steps in problem solving in mathematics

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Problem Solving in Mathematics

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The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication , or division? Then determine all the information that is being given to you in the question.

Mathematician George Pólya’s book, “ How to Solve It: A New Aspect of Mathematical Method ,” written in 1957, is a great guide to have on hand. The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya’s book and should help you untangle even the most complicated math problem.

Use Established Procedures

Learning how to solve problems in mathematics is knowing what to look for. Math problems often require established procedures and knowing what procedure to apply. To create procedures, you have to be familiar with the problem situation and be able to collect the appropriate information, identify a strategy or strategies, and use the strategy appropriately.

Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.

Look for Clue Words

Think of yourself as a math detective. The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop. If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation.

Common clue words for addition  problems:

Common clue words for  subtraction  problems:

• How much more

Common clue words for multiplication problems:

Common clue words for division problems:

Although clue words will vary a bit from problem to problem, you'll soon learn to recognize which words mean what in order to perform the correct operation.

Read the Problem Carefully

This, of course, means looking for clue words as outlined in the previous section. Once you’ve identified your clue words, highlight or underline them. This will let you know what kind of problem you’re dealing with. Then do the following:

• Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?
• What did you need to do in that instance?
• What facts are you given about this problem?
• What facts do you still need to find out about this problem?

Develop a Plan and Review Your Work

Based on what you discovered by reading the problem carefully and identifying similar problems you’ve encountered before, you can then:

• Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.
• If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.

If it seems like you’ve solved the problem, ask yourself the following:

• Does your solution seem probable?
• Does it answer the initial question?
• Did you answer using the language in the question?
• Did you answer using the same units?

If you feel confident that the answer is “yes” to all questions, consider your problem solved.

Tips and Hints

Some key questions to consider as you approach the problem may be:

• What are the keywords in the problem?
• Do I need a data visual, such as a diagram, list, table, chart, or graph?
• Is there a formula or equation that I'll need? If so, which one?
• Will I need to use a calculator? Is there a pattern I can use or follow?

Read the problem carefully, and decide on a method to solve the problem. Once you've finished working the problem, check your work and ensure that your answer makes sense and that you've used the same terms and or units in your answer.

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The easy 4 step problem-solving process (+ examples)

This is the 4 step problem-solving process that I taught to my students for math problems, but it works for academic and social problems as well.

Every problem may be different, but effective problem solving asks the same four questions and follows the same method.

• What’s the problem? If you don’t know exactly what the problem is, you can’t come up with possible solutions. Something is wrong. What are we going to do about this? This is the foundation and the motivation.
• What do you need to know? This is the most important part of the problem. If you don’t know exactly what the problem is, you can’t come up with possible solutions.
• What do you already know? You already know something related to the problem that will help you solve the problem. It’s not always obvious (especially in the real world), but you know (or can research) something that will help.
• What’s the relationship between the two? Here is where the heavy brainstorming happens. This is where your skills and abilities come into play. The previous steps set you up to find many potential solutions to your problem, regardless of its type.

When I used to tutor kids in math and physics , I would drill this problem-solving process into their heads. This methodology works for any problem, regardless of its complexity or difficulty. In fact, if you look at the various advances in society, you’ll see they all follow some variation of this problem-solving technique.

“The gap between understanding and misunderstanding can best be bridged by thought!” ― Ernest Agyemang Yeboah

Generally speaking, if you can’t solve the problem then your issue is step 3 or step 4; you either don’t know enough or you’re missing the connection.

Good problem solvers always believe step 3 is the issue. In this case, it’s a simple matter of learning more. Less skilled problem solvers believe step 4 is the root cause of their difficulties. In this instance, they simply believe they have limited problem-solving skills.

This is a fixed versus growth mindset and it makes a huge difference in the effort you put forth and the belief you have in yourself to make use of this step-by-step process. These two mindsets make a big difference in your learning because, at its core, learning is problem-solving.

Let’s dig deeper into the 4 steps. In this way, you can better see how to apply them to your learning journey.

Step 1: What’s the problem?

The ability to recognize a specific problem is extremely valuable.

Most people only focus on finding solutions. While a “solutions-oriented” mindset is a good thing, sometimes it pays to focus on the problem. When you focus on the problem, you often make it easier to find a viable solution to it.

When you know the exact nature of the problem, you shorten the time frame needed to find a solution. This reminds me of a story I was once told.

When does the problem-solving process start?

The process starts after you’ve identified the exact nature of the problem.

Homeowners love a well-kept lawn but hate mowing the grass.

Many companies and inventors raced to figure out a more time-efficient way to mow the lawn. Some even tried to design robots that would do the mowing. They all were chasing the solution, but only one inventor took the time to understand the root cause of the problem.

Most people figured that the problem was the labor required to maintain a lawn. The actual problem was just the opposite: maintaining a lawn was labor-intensive. The rearrangement seems trivial, but it reveals the true desire: a well-maintained lawn.

The best solution? Remove maintenance from the equation. A lawn made of artificial grass solved the problem . Hence, an application of Astroturf was discovered.

This way, the law always looked its best. Taking a few moments to apply critical thinking identified the true nature of the problem and yielded a powerful solution.

An example of choosing the right problem to work the problem-solving process on

One thing I’ve learned from tutoring high school students in math : they hate word problems.

This is because they make the student figure out the problem. Finding the solution to a math problem is already stressful. Forcing the student to also figure out what problem needs solving is another level of hell.

Word problems are not always clear about what needs to be solved. They also have the annoying habit of adding extraneous information. An ordinary math problem does not do this. For example, compare the following two problems:

What’s the height of h?

A radio station tower was built in two sections. From a point 87 feet from the base of the tower, the angle of elevation of the top of the first section is 25º, and the angle of elevation of the top of the second section is 40º. To the nearest foot, what is the height of the top section of the tower?

The first is a simple problem. The second is a complex problem. The end goal in both is the same.

