison problems and fractions, all the way into ratios and proportions in middle school. 

Step-by-Step Problem Solving

Read the problem. Then, have someone read it and repeat it, and every time a new piece of math information is presented, we’re going to put a chunk. So, as kids are reading the problem, they start to learn how to dissect what’s being asked. 

Not all first grade students will be able to read the story problem, but this process is modeled day after day after day in the first grade classroom, so eventually the child will become independent. 

I’m going to read a story problem: Mark has 9 strawberries, 6 of them are small. The rest are large. How many strawberries are large? 

Then, I’ll go back and read it in chunks: Mark has 9 strawberries . This is a new piece of mathematical information, so students will repeat that statement back and highlight or put a line there. The students also like to say chunk! Then we continue reading: Six of them were small. I’ll stop, repeat it, and the students say chunk! as they mark that chunk in their journals. Now we have two pieces of mathematical information. Let’s continue: The rest were large . Repeat and then chunk! So, we’ve got three sections of information that the problem has given us that we need to replicate in our visual model.  Finally, How many strawberries are large? Repeat that and then chunk!

By going through the problem slowly and methodically, students can really see these sections that they’re reading, and, as they’re going on to the subsequent steps of solving the problem, they can actually check off that they’ve included all the chunks of information in their visual model. 

In our problem, it asked me how many strawberries are large? To put it in a sentence form, I would say: Mark has ____ large strawberries. I like to say Hmm for the ____  as we’re reading it out loud.

In Kindergarten, we provide the sentence for students, leaving the blank space for their answer. But in 1st grade, we take some of the scaffolding away. It might say “There were _____ large ____” and the students have to fill in the blanks.

The sentence form is a great way to make sure that kids are comprehending what they’re reading. Generally, students in first grade have a difficult time trying to create a sentence form, because they aren’t yet developmentally ready to give you a complete answer in reading. But students will be required to do a sentence form in 2nd through 5th grade so we can be sure they understand the problems being asked, so it’s really great practice to start in 1st grade with the scaffolding.

Proportional model. We start the 1st grade year with a proportional model. We may scaffold here for the who or the what, and students will eventually start to learn what goes in that visual model. In this case, we’re talking about all of Mark’s strawberries, even though the question itself is only asking about how many of them are large.

In a proportional model, you might see the 9 squares. This is a missing addend problem so that title is going to have PWMA at the top, and there will be exactly nine squares. Some people might think that’s giving it away, but remember the goal of visual models? It’s not to solve the problem but understand what’s going on in the problem, so we’re more concerned about whether or not the student can label the drawing correctly. 

In this example, the student would total the bar at 9 and check off the first chunk of the problem that we read earlier –  Mark has nine strawberries.

The next part says “6 of them are small.” In 6 of my boxes, I’ll make six Xs, or I might make small circles, and at the top I can either write small or abbreviate with an s . 

Then it says “the rest are big.” I could label that other section of the boxes B for big, or write the whole word if I wanted. Then,  I need to put a question mark above that section between 9 (the total number of strawberries) and 6 (the number of small strawberries). That section represents the large strawberries, which is what my sentence form reminds me that I’m looking for. 

Technically, a student could just look at this easy proportional model and say there are 3 large strawberries because it’s right there in front of them. So some people might think this journal is just too easy, but at the end of the day, students are solidifying the process. They’re going back up to the problem and putting a check when they add Xs or circles for the six small strawberries. They’re putting in a check when they’ve talked about putting in the large strawberries. Then they put a question mark to show what we’re looking for. There’s a lot of detail that we’re looking for kids to have to interact with the text in math to show the comprehension. 

In some of our schools, we will do a unit bar at the bottom of the page. In the 1st grade journal we’ve created for Math4Littles, we’re going to leave the bar off and introduce the non-proportional bar a little bit later in the year. There is nothing wrong with having a model of the proportional bar and then underneath it having the non-proportional bar. In our journal, we plan to show the proportional bar, and then bring in both types of bars so that kids could see the relationship between the two. If where about this non proportional bar, where would I slice it to put the nine in? And then where’s my question mark? is it labeled? etc. 

The integral parts of visual models are: labelling the who or what, taking the bar and adjusting it based on the information that’s given, and writing in their question mark. Then it’s time to solve!

Computation. Although this step might not seem necessary because our sample problem is so simple, and to first graders after they do so many, it seems simple and both teachers and students might wonder why they’re even doing it, but I can promise that these problems will become more complex, very quickly. In our 1st grade journal, we will feature this look at the proportional bar, and then transition to having proportional and non proportional models, and then eventually just leaving it blank and having the student put in a non proportional bar to see that they can develop this progression. 

1st Grade Goals

The goal is, by the end of their first grade year, students should be able to solve problems with larger numbers and a non-proportional bar. You certainly don’t want to rush that progression. 1st grade is a really nice scaffold for students to get to that point of independence, because when we get to 2nd grade, we don’t do a whole lot of scaffolding. There are more open-ended sentences, more blanks, and students are doing more of the work.

Additionally, we want to mix up the types of problems we’re solving, give students time to understand them. You might do three days of part-whole addition to see if they can get it under their belt. Then do some part-whole subtraction, then mix the two to see if students are just following a pattern where we’re adding today or subtracting today. We want to know that they can really apply what they’re learning. Multi-step problems, where students have to add and then subtract, or vice versa, are next. Give students lots of good practice, and then mix it up again to see if they’re really following the words, or if they’re just learning a procedure. The last type of problem that we would integrate in the first grade is additive comparisons. 

Video Tutorials

In the video tutorials, you’ll see aspects of four different problems being displayed. Some will have the proportional bar, some will have the proportional and the non proportional and some just won’t have it just so you can get an overall idea of what this looks like as we go.

[yotuwp type=”playlist” id=”PL76vNL0J-a405ysBIwEwXfaMp5883yGh4″ ]

As you watch the videos, think about how you could set this up in your classroom, starting with some of the sample problems that we’re offering as a free download today. We will be releasing a full 1st grade journal soon, so stay tuned! 

Join us next week for problem solving in 2nd grade: What are the different problems that 2nd grade is going to encounter? How are journals coded?  As we start to look at how journals are coded, which you certainly could use these tutorial videos right away in your classroom or in your distance learning by thinking about story problems in a different way.

*Addition , *Subtraction , *Word Problems , Audience - Lower Elementary (K-2) , Series - Math4Littles | 0 comments

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Developing Problem-Solving Skills for Kids | Strategies & Tips

We've made teaching problem-solving skills for kids a whole lot easier! Keep reading and comment below with any other tips you have for your classroom!

Problem-Solving Skills for Kids: The Real Deal

Picture this: You've carefully created an assignment for your class. The step-by-step instructions are crystal clear. During class time, you walk through all the directions, and the response is awesome. Your students are ready! It's finally time for them to start working individually and then... 8 hands shoot up with questions. You hear one student mumble in the distance, "Wait, I don't get this" followed by the dreaded, "What are we supposed to be doing again?"

When I was a new computer science teacher, I would have this exact situation happen. As a result, I would end up scrambling to help each individual student with their problems until half the class period was eaten up. I assumed that in order for my students to learn best, I needed to be there to help answer questions immediately so they could move forward and complete the assignment.

Here's what I wish I had known when I started teaching coding to elementary students - the process of grappling with an assignment's content can be more important than completing the assignment's product. That said, not every student knows how to grapple, or struggle, in order to get to the "aha!" moment and solve a problem independently. The good news is, the ability to creatively solve problems is not a fixed skill. It can be learned by students, nurtured by teachers, and practiced by everyone!

Your students are absolutely capable of navigating and solving problems on their own. Here are some strategies, tips, and resources that can help:

Problem-Solving Skills for Kids: Student Strategies

These are strategies your students can use during independent work time to become creative problem solvers.

1. Go Step-By-Step Through The Problem-Solving Sequence 

Post problem-solving anchor charts and references on your classroom wall or pin them to your Google Classroom - anything to make them accessible to students. When they ask for help, invite them to reference the charts first.

2. Revisit Past Problems

If a student gets stuck, they should ask themself, "Have I ever seen a problem like this before? If so, how did I solve it?" Chances are, your students have tackled something similar already and can recycle the same strategies they used before to solve the problem this time around.

3. Document What Doesn’t Work

Sometimes finding the answer to a problem requires the process of elimination. Have your students attempt to solve a problem at least two different ways before reaching out to you for help. Even better, encourage them write down their "Not-The-Answers" so you can see their thought process when you do step in to support. Cool thing is, you likely won't need to! By attempting to solve a problem in multiple different ways, students will often come across the answer on their own.

4. "3 Before Me"

Let's say your students have gone through the Problem Solving Process, revisited past problems, and documented what doesn't work. Now, they know it's time to ask someone for help. Great! But before you jump into save the day, practice "3 Before Me". This means students need to ask 3 other classmates their question before asking the teacher. By doing this, students practice helpful 21st century skills like collaboration and communication, and can usually find the info they're looking for on the way.

Problem-Solving Skills for Kids: Teacher Tips

These are tips that you, the teacher, can use to support students in developing creative problem-solving skills for kids.

1. Ask Open Ended Questions

When a student asks for help, it can be tempting to give them the answer they're looking for so you can both move on. But what this actually does is prevent the student from developing the skills needed to solve the problem on their own. Instead of giving answers, try using open-ended questions and prompts. Here are some examples:

2. Encourage Grappling

Grappling  is everything a student might do when faced with a problem that does not have a clear solution. As explained in this article from Edutopia , this doesn't just mean perseverance! Grappling is more than that - it includes critical thinking, asking questions, observing evidence, asking more questions, forming hypotheses, and constructing a deep understanding of an issue.

