how to do maths problem solving

High Impact Tutoring Built By Math Experts

Personalized standards-aligned one-on-one math tutoring for schools and districts

Free ready-to-use math resources

Hundreds of free math resources created by experienced math teachers to save time, build engagement and accelerate growth

20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

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

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

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

What are problem-solving strategies?

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

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

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

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

The ultimate guide to problem solving techniques

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

20 Math Strategies For Problem-Solving

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

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

Strategies to understand the problem

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

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

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

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

1. Read the problem aloud

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

2. Highlight keywords 

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

3. Summarize the information

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

4. Determine the unknown

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

5. Make a plan

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

Strategies for solving the problem 

1. draw a model or diagram.

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

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

2. Act it out

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

3. Work backwards

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

For example,

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

4. Write a number sentence

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

5. Use a formula

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

Strategies for checking the solution 

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

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

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

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

2. Estimate to check for reasonableness

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

3. Plug-In Method

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

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

4. Peer Review

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

5. Use a Calculator

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

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

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

Polya’s 4 steps include:

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

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

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

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

How Third Space Learning improves problem-solving 

Resources .

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

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

One-on-one tutoring 

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

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

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

Problem-solving

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

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

READ MORE :

• 8 Common Core math examples
• Tier 3 Interventions: A School Leaders Guide
• Tier 2 Interventions: A School Leaders Guide
• Tier 1 Interventions: A School Leaders Guide

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

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

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

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

Related articles

Why Student Centered Learning Is Important: A Guide For Educators

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

Differentiated Instruction: 9 Differentiated Curriculum And Instruction Strategies For Teachers 

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

Ultimate Guide to Metacognition [FREE]

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

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

Privacy Overview

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

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.

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.

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.

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.

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

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

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

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. 

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

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.

Share this article

• Math for Kids
• Parenting Resources
• ELA for Kids
• Teaching Resources

Who Invented Math? History, Facts & More

How to Teach Number Recognition to Kids in 8 Easy Steps

How to Teach One to One Correspondence To Kids: 4 Easy Steps

How to Teach Odd and Even Numbers in 4 Easy Steps

How to Teach Long Division to Kids in 6 Easy Steps

8 Types of Preschool Programs for Kids in 2024

6-year-old Developmental Milestones Checklist

How to Prepare a Schedule for Kindergarten With Examples

How to Prepare a Schedule for Preschoolers With Sample

12 Best Funny Short Stories for Kids to Read in 2024

How to Make Reading Fun for Early Readers: 12 Best Ideas

13 Best Phoneme Segmentation Activities for Kids

How to Teach Decoding in 9 Simple Steps

300+ Halloween Words From A-Z for Kids [Free Downloadable]

17 Best Guided Reading Activities for Teachers

12 Best Strategies for Teaching English Grammar to Kids

11 Best Coloring Apps for Kids [Android & iOS]

12 Best Reading Bulletin Board Ideas for Your Classroom

15 Fun Summer Bulletin Board Ideas for 2024

13 Best Assessment Tools for Teachers in 2024

10 Best Strategies for Solving Math Word Problems

1. Understand the Problem by Paraphrasing

2. identify key information and variables, 3. translate words into mathematical symbols, 4. break down the problem into manageable parts, 5. draw diagrams or visual representations, 6. use estimation to predict answers, 7. apply logical reasoning for unknown variables, 8. leverage similar problems as templates, 9. check answers in the context of the problem, 10. reflect and learn from mistakes.

Have you ever observed the look of confusion on a student’s face when they encounter a math word problem ? It’s a common sight in classrooms worldwide, underscoring the need for effective strategies for solving math word problems . The main hurdle in solving math word problems is not just the math itself but understanding how to translate the words into mathematical equations that can be solved.

Math & ELA | PreK To Grade 5

Kids see fun ., you see real learning outcomes ..

Watch your kids fall in love with math & reading through our scientifically designed curriculum.

Generic advice like “read the problem carefully” or “practice more” often falls short in addressing students’ specific difficulties with word problems. Students need targeted math word problem strategies that address the root of their struggles head-on. 

A Guide on Steps to Solving Word Problems: 10 Strategies 

One of the first steps in tackling a math word problem is to make sure your students understand what the problem is asking. Encourage them to paraphrase the problem in their own words. This means they rewrite the problem using simpler language or break it down into more digestible parts. Paraphrasing helps students grasp the concept and focus on the problem’s core elements without getting lost in the complex wording.

Original Problem: “If a farmer has 15 apples and gives away 8, how many does he have left?”

Paraphrased: “A farmer had some apples. He gave some away. Now, how many apples does he have?”

This paraphrasing helps students identify the main action (giving away apples) and what they need to find out (how many apples are left).

Play these subtraction word problem games in the classroom for free:

Explore More

Students often get overwhelmed by the details in word problems. Teach them to identify key information and variables essential for solving the problem. This includes numbers , operations ( addition , subtraction , multiplication , division ), and what the question is asking them to find. Highlighting or underlining can be very effective here. This visual differentiation can help students focus on what’s important, ignoring irrelevant details.

• Encourage students to underline numbers and circle keywords that indicate operations (like ‘total’ for addition and ‘left’ for subtraction).
• Teach them to write down what they’re solving for, such as “Find: Total apples left.”

Problem: “A classroom has 24 students. If 6 more students joined the class, how many students are there in total?”

Key Information:

• Original number of students (24)
• Students joined (6)
• Looking for the total number of students

Here are some fun addition word problems that your students can play for free:

The transition from the language of word problems to the language of mathematics is a critical skill. Teach your students to convert words into mathematical symbols and equations. This step is about recognizing keywords and phrases corresponding to mathematical operations and expressions .

Common Translations:

• “Total,” “sum,” “combined” → Addition (+)
• “Difference,” “less than,” “remain” → Subtraction (−)
• “Times,” “product of” → Multiplication (×)
• “Divided by,” “quotient of” → Division (÷)
• “Equals” → Equals sign (=)

Problem: “If one book costs $5, how much would 4 books cost?”

Translation: The word “costs” indicates a multiplication operation because we find the total cost of multiple items. Therefore, the equation is 4 × 5 = $20

Complex math word problems can often overwhelm students. Incorporating math strategies for problem solving, such as teaching them to break down the problem into smaller, more manageable parts, is a powerful approach to overcome this challenge. This means looking at the problem step by step rather than simultaneously trying to solve it. Breaking it down helps students focus on one aspect of the problem at a time, making finding the solution more straightforward.

Problem: “John has twice as many apples as Sarah. If Sarah has 5 apples, how many apples do they have together?”

Steps to Break Down the Problem:

Find out how many apples John has: Since John has twice as many apples as Sarah, and Sarah has 5, John has 5 × 2 = 10

Calculate the total number of apples: Add Sarah’s apples to John’s to find the total,  5 + 10 = 15

By splitting the problem into two parts, students can solve it without getting confused by all the details at once.

Explore these fun multiplication word problem games:

Diagrams and visual representations can be incredibly helpful for students, especially when dealing with spatial or quantity relationships in word problems. Encourage students to draw simple sketches or diagrams to represent the problem visually. This can include drawing bars for comparison, shapes for geometry problems, or even a simple distribution to better understand division or multiplication problems .

Problem: “A garden is 3 times as long as it is wide. If the width is 4 meters, how long is the garden?”

Visual Representation: Draw a rectangle and label the width as 4 meters. Then, sketch the length to represent it as three times the width visually, helping students see that the length is 4 × 3 = 12

Estimation is a valuable skill in solving math word problems, as it allows students to predict the answer’s ballpark figure before solving it precisely. Teaching students to use estimation can help them check their answers for reasonableness and avoid common mistakes.

Problem: “If a book costs $4.95 and you buy 3 books, approximately how much will you spend?”

Estimation Strategy: Round $4.95 to the nearest dollar ($5) and multiply by the number of books (3), so 5 × 3 = 15. Hence, the estimated total cost is about $15.

Estimation helps students understand whether their final answer is plausible, providing a quick way to check their work against a rough calculation.

Check out these fun estimation and prediction word problem worksheets that can be of great help:

When students encounter problems with unknown variables, it’s crucial to introduce them to logical reasoning. This strategy involves using the information in the problem to deduce the value of unknown variables logically. One of the most effective strategies for solving math word problems is working backward from the desired outcome. This means starting with the result and thinking about the steps leading to that result, which can be particularly useful in algebraic problems.

Problem: “A number added to three times itself equals 32. What is the number?”

Working Backward:

Let the unknown number be x.

The equation based on the problem is  x + 3x = 32

Solve for x by simplifying the equation to 4x=32, then dividing by 4 to find x=8.

By working backward, students can more easily connect the dots between the unknown variable and the information provided.

Practicing problems of similar structure can help students recognize patterns and apply known strategies to new situations. Encourage them to leverage similar problems as templates, analyzing how a solved problem’s strategy can apply to a new one. Creating a personal “problem bank”—a collection of solved problems—can be a valuable reference tool, helping students see the commonalities between different problems and reinforcing the strategies that work.

Suppose students have solved a problem about dividing a set of items among a group of people. In that case, they can use that strategy when encountering a similar problem, even if it’s about dividing money or sharing work equally.

It’s essential for students to learn the habit of checking their answers within the context of the problem to ensure their solutions make sense. This step involves going back to the original problem statement after solving it to verify that the answer fits logically with the given information. Providing a checklist for this process can help students systematically review their answers.

Checklist for Reviewing Answers:

• Re-read the problem: Ensure the question was understood correctly.
• Compare with the original problem: Does the answer make sense given the scenario?
• Use estimation: Does the precise answer align with an earlier estimation?
• Substitute back: If applicable, plug the answer into the problem to see if it works.

Problem: “If you divide 24 apples among 4 children, how many apples does each child get?”

After solving, students should check that they understood the problem (dividing apples equally).

Their answer (6 apples per child) fits logically with the number of apples and children.

Their estimation aligns with the actual calculation.

Substituting back 4×6=24 confirms the answer is correct.

Teaching students to apply logical reasoning, leverage solved problems as templates, and check their answers in context equips them with a robust toolkit for tackling math word problems efficiently and effectively.

One of the most effective ways for students to improve their problem-solving skills is by reflecting on their errors, especially with math word problems. Using word problem worksheets is one of the most effective strategies for solving word problems, and practicing word problems as it fosters a more thoughtful and reflective approach to problem-solving

These worksheets can provide a variety of problems that challenge students in different ways, allowing them to encounter and work through common pitfalls in a controlled setting. After completing a worksheet, students can review their answers, identify any mistakes, and then reflect on them in their mistake journal. This practice reinforces mathematical concepts and improves their math problem solving strategies over time.

3 Additional Tips for Enhancing Word Problem-Solving Skills

Before we dive into the importance of reflecting on mistakes, here are a few impactful tips to enhance students’ word problem-solving skills further:

1. Utilize Online Word Problem Games

Incorporate online games that focus on math word problems into your teaching. These interactive platforms make learning fun and engaging, allowing students to practice in a dynamic environment. Games can offer instant feedback and adaptive challenges, catering to individual learning speeds and styles.

Here are some word problem games that you can use for free:

2. Practice Regularly with Diverse Problems

Consistent practice with a wide range of word problems helps students become familiar with different questions and mathematical concepts. This exposure is crucial for building confidence and proficiency.

Start Practicing Word Problems with these Printable Word Problem Worksheets:

3. Encourage Group Work

Solving word problems in groups allows students to share strategies and learn from each other. A collaborative approach is one of the best strategies for solving math word problems that can unveil multiple methods for tackling the same problem, enriching students’ problem-solving toolkit.

Conclusion 

Mastering math word problems is a journey of small steps. Encourage your students to practice regularly, stay curious, and learn from their mistakes. These strategies for solving math word problems are stepping stones to turning challenges into achievements. Keep it simple, and watch your students grow their confidence and skills, one problem at a time.

Frequently Asked Questions (FAQs)

How can i help my students stay motivated when solving math word problems.

Encourage small victories and use engaging tools like online games to make practice fun and rewarding.

What's the best way to teach beginners word problems?

Begin with simple problems that integrate everyday scenarios to make the connection between math and real-life clear and relatable.

How often should students practice math word problems?

Regular, daily practice with various problems helps build confidence and problem-solving skills over time.

• Pre-Kindergarten
• Kindergarten

Most Popular

76 Best Report Card Comments Samples for Teachers

117 Best Riddles for Kids (With Answers)

40 Best Good Vibes Quotes to Brighten Your Day

Recent posts.

17 Creative Art Activities for Preschoolers in 2024

Math & ELA | PreK To Grade 5

Kids see fun., you see real learning outcomes..

Watch your kids fall in love with math & reading through our scientifically designed curriculum.