The questions require the same knowledge (trigonometric functions), but the second is more difficult for students. Why? The second problem does not make it clear what the exact problem is. Before mathematics can even begin, you must know the problem, or else you risk solving the wrong one.

If you understand the problem, finding the solution is much easier. Understanding this, ironically, is the biggest problem for people.

Problem-solving is a universal language

Speaking of people, this method also helps settle disagreements.

When we disagree, we rarely take the time to figure out the exact issue. This happens for many reasons, but it always results in a misunderstanding. When each party is clear with their intentions, they can generate the best response.

Education systems fail when they don’t consider the problem they’re supposed to solve. Foreign language education in America is one of the best examples.

The problem is that students can’t speak the target language. It seems obvious that the solution is to have students spend most of their time speaking. Unfortunately, language classes spend a ridiculous amount of time learning grammar rules and memorizing vocabulary.

The problem is not that the students don’t know the imperfect past tense verb conjugations in Spanish. The problem is that they can’t use the language to accomplish anything. Every year, kids graduate from American high schools without the ability to speak another language, despite studying one for 4 years.

Well begun is half done

Before you begin to learn something, be sure that you understand the exact nature of the problem. This will make clear what you need to know and what you can discard. When you know the exact problem you’re tasked with solving, you save precious time and energy. Doing this increases the likelihood that you’ll succeed.

Step 2: What do you need to know?

All problems are the result of insufficient knowledge. To solve the problem, you must identify what you need to know. You must understand the cause of the problem. If you get this wrong, you won’t arrive at the correct solution.

Either you’ll solve what you thought was the problem, only to find out this wasn’t the real issue and now you’ve still got trouble or you won’t and you still have trouble. Either way, the problem persists.

If you solve a different problem than the correct one, you’ll get a solution that you can’t use. The only thing that wastes more time than an unsolved problem is solving the wrong one.

Imagine that your car won’t start. You replace the alternator, the starter, and the ignition switch. The car still doesn’t start. You’ve explored all the main solutions, so now you consider some different solutions.

Now you replace the engine, but you still can’t get it to start. Your replacements and repairs solved other problems, but not the main one: the car won’t start.

Then it turns out that all you needed was gas.

This example is a little extreme, but I hope it makes the point. For something more relatable, let’s return to the problem with language learning.

You need basic communication to navigate a foreign country you’re visiting; let’s say Mexico. When you enroll in a Spanish course, they teach you a bunch of unimportant words and phrases. You stick with it, believing it will eventually click.

When you land, you can tell everyone your name and ask for the location of the bathroom. This does not help when you need to ask for directions or tell the driver which airport terminal to drop you off at.

Finding the solution to chess problems works the same way

The book “The Amateur Mind” by IM Jeremy Silman improved my chess by teaching me how to analyze the board.

It’s only with a proper analysis of imbalances that you can make the best move. Though you may not always choose the correct line of play, the book teaches you how to recognize what you need to know . It teaches you how to identify the problem—before you create an action plan to solve it.

The problem-solving method always starts with identifying the problem or asking “What do you need to know?”. It’s only after you brainstorm this that you can move on to the next step.

Learn the method I used to earn a physics degree, learn Spanish, and win a national boxing title

• I was a terrible math student in high school who wrote off mathematics. I eventually overcame my difficulties and went on to earn a B.A. Physics with a minor in math
• I pieced together the best works on the internet to teach myself Spanish as an adult
• *I didn’t start boxing until the very old age of 22, yet I went on to win a national championship, get a high-paying amateur sponsorship, and get signed by Roc Nation Sports as a profession.

I’ve used this method to progress in mentally and physically demanding domains.

While the specifics may differ, I believe that the general methods for learning are the same in all domains.

This free e-book breaks down the most important techniques I’ve used for learning.

Step 3: What do you already know?

The only way to know if you lack knowledge is by gaining some in the first place. All advances and solutions arise from the accumulation and implementation of prior information. You must first consider what it is that you already know in the context of the problem at hand.

Isaac Newton once said, “If I have seen further, it is by standing on the shoulders of giants.” This is Newton’s way of explaining that his advancements in physics and mathematics would be impossible if it were not for previous discoveries.

Mathematics is a great place to see this idea at work. Consider the following problem:

What is the domain and range of y=(x^2)+6?

This simple algebra problem relies on you knowing a few things already. You must know:

• The definition of “domain” and “range”
• That you can never square any real number and get a negative

Once you know those things, this becomes easy to solve. This is also how we learn languages.

An example of the problem-solving process with a foreign language

Anyone interested in serious foreign language study (as opposed to a “crash course” or “survival course”) should learn the infinitive form of verbs in their target language. You can’t make progress without them because they’re the root of all conjugations. It’s only once you have a grasp of the infinitives that you can completely express yourself. Consider the problem-solving steps applied in the following example.

I know that I want to say “I don’t eat eggs” to my Mexican waiter. That’s the problem.

I don’t know how to say that, but last night I told my date “No bebo alcohol” (“I don’t drink alcohol”). I also know the infinitive for “eat” in Spanish (comer). This is what I already know.

Now I can execute the final step of problem-solving.

Step 4: What’s the relationship between the two?

I see the connection. I can use all of my problem-solving strategies and methods to solve my particular problem.

I know the infinitive for the Spanish word “drink” is “beber” . Last night, I changed it to “bebo” to express a similar idea. I should be able to do the same thing to the word for “eat”.

“No como huevos” is a pretty accurate guess.

In the math example, the same process occurs. You don’t know the answer to “What is the domain and range of y=(x^2)+6?” You only know what “domain” and “range” mean and that negatives aren’t possible when you square a real number.

A domain of all real numbers and a range of all numbers equal to and greater than six is the answer.

This is relating what you don’t know to what you already do know. The solutions appear simple, but walking through them is an excellent demonstration of the process of problem-solving.