There are lots of ways to provide opportunities for grappling. Anything that includes the Engineering Design Process is a good one! Examples include:

• Engineering or Art Projects
• Design-thinking challenges
• Computer science projects
• Science experiments

3. Emphasize Process Over Product

For elementary students, reflecting on the process of solving a problem helps them develop a growth mindset . Getting an answer "wrong" doesn't need to be a bad thing! What matters most are the steps they took to get there and how they might change their approach next time. As a teacher, you can support students in learning this reflection process.

4. Model The Strategies Yourself! 

As creative problem-solving skills for kids are being learned, there will likely be moments where they are frustrated or unsure. Here are some easy ways you can model what creative problem-solving looks and sounds like.

• Ask clarifying questions if you don't understand something
• Admit when don't know the correct answer
• Talk through multiple possible outcomes for different situations 
• Verbalize how you’re feeling when you find a problem

Practicing these strategies with your students will help create a learning environment where grappling, failing, and growing is celebrated!

Problem-Solving Skill for Kids

Did we miss any of your favorites? Comment and share them below!

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20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

Math strategies for problem-solving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.  

Problem-solving skills are essential to math in the general classroom and real-life. They require logical reasoning and critical thinking skills. Students must be equipped with strategies to help them find solutions to problems.

This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in real-life situations. 

What are problem-solving strategies?

Problem-solving strategies in math are methods students can use to figure out solutions to math problems. Some problem-solving strategies: 

• Draw a model
• Use different approaches
• Check the inverse to make sure the answer is correct

Students need to have a toolkit of math problem-solving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better. 

Strategies can help guide students to the solution when it is difficult ot know when to start.

The ultimate guide to problem solving techniques

Download these ready-to-go problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.

20 Math Strategies For Problem-Solving

Different problem-solving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem. 

Here are 20 strategies to help students develop their problem-solving skills. 

Strategies to understand the problem

Strategies that help students understand the problem before solving it helps ensure they understand: 

• The context
• What the key information is
• How to form a plan to solve it

Following these steps leads students to the correct solution and makes the math word problem easier .

Here are five strategies to help students understand the content of the problem and identify key information. 

1. Read the problem aloud

Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.

2. Highlight keywords 

When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed.  For example, if the word problem asks how many are left, the problem likely requires subtraction.  Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.

3. Summarize the information

Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary.  Summaries should include only the important information and be in simple terms that help contextualize the problem.

4. Determine the unknown

A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer.  Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.

5. Make a plan

Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it.  The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer.  Encourage students to make a list of each step they need to take to solve the problem before getting started.

Strategies for solving the problem 

1. draw a model or diagram.

Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process.  It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.

Similarly, you could draw a model to represent the objects in the problem:

2. Act it out

This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives .  When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts.  The examples below show how 1st-grade students could “act out” an addition and subtraction problem:

3. Work backwards

Working backwards is a popular problem-solving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution.  This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.

For example,

To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71.  Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.

4. Write a number sentence

When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved.  It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.

5. Use a formula

Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve.   Make sure students know the important mathematical formulas they will need in tests and real-life. It can help to display these around the classroom or, for those who need more support, on students’ desks.

Strategies for checking the solution 

Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense. 

There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work. 

2. Estimate to check for reasonableness

Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable.  Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten.  For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10.  When the estimate is clear the two numbers are close. This means your answer is reasonable. 

3. Plug-In Method

This method is particularly useful for algebraic equations. Specifically when working with variables.  To use the plug-in method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.

If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓

4. Peer Review

Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly.   Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixed-ability partners or similar-ability partners. In mixed-ability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills.  If partners have comparable ability levels and problem-solving skills, they may find that they approach problems differently or have unique insights to offer each other about the problem-solving process.

5. Use a Calculator

A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to real-world situations.  

Step-by-step problem-solving processes for your classroom

In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4-step process to solve problems. 

Polya’s 4 steps include:

• Understand the problem
• Devise a plan
• Carry out the plan

Today, in the style of George Polya, many problem-solving strategies use various acronyms and steps to help students recall. 

Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom. 

Here are 5 problem-solving strategies to introduce to students and use in the classroom.

How Third Space Learning improves problem-solving 

Resources .

Third Space Learning offers a free resource library is filled with hundreds of high-quality resources. A team of experienced math experts carefully created each resource to develop students mental arithmetic, problem solving and critical thinking. 

Explore the range of problem solving resources for 2nd to 8th grade students. 

One-on-one tutoring 

Third Space Learning offers one-on-one math tutoring to help students improve their math skills. Highly qualified tutors deliver high-quality lessons aligned to state standards. 

Former teachers and math experts write all of Third Space Learning’s tutoring lessons. Expertly designed lessons follow a “my turn, follow me, your turn” pedagogy to help students move from guided instruction and problem-solving to independent practice. 

Throughout each lesson, tutors ask higher-level thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problem-solving skills.

Problem-solving

Educators can use many different strategies to teach problem-solving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra. 

Teaching students how to choose and implement problem-solving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to real-life problem-solving.

READ MORE : 8 Common Core math examples

There are many different strategies for problem-solving; Here are 5 problem-solving strategies: • draw a model  • act it out  • work backwards  • write a number sentence • use a formula

Here are 10 strategies of problem-solving: • Read the problem aloud • Highlight keywords • Summarize the information • Determine the unknown • Make a plan • Draw a model  • Act it out  • Work backwards  • Write a number sentence • Use a formula

1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back

Some strategies you can use to solve challenging math problems are: breaking the problem into smaller parts, using diagrams or models, applying logical reasoning, and trying different approaches.

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Problem Solving Activities: 7 Strategies

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Problem solving can be a daunting aspect of effective mathematics teaching, but it does not have to be! In this post, I share seven strategic ways to integrate problem solving into your everyday math program.

In the middle of our problem solving lesson, my district math coordinator stopped by for a surprise walkthrough. 

I was so excited!

We were in the middle of what I thought was the most brilliant math lesson– teaching my students how to solve problem solving tasks using specific problem solving strategies. 

It was a proud moment for me!

Each week, I presented a new problem solving strategy and the students completed problems that emphasized the strategy. 

Genius right? 

After observing my class, my district coordinator pulled me aside to chat. I was excited to talk to her about my brilliant plan, but she told me I should provide the tasks and let my students come up with ways to solve the problems. Then, as students shared their work, I could revoice the student’s strategies and give them an official name. 

What a crushing blow! Just when I thought I did something special, I find out I did it all wrong. 

I took some time to consider her advice. Once I acknowledged she was right, I was able to make BIG changes to the way I taught problem solving in the classroom. 

When I Finally Saw the Light

To give my students an opportunity to engage in more authentic problem solving which would lead them to use a larger variety of problem solving strategies, I decided to vary the activities and the way I approached problem solving with my students. 

Problem Solving Activities

Here are seven ways to strategically reinforce problem solving skills in your classroom. 

Seasonal Problem Solving

Many teachers use word problems as problem solving tasks. Instead, try engaging your students with non-routine tasks that look like word problems but require more than the use of addition, subtraction, multiplication, and division to complete. Seasonal problem solving tasks and daily challenges are a perfect way to celebrate the season and have a little fun too!

Cooperative Problem Solving Tasks

Go cooperative! If you’ve got a few extra minutes, have students work on problem solving tasks in small groups. After working through the task, students create a poster to help explain their solution process and then post their poster around the classroom. Students then complete a gallery walk of the posters in the classroom and provide feedback via sticky notes or during a math talk session.

Notice and Wonder

Before beginning a problem solving task, such as a seasonal problem solving task, conduct a Notice and Wonder session. To do this, ask students what they notice about the problem. Then, ask them what they wonder about the problem. This will give students an opportunity to highlight the unique characteristics and conditions of the problem as they try to make sense of it. 

Want a better experience? Remove the stimulus, or question, and allow students to wonder about the problem. Try it! You’ll gain some great insight into how your students think about a problem.

Math Starters

Start your math block with a math starter, critical thinking activities designed to get your students thinking about math and provide opportunities to “sneak” in grade-level content and skills in a fun and engaging way. These tasks are quick, designed to take no more than five minutes, and provide a great way to turn-on your students’ brains. Read more about math starters here ! 

Create your own puzzle box! The puzzle box is a set of puzzles and math challenges I use as fast finisher tasks for my students when they finish an assignment or need an extra challenge. The box can be a file box, file crate, or even a wall chart. It includes a variety of activities so all students can find a challenge that suits their interests and ability level.

Calculators

Use calculators! For some reason, this tool is not one many students get to use frequently; however, it’s important students have a chance to practice using it in the classroom. After all, almost everyone has access to a calculator on their cell phones. There are also some standardized tests that allow students to use them, so it’s important for us to practice using calculators in the classroom. Plus, calculators can be fun learning tools all by themselves!

Three-Act Math Tasks

Use a three-act math task to engage students with a content-focused, real-world problem! These math tasks were created with math modeling in mind– students are presented with a scenario and then given clues and hints to help them solve the problem. There are several sites where you can find these awesome math tasks, including Dan Meyer’s Three-Act Math Tasks and Graham Fletcher’s 3-Acts Lessons . 

Getting the Most from Each of the Problem Solving Activities

When students participate in problem solving activities, it is important to ask guiding, not leading, questions. This provides students with the support necessary to move forward in their thinking and it provides teachers with a more in-depth understanding of student thinking. Selecting an initial question and then analyzing a student’s response tells teachers where to go next. 

Ready to jump in? Grab a free set of problem solving challenges like the ones pictured using the form below. 

Which of the problem solving activities will you try first? Respond in the comments below.