Parents, try for free Teachers, use for free

• Games for Kids
• Worksheets for Kids
• Math Worksheets
• ELA Worksheets
• Math Vocabulary
• Number Games
• Addition Games
• Subtraction Games
• Multiplication Games
• Division Games
• Addition Worksheets
• Subtraction Worksheets
• Multiplication Worksheets
• Division Worksheets
• Times Tables Worksheets
• Reading Games
• Writing Games
• Phonics Games
• Sight Words Games
• Letter Tracing Games
• Reading Worksheets
• Writing Worksheets
• Phonics Worksheets
• Sight Words Worksheets
• Letter Tracing Worksheets
• Prime Number
• Order of Operations
• Long multiplication
• Place value
• Parallelogram
• SplashLearn Success Stories
• SplashLearn Apps

© Copyright - SplashLearn

Back-to-School Learning Boost!

Turn play into progress., jumpstart learning now.

Explore 4,000+ games and 450+ lesson plans designed to make this school year the best one yet!

Parents, Try for Free Teachers, Use for Free

Try Math Solver

Get step-by-step explanations

Graph your math problems

Practice, practice, practice

Get math help in your language

Math Solver

Geogebra math solver.

Get accurate solutions and step-by-step explanations for algebra and other math problems, while enhancing your problem-solving skills!

Learn by   .css-1v0lc0l{color:var(--chakra-colors-blue-500);} doing

Guided interactive problem solving that’s effective and fun. Master concepts in 15 minutes a day.

Data Analysis

Computer Science

Programming & AI

Science & Engineering

Join over 10 million people learning on Brilliant

Master concepts in 15 minutes a day.

Whether you’re a complete beginner or ready to dive into machine learning and beyond, Brilliant makes it easy to level up fast with fun, bite-sized lessons.

Effective, hands-on learning

Visual, interactive lessons make concepts feel intuitive — so even complex ideas just click. Our real-time feedback and simple explanations make learning efficient.

Learn at your level

Students and professionals alike can hone dormant skills or learn new ones. Progress through lessons and challenges tailored to your level. Designed for ages 13 to 113.

Guided bite-sized lessons

We make it easy to stay on track, see your progress, and build your problem-solving skills one concept at a time.

Stay motivated

Form a real learning habit with fun content that’s always well-paced, game-like progress tracking, and friendly reminders.

© 2024 Brilliant Worldwide, Inc., Brilliant and the Brilliant Logo are trademarks of Brilliant Worldwide, Inc.

ChatGPT for Teachers

Trauma-informed practices in schools, teacher well-being, cultivating diversity, equity, & inclusion, integrating technology in the classroom, social-emotional development, covid-19 resources, invest in resilience: summer toolkit, civics & resilience, all toolkits, degree programs, trauma-informed professional development, teacher licensure & certification, how to become - career information, classroom management, instructional design, lifestyle & self-care, online higher ed teaching, current events, 10 ways to do fast math: tricks and tips for doing math in your head.

You don’t have to be a math teacher to know that a lot of students—and likely a lot of parents (it’s been awhile!)—are intimidated by math problems, especially if they involve large numbers. Learning techniques on how to do math quickly can help students develop greater confidence in math , improve math skills and understanding, and excel in advanced courses.

If it’s your job to teach those, here’s a great refresher.

Fast math tricks infographic

10 tricks for doing fast math

Here are 10 fast math strategies students (and adults!) can use to do math in their heads. Once these strategies are mastered, students should be able to accurately and confidently solve math problems that they once feared solving.

1. Adding large numbers

Adding large numbers just in your head can be difficult. This method shows how to simplify this process by making all the numbers a multiple of 10. Here is an example:

While these numbers are hard to contend with, rounding them up will make them more manageable. So, 644 becomes 650 and 238 becomes 240.

Now, add 650 and 240 together. The total is 890. To find the answer to the original equation, it must be determined how much we added to the numbers to round them up.

650 – 644 = 6 and 240 – 238 = 2

Now, add 6 and 2 together for a total of 8

To find the answer to the original equation, 8 must be subtracted from the 890.

890 – 8 = 882

So the answer to 644 +238 is 882.

2. Subtracting from 1,000

Here’s a basic rule to subtract a large number from 1,000: Subtract every number except the last from 9 and subtract the final number from 10

For example:

1,000 – 556

Step 1: Subtract 5 from 9 = 4

Step 2: Subtract 5 from 9 = 4

Step 3: Subtract 6 from 10 = 4

The answer is 444.

3. Multiplying 5 times any number

When multiplying the number 5 by an even number, there is a quick way to find the answer.

For example, 5 x 4 =

• Step 1: Take the number being multiplied by 5 and cut it in half, this makes the number 4 become the number 2.
• Step 2: Add a zero to the number to find the answer. In this case, the answer is 20.

When multiplying an odd number times 5, the formula is a bit different.

For instance, consider 5 x 3.

• Step 1: Subtract one from the number being multiplied by 5, in this instance the number 3 becomes the number 2.
• Step 2: Now halve the number 2, which makes it the number 1. Make 5 the last digit. The number produced is 15, which is the answer.

4. Division tricks

Here’s a quick way to know when a number can be evenly divided by these certain numbers:

• 10 if the number ends in 0
• 9 when the digits are added together and the total is evenly divisible by 9
• 8 if the last three digits are evenly divisible by 8 or are 000
• 6 if it is an even number and when the digits are added together the answer is evenly divisible by 3
• 5 if it ends in a 0 or 5
• 4 if it ends in 00 or a two digit number that is evenly divisible by 4
• 3 when the digits are added together and the result is evenly divisible by the number 3
• 2 if it ends in 0, 2, 4, 6, or 8

5. Multiplying by 9

This is an easy method that is helpful for multiplying any number by 9. Here is how it works:

Let’s use the example of 9 x 3.

Step 1 : Subtract 1 from the number that is being multiplied by 9.

3 – 1 = 2

The number 2 is the first number in the answer to the equation.

Step 2 : Subtract that number from the number 9.

9 – 2 = 7

The number 7 is the second number in the answer to the equation.

So, 9 x 3 = 27

6. 10 and 11 times tricks

The trick to multiplying any number by 10 is to add a zero to the end of the number. For example, 62 x 10 = 620.

There is also an easy trick for multiplying any two-digit number by 11. Here it is:

Take the original two-digit number and put a space between the digits. In this example, that number is 25.

Now add those two numbers together and put the result in the center:

2_(2 + 5)_5

The answer to 11 x 25 is 275.

If the numbers in the center add up to a number with two digits, insert the second number and add 1 to the first one. Here is an example for the equation 11 x 88

(8 + 1)_6_8

There is the answer to 11 x 88: 968

7. Percentage

Finding a percentage of a number can be somewhat tricky, but thinking about it in the right terms makes it much easier to understand. For instance, to find out what 5% of 235 is, follow this method:

• Step 1: Move the decimal point over by one place, 235 becomes 23.5.
• Step 2: Divide 23.5 by the number 2, the answer is 11.75. That is also the answer to the original equation.

8. Quickly square a two-digit number that ends in 5

Let’s use the number 35 as an example.

• Step 1: Multiply the first digit by itself plus 1.
• Step 2: Put a 25 at the end.

35 squared = [3 x (3 + 1)] & 25

[3 x (3 + 1)] = 12

12 & 25 = 1225

35 squared = 1225

9. Tough multiplication

When multiplying large numbers, if one of the numbers is even, divide the first number in half, and then double the second number. This method will solve the problem quickly. For instance, consider

Step 1: Divide the 20 by 2, which equals 10. Double 120, which equals 240.

Then multiply your two answers together.

10 x 240 = 2400

The answer to 20 x 120 is 2,400.

10. Multiplying numbers that end in zero

Multiplying numbers that end in zero is actually quite simple. It involves multiplying the other numbers together and then adding the zeros at the end. For instance, consider:

Step 1: Multiply the 2 times the 4

Step 2: Put all four of the zeros after the 8

200 x 400= 80,000

Practicing these fast math tricks can help both students and teachers improve their math skills and become secure in their knowledge of mathematics—and unafraid to work with numbers in the future.

You may also like to read

• Research-Based Math Teaching Strategies
• Tips in Teaching a Hands-On Math Curriculum
• 5 Tips to Help Get Students Engaged in High School Math
• 3 Tips for Running an Elementary School Math Workshop
• Seven Everyday Online Math Resources for Teachers
• Three Tips for Developing Elementary Math Tests

Categorized as: Tips for Teachers and Classroom Resources

Tagged as: Math and Science ,  Mathematics

• Master's in Trauma-Informed Education and Car...
• Online Associate's Degree Programs in Educati...
• 2020 Civics Engagement & Resilience: Tools fo...

How to Improve Problem-Solving Skills: Mathematics and Critical Thinking

In today’s rapidly changing world, problem-solving has become a quintessential skill. When we discuss the topic, it’s natural to ask, “What is problem-solving?” and “How can we enhance this skill, particularly in children?” The discipline of mathematics offers a rich platform to explore these questions. Through math, not only do we delve into numbers and equations, but we also explore how to improve problem-solving skills and how to develop critical thinking skills in math. Let’s embark on this enlightening journey together.

What is Problem-Solving?

At its core, problem-solving involves identifying a challenge and finding a solution. But it’s not always as straightforward as it sounds. So, what is problem-solving? True problem-solving requires a combination of creative thinking and logical reasoning. Mathematics, in many ways, embodies this blend. When a student approaches a math problem, they must discern the issue at hand, consider various methods to tackle it, and then systematically execute their chosen strategy.

But what is problem-solving in a broader context? It’s a life skill. Whether we’re deciding the best route to a destination, determining how to save for a big purchase, or even figuring out how to fix a broken appliance, we’re using problem-solving.

How to Develop Critical Thinking Skills in Math

Critical thinking goes hand in hand with problem-solving. But exactly how to develop critical thinking skills in math might not be immediately obvious. Here are a few strategies:

• Contextual Learning: Teaching math within a story or real-life scenario makes it relevant. When students see math as a tool to navigate the world around them, they naturally begin to think critically about solutions.
• Open-ended Questions: Instead of merely seeking the “right” answer, encourage students to explain their thought processes. This nudges them to think deeply about their approach.
• Group Discussions: Collaborative learning can foster different perspectives, prompting students to consider multiple ways to solve a problem.
• Challenging Problems: Occasionally introducing problems that are a bit beyond a student’s current skill level can stimulate critical thinking. They will have to stretch their understanding and think outside the box.

What are the Six Basic Steps of the Problem-Solving Process?

Understanding how to improve problem-solving skills often comes down to familiarizing oneself with the systematic approach to challenges. So, what are the six basic steps of the problem-solving process?

• Identification: Recognize and define the problem.
• Analysis: Understand the problem’s intricacies and nuances.
• Generation of Alternatives: Think of different ways to approach the challenge.
• Decision Making: Choose the most suitable method to address the problem.
• Implementation: Put the chosen solution into action.
• Evaluation: Reflect on the solution’s effectiveness and learn from the outcome.

By embedding these steps into mathematical education, we provide students with a structured framework. When they wonder about how to improve problem-solving skills or how to develop critical thinking skills in math, they can revert to this process, refining their approach with each new challenge.

Making Math Fun and Relevant

At Wonder Math, we believe that the key to developing robust problem-solving skills lies in making math enjoyable and pertinent. When students see math not just as numbers on a page but as a captivating story or a real-world problem to be solved, their engagement skyrockets. And with heightened engagement comes enhanced understanding.

As educators and parents, it’s crucial to continuously ask ourselves: how can we demonstrate to our children what problem-solving is? How can we best teach them how to develop critical thinking skills in math? And how can we instill in them an understanding of the six basic steps of the problem-solving process?

The answer, we believe, lies in active learning, contextual teaching, and a genuine passion for the beauty of mathematics.

The Underlying Beauty of Mathematics

Often, people perceive mathematics as a rigid discipline confined to numbers and formulas. However, this is a limited view. Math, in essence, is a language that describes patterns, relationships, and structures. It’s a medium through which we can communicate complex ideas, describe our universe, and solve intricate problems. Understanding this deeper beauty of math can further emphasize how to develop critical thinking skills in math.

Why Mathematics is the Ideal Playground for Problem-Solving

Math provides endless opportunities for problem-solving. From basic arithmetic puzzles to advanced calculus challenges, every math problem offers a chance to hone our problem-solving skills. But why is mathematics so effective in this regard?

• Structured Challenges: Mathematics presents problems in a structured manner, allowing learners to systematically break them down. This format mimics real-world scenarios where understanding the structure of a challenge can be half the battle.
• Multiple Approaches: Most math problems can be approached in various ways . This teaches learners flexibility in thinking and the ability to view a single issue from multiple angles.
• Immediate Feedback: Unlike many real-world problems where solutions might take time to show results, in math, students often get immediate feedback. They can quickly gauge if their approach works or if they need to rethink their strategy.

Enhancing the Learning Environment

To genuinely harness the power of mathematics in developing problem-solving skills, the learning environment plays a crucial role. A student who is afraid of making mistakes will hesitate to try out different approaches, stunting their critical thinking growth.

However, in a nurturing, supportive environment where mistakes are seen as learning opportunities, students thrive. They become more willing to take risks, try unconventional solutions, and learn from missteps. This mindset, where failure is not feared but embraced as a part of the learning journey, is pivotal for developing robust problem-solving skills.

