math problem solving strategies for middle school

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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.

Problem Solving Posters (Represent It! Bulletin Board)

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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!

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.

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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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Top 9 Math Strategies for Successful Learning (2021 and Beyond)

Written by Ashley Crowe

• Teaching Strategies
• Teacher Resources

• Why are effective Math strategies so important for students?
• Getting students excited about Math problems
• Top 9 math strategies for engaging lessons
• How teachers can refine math strategies

Math is an essential life skill. You use problem-solving every day. The math strategies you teach are needed, but many students have a difficult time making that connection between math and life.

Math isn’t just done with a pencil and paper. It’s not just solving word problems in a textbook. As an educator, you need fresh ways for math skills to stick while also keeping your students engaged. 

In this article, we’re sharing 9 engaging math strategies to boost your students’ learning . Show your students how fun math can be, and let’s freshen up those lesson plans!

Unlike other subjects, math builds on itself. You can’t successfully move forward without a strong understanding of previous materials. And this makes math instruction difficult.

To succeed in math, students need to do more than memorize formulas or drill times tables. They need to develop a full understanding of what their math lessons mean , and how they translate into the real world. To reach that level of understanding, you need a variety of teaching strategies. 

Conceptual understanding doesn’t just happen at the whiteboard. But it can be achieved by incorporating fun math activities into your lessons, including 

• Hands-on practice
• Collaborative projects
• Gamified or game-based learning

Repetition and homework are important. But for these lessons to really stick, your students need to find the excitement and wonder in math.

Creating excitement around math can be an uphill battle. But it’s one you and your students can win! 

Math is a challenging subject — both to teach and to learn. But it’s also one of the most rewarding. Finding the right mix of fun and learning can bring a lot of excitement to the classroom. 

Think about what your students already love doing. Video games? Legos? Use these passions to create exciting math lesson plans your students can relate to. 

Hands-on math practice can engage students that have disconnected from math. Putting away the pencils and textbooks and moving students out of their desks can re-energize your classroom.

If you’re teaching elementary or middle school math, find ways for your students to work together. Kids this age crave peer interaction. So don’t fight it — provide it! 

Play a variety of math games or puzzles . Give them a chance to problem-solve together. Build real-world skills in the classroom while also boosting student confidence. 

And be sure to celebrate all the wins! It is easy to get bogged down with instruction and testing. But even the smallest accomplishments are worth celebrating. And these rewarding moments will keep your students motivated and pushing forward.

Keep reading to uncover all of our top math strategies for keeping your students excited about math. 

1. Explicit instruction

You can’t always jump straight into the fun. Explicit instruction still provides the best foundation for the activities to come. 

Set up your lesson for the day at the whiteboard, along with materials to demonstrate the coming activities. Make sure to also focus on any new vocabulary and concepts. 

Tip: don't stay here for too long. Once the lesson is introduced, move on to the next fun strategy for the day!

2. Conceptual understanding

Helping your students understand the concept behind the lesson is crucial, but not always easy. Even your highest performing students may only be following a pattern to solve problems, without grasping the “why.”

Visual aids and math manipulatives are some of your best tools to increase conceptual understanding. Math is not a two dimensional subject. Even the best drawing of a cone isn’t going to provide the same experience as holding one. Find ways to let your students examine math from all sides.

Math manipulatives don’t need to be anything fancy. Basic wooden blocks, magnets, molding clay and other toys can create great hands-on lessons. No need to invest in expensive or hard-to-find materials. 

Math word problems are also a great time to break out a full-fledged demo. Hot Wheels cars can demonstrate velocity and acceleration. A tape measure is an interactive way to teach area and volume. These materials give your students a chance to bring math off the page and into real life. 

3. Using concepts in Math vocabulary

There’s more than one way to say something. And the more ways you can describe a mathematical concept, the better. Subtraction can also be described as taking away or removing. Memorizing multiplication facts is useful, but seeing these numbers used to calculate area gives them new meaning. 

Some math words are going to be unfamiliar. So to help students get comfortable with these concepts, demonstrate and label math ideas throughout your classroom . Understanding comes more easily when students are surrounded by new ideas. 

For example, create a division corner in your station rotations , with blocks to demonstrate the concept of one number going into another. Use baskets and labels to have students separate the blocks into each part of the division problem: dividend, divisor, quotient and remainder.  

Give students time to explore, and teach them big ideas with both academic and everyday terms. Demystify math and watch their confidence build!

