what is problem solving in mathematics definition

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What is problem solving?

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On this page we discuss "What is problem polving?" under three headings: introduction, four stages of problem solving, and the scientific approach.

Introduction

Naturally enough, problem solving is about solving problems. And we’ll restrict ourselves to thinking about mathematical problems here even though problem solving in school has a wider goal. When you think about it, the whole aim of education is to equip students to solve problems. 

But problem solving also contributes to mathematics itself. Mathematics consists of skills and processes. The skills are things that we are all familiar with. These include the basic arithmetical processes and the algorithms that go with them. They include algebra in all its levels as well as sophisticated areas such as the calculus. This is the side of the subject that is largely represented in the Strands of Number and Algebra, Geometry and Measurement and Statistics.

On the other hand, the processes of mathematics are the ways of using the skills creatively in new situations. Mathematical processes include problem solving, logic and reasoning, and communicating ideas. These are the parts of mathematics that enable us to use the skills in a wide variety of situations.

It is worth starting by distinguishing between the three words "method", "answer" and "solution". By "method" we mean the means used to get an answer. This will generally involve one or more Problem Solving Strategies . On the other hand, we use "answer" to mean a number, quantity or some other entity that the problem is asking for. Finally, a "solution" is the whole process of solving a problem, including the method of obtaining an answer and the answer itself.

method + answer = solution

But how do we do Problem Solving? There are four basic steps. Pólya enunciated these in 1945 but all of them were known and used well before then. Pólya’s four stages of problem solving are listed below.

Four stages of problem solving                             

1. Understand and explore the problem  2. Find a strategy  3. Use the strategy to solve the problem  4. Look back and reflect on the solution.

Although we have listed the four stages in order, for difficult problems it may not be possible to simply move through them consecutively to produce an answer. It is frequently the case that students move backwards and forwards between and across the steps.

You can't solve a problem unless you can first understand it. This requires not only knowing what you have to find but also the key pieces of information that need to be put together to obtain the answer.

Students will often not be able to absorb all the important information of a problem in one go. It will almost always be necessary to read a problem several times, both at the start and while working on it. With younger students it is worth repeating the problem and then asking them to put the question in their own words. Older students might use a highlighter to mark the important parts of the problem.

Finding a strategy tends to suggest that it is a simple matter to think of an appropriate strategy. However, for many problems students may find it necessary to play around with the information before they are able to think of a strategy that might produce a solution. This exploratory phase will also help them to understand the problem better and may make them aware of some piece of information that they had neglected after the first reading.

Having explored the problem and decided on a strategy, the third step, solve the problem , can be attempted. Hopefully now the problem will be solved and an answer obtained. During this phase it is important for the students to keep a track of what they are doing. This is useful to show others what they have done and it is also helpful in finding errors should the right answer not be found.

At this point many students, especially mathematically able ones, will stop. But it is worth getting them into the habit of looking back over what they have done. There are several good reasons for this. First of all it is good practice for them to check their working and make sure that they have not made any errors. Second, it is vital to make sure that the answer they obtained is in fact the answer to the problem. Third, in looking back and thinking a little more about the problem, students are often able to see another way of solving the problem. This new solution may be a nicer solution than the original and may give more insight into what is really going on. Finally, students may be able to generalise or extend the problem.

Generalising a problem means creating a problem that has the original problem as a special case. So a problem about three pigs may be changed into one which has any number of pigs.

In Problem 4 of What is a Problem? , there is a problem on towers. The last part of that problem asks how many towers can be built for any particular height. The answer to this problem will contain the answer to the previous three questions. There we were asked for the number of towers of height one, two and three. If we have some sort of formula, or expression, for any height, then we can substitute into that formula to get the answer for height three, for instance. So the "any" height formula is a generalisation of the height three case. It contains the height three case as a special example.

Extending a problem is a related idea. Here though, we are looking at a new problem that is somehow related to the first one. For instance, a problem that involves addition might be looked at to see if it makes any sense with multiplication. A rather nice problem is to take any whole number and divide it by two if it’s even and multiply it by three and add one if it’s odd. Keep repeating this manipulation. Is the answer you get eventually 1? We’ll do an example. Let’s start with 34. Then we get

34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

We certainly got to 1 then. Now it turns out that no one in the world knows if you will always get to 1 this way, no matter where you start. That’s something for you to worry about. But where does the extension come in? Well we can extend this problem, by just changing the 3 to 5. So this time instead of dividing by 2 if the number is even and multiplying it by three and adding one if it’s odd, try dividing by 2 if the number is even and multiplying it by 5 and adding one if it’s odd. This new problem doesn’t contain the first one as a special case, so it’s not a generalisation. It is an extension though – it’s a problem that is closely related to the original. 

It is by this method of generalisation and extension that mathematics makes great strides forward. Up until Pythagoras’ time, many right-angled triangles were known. For instance, it was known that a triangle with sides 3, 4 and 5 was a right-angled triangle. Similarly people knew that triangles with sides 5, 12 and 13, and 7, 24 and 25 were right angled. Pythagoras’ generalisation was to show that EVERY triangle with sides a, b, c was a right-angled triangle if and only if a 2 + b 2 = c 2 .

This brings us to an aspect of problem solving that we haven’t mentioned so far. That is justification (or proof). Your students may often be able to guess what the answer to a problem is but their solution is not complete until they can justify their answer.

Now in some problems it is hard to find a justification. Indeed you may believe that it is not something that any of the class can do. So you may be happy that the students can find an answer. However, bear in mind that this justification is what sets mathematics apart from every other discipline. Consequently the justification step is an important one that shouldn’t be missed too often.

Scientific approach                                   

Another way of looking at the Problem Solving process is what might be called the scientific approach. We show this in the diagram below.

Here the problem is given and initially the idea is to experiment with it or explore it in order to get some feeling as to how to proceed. After a while it is hoped that the solver is able to make a conjecture or guess what the answer might be. If the conjecture is true it might be possible to prove or justify it. In that case the looking back process sets in and an effort is made to generalise or extend the problem. In this case you have essentially chosen a new problem and so the whole process starts over again.

Sometimes, however, the conjecture is wrong and so a counter-example is found. This is an example that contradicts the conjecture. In that case another conjecture is sought and you have to look for a proof or another counterexample.

Some problems are too hard so it is necessary to give up. Now you may give up so that you can take a rest, in which case it is a ‘for now’ giving up. Actually this is a good problem solving strategy. Often when you give up for a while your subconscious takes over and comes up with a good idea that you can follow. On the other hand, some problems are so hard that you eventually have to give up ‘for ever’. There have been many difficult problems throughout history that mathematicians have had to give up on.

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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What Is Problem Solving In Mathematics?

Definition and importance of problem solving in mathematics

Problem solving in mathematics is more than just crunching numbers and arriving at an answer. It’s a fundamental part of the subject that encourages you to:

• Understand the problem : Grasp what is being asked. What are the unknowns, and what information do you have?
• Develop reasoning skills : Going beyond memorization, problem solving involves logical thinking and the ability to reason out various solutions.
• Apply knowledge creatively : Use what you know in new and varied contexts.
• Overcome challenges : Develop perseverance, as often the first approach may not lead to a solution.

This component of mathematics helps you in real-life situations because life , much like math, is full of problems waiting to be solved. It’s about being adaptable, methodical, and open-minded.

Key skills and strategies for effective problem solving

To master problem solving, certain skills and strategies are essential, including:

• Analytical thinking : Break down complex problems into manageable parts.
• Pattern recognition : Notice recurring themes that can simplify the process.
• Communication : Be able to articulate the problem and explain your solution to others.
• Critical thinking : Evaluate and improve your approach.

Some tried-and-true strategies that aid in problem solving are:

• Working backwards : Start from the desired outcome and trace steps back to the starting point.
• Making a list  or  table : Organize information systematically.
• Drawing a diagram or picture : Visual aids can make abstract problems more concrete.
• Guessing and checking : Make educated guesses and refine them based on feedback.
• Looking for analogies : Compare with problems you’ve already solved.

Remember, with practice and patience, you can strengthen your problem-solving muscles in mathematics and beyond. Each puzzle you encounter is a chance to learn and grow, so embrace the challenges!

Understanding the Problem

Read and analyze the problem statement.

