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5 Teaching Mathematics Through Problem Solving

Janet Stramel

Problem Solving

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.

Needlepoint of cats

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

Back of a needlepoint

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

Teacher teaching a math lesson

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?

problem solving methods math

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

Light bulb

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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10 Strategies for Problem Solving in Math

Created: December 25, 2023

Last updated: January 6, 2024

strategies for problem solving in math

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

What Are Problem Solving Strategies in Math?

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

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

Strategies for Problem-solving in Math

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

Understand the Problem

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

Math for Kids

Guess and Check

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

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

Work It Out

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

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

Work Backwards

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

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

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

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

Find a Pattern

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

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

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

Draw a Picture or Diagram

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

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

Trial and Error Method

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

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

Review Answers with Peers

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

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

Check out the Printable Math Worksheets for Your Kids!

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

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

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Problem Solving in Mathematics

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

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

Use Established Procedures

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

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

Look for Clue Words

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

Common clue words for addition  problems:

Common clue words for  subtraction  problems:

  • How much more

Common clue words for multiplication problems:

Common clue words for division problems:

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

Read the Problem Carefully

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

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

Develop a Plan and Review Your Work

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

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

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

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

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

Tips and Hints

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

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

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

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Study Smarter

17 maths problem solving strategies to boost your learning.

Worded problems getting the best of you? With this list of maths problem-solving strategies , you'll overcome any maths hurdle that comes your way.

student learning data and multiplication worksheets

Friday, 3rd June 2022

  • What are strategies?

Understand the problem

Devise a plan, carry out the plan, look back and reflect, practise makes progress.

Problem-solving is a critical life skill that everyone needs. Whether you're dealing with everyday issues or complex challenges, being able to solve problems effectively can make a big difference to your quality of life.

While there is no one 'right' way to solve a problem, having a toolkit of different techniques that you can draw upon will give you the best chance of success. In this article, we'll explore 17 different math problem-solving strategies you can start using immediately to deepen your learning and improve your skills.

What are maths problem-solving strategies?

Before we get into the strategies themselves, let's take a step back and answer the question: what are these strategies? In simple terms, these are methods we use to solve mathematical problems—essential for anyone learning how to study maths . These can be anything from asking open-ended questions to more complex concepts like the use of algebraic equations.

The beauty of these techniques is they go beyond strictly mathematical application. It's more about understanding a given problem, thinking critically about it and using a variety of methods to find a solution.

Polya's 4-step process for solving problems

We're going to use Polya's 4-step model as the framework for our discussion of problem-solving activities . This was developed by Hungarian mathematician George Polya and outlined in his 1945 book How to Solve It. The steps are as follows:

We'll go into more detail on each of these steps as well as take a look at some specific problem-solving strategies that can be used at each stage.

This may seem like an obvious one, but it's crucial that you take the time to understand what the problem is asking before trying to solve it. Especially with a math word problem , in which the question is often disguised in language, it's easy for children to misinterpret what's being asked.

confident student learning subtraction and counting in her head

Here are some questions you can ask to help you understand the problem:

Do I understand all the words used in the problem?

What am I asked to find or show?

Can I restate the problem in my own words?

Can I think of a picture or diagram that might help me understand the problem?

Is there enough information to enable me to find a solution?

Is there anything I need to find out first in order to find the answer?

What information is extra or irrelevant?

Once you've gone through these questions, you should have a good understanding of what the problem is asking. Now let's take a look at some specific strategies that can be used at this stage.

1. Read the problem aloud

This is a great strategy for younger students who are still learning to read. By reading the problem aloud, they can help to clarify any confusion and better understand what's being asked. Teaching older students to read aloud slowly is also beneficial as it encourages them to internalise each word carefully.

2. Summarise the information

Using dot points or a short sentence, list out all the information given in the problem. You can even underline the keywords to focus on the important information. This will help to organise your thoughts and make it easier to see what's given, what's missing, what's relevant and what isn't.

3. Create a picture or diagram

This is a no-brainer for visual learners. By drawing a picture, let's say with division problems, you can better understand what's being asked and identify any information that's missing. It could be a simple sketch or a more detailed picture, depending on the problem.

4. Act it out

Visualising a scenario can also be helpful. It can enable students to see the problem in a different way and develop a more intuitive understanding of it. This is especially useful for math word problems that are set in a particular context. For example, if a problem is about two friends sharing candy, kids can act out the problem with real candy to help them understand what's happening.

5. Use keyword analysis

What does this word tell me? Which operations do I need to use? Keyword analysis involves asking questions about the words in a problem in order to work out what needs to be done. There are certain key words that can hint at what operation you need to use.

How many more?

How many left?

Equal parts

Once you understand the problem, it's time to start thinking about how you're going to solve it. This is where having a plan is vital. By taking the time to think about your approach, you can save yourself a lot of time and frustration later on.

young student learning to recognise multiplication and number patterns

There are many methods that can be used to figure out a pathway forward, but the key is choosing an appropriate one that will work for the specific problem you're trying to solve. Not all students understand what it means to plan a problem so we've outlined some popular problem-solving techniques during this stage.

6. Look for a pattern

Sometimes, the best way to solve a problem is to look for a pattern. This could be a number, a shape pattern or even just a general trend that you can see in the information given. Once you've found it, you can use it to help you solve the problem.

7. Guess and check

While not the most efficient method, guess and check can be helpful when you're struggling to think of an answer or when you're dealing with multiple possible solutions. To do this, you simply make a guess at the answer and then check to see if it works. If it doesn't, you make another systematic guess and keep going until you find a solution that works.

8. Working backwards

Regressive reasoning, or working backwards, involves starting with a potential answer and working your way back to figure out how you would get there. This is often used when trying to solve problems that have multiple steps. By starting with the end in mind, you can work out what each previous step would need to be in order to arrive at the answer.

9. Use a formula

There will be some problems where a specific formula needs to be used in order to solve it. Let's say we're calculating the cost of flooring panels in a rectangular room (6m x 9m) and we know that the panels cost $15 per sq. metre.

installation of floor for area maths problem

There is no mention of the word 'area', and yet that is exactly what we need to calculate. The problem requires us to use the formula for the area of a rectangle (A = l x w) in order to find the total cost of the flooring panels.

10. Eliminate the possibilities

When there are a lot of possibilities, one approach could be to start by eliminating the answers that don't work. This can be done by using a process of elimination or by plugging in different values to see what works and what doesn't.

11. Use direct reasoning

Direct reasoning, also known as top-down or forward reasoning, involves starting with what you know and then using that information to try and solve the problem . This is often used when there is a lot of information given in the problem.

By breaking the problem down into smaller chunks, you can start to see how the different pieces fit together and eventually work out a solution.

12. Solve a simpler problem

One of the most effective methods for solving a difficult problem is to start by solving a simpler version of it. For example, in order to solve a 4-step linear equation with variables on both sides, you could start by solving a 2-step one. Or if you're struggling with the addition of algebraic fractions, go back to solving regular fraction addition first.

