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Module 1: Problem Solving Strategies

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Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

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!

Looking to add creative problem solving to your class?

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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.

K-5 Math Centers

K-5 math ideas, 3rd grade math, need help organizing your k-5 math block, math problem solving 101.

Have you ever given your students a money word problem where someone buys an item from a store, but your students come up with an answer where the person that bought the item ends up with more money than he or she came in with?

Word problem solving is one of those things that many of our children struggle with.  When used effectively, questioning and dramatization can be powerful tools for our students to use when solving these types of problems.

I came up with this approach after co-teaching a lesson with a 3rd grade teachers.  Her kids were having extreme difficulty comprehending a word problem she presented.  So we devised a lesson that would help students better understand problem solving.

The approach we took included the use of several literacy skills, like reading comprehension and writing. First, we started the lesson with a “think aloud” modeled by the teacher.We read and displayed the problem below but excluded ALL of the numbers. See the images below:

The purpose of reading the problem without the numbers is to get the students to understand what is actually happening in the problem.  Typically some students focus solely on keywords when solving word problems, but I do not advise using this approach exclusively.  With math problems, the context of the problem and actions in the problem determine how the child should go about solving it.

Read the Problem Without Numbers & Ask Questions:

After reading the problem (without numbers) to the students, I asked the following questions:

• Can you describe what is happening in your own words?
• What is the main idea of the problem?
• How could you act this out?

Make a Plan & Ask Questions:

After the students articulated what was happening in the problem, we made a plan to solve the problem.  I used the following guiding questions:

• Sample Answers include- We know that Kai has some goldfish. Kai donated or gave away some of the goldfish.
• Sample Answers include – We need to know how many goldfish Kai has.  We also need to know how many he gave anyway.  We also need to know how many bowls there are.
• Sample Answers include-   We need to find out how many fish belong in each bowl.

The class discussed the answers to the questions above. As we discussed the questions above the responses were written out on a problem solving template.

As part of this process, we clarified student understanding of the problem and determined what we needed to find and do to solve the problem.  Next, we walked the students through the process of showing their work using pictures.  Lastly, we checked our answers by writing an equation that matched the pictures to finally solve the problem.

Team Work Counts

After going through the process with the class, we decided to split the students into small groups of 3 and 4 to solve a math problem together.  The groups were expected to use the same process that we used to solve the problem.  It took a while but check out one of the final products below.

Benefits to Using this Process:

• Students understood what the problem is asking them to do
• Students are required to think and communicate as a team
• Students avoid making errors that can come with only using keywords
• Students are required to record their math reasoning using the problem solving template
• After using this process a couple of times, students get used to explaining and justifying their answers
• You become the facilitator of the learning by asking more questions, thereby making students independent thinkers

Things to Consider Include:

• This process in NOT quick.  It requires TIME.  You should not rush the process and expect to have it completed in 20 – 30 minutes in one day.
• This process is not a one time lesson.  Students may not get it the first time.  It should be seen a routine that can be used when solving word problems.

Be sure to let me know how this process works in your classroom in the comments below.

• Read more about: K-5 Math Ideas

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How to Use Real-World Problems to Teach Elementary School Math: 6 Tips

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When you think back on elementary school math, do you have fond memories of the countless worksheets you completed on adding fractions or solving division problems? Probably not.

Researchers and educators have been pushing for years for schools to move away from teaching math through a set of equations with no context around them, and towards an approach that pushes kids to use numerical reasoning to solve real problems, mirroring the way that they’ll encounter the use of math as adults.

The strategy is largely about setting kids up for success in the professional world, and educators can lay the groundwork decades earlier, even in kindergarten .

Here are some tips for using a real world problem-solving approach to teaching math to elementary school students.

1. There’s more than one right answer and more than one right method

A “real world task” can be as simple as asking students to think of equations that will get them to a particular “target” number, say, 14. Students could say 7 plus 7 is 14 or they could say 25 minus 11 is 14. Neither answer is better than the other, and that lesson teaches kids that there are multiple ways to use math to solve problems.

2. Give kids a chance to explain their thinking

The process you use to solve a real world math problem can be just as important as arriving at the correct answer, said Robbi Berry, who teaches 5th grade in Las Cruces, N.M. Her students have learned not to ask her if a particular answer is correct, she said, because she’ll turn the question back on them, asking them to explain how they know that it is right. She also gives her students a chance to explain to one another how they arrived at a particular solution, “We always share our strategies so that the kids can see the different ways” to arrive at an answer, she said. Students get excited, she said, when one of their classmates comes up with an approach they never would have thought of. “Math is creative,” Berry said. “It’s not just learning and memorizing.”

3. Be willing to deal with some off-the-wall answers

Problem solving does not necessarily mean going to the word problems in your textbook, said Latrenda Knighten, a mathematics instructional coach in Baton Rouge, La. For little kids, it can be as simple as showing a group of geometric shapes and asking what they have in common. Students may go off track a bit by talking about things like color, she said, but teachers can steer them towards thinking about things like how a rectangle differs from a triangle.

4. Let your students push themselves

Tackling these richer, real-world problems can be tougher than solving equations on a worksheet. And that is a good thing, said Jo Boaler, a professor at Stanford University and an expert on math education. “It’s really good for your brain to struggle,” she said. “We don’t want kids getting right answers all the time because that’s not giving their brains a really good workout.” These types of problems require collaboration, a skill that many don’t associate with math, but that is key to how math reasoning works beyond the classroom. The complexity and difficulty of the tasks means that students “have to talk to each other and really figure out what to do, what’s a good method?”

5. Celebrate ‘favorite mistakes’ to encourage intellectual risk taking

Wrong answers should be viewed as learning opportunities, Berry said. When one of her students makes an error, she asks if she can share it with the class as a “favorite mistake.” Most of the time, students are comfortable with that, and the class will work together to figure how the misstep happened.

6. Remember there’s no such thing as a being born with a ‘math brain’

Some teachers believe that certain students are just naturally good at math, and others are not, Boaler said. But that’s not true. “Brains are constantly shaping, changing, developing, connecting, and there is no fixed anything,” said Boaler, who often works alongside neuroscientists. What’s more, many elementary school teachers lack confidence in their own math abilities, she said. “They think they can’t do [math],” Boaler said. “And they often pass those ideas on” to their students.

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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.

Shametria Routt Banks

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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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A Blog About Parenting: Coping Skills, Behavior Management and Special Needs

25 Fun Problem Solving Activities for Kids

Problem-solving activities for kids : Explore 24 fun problem-solving games and activities, and learn effective tips and strategies to teach kids problem-solving skills. If you want to explore problem-solving strategies more in-depth, you can also grab our workbook “ Problem-Solving for Kids ” (printable resource).

Problem-solving is the cognitive process of finding solutions to challenges or complex situations.

A systematic approach to problem-solving tends to include defining the problem, gathering information and data, generating potential solutions, evaluating the pros and cons of each solution, making a decision, and implementing the chosen solution.

Effective problem-solving often requires critical thinking, a good dose of creativity, and the ability to consider multiple perspectives. It may also involve identifying patterns, breaking down a problem into manageable chunks, and applying our logic to develop solutions.

Problem-solving is present in everyday situations and across all fields: business, science, personal life, and education. There is not one single aspect in our lives where we don’t need to apply our problem-solving skills.

Table of Contents

• Problem-solving steps
• Development of problem-solving in childhood
• Benefits of developing problem-solving skills
• 10 Tips to teach kids problem-solving skills
• 10 Examples of problem-solving strategies
• 25 Problem-solving activities and games for kids

Problem-Solving Steps

Some key components of problem-solving include:

• Identifying the problem Recognizing and defining the issue or challenge that needs to be addressed.
• Analyzing the problem Investigating and understanding the underlying causes, factors, and relationships related to the problem.
• Generating solutions Generating potential solutions or strategies to address the problem.
• Evaluating all possible solutions (Pros and Cons Analysis) Assessing the feasibility, effectiveness, and potential consequences of each solution. Considering the positive and negative aspects of each solution.
• Decision-making Selecting the best solution based on our analysis and judgment.
• Implementing the best solution Actioning our chosen solution
• Monitoring progress and results
• Reflecting on the outcomes Reviewing and evaluating the outcomes of the implemented solution, learning from the experience, and making adjustments if necessary.

Development of Problem-Solving Skills in Childhood

Children begin to develop problem-solving skills from a very early age, and these skills continue to develop and refine throughout childhood and adolescence.

Babies soon learn about action and reaction. And, as early as eight months, they begin to acquire an understanding of cause and effect (they shake a rattle, it makes a sound; they push a toy, it falls)

Between 13 and 24 months, they start solving simple problems through trial and error and engage in symbolic play using their imagination.

As children progress into middle childhood (ages 7-11), they develop more advanced problem-solving skills. They become capable of understanding multiple perspectives and can consider multiple factors when solving problems. They start using logic and reasoning to solve increasingly complex problems.

During adolescence (ages 12 and up), problem-solving skills continue to develop. Teenagers can generate and test hypotheses and use deductive and inductive reasoning to arrive at solutions.

Each child will develop their problem-solving skills at their own pace. Some children may show advanced problem-solving abilities at an earlier age. Others may require more time and experience to develop these skills fully.

Benefits of Developing Problem-Solving Skills in Children

Problem-solving skills in children are crucial for children’s cognitive, social, and emotional development. It equips them to approach challenges, think critically, make informed decisions, and find creative solutions. 