In most cases, the solution won’t be this simple, but the process or finding it is the same. This may seem trivial, but this is a model for thinking that has served the greatest minds in history.

A recap of the 4 steps of the simple problem-solving process

• What’s the problem? There’s something wrong. There’s something amiss.
• What do you need to know? This is how to fix what’s wrong.
• What do you already know? You already know something useful that will help you find an effective solution.
• What’s the relationship between the previous two? When you use what you know to help figure out what you don’t know, there is no problem that won’t yield.

Learning is simply problem-solving. You’ll learn faster if you view it this way.

What was once complicated will become simple.

What was once convoluted will become clear.

Ed Latimore

I’m a writer, competitive chess player, Army veteran, physicist, and former professional heavyweight boxer. My work focuses on self-development, realizing your potential, and sobriety—speaking from personal experience, having overcome both poverty and addiction.

Follow me on Twitter.

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A Guide to Problem Solving

When confronted with a problem, in which the solution is not clear, you need to be a skilled problem-solver to know how to proceed. When you look at STEP problems for the first time, it may seem like this problem-solving skill is out of your reach, but like any skill, you can improve your problem-solving with practice. How do I become a better problem-solver? First and foremost, the best way to become better at problem-solving is to try solving lots of problems! If you are preparing for STEP, it makes sense that some of these problems should be STEP questions, but to start off with it's worth spending time looking at problems from other sources. This collection of NRICH problems  is designed for younger students, but it's very worthwhile having a go at a few to practise the problem-solving technique in a context where the mathematics should be straightforward to you. Then as you become a more confident problem-solver you can try more past STEP questions. One student who worked with NRICH said: "From personal experience, I was disastrous at STEP to start with. Yet as I persisted with it for a long time it eventually started to click - 'it' referring to being able to solve problems much more easily. This happens because your brain starts to recognise that problems fall into various categories and you subconsciously remember successes and pitfalls of previous 'similar' problems." A Problem-solving Heuristic for STEP Below you will find some questions you can ask yourself while you are solving a problem. The questions are divided into four phases, based loosely on those found in George Pólya's 1945 book "How to Solve It". Understanding the problem

• What area of mathematics is this?
• What exactly am I being asked to do?
• What do I know?
• What do I need to find out?
• What am I uncertain about?
• Can I put the problem into my own words?

Devising a plan

• Work out the first few steps before leaping in!
• Have I seen something like it before?
• Is there a diagram I could draw to help?
• Is there another way of representing?
• Would it be useful to try some suitable numbers first?
• Is there some notation that will help?

Carrying out the plan STUCK!

• Try special cases or a simpler problem
• Work backwards
• Guess and check
• Be systematic
• Work towards subgoals
• Imagine your way through the problem
• Has the plan failed? Know when it's time to abandon the plan and move on.

Looking back

• Have I answered the question?
• Sanity check for sense and consistency
• Check the problem has been fully solved
• Read through the solution and check the flow of the logic.

Throughout the problem solving process it's important to keep an eye on how you're feeling and making sure you're in control:

• Am I getting stressed?
• Is my plan working?
• Am I spending too long on this?
• Could I move on to something else and come back to this later?
• Am I focussing on the problem?
• Is my work becoming chaotic, do I need to slow down, go back and tidy up?
• Do I need to STOP, PEN DOWN, THINK?

Finally, don't forget that STEP questions are designed to take at least 30-45 minutes to solve, and to start with they will take you longer than that. As a last resort, read the solution, but not until you have spent a long time just thinking about the problem, making notes, trying things out and looking at resources that can help you. If you do end up reading the solution, then come back to the same problem a few days or weeks later to have another go at it.

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20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

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

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

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

What are problem-solving strategies?

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

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

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

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

The ultimate guide to problem solving techniques

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

20 Math Strategies For Problem-Solving

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

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

Strategies to understand the problem

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

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

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

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

1. Read the problem aloud

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

2. Highlight keywords 

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

3. Summarize the information

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

4. Determine the unknown

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

5. Make a plan

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

Strategies for solving the problem 

1. draw a model or diagram.

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

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

2. Act it out

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

3. Work backwards

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

For example,

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

4. Write a number sentence

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

5. Use a formula

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

Strategies for checking the solution 

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

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

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

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

2. Estimate to check for reasonableness

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

3. Plug-In Method

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

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

4. Peer Review

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

5. Use a Calculator

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

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

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

Polya’s 4 steps include:

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

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

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

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

How Third Space Learning improves problem-solving 

Resources .

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

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

One-on-one tutoring 

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

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

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

Problem-solving

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

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

READ MORE : 8 Common Core math examples

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

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

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

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

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Ultimate Guide to Metacognition [FREE]

Looking for a summary on metacognition in relation to math teaching and learning?

Check out this guide featuring practical examples, tips and strategies to successfully embed metacognition across your school to accelerate math growth.

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]

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

• Understand the problem.
• Devise a plan.
• Carry out the plan.
• Looking back.

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!

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

• What if the picture was different?
• What if the numbers were simpler?
• What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

Problem Solving Strategy 2 (Try Something!). If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Problem 2 (Payback)

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem?

This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

Problem Solving Strategy 3 (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, like this one, thinking visually can help! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

Problem Solving Strategy 4 (Make Up Numbers). Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

Problem 3 (Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64… It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”

Problem Solving Strategy 5 (Try a Simpler Problem). Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

Problem Solving Strategy 6 (Work Systematically). If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:

Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate). Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

Problem Solving Strategy 8 (Look for and Explain Patterns). Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

• Describe all of the patterns you see in the table.
• Can you explain and justify any of the patterns you see? How can you be sure they will continue?
• What calculation would you do to find the total number of squares on a 100 × 100 chess board?

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

Problem 4 (Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?