Once you've mastered the easier problem, you can then apply the same knowledge to the challenging one and see if it works.

13. Solve an equation

Another common problem-solving technique is setting up and solving an equation. For instance, let's say we need to find a number. We know that after it was doubled, subtracted from 32, and then divided by 4, it gave us an answer of 6. One method could be to assign this number a variable, set up an equation, and solve the equation by 'backtracking and balancing the equation'.

Now that you have a plan, it's time to implement it. This is where you'll put your problem-solving skills to the test and see if your solution actually works. There are a few things to keep in mind as you execute your plan:

14. Be systematic

When trying different methods or strategies, it's important to be systematic in your approach. This means trying one problem-solving strategy at a time and not moving on until you've exhausted all possibilities with that particular approach.

student practising word problems at home

15. Check your work

Once you think you've found a solution, it's important to check your work to make sure that it actually works. This could involve plugging in different values or doing a test run to see if your solution works in all cases.

16. Be flexible

If your initial plan isn't working, don't be afraid to change it. There is no one 'right' way to solve a problem, so feel free to try different things, seek help from different resources and continue until you find a more efficient strategy or one that works.

17. Don't give up

It's important to persevere when trying to solve a difficult problem. Just because you can't see a solution right away doesn't mean that there isn't one. If you get stuck, take a break and come back to the problem later with fresh eyes. You might be surprised at what you're able to see after taking some time away from it.

Once you've solved the problem, take a step back and reflect on the process that you went through. Most middle school students forget this fundamental step. This will help you to understand what worked well and what could be improved upon next time.

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Whether you do this after a math test or after an individual problem, here are some questions to ask yourself:

What was the most challenging part of the problem?

Was one method more effective than another?

Would you do something differently next time?

What have you learned from this experience?

By taking the time to reflect on your process you'll be able to improve upon it in future and become an even better problem solver. Make sure you write down any insights so that you can refer back to them later.

There is never only one way to solve math problems. But the best way to become a better problem solver is to practise, practise, practise! The more you do it, the better you'll become at identifying different strategies, and the more confident you'll feel when faced with a challenging problem.

The list we've covered is by no means exhaustive, but it's a good starting point for you to begin your journey. When you get stuck, remember to keep an open mind. Experiment with different approaches. Different word problems. Be prepared to go back and try something new. And most importantly, don't forget to have fun!

The essence and beauty of mathematics lies in its freedom. So while these strategies provide nice frameworks, the best work is done by those who are comfortable with exploration outside the rules, and of course, failure! So go forth, make mistakes and learn from them. After all, that's how we improve our problem-solving skills and ability.

Lastly, don't be afraid to ask for help. If you're struggling to solve math word problems, there's no shame in seeking assistance from a certified Melbourne maths tutor . In every lesson at Math Minds, our expert teachers encourage students to think creatively, confidently and courageously.

If you're looking for a mentor who can guide you through these methods, introduce you to other problem-solving activities and help you to understand Mathematics in a deeper way - get in touch with our team today. Sign up for your free online maths assessment and discover a world of new possibilities.

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How to Solve Math Problems Faster: 15 Techniques to Show Students

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Written by Marcus Guido

  • Teaching Strategies

“Test time. No calculators.”

You’ll intimidate many students by saying this, but teaching techniques to solve math problems with ease and speed can make it less daunting.

This can also  make math more rewarding . Instead of relying on calculators, students learn strategies that can improve their concentration and estimation skills while building number sense. And, while there are educators who  oppose math “tricks”  for valid reasons, proponents point to benefits such as increased confidence to handle difficult problems.

Here are 15 techniques to show students,  helping them solve math problems faster:

Addition and Subtraction

1. two-step addition.

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Many students struggle when learning to add integers of three digits or higher together, but changing the process’s steps can make it easier.

The first step is to  add what’s easy.  The second step is to  add the rest.

Let’s say students must find the sum of 393 and 89. They should quickly see that adding 7 onto 393 will equal 400 — an easier number to work with. To balance the equation, they can then subtract 7 from 89.

Broken down, the process is:

  • (393 + 7) + (89 – 7)

With this fast technique, big numbers won’t look as scary now.

2. Two-Step Subtraction

There’s a similar method for subtraction.

Remove what’s easy. Then remove what’s left.

Suppose students must find the difference of 567 and 153. Most will feel that 500 is a simpler number than 567. So, they just have to take away 67 from the minuend — 567 — and the subtrahend — 153 — before solving the equation.

Here’s the process:

  • (567 – 67) – (153 – 67)

Instead of two complex numbers, students will only have to tackle one.

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3. Subtracting from 1,000

You can  give students confidence  to handle four-digit integers with this fast technique.

To subtract a number from 1,000, subtract that number’s first two digits from 9. Then, subtract the final digit from 10.

Let’s say students must solve 1,000 – 438.  Here are the steps:

This also applies to 10,000, 100,000 and other integers that follow this pattern.

Multiplication and Division

4. doubling and halving.

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When students have to multiply two integers, they can speed up the process when one is an even number. They just need to  halve the even number and double the other number.

Students can stop the process when they can no longer halve the even integer, or when the equation becomes manageable.

Using 33 x 48 as an example,  here’s the process:

The only prerequisite is understanding the 2 times table.

5. Multiplying by Powers of 2

This tactic is a speedy variation of doubling and halving.

It simplifies multiplication if a number in the equation is a power of 2, meaning it works for 2, 4, 8, 16 and so on.

Here’s what to do:  For each power of 2 that makes up that number, double the other number.

For example, 9 x 16 is the same thing as 9 x (2 x 2 x 2 x 2) or 9 x 24. Students can therefore double 9 four times to reach the answer:

Unlike doubling and halving, this technique demands an understanding of exponents along with a strong command of the 2 times table.

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6. Multiplying by 9

For most students, multiplying by 9 — or 99, 999 and any number that follows this pattern — is difficult compared with multiplying by a power of 10.

But there’s an easy tactic to solve this issue, and  it has two parts.

First, students round up the 9  to 10. Second, after solving the new equation, they subtract the number they just multiplied by 10 from the answer.

For example, 67 x 9 will lead to the same answer as 67 x 10 – 67. Following the order of operations will give a result of 603. Similarly, 67 x 99 is the same as 67 x 100 – 67.

Despite more steps, altering the equation this way is usually faster.

7. Multiplying by 11

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There’s an easier way for multiplying two-digit integers by 11.

Let’s say students must find the product of 11 x 34.

The idea is to put a space between the digits, making it 3_4. Then, add the two digits together and put the sum in the space.

The answer is 374.

What happens if the sum is two digits? Students would put the second digit in the space and add 1 to the digit to the left of the space.  For example:

It’s multiplication without having to multiply.

8. Multiplying Even Numbers by 5

This technique only requires basic division skills.

There are two steps,  and 5 x 6 serves as an example. First, divide the number being multiplied by 5 — which is 6 — in half. Second, add 0 to the right of number.

The result is 30, which is the correct answer.

It’s an ideal, easy technique for students mastering the 5 times table.