4. Cooperative learning strategies

When students work together, it benefits everyone. More advanced students can lead, helping them solidify their knowledge. And they may have just the right words to describe an idea to others who are struggling.

It is rare in real-life situations for big problems to be solved alone. Cooperative learning allows students to view a problem from various angles. This can lead to more flexible, out-of-the-box thinking. 

After reviewing a word problem together as a class, ask small student groups to create their own problems. What is something they care about that they can solve with these skills? Involve them as much as possible in both the planning and solving. Encourage each student to think about what they bring to the group. There’s no better preparation for the future than learning to work as a team. 

5. Meaningful and frequent homework

When it comes to homework, it pays to think outside of textbooks and worksheets. Repetition is important, but how can you keep it fun?

Create more meaningful homework by including games in your curriculum plans. Encourage board game play or encourage families to play quiz-style games at home to improve critical thinking, problem solving and basic math skills. 

Sometimes you need homework that doesn’t put extra work onto the parents. The end of the day is already full for many families. To encourage practice and give parents a break, assign game-based options like Prodigy Math Game for homework. 

With Prodigy, students can enjoy a fun, video game experience that helps them stay excited and motivated to keep learning. They’ll practice math skills, while their parents have time to fix dinner. Plus, you’ll get progress reports that can help you plan future instruction . Win-win-win!    

Ready to make homework fun?

6. Puzzle pieces math instruction

Some kids excel at math. But others pull back and may rarely participate. That lack of confidence is hard to break through. How can you get your reluctant students to join in?

Try giving each student a piece of the puzzle. When you’re presenting your class with a problem, this creates necessary collaboration to get to the solution. 

Each student is given a piece of information needed to solve the problem. A number, a unit of measurement, or direction — break your problem into as many pieces as possible. 

If you have a large class, break down three or more problems at a time. The first task: find the other students who are working on your problem (try color-coding or using symbols to distinguish each problem’s parts). Then watch the learning happen as everyone plays their own important role. 

7. Verbalize math problems

There’s little time to slow down in the classroom. Instruction has to move fast to keep up with the expected standards. And students feel that, too. 

When possible, try to set aside some time to ask about your students’ math struggles. Make sure they know that they can come to you when they get stuck. Keep the conversation open to their questions as much as possible.

One great way to encourage questions is to address common troubles students have encountered in the past. Where have your past classes struggled? Point these out during your explicit instruction, and let your students know this is a tricky area. 

It’s always encouraging to know you’re not alone in finding something difficult. This also leaves the door open for questions, leading to more discovery and greater understanding.

8. Reflection time

Providing time to reflect gives the brain a chance to process the work completed. This can be done after both group and individual activities.

Group Reflection

After a collaborative activity, save some time for the group to discuss the project . Encourage them to ask:

• What worked?
• What didn’t work?
• Did I learn a new approach?
• What could we have done differently?
• Did someone share something I had never thought of before? 

These questions encourage critical thinking. They also show the value of working together with others to solve a problem. Everyone has different ways of approaching a problem, and they’re all valuable.

Individual Reflection

One way to make math more approachable is to show how often math is used. Journaling math encounters can be a great way for students to see that math is all around. 

Ask them to add a little bit to their journal every day, even just a line or two. Where did they encounter math outside of class? Or what have they learned in class that has helped them at home? 

Math skills easily transfer outside of the classroom. Help them see how much they have grown, both in terms of academics and social emotional learning .

9. Making Math facts fun

As a teacher, you know math is anything but boring. But transferring that passion to your students is a tricky task. So how can you make learning math facts fun?

Play games! Math games are great classroom activities. Here are a few examples:

• Design and play a board game.
• Build structures and judge durability.
• Divide into groups for a quiz or game show. 
• Get kids moving and measure speed or distance jumped.

Even repetitive tasks can be fun with the right tools. That’s why engaging games are a great way to help students build essential math skills. When students play Prodigy Math Game , for example, they learn curriculum-aligned math facts without things like worksheets or flashcards. This can help them become excited to play and learn! 

How teachers can refine Math strategies

Sometimes trying something new can make a huge difference for your students. But don’t stress and try to change too much at once. 

You know your classroom and students best. Pick a couple of your favorite strategies above and try them out. 

If you're looking to freshen up your math instruction, sign up for a free Prodigy teacher account. Your students can jump right into the magic of the Prodigy Math Game, and you’ll start seeing data on their progress right away! 

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Problem Solving Activities: 7 Strategies

• Critical Thinking

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.