When facing a new mathematical challenge, the first step is always to carefully read and analyze the problem statement. Don’t rush through this; take your time to fully grasp what the problem is about. It’s like being a detective – pay attention to every detail, as sometimes even a single word can alter the meaning of a problem. This careful reading sets the groundwork for your entire problem-solving approach.

• Look for key information : Pay close attention to facts, figures, and the question.
• Clarify the goals : Determine what the problem is asking you to find.
• Take notes : Write down important details that might help you later.

Identify the known and unknown quantities

Separating what you know from what you don’t know is like sorting puzzle pieces by color before you start assembling the picture. It’s a vital part of the process that gives you clear direction on where to focus your efforts.

• List what’s given : Jot down all the information provided in the problem.
• Pinpoint what’s missing : Recognize what you’re trying to find.
• Make assumptions, if necessary : Sometimes, you need to fill in gaps with logical guesses.

Remember, everyone makes mistakes, and that’s okay. Embrace them as learning opportunities. You’ve got this! By taking these initial steps, you’re laying a solid foundation to tackle mathematical problems with confidence. So, keep practicing, stay curious, and remind yourself that with each problem a little bit of persistence goes a long way. Good luck – you’re on your way to becoming a math problem-solving wizard!

Problem Solving Approaches

Trial and error method.

Have you ever heard of the trial and error method? It’s like trying on shoes until you find the perfect fit. Here’s how you can apply this technique to your math problems:

Experiment with different techniques: There’s more than one way to solve a problem and it’s totally okay to try them out until one sticks.

Learn from mistakes: Each error is a step closer to understanding what doesn’t work, narrowing down the right path for you.

Be patient and persistent: Stick with it, even if you don’t get it on the first try. Perseverance is your best friend in this case.

Remember, there’s no shame in stumbling along the way; it’s all part of the learning curve. You’re doing great by just giving it a shot!

Using a Systematic Approach

Systematic approaches are a bit like following a recipe. You have a step-by-step guide that helps you stay organized and focused. Try these strategies:

Outline a plan: Sketch out the steps you need to take to arrive at the solution. A plan can steer you in the right direction.

Break problems into smaller parts: Don’t overwhelm yourself; slice that bigger problem into manageable pieces.

Validate each step: As you work through the problem, check each part to make sure you’re on track.

Using a systematic approach can make complex problems feel more manageable, and before you know it, you’ll have arrived at the answer without feeling lost or frustrated. Keep going—you’ve got the tools you need to crack even the toughest problems!

Problem Solving Strategies

Working backwards method.

Sometimes the best way to tackle a problem is to start at the end and work your way back to the beginning. Here’s why this can be a game-changer for you:

Reverse-engineering the solution: If you know where you need to end up, try figuring out the steps that led there—it’s like solving a mystery backward!

Simplifies complex problems: Working backwards can often simplify the process since the endpoint is your guide.

Highlights necessary information: This method helps you identify the most crucial pieces of information needed to solve the problem.

Give this approach a try next time you’re confronted with a particularly tricky puzzle. Who knows, you may just uncover a shortcut to the solution!

Making a Table or Chart

Are you a visual learner? Making tables or charts can be particularly effective for you. They help organize information and can highlight patterns or relationships that aren’t immediately obvious. Consider the following advantages of this strategy:

Visual organization: A table or chart can make complex data more digestible and approachable.

Track progress: Seeing your data laid out can help you track where you are in the problem-solving process.

Identify patterns: Sometimes the key to solving a problem lies in a pattern that only becomes visible when data is laid out visually.

So, next time you feel overwhelmed by a bunch of numbers or data points, take a step back and try putting them into a chart or table. It might just provide the clarity you need to crack the code! Keep pushing forward; your problem-solving toolkit is getting more robust by the minute.

Problem Solving Techniques

Using diagrams or visual representations.

I have another neat trick up my sleeve to share with you, and it’s all about getting visual with diagrams! Here’s why you should consider drawing your way out of a problem:

Makes abstract ideas concrete: Sometimes, issues feel tangled and chaotic in your mind. Sketching them out can make the abstract tangible. 

Engages different parts of the brain: Using visuals can engage your brain’s right hemisphere, which is thought to be responsible for creativity and visualization.

Facilitates communication: When you’re working in a team, a diagram can be a universal language, making it easier for everyone to understand the issue at hand.

Go ahead, grab that pencil and start drawing. Whether it’s a flowchart, mind map, or simple doodle, it might just help you break down the walls between you and that elusive solution.

Using logical reasoning and critical thinking

You’re already using such great strategies, but let’s not forget about the power of good, old-fashioned logic and critical thinking. They are the backbone of effective problem solving. Here’s what they can do for you:

Breaks down complex problems: Logical reasoning enables you to dissect big issues into smaller, more manageable parts.

Facilitates decision making: By evaluating information critically, you can make more informed decisions.

Reduces bias: Critical thinking encourages you to question assumptions and explore multiple viewpoints, leading to less biased and more objective conclusions.

When your next challenge arises, remember to ask the tough questions and follow the trail of logic. Your keen sense of reason might just be the key to unlocking the solution!

Remember, every problem is different, and there’s no one-size-fits-all technique. Try blending these strategies with those you’re already rocking, and you’ll become an unstoppable problem-solving force! Keep it up!

Problem Solving in Different Mathematical Concepts

Problem solving in algebra.

Have you ever found yourself befuddled by x’s and y’s swarming in your textbook? Fear not, as algebra often becomes friendlier when you employ some problem-solving finesse. Here’s how you can tackle algebraic puzzles like a pro:

• Find common patterns : Algebra is all about recognizing patterns. Stay on the lookout for familiar equations or formulas that can simplify your work.
• Work backwards : Stuck on where to start? Try working backwards from the answer. This reverse-engineering approach can reveal insights into the steps needed.
• Use substitution : When equations get messy, substituting variables with numbers or simpler expressions can make life a whole lot easier.
• Implement these tips during your next algebra session, and watch those variables bow down to your newfound strategic prowess!

Problem Solving in Geometry

Shifting gears to geometry, where shapes and angles reign supreme, the right strategies can unveil the mysteries behind those theorems and postulates:

• Draw it out : Never underestimate the power of a good sketch. Often, drawing the problem can reveal solutions that aren’t immediately obvious in text form.
• Break it down : Complex geometric figures can be intimidating. Break them into smaller parts, and focus on solving each section at a time.
• Look for congruencies and similarities : Pattern recognition isn’t only useful in algebra. In geometry, identifying congruent angles and similar triangles can save you loads of time. Next time you’re faced with tangents and secants, remember that a ruler and compass are your trusty sidekicks on this geometric adventure.

Embrace these approaches when you next encounter mathematical head-scratchers. Each concept might require a unique touch, but your creative thinking and analytical skills will shine through no matter the equation or theorem!

Tips for Effective Problem Solving

Breaking down complex problems into smaller steps.

You’ve got this! When faced with a complicated problem, don’t let it overwhelm you. Take a deep breath and start by breaking it down into smaller, more manageable pieces. This approach makes the problem less intimidating and allows you to focus on one piece at a time.

Approach each step one at a time : Patience is key here. Concentrate on solving each individual part of the problem before trying to tackle it as a whole. This step-by-step process can prevent mistakes and lead you to the correct solution more efficiently.

Celebrate small victories : Every time you complete a step, give yourself a pat on the back. These mini wins help maintain your motivation and keep your spirits high.

Utilizing past problem solving experiences

Think back to similar challenges. Remember that algebra equation you conquered last week? Draw on the strategies that worked for you then. Your past successes are a treasure trove of wisdom.

Learn from previous mistakes : It’s okay if you didn’t get it right the first time. What’s important is to reflect on what went wrong and use that insight to inform your current approach.

Collaboration is your friend : If you’re truly stumped, don’t hesitate to ask for help. Maybe your peers or teachers have faced a similar issue before. Their experience could provide you the breakthrough you need.

Remember, every problem you face is an opportunity to sharpen your problem-solving skills. You’re capable of tackling each one, especially when you apply tried-and-true strategies tailored to the task at hand. Keep going, keep learning, and most importantly, keep believing in yourself!

Start Small: Don’t let a big problem scare you. Begin by splitting it up into bite-sized parts you can handle. Smaller issues are less daunting and easier to solve.