The benefits of good problem-solving skills in children include:

• Positive impact on self-esteem and confidence Identifying, analyzing, and solving their problems contributes to our kids’ sense of competence .
• Fosters Independence and Autonomy When our kids are able to problem-solve on their own, they take one more step toward independence
• Academic Success Problem-solving skills contribute to academic achievement, as they help students analyze and solve complex problems across various subjects.
• Cognitive Development Problem-solving fosters cognitive skills such as logical reasoning, analytical thinking, and abstract reasoning.
• Critical Thinking Problem-solving enhances critical thinking abilities, enabling children to evaluate information, identify biases, and make informed judgments.
• Creativity Problem-solving promotes creativity by encouraging children to think outside the box, generate innovative ideas, and explore multiple solutions.
• Emotional Resilience Problem-solving skills enhance emotional resilience by enabling children to manage and cope with challenges effectively, reducing stress and promoting well-being.
• Improved Social Interactions/Relationships Problem-solving abilities contribute to better social interactions, conflict resolution , and peer collaboration, promoting healthy relationships.
• Future career success Problem-solving skills are highly valued in the workplace and can positively influence future career success.

10+ Helpful Tips to Teach Kids Problem-Solving Skills

Teaching problem-solving skills to kids is an important part of their cognitive development. It helps them develop critical thinking, creativity, and resilience.

But how can we help our kids and students to develop this essential skill?

We can help our kids and students develop and improve their problem-solving skills in many ways.  These are some helpful tips that you could consider:

• Model problem-solving behavior When you see yourself in a problem-solving situation, verbalize your thought process: “I wonder how I should address this issue. I guess my alternatives could be… They all have positives and negatives….”
• Let them participate in the problem-solving situation “Could you help me solve this puzzle?”
• Provide real-life problem-solving situations Real-life scenarios make problem-solving more meaningful for kids. For example, discuss how to resolve a conflict with a sibling or how to make the morning routine smoother.
• Teach them how to break down problems Show them how to break down complex problems into manageable sub-problems.
• Practice brainstorming Create brainstorming situations where all the family (or the classroom) can contribute to solving a problem
• Teach the value of perseverance Sometimes, we must stick to a situation and persevere before finding a solution. Encourage kids to persevere through challenges and setbacks, emphasizing that mistakes and failures are opportunities for learning.
• Encourage critical thinking Encourage kids to analyze situations, consider different perspectives, and evaluate possible outcomes.
• How could we make your school lunch healthier but still yummy?
• How could we reuse/recycle all this paper?
• What could we do to help you remember all the steps in your night routine?
• Encourage reflection When they can find a solution for a problem, don’t jump to solve it for them. Encourage them to reflect on the problem and find and evaluate alternatives. And after a problem is solved, think about the whole process and the learnings. “How did this work?” “What did you learn” “Do you need to change anything?”
• Foster creativity Provide them with opportunities for imaginative play, creative projects, and brainstorming sessions.
• Teach the value of teamwork Teach kids the importance of working together to solve problems. Engage them in group activities or projects that require teamwork and collaboration. This helps kids learn the value of different perspectives and work together towards an objective while they practice their communication skills.
• Teach decision-making skills Teach kids how to approach problems systematically by going through the steps we have mentioned in our first section.
• Encourage both structured and free play. Structured play can help you create good problem-solving situations, while free play will foster creativity.

Developing problem-solving skills is an ongoing process that will also continue in adulthood. Provide your kids with guidance and support, and celebrate their efforts and achievements along the way.

10 Examples of Problem-Solving Strategies

There are different strategies that can help us solve a wide range of problems. Here are some commonly recognized problem-solving strategies:

1 . Trial and Error : This is the first problem strategy that we ever learn. We start using trial and error strategies in infancy, and it continues serving its purpose in many situations. This strategy involves trying different solutions or approaches and learning from the errors or failures until a successful solution is found.

2. Algorithm: An algorithm is a step-by-step procedure or a set of rules that guarantees a solution to a specific problem. It is a systematic approach to problem-solving that follows a predetermined set of instructions.

3. Heuristics: Heuristics are mental shortcuts or rules of thumb that help simplify problem-solving by providing quick and efficient strategies. While heuristics can be effective in many situations, they may also lead to biases and errors.

4. Divide and Conquer: This strategy involves breaking down a complex problem into smaller, more manageable chunks or steps that make the overall problem easier to tackle.

5. Working Backwards: This strategy involves starting from the desired outcome and working backward to determine the steps or actions needed to reach that outcome. We often use this problem-solving strategy when we set goals.

6. Analogical Reasoning: Analogical reasoning involves drawing parallels between the current problem and a similar problem that has been solved in the past. By applying the solution from the previous problem to the current one, individuals can find a solution more efficiently.

7. Brainstorming: Brainstorming gets lots of brains working on the same problem. It is a great collaborative problem-solving strategy that can bring different perspectives and experiences to the table and may result in lots of creative ideas and solutions. 

8. Decision Matrix: A decision matrix is a systematic approach to evaluating and comparing different options or solutions. It involves creating a matrix that lists alternatives and the criteria for evaluation. It assigns weights or scores to each criterion to come up with the optimal alternative.

9. Root Cause Analysis: Sometimes, we need to understand what is causing a problem before we can attempt to solve it, as different causes may require different approaches (for example, when you are sick, your doctor may need to understand what is causing the problem before prescribing a medicine)

10. Simulation and Modeling: Simulation involves creating a simplified representation or model of a problem situation to gain insights and test different scenarios.

Our choice of strategy will depend on the problem, available resources, and our own personal preferences and circumstances. We may also need to combine strategies or apply different ones to different aspects of a complex problem.

(Disclosure: We are a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for us to earn fees by linking to Amazon.com and affiliated sites. You can also read our Disclosure & Disclaimer policy  here )

Best Problem-Solving Activities for Kids

Play-based activities are centered around play and are designed to engage children in active learning and exploration. And fun problem-solving activities are a great way to develop children’s critical thinking, creativity, and decision-making skills.

In this section, we will review some problem-solving games and activities that will engage your kids’ critical-thinking skills and creativity.

1. Puzzle Games Puzzles are a fun activity for children of all ages. Young children will enjoy simple puzzles, while older children (and adults!) can have fun with more complex ones. Encourage them to use logical thinking and problem-solving strategies to complete the puzzles.

2. Crosswords A crossword is another fun type of puzzle and a good source of mental stimulation.

3. Sudoku Sudoku is a popular logic-based puzzle that involves filling a grid with numbers.

It can be extremely easy or very challenging, adaptable even for young learners.

Let’s go now for a couple of building challenges!

4. Build the Tallest Tower Give the child a set of materials (Legos, building blocks, wooden blocks, or other construction materials) and ask them to build the tallest tower they can. This simple game will encourage them to problem-solve as they build and figure out how to make the tower stable.

5. Build Towers with Different Materials Ask your child to build three different towers with different materials. Then assess how stable they are and how much weight they can hold. Analyze the pros and cons of using each type of material.

6. Treasure Hunt Set up a treasure hunt with clues leading to hidden objects or rewards. Children will have to follow the clues and solve puzzles to find the ultimate prize. This activity encourages problem-solving, critical thinking, and teamwork.

7. Scavenger Hunt Playing Scavenger Hunt can be a fun way for our kids to put their creative problem-solving skills to good use. Provide them with clues and puzzles that they must solve in order to find the next clue.

8. Mystery Bag Fill a bag with random objects and ask children to come up with creative uses for each item. Encourage them to think outside the box and find innovative solutions.

9. Memory Game While memory games primarily focus on memory retention and recall, they can indirectly contribute to problem-solving skills by developing cognitive abilities such as attention, information processing, and adjusting their strategies.

10. Role-Playing Scenarios Create role-playing scenarios where children have to solve a problem or make decisions. For example, pretend to be stranded on a desert island and ask them to decide what items they will take and how they will survive.

11. Role-Play Social Situations Work in developing social skills with social problem-solving situations.

12. Brainstorming Sessions Choose a topic or problem and hold brainstorming sessions where children can generate as many ideas as possible. Encourage them not to limit themselves (even if alternatives feel unfeasible!)

13. Team Building Activities and Games Engage children in team-building games like building a balloon tower. Each team member will need to collaborate, communicate, and problem-solve together to complete the project.

14. Escape Rooms An escape room is a super fun team problem-solving activity.

In an escape room, participants are locked inside a themed room and must work together to solve puzzles, find clues, and accomplish tasks within a given time limit in order to “escape” from the room.

15. Science Experiments Conduct simple science experiments that involve problem-solving. For example, in the classic “sink or float” experiment, children predict and test which objects will sink or float in water.

Problem-Solving Board Games

There are many board games that will test our kids problems solving activities. These are just a few examples:

16. Cluedo Players must solve a murder mystery by deducing the murderer, the weapon used, and the location of the crime. Players collect and examine clues to eliminate possibilities and make logical deductions.

17. Codenames Another classic game where players are split into two teams and must guess words based on clues from their teammates.

There are many codenames games available, including themes like Disney or Harry Potter.

18. Mastermind Game In this strategy game players take turns setting and solving secret codes

19. Scrabble Scrabble is a classic word game where players form words on a game board using letter tiles.

Kids must use their problem-solving skills to analyze the available letters, consider the best word combination and strategically place those words to score the highest points.

Learning Problem-Solving with Card Games

Card games provide opportunities for kids to develop problem-solving skills such as strategy, memory, pattern recognition, decision-making, and observation.

Just a couple of examples:

20. Uno Uno is a classic card game where kids match cards based on color or number. They need to assess their cards, strategize and make decisions about which cards to play to get rid of their cards while also considering the cards in their opponents’ hands.

21. Go Fish Go Fish is a classic card game where players try to collect sets of cards by asking other players if they have specific cards. Players need to remember which cards they have and make decisions about who to ask and what sets to pursue.

22. Coding Challenges Introduce children to coding activities using platforms like Scratch (or ScratchJr for younger kids), Code.org, or Tynker. Coding involves problem-solving and logical thinking, and children can create interactive stories, games, or animations.