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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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Math Problems In Middle School: How To Make Yours Fun, Engaging, And Curriculum-Focused

Andy Brighouse

Here, we look at how to plan for successful problem solving lessons with a range of tried and tested middle school math problems that are fun, engaging and curriculum-focused.

The notion of creating original math problems can strike fear deep into the heart of the uninitiated teacher. For some, simply delivering a lesson featuring untried problems is anxiety invoking.

Lessons based solely around problem solving activities need planning, and when doing this for the first time, it may seem like you are devising a strategic military operation with your lesson plan resembling the tactical objectives of a dawn raid. However, it doesn’t have to be like this; math is FUN (that’s why we teach it).

The art of teaching math strategies for problem solving in middle school, and the associated planning, is to focus on harnessing a particular mathematical strand. Ultimately, your problem solving lessons need to be targeting the skills that are necessary to make learners successful in the future.

As the teacher, your knowledge of math and the problems to be encountered will feed back into the planning so that the learning attributes gathered during problem solving can feed forward into exam success for the students.

How to teach problem solving

Understanding how to get students to solve problems (especially those unfamiliar, thought-provoking, and often slightly perplexing word problems) can be usefully reduced to getting students to consider just two questions…

• “ What do I know? ”  
•  “ What do I need? ”

These two questions are useful for us as educators, too (think data driven planning), but here, we are going to explore how to use them to drive forward our students’ ability and desire to find some solutions.

How to link problem solving to the curriculum

Your first task is to think carefully about which strand of math you want the problem to originate from. All strands of math are intrinsically linked, but if you’ve recently worked through an algebra unit, focus your math problem there.

Students can call upon skills from other strands, but their immediate thinking should be related to recent learnings or you will swiftly trigger their cognitive overload .

Don’t force a topic or real life context to fit a strand of math; consider the skills you want students to draw upon, and think about which sorts of numeracy questions would use those skills.

How to structure a problem solving session

Any problem solving session should commence with a hook; this could be a starter related to the problem at hand or a contextual discussion about why the problem is interesting.

Once you have devised a problem, attempt it yourself; this will show you what resources you need, and also the general thought processes and pitfalls your students are going to face.

It’s often a good idea to base your activities around topics that you know very well, and ones that you are confident your students have the relevant skills to be successful. Feel free to delay a particular lesson you have an idea for until you are comfortable with the subject matter.

Do some research and practice; this should be enjoyable because math is FUN. You’ll learn lots of new things, and become an even better teacher in the process.

Curriculum math problems

To get you started, below are some suggestions of the types of math problems and approaches you can take in each of the main math curriculum areas. You will clearly need to adjust your approach, and perhaps provide more or less differentiation and support at each stage depending on the age group you are teaching.

But with the right support, you’ll be amazed how far students will get trying to answer these math questions if they feel sufficiently motivated.

Number and place value math problems

Number theory is exciting. We know that, but our students don’t always, and it’s never too soon to introduce them to it. The plethora of conjectures, although fiendishly difficult to prove, are mostly based on concepts met in middle school.

Number problem example: Collatz conjecture

One of my “go to” examples of this would be the Collatz Conjecture. If you’re not familiar with this problem, I suggest looking it up; there are numerous great resources available suitable for all levels. The premise is quite simple…

• If the number is even, half it
• If the number is odd, triple it and then add one
• Take your answer, then repeat step 1 (i.e., either half it or triple it and add one).
• Continue until you think you have seen a pattern

The scope for the directions you can take in a lesson by using this problem and trying it out with smaller and larger numbers is enormous.

A well structured discussion with quality targeted questioning will conjure the ideas of mathematical algorithms, sequences, and patterns, through to the trickier concept of evidence versus proof.

Students can understand the problem easily. It’s straightforward to differentiate for ability and resource development is relatively minimal.

Watch the look on your students’ faces when you tell them that after so many years since its formulation, the Collatz Conjecture still has unclaimed prize money for a proof.

This instills the idea that even professional mathematicians struggle at times, and that problem solving is about investigating and building a toolkit of mathematical strategies.

Geometry math problems

Problem solving lessons focused on geometry and measurement lend themselves to practical activities; grab the colored paper, scissors, and glue, and then construct solutions.

This is where math can meet other disciplines, whether it’s engineering, art, architecture or even sport science. The reasons I do this are multiple:

• It demonstrates how applicable math is to EVERYTHING.
• It teaches students that math goes beyond the exercise book. Math is a way of thinking and is not “just about doing equations.”
• It allows students to be creative in math, which accesses more parts of the brain, and can give many students those magical lightbulb moments.