One Step at a Time: Tackle one piece of the puzzle before moving on to the next. This methodical strategy helps you stay focused and makes the overall problem solvable.

Celebrate Progress: Each completed step is a victory. Recognize these achievements to stay motivated on your problem-solving journey.

Recall Previous Successes: Think about how you’ve solved problems before. Use those strategies again—they’re part of your personal toolkit now.

Learn From Mistakes: It’s fine if things didn’t work out last time. Reflect on what happened and apply those lessons to your present challenge.

Seek Help When Needed: If you’re stuck, reach out. Others around you might have insights from their own experiences that can help you break through.

Remember that with every problem you come across, you get a chance to improve your problem-solving skills . You’ve got what it takes to overcome these challenges, especially when you apply proven strategies that suit the situation. Keep pushing forward, keep learning, and most importantly, keep believing in yourself!

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What IS Problem-Solving?

Ask teachers about problem-solving strategies, and you’re opening a can of worms! Opinions about the “best” way to teach problem-solving are all over the board. And teachers will usually argue for their process quite passionately.

When I first started teaching math over 25 years ago, it was very common to teach “keywords” to help students determine the operation to use when solving a word problem. For example, if you see the word “total” in the problem, you always add. Rather than help students become better problem solvers, the use of keywords actually resulted in students who don’t even feel the need to read and understand the problem–just look for the keywords, pick out the numbers, and do the operation indicated by the keyword.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

Another common strategy for teaching problem-solving is the use of acrostics that students can easily remember to perform the “steps” in problem-solving. CUBES is an example. Just as with keywords, however, students often follow the steps with little understanding. As an example, a common step is to underline or highlight the question. But if you ask students why they are underlining or highlighting the question, they often can’t tell you. The question is , in fact, super important, but they’ve not been told why. They’ve been told to underline the question, so they do.

The problem with both keywords and the rote-step strategies is that both methods try to turn something that is inherently messy into an algorithm! It’s way past time that we leave both methods behind.

First, we need to broaden the definition of problem-solving. Somewhere along the line, problem-solving became synonymous with “word problems.” In reality, it’s so much more. Every one of us solves dozens or hundreds of problems every single day, and most of us haven’t solved a word problem in years. Problem-solving is often described as  figuring out what to do when you don’t  know what to do.  My power went out unexpectedly this morning, and I have work to do. That’s a problem that I had to solve. I had to think about what the problem was, what my options were, and formulate a plan to solve the problem. No keywords. No acrostics. I’m using my phone as a hotspot and hoping my laptop battery doesn’t run out. Problem solved. For now.

If you want to get back to what problem-solving really is, you should consult the work of George Polya. His book, How to Solve It , which was first published in 1945, outlined four principles for problem-solving. The four principles are: understand the problem, devise a plan, carry out the plan, and look back. This document from UC Berkeley’s Mathematics department is a great 4-page overview of Polya’s process. You can probably see that the keyword and rote-steps strategies were likely based on Polya’s method, but it really got out of hand. We need to help students think , not just follow steps.

I created both primary and intermediate posters based on Polya’s principles. Grab your copies for free here !

I would LOVE to hear your comments about problem-solving!

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Do you tutor teachers?

I do professional development for district and schools, and I have online courses.

You make a great point when you mentioned that teaching students to look for “keywords” is not teaching students to become better problem solvers. I was once guilty of using the CUBES strategy, but have since learned to provide students with opportunity to grapple with solving a problem and not providing them with specified steps to follow.

I think we’ve ALL been there! We learn and we do better. 🙂

Love this article and believe that we can do so much better as math teachers than just teaching key words! Do you have an editable version of this document? We are wanting to use something similar for our school, but would like to tweak it just a bit. Thank you!

I’m sorry, but because of the clip art and fonts I use, I am not able to provide an editable version.

Hi Donna! I am working on my dissertation that focuses on problem-solving. May I use your intermediate poster as a figure, giving credit to you in my citation with your permission, for my section on Polya’s Traditional Problem-Solving Steps? You laid out the process so succinctly with examples that my research could greatly benefit from this image. Thank you in advance!

Absolutely! Good luck with your dissertation!

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem  in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems  includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving  focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

• The problem has important, useful mathematics embedded in it.
• The problem requires high-level thinking and problem solving.
• The problem contributes to the conceptual development of students.
• The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
• The problem can be approached by students in multiple ways using different solution strategies.
• The problem has various solutions or allows different decisions or positions to be taken and defended.
• The problem encourages student engagement and discourse.
• The problem connects to other important mathematical ideas.
• The problem promotes the skillful use of mathematics.
• The problem provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

Key features of a good mathematics problem includes:

• It must begin where the students are mathematically.
• The feature of the problem must be the mathematics that students are to learn.
• It must require justifications and explanations for both answers and methods of solving.

Problem solving is not a  neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

• Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
• What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
• Can the activity accomplish your learning objective/goals?

Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

• Allows students to show what they can do, not what they can’t.
• Provides differentiation to all students.
• Promotes a positive classroom environment.
• Advances a growth mindset in students
• Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

• YouCubed – under grades choose Low Floor High Ceiling
• NRICH Creating a Low Threshold High Ceiling Classroom
• Inside Mathematics Problems of the Month

Math in 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

Act Three is the “reveal.” Students share their thinking as well as their solutions.

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

• Dan Meyer’s Three-Act Math Tasks
• Graham Fletcher3-Act Tasks ]
• Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

Number Talks

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

• The teacher presents a problem for students to solve mentally.
• Provide adequate “ wait time .”
• The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
• For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
• Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

When the teacher says, “this is easy,” students may think,

• “Everyone else understands and I don’t. I can’t do this!”
• Students may just give up and surrender the mathematics to their classmates.
• Students may shut down.

Instead, you and your students could say the following:

• “I think I can do this.”
• “I have an idea I want to try.”
• “I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

• Provide your students a bridge between the concrete and abstract
• Serve as models that support students’ thinking
• Provide another representation
• Support student engagement
• Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

20 seconds to 2 minutes for students to make sense of questions

Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Problem Solving Skills: Meaning, Examples & Techniques

Table of Contents

26 January 2021

Reading Time: 2 minutes

Do your children have trouble solving their Maths homework?

Or, do they struggle to maintain friendships at school?

If your answer is ‘Yes,’ the issue might be related to your child’s problem-solving abilities. Whether your child often forgets his/her lunch at school or is lagging in his/her class, good problem-solving skills can be a major tool to help them manage their lives better.

Children need to learn to solve problems on their own. Whether it is about dealing with academic difficulties or peer issues when children are equipped with necessary problem-solving skills they gain confidence and learn to make healthy decisions for themselves. So let us look at what is problem-solving, its benefits, and how to encourage your child to inculcate problem-solving abilities

Problem-solving skills can be defined as the ability to identify a problem, determine its cause, and figure out all possible solutions to solve the problem.

• Trigonometric Problems

What is problem-solving, then? Problem-solving is the ability to use appropriate methods to tackle unexpected challenges in an organized manner. The ability to solve problems is considered a soft skill, meaning that it’s more of a personality trait than a skill you’ve learned at school, on-the-job, or through technical training. While your natural ability to tackle problems and solve them is something you were born with or began to hone early on, it doesn’t mean that you can’t work on it. This is a skill that can be cultivated and nurtured so you can become better at dealing with problems over time.

Problem Solving Skills: Meaning, Examples & Techniques are mentioned below in the Downloadable PDF. 

Benefits of learning problem-solving skills  

Promotes creative thinking and thinking outside the box.

Improves decision-making abilities.

Builds solid communication skills.

Develop the ability to learn from mistakes and avoid the repetition of mistakes.

Problem Solving as an ability is a life skill desired by everyone, as it is essential to manage our day-to-day lives. Whether you are at home, school, or work, life throws us curve balls at every single step of the way. And how do we resolve those? You guessed it right – Problem Solving.

Strengthening and nurturing problem-solving skills helps children cope with challenges and obstacles as they come. They can face and resolve a wide variety of problems efficiently and effectively without having a breakdown. Nurturing good problem-solving skills develop your child’s independence, allowing them to grow into confident, responsible adults. 