Shametria Routt Banks

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2 Responses

This is a very cool site. I hope it takes off and is well received by teachers. I work in mathematical problem solving and help prepare pre-service teachers in mathematics.

Thank you, Scott! Best wishes to you and your pre-service teachers this year!

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Problem-Solving Strategies

October 16, 2019

There are many different ways to solve a math problem, and equipping students with problem-solving strategies is just as important as teaching computation and algorithms. Problem-solving strategies help students visualize the problem or present the given information in a way that can lead them to the solution. Solving word problems using strategies works great as a number talks activity and helps to revise many skills.

Problem-solving strategies

1. create a diagram/picture, 2. guess and check., 3. make a table or a list., 4. logical reasoning., 5. find a pattern, 6. work backward, 1. create a diagram/draw a picture.

Creating a diagram helps students visualize the problem and reach the solution. A diagram can be a picture with labels, or a representation of the problem with objects that can be manipulated. Role-playing and acting out the problem like a story can help get to the solution.

Alice spent 3/4 of her babysitting money on comic books. She is left with $6. How much money did she make from babysitting?

2. Guess and check

Teach students the same strategy research mathematicians use.

With this strategy, students solve problems by making a reasonable guess depending on the information given. Then they check to see if the answer is correct and they improve it accordingly.  By repeating this process, a student can arrive at a correct answer that has been checked. It is recommended that the students keep a record of their guesses by making a chart, a table or a list. This is a flexible strategy that works for many types of problems. When students are stuck, guessing and checking helps them start and explore the problem. However, there is a trap. Exactly because it is such a simple strategy to use, some students find it difficult to consider other strategies. As problems get more complicated, other strategies become more important and more effective.

Find two numbers that have sum 11 and product 24.

Try/guess  5 and 6  the product is 30 too high

  adjust  to 4 and 7 with product 28 still high

  adjust  again 3 and 8 product 24

3. Make a table or a list

Carefully organize the information on a table or list according to the problem information. It might be a table of numbers, a table with ticks and crosses to solve a logic problem or a list of possible answers. Seeing the given information sorted out on a table or a list will help find patterns and lead to the correct solution.

To make sure you are listing all the information correctly read the problem carefully.

Find the common factors of 24, 30 and 18

Logical reasoning is the process of using logical, systemic steps to arrive at a conclusion based on given facts and mathematic principles. Read and understand the problem. Then find the information that helps you start solving the problem. Continue with each piece of information and write possible answers.

Thomas, Helen, Bill, and Mary have cats that are black, brown, white, or gray. The cats’ names are Buddy, Lucky, Fifi, and Moo. Buddy is brown. Thoma’s cat, Lucky, is not gray. Helen’s cat is white but is not named Moo. The gray cat belongs to Bill. Which cat belongs to each student, and what is its color?

A table or list is useful in solving logic problems.

Since Lucky is not gray it can be black or brown. However, Buddy is brown so Lucky has to be black.

Buddy is brown so it cannot be Helen’s cat. Helen’s cat cannot be Moo, Buddy or Lucky, so it is Fifi.

Therefore, Moo is Bill’s cat and Buddy is Mary’s cat.

5. Find a pattern.

Finding a pattern is a strategy in which students look for patterns in the given information in order to solve the problem. When the problem consists of data like numbers or events that are repeated then it can be solved using the “find a pattern” problem-solving strategy. Data can be organized in a table or a list to reveal the pattern and help discover the “rule” of the pattern.

 The “rule” can then be used to find the answer to the question and complete the table/list.

Shannon’s Pizzeria made 5 pizzas on Sunday, 10 pizzas on Monday, 20 pizzas on Tuesday, and 40 pizzas on Wednesday. If this pattern continues, how many pizzas will the pizzeria make on Saturday?

6. Working backward

Problems that can be solved with this strategy are the ones that  list a series of events or a sequence of steps .

In this strategy, the students must start with the solution and work back to the beginning. Each operation must be reversed to get back to the beginning. So if working forwards requires addition, when students work backward they will need to subtract. And if they multiply working forwards, they must divide when working backward.

Mom bought a box of candy. Mary took 5 of them, Nick took 4 of them and 31 were given out on Halloween night. The next morning they found 8 pieces of candy in the box. How many candy pieces were in the box when mom bought it.

For this problem, we know that the final number of candy was 8, so if we work backward to “put back” the candy that was taken from the box we can reach the number of candy pieces that were in the box, to begin with.

The candy was taken away so we will normally subtract them. However, to get back to the original number of candy we need to work backward and do the opposite, which is to add them.

8 candy pieces were left + the 31 given out + plus the ones Mary took + the ones Nick took

8+31+5+4= 48   Answer: The box came with 48 pieces of candy.

Selecting the best strategy for a problem comes with practice and often problems will require the use of more than one strategies.

Print and digital activities

I have created a collection of print and digital activity cards and worksheets with word problems (print and google slides) to solve using the strategies above. The collection includes 70 problems (5 challenge ones) and their solution s and explanations.

sample below

How to use the activity cards

Allow the students to use manipulatives to solve the problems. (counters, shapes, lego blocks, Cuisenaire blocks, base 10 blocks, clocks) They can use manipulatives to create a picture and visualize the problem. They can use counters for the guess and check strategy. Discuss which strategy/strategies are better for solving each problem. Discuss the different ways. Use the activities as warm-ups, number talks, initiate discussions, group work, challenge, escape rooms, and more.

Ask your students to write their own problems using the problems in this resource, and more, as examples. Start with a simple type. Students learn a lot when trying to compose a problem. They can share the problem with their partner or the whole class. Make a collection of problems to share with another class.

For the google slides the students can use text boxes to explain their thinking with words, add shapes and lines to create diagrams, and add (insert) tables and diagrams.

Many of the problems can be solved faster by using algebraic expressions. However, since I created this resource for grades 4 and up I chose to show simple conceptual ways of solving the problems using the strategies above. You can suggest different ways of solving the problems based on the grade level.

Find the free and premium versions of the resource below. The premium version includes 70 problems (challenge problems included) and their solutions

There are 2 versions of the resource

70 google slides with explanations + 70 printable task cards

70 google slides with explanations + 11 worksheets

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6 Tips for Teaching Math Problem-Solving Skills

Solving word problems is tougher than computing with numbers, but elementary teachers can guide students to do the deep thinking involved.

A growing concern with students is the ability to problem-solve, especially with complex, multistep problems. Data shows that students struggle more when solving word problems than they do with computation , and so problem-solving should be considered separately from computation. Why?

Consider this. When we’re on the way to a new destination and we plug in our location to a map on our phone, it tells us what lane to be in and takes us around any detours or collisions, sometimes even buzzing our watch to remind us to turn. When I experience this as a driver, I don’t have to do the thinking. I can think about what I’m going to cook for dinner, not paying much attention to my surroundings other than to follow those directions. If I were to be asked to go there again, I wouldn’t be able to remember, and I would again seek help.

If we can switch to giving students strategies that require them to think instead of giving them too much support throughout the journey to the answer, we may be able to give them the ability to learn the skills to read a map and have several ways to get there.

Here are six ways we can start letting students do this thinking so that they can go through rigorous problem-solving again and again, paving their own way to the solution. 

1. Link problem-solving to reading

When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools like counters or base 10 blocks, drawing a quick sketch of the problem, retelling the story in their own words, etc., can really help them to utilize the skills they already have to make the task less daunting.

We can break these skills into specific short lessons so students have a bank of strategies to try on their own. Here's an example of an anchor chart that they can use for visualizing . Breaking up comprehension into specific skills can increase student independence and help teachers to be much more targeted in their problem-solving instruction. This allows students to build confidence and break down the barriers between reading and math to see they already have so many strengths that are transferable to all problems.

2. Avoid boxing students into choosing a specific operation

It can be so tempting to tell students to look for certain words that might mean a certain operation. This might even be thoroughly successful in kindergarten and first grade, but just like when our map tells us where to go, that limits students from becoming deep thinkers. It also expires once they get into the upper grades, where those words could be in a problem multiple times, creating more confusion when students are trying to follow a rule that may not exist in every problem.

We can encourage a variety of ways to solve problems instead of choosing the operation first. In first grade, a problem might say, “Joceline has 13 stuffed animals and Jordan has 17. How many more does Jordan have?” Some students might choose to subtract, but a lot of students might just count to find the amount in between. If we tell them that “how many more” means to subtract, we’re taking the thinking out of the problem altogether, allowing them to go on autopilot without truly solving the problem or using their comprehension skills to visualize it. 

3. Revisit ‘representation’

The word “representation” can be misleading. It seems like something to do after the process of solving. When students think they have to go straight to solving, they may not realize that they need a step in between to be able to support their understanding of what’s actually happening in the problem first.

Using an anchor chart like one of these ( lower grade , upper grade ) can help students to choose a representation that most closely matches what they’re visualizing in their mind. Once they sketch it out, it can give them a clearer picture of different ways they could solve the problem.

Think about this problem: “Varush went on a trip with his family to his grandmother’s house. It was 710 miles away. On the way there, three people took turns driving. His mom drove 214 miles. His dad drove 358 miles. His older sister drove the rest. How many miles did his sister drive?”

If we were to show this student the anchor chart, they would probably choose a number line or a strip diagram to help them understand what’s happening.

If we tell students they must always draw base 10 blocks in a place value chart, that doesn’t necessarily match the concept of this problem. When we ask students to match our way of thinking, we rob them of critical thinking practice and sometimes confuse them in the process. 

4. Give time to process

Sometimes as educators, we can feel rushed to get to everyone and everything that’s required. When solving a complex problem, students need time to just sit with a problem and wrestle with it, maybe even leaving it and coming back to it after a period of time.