9. Multiplying Odd Numbers by 5

This is another time-saving tactic that works well when teaching students the 5 times table.

This one has three steps,  which 5 x 7 exemplifies.

First, subtract 1 from the number being multiplied by 5, making it an even number. Second, cut that number in half — from 6 to 3 in this instance. Third, add 5 to the right of the number.

The answer is 35.

Who needs a calculator?

10. Squaring a Two-Digit Number that Ends with 1

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Squaring a high two-digit number can be tedious, but there’s a shortcut if 1 is the second digit.

There are four steps to this shortcut,  which 812 exemplifies:

  • Subtract 1 from the integer: 81 – 1 = 80
  • Square the integer, which is now an easier number: 80 x 80 = 6,400
  • Add the integer with the resulting square twice: 6,400 + 80 + 80 = 6,560
  • Add 1: 6,560 + 1 = 6,561

This work-around eliminates the difficulty surrounding the second digit, allowing students to work with multiples of 10.

11. Squaring a Two-Digit Numbers that Ends with 5

Squaring numbers ending in 5 is easier, as there are  only two parts of the process.

First, students will always make 25 the product’s last digits.

Second, to determine the product’s first digits, students must multiply the number’s first digit — 9, for example — by the integer that’s one higher — 10, in this case.

So, students would solve 952 by designating 25 as the last two digits. They would then multiply 9 x 10 to receive 90. Putting these numbers together, the  result is 9,025.

Just like that, a hard problem becomes easy multiplication for many students.

12. Calculating Percentages

Cross-multiplication is an  important skill  to develop, but there’s an easier way to calculate percentages.

For example, if students want to know what 65% of 175 is, they can multiply the numbers together and move the decimal place two digits to the left.

The result is 113.75, which is indeed the correct answer.

This shortcut is a useful timesaver on tests and quizzes.

13. Balancing Averages

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To determine the average among a set of numbers, students can balance them instead of using a complex formula.

Suppose a student wants to volunteer for an average of 10 hours a week over a period of four weeks. In the first three weeks, the student worked for 10, 12 and 14 hours.

To determine the number of hours required in the fourth week, the student must  add how much he or she surpassed or missed the target average  in the other weeks:

  • 14 hours – 10 hours = 4 hours
  • 12 – 10 = 2
  • 10 – 10 = 0
  • 4 hours + 2 hours + 0 hours = 6 hours

To learn the number of hours for the final week, the student must  subtract the sum from the target average:

  • 10 hours – 6 hours = 4 hours

With practice, this method may not even require pencil and paper. That’s how easy it is. 

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Word Problems

14. identifying buzzwords.

Students who struggle to translate  word problems  into equations will benefit from learning how to spot buzzwords — phrases that indicate specific actions.

This isn’t a trick. It’s a tactic.

Teach students to look for these buzzwords,  and what skill they align with in most contexts:

Be sure to include buzzwords that typically appear in their textbooks (or other classroom  math books ), as well as ones you use on tests and assignments.

As a result, they should have an  easier time processing word problems .

15. Creating Sub-Questions

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For complex word problems, show students how to dissect the question by answering three specific sub-questions.

Each student should ask him or herself:

  • What am I looking for?  — Students should read the question over and over, looking for buzzwords and identifying important details.
  • What information do I need?  — Students should determine which facts, figures and variables they need to solve the question. For example, if they determine the question is rooted in subtraction, they need the minuend and subtrahend.
  • What information do I have?  — Students should be able to create the core equation using the information in the word problem, after determining which details are important.

These sub-questions help students avoid overload.

Instead of writing and analyzing each detail of the question, they’ll be able to identify key information. If you identify students who are struggling with these, you can use  peer learning  as needed.  

For more fresh approaches to teaching math in your classroom, consider treating your students to a range of  fun math activities .

Final Thoughts About these Ways to Solve Math Problems Faster

Showing these 15 techniques to students can give them the  confidence to tackle tough questions .

They’re also  mental math  exercises, helping them build skills related to focus, logic and critical thinking.

A rewarding class equals an  engaging class . That’s an easy equation to remember.

> Create or log into your teacher account on Prodigy  — a free, adaptive math game that adjusts content to accommodate player trouble spots and learning speeds. Aligned to US and Canadian curricula, it’s loved by more than 500,000 teachers and 15 million students.

Multiple Methods

  • Posted April 13, 2015
  • By Mary Tamer

UK Math

Do you remember watching your math teacher solve a problem on the blackboard and then diligently trying to copy her technique to solve the other problems on your worksheet? That’s the way many of us learned math. The problem is, we absorbed some counterproductive messages in the process. As it turns out, there isn’t always one best way to solve a given problem.

In his research, Associate Professor Jon Star is pushing hard to craft some new messages, by showing students how important it is to use multiple strategies when solving math problems.

“Math problems can be approached in many different ways,” says Star, an educational psychologist and former math teacher. “When a teacher insists that there is only one way, or only one best way, to solve a problem, students are missing out. There is great value in allowing them to explore and contrast many different ways to solve problems.”

Star and colleague Bethany Rittle-Johnson of Vanderbilt University have conducted a number of studies over the past decade that demonstrate the benefits of comparing a variety of problem-solving approaches for learning math, especially algebra. And their work has paid off: the US Department of Education’s Institute of Education Sciences echoed their findings in two recent publications by the What Works Clearinghouse : a new problem-solving guide for grades 4-8 and a new algebra practice guide for middle and high school students.

Building on this work, Star, Rittle-Johnson, and colleague  Kristie Newton of Temple University developed a set of curriculum materials  designed to be used in middle and high school algebra classrooms. The goal is to expose students to multiple problem-solving strategies and to build deep and flexible mathematical knowledge.

“In math class, you should have opportunities to talk about different approaches, and comparison helps us to think not only about what works in mathematics, but also about how and why things work,” says Star. “Our materials are designed to be used by algebra teachers to supplement their regular curriculum, to provide a stronger focus on the learning of multiple strategies.“ The curriculum materials were developed with middle and high schoolers in mind, but there are some applications for elementary schoolers as well. Educators can access the curriculum online at no cost.

In several recent studies, Star and his colleagues have studied the impact of teachers’ use of these materials on their students’ learning. He calls the results quite promising.

“Our research suggests that using our curriculum materials was not especially difficult for teachers, and that students enjoyed and benefited from the emphasis on multiple strategies,” says Star. “Many teachers already include multiple strategies for certain topics that they teach; our materials are designed to expand this focus across all topics in algebra.”

Additional Resources

  • Read " Developing Flexibility in Math Problem Solving ."
  • Visit the  Contrasting Cases website .
  • Use the  Contrasting Cases curriculum .
  • Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students

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6 Tips for Teaching Math Problem-Solving Skills

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

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

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

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

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

1. Link problem-solving to reading

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

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

2. Avoid boxing students into choosing a specific operation

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

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

3. Revisit ‘representation’

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

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

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

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

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

4. Give time to process

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

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

5. Ask questions that let Students do the thinking

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

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

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

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

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

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32 Mathematical Ideas: Problem-Solving Techniques

Jenna Lehmann

Solving Problems by Inductive Reasoning

Before we can talk about how to use inductive reasoning, we need to define it and distinguish it from deductive reasoning.