Children enjoy experimenting with a wide variety of situations as they develop their problem-solving skills through trial and error. A child’s action of sprinkling and pouring sand on their hands while playing in the ground, then finally mixing it all to eliminate the stickiness shows how fast their little minds work.

Sometimes children become frustrated when an idea doesn't work according to their expectations, they may even walk away from their project. They often become focused on one particular solution, which may or may not work.

However, they can be encouraged to try other methods of problem-solving when given support by an adult. The adult may give hints or ask questions in ways that help the kids to formulate their solutions. 

Encouraging Problem-Solving Skills in Kids

Practice problem solving through games.

Exposing kids to various riddles, mysteries, and treasure hunts, puzzles, and games not only enhances their critical thinking but is also an excellent bonding experience to create a lifetime of memories.

Create a safe environment for brainstorming

Welcome, all the ideas your child brings up to you. Children learn how to work together to solve a problem collectively when given the freedom and flexibility to come up with their solutions. This bout of encouragement instills in them the confidence to face obstacles bravely.

Invite children to expand their Learning capabilities

 Whenever children experiment with an idea or problem, they test out their solutions in different settings. They apply their teachings to new situations and effectively receive and communicate ideas. They learn the ability to think abstractly and can learn to tackle any obstacle whether it is finding solutions to a math problem or navigating social interactions.

Problem-solving is the act of finding answers and solutions to complicated problems. 

Developing problem-solving skills from an early age helps kids to navigate their life problems, whether academic or social more effectively and avoid mental and emotional turmoil.

Children learn to develop a future-oriented approach and view problems as challenges that can be easily overcome by exploring solutions. 

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Frequently Asked Questions (FAQs)

How do you teach problem-solving skills.

Model a useful problem-solving method. Problem solving can be difficult and sometimes tedious. ... 1. Teach within a specific context. ... 2. Help students understand the problem. ... 3. Take enough time. ... 4. Ask questions and make suggestions. ... 5. Link errors to misconceptions.

What makes a good problem solver?

Excellent problem solvers build networks and know how to collaborate with other people and teams. They are skilled in bringing people together and sharing knowledge and information. A key skill for great problem solvers is that they are trusted by others.

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Fluency, Reasoning and Problem Solving: What This Looks Like In Every Math Lesson

Neil Almond

Fluency, reasoning and problem solving are central strands of mathematical competency, as recognized by the National Council of Teachers of Mathematics (NCTM) and the National Research Council’s report ‘Adding It Up’.

They are key components to the Standards of Mathematical Practice, standards that are interwoven into every mathematics lesson. Here we look at how these three approaches or elements of math can be interwoven in a child’s math education through elementary and middle school.

We look at what fluency, reasoning and problem solving are, how to teach them, and how to know how a child is progressing in each – as well as what to do when they’re not, and what to avoid.

The hope is that this blog will help elementary and middle school teachers think carefully about their practice and the pedagogical choices they make around the teaching of what the common core refers to as ‘mathematical practices’, and reasoning and problem solving in particular.

Before we can think about what this would look like in Common Core math examples and other state-specific math frameworks, we need to understand the background to these terms.

What is fluency in math?

What is reasoning in math, what is problem solving in math, mathematical problem solving is a learned skill, performance vs learning: what to avoid when teaching fluency, reasoning, and problem solving.

• What IS ‘performance vs learning’?
• Teaching to “cover the curriculum” hinders development of strong problem solving skills.
• Fluency and reasoning – Best practice in a lesson, a unit, and a semester

Best practice for problem solving in a lesson, a unit, and a semester 

Fluency, reasoning and problem solving should not be taught by rote .

The Ultimate Guide to Problem Solving Techniques

Develop problem solving skills in the classroom with this free, downloadable worksheet

Fluency in math is a fairly broad concept. The basics of mathematical fluency – as defined by the Common Core State Standards for math – involve knowing key mathematical skills and being able to carry them out flexibly, accurately and efficiently.

But true fluency in math (at least up to middle school) means being able to apply the same skill to multiple contexts, and being able to choose the most appropriate method for a particular task.

Fluency in math lessons means we teach the content using a range of representations, to ensure that all students understand and have sufficient time to practice what is taught.

Read more: How the best schools develop math fluency

Reasoning in math is the process of applying logical thinking to a situation to derive the correct math strategy for problem solving  for a question, and using this method to develop and describe a solution.

Put more simply, mathematical reasoning is the bridge between fluency and problem solving. It allows students to use the former to accurately carry out the latter.

Read more: Developing math reasoning: the mathematical skills required and how to teach them .

It’s sometimes easier to start off with what problem solving is not. Problem solving is not necessarily just about answering word problems in math. If a child already has a readily available method to solve this sort of problem, problem solving has not occurred. Problem solving in math is finding a way to apply knowledge and skills you have to answer unfamiliar types of problems.

Read more: Math problem solving: strategies and resources for primary school teachers .

We are all problem solvers

First off, problem solving should not be seen as something that some students can do and some cannot. Every single person is born with an innate level of problem-solving ability.

Early on as a species on this planet, we solved problems like recognizing faces we know, protecting ourselves against other species, and as babies the problem of getting food (by crying relentlessly until we were fed).

All these scenarios are a form of what the evolutionary psychologist David Geary (1995) calls biologically primary knowledge. We have been solving these problems for millennia and they are so ingrained in our DNA that we learn them without any specific instruction.

Why then, if we have this innate ability, does actually teaching problem solving seem so hard?

As you might have guessed, the domain of mathematics is far from innate. Math doesn’t just happen to us; we need to learn it. It needs to be passed down from experts that have the knowledge to novices who do not.

This is what Geary calls biologically secondary knowledge. Solving problems (within the domain of math) is a mixture of both primary and secondary knowledge.

The issue is that problem solving in domains that are classified as biologically secondary knowledge (like math) can only be improved by practicing elements of that domain.

So there is no generic problem-solving skill that can be taught in isolation and transferred to other areas.

This will have important ramifications for pedagogical choices, which I will go into more detail about later on in this blog.

The educationalist Dylan Wiliam had this to say on the matter: ‘for…problem solving, the idea that students can learn these skills in one context and apply them in another is essentially wrong.’ (Wiliam, 2018) So what is the best method of teaching problem solving to elementary and middle school math students?

The answer is that we teach them plenty of domain specific biological secondary knowledge – in this case, math. Our ability to successfully problem solve requires us to have a deep understanding of content and fluency of facts and mathematical procedures.

Here is what cognitive psychologist Daniel Willingham (2010) has to say:

‘Data from the last thirty years leads to a conclusion that is not scientifically challengeable: thinking well requires knowing facts, and that’s true not simply because you need something to think about.

The very processes that teachers care about most—critical thinking processes such as reasoning and problem solving—are intimately intertwined with factual knowledge that is stored in long-term memory (not just found in the environment).’

Colin Foster (2019), a reader in Mathematics Education in the Mathematics Education Center at Loughborough University, UK, says, ‘I think of fluency and mathematical reasoning, not as ends in themselves, but as means to support students in the most important goal of all: solving problems.’

In that paper he produces this pyramid:

This is important for two reasons:

1)    It splits up reasoning skills and problem solving into two different entities

2)    It demonstrates that fluency is not something to be rushed through to get to the ‘problem solving’ stage but is rather the foundation of problem solving.

In my own work I adapt this model and turn it into a cone shape, as education seems to have a problem with pyramids and gross misinterpretation of them (think Bloom’s taxonomy).

Notice how we need plenty of fluency of facts, concepts, procedures and mathematical language.

Having this fluency will help with improving logical reasoning skills, which will then lend themselves to solving mathematical problems – but only if it is truly learnt and there is systematic retrieval of this information carefully planned across the curriculum.

I mean to make no sweeping generalization here; this was my experience both at university when training and from working in schools.

At some point, schools become obsessed with the ridiculous notion of moving students through content at an accelerated rate. I have heard it used in all manner of educational contexts while training and being a teacher. ‘You will need to show ‘accelerated progress in math’ in this lesson,’ ‘School officials will be looking for ‘accelerated progress’ etc.

I have no doubt that all of this came from a good place and from those wanting the best possible outcomes – but it is misguided.

I remember being told that we needed to get students onto the problem solving questions as soon as possible to demonstrate this mystical ‘accelerated progress’.