Incorporating Technology

In our digital age, technology offers innovative ways to explore math. Interactive apps and online platforms can provide dynamic problem-solving scenarios, making the process even more engaging. These tools can simulate real-world challenges, allowing students to apply their math skills in diverse contexts, further answering the question of how to improve problem-solving skills.

More than Numbers 

In summary, mathematics is more than just numbers and formulas—it’s a world filled with challenges, patterns, and beauty. By understanding its depth and leveraging its structured nature, we can provide learners with the perfect platform to develop critical thinking and problem-solving skills. The key lies in blending traditional techniques with modern tools, creating a holistic learning environment that fosters growth, curiosity, and a lifelong love for learning.

Join us on this transformative journey at Wonder Math. Let’s make math an adventure, teaching our children not just numbers and equations, but also how to improve problem-solving skills and navigate the world with confidence. Enroll your child today and witness the magic of mathematics unfold before your eyes!

FAQ: Mathematics and Critical Thinking

1. what is problem-solving in the context of mathematics.

Problem-solving in mathematics refers to the process of identifying a mathematical challenge and systematically working through methods and strategies to find a solution.

2. Why is math considered a good avenue for developing problem-solving skills?

Mathematics provides structured challenges and allows for multiple approaches to find solutions. This promotes flexibility in thinking and encourages learners to view problems from various angles.

3. How does contextual learning enhance problem-solving abilities?

By teaching math within a story or real-life scenario, it becomes more relevant for the learner. This helps them see math as a tool to navigate real-world challenges , thereby promoting critical thinking.

4. What are the six basic steps of the problem-solving process in math?

The six steps are: Identification, Analysis, Generation of Alternatives, Decision Making, Implementation, and Evaluation.

5. How can parents support their children in developing mathematical problem-solving skills?

Parents can provide real-life contexts for math problems , encourage open discussions about different methods, and ensure a supportive environment where mistakes are seen as learning opportunities.

6. Are there any tools or apps that can help in enhancing problem-solving skills in math?

Yes, there are various interactive apps and online platforms designed specifically for math learning. These tools provide dynamic problem-solving scenarios and simulate real-world challenges, making the learning process engaging.

7. How does group discussion foster critical thinking in math?

Group discussions allow students to hear different perspectives and approaches to a problem. This can challenge their own understanding and push them to think about alternative methods.

8. Is it necessary to always follow the six steps of the problem-solving process sequentially?

While the six steps provide a structured approach, real-life problem-solving can sometimes be more fluid. It’s beneficial to know the steps, but adaptability and responsiveness to the situation are also crucial.

9. How does Wonder Math incorporate active learning in teaching mathematics?

Wonder Math integrates mathematics within engaging stories and real-world scenarios, making it fun and relevant. This active learning approach ensures that students are not just passive recipients but active participants in the learning process.

10. What if my child finds a math problem too challenging and becomes demotivated?

It’s essential to create a supportive environment where challenges are seen as growth opportunities. Remind them that every problem is a chance to learn, and it’s okay to seek help or approach it differently.

Related posts

Summer Math Programs: How They Can Prevent Learning Loss in Young Students

As summer approaches, parents and educators alike turn their attention to how they can support young learners during the break. Summer is a time for relaxation, fun, and travel, yet it’s also a critical period when learning loss can occur. This phenomenon, often referred to as the “summer slide,” impacts students’ progress, especially in foundational subjects like mathematics. It’s reported…

Math Programs 101: What Every Parent Should Know When Looking For A Math Program

  As a parent, you know that a solid foundation in mathematics is crucial for your child’s success, both in school and in life. But with so many math programs and math help services out there, how do you choose the right one? Whether you’re considering Outschool classes, searching for “math tutoring near me,” or exploring tutoring services online, understanding…

📬 Sign Up for Our Amazing Newsletter!

Writing result-oriented ad copy is difficult, as it must appeal to, entice, and convince consumers to take action.

How to Work Through Hard Math Problems

parent of one of our students wrote today about his daughter’s occasional frustration with the difficulty of some of the problems in our courses. She does fantastic work in our courses , and was easily among the very top students in the class she took with me, and yet she still occasionally hits problems that she can’t solve.

Moreover, she has access to an excellent math teacher in her school who sometimes can’t help her get past these problems, either. (This is no slight to him—I have students bring me problems I can’t solve, too!) Her question: “Why does it have to be so hard?”

The Case for Doing Hard Things

We ask hard questions because so many of the problems worth solving in life are hard. If they were easy, someone else would have solved them before you got to them. This is why college classes at top-tier universities have tests on which nearly no one clears 70%, much less gets a perfect score. They’re training future researchers, and the whole point of research is to find and answer questions that have never been solved. You can’t learn how to do that without fighting with problems you can’t solve. If you are consistently getting every problem in a class correct, you shouldn’t be too happy — it means you aren’t learning efficiently enough. You need to find a harder class.

The problem with not being challenged sufficiently goes well beyond not learning math (or whatever) as quickly as you can. I think a lot of what we do at AoPS is preparing students for challenges well outside mathematics. The same sort of strategies that go into solving very difficult math problems can be used to tackle a great many problems. I believe we’re teaching students how to think, how to approach difficult problems, and that math happens to be the best way to do so for many people.

The first step in dealing with difficult problems is to accept and understand their importance. Don’t duck them. They will teach you a lot more than a worksheet full of easy problems. Brilliant “Aha!” moments almost always spring from minds cultivated by long periods of frustration. But without that frustration, those brilliant ideas never arise.

Strategies for Difficult Math Problems — and Beyond

Here are a few strategies for dealing with hard problems, and the frustration that comes with them:

Do something . Yeah, the problem is hard. Yeah, you have no idea what to do to solve it. At some point you have to stop staring and start trying stuff. Most of it won’t work. Accept that a lot of your effort will appear to have been wasted. But there’s a chance that one of your stabs will hit something, and even if it doesn’t, the effort may prepare your mind for the winning idea when the time comes.

We started developing an elementary school curriculum months and months before we had the idea that became Beast Academy . Our lead curriculum developer wrote 100–200 pages of content, dreaming up lots of different styles and approaches we might use. Not a one of those pages will be in the final work, but they spurred a great many ideas for content we will use. Perhaps more importantly, it prepared us so that when we finally hit upon the Beast Academy idea, we were confident enough to pursue it.

Simplify the problem . Try smaller numbers and special cases. Remove restrictions. Or add restrictions. Set your sights a little lower, then raise them once you tackle the simpler problem.

Reflect on successes . You’ve solved lots of problems. Some of them were even hard problems! How did you do it? Start with problems that are similar to the one you face, but also think about others that have nothing to do with your current problem. Think about the strategies you used to solve those problems, and you might just stumble on the solution.

A few months ago, I was playing around with some Project Euler problems, and I came upon a problem that (eventually) boiled down to generating integer solutions to c ² = a ² + b ² + ab in an efficient manner. Number theory is not my strength, but my path to the solution was to recall first the method for generating Pythagorean triples. Then, I thought about how to generate that method, and the path to the solution became clear. (I’m guessing some of our more mathematically advanced readers have so internalized the solution process for this type of Diophantine equation that you don’t have to travel with Pythagoras to get there!)

Focus on what you haven’t used yet . Many problems (particularly geometry problems) have a lot of moving parts. Look back at the problem, and the discoveries you have made so far and ask yourself: “What haven’t I used yet in any constructive way?” The answer to that question is often the key to your next step.

Work backwards . This is particularly useful when trying to discover proofs. Instead of starting from what you know and working towards what you want, start from what you want, and ask yourself what you need to get there.

Ask for help . This is hard for many outstanding students. You’re so used to getting everything right, to being the one everyone else asks, that it’s hard to admit you need help. When I first got to the Mathematical Olympiad Program (MOP) my sophomore year, I was in way over my head. I understood very little of anything that happened in class. I asked for help from the professor once — it was very hard to get up the courage to do so. I didn’t understand anything he told me during the 15 minutes he worked privately with me. I just couldn’t admit it and ask for more help, so I stopped asking. I could have learned much, much more had I just been more willing to admit to people that I just didn’t understand. (This is part of why our classes now have a feature that allows students to ask questions anonymously.) Get over it. You will get stuck. You will need help. And if you ask for it, you’ll get much farther than if you don’t.

Start early . This doesn’t help much with timed tests, but with the longer-range assignments that are parts of college and of life, it’s essential. Don’t wait until the last minute — hard problems are hard enough without having to deal with time pressure. Moreover, complex ideas take a long time to understand fully. The people you know who seem wicked smart, and who seem to come up with ideas much faster than you possibly could, are often people who have simply thought about the issues for much longer than you have. I used this strategy throughout college to great success — in the first few weeks of each semester, I worked far ahead in all of my classes. Therefore, by the end of the semester, I had been thinking about the key ideas for a lot longer than most of my classmates, making the exams and such at the end of the course a lot easier.

Take a break . Get away from the problem for a bit. When you come back to it, you may find that you haven’t entirely gotten away from the problem at all — the background processes of your brain have continued plugging away, and you’ll find yourself a lot closer to the solution. Of course, it’s a lot easier to take a break if you start early.

Start over . Put all your earlier work aside, get a fresh sheet of paper, and try to start from scratch. Your other work will still be there if you want to draw from it later, and it may have prepared you to take advantage of insights you make in your second go-round.

Give up . You won’t solve them all. At some point, it’s time to cut your losses and move on. This is especially true when you’re in training, and trying to learn new things. A single difficult problem is usually going to teach you more in the first hour or two than it will in the next six, and there are a lot more problems to learn from. So, set yourself a time limit, and if you’re still hopelessly stuck at the end of it, then read the solutions and move on.

Be introspective . If you do give up and read the solution, then read it actively, not passively. As you read it, think about what clues in the problem could have led you to this solution. Think about what you did wrong in your investigation. If there are math facts in the solution that you don’t understand, then go investigate. I was completely befuddled the first time I saw a bunch of stuff about “mod”s in an olympiad solution — we didn’t have the internet then, so I couldn’t easily find out how straightforward modular arithmetic is! You have the internet now, so you have no excuse. If you did solve the problem, don’t just pat yourself on the back. Think about the key steps you made, and what the signs were to try them. Think about the blind alleys you explored en route to the solution, and how you could have avoided them. Those lessons will serve you well later.

Come back . If you gave up and looked at the solutions, then come back and try the problem again a few weeks later. If you don’t have any solutions to look at, keep the problem alive. Store it away on paper or in your mind.

Richard Feynman once wrote that he would keep four or five problems active in the back of his mind. Whenever he heard a new strategy or technique, he would quickly run through his problems and see if he could use it to solve one of his problems. He credits this practice for some of the anecdotes that gave Feynman such a reputation for being a genius. It’s further evidence that being a genius is an awful lot about effort, preparation, and being comfortable with challenges.

Subscribe for news, tips and advice from AoPS

Richard rusczyk, related articles, the math of winning wordle: from letter distribution to first-word strategies, stop making silly mistakes, more articles, managing academic expectations, with charlene wang, the future of digital education, with matthew rascoff, the math of game shows: who wants to be a millionaire, navigating student-teacher conflict, with dr. stephanie mihalas, the math of adele’s album releases: can a hidden pattern help predict when she’ll release next, don’t fall into the calculus trap, more episodes, sapienship, with dr. jim clarke, wonder, with dr. frank keil, learning stem through fiction, with dr. pamela cosman, edtech at-home, with monica burns, learned helplessness, with vida john, receive weekly podcast summaries right to your inbox, get weekly podcast summaries/takeaways.

By clicking this button, I consent to receiving AoPS communications, and confirm that I am over 13, or under 13 and already a member of the Art of Problem Solving community. View our Privacy Policy .

Aops programs

• Solve equations and inequalities
• Simplify expressions
• Factor polynomials
• Graph equations and inequalities
• Advanced solvers
• All solvers
• Arithmetics
• Determinant
• Percentages
• Scientific Notation
• Inequalities

Math Topics

More solvers.

• Add Fractions
• Simplify Fractions

• Skip to main content
• Skip to primary sidebar
• Skip to footer

Additional menu

Khan Academy Blog

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

posted on September 20, 2023

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

Math Learning

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

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

Benefits of Math Learning

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

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

How to Know What Math You Need to Learn

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

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

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

Try Our Free Confidence Boosters

How to learn math.

Learning at School

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

Learning at Home

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

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

Learning Math with the Help of Artificial Intelligence (AI) 

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

Using Khan Academy’s AI Tutor, Khanmigo

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

Get Math Help with Khanmigo Right Now

You can learn anything .

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

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

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

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

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

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

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

Get Khanmigo

The best way to learn and teach with AI is here. Ace the school year with our AI-powered guide, Khanmigo. 

For learners     For teachers     For parents

• Math tutors
• Math classes
• 1st Grade Math Worksheets
• 2nd Grade Math Worksheets
• 3rd Grade Math Worksheets
• 4th Grade Math Worksheets
• 5th Grade Math Worksheets
• 6th Grade Math Worksheets
• 7th Grade Math Worksheets
• 8th Grade Math Worksheets
• Knowledge Base

10 Strategies for Problem-Solving in Math

reviewed by Jo-ann Caballes

Updated on August 21, 2024

It’s not surprising that kids who lack problem-solving skills feel stuck in math class. Students who are behind in problem-solving may have difficulties identifying and carrying out a plan of action to solve a problem. Math strategies for problem-solving allow children to use a range of approaches to work out math problems productively and with ease. This article explores math problem-solving strategies and how kids can use them both in traditional classes and in a virtual classroom. 