Problem Solving Strategy 9 (Find the Math, Remove the Context). Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

• What is the sum of all the numbers on the clock’s face?
• Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
• How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

Problem Solving Strategy 10 (Check Your Assumptions). When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

• Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons ↵

Mathematics for Elementary Teachers Copyright © 2018 by Michelle Manes is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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Problem Analysis in Math: Using the 5-Step Problem-Solving Approach This blog will explore the ways in which problem analysis of student mathematics difficulties can be applied within a problem-solving approach. We will review the core principles of a multi-tier system of support (MTSS) framework, identify the steps within a problem-solving approach, and explore the ways in which problem analysis helps to inform intervention development within the context of mathematics instruction. Core Principles of an MTSS Framework A variety of definitions of an MTSS framework exist within the field of education; however, several common principles are apparent and have helped to shape much of the work within this area. The National Association of State Directors of Special Education (Batsche et al., 2006) define eight core principles that capture some of the most important aspects and core beliefs of an MTSS framework: We can effectively teach all children. Intervene early. Use a multi-tier model of service delivery. Use a problem-solving model to make decisions within a multi-tier model. Use scientific, research-based validated intervention and instruction to the extent available. Monitor student progress to inform instruction. Use data to make decisions. Use assessment for screening, diagnostics, and progress monitoring. The fourth core principle refers to utilizing a problem-solving model to make decisions. More specifically, educators and administrators should use a clearly defined problem-solving process that guides their team in identifying the problem, analyzing the size and effect of the problem, developing a plan for intervention to address the problem, implementing the plan, and examining the effectiveness of the intervention plan. 5-Step Problem-Solving Approach The problem-solving approach utilized by the FastBridge Learning ® system includes the following five steps: Problem identification Problem analysis Plan development Plan implementation Plan evaluation Following this 5-step problem-solving approach helps to guide school teams of educators and administrators in engaging in data-based decision making: a core principle of the MTSS framework. The first step in this problem-solving approach is problem identification. Christ and Arañas (2014) define a problem as a discrepancy between observed and expected performance. Regularly scheduled universal screening plays an important role in problem identification. The FastBridge Learning ® system offers a variety of screening measures for mathematics, which are summarized in the table below. While the results of regularly scheduled universal screening help to inform whether or not a discrepancy between observed and expected performance exists, problem analysis aims to identify the size and effects of the problem. Problem Analysis After a problem has been identified, the problem must be defined through the method of problem analysis. Only after a problem has been sufficiently analyzed and defined, can the significance of a problem be understood (Brown-Chidsey & Bickford, 2016). Problem analysis can occur at both the individual level and the group level. Problem Analysis at the Individual Leve l At the individual level, a discrepancy between observed and expected performance may appear within an individual’s universal screening results. FastBridge Learning ® Individual Skills Reports can provide detailed information to support problem analysis. The Individual Skills Report provides a snapshot of a given student’s risk, relative to benchmark goals, and also provides detailed results on an item-by-item basis. This item-by-item analysis provides insight into the skills the student demonstrates and which skills may require additional instructional support. If the student’s performance is relatively close to the benchmark goal, the problem is likely to be understood as a minor problem. Item-by-item analysis can help a teacher determine if targeted reteaching of specific content may help the student to reach the goal, or if more intensive intervention may be necessary. An example of a FastBridge Learning ® Individual Skills Report for the earlyMath subtest Numeral Identification (Kindergarten) may be seen below. The above report suggests that the sample student is at “some risk” for difficulty with numeral identification. It also reveals that the student was able to identify the given numerals with 91% accuracy. Additionally, the item-by-item analysis indicates that the student struggled to identify the following numerals: 12, 14, and 19. In this case, problem analysis may suggest that this is a minor problem, which may be remedied through targeted reteaching of the numerals 12, 14, and 19. In contrast, if during problem analysis, an Individual Skills Report for Numeral Identification indicated very low accuracy and a substantial number of misidentified numerals, the results would suggest a more significant problem. Significant problems warrant planning for more intensive intervention. Problem analysis at the group level At the group level, problem analysis seeks to determine the size and effects of a problem at the class, grade, or school-wide level. FastBridge Learning ® Group Screening Reports and Group Skills Reports help to provide insight about groups of students at risk for learning difficulties. An example of a Group Screening Report for earlyMath may be seen below. The Group Screening Report pictured above indicates that approximately 77% of Ms. Horst’s kindergarten class scored at or above the benchmark goal on the earlyMath Composite during the fall benchmark period, while 6% and 18% were identified as being at “some risk” and “high risk” respectively. All students should demonstrate growth in math achievement throughout the school year. If there are students who met the fall benchmark for mathematics, but not the winter and/or spring benchmarks, problem analysis must occur. In Ms. Horst’s class, we can see by the spring benchmark period, only 12% of her kindergarten class scored at or above the benchmark goal on the earlyMath composite, while 18% and 71% were identified as being at “some risk” and “high risk” respectively. Problem analysis at the group level may aim to answer questions such as the following: What type of mathematics instruction was provided for these students? Did other kindergarten classrooms within the school experience a similar pattern of mathematics difficulty? With which specific mathematics skills (e.g., decomposing, number sequencing, numeral identification) did the group demonstrate success and difficulty? How close to the end-of-year mathematics goals are the students now? In