This makes sense; you have a group of students and you have taken them from not knowing something to working out pretty sophisticated 2-step or multi-step word problems within an hour. How is that not ‘accelerated progress?’

This was a frequent feature of my lessons up until last academic year: teach a mathematical procedure; get the students to do about 10 of them in their books; mark these and if the majority were correct, model some reasoning/problem solving questions from the same content as the fluency content; give the students some reasoning and word problem questions and that was it.

I wondered if I was the only one who had been taught this while at university so I did a quick poll on Twitter and found that was not the case.

I know these numbers won’t be big enough for a representative sample but it still shows that others are familiar with this approach.

The issue with the lesson framework I mentioned above is that it does not take into account ‘performance vs learning.’

What IS ‘performance vs learning’?

The premise is that performance in a lesson is not a good proxy for learning.

Yes, those students were performing well after I had modeled a mathematical procedure for them, and managed to get questions correct.

But if problem solving depends on a deep knowledge of mathematics, this approach to lesson structure is going to be very ineffective.

As mentioned earlier, the reasoning and problem solving questions were based on the same math content as the fluency exercises, making it more likely that students would solve problems correctly whether they fully understood them or not.

Chances are that all they’d need to do is find the numbers in the questions and use the same method they used in the fluency section to get their answers (a process referred to as “number plucking”) – not exactly high level problem solving skills.

Teaching to “cover the curriculum” hinders development of strong problem solving skills.

This is one of my worries with ‘math mastery schemes’ that block content so that, in some circumstances, it is not looked at again until the following year (and with new objectives).

The pressure for teachers to ‘get through the curriculum’ results in many opportunities to revisit content being missed in the classroom.

Students are unintentionally forced to skip ahead in the fluency, reasoning, problem solving chain without proper consolidation of the earlier processes.

As David Didau (2019) puts it, ‘When novices face a problem for which they do not have a conveniently stored solution, they have to rely on the costlier means-end analysis.

This is likely to lead to cognitive overload because it involves trying to work through and hold in mind multiple possible solutions.

It’s a bit like trying to juggle five objects at once without previous practice. Solving problems is an inefficient way to get better at problem solving.’

Fluency and reasoning – Best practice in a lesson, a unit, and a semester

By now I hope you have realized that when it comes to problem solving, fluency is king. As such we should look to mastery math based teaching to ensure that the fluency that students need is there.

The answer to what fluency looks like will obviously depend on many factors, including the content being taught and the grade you find yourself teaching.

But we should not consider rushing them on to problem solving or logical reasoning in the early stages of this new content as it has not been learnt, only performed.

I would say that in the early stages of learning, content that requires the end goal of being fluent should take up the majority of lesson time – approximately 60%. The rest of the time should be spent rehearsing and retrieving other knowledge that is at risk of being forgotten about.

This blog on mental math strategies students should learn at each grade level is a good place to start when thinking about the core aspects of fluency that students should achieve.

Little and often is a good mantra when we think about fluency, particularly when revisiting the key mathematical skills of number bond fluency or multiplication fluency. So when it comes to what fluency could look like throughout the day, consider all the opportunities to get students practicing.

They could chant multiplication facts when transitioning. If a lesson in another subject has finished earlier than expected, use that time to quiz students on number bonds. Have fluency exercises as part of the morning work.

Read more: How to teach multiplication for instant recall

What about best practice over a longer period?

Thinking about what fluency could look like across a unit of work would again depend on the unit itself.

Look at this unit below from a popular scheme of work.

They recommend 20 days to cover 9 objectives. One of these specifically mentions problem solving so I will forget about that one at the moment – so that gives 8 objectives.

I would recommend that the fluency of this unit look something like this:

This type of structure is heavily borrowed from Mark McCourt’s phased learning idea from his book ‘Teaching for Mastery.’

This should not be seen as something set in stone; it would greatly depend on the needs of the class in front of you. But it gives an idea of what fluency could look like across a unit of lessons – though not necessarily all math lessons.

When we think about a semester, we can draw on similar ideas to the one above except that your lessons could also pull on content from previous units from that semester.

So lesson one may focus 60% on the new unit and 40% on what was learnt in the previous unit.

The structure could then follow a similar pattern to the one above.

When an adult first learns something new, we cannot solve a problem with it straight away. We need to become familiar with the idea and practice before we can make connections, reason and problem solve with it.

The same is true for students. Indeed, it could take up to two years ‘between the mathematics a student can use in imitative exercises and that they have sufficiently absorbed and connected to use autonomously in non-routine problem solving.’ (Burkhardt, 2017).

Practice with facts that are secure

So when we plan for reasoning and problem solving, we need to be looking at content from 2 years ago to base these questions on.

You could get students in 3rd grade to solve complicated place value problems with the numbers they should know from 1st or 2nd grade. This would lessen the cognitive load , freeing up valuable working memory so they can actually focus on solving the problems using content they are familiar with.

Increase complexity gradually

Once they practice solving these types of problems, they can draw on this knowledge later when solving problems with more difficult numbers.

This is what Mark McCourt calls the ‘Behave’ phase. In his book he writes:

‘Many teachers find it an uncomfortable – perhaps even illogical – process to plan the ‘Behave’ phase as one that relates to much earlier learning rather than the new idea, but it is crucial to do so if we want to bring about optimal gains in learning, understanding and long term recall.’  (Mark McCourt, 2019)

This just shows the fallacy of ‘accelerated progress’; in the space of 20 minutes some teachers are taught to move students from fluency through to non-routine problem solving, or we are somehow not catering to the needs of the child.

When considering what problem solving lessons could look like, here’s an example structure based on the objectives above.

It is important to reiterate that this is not something that should be set in stone. Key to getting the most out of this teaching for mastery approach is ensuring your students (across abilities) are interested and engaged in their work.

Depending on the previous attainment and abilities of the children in your class, you may find that a few have come across some of the mathematical ideas you have been teaching, and so they are able to problem solve effectively with these ideas.

Equally likely is encountering students on the opposite side of the spectrum, who may not have fully grasped the concept of place value and will need to go further back than 2 years and solve even simpler problems.

In order to have the greatest impact on class performance, you will have to account for these varying experiences in your lessons.

Read more: 

• Math Mastery Toolkit : A Practical Guide To Mastery Teaching And Learning
• Problem Solving and Reasoning Questions and Answers
• Get to Grips with Math Problem Solving For Elementary Students
• Mixed Ability Teaching for Mastery: Classroom How To
• 21 Math Challenges To Really Stretch Your More Able Students
• Why You Should Be Incorporating Stem Sentences Into Your Elementary Math Teaching

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The content in this article was originally written by primary school lead teacher Neil Almond and has since been revised and adapted for US schools by elementary math teacher Jaclyn Wassell.

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Problem Solving

• Reference work entry
• pp 2680–2683
• Cite this reference work entry

• David H. Jonassen 2 &
• Woei Hung 3  

1934 Accesses

12 Citations

Cognition ; Problem typology ; Problem-based learning ; Problems ; Reasoning

Problem solving is the process of constructing and applying mental representations of problems to finding solutions to those problems that are encountered in nearly every context.

Theoretical Background

Problem solving is the process of articulating solutions to problems. Problems have two critical attributes. First, a problem is an unknown in some context. That is, there is a situation in which there is something that is unknown (the difference between a goal state and a current state). Those situations vary from algorithmic math problems to vexing and complex social problems, such as violence in society (see Problem Typology ). Second, finding or solving for the unknown must have some social, cultural, or intellectual value. That is, someone believes that it is worth finding the unknown. If no one perceives an unknown or a need to determine an unknown, there is no perceived problem. Finding...

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Jonassen, D.H., Hung, W. (2012). Problem Solving. In: Seel, N.M. (eds) Encyclopedia of the Sciences of Learning. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1428-6_208

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Word Problems 

Addition and subtraction, multiplication and division, equation-related mathematical problems, problem|definition & meaning.

A mathematical problem is an unsolved question . These problems usually provide some values and ask to find some unknown value . For example, if you cycled a total distance of 16 km in one hour, then what was your average speed? In other problems, you might have an equation already and need to solve it for the unknown variable e.g., what is the value of x in x + 6 = 21?