This might mean we need to give them fewer problems but go deeper with those problems we give them. We can also speed up processing time when we allow for collaboration and talk time with peers on problem-solving tasks. 

5. Ask questions that let Students do the thinking

Questions or prompts during problem-solving should be very open-ended to promote thinking. Telling a student to reread the problem or to think about what tools or resources would help them solve it is a way to get them to try something new but not take over their thinking.

These skills are also transferable across content, and students will be reminded, “Good readers and mathematicians reread.” 

6. Spiral concepts so students frequently use problem-solving skills

When students don’t have to switch gears in between concepts, they’re not truly using deep problem-solving skills. They already kind of know what operation it might be or that it’s something they have at the forefront of their mind from recent learning. Being intentional within their learning stations and assessments about having a variety of rigorous problem-solving skills will refine their critical thinking abilities while building more and more resilience throughout the school year as they retain content learning in the process. 

Problem-solving skills are so abstract, and it can be tough to pinpoint exactly what students need. Sometimes we have to go slow to go fast. Slowing down and helping students have tools when they get stuck and enabling them to be critical thinkers will prepare them for life and allow them multiple ways to get to their own destination.

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Maneuvering the Middle

Student-Centered Math Lessons

Math Problem Solving Strategies

How many times have you been teaching a concept that students are feeling confident in, only for them to completely shut down when faced with a word problem?  For me, the answer is too many to count.  Word problems require problem solving strategies. And more than anything, word problems require decoding, eliminating extra information, and opportunities for students to solve for something that the question is not asking for .  There are so many places for students to make errors! Let’s talk about some problem solving strategies that can help guide and encourage students!

1. C.U.B.E.S.

C.U.B.E.S stands for circle the important numbers, underline the question, box the words that are keywords, eliminate extra information, and solve by showing work.  

• Why I like it: Gives students a very specific ‘what to do.’
• Why I don’t like it: With all of the annotating of the problem, I’m not sure that students are actually reading the problem.  None of the steps emphasize reading the problem but maybe that is a given.

2. R.U.N.S.

R.U.N.S. stands for read the problem, underline the question, name the problem type, and write a strategy sentence. 

• Why I like it: Students are forced to think about what type of problem it is (factoring, division, etc) and then come up with a plan to solve it using a strategy sentence.  This is a great strategy to teach when you are tackling various types of problems.
• Why I don’t like it: Though I love the opportunity for students to write in math, writing a strategy statement for every problem can eat up a lot of time.

3. U.P.S. CHECK

U.P.S. Check stands for understand, plan, solve, and check.

• Why I like it: I love that there is a check step in this problem solving strategy.  Students having to defend the reasonableness of their answer is essential for students’ number sense.
• Why I don’t like it: It can be a little vague and doesn’t give concrete ‘what to dos.’ Checking that students completed the ‘understand’ step can be hard to see.

4. Maneuvering the Middle Strategy AKA K.N.O.W.S.

Here is the strategy that I adopted a few years ago.  It doesn’t have a name yet nor an acronym, (so can it even be considered a strategy…?)

UPDATE: IT DOES HAVE A NAME! Thanks to our lovely readers, Wendi and Natalie!

• Know: This will help students find the important information.
• Need to Know: This will force students to reread the question and write down what they are trying to solve for.
• Organize:   I think this would be a great place for teachers to emphasize drawing a model or picture.
• Work: Students show their calculations here.
• Solution: This is where students will ask themselves if the answer is reasonable and whether it answered the question.

Ideas for Promoting Showing Your Work

• White boards are a helpful resource that make (extra) writing engaging!
• Celebrating when students show their work. Create a bulletin board that says ***I showed my work*** with student exemplars.
• Take a picture that shows your expectation for how work should look and post it on the board like Marissa did here.

Show Work Digitally

Many teachers are facing how to have students show their work or their problem solving strategy when tasked with submitting work online. Platforms like Kami make this possible. Go Formative has a feature where students can use their mouse to “draw” their work. 

If you want to spend your energy teaching student problem solving instead of writing and finding math problems, look no further than our All Access membership . Click the button to learn more. 

Students who plan succeed at a higher rate than students who do not plan.   Do you have a go to problem solving strategy that you teach your students? 

Editor’s Note: Maneuvering the Middle has been publishing blog posts for nearly 8 years! This post was originally published in September of 2017. It has been revamped for relevancy and accuracy.

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Reader Interactions

18 comments.

October 4, 2017 at 7:55 pm

As a reading specialist, I love your strategy. It’s flexible, “portable” for any problem, and DOES get kids to read and understand the problem by 1) summarizing what they know and 2) asking a question for what they don’t yet know — two key comprehension strategies! How about: “Make a Plan for the Problem”? That’s the core of your rationale for using it, and I bet you’re already saying this all the time in class. Kids will get it even more because it’s a statement, not an acronym to remember. This is coming to my reading class tomorrow with word problems — thank you!

October 4, 2017 at 8:59 pm

Hi Nora! I have never thought about this as a reading strategy, genius! Please let me know how it goes. I would love to hear more!

December 15, 2017 at 7:57 am

Hi! I am a middle school teacher in New York state and my district is “gung ho” on CUBES. I completely agree with you that kids are not really reading the problem when using CUBES and only circling and boxing stuff then “doing something” with it without regard for whether or not they are doing the right thing (just a shot in the dark!). I have adopted what I call a “no fear word problems” procedure because several of my students told me they are scared of word problems and I thought, “let’s take the scary out of it then by figuring out how to dissect it and attack it! Our class strategy is nearly identical to your strategy:

1. Pre-Read the problem (do so at your normal reading speed just so you basically know what it says) 2. Active Read: Make a short list of: DK (what I Definitely Know), TK (what I Think I Know and should do), and WK (what I Want to Know– what is the question?) 3. Draw and Solve 4. State the answer in a complete sentence.

This procedure keep kids for “surfacely” reading and just trying something that doesn’t make sense with the context and implications of the word problem. I adapted some of it from Harvey Silver strategies (from Strategic Teacher) and incorporated the “Read-Draw-Write” component of the Eureka Math program. One thing that Harvey Silver says is, “Unlike other problems in math, word problems combine quantitative problem solving with inferential reading, and this combination can bring out the impulsive side in students.” (The Strategic Teacher, page 90, Silver, et al.; 2007). I found that CUBES perpetuates the impulsive side of middle school students, especially when the math seems particularly difficult. Math word problems are packed full of words and every word means something to about the intent and the mathematics in the problem, especially in middle school and high school. Reading has to be done both at the literal and inferential levels to actually correctly determine what needs to be done and execute the proper mathematics. So far this method is going really well with my students and they are experiencing higher levels of confidence and greater success in solving.

October 5, 2017 at 6:27 am

Hi! Another teacher and I came up with a strategy we call RUBY a few years ago. We modeled this very closely after close reading strategies that are language arts department was using, but tailored it to math. R-Read the problem (I tell kids to do this without a pencil in hand otherwise they are tempted to start underlining and circling before they read) U-Underline key words and circle important numbers B-Box the questions (I always have student’s box their answer so we figured this was a way for them to relate the question and answer) Y-You ask yourself: Did you answer the question? Does your answer make sense (mathematically)

I have anchor charts that we have made for classrooms and interactive notebooks if you would like them let me me know….

October 5, 2017 at 9:46 am

Great idea! Thanks so much for sharing with our readers!

October 8, 2017 at 6:51 pm

LOVE this idea! Will definitely use it this year! Thank you!

December 18, 2019 at 7:48 am

I would love an anchor chart for RUBY

October 15, 2017 at 11:05 am

I will definitely use this concept in my Pre-Algebra classes this year; I especially like the graphic organizer to help students organize their thought process in solving the problems too.

April 20, 2018 at 7:36 am

I love the process you’ve come up with, and think it definitely balances the benefits of simplicity and thoroughness. At the risk of sounding nitpicky, I want to point out that the examples you provide are all ‘processes’ rather than strategies. For the most part, they are all based on the Polya’s, the Hungarian mathematician, 4-step approach to problem solving (Understand/Plan/Solve/Reflect). It’s a process because it defines the steps we take to approach any word problem without getting into the specific mathematical ‘strategy’ we will use to solve it. Step 2 of the process is where they choose the best strategy (guess and check, draw a picture, make a table, etc) for the given problem. We should start by teaching the strategies one at a time by choosing problems that fit that strategy. Eventually, once they have added multiple strategies to their toolkit, we can present them with problems and let them choose the right strategy.

June 22, 2018 at 12:19 pm

That’s brilliant! Thank you for sharing!

May 31, 2018 at 12:15 pm

Mrs. Brack is setting up her second Christmas tree. Her tree consists of 30% red and 70% gold ornaments. If there are 40 red ornaments, then how many ornaments are on the tree? What is the answer to this question?

June 22, 2018 at 10:46 am

Whoops! I guess the answer would not result in a whole number (133.333…) Thanks for catching that error.

July 28, 2018 at 6:53 pm

I used to teach elementary math and now I run my own learning center, and we teach a lot of middle school math. The strategy you outlined sounds a little like the strategy I use, called KFCS (like the fast-food restaurant). K stands for “What do I know,” F stands for “What do I need to Find,” C stands for “Come up with a plan” [which includes 2 parts: the operation (+, -, x, and /) and the problem-solving strategy], and lastly, the S stands for “solve the problem” (which includes all the work that is involved in solving the problem and the answer statement). I find the same struggles with being consistent with modeling clearly all of the parts of the strategy as well, but I’ve found that the more the student practices the strategy, the more intrinsic it becomes for them; of course, it takes a lot more for those students who struggle with understanding word problems. I did create a worksheet to make it easier for the students to follow the steps as well. If you’d like a copy, please let me know, and I will be glad to send it.