Inductive reasoning is when one makes generalizations based on repeated observations of specific examples. For instance, if I have only ever had mean math teachers, I might draw the conclusion that all math teachers are mean. Because I witnessed multiple instances of mean math teachers and only mean math teachers, I’ve drawn this conclusion. That being said, one of the downfalls of inductive reasoning is that it only takes meeting one nice math teacher for my original conclusion to be proven false. This is called a counterexample . Since inductive reasoning can so easily be proven false with one counterexample, we don’t say that a conclusion drawn from inductive reasoning is the absolute truth unless we can also prove it using deductive reasoning. With inductive reasoning, we can never be sure that what is true in a specific case will be true in general, but it is a way of making an educated guess.

Deductive reasoning depends on a hypothesis that is considered to be true. In other words, if X = Y and Y = Z, then we can deduce that X = Z. An example of this might be that if we know for a fact that all dogs are good, and Lucky is a dog, then we can deduce that Lucky is good.

Strategies for Problem Solving

No matter what tool you use to solve a problem, there is a method for going about solving the problem.

  • Understand the Problem: You may need to read a problem several times before you can conceptualize it. Don’t become frustrated, and take a walk if you need to. It might take some time to click.
  • Devise a Plan: There may be more than one way to solve the problem. Find the way which is most comfortable for you or the most practical.
  • Carry Out the Plan: Try it out. You may need to adjust your plan if you run into roadblocks or dead ends.
  • Look Back and Check: Make sure your answer gives sense given the context.

There are several different ways one might go about solving a problem. Here are a few:

  • Tables and Charts: Sometimes you’ll be working with a lot of data or computing a problem with a lot of different steps. It may be best to keep it organized in a table or chart so you can refer back to previous work later.
  • Working Backward: Sometimes you’ll be given a word problem where they describe a series of algebraic functions that took place and then what the end result is. Sometimes you’ll have to work backward chronologically.
  • Using Trial and Error: Sometimes you’ll know what mathematical function you need to use but not what number to start with. You may need to use trial and error to get the exact right number.
  • Guessing and Checking: Sometimes it will appear that a math problem will have more than one correct answer. Be sure to go back and check your work to determine if some of the answers don’t actually work out.
  • Considering a Similar, Simpler Problem: Sometimes you can use the strategy you think you would like to use on a simpler, hypothetical problem first to see if you can find a pattern and apply it to the harder problem.
  • Drawing a Sketch: Sometimes—especially with geometrical problems—it’s more helpful to draw a sketch of what is being asked of you.
  • Using Common Sense: Be sure to read questions very carefully. Sometimes it will seem like the answer to a question is either too obvious or impossible. There is usually a phrasing of the problem which would lead you to believe that the rules are one way when really it’s describing something else. Pay attention to literal language.

This chapter was originally posted to the Math Support Center blog at the University of Baltimore on November 6, 2019.

Math and Statistics Guides from UB's Math & Statistics Center Copyright © by Jenna Lehmann is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Teaching Problem Solving in Math

  • Freebies , Math , Planning

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Every year my students can be fantastic at math…until they start to see math with words. For some reason, once math gets translated into reading, even my best readers start to panic. There is just something about word problems, or problem-solving, that causes children to think they don’t know how to complete them.

Every year in math, I start off by teaching my students problem-solving skills and strategies. Every year they moan and groan that they know them. Every year – paragraph one above. It was a vicious cycle. I needed something new.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

I put together a problem-solving unit that would focus a bit more on strategies and steps in hopes that that would create problem-solving stars.

The Problem Solving Strategies

First, I wanted to make sure my students all learned the different strategies to solve problems, such as guess-and-check, using visuals (draw a picture, act it out, and modeling it), working backward, and organizational methods (tables, charts, and lists). In the past, I had used worksheet pages that would introduce one and provide the students with plenty of problems practicing that one strategy. I did like that because students could focus more on practicing the strategy itself, but I also wanted students to know when to use it, too, so I made sure they had both to practice.

I provided students with plenty of practice of the strategies, such as in this guess-and-check game.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

There’s also this visuals strategy wheel practice.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

I also provided them with paper dolls and a variety of clothing to create an organized list to determine just how many outfits their “friend” would have.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Then, as I said above, we practiced in a variety of ways to make sure we knew exactly when to use them. I really wanted to make sure they had this down!

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Anyway, after I knew they had down the various strategies and when to use them, then we went into the actual problem-solving steps.

The Problem Solving Steps

I wanted students to understand that when they see a story problem, it isn’t scary. Really, it’s just the equation written out in words in a real-life situation. Then, I provided them with the “keys to success.”

S tep 1 – Understand the Problem.   To help students understand the problem, I provided them with sample problems, and together we did five important things:

  • read the problem carefully
  • restated the problem in our own words
  • crossed out unimportant information
  • circled any important information
  • stated the goal or question to be solved

We did this over and over with example problems.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Once I felt the students had it down, we practiced it in a game of problem-solving relay. Students raced one another to see how quickly they could get down to the nitty-gritty of the word problems. We weren’t solving the problems – yet.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Then, we were on to Step 2 – Make a Plan . We talked about how this was where we were going to choose which strategy we were going to use. We also discussed how this was where we were going to figure out what operation to use. I taught the students Sheila Melton’s operation concept map.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

We talked about how if you know the total and know if it is equal or not, that will determine what operation you are doing. So, we took an example problem, such as:

Sheldon wants to make a cupcake for each of his 28 classmates. He can make 7 cupcakes with one box of cupcake mix. How many boxes will he need to buy?

We started off by asking ourselves, “Do we know the total?” We know there are a total of 28 classmates. So, yes, we are separating. Then, we ask, “Is it equal?” Yes, he wants to make a cupcake for EACH of his classmates. So, we are dividing: 28 divided by 7 = 4. He will need to buy 4 boxes. (I actually went ahead and solved it here – which is the next step, too.)

Step 3 – Solving the problem . We talked about how solving the problem involves the following:

  • taking our time
  • working the problem out
  • showing all our work
  • estimating the answer
  • using thinking strategies

We talked specifically about thinking strategies. Just like in reading, there are thinking strategies in math. I wanted students to be aware that sometimes when we are working on a problem, a particular strategy may not be working, and we may need to switch strategies. We also discussed that sometimes we may need to rethink the problem, to think of related content, or to even start over. We discussed these thinking strategies:

  • switch strategies or try a different one
  • rethink the problem
  • think of related content
  • decide if you need to make changes
  • check your work
  • but most important…don’t give up!