Some mathematical Problems include the usage of words , such as, “Jim maintained a speed of 30 kilometers per hour during the entire hour . How much ground did he cover?” Others use equations such as  “If x + 19 = 78, what is x?” In this article, we shall talk about them in great detail.

Figure 1 – Problems in Mathematics

Word problems are notoriously difficult for pupils to master. Because there are so many moving parts in the process of solving word problems , it can be difficult to isolate the specific factor that is making things difficult for students.

Types of Word Problems

Figure 2 – Types of Word Problems

There are three primary categories of problems involving addition and subtraction:

• joining problems
• separating problems
• comparing problems

Any problem in which you begin with one quantity , acquire some more, and then finish with a greater quantity is said to be a joining problem. Any situation in which you begin with one quantity , remove some of that quantity , and then finish up with a smaller quantity is known as a separating problem. Take, for instance:

• Joining: make sure you have 4 orange slices. Another 5 orange slices were handed to me by my brother. How many orange slices do I still have in my possession right now?
• Separating: Count nine blueberries before you separate them. I eat four of them in total. How many blueberries are there still available?

In each of these scenarios, the end outcome is unknown because we know how much we begin with, we know how much is joined or separated (the change), and now we need to determine how much is left over after the process . Both of these issues may be solved using the following straightforward pattern:

Students need to solve issues in which either the result, the change, or the starting point is unknown .

• Make a tally of the first set.
• Add some of them to the original set, or take some of them away from it.
• Determine the updated total when the event has been completed.

You’ll find that there are innumerable opportunities throughout the day to utilize the terminology of joining and separating! The most obvious of them is eating because your child is always considering joining other children in order to grab more of their food and then consuming that food after it has been obtained (separating).

When you play a game with your child that involves blocks, ask them to count how many are in their tower. Then you should ask them, “ Now I’ve added three more blocks to your tower . How many blocks do you currently have on your tower?” Have your child count the number of cars in a row while they see vehicles parked in a parking lot. Then pose the following question: “If two vehicles depart the parking lot, how many vehicles will be left?”

If your child is having trouble, you shouldn’t give them the solution. Instead, you should assist them in carrying out the action.

“You have 9 blueberries. Now you are going to devour four of them. When you consume those blueberries, what happens to the blueberries? Your kid might remark something along the lines of “They went in my stomach!” And you are able to reply, “Good.” So, those blueberries are still sitting there on your plate, are they?”

They will respond with a negative, at which point you can say, “Ok. So please display those blueberries that are disappearing from your dish. When performing an additive comparison, the problem might have the following forms, where x could be any whole number.

• How much further is it?
• How much less do you want?

There are three primary kinds of word problems involving multiplication and division

• Equal groups
• Comparing problems

Equal Groups

When there is the same number of items in two different groups, we refer to such groupings as “equal groups.” Therefore, an equal number of items or things are grouped together in each equal group.

For example: If there are three boxes and you put five candies in each box, then there will be an equal number of candies in each box. At this point, we will assume that there ar e three equal groupings , each containing five candies .

The operations of multiplication and division can be represented by using rows and columns in an array. The rows denote the number of different categories. The number of items in each category, as well as their respective sizes, are denoted by the columns, although this is not necessarily a strict rule, and the two can be swapped.

It is essential for one to keep in mind that rows, which represent groups, are drawn horizontally, and columns, which represent the number of items in each group, are drawn vertically.

Comparing Problems

When doing a multiplicative comparison, the problem can involve expressions such as where x can be any whole number.

• x multiples as many as that
• x multiplied in comparison to

It is possible that the product, the size of the group, or the number of groups will all be unknown within each type of problem. Once again, we make use of counters to guide the students through the process of selecting the appropriate equation to apply while trying to solve the various kinds of issues.

The difference between issues involving addition and subtraction and problems involving multiplication and division should be brought to the attention of the students.

Students’ thinking and ability to solve problems are considerably improved when they are required to compose word problems in a fashion that is consistent with a certain word problem style. Because this is a considerably more difficult ability, the first time we practice composing word problems, we usually do so under the supervision of an instructor.

Equations are used to solve issues , and in order to solve a problem using equations, we must do two things:

• Translate the expression of the question from the common language into the algebraic language to get an equation.
• Reduce this equation so that the unknown quantity will stand by itself , and its value will be stated in terms of known quantities on the opposite side .

One of the most notable characteristics of an algebraic solution is that the quantity that is being sought is incorporated into the very operation that is being performed. Because of this, we are able to construct a statement of the conditions in the same form as if the problem had already been solved.

Figure 3 – Equation Related Mathematical Problems

Nothing else has to be done at this point other than to simplify the equation and determine the total sum of the quantities that are already known. Because they are equivalent to the unknown quantity on the opposite side of the equation , the value of that is likewise determined, which means that the problem has been solved as a result.

Examples of Problems

These examples are quite literally examples of problems and illustrate the process of translating words into equations and then simplifying them to find the value of the unknown variable.

When a given integer is divided by 10, the sum of the quotient, dividend, and divisor equals 54. Determine the number that satisfies it.

Let x equal the desired number. Then:

(x / 10) + x + 10 = 54

x + 10x + 100 = 540

11x = 540 – 100

When a deal is made, a particular amount of profit or loss is realized by a merchant. In the second deal, he makes a profit of 250 dollars, but in the third, he loses 50 dollars. In the end, he determines that the three transactions resulted in a profit of one hundred dollars for him. In comparison to the first, how much ground did he make or lose?

In this particular illustration, the profit and the loss are of opposing characters, so it is necessary to differentiate between them using signs that are the opposite of one another. When the profit is a plus sign (+), the loss should be a minus sign (-).

Let’s say x equals the total amount needed.

The conclusion that follows from this is that x plus 250 minus 50 equals 100.

So, x = -100 .

The fact that the answer has a negative sign attached to it demonstrates that there was a loss incurred in the initial transaction; hence, the correct sign for x is also a negative sign. However, because this is dependent on the response, leaving it out of the calculation won’t result in an error at all.

All images were created with GeoGebra.

Prism Definition < Glossary Index > Profit Definition

Mathematics as a Complex Problem-Solving Activity

By jacob klerlein and sheena hervey, generation ready.

By the time young children enter school they are already well along the pathway to becoming problem solvers. From birth, children are learning how to learn: they respond to their environment and the reactions of others. This making sense of experience is an ongoing, recursive process. We have known for a long time that reading is a complex problem-solving activity. More recently, teachers have come to understand that becoming mathematically literate is also a complex problem-solving activity that increases in power and flexibility when practiced more often. A problem in mathematics is any situation that must be resolved using mathematical tools but for which there is no immediately obvious strategy. If the way forward is obvious, it’s not a problem—it is a straightforward application.

Mathematicians have always understood that problem-solving is central to their discipline because without a problem there is no mathematics. Problem-solving has played a central role in the thinking of educational theorists ever since the publication of Pólya’s book “How to Solve It,” in 1945. The National Council of Teachers of Mathematics (NCTM) has been consistently advocating for problem-solving for nearly 40 years, while international trends in mathematics teaching have shown an increased focus on problem-solving and mathematical modeling beginning in the early 1990s. As educators internationally became increasingly aware that providing problem-solving experiences is critical if students are to be able to use and apply mathematical knowledge in meaningful ways (Wu and Zhang 2006) little changed at the school level in the United States.

“Problem-solving is not only a goal of learning mathematics, but also a major means of doing so.”

(NCTM, 2000, p. 52)

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. For many teachers of mathematics this was the first time they had been expected to incorporate student collaboration and discourse with problem-solving. This practice requires teaching in profoundly different ways as schools moved from a teacher-directed to a more dialogic approach to teaching and learning. The challenge for teachers is to teach students not only to solve problems but also to learn about mathematics through problem-solving. While many students may develop procedural fluency, they often lack the deep conceptual understanding necessary to solve new problems or make connections between mathematical ideas.

“A problem-solving curriculum, however, requires a different role from the teacher. Rather than directing a lesson, the teacher needs to provide time for students to grapple with problems, search for strategies and solutions on their own, and learn to evaluate their own results. Although the teacher needs to be very much present, the primary focus in the class needs to be on the students’ thinking processes.”