February 3, 2019 at 3:56 pm

This is a supportive and encouraging site. Several of the comments and post are spot on! Especially, the “What I like/don’t like” comparisons.

March 7, 2019 at 6:59 am

Have you named your unnamed strategy yet? I’ve been using this strategy for years. I think you should call it K.N.O.W.S. K – Know N – Need OW – (Organise) Plan and Work S – Solution

September 2, 2019 at 11:18 am

Going off of your idea, Natalie, how about the following?

K now N eed to find out O rganize (a plan – may involve a picture, a graphic organizer…) W ork S ee if you’re right (does it make sense, is the math done correctly…)

I love the K & N steps…so much more tangible than just “Read” or even “Understand,” as I’ve been seeing is most common in the processes I’ve been researching. I like separating the “Work” and “See” steps. I feel like just “Solve” May lead to forgetting the checking step.

March 16, 2020 at 4:44 pm

I’m doing this one. Love it. Thank you!!

September 17, 2019 at 7:14 am

Hi, I wanted to tell you how amazing and kind you are to share with all of us. I especially like your word problem graphic organizer that you created yourself! I am adopting it this week. We have a meeting with all administrators to discuss algebra. I am going to share with all the people at the meeting.

I had filled out the paperwork for the number line. Is it supposed to go to my email address? Thank you again. I am going to read everything you ahve given to us. Have a wonderful Tuesday!

Find Fun Summer Practice in the shop! →

Tips for Teaching Word Problems in 1st Grade

Learning how to solve word problems is an essential skill for first graders!  It’s also one of the more challenging math concepts that our young learners practice in first grade.  In this post, I want to share some helpful tips for teaching word problems in 1st grade.  With these tips and resources, you’ll be able to keep your students engaged in Word Problem Practice all year long!

Teaching Word Problems: 1st Grade Classroom

Understanding and solving word problems can help your students see that math is all around them every day.  Plus, there are many other skills that your first graders can practice as they solve story problems.

• Reading Skills:   First graders are growing in their reading skills by leaps and bounds all year!  Word problems are an excellent way to add some reading practice to other pockets of your daily routine. Not only are students putting their decoding skills to work, but they are also stretching those reading comprehension muscles!
• Fine Motor Practice:  While your young students have come very far in their fine motor skills since entering kindergarten, they likely need continued practice in this area.  Using manipulatives to model and solve story problems can be a great way for students to practice their hand-eye coordination, pincer grasp, and more!
• Math Strategies:   Students learn many addition and subtraction strategies in kindergarten!  However, if they don’t continue to use those strategies in first grade, they might forget about them!  Word problem practice is a great time to remind students to try different math strategies.

As teachers, we can see the importance of word problem practice for our young learners.  However, they might not be as excited about them! Here are some tips for teaching word problems in a way that keeps your students engaged in this important practice.

1. Dissect Word Problems Together

Since first graders are still developing their reading comprehension skills, they might draw a total blank after reading a math word problem on their own.  Instead, be sure to practice dissecting word problems together as a class!  Highlighters, crayons, or even see-through colored math counters can mark the important information in the problem.  You could also teach students hand signals to hold up when they hear important information in a word problem that’s being read as a whole class.  There are many ways that you can make the process of dissecting word problems more engaging and memorable for your students!

2. Spread It Out

A great way to bring variety to your word problem practice is to incorporate math story problems throughout your daily routine!  You could have a story problem written on the whiteboard for your morning meeting. You can also incorporate story problem practice into your math rotations!  When you find yourself with extra time before your next transition, you could read and dissect a story problem together as a class. You might even see an opportunity to do a quick math story problem from a book you’re reading out loud to your students! 

3. Incorporate Movement

Story problems can also be used for meaningful movement breaks in the classroom.  You could display a word problem and have students take turns playing the part of the objects.  Students love to be seashells, bubbles, and butterflies! 

4. Have Students Write Their Own Word Problems

As your first graders become more familiar with the structure of story problems, they’ll be ready to write their own!  You can use a template with sentence frames for students who need that level of support.  As students grow and develop their writing skills, they might want to try their hand at writing and illustrating their own word problems.  

No matter which method you use to have students write their own 1st grade word problems, be sure to use the final product in some way.  You could put the papers in a pile and choose a few at a time to discuss as a class.  Depending on your students, you could also recirculate the completed papers and have each student solve a word problem written by a classmate. You could even put together a class book of student-generated word problems! Just put the pages in sheet protectors (or laminate) and have students use dry-erase markers to solve the problem on each page.

5. Use Seasonal Story Problems

One of my favorite ways to keep students engaged in story problem practice is to use seasonal problems.  Not only do students find these problems more interesting, but they also support comprehension!  When students are currently experiencing a season, it is easier for them to visualize and understand what the story problem is talking about.

Another fun reason to use seasonal story problems is that you can pull out all of those holiday mini-erasers and other seasonal manipulatives that you have on hand!  This is also a great way to help students channel their excitement about upcoming holidays into a meaningful learning activity.

Seasonal First Grade Math Word Problems

If you could like to use seasonal word problems in your first grade classroom, I have the perfect solution for you!  I have put together a resource with over 150 seasonal story problems that you can use all year long! Each page comes with coordinating manipulatives that students can cut out to help them model the word problem.  This is great for fine motor practice, as well! 

Since everything you need for each word problem is included on the page, this is a no-prep resource that you can use to supplement your math practice.  These story problem worksheets are perfect for small group intervention, whole group instruction, centers, morning work, or even for homework!  If you’d like to take a closer look at this resource, you can find it in my shop .

Math Story Problems Seasonal Addition and Subtraction Pack

Save these tips for teaching word problems in first grade.

I hope that this post has given you some inspiration for teaching word problems in first grade!  If you’d like to come back to this post later, just add the pin below to your favorite teaching board on Pinterest.  You’ll be able to quickly find these tips and resources whenever you’re feeling unsure of how to teach word problems to first graders!

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Teaching Addition Strategies in 1st Grade

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Tips and tricks for teaching Addition Strategies in 1st grade! Have you introduced addition in your first grade classroom yet? Addition is the standard that we spend the most time on throughout the year is 1st grade! Our goal by the end of the year is for our first graders to be able to add fluently and automatically within 10 and to be able to flexibly use strategies for addition within 20.

So how do we get them there? In this post, I am going to share my favorite tips and tricks for introducing Addition Strategies in 1st grade! You can also grab my favorite Addition Math Mats for hands-on small group learning for Free! 🎉

You can find all of the lessons, activities, worksheets, and anchor charts from this post in my Addition Strategies Math Unit here! It is part of my Happy Math Curriculum for 1st Grade!

What are Addition Strategies?

Teaching a variety of Addition Strategies is so important to create flexible problem solvers and build fact fluency!

I tell my students that our addition strategies are like tools in our Strategy Toolbox. When we need to add, we can pull out a tool or strategy to help us!

Addition in 1st Grade – Where do we start?

Before students are able to add fluently and automatically, they need to have a concrete understanding of what addition is and how it works. Building conceptual understanding is essential with any new math skill and it is always where I start!

We want our students to understand that Addition means to join together parts to make a whole. The first strategy I teach my students is using manipulatives to Make a Model. This is simply using manipulatives to act out an addition story. We use a variety of tools such as cubes and counters on a ten frame to model an addition problem.

I also like to have my students get up and act out real world story problems during our whole group mini lessons. 

Introducing Addition Strategies

I introduce each of our Addition Strategies during our whole group math mini lesson using our interactive mini lesson slides. Students usually bring a dry erase board to the rug so that they are able to actively participate in the lesson.

During each mini lesson I:

• ⭐️ Introduce the strategy
• ⭐️ Model the strategy
• ⭐️ Give my students an opportunity for quick guided practice

After learning a new strategy, we add our Addition Strategy Poster to our strategy wall!

Hands-On Practice with Addition Strategies

After our mini lesson, we begin our math rotations. Students visit the teacher table for small group instruction daily! At the teacher table, I give my students hands-on practice with our addition strategy.

My favorite Small Group Activity is using Math Mats! I simply print and slip them into a dry erase pocket. Then I just add manipulatives for engaging, hands-on practice of math skills. You can grab some of my favorite Addition Math Mats for FREE here!

During our small group time, I keep these mini addition strategy posters handy at my teacher table for us to reference during the lesson.

What other Addition Strategies should I teach?

With any math skill, I always apply the CRA model: Concrete, Representational, Abstract. I use this as a guide to teach addition strategies in a systematic way. I always start with the most concrete (using manipulatives or acting out addition situations), then move to representational strategies (using a number line, counting on with fingers, drawing pictures), then finally I move on to teaching mental math strategies. These include adding 0 and 1, doubles facts, Friends of 10, and more!

Keep reading for tips for teaching each strategy! 🥰

Concrete Addition Strategies

💗 making a model.

Students learn to use manipulatives such as cubes and counters to add. I always keep manipulatives available for students to use when working independently.

💗 Using a Number Bond

I LOVE number bonds for building conceptual understanding of addition. They are amazing for students to be able to visualize the parts joining together to make the whole and the relationship between the parts and the whole! Building this understanding is essential for skills that come later such as subtraction and missing addends.

We spend a lot of time in small groups with our Part Part Whole Math Mats. We practice joining parts to make a whole and also decomposing numbers into number partners. I like to give my students a set of cubes (I start with 6!) I challenge students to break that stack of cubes apart in different ways.