What are problem-solving strategies in math?

Problem-solving strategies for math make it easier to tackle math and work up an effective solution. When we face any kind of problem, it’s usually impossible to solve it without carrying out a good plan.In other words, these strategies were designed to make math for kids easier and more manageable. Another great benefit of these strategies is that kids can spend less time cracking math problems. 

Here are some problem-solving methods:

• Drawing a picture or diagram (helps visualize the problem)
• Breaking the problem into smaller parts (to keep track of what has been done)
• Making a table or a list (helps students to organize information)

When children have a toolkit of math problem-solving strategies at hand, it makes it easier for them to excel in math and progress faster. 

How to solve math problems?

To solve math problems, it’s worth having strategies for math problem-solving that include several steps, but it doesn’t necessarily mean they are failproof. They serve as a guide to the solution when it’s difficult to decide where and how to start. Research suggests that breaking down complex problems into smaller stages can reduce cognitive load and make it easier for students to solve problems. Essentially, a suitable strategy can help kids to find the right answers fast. 

Here are 5 math problem-solving strategies for kids:

• Recognize the Problem  
• Work up a Plan  
• Carry Out the Plan  
• Review the Work  
• Reflect and Analyze  

Understanding the Problem 

Understanding the problem is the first step in the journey of solving it. Without doing this, kids won’t be able to address it in any way. In the beginning, it’s important to read the problem carefully and make sure to understand every part of it. Next, when kids know what they are asked to do, they have to write down the information they have and determine what essentially they need to solve. 

Work Out a Plan 

Working out a plan is one of the most important steps to solving math problems. Here, the kid has to choose a good strategy that will help them with a specific math problem. Outline these steps either in mind or on paper.

Carry Out the Plan 

Being methodical at that stage is key. It involves following the plan and performing calculations with the correct operations and rules. Finally, when the work is done, the child can review and show their work to a teacher or tutor.

Review the Work

This is where checking if the answer is correct takes place. If time allows, children and the teacher can choose other methods and try to solve the same problem again with a different approach. 

Reflect and Analyze

This stage is a great opportunity to think about how the problem was solved: did any part cause confusion? Was there a more efficient method? It’s important to let the child know that they can use the insights gained for future reference. 

Ways to solve math problems

The ways to solve math problems for kids are numerous, but it doesn’t mean they all work the same for everybody. For example, some children may find visual strategies work best for them; some prefer acting out the problem using movements. Finding what kind of method or strategy works best for your kid will be extremely beneficial both for school performance and in real-life scenarios where they can apply problem-solving. 

Online tutoring platforms like Brighterly offer personalized assistance, interactive tools, and access to resources that help to determine which strategies are best for your child. Expertise-driven tutors know how to guide your kid so they won’t be stuck with the same fallacies that interfere with effective problem-solving. For example, tutors can assist kids with drawing a diagram, acting a problem out with movement, or working backward. All of these ways are highly effective, especially with a trusted supporter by your side. 

Your child will fall in love with math 
after just one lesson!

Choose 1:1 online math tutoring.

What are 10 strategies for solving math problems?

There are plenty of different problem-solving strategies for mathematical problems to help kids discover answers. Let’s explore 10 popular problem-solving strategies:

Understand the Problem

Figuring out the nature of math problems is the key to solving them. Kids need to identify what kind of issue this is (fraction problem, word problem, quadratic equation, etc.) and work up a plan to solve it. 

Guess and Check

With this approach, kids simply need to keep guessing until they get the answer right. While this approach may seem irrelevant, it illustrates what the kid’s thinking process is. 

Work It Out

This method encourages students to write down or say their problem-solving process instead of going straight to solving it without preparation. This minimizes the probability of mistakes. 

Work Backwords 

Working backward is a great problem-solving strategy to acquire a fresh perspective. It requires one to come up with a probable solution and decide which step to take to come to that solution. 

A visual representation of a math problem may help kids to understand it in full. One way to visualize a problem is to use a blank piece of paper and draw a picture, including all of the aspects of the issue. 

Find a Pattern 

By helping students see patterns in math problems, we help them to extract and list relevant details. This method is very effective in learning shapes and other topics that need repetition. 

It may be self-explanatory, but it’s quite helpful to ask, “What are some possible solutions to this issue?”. By giving kids time to think and reflect, we help them to develop creative and critical thinking.

Draw a Picture or Diagram

Instead of drawing the math problem yourself, ask the kid to draw it themselves. They can draw pictures of the ideas they have been taught to help them remember the concepts better.

Trial and Error Method

Not knowing clear formulas or instructions, kids won’t be able to solve anything. Ask them to make a list of possible answers based on rules they already know. Let them learn by making mistakes and trying to find a better solution. 

Review Answers with Peers

It’s so fun to solve problems alongside your peers. Kids can review their answers together and share ideas on how each problem can be solved. 

Help your kid achieve their full math potential

The best Brighterly tutors are ready to help with that.

Math problem-solving strategies for elementary students

5 problem-solving strategies for elementary students include:

Using Simple Language 

Ask students to explain the problem in their own words to make sure they understand the problem correctly.

Using Visuals and Manipulatives 

Using drawing and manipulatives like counters, blocks, or beads can help students grasp the issue faster. 

Simplifying the Problem

Breaking the problem into a step-by-step process and smaller, manageable steps will allow students to find the solution faster. 

Looking for Patterns 

Identifying patterns in numbers and operations is a great strategy to help students gain more confidence along the way. 

Using Stories

Turning math problems into stories will surely engage youngsters and make them participate more actively. 

To recap, students need to have effective math problem-solving strategies up their sleeves. Not only does it help them in the classroom, but it’s also an essential skill for real-life situations. Productive problem-solving strategies for math vary depending on the grade. But what they have in common is that kids have to know how to break the issue into smaller parts and apply critical and creative thinking to solve it. 

If you want your kid to learn how to thrive in STEM and apply problem-solving strategies to both math and real life, book a free demo lesson with Brighterly today! Make your child excited about math!

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.

As adults, we take numbers for granted, but preschoolers and kindergartners have no idea what these symbols mean. Yet, we often demand instant understanding and flawless performance when we start teaching numbers to our children. If you don’t have a clue about how to teach numbers for kids, browse no more. You will get four […]

May 19, 2022

Teaching strategies aren’t something that is set in stone and continue to evolve every year. Even though traditional educational strategies like teachers teaching in front of the classroom seemed to work for decades with little to no adjustments, the digital age has brought along numerous challenges. Teaching methods for kids require new strategies, so educators […]

Math homework help for grade 1-8

Get the homework support your child needs!

Choose kid's grade

Math Program Boost Your Child's Math Abilities! Ideal for 1st-8th Graders, Perfectly Synced with School Curriculum!

After-school math classes for grades 1-8

Personalized lessons for kids & teens.

Our after-school math curriculum includes

Related posts.

15 Thanksgiving Math Activities

On Thanksgiving, we pause to reflect on the blessings in our lives, including our loved ones, our accomplishments, and the gift of life itself. So, why not set up math-themed games for your little one while shopping for the Thanksgiving turkey, mashed potatoes, and pumpkins? In this post, we’ll review some fun math things kids […]

Apr 05, 2022

30 Basic Classroom Rules For Kids

Picture this: you’re in the middle of explaining trigonometry, working hard to keep everyone engaged, when suddenly, Casey decides to break Mason’s ruler. Chaos ensues, and before you know it, your entire class is fixated on this mini-drama, forgetting all about trigonometry. This is just one example, but let’s be honest; there are plenty more […]

Nov 06, 2023

11 Mental Math Practice Tips

Fans of AI tools often call mental math an outdated practice. However, it is still important both for kids and adults since it gives you more than just easy device-less calculations. This practice sharpens cognitive skills and boosts overall confidence. That’s why you can still see dozens of topics like “How to improve my mental […]

May 06, 2022

Want your kid to excel in math?

Kid’s grade

We use cookies to help give you the best service possible. If you continue to use the website we will understand that you consent to the Terms and Conditions. These cookies are safe and secure. We will not share your history logs with third parties. Learn More

• PRO Courses Guides New Tech Help Pro Expert Videos About wikiHow Pro Upgrade Sign In
• EDIT Edit this Article
• EXPLORE Tech Help Pro About Us Random Article Quizzes Request a New Article Community Dashboard This Or That Game Happiness Hub Popular Categories Arts and Entertainment Artwork Books Movies Computers and Electronics Computers Phone Skills Technology Hacks Health Men's Health Mental Health Women's Health Relationships Dating Love Relationship Issues Hobbies and Crafts Crafts Drawing Games Education & Communication Communication Skills Personal Development Studying Personal Care and Style Fashion Hair Care Personal Hygiene Youth Personal Care School Stuff Dating All Categories Arts and Entertainment Finance and Business Home and Garden Relationship Quizzes Cars & Other Vehicles Food and Entertaining Personal Care and Style Sports and Fitness Computers and Electronics Health Pets and Animals Travel Education & Communication Hobbies and Crafts Philosophy and Religion Work World Family Life Holidays and Traditions Relationships Youth
• Browse Articles
• Learn Something New
• Quizzes Hot
• Happiness Hub
• This Or That Game
• Train Your Brain
• Explore More
• Support wikiHow
• About wikiHow
• Log in / Sign up
• Education and Communications
• Mathematics

How to Improve Mental Math Skills

Last Updated: August 2, 2024 Approved

This article was co-authored by Daron Cam . Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College. There are 8 references cited in this article, which can be found at the bottom of the page. wikiHow marks an article as reader-approved once it receives enough positive feedback. In this case, 84% of readers who voted found the article helpful, earning it our reader-approved status. This article has been viewed 336,815 times.

Eventually, you'll find yourself in a situation where you'll have to solve a math problem without a calculator. Trying to imagine a pen and paper in your head often doesn't help much. Fortunately there are faster and easier ways to do calculations in your head—and they often break down a problem in a way that makes more sense than what you learned in school. Whether you're a stressed-out student or a math wizard looking for even faster tricks, there's something for everyone to learn.

Break addition and subtraction problems into parts.

• 712 + 281 → "700 + 200," "10 + 80," and "2 + 1"
• 700 + 200 = 9 00, then 10 + 80 = 9 0, then 2 + 1 = 3
• 900 + 90 + 3 = 993 .
• Thinking in "hundreds" or "tens" instead of single digits will make it easier to keep track when digits sum to more than ten. For example, for 37 + 45, think "30 + 40 = 70" and "7 + 5 = 12". Then add 70 + 12 to get 82.

Change the problem to make round numbers.

• Addition : For 596 + 380 , realize that you can add 4 to 596 to round it to 600, then add 600 + 380 to get 980. Undo the rounding by subtracting 4 from 980 to get 976 .
• Subtraction : For 815 - 521 , break it up into 800 - 500, 10 - 20, and 5 - 1. To turn the awkward "10 - 20" into "20 - 20", add 10 to 815 to get 825. Now solve to get 304, then undo the rounding by subtracting 10 to get 294 .
• Multiplication : For 38 x 3 , you can add 2 to 38 to make the problem 40 x 3, which is 120. Since the 2 you added got multiplied by three, you need to undo the rounding by subtracting 2 x 3 = 6 at the end to get 120 - 6 = 114 .

Learn to add many numbers at once.

• For example, 7 + 4 + 9 + 13 + 6 + 51 can be reorganized to (7 + 13) + (9 + 51) + (6 + 4) = 20 + 60 + 10 = 90.

Multiply from left to right.

• For 453 x 4 , start with 400 x 4 = 1600, then 50 x 4 = 200, then 3 x 4 = 12. Add them all together to get 1812 .
• If both numbers have more than one digit, you can break it into parts. Each digit has to multiply with each other digit, so it can be tough to keep track of it all. 34 x 12 = (34 x 10) + (34 x 2) , which you can break down further into (30 x 10) + (4 x 10) + (30 x 2) + (4 x 2) = 300 + 40 + 60 + 8 = 408 .

Try a fast multiplication trick best for numbers 11 through 19.

• Let's look at numbers close to 10, like 13 x 15 . Subtract 10 from the second number, then add your answer to the first: 15 - 10 = 5, and 13 + 5 = 18.
• Multiply your answer by ten: 18 x 10 = 180.
• Next, subtract ten from both sides and multiply the results: 3 x 5 = 15.
• Add your two answers together to get the final answer: 180 + 15 = 195 .
• Careful with smaller numbers! For 13 x 8, you start with "8 - 10 = -2", then "13 + -2 = 11". If it's hard to work with negative numbers in your head, try a different method for problems like this.
• For larger numbers, it will be easier to use a "base number" like 20 or 30 instead of 10. If you try this, make sure you use that number everywhere that 10 is used above. [3] X Research source For example, for 21 x 24, you start by adding 21 + 4 to get 25. Now multiply 25 by 20 (instead of ten) to get 500, and add 1 x 4 = 4 to get 504.