To make sure they were getting in practice utilizing these thinking strategies, I gave each group chart paper with a letter from a fellow “student” (not a real student), and they had to give advice on how to help them solve their problem using the thinking strategies above.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Finally, Step 4 – Check It.   This is the step that students often miss. I wanted to emphasize just how important it is! I went over it with them, discussing that when they check their problems, they should always look for these things:

  • compare your answer to your estimate
  • check for reasonableness
  • check your calculations
  • add the units
  • restate the question in the answer
  • explain how you solved the problem

Then, I gave students practice cards. I provided them with example cards of “students” who had completed their assignments already, and I wanted them to be the teacher. They needed to check the work and make sure it was completed correctly. If it wasn’t, then they needed to tell what they missed and correct it.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

To demonstrate their understanding of the entire unit, we completed an adorable lap book (my first time ever putting together one or even creating one – I was surprised how well it turned out, actually). It was a great way to put everything we discussed in there.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Once we were all done, students were officially Problem Solving S.T.A.R.S. I just reminded students frequently of this acronym.

Stop – Don’t rush with any solution; just take your time and look everything over.

Think – Take your time to think about the problem and solution.

Act  – Act on a strategy and try it out.

Review – Look it over and see if you got all the parts.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Wow, you are a true trooper sticking it out in this lengthy post! To sum up the majority of what I have written here, I have some problem-solving bookmarks FREE to help you remember and to help your students!

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

You can grab these problem-solving bookmarks for FREE by clicking here .

You can do any of these ideas without having to purchase anything. However, if you are looking to save some time and energy, then they are all found in my Math Workshop Problem Solving Unit . The unit is for grade three, but it  may work for other grade levels. The practice problems are all for the early third-grade level.

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Problem Solving Methods for Mathematical Modelling

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Providing a method for problem solving can support students working on modelling tasks. A few candidate methods are presented here. In a qualitative study, one of these problem solving methods was introduced to students in grades 4 and 6 (Germany), to be used in their work on modelling tasks. The students were observed as they worked and were subsequently interviewed. The results reveal differences between grades, and widely varying problem solving processes. The differences in written final solutions are considerably less pronounced.

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Greefrath, G. (2015). Problem Solving Methods for Mathematical Modelling. In: Stillman, G., Blum, W., Salett Biembengut, M. (eds) Mathematical Modelling in Education Research and Practice. International Perspectives on the Teaching and Learning of Mathematical Modelling. Springer, Cham. https://doi.org/10.1007/978-3-319-18272-8_13

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3.03: Bisection Methods for Solving a Nonlinear Equation

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Lesson 1: Background of Bisection Method

Learning objectives.

After successful completion of this lesson, you should be able to

1) articulate the background to the bisection method

Introduction

One of the first numerical methods developed to find the root of a nonlinear equation \(f(x) = 0\) was the bisection method (also called the binary-search method). The procedure is based on the following theorem.

An equation \(f(x) = 0\) , where \(f(x)\) is a real continuous function, has at least one root between \(x_{l}\) and \(x_{u}\) if \(f(x_{l})f(x_{u}) < 0\) (See Figure \(\PageIndex{1.1}\)).

Note that if \(f(x_{l})f(x_{u}) > 0\) , there may or may not be any root between \(x_{l}\) and \(x_{u}\) (Figures \(\PageIndex{1.2}\) and \(\PageIndex{1.3}\)). If \(f(x_{l})f(x_{u}) < 0\) , then there may be more than one root between \(x_{l}\) and \(x_{u}\) (Figure \(\PageIndex{1.4}\)). So the theorem only guarantees one root between \(x_{l}\) and \(x_{u}\) .

Since the method is based on finding the root between two points, the technique falls under bracketing methods.

Since the root is bracketed between two points, \(x_{l}\) and \(x_{u}\) , one can find the midpoint \(x_{m}\) between \(x_{l}\) and \(x_{u}\) . We hence get two new intervals \(x_{l}\) and \(x_{m}\) , and \(x_{m}\) and \(x_{u}\) .

A real, continuous function f(x) changes sign between points x_l and x_u, meaning at least one root exists between those points.

Is the root now between \(x_{l}\) and \(x_{m}\) or between \(x_{m}\) and \(x_{u}\) ? Well, one can find the sign of \(f(x_{l})f(x_{m})\) , and if \(f(x_{l})f(x_{m}) < 0\) then the new bracket is between \(x_{l}\) and \(x_{m}\); otherwise, it is between \(x_{m}\) and \(x_{u}\) . So, you can see that you are halving the interval. As one repeats this process, the width of the interval \(\left\lbrack x_{l},x_{u} \right\rbrack\) becomes smaller and smaller, and you can zero on to the root of the equation \(f(x) = 0\) .

Audiovisual Lectures

Title: Bisection Method - Introduction

Summary: Learn what the bisection method of solving nonlinear equations is based on.

Lesson 2: Bisection Method Algorithm

1) write the algorithm for the bisection method of solving a nonlinear equation.

What is the bisection method, and what is it based on?

An equation \(f(x) = 0\) , where \(f(x)\) is a real continuous function, has at least one root between \(x_{l}\) and \(x_{u}\) if \(f(x_{l})f(x_{u}) < 0\) (See Figure \(\PageIndex{2.1}\)).

Bisection method

Since the method is based on finding the root between two points, the technique falls under the category of bracketing methods.

Since the root is bracketed between two points, \(x_{l}\) and \(x_{u}\) , one can find the mid-point, \(x_{m}\) between \(x_{l}\) and \(x_{u}\) . This gives us two new intervals \(x_{l}\) and \(x_{m}\) , and \(x_{m}\) and \(x_{u}\) .

Is the root now between \(x_{l}\) and \(x_{m}\) or between \(x_{m}\) and \(x_{u}\) ? Well, one can find the sign of \(f(x_{l})f(x_{m})\) , and if \(f(x_{l})f(x_{m}) < 0\) then the new bracket is between \(x_{l}\) and \(x_{m}\) , otherwise, it is between \(x_{m}\) and \(x_{u}\) . So, you can see that you are literally halving the interval. As one repeats this process, the width of the interval \(\left\lbrack x_{l},x_{u} \right\rbrack\) becomes smaller and smaller, and you can zero into the root of the equation \(f(x) = 0\) . The algorithm for the bisection method is given as follows.

Algorithm for the bisection method

The steps to apply the bisection method to find the root of the equation \(f(x) = 0\) are

Choose \(x_{l}\) and \(x_{u}\) as two guesses for the root such that \(f(x_{l})f(x_{u}) < 0\) , or in other words, \(f(x)\) changes sign between \(x_{l}\) and \(x_{u}\) .

Estimate the root \(x_{m}\) of the equation \(f(x) = 0\) as the mid-point between \(x_{l}\) and \(x_{u}\) as

\[x_{m} = \frac{x_{l} + \ x_{u}}{2} \nonumber\]

Now check the following.

If \(f(x_{l})f(x_{m}) < 0\) , then the root lies between \(x_{l}\) and \(x_{m}\) ; then \(x_{l} = x_{l}\) and \(x_{u} = x_{m}\) .

If \(f(x_{l})f(x_{m}) > 0\) , then the root lies between \(x_{m}\) and \(x_{u}\) ; then \(x_{l} = x_{m}\) and \(x_{u} = x_{u}\) .