23. Outdoor Problem Solving Take children outside and present them with challenges that require problem-solving, such as building a shelter using natural materials or finding their way through an obstacle course.

24. Problem-Solving Worksheets Help your child follow a systematic approach to problem-solving with these helpful worksheets

25. Goal-Setting Activities for Kids Learning to set goals and make plans to achieve them is also a problem-solving activity. I have several resources to teach kids about goal-setting that I will list below:

• Goal-Setting Activities for Kids
• SMART Goals for Kids
• Goal Tracker Thermometer

Remember to provide guidance and support during these activities while encouraging children to think independently and come up with their own solutions.

Problem-Solving Worksheets

Looking for kid-friendly examples of problem-solving strategies ?

This workbook explores the following  problem-solving strategies  (with child-friendly examples and activities):

• Trial and Error
• Heuristics (Clever shortcuts)
• Divide and Conquer
• Working Backwards
• Brainstorming
• Decision Matrix
• Root Cause Analysis
• Systematic problem-solving

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120 Math Word Problems To Challenge Students Grades 1 to 8

Written by Marcus Guido

Hey teachers! 👋

Use Prodigy to spark a love for math in your students – including when solving word problems!

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You sit at your desk, ready to put a math quiz, test or activity together. The questions flow onto the document until you hit a section for word problems.

A jolt of creativity would help. But it doesn’t come.

Whether you’re a 3rd grade teacher or an 8th grade teacher preparing students for high school, translating math concepts into real world examples can certainly be a challenge.

This resource is your jolt of creativity. It provides examples and templates of math word problems for 1st to 8th grade classes.

There are 120 examples in total.

The list of examples is supplemented by tips to create engaging and challenging math word problems.

120 Math word problems, categorized by skill

Addition word problems.

Best for: 1st grade, 2nd grade

1. Adding to 10: Ariel was playing basketball. 1 of her shots went in the hoop. 2 of her shots did not go in the hoop. How many shots were there in total?

2. Adding to 20: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store to get 3 more pieces of gum. How many pieces of gum does Adrianna have now?

3. Adding to 100: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store and got 70 pieces of strawberry gum and 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

4. Adding Slightly over 100: The restaurant has 175 normal chairs and 20 chairs for babies. How many chairs does the restaurant have in total?

5. Adding to 1,000: How many cookies did you sell if you sold 320 chocolate cookies and 270 vanilla cookies?

6. Adding to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In June, the hobby store sold 15,498 more trading cards than normal. In total, how many trading cards did the hobby store sell in June?

7. Adding 3 Numbers: Billy had 2 books at home. He went to the library to take out 2 more books. He then bought 1 book. How many books does Billy have now?

8. Adding 3 Numbers to and over 100: Ashley bought a big bag of candy. The bag had 102 blue candies, 100 red candies and 94 green candies. How many candies were there in total?

Subtraction word problems

Best for: 1st grade, second grade

9. Subtracting to 10: There were 3 pizzas in total at the pizza shop. A customer bought 1 pizza. How many pizzas are left?

10. Subtracting to 20: Your friend said she had 11 stickers. When you helped her clean her desk, she only had a total of 10 stickers. How many stickers are missing?

11. Subtracting to 100: Adrianna has 100 pieces of gum to share with her friends. When she went to the park, she shared 10 pieces of strawberry gum. When she left the park, Adrianna shared another 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

Practice math word problems with Prodigy Math

Join millions of teachers using Prodigy to make learning fun and differentiate instruction as they answer in-game questions, including math word problems from 1st to 8th grade!

12. Subtracting Slightly over 100: Your team scored a total of 123 points. 67 points were scored in the first half. How many were scored in the second half?

13. Subtracting to 1,000: Nathan has a big ant farm. He decided to sell some of his ants. He started with 965 ants. He sold 213. How many ants does he have now?

14. Subtracting to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In July, the hobby store sold a total of 20,777 trading cards. How many more trading cards did the hobby store sell in July compared with a normal month?

15. Subtracting 3 Numbers: Charlene had a pack of 35 pencil crayons. She gave 6 to her friend Theresa. She gave 3 to her friend Mandy. How many pencil crayons does Charlene have left?

16. Subtracting 3 Numbers to and over 100: Ashley bought a big bag of candy to share with her friends. In total, there were 296 candies. She gave 105 candies to Marissa. She also gave 86 candies to Kayla. How many candies were left?

Multiplication word problems

Best for: 2nd grade, 3rd grade

17. Multiplying 1-Digit Integers: Adrianna needs to cut a pan of brownies into pieces. She cuts 6 even columns and 3 even rows into the pan. How many brownies does she have?

18. Multiplying 2-Digit Integers: A movie theatre has 25 rows of seats with 20 seats in each row. How many seats are there in total?

19. Multiplying Integers Ending with 0: A clothing company has 4 different kinds of sweatshirts. Each year, the company makes 60,000 of each kind of sweatshirt. How many sweatshirts does the company make each year?

20. Multiplying 3 Integers: A bricklayer stacks bricks in 2 rows, with 10 bricks in each row. On top of each row, there is a stack of 6 bricks. How many bricks are there in total?

21. Multiplying 4 Integers: Cayley earns $5 an hour by delivering newspapers. She delivers newspapers 3 days each week, for 4 hours at a time. After delivering newspapers for 8 weeks, how much money will Cayley earn?

Division word problems

Best for: 3rd grade, 4th grade, 5th grade

22. Dividing 1-Digit Integers: If you have 4 pieces of candy split evenly into 2 bags, how many pieces of candy are in each bag?

23. Dividing 2-Digit Integers: If you have 80 tickets for the fair and each ride costs 5 tickets, how many rides can you go on?

24. Dividing Numbers Ending with 0: The school has $20,000 to buy new computer equipment. If each piece of equipment costs $50, how many pieces can the school buy in total?

25. Dividing 3 Integers: Melissa buys 2 packs of tennis balls for $12 in total. All together, there are 6 tennis balls. How much does 1 pack of tennis balls cost? How much does 1 tennis ball cost?

26. Interpreting Remainders: An Italian restaurant receives a shipment of 86 veal cutlets. If it takes 3 cutlets to make a dish, how many cutlets will the restaurant have left over after making as many dishes as possible?

Mixed operations word problems

27. Mixing Addition and Subtraction: There are 235 books in a library. On Monday, 123 books are taken out. On Tuesday, 56 books are brought back. How many books are there now?

28. Mixing Multiplication and Division: There is a group of 10 people who are ordering pizza. If each person gets 2 slices and each pizza has 4 slices, how many pizzas should they order?

29. Mixing Multiplication, Addition and Subtraction: Lana has 2 bags with 2 marbles in each bag. Markus has 2 bags with 3 marbles in each bag. How many more marbles does Markus have?

30. Mixing Division, Addition and Subtraction: Lana has 3 bags with the same amount of marbles in them, totaling 12 marbles. Markus has 3 bags with the same amount of marbles in them, totaling 18 marbles. How many more marbles does Markus have in each bag?

Ordering and number sense word problems

31. Counting to Preview Multiplication: There are 2 chalkboards in your classroom. If each chalkboard needs 2 pieces of chalk, how many pieces do you need in total?

32. Counting to Preview Division: There are 3 chalkboards in your classroom. Each chalkboard has 2 pieces of chalk. This means there are 6 pieces of chalk in total. If you take 1 piece of chalk away from each chalkboard, how many will there be in total?

33. Composing Numbers: What number is 6 tens and 10 ones?

34. Guessing Numbers: I have a 7 in the tens place. I have an even number in the ones place. I am lower than 74. What number am I?

35. Finding the Order: In the hockey game, Mitchell scored more points than William but fewer points than Auston. Who scored the most points? Who scored the fewest points?

Fractions word problems

Best for: 3rd grade, 4th grade, 5th grade, 6th grade

36. Finding Fractions of a Group: Julia went to 10 houses on her street for Halloween. 5 of the houses gave her a chocolate bar. What fraction of houses on Julia’s street gave her a chocolate bar?

37. Finding Unit Fractions: Heather is painting a portrait of her best friend, Lisa. To make it easier, she divides the portrait into 6 equal parts. What fraction represents each part of the portrait?

38. Adding Fractions with Like Denominators: Noah walks ⅓ of a kilometre to school each day. He also walks ⅓ of a kilometre to get home after school. How many kilometres does he walk in total?

39. Subtracting Fractions with Like Denominators: Last week, Whitney counted the number of juice boxes she had for school lunches. She had ⅗ of a case. This week, it’s down to ⅕ of a case. How much of the case did Whitney drink?

40. Adding Whole Numbers and Fractions with Like Denominators: At lunchtime, an ice cream parlor served 6 ¼ scoops of chocolate ice cream, 5 ¾ scoops of vanilla and 2 ¾ scoops of strawberry. How many scoops of ice cream did the parlor serve in total?

41. Subtracting Whole Numbers and Fractions with Like Denominators: For a party, Jaime had 5 ⅓ bottles of cola for her friends to drink. She drank ⅓ of a bottle herself. Her friends drank 3 ⅓. How many bottles of cola does Jaime have left?

42. Adding Fractions with Unlike Denominators: Kevin completed ½ of an assignment at school. When he was home that evening, he completed ⅚ of another assignment. How many assignments did Kevin complete?

43. Subtracting Fractions with Unlike Denominators: Packing school lunches for her kids, Patty used ⅞ of a package of ham. She also used ½ of a package of turkey. How much more ham than turkey did Patty use?

44. Multiplying Fractions: During gym class on Wednesday, the students ran for ¼ of a kilometre. On Thursday, they ran ½ as many kilometres as on Wednesday. How many kilometres did the students run on Thursday? Write your answer as a fraction.