Simplify problems with numbers ending in zero.

• Addition : If all numbers have zeroes at the end, you can ignore the zeroes they have in common and restore them at the end. 85 0 + 12 0 → 85 + 12 = 97, then restore the shared zero: 97 0 .
• Subtraction works the same way: 10 00 - 7 00 → 10 - 7 = 3, then restore the two shared zeroes to get 3 00 . Notice that you can only remove the two zeroes the numbers have in common, and must keep the third zero in 1000.
• Multiplication : ignore all the zeroes, then restore each one individually. 3 000 x 5 0 → 3 x 5 = 15, then restore all four zeroes to get 15 0 , 00 0 .
• Division : you can remove all shared zeroes and the answer will be the same. 60, 000 ÷ 12, 000 = 60 ÷ 12 = 5 . Don't add any zeroes back on.

Easily multiply by 4, 5, 8, or 16.

• To multiply by 5, instead multiply by 10, then divide by 2.
• To multiply by 4, instead double the number, then double it again.
• For 8, 16, 32, or even higher powers of two, just keep doubling. For example, 13 x 8 = 13 x 2 x 2 x 2, so double 13 three times: 13 → 26 → 52 → 104 .

Memorize the 11s trick.

• What is 7 2 x 11?
• Add the two digits together: 7 + 2 = 9.
• Put the answer in between the original digits: 7 2 x 11 = 7 9 2 .
• If the sum is more than 10, place only the final digit and carry the one: 5 7 x 11 = 6 2 7 , because 5 + 7 = 12. The 2 goes in the middle and the 1 gets added to the 5 to make 6.

Turn percentages into easier problems.

• 79% of 10 is the same as 10% of 79. This is true of any two numbers. If you can't find the answer to a percentage problem, try switching it around.
• To find 10% of a number, move the decimal one place to the left (10% of 65 is 6.5). To find 1% of a number, move the decimal two places to the left (1% of 65 is 0.65).
• Use these rules for 10% and 1% to help you with more difficult percentages. For example, 5% is ½ of 10%, so 5% of 80 = (10% of 80) x ½ = 8 x ½ = 4 .
• Break percentages into easier parts: 30% of 900 = (10% of 900) x 3 = 90 x 3 = 270 .

Memorize advanced multiplication shortcuts for specific problems.

• For problems like 84 x 86 , where the tens place is the same and the ones place digits sum to exactly 10, the first digits of the answer are (8 + 1) x 8 = 72 and the last digits are 4 x 6 = 24, for an answer of 7224 . That is, for a problem AB x AC, if B + C = 10, the answer starts with A(A+1) and ends with BC. This also works for larger numbers if all digits besides the ones place are identical. [6] X Research source
• You can rewrite the powers of five (5, 25, 125, 625, ...) as powers of 10 divided by an integer (10 / 2, 100 / 4, 1000 / 8, 10000 / 16, ...). [7] X Research source So 88 x 125 becomes 88 x 1000 ÷ 8 = 88000 ÷ 8 = 11000 .

Memorize squares charts.

• Memorize the squares from 1 to 20 (or higher, if you're ambitious). (That is, 1 x 1 = 1; 2 x 2 = 4; 3 x 3 = 9, and so on.)
• To multiply two numbers, first find their average (the number exactly between them). For example, the average of 18 and 14 is 16.
• Square this answer. Once you've memorized the squares chart, you'll know that 16 x 16 is 256.
• Next, look at the difference between the original numbers and their average: 18 - 16 = 2. (Always use a positive number here.)
• Square this number as well: 2 x 2 = 4.
• To get your final answer, take the first square and subtract the second: 256 - 4 = 252 .

Find useful ways to practice your mental math.

• Flashcards are great for memorizing multiplication and division tables, or for getting used to tricks for specific kinds of problems. Write the problem on one side and the answer on the other, and quiz yourself daily until you get them all right.
• Online math quizzes are another way to test your ability. Look for a well-reviewed app or website made by an educational program.
• Practice in everyday situations. You could add together the total of items you buy as you shop, or multiply the gas cost per volume by your car's tank size to find the total cost. The more of a habit this becomes, the easier it will be.

Joseph Meyer

Exercise your mental math muscles. Improve your math skills by solving daily math problems without using calculators, paper, or counting aids. By solely using your mind and getting into math discussions with your classmates, you will refine your skills and discover new approaches to problem-solving.

Practice Problems and Answers

Community Q&A

• In the real world, you don't always need to know the exact answer. If you're at the grocery store and trying to add 7.07 + 8.95 + 10.09, you could round to the closest whole numbers and estimate that the total is roughly 7 + 9 + 10 = 26. Thanks Helpful 13 Not Helpful 3
• Some people find it easier to think in money than abstract numbers. Instead of 100 - 55, try thinking of a dollar minus a 50¢ coin and a 5¢ coin. Thanks Helpful 7 Not Helpful 9

You Might Also Like

• ↑ http://gizmodo.com/10-tips-to-improve-your-mental-math-ability-1792597814
• ↑ https://www.youtube.com/watch?v=Rgw9Ik5ZGaY
• ↑ https://www.youtube.com/watch?v=SV1dC1KAl_U
• ↑ https://www.youtube.com/watch?v=1JW9BA57aR8
• ↑ http://www.wired.co.uk/article/master-mental-maths
• ↑ https://www.youtube.com/watch?v=YCBTw8KAqkw
• ↑ https://www.scientificamerican.com/article/5-tips-faster-mental-multiplication/
• ↑ Daron Cam. Academic Tutor. Expert Interview. 29 May 2020.

About This Article

One way to improve your mental math skills is to memorize your multiplication and division tables, so you always have the answer to those problems instantly. If you have trouble memorizing the numbers, try creating your own flash cards with blank notecards and asking a friend to help you practice. Another good way to practice your mental math skills is to add up the prices of your items when you’re at the store, and check to make sure you added correctly once the cashier rings you up. You can also try downloading a mental math app like Luminosity to keep your math skills sharp. To learn how to visualize an equation in your head, read on! Did this summary help you? Yes No

• Send fan mail to authors

Reader Success Stories

May 23, 2022

Did this article help you?

Feb 12, 2023

Apr 11, 2023

Ariel Arnaiz

Nov 22, 2022

Deborah Yeary

Jun 8, 2018

Featured Articles

Trending Articles

Solving Word Questions

With LOTS of examples!

In Algebra we often have word questions like:

Example: Sam and Alex play tennis.

On the weekend Sam played 4 more games than Alex did, and together they played 12 games.

How many games did Alex play?

How do we solve them?

The trick is to break the solution into two parts:

Turn the English into Algebra.

Then use Algebra to solve.

Turning English into Algebra

To turn the English into Algebra it helps to:

• Read the whole thing first
• Do a sketch if possible
• Assign letters for the values
• Find or work out formulas

You should also write down what is actually being asked for , so you know where you are going and when you have arrived!

Also look for key words:

Thinking Clearly

Some wording can be tricky, making it hard to think "the right way around", such as:

Example: Sam has 2 dollars less than Alex. How do we write this as an equation?

• Let S = dollars Sam has
• Let A = dollars Alex has

Now ... is that: S − 2 = A

or should it be: S = A − 2

or should it be: S = 2 − A

The correct answer is S = A − 2

( S − 2 = A is a common mistake, as the question is written "Sam ... 2 less ... Alex")

Example: on our street there are twice as many dogs as cats. How do we write this as an equation?

• Let D = number of dogs
• Let C = number of cats

Now ... is that: 2D = C

or should it be: D = 2C

Think carefully now!

The correct answer is D = 2C

( 2D = C is a common mistake, as the question is written "twice ... dogs ... cats")

Let's start with a really simple example so we see how it's done:

Example: A rectangular garden is 12m by 5m, what is its area ?

Turn the English into Algebra:

• Use w for width of rectangle: w = 12m
• Use h for height of rectangle: h = 5m

Formula for Area of a Rectangle : A = w × h

We are being asked for the Area.

A = w × h = 12 × 5 = 60 m 2

The area is 60 square meters .

Now let's try the example from the top of the page:

Example: Sam and Alex play Tennis. On the weekend Sam played 4 more games than Alex did, and together they played 12 games. How many games did Alex play?

• Use S for how many games Sam played
• Use A for how many games Alex played

We know that Sam played 4 more games than Alex, so: S = A + 4

And we know that together they played 12 games: S + A = 12

We are being asked for how many games Alex played: A

Which means that Alex played 4 games of tennis.

Check: Sam played 4 more games than Alex, so Sam played 8 games. Together they played 8 + 4 = 12 games. Yes!

A slightly harder example:

Example: Alex and Sam also build tables. Together they make 10 tables in 12 days. Alex working alone can make 10 in 30 days. How long would it take Sam working alone to make 10 tables?

• Use a for Alex's work rate
• Use s for Sam's work rate

12 days of Alex and Sam is 10 tables, so: 12a + 12s = 10

30 days of Alex alone is also 10 tables: 30a = 10

We are being asked how long it would take Sam to make 10 tables.

30a = 10 , so Alex's rate (tables per day) is: a = 10/30 = 1/3

Which means that Sam's rate is half a table a day (faster than Alex!)

So 10 tables would take Sam just 20 days.

Should Sam be paid more I wonder?

And another "substitution" example:

Example: Jenna is training hard to qualify for the National Games. She has a regular weekly routine, training for five hours a day on some days and 3 hours a day on the other days. She trains altogether 27 hours in a seven day week. On how many days does she train for five hours?

• The number of "5 hour" days: d
• The number of "3 hour" days: e

We know there are seven days in the week, so: d + e = 7

And she trains 27 hours in a week, with d 5 hour days and e 3 hour days: 5d + 3e = 27

We are being asked for how many days she trains for 5 hours: d

The number of "5 hour" days is 3

Check : She trains for 5 hours on 3 days a week, so she must train for 3 hours a day on the other 4 days of the week.

3 × 5 hours = 15 hours, plus 4 × 3 hours = 12 hours gives a total of 27 hours

Some examples from Geometry:

Example: A circle has an area of 12 mm 2 , what is its radius?

• Use A for Area: A = 12 mm 2
• Use r for radius

And the formula for Area is: A = π r 2

We are being asked for the radius.

We need to rearrange the formula to find the area

Example: A cube has a volume of 125 mm 3 , what is its surface area?

Make a quick sketch:

• Use V for Volume
• Use A for Area
• Use s for side length of cube
• Volume of a cube: V = s 3
• Surface area of a cube: A = 6s 2

We are being asked for the surface area.

First work out s using the volume formula:

Now we can calculate surface area:

An example about Money:

Example: Joel works at the local pizza parlor. When he works overtime he earns 1¼ times the normal rate. One week Joel worked for 40 hours at the normal rate of pay and also worked 12 hours overtime. If Joel earned $660 altogether in that week, what is his normal rate of pay?

• Joel's normal rate of pay: $N per hour
• Joel works for 40 hours at $N per hour = $40N
• When Joel does overtime he earns 1¼ times the normal rate = $1.25N per hour
• Joel works for 12 hours at $1.25N per hour = $(12 × 1¼N) = $15N
• And together he earned $660, so:

$40N + $(12 × 1¼N) = $660

We are being asked for Joel's normal rate of pay $N.

So Joel’s normal rate of pay is $12 per hour

Joel’s normal rate of pay is $12 per hour, so his overtime rate is 1¼ × $12 per hour = $15 per hour. So his normal pay of 40 × $12 = $480, plus his overtime pay of 12 × $15 = $180 gives us a total of $660

More about Money, with these two examples involving Compound Interest

Example: Alex puts $2000 in the bank at an annual compound interest of 11%. How much will it be worth in 3 years?

This is the compound interest formula:

So we will use these letters:

• Present Value PV = $2,000
• Interest Rate (as a decimal): r = 0.11
• Number of Periods: n = 3
• Future Value (the value we want): FV

We are being asked for the Future Value: FV

Example: Roger deposited $1,000 into a savings account. The money earned interest compounded annually at the same rate. After nine years Roger's deposit has grown to $1,551.33 What was the annual rate of interest for the savings account?

The compound interest formula:

• Present Value PV = $1,000
• Interest Rate (the value we want): r
• Number of Periods: n = 9
• Future Value: FV = $1,551.33

We are being asked for the Interest Rate: r

So the annual rate of interest is 5%

Check : $1,000 × (1.05) 9 = $1,000 × 1.55133 = $1,551.33

And an example of a Ratio question:

Example: At the start of the year the ratio of boys to girls in a class is 2 : 1 But now, half a year later, four boys have left the class and there are two new girls. The ratio of boys to girls is now 4 : 3 How many students are there altogether now?

• Number of boys now: b
• Number of girls now: g

The current ratio is 4 : 3

Which can be rearranged to 3b = 4g

At the start of the year there was (b + 4) boys and (g − 2) girls, and the ratio was 2 : 1

b + 4 g − 2 = 2 1

Which can be rearranged to b + 4 = 2(g − 2)

We are being asked for how many students there are altogether now: b + g

There are 12 girls !