order to complete a thorough problem analysis, supplemental data about student performance and instructional practices may need to be collected. For instance, classroom observations and/or a teacher interview would provide insight into the first question. Group Screening Reports for the other kindergarten classrooms in the school could help identify if this is a schoolwide pattern of difficulty or a classroom-specific pattern of difficulty. An analysis of the Group Skills Report for Ms. Horst’s class would provide insight into students’ strengths and weaknesses across the earlyMath Composite’s subtests. Additional, targeted, follow-up assessment using selected FastBridge Learning ® earlyMath subtests could help to gather information about the final question. The goal of problem analysis is to determine the significance of a problem and to develop a hypothesis about why a problem has occurred. The information gathered during the problem analysis stage is then used to inform the third step of the 5-step problem-solving approach: plan development. Batsche, G., Elliott, J., Graden, J. L., Grimes, J., Kovaleski, J. F., Prasse, D., Schrag, J., & Tilly, W.D. (2006). Response to intervention: Policy considerations and implementation. Alexandria, VA: National Association of State Directors of Special Education, Inc. Brown-Chidsey, R. and Bickford, R. (2016). Practical handbook of multi-tiered systems of support: Building academic and behavioral success in schools. New York, NY: The Guilford Press. Christ, T.J., & Arañas, Y.A. (2014). Best practices in problem analysis. In A. Thomas & J. Grimes (Eds.), Best Practices in School Psychology VI . Bethesda, MD: National Association of School Psychologists. Want Better Math Grades? ✅ Unlimited Solutions ✅ Step-by-Step Answers ✅ Available 24/7 ➕ Free Bonuses ($1085 value!) On this page Search IntMath Math interactives About (site info) Uses of Trignometry ASCIIMath input, KaTeX output ASCIIMath input, LaTeX and KaTeX output Send Math in emails Syntax for ASCIIMathML Math Display Experiments Scientific Notebook Math Problem Solver Related Sections Math Tutoring Need help? Chat with a tutor anytime, 24/7. This tool combines the power of mathematical computation engine that excels at solving mathematical formulas with the power of artificial intelligence large language models to parse and generate natural language answers. This creates a math problem solver that's more accurate than ChatGPT, more flexible than a math calculator, and provides answers faster than a human tutor. Sign up for free here . Problem Solver Subjects Our math problem solver that lets you input a wide variety of math math problems and it will provide a step by step answer. This math solver excels at math word problems as well as a wide range of math subjects. Math Word Problems Pre-Algebra Geometry Graphing Trigonometry Precalculus Finite Math Linear Algebra Here are example math problems within each subject that can be input into the calculator and solved. This list is constanstly growing as functionality is added to the calculator. Basic Math Solutions Below are examples of basic math problems that can be solved. Long Arithmetic Rational Numbers Operations with Fractions Ratios, Proportions, Percents Measurement, Area, and Volume Factors, Fractions, and Exponents Unit Conversions Data Measurement and Statistics Points and Line Segments Math Word Problem Solutions Math word problems require interpreting what is being asked and simplifying that into a basic math equation. Once you have the equation you can then enter that into the problem solver as a basic math or algebra question to be correctly solved. Below are math word problem examples and their simplified forms. Word Problem: Rachel has 17 apples. She gives some to Sarah. Sarah now has 8 apples. How many apples did Rachel give her? Simplified Equation: 17 - x = 8 Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have? Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b) Simplified Equation: {r = d + 12, d = b + 6, r = 2 �� b} Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left? Simplified: 40 - 10 - 5 Pre-Algebra Solutions Below are examples of Pre-Algebra math problems that can be solved. Variables, Expressions, and Integers Simplifying and Evaluating Expressions Solving Equations Multi-Step Equations and Inequalities Ratios, Proportions, and Percents Linear Equations and Inequalities Algebra Solutions Below are examples of Algebra math problems that can be solved. Algebra Concepts and Expressions Points, Lines, and Line Segments Simplifying Polynomials Factoring Polynomials Linear Equations Absolute Value Expressions and Equations Radical Expressions and Equations Systems of Equations Quadratic Equations Inequalities Complex Numbers and Vector Analysis Logarithmic Expressions and Equations Exponential Expressions and Equations Conic Sections Vector Spaces 3d Coordinate System Eigenvalues and Eigenvectors Linear Transformations Number Sets Analytic Geometry Trigonometry Solutions Below are examples of Trigonometry math problems that can be solved. Algebra Concepts and Expressions Review Right Triangle Trigonometry Radian Measure and Circular Functions Graphing Trigonometric Functions Simplifying Trigonometric Expressions Verifying Trigonometric Identities Solving Trigonometric Equations Complex Numbers Analytic Geometry in Polar Coordinates Exponential and Logarithmic Functions Vector Arithmetic Precalculus Solutions Below are examples of Precalculus math problems that can be solved. Operations on Functions Rational Expressions and Equations Polynomial and Rational Functions Analytic Trigonometry Sequences and Series Analytic Geometry in Rectangular Coordinates Limits and an Introduction to Calculus Calculus Solutions Below are examples of Calculus math problems that can be solved. Evaluating Limits Derivatives Applications of Differentiation Applications of Integration Techniques of Integration Parametric Equations and Polar Coordinates Differential Equations Statistics Solutions Below are examples of Statistics problems that can be solved. Algebra Review Average Descriptive Statistics Dispersion Statistics Probability Probability Distributions Frequency Distribution Normal Distributions t-Distributions Hypothesis Testing Estimation and Sample Size Correlation and Regression Finite Math Solutions Below are examples of Finite Math problems that can be solved. Polynomials and Expressions Equations and Inequalities Linear Functions and Points Systems of Linear Equations Mathematics of Finance Statistical Distributions Linear Algebra Solutions Below are examples of Linear Algebra math problems that can be solved. Introduction to Matrices Linear Independence and Combinations Chemistry Solutions Below are examples of Chemistry problems that can be solved. Unit Conversion Atomic Structure Molecules and Compounds Chemical Equations and Reactions Behavior of Gases Solutions and Concentrations Physics Solutions Below are examples of Physics math problems that can be solved. Static Equilibrium Dynamic Equilibrium Kinematics Equations Electricity Thermodymanics Geometry Graphing Solutions Below are examples of Geometry and graphing math problems that can be solved. Step By Step Graphing Linear Equations and Functions Polar Equations Looking for the old Mathway Calculator? We've moved it to here . Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class. Email Address Sign Up 4 Best Steps To Problem Solving in Math That Lead to Results Eastern Shore Math Teacher What does problem solving in math mean, and how to develop these skills in students?  Problem solving involves tasks that are challenging and make students think.  In teaching through problem solving, learning takes place while trying to solve problems with specific concepts and skills. Therefore, teachers need to provide safe learning spaces that foster a growth mindset in math in order for students to take risks to solve problems.   In addition, providing students with problem solving steps in math builds success in solving problems. By providing rich mathematical tasks and engaging puzzles, students improve their number sense and mindset about mathematics.  Click Here to get this Freebie of 71 Math Number Puzzles delivered to your inbox to use with your students.  Students who feel successful in math class are happier and more engaged in learning.  Check out  The Bonus Guide for Creating a Growth Mindset Classroom and Students Who Love Math for ideas, lessons, and mindset surveys for students to use in your classroom to cultivate a positive classroom community in mathematics.    You can also sign up for other freebies from me Here at Easternshoremathteacher.com . Have you ever given students a word problem or rich task, and they froze?  They have no idea how to tackle the problem, even if it is a concept they are successful with.   This is because they need problem solving strategies.  I started to incorporate more problem solving tasks into my teaching in addition to making the 4 steps for problem solving a school-wide initiative and saw results.   What is Problem Solving in Math? When educators use the term problem solving , they are referring to mathematical tasks that are challenging and require students to think.   Such tasks or problems can promote students’ conceptual understanding, foster their ability to reason and communicate mathematically, and capture their interests and curiosity (Hiebert & Wearne, 1993; Marcus & Fey, 2003; NCTM, 1991; van de Walle, 2003). How Should Problem Solving For Math Be Taught? Problem solving should not be done in isolation.  In the past, we would teach the concepts and procedures and then assign one-step “story” problems designed to provide practice on the content. Next, we would teach problem solving as a collection of strategies such as “draw a picture” or “guess and check.”  Eventually, students would be given problems to apply the skills and strategies.  Instead, we need to make problem solving an integral part of mathematics learning.  In teaching through problem solving, learning takes place while trying to solve problems with specific concepts and skills. As students solve problems, they can use any strategy. Then, they justify their solutions with their classmates and learn new ways to solve problems.  Students do not need every task to involve problem solving.  Sometimes the goal is to just learn a skill or strategy.    Criteria for Problem Solving Math  Lappan and Phillips (1998) developed a set of criteria for a good problem that they used to develop their middle school mathematics curriculum (Connected Mathematics). The problem: has important, useful mathematics embedded in it. requires higher-level thinking and problem solving. contributes to the conceptual development of students. creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty. can be approached by students in multiple ways using different solution strategies. has various solutions or allows different decisions or positions to be taken and defended. encourages student engagement and discourse. connects to other important mathematical ideas. promotes the skillful use of mathematics. provides an opportunity to practice important skills. Of course, not every problem will include all of the above. However, the first four are essential.  Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill. The real value of these criteria is that they provide teachers with guidelines for making decisions about how to make problem solving a central aspect of their instruction.  Read more at NCTM . Problem Solving Teaching Methods Teaching students these 4 steps for solving problems allows them to have a process for unpacking difficult problems.   As you teach, model the process of using these 4 steps to solve problems.   Then, encourage students to use these steps as they solve problems.   Click here for Posters, Bookmarks, and Labels to use in your classroom to promote the use of the problem solving steps in math.   How Problem Solving Skills Develop Problem solving skills are developed over time and are improved with effective teaching practices.  In addition, teachers need to select rich tasks that focus on the math concepts the teacher wants their students to explore.  Problem Solving 4 Steps Understand the problem.  Read & Think Circle the needed information and underline the question.  Write an answer STEM sentence.  There are_____ pages left to read.  Plan Out How to Solve the Problem Make a Plan Use a strategy.  (Draw a Picture, Work Backwards, Look for a Pattern, Create a Table, Bar Model) Use math tools. Do the Problem Solve the Problem Show your work to solve the problem.  This could include an equation.  Check Your Work on the Problem Answer & Check Write the answer into the answer stem. Does your answer make sense? Check your work using a different strategy. Check out these Printables for Problem Solving Steps in Math . Teaching Problem Solving Strategies A problem solving strategy is a plan used to find a solution.  Understanding how a variety of problem solving strategies work is important because different problems require you to approach them in different ways to find the best solution. By mastering several problem-solving strategies, you can select the right plan for solving a problem.  Here are a few strategies to use with students: Draw a Picture Work Backwards Look for a Pattern Create a Table  Why is Using Problem Solving Steps For Math Important? Problem solving allows students to develop an understanding of concepts rather than just memorizing a set of procedures to solve a problem.  In addition, it fosters collaboration and communication when students explain the processes they used to arrive at a solution. Through problem-solving, students develop a deeper understanding of mathematical concepts, become more engaged, and see the importance of mathematics in their lives.  NCTM Process Standards In 2011 the Common Core State Standards incorporated the NCTM Process Standards of problem-solving, reasoning and proof, communication, representation, and connections into the Standards for Mathematical Practice.  With these process standards, the focus became more on mathematics through problem solving.   Students could no longer just develop procedural fluency, they needed to develop conceptual understanding in order to solve new problems and make connections between mathematical ideas.  Engaging Students to Learn in Mathematics Class Engaging students to learn in math class will help students to love math.  