If \(f(x_{l})f(x_{m}) = 0\) ; then the root is \(x_{m}\) . Stop the algorithm if this is true.

Find the new estimate of the root

Find the absolute relative approximate error as

\[\left| \in_{a} \right| = \left| \frac{x_{m}^{\text{new}} - x_{m}^{\text{old}}}{x_{m}^{\text{new}}} \right|\ \times 100 \nonumber\]

\[x_{m}^{\text{new}} = \text{estimated root from the present iteration} \nonumber\]

\[x_{m}^{\text{old}}= \text{estimated root from the previous iteration} \nonumber\]

Compare the absolute relative approximate error \(\left| \in_{a} \right|\) with the pre-specified relative error tolerance \(\in_{s}\) . If \(\left| \in_{a} \right| > \in_{s}\) , then go to Step 3, else stop the algorithm. Note one should also check whether the number of iterations is more than the maximum number of iterations allowed. If so, one needs to terminate the algorithm and notify the user about it.

Title: Bisection Method - Algorithm

Summary: Learn the algorithm of the bisection method of solving nonlinear equations of the form \(f(x) = 0\) .

Lesson 3: Application of Bisection Method

After successful completion of this lesson, you should be able to:

1) apply the bisection method to solve for roots of a nonlinear equation.

Applications

In the previous lesson, you learned the theory of the bisection method of solving a nonlinear equation. In this lesson, we apply the algorithm of the bisection method to solve a nonlinear equation.

Example \(\PageIndex{3.1}\)

Use the bisection method to find the root of the nonlinear equation

\[x^{3} = 20 \nonumber\]

Use initial lower and upper guesses of \(1\) and \(4\) , respectively.

Conduct three iterations to estimate the root of the equation.

Find the absolute relative approximate error at the end of each iteration.

Find the number of significant digits that are at least correct at the end of each iteration.

Rewrite the equation \(x^3=20\) in the form \(f(x) = 0\) that gives

\[f(x) = x^{3} - 20 = 0 \nonumber\]

Check if the function changes the sign between the two initial guesses, \(x_{l}\) and \(x_{u}\) . The initial guesses are given as \(x_{l} = 1\) and \(x_{u} = 4\)

\[\begin{split} f(x_{l}) &= f(1)\\ &= 1^{3} - 20\\ &= - 19\end{split} \nonumber\]

\[\begin{split} f(x_{u}) &= f(4)\\ &= 4^{3} - 20\\ &= 44\end{split} \nonumber\]

\[\begin{split} f(x_{l})f(x_{u}) &= f(1)f(4)\\ &= ( - 19)(44) < 0\end{split} \nonumber\]

This change in sign tells us that the initial bracket of \([1,4]\) given to us is valid.

Iteration 1

\[x_{l} = 1,\ x_{u} = 4 \nonumber\]

The estimate of the root is

\[\begin{split} x_{m} &= \frac{x_{l} + x_{u}}{2}\\ &= \frac{1 + 4}{2}\\ &= 2.5\end{split} \nonumber\]

Iteration 2

Find the value of the function at the midpoint from the previous iteration and use it to determine the new bracket.

\[\begin{split} f(x_{m}) &= f(2.5)\\ &= (2.5)^{3} - 20\\ &= - 4.375\end{split} \nonumber\]

\[\begin{split} f(x_{l})f(x_{m}) &= f(1)f(2.5)\\ &= ( - 19)( - 4.375) > 0\end{split} \nonumber\]

Since \(f(x_{l})f(x_{m}) > 0\) , the root does not lie between \(x_{l}\) and \(x_{m}\) , but between \(x_{m}\) and \(x_{u}\) , that is, \(2.5\) and \(4\) .

\[x_{l} = 2.5,\ x_{u} = 4 \nonumber\]

\[\begin{split} x_{m} &= \frac{x_{l} + x_{u}}{2}\\ &= \frac{2.5 + 4}{2}\\ &= 3.25\end{split} \nonumber\]

The absolute relative approximate error \(\left| \varepsilon_{a} \right|\) at the end of Iteration 2 is

\[\begin{split} \left| \varepsilon_{a} \right| &= \left| \frac{x_{m}^{\text{new}} - x_{m}^{\text{old}}}{x_{m}^{\text{new}}} \right| \times 100\\ &= \left| \frac{3.25 - 2.5}{3.25} \right| \times 100\\ &= 23.1\%\end{split} \nonumber\]

None of the significant digits are at least correct in the estimated root of the equation because the absolute relative approximate error is greater than \(5\%\) .

Iteration 3

\[\begin{split} f(x_{m}) &= f(3.25)\\ &= (3.25)^{3} - 20\\ &= 14.3281\end{split} \nonumber\]

\[\begin{split} f(x_{l})f(x_{m}) &= f(2.5)f(3.25)\\ &= ( - 4.375)(14.3281) < 0 \end{split} \nonumber\]

Since \(f(x_{l})f(x_{m}) < 0\) , the root does lie between \(x_{l}\) and \(x_{m}\) , that is, \(2.5\) and \(3.25\) .

\[x_{l} = 2.5,x_{u} = 3.25 \nonumber\]

\[\begin{split} x_{m} &= \frac{x_{l} + x_{u}}{2}\\ &= \frac{2.5 + 3.25}{2}\\ &= 2.875\end{split} \nonumber\]

The absolute relative approximate error \(\left| \varepsilon_{a} \right|\) at the end of Iteration 3 is

\[\begin{split} \left| \epsilon_{a} \right| &= \left| \frac{x_{m}^{\text{new}} - x_{m}^{\text{old}}}{x_{m}^{\text{new}}} \right| \times 100\\ &= \left| \frac{2.875 - 2.5}{2.875} \right| \times 100\\ &= 13.0\%\end{split} \nonumber\]

Still, none of the significant digits are at least correct in the estimated root of the equation as the absolute relative approximate error is greater than \(5\%\) .

Example \(\PageIndex{3.2}\)

You are working for ‘DOWN THE TOILET COMPANY’ that makes floats for ABC commodes. The floating ball has a specific gravity of \(0.6\) and has a radius of \(5.5\ \text{cm}\) . You are asked to find the depth to which the ball is submerged when floating in the water.

The equation that gives the depth \(x\) to which the ball is submerged underwater is given by

\[x^{3} - 0.165x^{2} + 3.993 \times 10^{- 4} = 0 \nonumber\]

Use the bisection method of finding roots of equations to find the depth \(x\) to which the ball is submerged underwater. Conduct three iterations to estimate the root of the above equation. Find the absolute relative approximate error at the end of each iteration, and the number of significant digits at least correct at the end of each iteration.

From the physics of the problem, the ball would be submerged somewhere between \(x = 0\) and \(x = 2R\) ,

\[R = \text{radius of the ball,} \nonumber\]

\[0 \leq x \leq 2R \nonumber\]

\[0 \leq x \leq 2(0.055) \nonumber\]\[0 \leq x \leq 0.11 \nonumber\]

A ball of radius R floats in water, with a region of the ball of height x submerged.