Representational Addition Strategies

💗 using a number line.

Using a number line challenges students to think about adding numbers in a different way. We learn that when we add, we are actually counting forward. We can use the number line as a tool to help us keep track of numbers as we count forward to find the sum.

Using mini erasers with our math mats is always a hit during small group time!

💗 Counting On

After we learn to count on using a number line, we learn that we can also count on by locking the bigger addend in our brain and counting on the smaller addend. Students learn that they can use their brains or math tools to help count on.

💗 Drawing a Math Picture to Add

Finally, students learn how to draw math pictures to help them add. I teach my students that math pictures need to be clear and they need to be simple (we talk about how they are different than the pictures they draw in art class 😂)

We practice solving addition story problems by drawing a picture. I love using Numberless Word Problems during small group to promote problem solving!

Abstract: Mental Math Addition Strategies

In 1st grade, we are beginning to work toward fact fluency. By the end of the year, we want students to be fluent with facts up to 10. Instead of memorizing random facts, teaching strategies helps students develop fact fluency.

I like to teach students about Snap Facts – or facts we know in a snap! I tell them we want to add Snap Facts to our brain! The more we practice, the easier it is to keep Snap Facts in our brain!

💗 Adding 0 and Adding 1

These are the first mental math strategies we learn in 1st grade. Students learn that when we add 0 to a number, the number stays the same! When we add 1, we just count forward 1!

💗 Commutative Property

The Commutative Property tells us that the order of the addends in an equation doesn’t matter. Even if the addends switch places (or flip flop!) the sum stays the same. This is important because it allows us to be more flexible when adding. If we know that 5+1 is 6, then 1 +5 also has to be 6!

We practice this in small group with dominos. Students pick a domino and write the equation, then flip it and write the related fact!

💗 Friends of 10

Next, we introduce ways to make 10. Knowing the “Friends of 10” is so important for our learners. 10 is a benchmark number in first grade. Knowing combinations of 10 helps increase students’ number sense and fact fluency. Building a solid foundation of making 10 will help with future math skills such as adding larger numbers, subtraction, and missing addends.

Shake and Spill is one of our favorite activities for exploring the Friends of Ten. I give each student in my small group a cup with ten 2-sided counters. Students shake, spill, and count how many of each color. We try to discover all of the different ways to make ten!

In small groups, we also use cube trains to practice making different combinations of ten.

💗 Doubles Facts

Next, we work on learning our doubles facts! A Doubles Fact is a number sentence where two of the same number are added together. Knowing the doubles facts automatically is important for fact fluency and helps students to know many other facts as well.

It will take time and practice for students to learn their doubles facts by heart. I suggest songs and movement to help! There are great YouTube doubles facts songs if you are able to play these for your students. I like to concorporate these math fact songs into daily movement breaks. A few favorites are the Doubles Number Rock (all of these songs are so good!) and Doubles Doubles! By Harry Kindergarten.

In small group, we practice making doubles facts with manipulatives and playing doubles facts games!

💗 Doubles + 1 Facts

After practicing our doubles facts, we work on doubles + 1 or near doubles facts! For example, if I know that 5+5=10, then 5+6=11 because I just add one more. To recognize doubles +1 facts, I like to teach students to look for “neighbor” numbers. 5 and 6 aren’t the same but they are right next to each other. That means that 6 is just 1 more than 5.

💗 10 + More

The next strategy we learn is 10+ More! This is when we add on numbers to ten to make a teen number. In our mini lesson, students look for patterns when we add numbers to 10. when we add a number to 10, we keep the 1 and replace the 0 with whatever number we are adding (10+3=13, 10+4=14, 10+5=15, etc.) Learning this pattern will help students answer 10+ more facts in a snap. This lesson introduces the concept that our sum is made up of a group of 10 and extra ones.

💗 Make a 10 to Add

This is a complex mental math strategy that requires us to shift the amounts of our addends to make numbers easier to add quickly. Instead of thinking, ’what’s 8+5?’ we can think ‘10+3 is 13, so I know 8+5 is also 13’.

In order for our students to be able to use this strategy there are a few skills they need to have down first:

• They need to have basic fact fluency within 10 in order to break apart the number
• They need to be fluent with the Friends of Ten
• They need to be able to use the Ten +More strategy

When I first introduce this strategy, not all of my students are ready for it, and that’s okay! We work on the strategy in a very concrete way by using two ten frames to move counters to make a ten.

In small groups, we use math mats and manipulatives to practice this strategy in a concrete way. Some students are able to apply this strategy to equations without using the manipulatives and others aren’t there yet, and that’s okay! I always revisit this strategy later in the year when our fact fluency is stronger.

Addition Strategies Math Centers and Activities

I love to give my students a lot of opportunities to practice adding with games and activities! Here are a few of our favorite!

⭐️ Add and Cover

⭐️ Addition Write and Wipe Cards

These cards are perfect for reviewing during small group time.

⭐️ Addition Strategy Sort

This is an amazing activity for critical thinking and developing fact fluency!

I like to do this during small groups to review strategies. Then, I will use it again as a math center. This is also perfect for a whole group review!

Go over each strategy you have taught your students. You can choose to working on only a few types of facts at a time and gradually add more. Or you might wait until all strategies have been taught and do it all together.

Show students each of the headers and have them take turns choosing a card and deciding where it will go. You can also have students say the sum as they sort.

⭐️ Addition Strategy Book

This book is perfect for independent work at the end of a math lesson. After teaching the strategy, have students complete the page in the book. I love sharing these books with families so that they can see examples of the strategies students are learning.

Solving Addition Word Problems using Strategies

Word problems are so important in first grade! We want our students to develop into problem solvers who think deeply and use different strategies to solve.

I love using Numberless Word Problems to promote problem solving! These slides make a great math warm up before starting our mini lesson!

Addition Worksheets for Grade 1

For independent practice, I included addition worksheets. Students will practice applying the addition strategies we’ve learned.

These can be used during math centers, for morning work, or sent home as extra practice!

These could also be used to put together Math Folders or packets for independent work time during math rotations.

How do I assess when teaching Addition Strategies to first graders?

Assessment is an important part of guided math! It not only tells us if students have mastered concepts at the end of a unit, but it also helps inform teaching decisions during a unit. There are 3 main ways that I assess in my guided math block. 

• Exit Slips: Giving quick exit slips at the end of a small group session is perfect for checking students understanding during a math unit.
• Observation: Along with exit slips, I use a lot of observation as a formative assessment. I keep anecdotal notes in my guided math planning binder! You can grab it for free here!
• Brain Check: At the end of a unit, I give a summative assessment over the standards we covered! In the unit, I included two versions of the brain check so that one could be given as a pre-test and one as a post-test. 

Celebrating the end of our Addition Strategies Guided Math Unit in 1st Grade

We celebrate the end of our unit with this Addition Strategy Toolbox Math Craft! These are perfect to display in the hallway or on a bulletin board to share learning!

I hope this post gave you tons of ideas for Addition Strategies activities and lessons for first grade!

Want your Addition Strategies Unit planned for you? 

You can grab this 800+ page resource that includes:

• ⭐️ Ready-To-Use Mini Lesson Slides (15 days of lessons)
• ⭐️ Hands-on Activities to use during small group instruction, whole group, and math centers
• ⭐️  Addition Strategy Posters and Mini Posters 
• ⭐️ Daily Addition Word Problems
• ⭐️  Worksheets and Independent Practice
• ⭐️ Assessment 

FIND IT HERE!

More posts you might be interested in:

• Guided Math in First Grade
• Tips for Develop Fact Fluency
• Ideas to Teach Number Sense

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I am a first grade Teacher. Your worksheets and strategies are very helpful.

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1st Grade Math Word Problems Worksheets

Grade 1 word problems.

These grade 1 word problem worksheets relate first grade math concepts to the real world. The word problems cover addition, subtraction, time, money, fractions and lengths.

We encourage students to think about the problems carefully by:

• providing a number of mixed word problem worksheets;
• sometimes including irrelevant data within word problems.

Addition word problems

Single digit addition

Addition with sums 50 or less

3 or more numbers added together

Subtraction word problems

Subtracting single digit numbers

Subtracting numbers under 50

Mixed addition and subtraction word problems

Add / subtract word problems with mostly single digit numbers

Add / subtract word problems with numbers under 50

Time word problems

Time and elapsed time problems (whole hours)

Money word problems

Counting money (coins only)

Measurement word problems

Combining and comparing lengths (inches)

Combining and comparing lengths (cm)

Fraction word problems

Write the fraction from the story (parts of whole, parts of group)

Mixed word problems

Addition, subtraction, money, time, fractions and length word problems mixed  

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Problem and Solution for First Grade

• Read Alouds , Reading & Literacy

Being able to easily identify the problem and solution in a story is a foundational reading comprehension skill for first grade students.

It's an activity that can be done with every single read aloud.

It can (and should) be done whole group, in small group, with partners, in literacy stations and during independent reading.

It's also a great skill for parents to work on at home. 

Describing the problem includes:

• Identifying the problem the character(s) is experiencing in the beginning of the story
• Asking questions and making predictions about possible upcoming events and about possible ways to solve the problem
• Understanding how the events and characters influence the problem throughout the story

Describing the solution includes: 

• Confirming or revising predictions about the resolution
• Explaining how the resolution solves the problem
• Describing what happens to the character(s) because of the resolution

Examples of Problem and Solution

You might be totally comfortable with the topic of problem and solution, but when we are in the moment in front of 22 little faces, sometime we blank! 

It's time to give our students examples of what Problem and Solution is…

but we've got nothing! 

It helps to think of a few examples ahead of time and jot them down. 