(Burns, 2000, p. 29)

Learning to problem solve

To understand how students become problem solvers we need to look at the theories that underpin learning in mathematics. These include recognition of the developmental aspects of learning and the essential fact that students actively engage in learning mathematics through “doing, talking, reflecting, discussing, observing, investigating, listening, and reasoning” (Copley, 2000, p. 29). The concept of co-construction of learning is the basis for the theory. Moreover, we know that each student is on their unique path of development.

Beliefs underpinning effective teaching of mathematics

• Every student’s identity, language, and culture need to be respected and valued.
• Every student has the right to access effective mathematics education.
• Every student can become a successful learner of mathematics.

Children arrive at school with intuitive mathematical understandings. A teacher needs to connect with and build on those understandings through experiences that allow students to explore mathematics and to communicate their ideas in a meaningful dialogue with the teacher and their peers.

Learning takes place within social settings (Vygotsky, 1978). Students construct understandings through engagement with problems and interaction with others in these activities. Through these social interactions, students feel that they can take risks, try new strategies, and give and receive feedback. They learn cooperatively as they share a range of points of view or discuss ways of solving a problem. It is through talking about problems and discussing their ideas that children construct knowledge and acquire the language to make sense of experiences.

Students acquire their understanding of mathematics and develop problem-solving skills as a result of solving problems, rather than being taught something directly (Hiebert1997). The teacher’s role is to construct problems and present situations that provide a forum in which problem-solving can occur.

Why is problem-solving important?

Our students live in an information and technology-based society where they need to be able to think critically about complex issues, and “analyze and think logically about new situations, devise unspecified solution procedures, and communicate their solution clearly and convincingly to others” (Baroody, 1998). Mathematics education is important not only because of the “gatekeeping role that mathematics plays in students’ access to educational and economic opportunities,” but also because the problem-solving processes and the acquisition of problem-solving strategies equips students for life beyond school (Cobb, & Hodge, 2002).

The importance of problem-solving in learning mathematics comes from the belief that mathematics is primarily about reasoning, not memorization. Problem-solving allows students to develop understanding and explain the processes used to arrive at solutions, rather than remembering and applying a set of procedures. It is through problem-solving that students develop a deeper understanding of mathematical concepts, become more engaged, and appreciate the relevance and usefulness of mathematics (Wu and Zhang 2006). Problem-solving in mathematics supports the development of:

• The ability to think creatively, critically, and logically
• The ability to structure and organize
• The ability to process information
• Enjoyment of an intellectual challenge
• The skills to solve problems that help them to investigate and understand the world

Problem-solving should underlie all aspects of mathematics teaching in order to give students the experience of the power of mathematics in the world around them. This method allows students to see problem-solving as a vehicle to construct, evaluate, and refine their theories about mathematics and the theories of others.

Problems that are “Problematic”

The teacher’s expectations of the students are essential. Students only learn to handle complex problems by being exposed to them. Students need to have opportunities to work on complex tasks rather than a series of simple tasks devolved from a complex task. This is important for stimulating the students’ mathematical reasoning and building durable mathematical knowledge (Anthony and Walshaw, 2007). The challenge for teachers is ensuring the problems they set are designed to support mathematics learning and are appropriate and challenging for all students.  The problems need to be difficult enough to provide a challenge but not so difficult that students can’t succeed. Teachers who get this right create resilient problem solvers who know that with perseverance they can succeed. Problems need to be within the students’ “Zone of Proximal Development” (Vygotsky 1968). These types of complex problems will provide opportunities for discussion and learning.

Students will have opportunities to explain their ideas, respond to the ideas of others, and challenge their thinking. Those students who think math is all about the “correct” answer will need support and encouragement to take risks. Tolerance of difficulty is essential in a problem-solving disposition because being “stuck” is an inevitable stage in resolving just about any problem. Getting unstuck typically takes time and involves trying a variety of approaches. Students need to learn this experientially. Effective problems:

• Are accessible and extendable
• Allow individuals to make decisions
• Promote discussion and communication
• Encourage originality and invention
• Encourage “what if?” and “what if not?” questions
• Contain an element of surprise (Adapted from Ahmed, 1987)

“Students learn to problem solve in mathematics primarily through ‘doing, talking, reflecting, discussing, observing, investigating, listening, and reasoning.”

(Copley, 2000, p. 29)

“…as learners investigate together. It becomes a mini- society – a community of learners engaged in mathematical activity, discourse and reflection. Learners must be given the opportunity to act as mathematicians by allowing, supporting and challenging their ‘mathematizing’ of particular situations. The community provides an environment in which individual mathematical ideas can be expressed and tested against others’ ideas.…This enables learners to become clearer and more confident about what they know and understand.”

(Fosnot, 2005, p. 10)

Research shows that ‘classrooms where the orientation consistently defines task outcomes in terms of the answers rather than the thinking processes entailed in reaching the answers negatively affects the thinking processes and mathematical identities of learners’ (Anthony and Walshaw, 2007, page 122).

Effective teachers model good problem-solving habits for their students. Their questions are designed to help children use a variety of strategies and materials to solve problems. Students often want to begin without a plan in mind. Through appropriate questions, the teacher gives students some structure for beginning the problem without telling them exactly what to do. In 1945 Pólya published the following four principles of problem-solving to support teachers with helping their students.

• Understand and explore the problem
• Find a strategy
• Use the strategy to solve the problem
• Look back and reflect on the solution

Problem-solving is not linear but rather a complex, interactive process. Students move backward and forward between and across Pólya’s phases. The Common Core State Standards describe the process as follows:

“Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary”. (New York State Next Generation Mathematics Learning Standards 2017).

Pólya’s Principals of Problem-Solving

Students move forward and backward as they move through the problem-solving process.

The goal is for students to have a range of strategies they use to solve problems and understand that there may be more than one solution. It is important to realize that the process is just as important, if not more important, than arriving at a solution, for it is in the solution process that students uncover the mathematics. Arriving at an answer isn’t the end of the process. Reflecting on the strategies used to solve the problem provides additional learning experiences. Studying the approach used for one problem helps students become more comfortable with using that strategy in a variety of other situations.

When making sense of ideas, students need opportunities to work both independently and collaboratively. There will be times when students need to be able to work independently and other times when they will need to be able to work in small groups so that they can share ideas and learn with and from others.

Getting real

Effective teachers of mathematics create purposeful learning experiences for students through solving problems in relevant and meaningful contexts. While word problems are a way of putting mathematics into contexts, it doesn’t automatically make them real. The challenge for teachers is to provide students with problems that draw on their experience of reality, rather than asking them to suspend it. Realistic does not mean that problems necessarily involve real contexts, but rather they make students think in “real” ways.

Planning for talk

By planning for and promoting discourse, teachers can actively engage students in mathematical thinking. In discourse-rich mathematics classes, students explain and discuss the strategies and processes they use in solving mathematical problems, thereby connecting their everyday language with the specialized vocabulary of mathematics.

Students need to understand how to communicate mathematically, give sound mathematical explanations, and justify their solutions. Effective teachers encourage their students to communicate their ideas orally, in writing, and by using a variety of representations. Through listening to students, teachers can better understand what their students know and misconceptions they may have. It is the misconceptions that provide a window into the students’ learning process. Effective teachers view thinking as “the process of understanding,” they can use their students’ thinking as a resource for further learning. Such teachers are responsive both to their students and to the discipline of mathematics.

“Mathematics today requires not only computational skills but also the ability
to think and reason mathematically in order to solve the new problems and learn the new ideas that students will face in the future. Learning is enhanced in classrooms where students are required to evaluate their own ideas and those of others, are encouraged to make mathematical conjectures and test them, and are helped to develop their reasoning skills.”

(John Van De Walle)

“Equity. Excellence in mathematics education requires equity—high expectations and strong support for all students.”

How teachers organize classroom instruction is very much dependent on what they know and believe about mathematics and on what they understand about mathematics teaching and learning. Teachers need to recognize that problem-solving processes develop over time and are significantly improved by effective teaching practices. The teacher’s role begins with selecting rich problem-solving tasks that focus on the mathematics the teacher wants their students to explore. A problem-solving approach is not only a way for developing students’ thinking, but it also provides a context for learning mathematical concepts. Problem-solving allows students to transfer what they have already learned to unfamiliar situations. A problem-solving approach provides a way for students to actively construct their ideas about mathematics and to take responsibility for their learning. The challenge for mathematics teachers is to develop the students’ mathematical thinking process alongside the knowledge and to create opportunities to present even routine mathematics tasks in problem-solving contexts.