Let’s us assume

\[x_{l} = 0,\ x_{u} = 0.11 \nonumber\]

Check if the function changes sign between \(x_{l}\) and \(x_{u}\) .

\[\begin{split} f(x_{l}) &= f(0) \\&= (0)^{3} - 0.165(0)^{2} + 3.993 \times 10^{- 4} \\&= 3.993 \times 10^{- 4} \end{split} \nonumber\]

\[\begin{split} f(x_{u}) &= f(0.11)\\ &= (0.11)^{3} - 0.165(0.11)^{2} + 3.993 \times 10^{- 4} \\&= - 2.662 \times 10^{- 4} \end{split} \nonumber\]

\[\begin{split} f(x_{l})f(x_{u}) &= f(0)f(0.11) \\&= (3.993 \times 10^{- 4})( - 2.662 \times 10^{- 4}) \\&< 0 \end{split} \nonumber\]

So, there is at least one root between \(x_{l}\) and \(x_{u}\) , that is between \(0\) and \(0.11\) .

\[\begin{split} x_{m} &= \frac{x_{l} + x_{u}}{2}\\ &= \frac{0 + 0.11}{2}\\ &= 0.055 \end{split} \nonumber\]

\[\begin{split} f\left(x_{m}\right)&=f(0.055)\\&=(0.055)^{3}-0.165(0.055)^{2}+3.993 \times 10^{-4}\\&=6.655 \times 10^{-5} \end{split} \nonumber\]

\[\begin{split} f(x_{l})f(x_{m}) &= f(0)f(0.055) \\&= \left( 3.993 \times 10^{- 4} \right)\left( 6.655 \times 10^{- 4} \right) \\&> 0 \end{split} \nonumber\]

Hence the root is bracketed between \(x_{m}\) and \(x_{u}\) , that is, between \(0.055\) and \(0.11\) . So, the lower and upper limit of the new bracket is

\[x_{l} = 0.055,\ x_{u} = 0.11 \nonumber\]

At this point, the absolute relative approximate error \(\left| \epsilon_{a} \right|\) cannot be calculated as we do not have a previous approximation.

\[\begin{split} x_{m} &= \frac{x_{l} + x_{u}}{2}\\ &= \frac{0.055 + 0.11}{2}\\ &= 0.0825 \end{split} \nonumber\]

\[\begin{split} f(x_{m}) &= f(0.0825) \\&= (0.0825)^{3} - 0.165(0.0825)^{2} + 3.993 \times \text{1}\text{0}^{- 4}\\ &= - 1.622 \times \text{1}\text{0}^{- 4}\end{split} \nonumber\]

\[\begin{split} f\left( x_{l} \right)f\left( x_{m} \right) &= f\left( 0.055 \right)f\left( 0.0825 \right) \\&= \left( 6.655 \times 10^{- 5} \right) \times \left( - 1.622 \times 10^{- 4} \right) \\&<0 \end{split} \nonumber\]

Hence, the root is bracketed between \(x_{l}\) and \(x_{m}\) , that is, between \(0.055\) and \(0.0825\) . So, the lower and upper limit of the new bracket is

\[x_{l} = 0.055,\ x_{u} = 0.0825 \nonumber\]

The absolute relative approximate error \(\left| \epsilon_{a} \right|\) at the end of Iteration 2 is

\[\begin{split} \left| \epsilon_{a} \right| &= \left| \frac{x_{m}^{\text{new}} - x_{m}^{\text{old}}}{x_{m}^{\text{new}}} \right| \times 100\\ &= \left| \frac{0.0825 - 0.055}{0.0825} \right| \times 100\\ &= 33.33\% \end{split} \nonumber\]

None of the significant digits are at least correct in the estimated root of \(x_{m} = 0.0825\) because the absolute relative approximate error is greater than \(5\%\) .

\[\begin{split} x_{m} &= \frac{x_{l} + x_{u}}{2}\\ &= \frac{0.055 + 0.0825}{2}\\ &= 0.06875 \end{split} \nonumber\]

\[\begin{split} f\left( x_{m} \right) &= f(0.06875) \\&= (0.06875)^{3} - 0.165(0.06875)^{2} + 3.993 \times {1}{0}^{- 4}\\&= - 5.563 \times {1}{0}^{- 5} \end{split} \nonumber\]

\[\begin{split} f(x_{l})f(x_{m}) &= f(0.055)f(0.06875) \\&= (6.655 \times 10^{5}) \times ( - \text{5.563} \times 10^{- 5}) \\&< 0 \end{split} \nonumber\]

Hence, the root is bracketed between \(x_{l}\) and \(x_{m}\) , that is, between \(0.055\) and \(0.06875\) . So the lower and upper limit of the new bracket is

\[x_{l} = 0.055,\ x_{u} = 0.06875 \nonumber\]

The absolute relative approximate error \(\left| \epsilon_{a} \right|\) at the ends of Iteration 3 is

\[\begin{split} \left| \epsilon_{a} \right| &= \left| \frac{x_{m}^{\text{new}} - x_{m}^{\text{old}}}{x_{m}^{\text{new}}} \right| \times 100\\ &= \left| \frac{0.06875 - 0.0825}{0.06875} \right| \times 100\\ &= 20\% \end{split} \nonumber\]

Seven more iterations were conducted, and these iterations are shown in Table \(\PageIndex{3.1}\).

At the end of the \(10^{\text{th}}\) iteration,

\[\left| \epsilon_{a} \right| = 0.1721\% \nonumber\]

Hence the number of significant digits at least correct is given by the largest value of \(m\) for which

\[\left| \epsilon_{a} \right| \leq 0.5 \times 10^{2 - m} \nonumber\]

\[0.1721 \leq 0.5 \times 10^{2 - m} \nonumber\]

\[0.3442 \leq 10^{2 - m} \nonumber\]

\[\log(0.3442) \leq 2 - m \nonumber\]

\[m \leq 2 - \log(0.3442) = 2.463 \nonumber\]

\[m = 2 \nonumber\]

The number of significant digits at least correct in the estimated root of \(0.06241\) at the end of the \(10^{\text{th}}\) iteration is \(2\) .

Lesson 4: Advantages and Pitfalls of Bisection Method

1) enumerate the advantages of the bisection method and the reasoning behind them.

2) list the drawbacks of the bisection method and the reason behind them.

Advantages of the bisection method

a) The bisection method is always convergent. Since the technique brackets the root, the procedure is guaranteed to converge.

b) As iterations are conducted, the interval gets halved. So, one can guarantee the error in the solution of the equation.