Children develop a dislike of math early on and end up resenting it into adult life.   Even in the real world, students will likely have to do some form of mathematics in their personal or working life.  So how can teachers make math more interesting to engage students in the subject? Read more at 5 Best Strategies for Engaging Students to Learn in Mathematics Class Teachers can promote number sense by providing rich mathematical tasks and encouraging students to make connections to their own experiences and previous learning. Sign up on my webpage to get this Freebie of 71 Math Number Puzzles delivered to your inbox to use with your students.  Providing opportunities to do math puzzles daily is one way to help students develop their number sense.  CLICK Here to sign up for  71 Math Number Puzzles and check out my website. Promoting a Growth Mindset Research shows that there is a link between a growth mindset and success. In addition, kids who have a growth mindset about their abilities perform better and are more engaged in the classroom.  Students need to be able to preserve and make mistakes when problem solving.   Read more … 5 Powerful and Easy Lessons Teaching Students How to Get a Growth Mindset Here are some Resources to Use to Grow a Growth Mindset Free Mindset Survey Growth Mindset Classroom Display Free Growth Mindset Lessons Using Word Problems Story Problems and word problems are one way to promote problem solving.   In addition, they provide great practice in using the 4 steps of solving problems.   Then, students are ready for more challenging problems.   For Kindergarten Subtraction within 5 For First Grade Word Problems to 20 Word Problems of Subtraction For Second Grade Two Step Word Problems with Addition and Subtraction Grade 2 Addition and Subtraction Word Problems Word Problems with Subtraction  For Third Grade Word Problems Division and Multiplication Multiplication Word Problems For Fourth Grade Multiplication Area Model Multiplicative Comparison Word Problems Resources for Problem Solving 3 Act Tasks What’s the Best Proven Way to Teach Word Problems with Two Step Equations? 5 Powerful and Easy Lessons Teaching Students How to Get a Growth Mindset 5 Powerful Ideas to Help Students Develop a Growth Mindset in Mathematics Problem Solving Steps For Math  In mathematics, problem solving is one of the most important topics to teach.  Learning to problem solve helps students apply mathematics to real-world situations. In addition, it is used for a deeper understanding of mathematical concepts.  By providing rich mathematical tasks and engaging puzzles, students improve their number sense and mindset about mathematics.  Click Here to get this Freebie of 71 Math Number Puzzles delivered to your inbox to use with your students.  Check out  The Free Ultimate Guide for Creating a Growth Mindset Classroom and Students Who Love Math for ideas, lessons, and mindset surveys to use to cultivate a growth mindset classroom. Start by modeling using the problem solving steps in math and allowing opportunities for students to use the steps to solve problems.   As students become more comfortable with using the steps and have some strategies to use,  provide more challenging tasks.  Then, students will begin to see the importance of problem solving in math and connecting their learning to real-world situations.  3 Responses Pingback: How to Successfully and Easily Teach Word Problems in Multiplication and Division - Eastern Shore Math Teacher Pingback: Top 7 Must Have Best Beginning of The Year Bulletin Board Ideas For Math - Eastern Shore Math Teacher Pingback: 3 Brilliant Math Word Problem Solving Strategies To Use With Students - Eastern Shore Math Teacher Leave a Reply Cancel reply Your email address will not be published. Required fields are marked * Save my name, email, and website in this browser for the next time I comment. More blog posts How to Effectively Teach Place Value 2 Digits for Understanding of Numbers In math class, place value 2 digits practice builds the foundation for our number system.  We How to Make Math More Engaging in 3 Simple and Effective Steps Are you wondering how to make math more engaging?  Do you sometimes look around your classroom 3 Best Missing Numbers Activities and Games for Numbers to 100 I am always looking for new and fun missing numbers activities. These activities help students practice 3 Terrific Time-Saving Number Activities to Understand Teen Numbers To understand teen numbers, kids need to understand place value.  Kids learn to count to 10 How to Master Counting Numbers to 20: Best Activities for Success! Counting numbers to 20 doesn’t sound like a complicated skill however, there are several components to From Zero to Hero: What are Proven Best Strategies for Teaching Addition? What are the best strategies for teaching addition?   Computational fluency or addition and subtraction facts Hi, I'm Eastern Shore Math Teacher! I have been teaching for over 22 years in an elementary school.  I help educators plan engaging math lessons and cultivate a positive math culture in their classrooms.   Sign up and I will send you the growth mindset classroom guide and I will help you get your elementary students to love math. Copyright 2021 | Eastern Shore Math Teacher| All Rights Reserved Directories Search form You are here. Autumn 2024 MATH 120 AH: Precalculus   News Feed   Alumni Update   Mailing List Ai Tutor : Question & Answer 4+ Build on chatgpt api & gpt-4o, kiran madad. 4.4 • 96 Ratings Offers In-App Purchases Screenshots Description. Ai Tutor Math Solver, the all-in-one app that empowers you to conquer any mathematical challenge. Don't just get an answer, understand the "why" behind it. 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Visualize concepts with graphing tools, manipulate equations to see how they change, and practice your skills with built-in quizzes. From basic arithmetic (addition, subtraction, multiplication, division) to complex calculus (integrations, derivatives), Ai Tutor Math Solver tackles a vast array of mathematical topics. Need help with algebra, geometry, trigonometry, or statistics? Ai Tutor Math Solver has your back! https://sites.google.com/view/sublimationtermofuse/home https://sites.google.com/view/sublimationprivacypolicy/home Version 1.1 Tutor Ai: Now powered by ChatGPT API & GPT-4 - Performance Enhancement - Bugs Fixed Ratings and Reviews Extremely helpful. This app is extremely helpful. It breaks down problems into easy-to-understand steps. A game-changer for math learning This app has transformed how I approach learning math. It tailors the difficulty to my level and helps build my confidence with every session. 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You Might Also Like Math Notes: AI Homework Helper Ai Tutor : Math Solver & Scan Photo Math AI Homework Helper Homework : AI Math Calculator AI Math - Homework Helper AI Math Problem Solver: Photo 373 IMAGES Classroom Poster: 4 Steps to Solve Any Math Problem Math Problem Solving 101 What IS Problem-Solving? What IS Problem-Solving? Problem Solving Strategies (Maths) Problem Solving Strategies VIDEO Mastering Two-Step Equations: Problem-Solving Techniques Problem Solving and Reasoning: Polya's Steps and Problem Solving Strategies Solving Math Problems in 2024 #shorts #short #reels #viral #viralvideo #maths #study #students what is problem Types of Problem solving And purpose The Problem-Solving Process & Strategies COMMENTS 1.5: Problem Solving Solution. 