Given the efforts to date to include problem-solving as an integral component of the mathematics curriculum and the limited implementation in classrooms, it will take more than rhetoric to achieve this goal. While providing valuable professional learning, resources, and more time are essential steps, it is possible that problem-solving in mathematics will only become valued when high-stakes assessment reflects the importance of students’ solving of complex problems.

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What Is Translation in Math? Definition, Examples & How-to Guide

Whether you have just started learning about translation, are preparing for a test, or simply looking to refresh your knowledge, this middle-school-friendly guide is for you.

Read on to find easy-to-follow definitions, a simple how-to, and solved examples we prepared to help you master translations in math.

What Is Translation in Math?

Translation is a geometric transformation in which we move every point of a figure the same distance in the same direction, without changing its size, shape, or orientation.

When we translate a figure, we move it across the coordinate plane in a straight line, either horizontally, vertically, or diagonally.

Translation is one of the 4 types of geometric transformations which also include:

• Dilation : Enlarging or shrinking a figure evenly while keeping the shape and orientation the same
• Reflection : Flipping a figure over a line to create a mirror image
• Rotation : Turning a figure around a fixed point

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Translations in Everyday Life

Examples of translation are all around us.

Pushing a toy car in a straight line or sliding a chess piece across the board are great examples of translation because both figures preserve their size, shape, and orientation as you move them from one point to another.

Similarly, when you push a shopping cart in a straight line across a supermarket aisle, you are translating the cart's position.

  But , if a wheel of your shopping cart comes off, that wouldn't be called a translation anymore because the cart wouldn't keep its shape and parts together as it moves.

Pushing a toy car down a straight path is a great example of translation in real life.

How to Translate Figures on the Coordinate Plane

Translating figures means moving them across the coordinate plane in a straight line, whether horizontally, vertically, or diagonally.

When we translate a figure, every point on the figure moves by the same number of units or translation distance so that the figure keeps the same size and shape.

We often use the variable "k" to represent translation distance. If our scenario has more than one type of translation, we may also use other letters to represent different translation distances.

The original figure before the transformation is called the " pre-image ," and the resulting figure after the transformation is called the " image ."

So, to begin, we will create a pre-image of a rectangle ABCD on a coordinate plane.

Now, let’s translate this figure by 5 units to the right and 1 unit up, where each square on the grid represents one unit.

First, we need to locate each corner of the pre-image (A, B, C, and D).

For each corner, we plot a new point that’s 5 units to the right and 1 unit up.

Next, connect the four new points to make the image (A'B'C'D').

Lastly, determine the x and y coordinates of the pre-image and the image.

Those would be:

• A (-4,1), B (-4,3), C (-1,3), and D (-1,1)
• A' (1, 2), B' (1,4), C' (4,4), and D' (4,2)

If we compare the coordinates of the pre-image and the image, we see that all the x-coordinates increased by 5 units and all y-coordinates increased by 1 unit.

We can conclude that when translating a figure to the right , we move it along the x-axis towards larger positive values, so we add the translation distance (k ) to each x-coordinate (x + k).

Similarly,  when translating a figure up , we move it along the y-axis towards larger positive values, so we add the translation distance (j ) to each y-coordinate (y + j).

When translating a figure to the left , we move it along the x-axis towards smaller negative values, so we subtract the horizontal translation distance (k ) from each x-coordinate ( x – k).

When translating a figure down , we move it along the y-axis towards smaller negative values, so we subtract the vertical translation distance (j ) from each y-coordinate (y – j). 

Makes sense?

Let’s recap:

• Moving right : Add the translation distance to each x-coordinate:
• Moving up: Add the translation distance to each y-coordinate:
• Moving left : Subtract the translation distance from each x-coordinate:
• Moving down:  Subtract translation distance from each y-coordinate:

Let’s look at this example. The graph shows the pre-image of a rectangle (ABCD).

The rectangle should be translated 6 units to the right and 1 unit up.

Firstly, we locate the corners of the rectangle (A, B, C, and D).

Next, we plot the new points by moving each corner 6 units to the right and 1 unit up.

Now, we simply connect the points to create the image.

Since we moved the rectangle 6 units to the right along the x-axis and 1 unit up along the y-axis, we can find the image coordinates using this formula:

(x, y) —> (x + 6, y + 1)

The coordinates of the pre-image are:

A (-4,1), B (-4,4), C (-2,4), D (-2,1)

Let’s calculate the coordinates of the image:

• A' (–4 + 6, 1 + 1) = (2, 2)
• B' (–4 + 6, 4 + 1) = (2, 5)
• C' (–2 + 6, 4 + 1) = (4,5)
• D' (–2 + 6, 1 +1) = (4,2) 

Not so scary, right?

Solved Examples of Translation on a Coordinate Plane

Now, let's apply what we've learned with a few practical examples.

The pre-image triangle (ABC) on the coordinate plane needs to be translated 4 units to the left and 1 unit downward.

As we’ve done before, we first locate each corner of the pre-image.

For each corner, we plot a new point that’s 4 units to the left and 1 unit downward.

We then connect the new points and make the image (A'B'C').

A (1,2), B (1,5), C (3,4)

Since we’re moving the figure 4 units left and 1 unit down, subtract 4 units from each x-coordinate and 1 from each y-coordinate to find the coordinates of the image.

(x, y) —> (x – 4, y – 1)

Let’s do the math:

• A' = (1 – 4, 2 – 1) = (–3, 1)
• B' = (1 – 4, 5 – 1) = (–3, 4)
• C' = (3 – 4, 4 – 1) = (–1, 3) 

The coordinates for the image are: A' (-3, 1), B' (-3, 4), C' (-1, 3)

Let’s translate this rhombus (ABCD) and move it 1 unit left and 4 units down.

We locate each corner of the rhombus.

Then, we plot the new points by moving each corner 1 unit left and 4 down.

We connect the new points and create the image.

The rhombus’ coordinates are:

A (-2, 3), B (-3, 2), C (-2, 1), D (-1, 2)

To get the coordinates of the image, we subtract 1 from each x-coordinate and 4 from each y-coordinate.

(x, y) —> (x – 1, y – 4)

Let’s calculate:

• A' = (–2 – 1, 3 – 4) = (–3, –1)
• B' = (–3 – 1, 2 – 4) = (–4, –2)
• C' = (–2 – 1, 1 – 4) = (–3, –3)
• D' = (–1 – 1, 2 – 4) = (–2, –2) 

The coordinates for the image will be: A' (-3, -1), B' (-4, -2), C' (-3, -3), D' (-2, -2)

Our next task is to translate this rectangle (ABCD) in the coordinate plane below by moving it 6 units to the right and 2 units down.

We follow the same steps and locate the corners of the pre-image (ABCD).

We shift each point 6 units to the right and 2 units down. 

We connect the new points to produce the image.

The pre-image coordinates are:

A (-5, 4), B (-2, 4), C (-2, -2), D (-5, -2)

To get the coordinates for the image, we add 6 to each x-coordinate and subtract 2 from each y-coordinate.

(x, y) —> (x + 6, y – 2)

Let’s calculate the new coordinates:

• A' = (–5 + 6, 4 – 2) = (1, 2)
• B' = (–2 + 6, 4 – 2) = (4, 2)
• C' = (–2 + 6, –2 – 2) = (4, –4)
• D' = (–5 + 6, –2 – 2) = (1, –4)

The coordinates for the image will be:

A' (1,2), B' (4,2) C' (4, -4), D' (1, -4)

Frequently Asked Questions About Translation

Here are the most common questions Mathnasium’s geometry tutors get about translation in math.

1) What are the key characteristics of a translation?

In translation, our figure preserves the size, shape, and orientation as we move along the coordinate plane.

2)   Is there a limit to the distance a figure can be translated?

There is no limit to how far a figure can be translated in mathematics.

However, in practice, how far you can translate a figure may be limited by how big your coordinate plane (or graph paper) is.

3)   Is translation and transformation the same thing in math?

Yes and no. Translation is one of 4 geometric transformations in math. The other 3 types of geometric transformations are reflection, rotation, and dilation. 

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