Drawbacks of bisection method

a) The convergence of the bisection method is slow as it is based on halving the interval.

b) If one of the initial guesses is closer to the root, it will take a larger number of iterations to reach the root.

c) If a function \(f(x)\) is such that it just touches the x-axis (Figure \(\PageIndex{4.1}\)) such as

\[f(x) = x^{2} = 0 \nonumber\]

it will be unable to find the lower guess, \(x_{l}\) , and upper guess, \(x_{u}\) , such that

\[f(x_{l})f(x_{u}) < 0 \nonumber\]

d) For functions \(f(x)\) where there is a singularity, and it reverses the sign at the singularity, the bisection method may converge on the singularity (Figure \(\PageIndex{4.2}\)). An example includes

\[f(x) = \frac{1}{x} \nonumber\]

where \(x_{l} = - 2\) , \(x_{u} = 3\) are valid initial guesses which satisfy

However, the function is not continuous, and the theorem that a root exists is also not applicable.

The equation f(x)=x^2 has a single root at x=0 that cannot be bracketed.

Audiovisual Lecture

Title: Bisection Method: Advantages and Drawbacks

Summary : This video discusses the advantages and drawbacks of the bisection method - a numerical method to find roots of a nonlinear equation.

Multiple Choice Test

(1). The bisection method of finding roots of nonlinear equations falls under the category of a(n) _________ method.

(B) bracketing

(D) graphical

(2). If \(f(a)f\left( b \right) < 0\) for a real continuous function \(f\left( x \right)\) , then in the domain of \(\left\lbrack a,b \right\rbrack\) for \(f\left( x \right) = 0\) , there is (are)

(A) one root

(B) an undeterminable number of roots

(C) no root

(D) at least one root

(3). Assuming an initial bracket of \(\left\lbrack 1,5 \right\rbrack\) , the second (at the end of 2 iterations) iterative value of the root of \(te^{- t} - 0.3 = 0\) using the bisection method is

(B) \(1.5\)

(4). To find the root of \(f(x) = 0\) , a scientist is using the bisection method. At the beginning of an iteration, the lower and upper guesses of the root are \(x_{l}\) and \(x_{u}\) . At the end of the iteration, the absolute relative approximate error in the estimated value of the root would be

(A) \(\displaystyle \left| \frac{x_{u}}{x_{u} + x_{l}} \right|\)

(B) \(\displaystyle \left| \frac{x_{l}}{x_{u} + x_{l}} \right|\)

(C) \(\displaystyle \left| \frac{x_{u} - x_{l}}{x_{u} + x_{l}} \right|\)

(D) \(\displaystyle \left| \frac{x_{u} + x_{l}}{x_{u} - x_{l}} \right|\)

(5). For an equation like \(x^{2} = 0\) , a root exists at \(x = 0\) . The bisection method cannot be adopted to solve this equation in spite of the root existing at \(x = 0\) because the function \(f\left( x \right) = x^{2}\)

(A) is a polynomial

(B) has repeated zeros at \(x = 0\)

(C) is always non-negative

(D) has a slope equal to zero at \(x = 0\)

(6). The ideal gas law is given by

\[pv = RT \nonumber\]

where \(p\) is the pressure, \(v\) is the specific volume, \(R\) is the universal gas constant, and \(T\) is the absolute temperature. This equation is only accurate for a limited range of pressure and temperature. Vander Waals came up with an equation that was accurate for larger ranges of pressure and temperature given by

\[\displaystyle \left( p + \frac{a}{v^{2}} \right)\left( v - b \right) = RT \nonumber\]

where \(a\) and \(b\) are empirical constants dependent on a particular gas. Given the value of \(R = 0.08\) , \(a = 3.592\) , \(b = 0.04267\) , \(p = 10\) and \(T = 300\) (assume all units are consistent), one is going to find the specific volume, \(v\) , for the above values. Without finding the solution from the Vander Waals equation, what would be a good initial guess for \(v\) ?

(B) \(1.2\)

(C) \(2.4\)

(D) \(3.6\)

For complete solution, go to

http://nm.mathforcollege.com/mcquizzes/03nle/quiz_03nle_bisection_solution.pdf

Problem Set

(1). Find the estimate of the root of \(x^{2} - 4 = 0\) by using the bisection method. Use initial guesses of \(1.7\) and \(2.4\) . Conduct three iterations, and calculate the approximate error, true error, absolute relative approximate error, and absolute relative true error at the end of each iteration.

(2). You are working for DOWN THE TOILET COMPANY that makes floats for ABC commodes. The ball has a specific gravity of 0.6 and has a radius of 5.5 cm. You are asked to find the depth to which the ball will get submerged when floating in the water.

The equation that gives the depth \(x\) (unit of \(x\) is meters) to which the ball is submerged under water is given by

Solving it exactly would require some effort. However, using numerical techniques such as the bisection method, we can solve this equation and any other equation of the form \(f(x) = 0\) . Solve the above equation by the bisection method and do the following. Use initial guesses of \(x = 0\) and \(x = 0.11\) .

a) Conduct three iterations.

b) Calculate the absolute relative approximate error at the end of each step.

c) Find the number of significant digits at least correct at the end of each iteration.

d) How can you use the knowledge of the physics of the problem to develop initial guesses for the bisection method?

\(a)\ 0.055, 0.0825, 0.06875\)

\(b)\ n/a, 33.333\%, 20.00\%\)

\(c)\ n/a, 0, 0\)

\(d)\) Since the diameter is \(0.11\ \text{m}\) , initial guesses of \(0\) and \(0.11\) are reasonable starting values

(3). The velocity of a body is given by the following equation

\[v(t) = te^{- t} + \frac{1}{t} \nonumber\]

\(t\) is given in seconds and \(v\) is given in m/s.

Find the time when the velocity of the body will be \(0.35\ \text{m/s}\) . Use bisection method and conduct three iterations. Use initial bracketing guess of \([1,8]\) . Show all steps in calculating the estimated root, absolute relative approximate error and the velocity of the body for each iteration. Also, tabulate your answers as iteration number, estimated root, absolute relative approximate error, and velocity of the body.

(4). Enumerate the drawbacks of the bisection method of solving nonlinear equations.

(5). The velocity of a body is given by

\[v = 5e^{- t} + 4 + \sin(t), \nonumber\]

\(v\) is given in m/s,

\(t\) is given in s.

Derive the nonlinear equation that you will need to solve to find when the acceleration of the body would be \(1.54 \ \text{m/s}^{2}\) . Find the solution of the equation by using three iterations of the bisection method.

The equation has no real roots. Note the question asks when acceleration is \(1.54 \ \text{m/s}^{2}\)

(6). To solve the equation \(f(x) = 0\) , an engineer is using the bisection method. He starts with an initial valid bracket of \([2, 7]\) . Find the maximum possible absolute true error in his estimate of the root at the end of two iterations. Show your reasoning clearly for your answer.

(7). Estimate the next guess for the root of \(x^{2} - 16 = 0\) by using a modified bisection method as explained below. The initial bracket of \([1,8]\) is found as a valid bracket. In the case of bisection method, the root estimated at the end of the first iteration is the midpoint between \(1\) and \(8\) . Instead, in the modified bisection method, the root estimated at the end of the first iteration would be the point where the straight line drawn from the function at \(x = 1\) to the function at \(x = 8\) crosses the \(x\) -axis. What is this estimate of the root?

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