45. Dividing Fractions: A clothing manufacturer uses ⅕ of a bottle of colour dye to make one pair of pants. The manufacturer used ⅘ of a bottle yesterday. How many pairs of pants did the manufacturer make?

46. Multiplying Fractions with Whole Numbers: Mark drank ⅚ of a carton of milk this week. Frank drank 7 times more milk than Mark. How many cartons of milk did Frank drink? Write your answer as a fraction, or as a whole or mixed number.

Decimals word problems

Best for: 4th grade, 5th grade

47. Adding Decimals: You have 2.6 grams of yogurt in your bowl and you add another spoonful of 1.3 grams. How much yogurt do you have in total?

48. Subtracting Decimals: Gemma had 25.75 grams of frosting to make a cake. She decided to use only 15.5 grams of the frosting. How much frosting does Gemma have left?

49. Multiplying Decimals with Whole Numbers: Marshall walks a total of 0.9 kilometres to and from school each day. After 4 days, how many kilometres will he have walked?

50. Dividing Decimals by Whole Numbers: To make the Leaning Tower of Pisa from spaghetti, Mrs. Robinson bought 2.5 kilograms of spaghetti. Her students were able to make 10 leaning towers in total. How many kilograms of spaghetti does it take to make 1 leaning tower?

51. Mixing Addition and Subtraction of Decimals: Rocco has 1.5 litres of orange soda and 2.25 litres of grape soda in his fridge. Antonio has 1.15 litres of orange soda and 0.62 litres of grape soda. How much more soda does Rocco have than Angelo?

52. Mixing Multiplication and Division of Decimals: 4 days a week, Laura practices martial arts for 1.5 hours. Considering a week is 7 days, what is her average practice time per day each week?

Comparing and sequencing word problems

Best for: Kindergarten, 1st grade, 2nd grade

53. Comparing 1-Digit Integers: You have 3 apples and your friend has 5 apples. Who has more?

54. Comparing 2-Digit Integers: You have 50 candies and your friend has 75 candies. Who has more?

55. Comparing Different Variables: There are 5 basketballs on the playground. There are 7 footballs on the playground. Are there more basketballs or footballs?

56. Sequencing 1-Digit Integers: Erik has 0 stickers. Every day he gets 1 more sticker. How many days until he gets 3 stickers?

57. Skip-Counting by Odd Numbers: Natalie began at 5. She skip-counted by fives. Could she have said the number 20?

58. Skip-Counting by Even Numbers: Natasha began at 0. She skip-counted by eights. Could she have said the number 36?

59. Sequencing 2-Digit Numbers: Each month, Jeremy adds the same number of cards to his baseball card collection. In January, he had 36. 48 in February. 60 in March. How many baseball cards will Jeremy have in April?

Time word problems

66. Converting Hours into Minutes: Jeremy helped his mom for 1 hour. For how many minutes was he helping her?

69. Adding Time: If you wake up at 7:00 a.m. and it takes you 1 hour and 30 minutes to get ready and walk to school, at what time will you get to school?

70. Subtracting Time: If a train departs at 2:00 p.m. and arrives at 4:00 p.m., how long were passengers on the train for?

71. Finding Start and End Times: Rebecca left her dad’s store to go home at twenty to seven in the evening. Forty minutes later, she was home. What time was it when she arrived home?

Money word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade, 5th grade

60. Adding Money: Thomas and Matthew are saving up money to buy a video game together. Thomas has saved $30. Matthew has saved $35. How much money have they saved up together in total?

61. Subtracting Money: Thomas has $80 saved up. He uses his money to buy a video game. The video game costs $67. How much money does he have left?

62. Multiplying Money: Tim gets $5 for delivering the paper. How much money will he have after delivering the paper 3 times?

63. Dividing Money: Robert spent $184.59 to buy 3 hockey sticks. If each hockey stick was the same price, how much did 1 cost?

64. Adding Money with Decimals: You went to the store and bought gum for $1.25 and a sucker for $0.50. How much was your total?

65. Subtracting Money with Decimals: You went to the store with $5.50. You bought gum for $1.25, a chocolate bar for $1.15 and a sucker for $0.50. How much money do you have left?

67. Applying Proportional Relationships to Money: Jakob wants to invite 20 friends to his birthday, which will cost his parents $250. If he decides to invite 15 friends instead, how much money will it cost his parents? Assume the relationship is directly proportional.

68. Applying Percentages to Money: Retta put $100.00 in a bank account that gains 20% interest annually. How much interest will be accumulated in 1 year? And if she makes no withdrawals, how much money will be in the account after 1 year?

Physical measurement word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade

72. Comparing Measurements: Cassandra’s ruler is 22 centimetres long. April’s ruler is 30 centimetres long. How many centimetres longer is April’s ruler?

73. Contextualizing Measurements: Picture a school bus. Which unit of measurement would best describe the length of the bus? Centimetres, metres or kilometres?

74. Adding Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Last year, Micha’s dad used 100 litres of gas. This year, her dad used 90 litres of gas. How much gas did he use in total for the two years?

75. Subtracting Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Over the past two years, Micha’s dad used 200 litres of gas. This year, he used 100 litres of gas. How much gas did he use last year?

76. Multiplying Volume and Mass: Kiera wants to make sure she has strong bones, so she drinks 2 litres of milk every week. After 3 weeks, how many litres of milk will Kiera drink?

77. Dividing Volume and Mass: Lillian is doing some gardening, so she bought 1 kilogram of soil. She wants to spread the soil evenly between her 2 plants. How much will each plant get?

78. Converting Mass: Inger goes to the grocery store and buys 3 squashes that each weigh 500 grams. How many kilograms of squash did Inger buy?

79. Converting Volume: Shad has a lemonade stand and sold 20 cups of lemonade. Each cup was 500 millilitres. How many litres did Shad sell in total?

80. Converting Length: Stacy and Milda are comparing their heights. Stacy is 1.5 meters tall. Milda is 10 centimetres taller than Stacy. What is Milda’s height in centimetres?

81. Understanding Distance and Direction: A bus leaves the school to take students on a field trip. The bus travels 10 kilometres south, 10 kilometres west, another 5 kilometres south and 15 kilometres north. To return to the school, in which direction does the bus have to travel? How many kilometres must it travel in that direction?

Ratios and percentages word problems

Best for: 4th grade, 5th grade, 6th grade

82. Finding a Missing Number: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. Jenny has 28 trophies. How many does Meredith have?

83. Finding Missing Numbers: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. The difference between the numbers is 12. What are the numbers?

84. Comparing Ratios: The school’s junior band has 10 saxophone players and 20 trumpet players. The school’s senior band has 18 saxophone players and 29 trumpet players. Which band has the higher ratio of trumpet to saxophone players?

85. Determining Percentages: Mary surveyed students in her school to find out what their favourite sports were. Out of 1,200 students, 455 said hockey was their favourite sport. What percentage of students said hockey was their favourite sport?

86. Determining Percent of Change: A decade ago, Oakville’s population was 67,624 people. Now, it is 190% larger. What is Oakville’s current population?

87. Determining Percents of Numbers: At the ice skate rental stand, 60% of 120 skates are for boys. If the rest of the skates are for girls, how many are there?

88. Calculating Averages: For 4 weeks, William volunteered as a helper for swimming classes. The first week, he volunteered for 8 hours. He volunteered for 12 hours in the second week, and another 12 hours in the third week. The fourth week, he volunteered for 9 hours. For how many hours did he volunteer per week, on average?

Probability and data relationships word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade

89. Understanding the Premise of Probability: John wants to know his class’s favourite TV show, so he surveys all of the boys. Will the sample be representative or biased?

90. Understanding Tangible Probability: The faces on a fair number die are labelled 1, 2, 3, 4, 5 and 6. You roll the die 12 times. How many times should you expect to roll a 1?

91. Exploring Complementary Events: The numbers 1 to 50 are in a hat. If the probability of drawing an even number is 25/50, what is the probability of NOT drawing an even number? Express this probability as a fraction.

92. Exploring Experimental Probability: A pizza shop has recently sold 15 pizzas. 5 of those pizzas were pepperoni. Answering with a fraction, what is the experimental probability that he next pizza will be pepperoni?

93. Introducing Data Relationships: Maurita and Felice each take 4 tests. Here are the results of Maurita’s 4 tests: 4, 4, 4, 4. Here are the results for 3 of Felice’s 4 tests: 3, 3, 3. If Maurita’s mean for the 4 tests is 1 point higher than Felice’s, what’s the score of Felice’s 4th test?

94. Introducing Proportional Relationships: Store A is selling 7 pounds of bananas for $7.00. Store B is selling 3 pounds of bananas for $6.00. Which store has the better deal?

95. Writing Equations for Proportional Relationships: Lionel loves soccer, but has trouble motivating himself to practice. So, he incentivizes himself through video games. There is a proportional relationship between the amount of drills Lionel completes, in x , and for how many hours he plays video games, in y . When Lionel completes 10 drills, he plays video games for 30 minutes. Write the equation for the relationship between x and y .

Geometry word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade, 8th grade

96. Introducing Perimeter:  The theatre has 4 chairs in a row. There are 5 rows. Using rows as your unit of measurement, what is the perimeter?

97. Introducing Area: The theatre has 4 chairs in a row. There are 5 rows. How many chairs are there in total?

98. Introducing Volume: Aaron wants to know how much candy his container can hold. The container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. What is the container’s volume?

99. Understanding 2D Shapes: Kevin draws a shape with 4 equal sides. What shape did he draw?

100. Finding the Perimeter of 2D Shapes: Mitchell wrote his homework questions on a piece of square paper. Each side of the paper is 8 centimetres. What is the perimeter?