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. Step-by-Step Calculator To solve math problems step-by-step start by reading the problem carefully and understand what you are being asked to find. Next, identify the relevant information, define the variables, and plan a strategy for solving the problem. en. Related Symbolab blog posts. Practice Makes Perfect. GeoGebra Math Solver Get accurate solutions and step-by-step explanations for algebra and other math problems with the free GeoGebra Math Solver. Enhance your problem-solving skills while learning how to solve equations on your own. Try it now! Microsoft Math Solver Get math help in your language. Works in Spanish, Hindi, German, and more. Online math solver with free step by step solutions to algebra, calculus, and other math problems. Get help on the web or with our math app. Module 1: Problem Solving Strategies 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. ... Make sure you use Polya's 4 problem solving steps. (12 points) Problem Solving Strategy 2 (Draw a Picture). Step-by-Step Math Problem Solver QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students. The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and ... Solve equations and systems with Step-by-Step Math Problem Solver An identity is an equation that is satisfied by all numbers from its replacement set. Example 1 Consider the equation 2x-1 = x+2. The replacement set here is the set of all real numbers. The equation is conditional since, for example, 1 is a member of the replacement set but not of the solution set. Example 2 Consider the equation (x-1) (x+1 ... Problem Solving in Mathematics Mathematician George Pólya's book, "How to Solve It: A New Aspect of Mathematical Method," written in 1957, is a great guide to have on hand.The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya's book and should help you untangle even the most complicated math problem. Step-by-Step Solutions Starting at $5.00 /month. Get step-by-step answers and hints for your math homework problems. Learn the basics, check your work, gain insight on different ways to solve problems. For chemistry, calculus, algebra, trigonometry, equation solving, basic math and more. Step-by-Step Math Problem Solver QuickMath offers a step-by-step math problem solver for various equations and expressions, simplifying complex math tasks. The easy 4 step problem-solving process (+ examples) Consider the problem-solving steps applied in the following example. I know that I want to say "I don't eat eggs" to my Mexican waiter. That's the problem. I don't know how to say that, but last night I told my date "No bebo alcohol" ("I don't drink alcohol"). I also know the infinitive for "eat" in Spanish (comer). A Guide to Problem Solving A Guide to Problem Solving. When confronted with a problem, in which the solution is not clear, you need to be a skilled problem-solver to know how to proceed. When you look at STEP problems for the first time, it may seem like this problem-solving skill is out of your reach, but like any skill, you can improve your problem-solving with practice. 1.3: Problem Solving Strategies 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. Mathway Free math problem solver answers your algebra homework questions with step-by-step explanations. Mathway. Visit Mathway on the web. Start 7-day free trial on the app. Start 7-day free trial on the app. Download free on Amazon. Download free in Windows Store. 20 Effective Math Strategies For Problem Solving 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. Symbolab Provide step-by-step solutions to math word problems Graphing Plot and analyze functions and equations with detailed steps Geometry Solve geometry problems, proofs, and draw geometric shapes Math Help Tailored For You Practice Practice and improve your math skills through interactive personalized exercises and quizzes Problem Solving Strategies Problem Solving Strategy 6 (Work Systematically). If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated. For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how ... Polya's Problem Solving Process Polya's four step method for problem solving is. 1) Understand the Problem-Make sure you understand what the question is asking and what information will be used to solve the problem. 2) Devise a ... Cymath Join millions of users in problem solving! +. > < ... Problem Analysis in Math: Using the 5-Step Problem-Solving Approach Problem identification. Problem analysis. Plan development. Plan implementation. Plan evaluation. Following this 5-step problem-solving approach helps to guide school teams of educators and administrators in engaging in data-based decision making: a core principle of the MTSS framework. Math Problem Solver This creates a math problem solver that's more accurate than ChatGPT, more flexible than a math calculator, and provides answers faster than a human tutor. Sign up for free here. Problem Solver Subjects. Our math problem solver that lets you input a wide variety of math math problems and it will provide a step by step answer. 10.1: George Polya's Four Step Problem Solving Process Is this problem similar to another problem you have solved? Step 2: Devise a Plan: Below are some strategies one might use to solve a problem. Can one (or more) of the following strategies be used? (A strategy is defined as an artful means to an end.) 1. 4 Steps in Solving Problems Learn the steps you can follow to solve any math word problem.We hope you are enjoying this video! For more in-depth learning, check out Miacademy.co (https:... 4 Best Steps To Problem Solving in Math That Lead to Results In the past, we would teach the concepts and procedures and then assign one-step "story" problems designed to provide practice on the content. Next, we would teach problem solving as a collection of strategies such as "draw a picture" or "guess and check.". Eventually, students would be given problems to apply the skills and strategies. MATH 120 AH: Precalculus Basic properties of functions, graphs; with emphasis on linear, quadratic, trigonometric, exponential functions and their inverses. Emphasis on multi-step problem solving. Recommended: completion of Department of Mathematics' Guided Self-Placement. Offered: AWSpS. NYC Solves About the Curricula. A curriculum is how standards, or learning goals, for every grade and subject are translated into day-to-day activities. As part of the NYC Solves initiative, all high schools will use Illustrative Mathematics and districts will choose a comprehensive, evidence-based curricula for middle school math instruction from an approved list. ‎Ai Tutor : Question & Answer on the App Store Ai Tutor Math Solver, the all-in-one app that empowers you to conquer any mathematical challenge. Don't just get an answer, understand the "why" behind it. Ai Tutor Math Solver breaks down each problem into clear, step-by-step explanations, allowing you to grasp the concepts and solve similar problems independently. case study cover letter presentation problem solving rewiew prompts websites tips