Examples of problem and solution work best if the are real life examples that the students can relate to! 

Here's a few examples from the school day: 

• Your pencil breaks
• You can't find your book
• You forgot your lunch
• You can't remember the directions
• You left your jacket on the playground
• You don't know how to tie your shoes
• You need a supply that another student is using

By using these real-life examples, you are not only teaching problem and solution, but you are reinforcing your classroom management as well!

How to Introduce Problem and Solution for First Grade

If you teach lower elementary, then you know that there is SO much that goes into the comprehension of a book. First grade students are learning so much at this age and even listening comprehension requires their little brains to work so hard.

Why do I bring this up?

Because I want you to think about how hard they are ALREADY working when they are listening to a read aloud and when you use a mentor text to introduce a NEW SKILL, most students’ brains go into overload!

So what should we do instead?

Start with a non-text activity. Let me introduce you to a new kind of “slideshow”. 

I like to use interactive slideshows/powerpoints. My slideshows always follow this order: 

• Teaching Slides: Introduces students to WHAT the skill is
• Guided Practice: Introduces students to HOW to apply the skill
• Interactive Practice: Gives students an example and allows them to PRACTICE the skill in an easy and concise way

Problem and Solution Non-Text Activities for First Grade

After we practice the slideshow, then we practice building our problem and solution muscles with some guided and independent practice using a NON-TEXT ACTIVITY such as a station game.

Non-text activities are a HUGE asset to students because it allows them to build and flex their problem and solution muscles before we ask them to apply those skills to a text! 

How to Complete the Activity

We will do this activity together as a group first, then it will move into our guided reading stations where students will complete it independently!

I've found the best and easiest way to do this is with images of problems/solutions that my first grade students know a lot about. 

My students do this Problem and Solution activity (seen on the right). We do one together as a class for guided practice. 

For this activity, students must correctly match the problem and solution. Then I have students write about the problem and solution with an emphasis on why the solution actually works for this problem.

After we do this activity together, it goes into their stations for Guided Reading. 

The best part about this activity is that you can leave it out for several weeks because each time students can choose a new set of pictures! 

Mentor Texts for Problem and Solution

A mentor text is an incredibly powerful tool for teaching reading comprehension skills! 

(make sure you keep reading to the end of this post to see a list of my favorite mentor texts for problem & solution)

The problem that many teachers run into with mentor texts is that there are SO MANY different skills you can teach with the same mentor text….

Sometimes we try to do TOO MUCH and we overwhelm our students! 

Let me introduce you to a Comprehension Focus Question (CFQ). 

A CFQ is one question that you focus on through the entire text! 

It simplifies things for you and your students. (more on that later)…

But let's take a minute to dispel so myths about mentor texts..

A mentor text is NOT a book that you read once and put it away.

A mentor text is a book that you read once, then refer back to again and again and again.

The greatest benefit of a good mentor text is that after you have read it once, when you refer back to it, you aren’t reading the entire book again, you are simply referring back to one or two pages.

It will save you SO much time.

AND students are already familiar with the story line meaning that already have a foundation for whatever comprehension skill you are about to dive into!

How to Boost Comprehension for Problem & Solution

Comprehension Focus Questions

As I mentioned, a Comprehension Focus Question (CFQ) is a very focused and intentional comprehension goal for an activity, a week or even a unit.

If you have done your research and you understand your learning standard, the vocabulary and what students need to know…

Then it becomes very easy to choose a goal (or a comprehension focus question).

But, why do you need a comprehension goal?

To stay FOCUSED!

Not just for you, but for your students also!

Let’s look at an example. Let’s say that this week you are focusing on how to make an inference. Well, there are about a hundred different ways you can make an inference and a CFQ allows you to focus on one area at a time.

Example CFQ: “How Did The Character Change From ___ To ____?”

In this comprehension focus question, you and your students are focusing in on the characters of the story.

The great thing about CFQ’s is that the next time you pull out this mentor text, you can choose a different CFQ to focus on while still practicing how to make an inference!

Sentence Stems

Another great strategy that falls right in long with mentor texts and comprehension focus questions is sentence stems.

A sentence stem is a phrase that your first grade students will use to answer a comprehension question.

Sentence stems are designed to get students to answer comprehension questions more fully rather than giving one word answers.

Sentence stems encourage students to explain their thinking.

I like to have a list of sentence stems next to my table that are specific to each comprehension skill. I stick to one or two stems per skill for the entire year because I want my students to be consistent. (This also makes it a lot easier for them)

If we are sticking with our Make an Inference example, I would use the following sentence stems:

I Think ___ Because____.   

I Read This ____ So I Think _____

Problem and Solution Activities for First Grade

All of the activities that you found in this post, both printable and digital, along with UNIT LESSON PLANS can be found in my Problem and Solution Bundle here.

You can save up to 20% by purchasing the items together, but you can also purchase individual items to better fit your needs!

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I am a current Elementary Librarian and Enrichment Teacher, mother of two, follower of Christ and Texas native. In my own classroom, I love to save time by finding unique ways to integrate writing, social studies and science into all parts of my day. I also love all things organization!

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10 Strategies for Problem Solving in Math

Created on May 19, 2022

Updated on January 6, 2024

When faced with problem-solving, children often get stuck. Word puzzles and math questions with an unknown variable, like x, usually confuse them. Therefore, this article discusses math strategies and how your students may use them since instructors often have to lead students through this problem-solving maze.

What Are Problem Solving Strategies in Math?

If you want to fix a problem, you need a solid plan. Math strategies for problem solving are ways of tackling math in a way that guarantees better outcomes. These strategies simplify math for kids so that less time is spent figuring out the problem. Both those new to mathematics and those more knowledgeable about the subject may benefit from these methods.

There are several methods to apply problem-solving procedures in math, and each strategy is different. While none of these methods failsafe, they may help your student become a better problem solver, particularly when paired with practice and examples. The more math problems kids tackle, the more math problem solving skills they acquire, and practice is the key.

Strategies for Problem-solving in Math

Even if a student is not a math wiz, a suitable solution to mathematical problems in math may help them discover answers. There is no one best method for helping students solve arithmetic problems, but the following ten approaches have shown to be very effective.

Understand the Problem

Understanding the nature of math problems is a prerequisite to solving them. They need to specify what kind of issue it is ( fraction problem , word problem, quadratic equation, etc.). Searching for keywords in the math problem, revisiting similar questions, or consulting the internet are all great ways to strengthen their grasp of the material. This step keeps the pupil on track.

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Guess and check.

One of the time-intensive strategies for resolving mathematical problems is the guess and check method. In this approach, students keep guessing until they get the answer right.

After assuming how to solve a math issue, students should reintroduce that assumption to check for correctness. While the approach may appear cumbersome, it is typically successful in revealing patterns in a child’s thought process.

Work It Out

Encourage pupils to record their thinking process as they go through a math problem. Since this technique requires an initial comprehension of the topic, it serves as a self-monitoring method for mathematics students. If they immediately start solving the problem, they risk making mistakes.

Students may keep track of their ideas and fix their math problems as they go along using this method. A youngster may still need you to explain their methods of solving the arithmetic questions on the extra page. This confirmation stage etches the steps they took to solve the problem in their minds.

Work Backwards

In mathematics, a fresh perspective is sometimes the key to a successful solution. Young people need to know that the ability to recreate math problems is valuable in many professional fields, including project management and engineering.

Students may better prepare for difficulties in real-world circumstances by using the “Work Backwards” technique. The end product may be used as a start-off point to identify the underlying issue.

In most cases, a visual representation of a math problem may help youngsters understand it better. Some of the most helpful math tactics for kids include having them play out the issue and picture how to solve it.

One way to visualize a workout is to use a blank piece of paper to draw a picture or make tally marks. Students might also use a marker and a whiteboard to draw as they demonstrate the technique before writing it down.

Find a Pattern

Kids who use pattern recognition techniques can better grasp math concepts and retain formulae. The most remarkable technique for problem solving in mathematics is to help students see patterns in math problems by instructing them how to extract and list relevant details. This method may be used by students when learning shapes and other topics that need repetition.

Students may use this strategy to spot patterns and fill in the blanks. Over time, this strategy will help kids answer math problems quickly.

When faced with a math word problem, it might be helpful to ask, “What are some possible solutions to this issue?” It encourages you to give the problem more thought, develop creative solutions, and prevent you from being stuck in a rut. So, tell the pupils to think about the math problems and not just go with the first solution that comes to mind.

Draw a Picture or Diagram

Drawing a picture of a math problem can help kids understand how to solve it, just like picturing it can help them see it. Shapes or numbers could be used to show the forms to keep things easy. Kids might learn how to use dots or letters to show the parts of a pattern or graph if you teach them.

Charts and graphs can be useful even when math isn’t involved. Kids can draw pictures of the ideas they read about to help them remember them after they’ve learned them. The plan for how to solve the mathematical problem will help kids understand what the problem is and how to solve it.

Trial and Error Method

The trial and error method may be one of the most common problem solving strategies for kids to figure out how to solve problems. But how well this strategy is used will determine how well it works. Students have a hard time figuring out math questions if they don’t have clear formulas or instructions.

They have a better chance of getting the correct answer, though, if they first make a list of possible answers based on rules they already know and then try each one. Don’t be too quick to tell kids they shouldn’t learn by making mistakes.

Review Answers with Peers

It’s fun to work on your math skills with friends by reviewing the answers to math questions together. If different students have different ideas about how to solve the same problem, get them to share their thoughts with the class.

During class time, kids’ ways of working might be compared. Then, students can make their points stronger by fixing these problems.

Check out the Printable Math Worksheets for Your Kids!