Geometry problem example: Packing boxes

A great idea for a very practical problem solving lesson would be looking at how objects pack into boxes.

This could be done using concrete resources such as tennis balls in various sized boxes, but with planning could involve also converting the problem to two dimensions, using circles with different shape mats to investigate layout configurations.

Students can then be encouraged to compare how the 2D and 3D are dealt with. For teachers wishing to brush up on the theory behind this type of work, we would need to look at the differences between geometry and topology .

Data handling and probability math problems

Data handling and probability are far more contemporary than the other strands we teach in school, so I would let the problem solving reflect this; choose very modern problems to focus on. In terms of the real world aspect, data and probability have huge implications regarding human actions.

This indicates that a good hook for the students is to have them use their knowledge of data and probability to solve a human problem.

Think about issues that are big in the news or that are prevalent in other subjects. There are great activities that can be created from issues involving the climate and environment.

A well written brief could not only boost the students’ mathematical problem solving skills, but could also lead to the solution of wider problems with the school or local area.

Data handling problem example: Recycling around the school

Why not get your students to solve the school’s recycling problems using math? Ask your students to analyze the locations and number of waste bins and recycling bins around the school building.

Automatically, this allows for rehearsal of the concepts from the data handling cycle, but also provides raw data for other problem solving activities linked to the overarching theme.

Students could be asked to consider the probability that upon leaving a cafeteria with food, they will pass a trash can of the necessary type before arriving at their next class (while following a direct route).

The scope here is huge, as the problem solving process includes use of estimation, modeling and measurement (of distance and time).

Algebra math problems

Designing problem solving lessons based around algebra may seem scary, but we are not setting out to prove Fermat’s Last Theorem. Start with the basics; building and using expressions.

You could use shapes made from paper, denoting side lengths with variables; as new shapes are created, students use the variables to determine linear expressions for perimeter and quadratic expressions for area (if they have met those concepts).

What we are looking for is any activity that allows students to use algebra to generalize.

We now come back to those important questions I mentioned:

“ What do I know? ”  and  “ What do I need to know? ”

Algebra problem example: Definitions and representations using algebra

When it comes to problem solving with algebra, students should build a “tool kit.” I usually begin by asking how we define an even number, an odd number; how do we represent this using algebra?

Then, move on to a square number, one more than a cube number, etc. Students can record their results and create a dictionary of algebraic phrases. Later on, we can use these to look at whether the square of an odd number is always odd, or why the square of an even number is a multiple of four.

Students can use these terms to create and manipulate a variety of polynomial expressions through addition and subtraction. They can go on to form and solve equations through inverse operations and even explore decimal and fractional terms. 

These investigations can use simple or more complex numbers and be tailored to the abilities of your class.

How to lead a problem solving lesson step by step

For any math problem, but particularly in your lessons specifically focused on problem solving, students need to be coaxed into realizing how much they know about the problem already; combining this with what they need as an outcome should create a journey that contains the steps of a solution.

I’ll use an example to illustrate this.

Problem solving example: Waring’s prime number conjecture

Waring’s prime number conjecture states that every odd number (excluding 1) is a prime or the sum of three primes.

Your students need to know certain things in order to look at this:

• The definition of an odd number
• What a prime number is

Ideally, your starter activity will include some assessment for learning (for more on these see this article on teaching strategies) and a discussion to bring these ideas to the forefront of your students’ minds.

Save mentioning that this is a named conjecture until the end of the lesson.

A possible line of inquiry could be

 “Which odd numbers less than 50 can be written as the sum of three primes?”

Depending on your class, you may need a more open question, or you may need to scaffold the problem into steps. Using sentence stems can help students justify their reasoning if they are struggling.

Ultimately, as long as students are encouraged to consider what they know about the problem already, and what their objective is, they will be able to actively engage in the problem solving process.

If you’ve taken my advice, you will have already attempted this problem; you know which numbers are tricky and which numbers have several solutions.

By the way, have you worked out why some numbers have a unique three prime sum and why some don’t? Maybe you should investigate this problem; remember math is FUN.

If, during an activity, you are asked if/why/how something works and you don’t know; be honest. Students tend to welcome honesty from their teachers. Sit down with the students and try and work it out; this is great for building relationships and encourages a collaborative approach.