101. Determining the Area of 2D Shapes: A single trading card is 9 centimetres long by 6 centimetres wide. What is its area?

102. Understanding 3D Shapes: Martha draws a shape that has 6 square faces. What shape did she draw?

103. Determining the Surface Area of 3D Shapes: What is the surface area of a cube that has a width of 2cm, height of 2 cm and length of 2 cm?

104. Determining the Volume of 3D Shapes: Aaron’s candy container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. Bruce’s container is 25 centimetres tall, 9 centimetres long and 9 centimetres wide. Find the volume of each container. Based on volume, whose container can hold more candy?

105. Identifying Right-Angled Triangles: A triangle has the following side lengths: 3 cm, 4 cm and 5 cm. Is this triangle a right-angled triangle?

106. Identifying Equilateral Triangles: A triangle has the following side lengths: 4 cm, 4 cm and 4 cm. What kind of triangle is it?

107. Identifying Isosceles Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 5 cm. What kind of triangle is it?

108. Identifying Scalene Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 6 cm. What kind of triangle is it?

109. Finding the Perimeter of Triangles: Luigi built a tent in the shape of an equilateral triangle. The perimeter is 21 metres. What is the length of each of the tent’s sides?

110. Determining the Area of Triangles: What is the area of a triangle with a base of 2 units and a height of 3 units?

111. Applying Pythagorean Theorem: A right triangle has one non-hypotenuse side length of 3 inches and the hypotenuse measures 5 inches. What is the length of the other non-hypotenuse side?

112. Finding a Circle’s Diameter: Jasmin bought a new round backpack. Its area is 370 square centimetres. What is the round backpack’s diameter?

113. Finding a Circle's Area: Captain America’s circular shield has a diameter of 76.2 centimetres. What is the area of his shield?

114. Finding a Circle’s Radius: Skylar lives on a farm, where his dad keeps a circular corn maze. The corn maze has a diameter of 2 kilometres. What is the maze’s radius?

Variables word problems

Best for: 6th grade, 7th grade, 8th grade

115. Identifying Independent and Dependent Variables: Victoria is baking muffins for her class. The number of muffins she makes is based on how many classmates she has. For this equation, m is the number of muffins and c is the number of classmates. Which variable is independent and which variable is dependent?

116. Writing Variable Expressions for Addition: Last soccer season, Trish scored g goals. Alexa scored 4 more goals than Trish. Write an expression that shows how many goals Alexa scored.

117. Writing Variable Expressions for Subtraction: Elizabeth eats a healthy, balanced breakfast b times a week. Madison sometimes skips breakfast. In total, Madison eats 3 fewer breakfasts a week than Elizabeth. Write an expression that shows how many times a week Madison eats breakfast.

118. Writing Variable Expressions for Multiplication: Last hockey season, Jack scored g goals. Patrik scored twice as many goals than Jack. Write an expression that shows how many goals Patrik scored.

119. Writing Variable Expressions for Division: Amanda has c chocolate bars. She wants to distribute the chocolate bars evenly among 3 friends. Write an expression that shows how many chocolate bars 1 of her friends will receive.

120. Solving Two-Variable Equations: This equation shows how the amount Lucas earns from his after-school job depends on how many hours he works: e = 12h . The variable h represents how many hours he works. The variable e represents how much money he earns. How much money will Lucas earn after working for 6 hours?

How to easily make your own math word problems & word problems worksheets

Armed with 120 examples to spark ideas, making your own math word problems can engage your students and ensure alignment with lessons. Do:

• Link to Student Interests:  By framing your word problems with student interests, you’ll likely grab attention. For example, if most of your class loves American football, a measurement problem could involve the throwing distance of a famous quarterback.
• Make Questions Topical:  Writing a word problem that reflects current events or issues can engage students by giving them a clear, tangible way to apply their knowledge.
• Include Student Names:  Naming a question’s characters after your students is an easy way make subject matter relatable, helping them work through the problem.
• Be Explicit:  Repeating keywords distills the question, helping students focus on the core problem.
• Test Reading Comprehension:  Flowery word choice and long sentences can hide a question’s key elements. Instead, use concise phrasing and grade-level vocabulary.
• Focus on Similar Interests:  Framing too many questions with related interests -- such as football and basketball -- can alienate or disengage some students.
• Feature Red Herrings:  Including unnecessary information introduces another problem-solving element, overwhelming many elementary students.

A key to differentiated instruction , word problems that students can relate to and contextualize will capture interest more than generic and abstract ones.

Final thoughts about math word problems

You’ll likely get the most out of this resource by using the problems as templates, slightly modifying them by applying the above tips. In doing so, they’ll be more relevant to -- and engaging for -- your students.

Regardless, having 120 curriculum-aligned math word problems at your fingertips should help you deliver skill-building challenges and thought-provoking assessments.

The result?

A greater understanding of how your students process content and demonstrate understanding, informing your ongoing teaching approach.

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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.

Related articles

Why Student Centered Learning Is Important: A Guide For Educators

13 Effective Learning Strategies: A Guide to Using them in your Math Classroom

Differentiated Instruction: 9 Differentiated Curriculum And Instruction Strategies For Teachers 

5 Math Mastery Strategies To Incorporate Into Your 4th and 5th Grade Classrooms

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Home » SEL Implementation » Developing Critical Thinking: Problem Solving Scenarios for Elementary Students

Developing Critical Thinking: Problem Solving Scenarios for Elementary Students

Key takeaways.

• Developing critical thinking skills in elementary students enhances their cognitive abilities, academic performance, and prepares them for future challenges.
• Problem-solving scenarios are effective tools for fostering critical thinking and can be categorized into everyday life, academic, and social scenarios.
• For successful implementation, scenarios should be relatable, age-appropriate, encourage open-ended thinking, and promote collaboration.
• Incorporating problem-solving scenarios into the curriculum improves students’ analytical thinking, problem-solving abilities, empathy, and social skills.

Introduction: Developing Critical Thinking: Problem-Solving Scenarios for Elementary Students

In today’s post, we will explore the importance of developing critical thinking skills in elementary students and how problem-solving scenarios can be an effective tool for fostering these skills. As educators and parents, it is crucial to provide opportunities for children to think critically and solve problems independently. Let’s dive in!

Understanding Critical Thinking

Before we delve into problem-solving scenarios, let’s first understand what critical thinking is and why it is essential for elementary students. Critical thinking is the ability to analyze information, evaluate arguments, and make reasoned decisions. It involves skills such as logical reasoning, problem-solving, and decision-making.

Developing critical thinking skills in elementary students has numerous benefits. It enhances their cognitive abilities, improves academic performance, and prepares them for future challenges. Critical thinkers are more likely to excel in problem-solving tasks, communicate effectively, and exhibit creativity.

Problem Solving Scenarios for Elementary Students

Problem-solving scenarios are real-life or hypothetical situations that require students to think critically and come up with solutions. These scenarios provide a practical and engaging way for students to apply their critical thinking skills. By presenting them with challenging problems, we can encourage them to think deeply, analyze information, and make informed decisions.

When creating problem-solving scenarios, keep in mind the following tips:

• Make the scenarios relatable to students’ lives and experiences.
• Ensure the scenarios are age-appropriate and aligned with the curriculum.
• Encourage open-ended thinking by avoiding simple “yes” or “no” answers.
• Provide opportunities for collaboration and discussion among students.

Types of Problem-Solving Scenarios

Problem-solving scenarios can be categorized into three main types: everyday life scenarios, academic scenarios, and social scenarios. Let’s explore each type and discuss how they promote critical thinking in elementary students.

Everyday Life Scenarios

Everyday life scenarios are situations that students encounter in their daily lives. These scenarios can range from simple problems, such as deciding what to wear on a rainy day, to more complex issues, such as resolving conflicts with friends. By presenting students with everyday life scenarios, we can help them develop problem-solving skills that are applicable to real-life situations.

Here are a few examples of everyday life scenarios for elementary students:

• You are at the grocery store, and you realize you don’t have enough money to buy everything on your list. What do you do?
• Your friend is upset because someone in your class said something mean to them. How can you help your friend feel better?
• You are playing a game with your friends, and they want to change the rules. What should you do?

To encourage students to analyze and solve everyday problems, ask open-ended questions such as:

• What are the possible solutions to this problem?
• What are the consequences of each solution?
• Which solution do you think is the best, and why?

Academic Scenarios

Academic scenarios focus on challenges related to school subjects and learning. These scenarios help students apply critical thinking skills to academic tasks and develop a deeper understanding of the subject matter. By engaging students in academic problem-solving, we can enhance their analytical thinking and problem-solving abilities.

Here are a few examples of academic scenarios for elementary students:

• You are given a math problem that you don’t understand. How can you figure it out?
• You are reading a story, and there is a word you don’t know. What strategies can you use to figure out the meaning?
• You are working on a science experiment, and your results are different from what you expected. How can you troubleshoot the problem?

To guide students in applying critical thinking to academic challenges, encourage them to:

• Break down the problem into smaller parts.
• Identify relevant information and resources.
• Generate multiple possible solutions.
• Evaluate the effectiveness of each solution.

Social Scenarios

Social scenarios involve situations that require students to navigate social interactions and relationships. These scenarios help students develop empathy, perspective-taking, and conflict resolution skills. By engaging students in social problem-solving, we can foster their emotional intelligence and promote positive social behaviors.

Here are a few examples of social scenarios for elementary students:

• You and your friend want to play with the same toy. How can you solve this problem without fighting?
• You see someone being left out during recess. What can you do to include them?
• Your classmate is being teased by another student. How can you support your classmate?

To teach students to navigate social situations using critical thinking, encourage them to:

• Consider the feelings and perspectives of others.
• Brainstorm possible solutions that are fair and respectful.
• Reflect on the consequences of their actions.
• Seek guidance from trusted adults when needed.