And 3b = 4g , so b = 4g/3 = 4 × 12 / 3 = 16 , so there are 16 boys

So there are now 12 girls and 16 boys in the class, making 28 students altogether .

There are now 16 boys and 12 girls, so the ratio of boys to girls is 16 : 12 = 4 : 3 At the start of the year there were 20 boys and 10 girls, so the ratio was 20 : 10 = 2 : 1

And now for some Quadratic Equations :

Example: The product of two consecutive even integers is 168. What are the integers?

Consecutive means one after the other. And they are even , so they could be 2 and 4, or 4 and 6, etc.

We will call the smaller integer n , and so the larger integer must be n+2

And we are told the product (what we get after multiplying) is 168, so we know:

n(n + 2) = 168

We are being asked for the integers

That is a Quadratic Equation , and there are many ways to solve it. Using the Quadratic Equation Solver we get −14 and 12.

Check −14: −14(−14 + 2) = (−14)×(−12) = 168 YES

Check 12: 12(12 + 2) = 12×14 = 168 YES

So there are two solutions: −14 and −12 is one, 12 and 14 is the other.

Note: we could have also tried "guess and check":

• We could try, say, n=10: 10(12) = 120 NO (too small)
• Next we could try n=12: 12(14) = 168 YES

But unless we remember that multiplying two negatives make a positive we might overlook the other solution of (−14)×(−12).

Example: You are an Architect. Your client wants a room twice as long as it is wide. They also want a 3m wide veranda along the long side. Your client has 56 square meters of beautiful marble tiles to cover the whole area. What should the length of the room be?

Let's first make a sketch so we get things right!:

• the length of the room: L
• the width of the room: W
• the total Area including veranda: A
• the width of the room is half its length: W = ½L
• the total area is the (room width + 3) times the length: A = (W+3) × L = 56

We are being asked for the length of the room: L

This is a quadratic equation , there are many ways to solve it, this time let's use factoring :

And so L = 8 or −14

There are two solutions to the quadratic equation, but only one of them is possible since the length of the room cannot be negative!

So the length of the room is 8 m

L = 8, so W = ½L = 4

So the area of the rectangle = (W+3) × L = 7 × 8 = 56

There we are ...

... I hope these examples will help you get the idea of how to handle word questions. Now how about some practice?

Get step-by-step solutions with MathGPT

Upload a screenshot or picture of your question and get instant help from your personal ai math tutor.

Click here to take a picture or select an image

Drag & drop or click here to upload an image of your problem

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.

Your content has been saved!

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.

Over 5 Billion Problems Solved

Step-by-step examples.