It also instills in students that, as mathematicians, we try to solve problems because we don’t yet have all the answers.

Read more: Collaborative Lesson Planning

Reviewing the problem solving process

I now return to my military operation analogy: the mission debrief. Make sure you have plenty of time at the end of problem solving activities for a rich and lively discussion, with all your students involved.

Plan your questions well in advance; I think of my questions when I am having my own attempts at solving the problem. Your questioning strategy should draw out if a solution was found, and how. If a solution eluded everyone, discuss why.

Is a solution possible? Was something else needed? Is an approximate solution the best we could hope for?

It‘s well worth explaining to your students that not all problems have exact solutions; sometimes we have to optimize or estimate as best we can, and that is our solution.

Growing resilience through problem solving

The key thing is to keep reminding your students of the math skills and strategies they are using. Resilience should build over time as students encounter a wider range of problems and have to deploy their skills in different ways. It is this resilience, and lack of fear, when faced with the unfamiliar that gives students the confidence to pause and think…

• “What do I know?”        
• “What do I need?”

By asking students to consider these two questions, and using your love of math and a variety of instructional strategies to inject some fun into the solution process, you can create an environment where students engage with unfamiliar and challenging problems. Your students will become the problem solvers you want them to be.

Students will make mistakes, they will struggle, and occasionally, they will complain. It is through talking to your students about these difficulties and how to overcome them that they become stronger; this is the idea of a growth mindset .

Discuss what worked, what didn’t, mind-map strategies in groups, incorporate tasks based on collaboration and then next time, use competition. As well as improving engagement, your students will adapt to new scenarios with greater ease.

Resources to support problem solving in middle school

If your middle school students need support to apply any topics your math problems investigate, you can use Third Space Learning’s free step by step teaching and learning guides. These are available in the resource library alongside practice math questions and downloadable worksheets on topics from factorizing to fractions.

Your middle school students come to you as new recruits; they have a fear of the unknown and may lack confidence. Armed with the ideas in this article, as well as the resources and worksheets available, you will soon find your ability to build them up into problem solving soldiers that can tackle any problem on the mathematical battlefield.

READ MORE :

• 6th Grade Math Problems
• 7th Grade Math Problems
• 8th grade math problems

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

The content in this article was originally written by math teacher and author Andy Brighouse and has since been revised and adapted for US schools by elementary math teacher Katie Keeton.

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Khan Academy Blog

Unlocking the Power of Math Learning: Strategies and Tools for Success

posted on September 20, 2023

Mathematics, the foundation of all sciences and technology, plays a fundamental role in our everyday lives. Yet many students find the subject challenging, causing them to shy away from it altogether. This reluctance is often due to a lack of confidence, a misunderstanding of unclear concepts, a move ahead to more advanced skills before they are ready, and ineffective learning methods. However, with the right approach, math learning can be both rewarding and empowering. This post will explore different approaches to learning math, strategies for success, and cutting-edge tools to help you achieve your goals.

Math Learning

Math learning can take many forms, including traditional classroom instruction, online courses, and self-directed learning. A multifaceted approach to math learning can improve understanding, engage students, and promote subject mastery. A 2014 study by the National Council of Teachers of Mathematics found that the use of multiple representations, such as visual aids, graphs, and real-world examples, supports the development of mathematical connections, reasoning, and problem-solving skills.

Moreover, the importance of math learning goes beyond solving equations and formulas. Advanced math skills are essential for success in many fields, including science, engineering, finance, health care, and technology. In fact, a report by Burning Glass Technologies found that 71% of high-salary, entry-level positions require advanced math skills.

Benefits of Math Learning

In today’s 21st-century world, having a broad knowledge base and strong reading and math skills is essential. Mathematical literacy plays a crucial role in this success. It empowers individuals to comprehend the world around them and make well-informed decisions based on data-driven understanding. More than just earning good grades in math, mathematical literacy is a vital life skill that can open doors to economic opportunities, improve financial management, and foster critical thinking. We’re not the only ones who say so:

• Math learning enhances problem-solving skills, critical thinking, and logical reasoning abilities. (Source: National Council of Teachers of Mathematics )
• It improves analytical skills that can be applied in various real-life situations, such as budgeting or analyzing data. (Source: Southern New Hampshire University )
• Math learning promotes creativity and innovation by fostering a deep understanding of patterns and relationships. (Source: Purdue University )
• It provides a strong foundation for careers in fields such as engineering, finance, computer science, and more. These careers generally correlate to high wages. (Source: U.S. Bureau of Labor Statistics )
• Math skills are transferable and can be applied across different academic disciplines. (Source: Sydney School of Education and Social Work )

How to Know What Math You Need to Learn

Often students will find gaps in their math knowledge; this can occur at any age or skill level. As math learning is generally iterative, a solid foundation and understanding of the math skills that preceded current learning are key to success. The solution to these gaps is called mastery learning, the philosophy that underpins Khan Academy’s approach to education .