Implementing Problem-Solving Scenarios in the Classroom

Now that we understand the different types of problem-solving scenarios, let’s explore how we can effectively implement them in the classroom.

Integrating problem-solving scenarios into the curriculum can be done through various methods:

• Designing specific lessons or activities around problem-solving scenarios.
• Incorporating scenarios into existing subjects or projects.
• Using scenarios as prompts for class discussions or debates.

Strategies for facilitating student engagement and participation include:

• Encouraging collaboration and group work.
• Providing scaffolding and support as needed.
• Allowing for student choice and autonomy in problem-solving approaches.

Assessing and evaluating students’ critical thinking skills through scenarios can be done through:

• Observation and anecdotal notes.
• Student self-reflection and self-assessment.
• Performance-based assessments, such as presentations or projects.

Developing critical thinking skills in elementary students is crucial for their academic and personal growth. Problem-solving scenarios provide an effective and engaging way to foster these skills. By incorporating everyday life, academic, and social scenarios into the curriculum, we can help students develop the ability to think critically, solve problems independently, and make informed decisions.

I encourage educators and parents to incorporate problem-solving scenarios into their teaching and parenting practices. By nurturing critical thinking in elementary students, we are equipping them with lifelong skills that will benefit them in all areas of life.

How Can We Further Develop Critical Thinking and Problem-Solving Skills in Elementary Students?

Ready to get started? Start your Everyday Speech Free trial today and explore a wide range of problem-solving scenarios for elementary students. Let’s empower our children to become confident and independent critical thinkers!

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Problem-solving skills are necessary in all areas of life, and classroom problem solving activities can be a great way to get students prepped and ready to solve real problems in real life scenarios. Whether in school, work or in their social relationships, the ability to critically analyze a problem, map out all its elements and then prepare a workable solution is one of the most valuable skills one can acquire in life.

Educating your students about problem solving skills from an early age in school can be facilitated through classroom problem solving activities. Such endeavors encourage cognitive as well as social development, and can equip students with the tools they’ll need to address and solve problems throughout the rest of their lives. Here are five classroom problem solving activities your students are sure to benefit from as well as enjoy doing:

1. Brainstorm bonanza

Having your students create lists related to whatever you are currently studying can be a great way to help them to enrich their understanding of a topic while learning to problem-solve. For example, if you are studying a historical, current or fictional event that did not turn out favorably, have your students brainstorm ways that the protagonist or participants could have created a different, more positive outcome. They can brainstorm on paper individually or on a chalkboard or white board in front of the class.

2. Problem-solving as a group

Have your students create and decorate a medium-sized box with a slot in the top. Label the box “The Problem-Solving Box.” Invite students to anonymously write down and submit any problem or issue they might be having at school or at home, ones that they can’t seem to figure out on their own. Once or twice a week, have a student draw one of the items from the box and read it aloud. Then have the class as a group figure out the ideal way the student can address the issue and hopefully solve it.

3. Clue me in

This fun detective game encourages problem-solving, critical thinking and cognitive development. Collect a number of items that are associated with a specific profession, social trend, place, public figure, historical event, animal, etc. Assemble actual items (or pictures of items) that are commonly associated with the target answer. Place them all in a bag (five-10 clues should be sufficient.) Then have a student reach into the bag and one by one pull out clues. Choose a minimum number of clues they must draw out before making their first guess (two- three). After this, the student must venture a guess after each clue pulled until they guess correctly. See how quickly the student is able to solve the riddle.

4. Survivor scenarios

Create a pretend scenario for students that requires them to think creatively to make it through. An example might be getting stranded on an island, knowing that help will not arrive for three days. The group has a limited amount of food and water and must create shelter from items around the island. Encourage working together as a group and hearing out every child that has an idea about how to make it through the three days as safely and comfortably as possible.

5. Moral dilemma

Create a number of possible moral dilemmas your students might encounter in life, write them down, and place each item folded up in a bowl or bag. Some of the items might include things like, “I saw a good friend of mine shoplifting. What should I do?” or “The cashier gave me an extra $1.50 in change after I bought candy at the store. What should I do?” Have each student draw an item from the bag one by one, read it aloud, then tell the class their answer on the spot as to how they would handle the situation.

Classroom problem solving activities need not be dull and routine. Ideally, the problem solving activities you give your students will engage their senses and be genuinely fun to do. The activities and lessons learned will leave an impression on each child, increasing the likelihood that they will take the lesson forward into their everyday lives.

You may also like to read

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• Activities for Teaching Tolerance in the Classroom
• 5 Problem-Solving Activities for Elementary Classrooms
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71+ Free Social Problem-Solving Scenarios

Do you have kiddos who struggle with their social problem-solving skills? Teach your students the simple process of how to solve a problem along with having them review how well their solution worked or didn’t work.

Why Teach Problem Solving Skills?

Learning to problem solve is an essential skill that is used not only throughout childhood but also into adulthood. Social problem solving is the ability to change or adapt to undesirable situations that arise throughout our day.

On a daily basis, a child will encounter social problems that they will need to solve.

Anything from:

• arguing with another student
• to hurting a friend’s feelings
• to having a difficult conversation
• working with others

Start with Small Problems

Many of the “problems” children encounter are often small problems which the child may be over-reacting to, such as wanting a different coloring crayon or wanting to be first in line, however, these small problems are still very real to the child.

Practicing problem-solving with these small problems can be a great learning opportunity. Children can practice problem-solving with a small problem which can help them learn how to handle bigger problems in the future.

Problem Solving Importance

Social problem-solving skills are critical to a child’s social interactions, personal and professional relationships. A child’s ability to handle change, cope with stress, and handle challenges improves with a child’s ability to successfully solve social problems.

The ultimate goal is that the child will be able to solve social problems all on their own, but until they can independently solve a problem they will need to learn how to communicate and self-advocate to positively solve their problems.  

Steps to Problem Solving

Children can be taught how to problem solve through a guided process of breaking down the problem and using simple steps to solve the problem.

Learning specific steps to problem-solving can allow children to remember how to solve a problem when they become overwhelmed or stressed.

Although learning to solve a problem independently can take some time and practice it is well worth the investment to have a child who can eventually solve most social situations in a positive manner on their own.

What we learnt about solving problems is don't freak out, if one thing doesn't work , try something else out. And work together as a team. #melthammathsweek #MELTHAMPUPILVOICE @problemsolveit pic.twitter.com/iVm1Im4Aue — yr6melthamce (@yr6melthamce) February 4, 2019

Problem Solving Form

Teach your students the 4 steps to becoming a social problem-solver.

• Identify the problem. For instance, start by having your student identify the social problem.
• Create three solutions. Also, have your student come up with three different solutions that they could use to solve the problem that they identified.
• Identify the consequences. Then, identify the consequence for each individual solution.
• Pick the best solution.  Lastly, have your student identify which of their three solutions is the best choice Then have your student put into words why they think that solution is the best solution.

Problem Solving Review Form

After your students go through the social problem-solver have them use the social problem-solving review form.

• What happened.  For instance, after your student tried their solution have them explain what happened next.
• Review the results. Also, have your student identify whether or not their solution got them the results they wanted.
• Use this solution again. Furthermore, have your student identify whether or not they would use this solution again in the future to solve the same or similar problem.
• What would you do differently? Finally, have your student explain what they would do differently if they didn’t get the results they wanted or if they wouldn’t use that solution again in the future.

71+ Social Problem Scenarios + 6 Blank Scenarios

Use the 71 social problem-solving scenarios to have your students get great experience practicing how to solve a social problem.

Also, included are 6 blank scenarios. Then laminate them so you can use them over and over again. Therefore, create social problems that the student experiences and needs help solving.

Wordless Video teaching Problem Solving

Watch this super cute wordless animation with your students and have them discuss the problem they see and how to best solve the problem.

Use this as a fun practice example to get your students started towards learning how to problem-solve.

Demonstrate Through Modeling

Model and discuss empathy.

First and foremost, children need to understand how another person might be feeling in a given situation in order to become a good social problem solver. The student needs to learn how to “stand in someone else’s shoes” for a little bit.

One way you can work on this skill is during the reading time you can focus on how a particular character in the story might be feeling.

Ask questions, such as:

• “How do they feel right now?”
• “How would you feel in that same situation?”
• “Why do you think they feel that way?”

Model Problem-Solving Skills as the Teacher

When you are faced with a problem you can solve the problem by thinking aloud for the students to hear how you solve a problem.

You can state the problem, then come up with possible solutions, then identify the possible consequences to each solution, then pick and explain why a solution is the best option.

For example, you could say, “I was hoping to take the class outside for a stress walk around the track before the reading test, but the problem is that it is raining outside. I could still take you outside, but then you will get wet, or we could walk the halls, but then we’d have to be really quiet because there are other classes learning, or we could just skip the walk and take the reading test, but then you might not do as well on the test. I think based on all of those solutions the best solution will be to walk the hallway, but you guys will have to promise to be quiet so that we don’t disrupt other classes.

Modeling the problem-solving process can be very helpful for the students to watch, observe, and later implement themselves.

Teach Communication

Have students communicate how they are feeling.

Teaching your students to share their emotions in a respectful way can improve their ability to problem-solve.

Have students use an “I” sentence frame, such as, “I feel _____ (insert feeling word) when _____ (identify what made you feel that way).”

For example, “I felt sad when Jackson broke my favorite pencil” or “I was mad when I wasn’t picked to be first in line.”

This way students can communicate how they are feeling using honest and open communication. Teaching students to appropriately communicate their emotions can help solve some social problems from the beginning.

Encourage Independency

Encourage your student to problem solve.