• Adding Using Long Addition
• Long Subtraction
• Long Multiplication
• Long Division
• Dividing Using Partial Quotients Division
• Converting Regular to Scientific Notation
• Arranging a List in Order
• Expanded Notation
• Prime or Composite
• Comparing Expressions
• Converting to a Percentage
• Finding the Additive Inverse
• Finding the Multiplicative Inverse
• Reducing Fractions
• Finding the Reciprocal
• Converting to a Decimal
• Converting to a Mixed Number
• Adding Fractions
• Subtracting Fractions
• Multiplying Fractions
• Dividing Fractions
• Converting Ratios to Fractions
• Converting Percents to Decimal
• Converting Percents to Fractions
• Converting the Percent Grade to Degree
• Converting the Degree to Percent Grade
• Finding the Area of a Rectangle
• Finding the Perimeter of a Rectangle
• Finding the Area of a Square
• Finding the Perimeter of a Square
• Finding the Area of a Circle
• Finding the Circumference of a Circle
• Finding the Area of a Triangle
• Finding the Area of a Trapezoid
• Finding the Volume of a Box
• Finding the Volume of a Cylinder
• Finding the Volume of a Cone
• Finding the Volume of a Pyramid
• Finding the Volume of a Sphere
• Finding the Surface Area of a Box
• Finding the Surface Area of a Cylinder
• Finding the Surface Area of a Cone
• Finding the Surface Area of a Pyramid
• Converting to a Fraction
• Simple Exponents
• Prime Factorizations
• Finding the Factors
• Simplifying Fractions
• Converting Grams to Kilograms
• Converting Grams to Pounds
• Converting Grams to Ounces
• Converting Feet to Inches
• Converting to Meters
• Converting Feet to Miles
• Converting Feet to Yards
• Converting to Feet
• Converting to Yards
• Converting Miles to Feet
• Converting Miles to Kilometers
• Converting Miles to Yards
• Converting Kilometers to Miles
• Converting Kilometers to Meters
• Converting Meters to Feet
• Converting Meters to Inches
• Converting Ounces to Grams
• Converting Ounces to Pounds
• Converting Ounces to Tons
• Converting Pounds to Grams
• Converting Pounds to Ounces
• Converting Pounds to Tons
• Converting Yards to Feet
• Converting Yards to Millimeters
• Converting Yards to Inches
• Converting Yards to Miles
• Converting Yards to Meters
• Converting Fahrenheit to Celsius
• Converting Celsius to Fahrenheit
• Finding the Median
• Finding the Mean (Arithmetic)
• Finding the Mode
• Finding the Minimum
• Finding the Maximum
• Finding the Lower or First Quartile
• Finding the Upper or Third Quartile
• Finding the Five Number Summary
• Finding a Point's Quadrant
• Finding the Midpoint of a Line Segment
• Distance Formula
• Arithmetic Operations
• Combining Like Terms
• Determining if the Expression is a Polynomial
• Distributive Property
• Simplifying
• Multiplication
• Polynomial Addition
• Polynomial Subtraction
• Polynomial Multiplication
• Polynomial Division
• Simplifying Expressions
• Evaluate the Expression Using the Given Values
• Multiplying Polynomials Using FOIL
• Identifying Degree
• Operations on Polynomials
• Negative Exponents
• Evaluating Radicals
• Solving by Adding/Subtracting
• Solving by Multiplying/Dividing
• Solving Containing Decimals
• Solving for a Variable
• Solving Linear Equations
• Solving Linear Inequalities
• Finding the Quadratic Constant of Variation
• Converting the Percent Grade to Slope
• Converting the Slope to Percent Grade
• Finding Equations Using Slope-Intercept
• Finding the Slope
• Finding the y Intercept
• Calculating Slope and y-Intercept
• Rewriting in Slope-Intercept Form
• Finding Equations Using the Slope-Intercept Formula
• Finding Equations Using Two Points
• Finding a Perpendicular Line Containing a Given Point
• Finding a Parallel Line Containing a Given Point
• Finding a Parallel Line to the Given Line
• Finding a Perpendicular Line to the Given Line
• Finding Ordered Pair Solutions
• Using a Table of Values to Graph an Equation
• Finding the Equation Using Point-Slope Form
• Finding the Surface Area of a Sphere
• Solving by Graphing
• Finding the LCM of a List of Expressions
• Finding the LCD of a List of Expressions
• Determining if the Number is a Perfect Square
• Finding the Domain
• Evaluating the Difference Quotient
• Solving Using the Square Root Property
• Determining if True
• Finding the Holes in a Graph
• Finding the Common Factors
• Expand a Trinomial with the Trinomial Theorem
• Finding the Start Point Given the Mid and End Points
• Finding the End Point Given the Start and Mid Points
• Finding the Slope and y-Intercept
• Finding the Equation of the Parabola
• Finding the Average Rate of Change
• Finding the Slope of the Perpendicular Line to the Line Through the Two Points
• Rewriting Using Negative Exponents
• Synthetic Division
• Maximum Number of Real Roots/Zeros
• Finding All Possible Roots/Zeros (RRT)
• Finding All Roots with Rational Root Test (RRT)
• Finding the Remainder
• Finding the Remainder Using Long Polynomial Division
• Reordering the Polynomial in Ascending Order
• Reordering the Polynomial in Descending Order
• Finding the Leading Term
• Finding the Leading Coefficient
• Finding the Degree, Leading Term, and Leading Coefficient
• Finding the GCF of a Polynomial
• Factoring Out Greatest Common Factor (GCF)
• Identifying the Common Factors
• Cancelling the Common Factors
• Finding the LCM using GCF
• Finding the GCF
• Factoring Trinomials
• Trinomial Squares
• Factoring Using Any Method
• Factoring a Difference of Squares
• Factoring a Sum of Cubes
• Factoring by Grouping
• Factoring a Difference of Cubes
• Determine if an Expression is a Factor
• Determining if Factor Using Synthetic Division
• Find the Factors Using the Factor Theorem
• Determining if Polynomial is Prime
• Determining if the Polynomial is a Perfect Square
• Expand using the Binomial Theorem
• Factoring over the Complex Numbers
• Finding All Integers k Such That the Trinomial Can Be Factored
• Determining if Linear
• Rewriting in Standard Form
• Finding x and y Intercepts
• Finding Equations Using the Point Slope Formula
• Finding Equations Given Point and y-Intercept
• Finding the Constant Using Slope
• Finding the Slope of a Parallel Line
• Finding the Slope of a Perpendicular Line
• Simplifying Absolute Value Expressions
• Solving with Absolute Values
• Finding the Vertex for the Absolute Value
• Rewriting the Absolute Value as Piecewise
• Calculating the Square Root
• Simplifying Radical Expressions
• Rationalizing Radical Expressions
• Solving Radical Equations
• Rewriting with Rational (Fractional) Exponents
• Finding the Square Root End Point
• Operations on Rational Expressions
• Determining if the Point is a Solution
• Solving over the Interval
• Finding the Range
• Finding the Domain and Range
• Solving Rational Equations
• Adding Rational Expressions
• Subtracting Rational Expressions
• Multiplying Rational Expressions
• Finding the Equation Given the Roots
• Finding the Asymptotes
• Finding the Constant of Variation
• Finding the Equation of Variation
• Substitution Method
• Addition/Elimination Method
• Graphing Method
• Determining Parallel Lines
• Determining Perpendicular Lines
• Dependent, Independent, and Inconsistent Systems
• Finding the Intersection (and)
• Using the Simplex Method for Constraint Maximization
• Using the Simplex Method for Constraint Minimization
• Finding the Union (or)
• Finding the Equation with Real Coefficients
• Solving in Terms of the Arbitrary Variable
• Finding a Direct Variation Equation
• Finding the Slope for Every Equation
• Finding a Variable Using the Constant of Variation
• Quadratic Formula
• Solving by Factoring
• Solve by Completing the Square
• Finding the Perfect Square Trinomial
• Finding the Quadratic Equation Given the Solution Set
• Finding a,b, and c in the Standard Form
• Finding the Discriminant
• Finding the Zeros by Completing the Square
• Quadratic Inequalities
• Rational Inequalities
• Converting from Interval to Inequality
• Converting to Interval Notation
• Rewriting as a Single Interval
• Determining if the Relation is a Function
• Finding the Domain and Range of the Relation
• Finding the Inverse of the Relation
• Finding the Inverse
• Determining if One Relation is the Inverse of Another
• Determining if Surjective (Onto)
• Determining if Bijective (One-to-One)
• Determining if Injective (One to One)
• Rewriting as an Equation
• Rewriting as y=mx+b
• Solving Function Systems
• Find the Behavior (Leading Coefficient Test)
• Determining Odd and Even Functions
• Describing the Transformation
• Finding the Symmetry
• Arithmetic of Functions
• Domain of Composite Functions
• Finding Roots Using the Factor Theorem
• Determine if Injective (One to One)
• Determine if Surjective (Onto)
• Finding the Vertex
• Finding the Sum
• Finding the Difference
• Finding the Product
• Finding the Quotient
• Finding the Domain of the Sum of the Functions
• Finding the Domain of the Difference of the Functions
• Finding the Domain of the Product of the Functions
• Finding the Domain of the Quotient of the Functions
• Finding Roots (Zeros)
• Identifying Zeros and Their Multiplicities
• Finding the Bounds of the Zeros
• Proving a Root is on the Interval
• Finding Maximum Number of Real Roots
• Function Composition
• Rewriting as a Function
• Determining if a Function is Rational
• Determining if a Function is Proper or Improper
• Maximum/Minimum of Quadratic Functions
• Finding All Complex Number Solutions
• Rationalizing with Complex Conjugates
• Vector Arithmetic
• Finding the Complex Conjugate
• Finding the Magnitude of a Complex Number
• Simplifying Logarithmic Expressions
• Expanding Logarithmic Expressions
• Evaluating Logarithms
• Rewriting in Exponential Form
• Converting to Logarithmic Form
• Exponential Expressions
• Exponential Equations
• Converting to Radical Form
• Find the Nth Root of the Given Value
• Simplifying Matrices
• Finding the Variables
• Solving the System of Equations Using an Inverse Matrix
• Finding the Dimensions
• Multiplication by a Scalar
• Subtraction
• Finding the Determinant of the Resulting Matrix
• Finding the Inverse of the Resulting Matrix
• Finding the Identity Matrix
• Finding the Scalar multiplied by the Identity Matrix
• Simplifying the Matrix Operation
• Finding the Determinant of a 2x2 Matrix
• Finding the Determinant of a 3x3 Matrix
• Finding the Determinant of Large Matrices
• Inverse of a 2x2 Matrix
• Inverse of an nxn Matrix
• Finding Reduced Row Echelon Form
• Finding the Transpose
• Finding the Adjoint
• Finding the Cofactor Matrix
• Finding the Pivot Positions and Pivot Columns
• Finding the Basis and Dimension for the Row Space of the Matrix
• Finding the Basis and Dimension for the Column Space of the Matrix
• Finding the LU Decomposition of a Matrix
• Identifying Conic Sections
• Identifying Circles
• Finding a Circle Using the Center and Another Point
• Finding a Circle by the Diameter End Points
• Finding the Parabola Equation Using the Vertex and Another Point
• Finding the Properties of the Parabola
• Finding the Vertex Form of the Parabola
• Finding the Vertex Form of an Ellipse
• Finding the Vertex Form of a Circle
• Finding the Vertex Form of a Hyperbola
• Finding the Standard Form of a Parabola
• Finding the Expanded Form of an Ellipse
• Finding the Expanded Form of a Circle
• Finding the Expanded Form of a Hyperbola
• Vector Addition
• Vector Subtraction
• Vector Multiplication by a Scalar
• Finding the Length
• Finding the Position Vector
• Determining Column Spaces
• Finding an Orthonormal Basis by Gram-Schmidt Method
• Rewrite the System as a Vector Equality
• Finding the Rank
• Finding the Nullity
• Finding the Distance
• Finding the Plane Parallel to a Line Given four 3d Points
• Finding the Intersection of the Line Perpendicular to Plane 1 Through the Origin and Plane 2
• Finding the Eigenvalues
• Finding the Characteristic Equation
• Finding the Eigenvectors/Eigenspace of a Matrix
• Proving a Transformation is Linear
• Finding the Kernel of a Transformation
• Projecting Using a Transformation
• Finding the Pre-Image
• Finding the Intersection of Sets
• Finding the Union of Number Sets
• Determining if a Set is a Subset of Another Set
• Determining if Two Sets are Mutually Exclusive
• Finding the Set Complement of Two Sets
• Finding the Power Set
• Finding the Cardinality
• Finding the Cartesian Product of Two Sets
• Determining if a Set is a Proper Subset of Another Set
• Finding the Function Rule
• Finding the Square or Rectangle Area Given Four Points
• Finding the Square or Rectangle Perimeter Given Four Points
• Finding the Square or Rectangle Area Given Three Points
• Finding the Square or Rectangle Perimeter Given Three Points
• Finding the Equation of a Circle
• Finding the Equation of a Hyperbola
• Finding the Equation of an Ellipse
• Partial Fraction Decomposition
• Finding an Angle Using another Angle
• Pythagorean Theorem
• Finding the Sine
• Finding the Cosine
• Finding the Tangent
• Finding the Trig Value
• Converting to Degrees, Minutes, and Seconds
• Finding Trig Functions Using Identities
• Finding Trig Functions Using the Right Triangle
• Converting Radians to Degrees
• Converting Degrees to Radians
• Finding a Reference Angle
• Finding a Supplement
• Finding a Complement
• Converting RPM to Radians per Second
• Finding the Quadrant of the Angle
• Graphing Sine & Cosine Functions
• Graphing Other Trigonometric Functions
• Amplitude, Period, and Phase Shift
• Finding the Other Trig Values in a Quadrant
• Finding the Exact Value
• Finding the Value Using the Unit Circle
• Expanding Trigonometric Expressions
• Expanding Using Double-Angle Formulas
• Expanding Using Triple-Angle Formulas
• Expanding Using Sum/Difference Formulas
• Simplify Using Pythagorean Identities
• Simplify by Converting to Sine/Cosine
• Inverting Trigonometric Expressions
• Finding the Trig Value of an Angle
• Expanding Using De Moivre's Theorem
• Verifying Trigonometric Identities
• Using Fundamental Identities
• Solving Standard Angle Equations
• Complex Trigonometric Equations
• Solving the Triangle
• Find the Roots of a Complex Number
• Complex Operations
• Trigonometric Form of a Complex Number
• Converting to Polar Coordinates
• Identifying and Graphing Circles
• Identifying and Graphing Limacons
• Identifying and Graphing Roses
• Identifying and Graphing Cardioids
• Difference Quotient
• Finding Upper and Lower Bounds
• Evaluating Functions
• Right Triangle Trigonometry
• Arithmetic Sequences/Progressions
• Geometric Sequences/Progressions
• Finding the Next Term of the Sequence
• Finding the nth Term Given a List of Numbers
• Finding the nth Term
• Finding the Sum of First n Terms
• Expanding Series Notation
• Finding the Sum of the Series
• Finding the Sum of the Infinite Geometric Series
• Converting to Rectangular Coordinates
• Evaluating Limits Approaching a Value
• Evaluating Limits Approaching Infinity
• Finding the Angle Between the Vectors
• Determining if the Point is on the Graph
• Finding the Antiderivative
• Checking if Continuous Over an Interval
• Determining if a Series is Divergent
• Using the Integral Test for Convergence
• Determining if an Infinite Series is Convergent Using Cauchy's Root Test
• Using the Limit Definition to Find the Tangent Line at a Given Point
• Finding the nth Derivative
• Finding the Derivative Using Product Rule
• Finding the Derivative Using Quotient Rule
• Finding the Derivative Using Chain Rule
• Use Logarithmic Differentiation to Find the Derivative
• Finding the Derivative
• Implicit Differentiation
• Using the Limit Definition to Find the Derivative
• Evaluating the Derivative
• Finding Where dy/dx is Equal to Zero
• Finding the Linearization
• Finding a Tangent Line to a Curve
• Checking if Differentiable Over an Interval
• The Mean Value Theorem
• Finding the Inflection Points
• Find Where the Function Increases/Decreases
• Finding the Critical Points of a Function
• Find Horizontal Tangent Line
• Evaluating Limits with L'Hospital Rule
• Local Maxima and Minima
• Finding the Absolute Maximum and Minimum on the Given Interval
• Finding Concavity using the Second Derivative
• Finding the Derivative using the Fundamental Theorem of Calculus
• Find the Turning Points
• Finding the Integral
• Evaluating Definite Integrals
• Evaluating Indefinite Integrals
• Substitution Rule
• Finding the Arc Length
• Finding the Average Value of the Derivative
• Finding the Average Value of the Equation
• Finding Area Between Curves
• Finding the Volume
• Finding the Average Value of the Function
• Finding the Root Mean Square
• Integration by Parts
• Trigonometric Integrals
• Trigonometric Substitution
• Integration by Partial Fractions
• Eliminating the Parameter from the Function
• Verify the Solution of a Differential Equation
• Solve for a Constant Given an Initial Condition
• Find an Exact Solution to the Differential Equation
• Verify the Existence and Uniqueness of Solutions for the Differential Equation
• Solve for a Constant in a Given Solution
• Solve the Bernoulli Differential Equation
• Solve the Linear Differential Equation
• Solve the Homogeneous Differential Equation
• Solve the Exact Differential Equation
• Approximate a Differential Equation Using Euler's Method
• Finding Elasticity of Demand
• Finding the Consumer Surplus
• Finding the Producer Surplus
• Finding the Gini Index
• Finding the Geometric Mean
• Finding the Quadratic Mean (RMS)
• Find the Mean Absolute Deviation
• Finding the Mid-Range (Mid-Extreme)
• Finding the Interquartile Range (H-Spread)
• Finding the Midhinge
• Finding the Standard Deviation
• Finding the Skew of a Data Set
• Finding the Range of a Data Set
• Finding the Variance of a Data Set
• Finding the Class Width
• Solving Combinations
• Solving Permutations
• Finding the Probability of Both Independent Events
• Finding the Probability of Both Dependent Events
• Finding the Probability for Both Mutually Exclusive Events
• Finding the Conditional Probability for Independent Events
• Determining if Given Events are Independent/Dependent Events
• Determining if Given Events are Mutually Exclusive Events
• Finding the Probability of Both not Mutually Exclusive Events
• Finding the Conditional Probability Using Bayes' Theorem
• Finding the Probability of the Complement
• Describing Distribution's Two Properties
• Finding the Expectation
• Finding the Variance
• Finding the Probability of a Binomial Distribution
• Finding the Probability of the Binomial Event
• Finding the Mean
• Finding the Relative Frequency
• Finding the Percentage Frequency
• Finding the Upper and Lower Class Limits of the Frequency Table
• Finding the Class Boundaries of the Frequency Table
• Finding the Class Width of the Frequency Table
• Finding the Midpoints of the Frequency Table
• Finding the Mean of the Frequency Table
• Finding the Variance of the Frequency Table
• Finding the Standard Deviation of the Frequency Table
• Finding the Cumulative Frequency of the Frequency Table
• Finding the Relative Frequency of the Frequency Table
• Finding the Median Class Interval of the Frequency Table
• Finding the Modal Class of the Frequency Table
• Creating a Grouped Frequency Distribution Table
• Finding the Data Range
• Finding a z-Score for a Normal Distribution
• Approximating Using Normal Distribution
• Finding the Probability of the z-Score Range
• Finding the Probability of a Range in a Nonstandard Normal Distribution
• Finding the z-Score Using the Table
• Finding the z-Score
• Testing the Claim
• Finding a t-Value for a Confidence Level
• Finding the Critical t-Value
• Setting the Alternative Hypothesis
• Setting the Null Hypothesis
• Determining if Left, Right, or Two Tailed Test Given the Null Hypothesis
• Determining if Left, Right, or Two Tailed Test Given the Alternative Hypothesis
• Finding Standard Error
• Finding the Linear Correlation Coefficient
• Determining if the Correlation is Significant
• Finding a Regression Line
• Cramer's Rule
• Solving using Matrices by Elimination
• Solving using Matrices by Row Operations
• Solving using an Augmented Matrix
• Finding the Simple Interest Received
• Finding the Present Value with Compound Interest
• Finding the Simple Interest Future Value
• Finding the Future Value with Continuous Interest
• Finding the Norm in Real Vector Space
• Finding the Direction Angle of the Vector
• Finding the Cross Product of Vectors
• Finding the Dot Product of Vectors
• Determining if Vectors are Orthogonal
• Finding the Distance Between the Vectors
• Finding a Unit Vector in the Same Direction as the Given Vector
• Finding the Angle Between Two Vectors Using the Cross Product
• Finding the Angle Between Two Vectors Using the Dot Product
• Finding the Projection of One Vector Onto another Vector
• Matrices Addition
• Matrices Subtraction
• Matrices Multiplication
• Finding the Trace
• Finding the Basis
• Matrix Dimension
• Convert to a Linear System
• Diagonalizing a Matrix
• Determining the value of k for which the system has no solutions
• Linear Independence of Real Vector Spaces
• Finding the Null Space
• Determining if the Vector is in the Span of the Set
• Finding the Number of Protons
• Finding the Number of Electrons
• Finding the Number of Neutrons
• Finding the Mass of a Single Atom
• Finding the Electron Configuration
• Finding the Atomic Mass
• Finding the Atomic Number
• Finding the Mass Percentages
• Finding Oxidation Numbers
• Balancing Chemical Equations
• Balancing Burning Reactions
• Finding the Density at STP
• Determining if the Compound is Soluble in Water
• Finding Mass
• Finding Density
• Finding Weight
• Finding Force
• Finding the Work Done
• Finding Angular Velocity
• Finding Centripetal Acceleration
• Finding Final Velocity
• Finding Average Acceleration
• Finding Displacement
• Finding Voltage Using the Ohm's Law
• Finding Electrical Power
• Finding Kinetic Energy
• Finding Power
• Finding Wavelength
• Finding Frequency
• Finding Pressure of the Gas

Popular Problems

• Pre-Algebra
• Trigonometry
• Precalculus
• Finite Math
• Linear Algebra
• Recent Popular Problems

Calculator Pages

Formula pages.

• Algebra Formulas
• Trigonometry Formulas
• Geometric Formulas
• Terms ( Premium )
• DO NOT SELL MY INFO
• Mathway © 2024

Please ensure that your password is at least 8 characters and contains each of the following:

• a special character: @$#!%*?&

You are using an outdated browser. Upgrade your browser today or install Google Chrome Frame to better experience this site.

Opening up students’ mathematical thinking using open middle math problems

When paired with number talks , open middle problems can help teachers embed rich conversation throughout math class.

Defining the problem

So, what exactly are open middle math problems? In an open middle problem, students are given starting information and either an answer or a definition of the answer. By “definition of the answer,” we mean that students are given a description of the type of answer toward which they are driving. For example, many open middle math problems ask students to find the least or greatest solution or a number that is closest to a given number. How they get from start to end is up to them. Rather than following a solution path that was stepped out for them by the teacher, the path is completely under their control.

The best way to understand an open middle math problem may be to compare one to a typical textbook problem for the same topic. For a lesson on three-digit subtraction, a standard textbook might ask students to find 539 – 286 either as a straight computation item or embedded within a word problem. By contrast, an open middle math problem for the same topic asks students to use the digits 1–9, at most one time each, to make two three-digit numbers with a difference as close to 329 as possible.