Mastery learning is an educational philosophy that emphasizes the importance of a student fully understanding a concept before moving on to the next one. Rather than rushing students through a curriculum, mastery learning asks educators to ensure that learners have “mastered” a topic or skill, showing a high level of proficiency and understanding, before progressing. This approach is rooted in the belief that all students can learn given the appropriate learning conditions and enough time, making it a markedly student-centered method. It promotes thoroughness over speed and encourages individualized learning paths, thus catering to the unique learning needs of each student.

Students will encounter mastery learning passively as they go through Khan Academy coursework, as our platform identifies gaps and systematically adjusts to support student learning outcomes. More details can be found in our Educators Hub . 

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How to learn math.

Learning at School

One of the most common methods of math instruction is classroom learning. In-class instruction provides students with real-time feedback, practical application, and a peer-learning environment. Teachers can personalize instruction by assessing students’ strengths and weaknesses, providing remediation when necessary, and offering advanced instruction to students who need it.

Learning at Home

Supplemental learning at home can complement traditional classroom instruction. For example, using online resources that provide additional practice opportunities, interactive games, and demonstrations, can help students consolidate learning outside of class. E-learning has become increasingly popular, with a wealth of online resources available to learners of all ages. The benefits of online learning include flexibility, customization, and the ability to work at one’s own pace. One excellent online learning platform is Khan Academy, which offers free video tutorials, interactive practice exercises, and a wealth of resources across a range of mathematical topics.

Moreover, parents can encourage and monitor progress, answer questions, and demonstrate practical applications of math in everyday life. For example, when at the grocery store, parents can ask their children to help calculate the price per ounce of two items to discover which one is the better deal. Cooking and baking with your children also provides a lot of opportunities to use math skills, like dividing a recipe in half or doubling the ingredients. 

Learning Math with the Help of Artificial Intelligence (AI) 

AI-powered tools are changing the way students learn math. Personalized feedback and adaptive practice help target individual needs. Virtual tutors offer real-time help with math concepts while AI algorithms identify areas for improvement. Custom math problems provide tailored practice, and natural language processing allows for instant question-and-answer sessions. 

Using Khan Academy’s AI Tutor, Khanmigo

Transform your child’s grasp of mathematics with Khanmigo , the 24/7 AI-powered tutor that specializes in tailored, one-on-one math instruction. Available at any time, Khanmigo provides personalized support that goes beyond mere answers to nurture genuine mathematical understanding and critical thinking. Khanmigo can track progress, identify strengths and weaknesses, and offer real-time feedback to help students stay on the right track. Within a secure and ethical AI framework, your child can tackle everything from basic arithmetic to complex calculus, all while you maintain oversight using robust parental controls.

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You can learn anything .

Math learning is essential for success in the modern world, and with the right approach, it can also be enjoyable and rewarding. Learning math requires curiosity, diligence, and the ability to connect abstract concepts with real-world applications. Strategies for effective math learning include a multifaceted approach, including classroom instruction, online courses, homework, tutoring, and personalized AI support. 

So, don’t let math anxiety hold you back; take advantage of available resources and technology to enhance your knowledge base and enjoy the benefits of math learning.

National Council of Teachers of Mathematics, “Principles to Actions: Ensuring Mathematical Success for All” , April 2014

Project Lead The Way Research Report, “The Power of Transportable Skills: Assessing the Demand and Value of the Skills of the Future” , 2020

Page. M, “Why Develop Quantitative and Qualitative Data Analysis Skills?” , 2016

Mann. EL, Creativity: The Essence of Mathematics, Journal for the Education of the Gifted. Vol. 30, No. 2, 2006, pp. 236–260, http://www.prufrock.com ’

Nakakoji Y, Wilson R.” Interdisciplinary Learning in Mathematics and Science: Transfer of Learning for 21st Century Problem Solving at University ”. J Intell. 2020 Sep 1;8(3):32. doi: 10.3390/jintelligence8030032. PMID: 32882908; PMCID: PMC7555771.

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