If your student is struggling to problem solve independently encourage them to do so using open-ended questions.

• “How could you fix this problem?”
• “What would be a fair solution?”
• “What would happen if you used that solution?”

Let the Student try to Problem Solve Independently

Give your students the space to try and solve their own problems using the guided strategies. Try not to come running to their rescue for every little problem.

Some problems are small and a great opportunity for the student to learn and practice. If an adult does all of the problem solving for a student then what are they really learning?

Give your students the time and space they need to practice solving small problems on their own. Of course, if it is a bigger or more serious problem then have an adult help guide the problem-solving process.

Tell an Adult

Remind your students that there are still some problems that are too big for them to solve on their own and that it is okay to get help from an adult to solve big problems.

For example, if the student doesn’t feel safe, someone is being hurt physically or emotionally, or if they tried to solve a problem independently but it didn’t work and they need help. Let them know that it’s okay to tell an adult.

Teach How to Disagree and How to Make Up

Discuss how to disagree respectfully.

Remind your student that they won’t always agree with their teacher, friends, classmate, or parents and that’s okay. Even the people we like might have different opinions, interests, and likes than we do.

However, even if we disagree with someone we should still treat them with respect. Treating someone with respect means to not call them names, ignore them, yell or hit them. It means that you do try to create solutions that both parties can agree with and to apologize when we hurt others’ feelings.

Role-Play How to Make Up

Practice in everyday life how to make up after a social problem .

Students are really having to stretch their brains today. It's @NSPCC #NumberDay and @problemsolveit are challenging Y9 and 10 to solve the escape room boxes. It's not as easy as it looks! The promise of a few sweet treats for the winners seems to be helping though! pic.twitter.com/AxRRJnJIv2 — CongletonHS (@CongletonHS) February 2, 2018

Be sure to get your free social problem solver today below! I hope you and your students love this freebie.

Have your students use task card scenarios to help them identify how they and others might feel in different social scenarios. Be sure to discuss the problem, identify possible solutions, identify the consequences of those possible solutions, and then based on those consequences pick the best solution.

Make social problem-solving a game by telling the students that they are social detectives and that it is their job to use what they know about social rules to help them identify the possible and best solutions.

Start practicing today with 71+ free social problem social task cards! Do your students need more practice?

Be sure to check out my other freebie for 31 wordless animated videos to teach problem-solving and so much more.

Make Problem Solving Easier with this Freebie!

Download yours today to get started.

Get More Problem Solving Time Saving Materials

Next, be sure to check out the following time-saving materials to continue to teach your students how to solve their social problems in addition to this freebie.

Weekly Social Pragmatics Homework

• Weekly problem-solving.   Send home a  weekly homework page  that includes a problem-solving scenario plus an idiom and a conversational practice scenario.

Restorative Justice Problem Solving Flip Book

• Restorative justice graphic visual.  Use this graphic visual to help your student  restore a social relationship  after a social problem.

Self-Advocating Role-Play Scenarios

• Self-advocating in high school.  Teach your high schoolers the process to  self-advocate  for what they need.

5th-12th Grade Life Skills Problem Solving

• Life skills problem-solving.  In addition, this  life skills differentiated bundle  includes a problem-solving lesson plan.

I recommend you read Problem Solving Wheel: Help Kids Solve Their Own Problems , 61+ Free Fillable SLP Planner Pages 2020-2021 , 430+ Free Multisyllabic Words List Activity Bundle , or 432+ Free IEP Goal Bank to Save You Time posts because they include freebies as well and who doesn’t want more freebies!

Got questions? Leave a comment. Let’s chat!

Monday 30th of January 2023

Hello! I have entered my name and email twice (yesterday & today) to receive to 71+ Free Social Problem-Solving Senarios, but I have not received anything yet. Not even an email back to mine in order to subcribe. Thanks for your help! Tracy

Melissa Berg

Tuesday 31st of January 2023

Hi Tracy, Thanks so much for reaching out! Sorry about that. We went ahead and sent you an email with the PDF attached. Wishing you all my best, Melissa

Problem Solving Skills

Tuesday 30th of August 2022

I truly love your site. Excellent colors, theme and writing. Thanks for sharing.

Laura Ricca

Monday 11th of April 2022

Tuesday 12th of April 2022

Hi Laura, I'm glad you found this resource helpful. Melissa

Modified Mental Health and Suicide Prevention - Speech Therapy Store

Monday 11th of May 2020

[…] 71+ FREE SOCIAL PROBLEM-SOLVING SCENARIOS […]

Problem Solving Wheel: Help Kids Solve Their Own Problems - Speech Therapy Store

Monday 4th of May 2020

[…] 71+ Free Social Problem Solving Task Cards Scenarios […]

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30 Problem Solving Scenarios for Speech Therapy Practice

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Problem solving scenarios.

• Your friends came over to your house for a movie night. One of your friends brought another friend so there are more people than you planned for. You want to pass out the drinks but you only have five cans of soda and you need 6 for everyone to have one. What could you do?
• After basketball practice you go back to the locker room with your team to shower and change. When you are done dressing, you can't find your shoes. What could you do?
• You have been waiting all day for lunch to come because you are starving. Finally class gets over and you get to go to lunch. Except when you go to get to your lunch, it's not there. You probably left it at home. What could you do?
• There is a guy in your class who is always mean to you. He always bumps you when he walks by and he calls you names. He knocks stuff out of your hands and makes you feel stupid. You don't think you can take it anymore. What could you do?
• You really want to invite this new girl/guy to come to your birthday party, but you have never talked to them before. You are worried they will say no. What could you do?
• You rode the bus to school today and on the way in people are pointing and laughing at you. You go in the bathroom and see that you have pink gum all over the back of your pants. What could you do?
• You wake up and see that your alarm never went off. So you are starting your morning 15 minutes later than you planned. It is a really important day at school and you cannot be late. What could you do?
• You are giving a group presentation in front of class and it's your turn to talk. All of the sudden you sneeze. You cover it with your hand, but now your hand is full of stuff you sneezed out. What could you do?
• You are eating dinner at a fancy restaurant with your parents and their friends. You have a really messy dinner and accidentally flip a noodle into the lady's lap. They are busy talking and don't notice it. What could you do?
• You are taking a test and there is no talking allowed. You are writing your answers on the paper and your pencil breaks. What could you do?
• You are taking a test and the guy behind you asks you for help. He wants to know what you put for question number two. What could you do?
• You are at a birthday party and you have waited in line for a long time for your turn to hit the pinata. It is finally going to be your turn and it looks like the next hit will break the pinata. But you suddenly have to go to the bathroom. What could you do?
• You are hanging outside with your friend and she decides to pick your neighbor's flowers. She gives you the pretty handful of flowers and right then your neighbor opens the door. She asks you why you picked her flowers. What could you do?
• You borrowed your sister's skates one day without asking and they broke while you were using them. What could you do?
• You are eating at a friend's house and the mom piles your plate full of food. It looks really good and you want to eat it all but you can't because you just ate a snack. What could you do so you don't hurt her feelings?

SEE ALSO:   The Best Free App for Speech Therapy

• Your teacher was working at her desk.  You wanted to ask her a question, but she didn't see your hand raised. What should you do?
• You started to do your work, but you weren't sure if you were doing it right. What should you do?
• You were playing tether-ball and were the champion so far.  In the next game, you slightly touched the rope.  Only one student saw you touch the rope. What will you do?
• The teacher is giving directions, but your friend sitting next to you keeps talking.  You can't hear the directions. What should you do?
• You didn't do your homework.  Your teacher was upset with you. What should you do?
• You finished eating and felt a burp coming. What are you going to do?
• You were waiting to swing.  When it was your turn, another boy jumped in front of you and took the swing. What would you do?
• You waited a long time, but your mom didn't come to pick you up after school. What should you do?
• A bully threatened to beat you up after school. What should you do?
• A boy on the playground keeps pushing you and making you mad. What would you do?
• You were sitting in class doing your work and you hear the fire alarm. What should you do?
• An adult you didn't know came on to the playground and asked if you would help look for his lost dog. What would you do?
• You forgot your lunch at home. What would you do?
• The person sitting behind you keeps tapping your chair with his foot. What should you do?
• You finished your work early. What should you do?

This list of functional words was professionally selected to be the most useful for a child or adult who has difficulty with problem solving scenarios.

We encourage you to use this list when practicing at home.

Home practice will make progress toward meeting individual language goals much faster.

Speech-Language Pathologists (SLPs) are only able to see students/clients 30-60 mins (or less) per week. This is not enough time or practice for someone to handle Problem solving scenarios.

Every day that your loved one goes without practice it becomes more difficult to help them. 

SEE ALSO:   The Best Books for Speech Therapy Practice

We know life is busy , but if you're reading this you're probably someone who cares about helping their loved one as much as you can.

Practice 5-10 minutes whenever you can, but try to do it on a consistent basis (daily).

Please, please, please use this list to practice.

It will be a great benefit to you and your loved one's progress.

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CDC Resources for the Classroom

Teaching students using real-world public health problems can help improve their understanding of science. It allows them to apply knowledge and skills from a variety of STEM subjects. It can also help them develop essential career skills, such as problem-solving, critical thinking, innovation, collaboration, and communication. Learning about public health can also help students develop new ideas for career paths.

CDC resources for K–12 teachers are designed to:

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Watch CBS News

How two high school students solved a 2,000-year-old math puzzle

By Bill Whitaker , Aliza Chasan , Sara Kuzmarov, Mariah Campbell

May 5, 2024 / 7:00 PM EDT / CBS News

A high school math teacher at St. Mary's Academy in New Orleans, Michelle Blouin Williams, was looking for ingenuity when she and her colleagues set a school-wide math contest with a challenging bonus question. That bonus question asked students to create a new proof for the Pythagorean Theorem, a fundamental principle of geometry, using trigonometry. The teachers weren't necessarily expecting anyone to solve it, as proofs of the Pythagorean Theorem using trigonometry were believed to be impossible for nearly 2,000 years.