There are different ways to solve problems that can affect how fast and well students do on math tests. That’s why they need to learn the best ways to do things. If students follow the steps in this piece, they will have better experiences with solving math questions.

Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.

As adults, we take numbers for granted, but preschoolers and kindergartners have no idea what these symbols mean. Yet, we often demand instant understanding and flawless performance when we start teaching numbers to our children. If you don’t have a clue about how to teach numbers for kids, browse no more. You will get four […]

May 19, 2022

Teaching children is a complex process because they require more attention than an adult person. You may need to employ different teaching strategies when teaching kids. But what are teaching strategies? Teaching strategies are the methods to ensure your kids or students learn efficiently. But not all strategies yield similarly, and if the one you […]

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Standard 1: Make Sense of Problems & Persevere in Solving Them

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Classroom Observations

Teachers who are developing students’ capacity to “make sense of problems and persevere in solving them” develop ways of framing mathematical challenges that are clear and explicit, and then check-in repeatedly with students to help them clarify their thinking and their process. An early childhood teacher might ask her students to work in pairs to evaluate their approach to a problem, telling a partner to describe their process, saying “what [they] did, and what [they] might do next time.” A middle childhood teacher might post a set of different approaches to a solution, asking students to identify “what this mathematician was thinking or trying out” and evaluating the success of the strategy. An early adolescence teacher might have students articulate a specific way of laying out the terrain of a problem and evaluating different starting points for solving. A teacher of adolescents and young adults might frame the task as a real-world design conundrum, inviting students to engage in a “tinkering” process of working toward mathematical proof, changing course as necessary as they develop their thinking. Visit the video excerpts below to view multiple examples of teachers engaging students in sense-making and mathematical perseverance.

The Standard

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Practice Standards

• Make sense of problems & persevere in solving them
• Reason abstractly & quantitatively
• Construct viable arguments & critique the reasoning of others
• Model with mathematics
• Use appropriate tools strategically
• Attend to precision
• Look for & make use of structure
• Look for & express regularity in repeated reasoning

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Problem-Solving Strategies and Obstacles

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Sean is a fact-checker and researcher with experience in sociology, field research, and data analytics.

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From deciding what to eat for dinner to considering whether it's the right time to buy a house, problem-solving is a large part of our daily lives. Learn some of the problem-solving strategies that exist and how to use them in real life, along with ways to overcome obstacles that are making it harder to resolve the issues you face.

What Is Problem-Solving?

In cognitive psychology , the term 'problem-solving' refers to the mental process that people go through to discover, analyze, and solve problems.

A problem exists when there is a goal that we want to achieve but the process by which we will achieve it is not obvious to us. Put another way, there is something that we want to occur in our life, yet we are not immediately certain how to make it happen.

Maybe you want a better relationship with your spouse or another family member but you're not sure how to improve it. Or you want to start a business but are unsure what steps to take. Problem-solving helps you figure out how to achieve these desires.

The problem-solving process involves:

• Discovery of the problem
• Deciding to tackle the issue
• Seeking to understand the problem more fully
• Researching available options or solutions
• Taking action to resolve the issue

Before problem-solving can occur, it is important to first understand the exact nature of the problem itself. If your understanding of the issue is faulty, your attempts to resolve it will also be incorrect or flawed.

Problem-Solving Mental Processes

Several mental processes are at work during problem-solving. Among them are:

• Perceptually recognizing the problem
• Representing the problem in memory
• Considering relevant information that applies to the problem
• Identifying different aspects of the problem
• Labeling and describing the problem

Problem-Solving Strategies

There are many ways to go about solving a problem. Some of these strategies might be used on their own, or you may decide to employ multiple approaches when working to figure out and fix a problem.

An algorithm is a step-by-step procedure that, by following certain "rules" produces a solution. Algorithms are commonly used in mathematics to solve division or multiplication problems. But they can be used in other fields as well.

In psychology, algorithms can be used to help identify individuals with a greater risk of mental health issues. For instance, research suggests that certain algorithms might help us recognize children with an elevated risk of suicide or self-harm.

One benefit of algorithms is that they guarantee an accurate answer. However, they aren't always the best approach to problem-solving, in part because detecting patterns can be incredibly time-consuming.

There are also concerns when machine learning is involved—also known as artificial intelligence (AI)—such as whether they can accurately predict human behaviors.

Heuristics are shortcut strategies that people can use to solve a problem at hand. These "rule of thumb" approaches allow you to simplify complex problems, reducing the total number of possible solutions to a more manageable set.

If you find yourself sitting in a traffic jam, for example, you may quickly consider other routes, taking one to get moving once again. When shopping for a new car, you might think back to a prior experience when negotiating got you a lower price, then employ the same tactics.

While heuristics may be helpful when facing smaller issues, major decisions shouldn't necessarily be made using a shortcut approach. Heuristics also don't guarantee an effective solution, such as when trying to drive around a traffic jam only to find yourself on an equally crowded route.

Trial and Error

A trial-and-error approach to problem-solving involves trying a number of potential solutions to a particular issue, then ruling out those that do not work. If you're not sure whether to buy a shirt in blue or green, for instance, you may try on each before deciding which one to purchase.

This can be a good strategy to use if you have a limited number of solutions available. But if there are many different choices available, narrowing down the possible options using another problem-solving technique can be helpful before attempting trial and error.

In some cases, the solution to a problem can appear as a sudden insight. You are facing an issue in a relationship or your career when, out of nowhere, the solution appears in your mind and you know exactly what to do.

Insight can occur when the problem in front of you is similar to an issue that you've dealt with in the past. Although, you may not recognize what is occurring since the underlying mental processes that lead to insight often happen outside of conscious awareness .

Research indicates that insight is most likely to occur during times when you are alone—such as when going on a walk by yourself, when you're in the shower, or when lying in bed after waking up.

How to Apply Problem-Solving Strategies in Real Life

If you're facing a problem, you can implement one or more of these strategies to find a potential solution. Here's how to use them in real life:

• Create a flow chart . If you have time, you can take advantage of the algorithm approach to problem-solving by sitting down and making a flow chart of each potential solution, its consequences, and what happens next.
• Recall your past experiences . When a problem needs to be solved fairly quickly, heuristics may be a better approach. Think back to when you faced a similar issue, then use your knowledge and experience to choose the best option possible.
• Start trying potential solutions . If your options are limited, start trying them one by one to see which solution is best for achieving your desired goal. If a particular solution doesn't work, move on to the next.
• Take some time alone . Since insight is often achieved when you're alone, carve out time to be by yourself for a while. The answer to your problem may come to you, seemingly out of the blue, if you spend some time away from others.

Obstacles to Problem-Solving

Problem-solving is not a flawless process as there are a number of obstacles that can interfere with our ability to solve a problem quickly and efficiently. These obstacles include:

• Assumptions: When dealing with a problem, people can make assumptions about the constraints and obstacles that prevent certain solutions. Thus, they may not even try some potential options.
• Functional fixedness : This term refers to the tendency to view problems only in their customary manner. Functional fixedness prevents people from fully seeing all of the different options that might be available to find a solution.
• Irrelevant or misleading information: When trying to solve a problem, it's important to distinguish between information that is relevant to the issue and irrelevant data that can lead to faulty solutions. The more complex the problem, the easier it is to focus on misleading or irrelevant information.
• Mental set: A mental set is a tendency to only use solutions that have worked in the past rather than looking for alternative ideas. A mental set can work as a heuristic, making it a useful problem-solving tool. However, mental sets can also lead to inflexibility, making it more difficult to find effective solutions.

How to Improve Your Problem-Solving Skills

In the end, if your goal is to become a better problem-solver, it's helpful to remember that this is a process. Thus, if you want to improve your problem-solving skills, following these steps can help lead you to your solution:

• Recognize that a problem exists . If you are facing a problem, there are generally signs. For instance, if you have a mental illness , you may experience excessive fear or sadness, mood changes, and changes in sleeping or eating habits. Recognizing these signs can help you realize that an issue exists.
• Decide to solve the problem . Make a conscious decision to solve the issue at hand. Commit to yourself that you will go through the steps necessary to find a solution.
• Seek to fully understand the issue . Analyze the problem you face, looking at it from all sides. If your problem is relationship-related, for instance, ask yourself how the other person may be interpreting the issue. You might also consider how your actions might be contributing to the situation.
• Research potential options . Using the problem-solving strategies mentioned, research potential solutions. Make a list of options, then consider each one individually. What are some pros and cons of taking the available routes? What would you need to do to make them happen?
• Take action . Select the best solution possible and take action. Action is one of the steps required for change . So, go through the motions needed to resolve the issue.
• Try another option, if needed . If the solution you chose didn't work, don't give up. Either go through the problem-solving process again or simply try another option.

You can find a way to solve your problems as long as you keep working toward this goal—even if the best solution is simply to let go because no other good solution exists.

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Stewart SL, Celebre A, Hirdes JP, Poss JW. Risk of suicide and self-harm in kids: The development of an algorithm to identify high-risk individuals within the children's mental health system . Child Psychiat Human Develop . 2020;51:913-924. doi:10.1007/s10578-020-00968-9

Rosenbusch H, Soldner F, Evans AM, Zeelenberg M. Supervised machine learning methods in psychology: A practical introduction with annotated R code . Soc Personal Psychol Compass . 2021;15(2):e12579. doi:10.1111/spc3.12579

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Chrysikou EG, Motyka K, Nigro C, Yang SI, Thompson-Schill SL. Functional fixedness in creative thinking tasks depends on stimulus modality .  Psychol Aesthet Creat Arts . 2016;10(4):425‐435. doi:10.1037/aca0000050

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By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."