But then, in December 2022, Calcea Johnson and Ne'Kiya Jackson, seniors at St. Mary's Academy, stepped up to the challenge. The $500 prize money was a motivating factor.

After months of work, they submitted their innovative proofs to their teachers. With the contest behind them, their teachers encouraged the students to present at a mathematics conference, and then to seek to publish their work. And even today, they're not done. Now in college, they've been working on further proofs of the Pythagorean Theorem and believe they have found five more proofs. Amazingly, despite their impressive achievements, they insist they're not math geniuses.

"I think that's a stretch," Calcea said.

The St. Mary's math contest

When the pair started working on the math contest they were familiar with the Pythagorean Theorem's equation: A² + B² = C², which explains that by knowing the length of two sides of a right triangle, it's possible to figure out the length of the third side.

When Calcea and Ne'Kiya set out to create a new Pythagorean Theorem proof, they didn't know that for thousands of years, one using trigonometry was thought to be impossible.  In 2009, mathematician Jason Zimba submitted one, and now Calcea and Ne'Kiya are adding to the canon.

Calcea and Ne'Kiya had studied geometry and some trigonometry when they started working on their proofs, but said they didn't feel math was easy. As the contest went on, they spent almost all their free time developing their ideas.

"The garbage can was full of papers, which she would, you know, work out the problems and if that didn't work, she would ball it up, throw it in the trash," Cal Johnson, Calcea's dad, said.

Neliska Jackson, Ne'Kiya's mother, says lightheartedly, that most of the time, her daughter's work was beyond her. 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

"Well, our teacher approached us and was like, 'Hey, you might be able to actually present this,'" Ne'Kiya said. "I was like, 'Are you joking?' But she wasn't. So we went. I got up there. We presented and it went well, and it blew up."

Why Calcea' and Ne'kiya's work "blew up"

The reaction was insane and unexpected, Calcea said. News of their accomplishment spread around the world. The pair got a write-up in South Korea and a shoutout from former first lady Michelle Obama. They got a commendation from the governor and keys to the city of New Orleans. 

Calcea and Ne'Kiya said they think there's several reasons why people found their work so impressive. 

"Probably because we're African American, one," Ne'Kiya said. "And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part."

Ne'Kiya said she'd like their accomplishment to be celebrated for what it is: "a great mathematical achievement."

In spite of the community's celebration of the students' work, St. Mary's Academy president and interim principal Pamela Rogers said that with recognition came racist calls and comments. 

"[People said] 'they could not have done it. African Americans don't have the brains to do it.' Of course, we sheltered our girls from that," Rogers said. "But we absolutely did not expect it to come in the volume that it came."

Rogers said too often society has a vision of who can be successful.

"To some people, it is not always an African American female," Rogers said. "And to us, it's always an African American female."

Success at St. Marys 

St. Mary's, a private Catholic elementary and high school, was started for young Black women just after the Civil War. Ne'Kiya and Calcea follow a long line of barrier-breaking graduates. Leah Chase , the late queen of Creole cuisine, was an alum. So was Michelle Woodfork, the first African American female New Orleans police chief, and Dana Douglas, a judge for the Fifth Circuit Court of Appeals. 

Math teacher Michelle Blouin Williams, who initiated the math contest, said Calcea and Ne'Kiya are typical St. Mary's students. She said if they're "unicorns," then every student who's matriculated through the school is a "beautiful, Black unicorn."

Students hear that message from the moment they walk in the door, Rogers said. 

"We believe all students can succeed, all students can learn," the principal said. "It does not matter the environment that you live in."

About half the students at St. Mary's get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. There's no test to get in, but expectations are high and rules are strict: cellphones are not allowed and modest skirts and hair in its natural color are required. 

Students said they appreciate the rules and rigor.

"Especially the standards that they set for us," junior Rayah Siddiq said. "They're very high. And I don't think that's ever going to change." 

What's next for Ne'Kiya and Calcea

Last year when Ne'Kiya and Calcea graduated, all their classmates were accepted into college and received scholarship offers. The school has had a 100% graduation rate and a 100% college acceptance rate for 17 years, according to Rogers.

Ne'Kiya got a full ride in the pharmacy department at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University. Neither one is pursuing a career in math, though Calcea said she may minor in math.

"People might expect too much out of me if I become a mathematician," Ne'Kiya said wryly. 

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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Facility for Rare Isotope Beams

At michigan state university, international research team uses wavefunction matching to solve quantum many-body problems, new approach makes calculations with realistic interactions possible.

FRIB researchers are part of an international research team solving challenging computational problems in quantum physics using a new method called wavefunction matching. The new approach has applications to fields such as nuclear physics, where it is enabling theoretical calculations of atomic nuclei that were previously not possible. The details are published in Nature (“Wavefunction matching for solving quantum many-body problems”) .

Ab initio methods and their computational challenges

An ab initio method describes a complex system by starting from a description of its elementary components and their interactions. For the case of nuclear physics, the elementary components are protons and neutrons. Some key questions that ab initio calculations can help address are the binding energies and properties of atomic nuclei not yet observed and linking nuclear structure to the underlying interactions among protons and neutrons.

Yet, some ab initio methods struggle to produce reliable calculations for systems with complex interactions. One such method is quantum Monte Carlo simulations. In quantum Monte Carlo simulations, quantities are computed using random or stochastic processes. While quantum Monte Carlo simulations can be efficient and powerful, they have a significant weakness: the sign problem. The sign problem develops when positive and negative weight contributions cancel each other out. This cancellation results in inaccurate final predictions. It is often the case that quantum Monte Carlo simulations can be performed for an approximate or simplified interaction, but the corresponding simulations for realistic interactions produce severe sign problems and are therefore not possible.

Using ‘plastic surgery’ to make calculations possible

The new wavefunction-matching approach is designed to solve such computational problems. The research team—from Gaziantep Islam Science and Technology University in Turkey; University of Bonn, Ruhr University Bochum, and Forschungszentrum Jülich in Germany; Institute for Basic Science in South Korea; South China Normal University, Sun Yat-Sen University, and Graduate School of China Academy of Engineering Physics in China; Tbilisi State University in Georgia; CEA Paris-Saclay and Université Paris-Saclay in France; and Mississippi State University and the Facility for Rare Isotope Beams (FRIB) at Michigan State University (MSU)—includes  Dean Lee , professor of physics at FRIB and in MSU’s Department of Physics and Astronomy and head of the Theoretical Nuclear Science department at FRIB, and  Yuan-Zhuo Ma , postdoctoral research associate at FRIB.

“We are often faced with the situation that we can perform calculations using a simple approximate interaction, but realistic high-fidelity interactions cause severe computational problems,” said Lee. “Wavefunction matching solves this problem by doing plastic surgery. It removes the short-distance part of the high-fidelity interaction, and replaces it with the short-distance part of an easily computable interaction.”

This transformation is done in a way that preserves all of the important properties of the original realistic interaction. Since the new wavefunctions look similar to that of the easily computable interaction, researchers can now perform calculations using the easily computable interaction and apply a standard procedure for handling small corrections called perturbation theory.  A team effort

The research team applied this new method to lattice quantum Monte Carlo simulations for light nuclei, medium-mass nuclei, neutron matter, and nuclear matter. Using precise ab initio calculations, the results closely matched real-world data on nuclear properties such as size, structure, and binding energies. Calculations that were once impossible due to the sign problem can now be performed using wavefunction matching.

“It is a fantastic project and an excellent opportunity to work with the brightest nuclear scientist s in FRIB and around the globe,” said Ma. “As a theorist , I'm also very excited about programming and conducting research on the world's most powerful exascale supercomputers, such as Frontier , which allows us to implement wavefunction matching to explore the mysteries of nuclear physics.”

While the research team focused solely on quantum Monte Carlo simulations, wavefunction matching should be useful for many different ab initio approaches, including both classical and  quantum computing calculations. The researchers at FRIB worked with collaborators at institutions in China, France, Germany, South Korea, Turkey, and United States.

“The work is the culmination of effort over many years to handle the computational problems associated with realistic high-fidelity nuclear interactions,” said Lee. “It is very satisfying to see that the computational problems are cleanly resolved with this new approach. We are grateful to all of the collaboration members who contributed to this project, in particular, the lead author, Serdar Elhatisari.”

This material is based upon work supported by the U.S. Department of Energy, the U.S. National Science Foundation, the German Research Foundation, the National Natural Science Foundation of China, the Chinese Academy of Sciences President’s International Fellowship Initiative, Volkswagen Stiftung, the European Research Council, the Scientific and Technological Research Council of Turkey, the National Natural Science Foundation of China, the National Security Academic Fund, the Rare Isotope Science Project of the Institute for Basic Science, the National Research Foundation of Korea, the Institute for Basic Science, and the Espace de Structure et de réactions Nucléaires Théorique.

Michigan State University operates the Facility for Rare Isotope Beams (FRIB) as a user facility for the U.S. Department of Energy Office of Science (DOE-SC), supporting the mission of the DOE-SC Office of Nuclear Physics. Hosting what is designed to be the most powerful heavy-ion accelerator, FRIB enables scientists to make discoveries about the properties of rare isotopes in order to better understand the physics of nuclei, nuclear astrophysics, fundamental interactions, and applications for society, including in medicine, homeland security, and industry.

The U.S. Department of Energy Office of Science is the single largest supporter of basic research in the physical sciences in the United States and is working to address some of today’s most pressing challenges. For more information, visit energy.gov/science.