Although understanding of place value, regrouping, the relative magnitude of numbers, and the relationship between addition and subtraction underlie both types of problems, with the more traditional problem, a student can answer correctly by following a set of learned steps without truly understanding the conceptual underpinnings. However, students must actively engage with these concepts to solve the open middle math problem. The open structure forces students to consider these concepts and constantly make decisions and adjust their solution path. Here are some of the questions students might consider when solving the open middle math problem:

• What strategies can be applied to solve? Should I subtract? Should I add up? Can estimation help?
• Should I focus on a certain place value for each number first, or should I create one number completely?
• Is it possible to make two numbers where regrouping will not be required?
• Is it possible to get exactly 329?

Open middle problems can also be word problems . While some open middle math problems are structured this way, others can easily be turned into word problems. For example, the three-digit subtraction problem above can be rewritten as a word problem such as this: Jared and Suni are playing a video game. Each of their scores were three-digit numbers. Jared’s score was 329 less than Suni’s. Use the digits 1–9, at most one time each, to make two three-digit numbers that could be Jared and Suni’s scores.

It’s the journey, not the destination

Because of the way they are designed, open middle math problems emphasize process over product. While students are still expected to arrive at a defensible answer, open middle problems push students to utilize a variety of mathematical skills and understandings to wrestle with the problem. The goal is not to be the quickest to follow a predetermined solution path but, rather, to allow for a more iterative approach where students continually test and revise strategies. Dan Meyer terms this “patient problem solving,” where students bring all their mathematical knowledge and intuition to bear on solving non-routine problems. The active engagement that this engenders supports rich mathematical thinking and discussion amongst students.

One of the co-developers of open middle problems, Nanette Johnson, describes their value this way: “This structure requires students to prove to themselves and others that they have truly found the best possible answer. Rather than putting down their pencils and saying, ‘I’m done,’ students continue to think, argue, and work. They develop the habit of making multiple attempts to solve the problem, each time wondering if they can come up with an even better solution than their last.”

The fact that students can arrive at different answers naturally prompts discussion between students of how they approached the problem, why they think their answer best meets the criteria, and whether other answers could get closer to the goal. As Johnson states, this serves to foster increased engagement and persistence. It also provides a vehicle for the rich mathematical discourse we were seeking in our districts.

Low floors, high ceilings, and big concepts

By design, open middle math problems allow easy entry for all students, but they also provide the flexibility to allow students to stretch themselves. For this reason, open middle problems have been described as easy in terms of getting an answer, but not so easy in terms of getting the ideal or “best” answer. Take, for example, that problem that challenges students to use the digits 1–9, at most one time each, to create two mixed numbers with the least possible difference. Students are given a template like the one on the Open Middle website in which to write their numbers.

Students who are still developing their understanding of fraction operations might take a simple approach of having the digits in the second mixed number be one less than the digits in the first mixed number: 9 5/7 – 8 4/6. This results in a difference of 1 1/21. Another student might opt to use the greatest digits available, 8 and 9, to make the denominators of the fractions, knowing that the greater number of parts a fraction is divided into, the smaller it is. Perhaps they create the expression 2 7/9 – 1 6/8 , which results in a difference of 1 1/36. At this point, the teacher might ask these two students which of their answers is the least, sparking a conversation about the relative magnitude of the numbers, benchmark fractions, and estimation. The teacher could then push students by challenging them to see if it is possible to get an answer that is less than one.

A student further along in their understanding of fractions might recognize that the need to regroup could reduce the size of the final number. Perhaps they create the problem 3 1/9 – 2 7/8 with the result of 17/72. Student interactions or well-placed teacher questions, such as, “Is it possible to get an answer of 0?” might prompt some students to realize using improper fractions can further reduce the magnitude of the answer, indeed all the way down to zero.

The nature of the problem allows all students entry without limiting students with more advanced knowledge of the involved concepts. Rich conversations and strategic teacher questions help all students expand and grow their understanding of concepts beyond the idea of fraction subtraction, while providing teachers insights into each students’ level of understanding of these concepts. In the iterative process of testing ideas, discussing different approaches, and trying new solution paths, students explore far deeper concepts of the meaning of fractions, number magnitude, and equivalent forms.

Impact of using open middle math problems

While there is no research that specifically examines open middle problems, there are studies that show the efficacy of components of the approach. In a research paper on equitable teaching approaches to math , Jo Boaler and Megan Staples call out that several studies have demonstrated that “conceptually oriented mathematics materials, taught well and with consistency, have shown higher and more equitable results for participating students than procedure-oriented curricula taught using a demonstration and practice approach.” Open middle math problems support such a conceptual orientation and, by design, break the “demonstration and practice” model.

Having students discuss and compare different solution approaches is a critical component of open math problems. Although students are not working in groups, they are also not working in isolation. Part of the iterative process of using open middle math problems includes time to share and discuss the relative merits of various solution approaches and use these to iterate new approaches. Actively comparing solution methods has been shown to increase procedural understanding , particularly for lower achieving students. Another study found that students who learned by comparing alternative solution approaches demonstrated greater conceptual knowledge and more flexible problem-solving than students who reflected on different approaches one at a time.

Open middle math problems are well aligned to NCTM’s process standards . They provide rich opportunities to practice:

• Building new mathematical knowledge through problem-solving
• Monitoring and reflecting on the problem-solving process
• Making and exploring mathematical conjectures
• Developing, evaluating, and communicating mathematical reasoning and analyzing the reasoning of others

Getting started

When working on incorporating using open middle math problems in your classroom, we encourage you to determine how to integrate them into existing math lesson plans so they aren’t one more thing you feel you have to do. Consider, for example, how open middle math tasks build off number talks but take you and your students to the next level.

Both open middle math problems and number talks encourage rich dialogue between students. While number talks are conversation-base only, open middle math problems involve both conversation and written responses. Number talks are brief 10–15-minute warmups, while open middle math problems give students more time to explore their thinking and the thinking of others. They also provide teachers with a written record of this thinking and how it evolves over the course of iterative problem-solving.

You might also try working on open middle problems yourself, which can help you understand the power of the approach.

Adding doesn’t necessarily mean more

We also encourage you to focus on potential roadblocks to making open middle math problems a part of your classroom. In our experience, time can be a big concern. If that’s worrying you as well, we suggest you focus on the relative merit of having students deeply engage in a single rich mathematical problem in place of completing 10 rote problems. This can help you move away from a “mile-wide, inch deep” approach to math.

Remember, too, that these problems do not need to be completed in a day. Take the time to let students work through possible solutions, share, discuss, and revise over the course of one or more days.

Finally, time you previously spent looking for better resources could be repurposed into unpacking open middle math problems to prepare you for facilitating a rich classroom discussion. Here are some questions to get you started:

• What big mathematical ideas do you hope students will bring into discussions?
• What models or mathematical terms might students use in their explanations?
• What are examples of clear justification or correct reasoning you hope to see?
• What are some possible solution approaches and what do each of them reveal about students’ understanding of the underlying concepts?

Give it a try!

Adding open middle tasks to your math routines can truly deepen both math conversations and student engagement in your classroom. Ready to try them yourself? Use these links to get more information about open middle math problems and supporting mathematical writing and discourse.

Open middle resources

• Open Middle This site, created by Open Middle co-creators Nanette Johnson and Robert Kaplinsky, contains a wealth of open middle tasks organized by grade and domain as well as worksheets in English, French, and Spanish and printable number tiles.
• Open Middle Exercises GeoGebra has converted many of the paper versions of open middle exercises into interactive versions.
• Open Middle Math: Problems that Unlock Student Thinking, Grades 6–12 You can preview Robert Kaplinsky’s book about open middle math here.

Classroom discussion and mathematical writing resources

• “4 ways to engage students with writing in math class” This article delves into a different way to support and encourage student writing about math, including how to write high-quality explanations for how they solved a problem.
• “Eliciting, supporting, and guiding the math: Three key functions of the teacher’s role in facilitating meaningful mathematical discourse” This piece defines what makes math discourse meaningful, why it’s important, and what the teacher’s role is in facilitating such discourse.
• “Mathematical discourse in the classroom: A prime time for discussion” This article provides strategies for facilitating meaningful mathematical discussion with the goal of helping students develop a deeper understanding of mathematical concepts.
• “Using the 5 practices in mathematics teaching” This article provides an overview of how teachers can successfully orchestrate classroom discussions by anticipating possible student solutions, monitoring their work, selecting and sequencing how students share their work, and asking questions that help students make connections between mathematical ideas.

Recommended for you

Math enrichment for all: 3 ways to engage all learners in deep mathematical thinking

6 tips for supporting problem-based learning in your math classroom

7 tips for encouraging student discourse about math with number talks

• View all posts

Reading differentiation made easy

MAP Reading Fluency now includes Coach, a virtual tutor designed to help students strengthen reading skills in as little as 30 minutes a week.

Helping students grow

Students continue to rebound from pandemic school closures. NWEA® and Learning Heroes experts talk about how best to support them here on our blog, Teach. Learn. Grow.

See the post

Put the science of reading into action

The science of reading is not a buzzword. It’s the converging evidence of what matters and what works in literacy instruction. We can help you make it part of your practice.

Get the guide

Support teachers with PL

High-quality professional learning can help teachers feel invested—and supported—in their work.

Read the article

Content disclaimer:

Teach. Learn. Grow. includes diverse perspectives that are meant to be a resource to educators and leaders across the country and around the world. The views expressed are those of the authors and do not necessarily represent those of NWEA.

• Timely Resources
• Innovative Teaching Ideas
• Classroom Voices
• Education Trends

Four Principles for Effective Math Intervention

The Institute for Education Sciences, an authority on Response to Intervention (RtI) for math published a groundbreaking report on RtI that outlines a series of intervention recommendations. The report Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle schools  covers recommendations that were judged to have strong or moderate evidence to support RtI and provide the foundation for effective math intervention. Here are specific strategies that fall under these four key recommendations:

1. Instruction during the intervention should be explicit and systematic.

This recommendation from the report includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.

Students who have been classified as Tier 2 or 3 in RtI need their instruction to be organized and scaffolded. They lack the numeracy skills and background knowledge to engage in theoretical exercises with math

Nothing is more systematic than the four-step approach to problem solving, first outlined by educator George Polly in 1945:

•  Understand the problem. Restate the problem, and then identify the information given and the information that needs to be determined.
• Make a plan. Relate the problem to similar problems solved in the past. Consider possible strategies, and then select a strategy or a combination of strategies
• Carry out the plan. Execute the chosen strategy and perform the necessary calculations. Revise and apply different strategies as necessary. 
• Look back at the solution. Evaluate the strategy/strategies used for problem solving and then assess if there is a better way to approach the problem.

2. Interventions should include instruction on solving word problems that is based on common underlying structures.

A stripped-down version of the Gradual Release Model—the “I do, We do, You do” strategy—is effective in all levels of education. As RtI students require as much structure as possible, the strategy gives them an effective way to know what to expect from a lesson. In word problems, not only can “I do, We do, You do” be used to solve problems, it can also be used to have students create their own word problems, reaching synthesis, a higher level of taxonomy. Students should start with sentences that involve a specific math operation and build from there. (McCarney, S. B., Cummins Wunderlich, K., Bauer, A. The Teacher’s Resource Guide.)

3. Intervention materials should include opportunities for students to work with visual representations of mathematical ideas.

This recommendation from the RtI report goes on to say that interventionists should be proficient in the use of visual representations of mathematical ideas. Graphics are important in math instruction, especially as the curriculum becomes more data=based under the Common Core State Standards. RtI students are not excluded from having to be able to read charts, graphs, and other math graphics. Intervention Central , which provides educators with free RtI resources, has a great intervention that uses the Question-Answer Relationship (QAR) to help students break down math graphics. In short, QARs come in four types:

•  “Right There” questions are found explicitly in the graphic.
•  “Think and Search” questions are not quite as explicit, but still can be found in the graphic with some close analysis.
• “Author and You” questions ask students to compare the data with their own life experiences and opinions.
• “On My Own” questions require only the student’s own knowledge and experiences to answer.

4. Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.

Rehearsal of math facts is a key step in the RtI process because many of the students who fall into Tier 2 or 3 status are missing key pieces of background information, usually in the form of math facts.  

Taking ten minutes each day to review facts is typically done with flash cards, which can be tedious and, therefore, does not always prove to be effective. Another strategy, described by Intervention Central, uses flash-card practice that balances “known” facts with “unknown” facts. Unknown facts are modeled by a teacher or tutor and then presented with known facts—those already mastered by the student—in a sequence. Not only does this systematically build recall, it also builds confidence in the student because they are consistently getting cards correct throughout the process.  

Using Technology to Boost RtI for Math

Digital curriculum, at its best, incorporates the scaffolds and formative assessment that define successful RtI for math. Personalized learning and ongoing, robust data collection provide a meaningful feedback loop for both the student and teacher that leads to deeper learning and higher achievement. Real-time feedback—while learning is happening—is critical so that students don’t practice new math skills, again and again , incorrectly.  

Technological interventions to boost math achievement can be an important part of a successful math intervention , but a commitment to the process is also a necessary part of helping students succeed in math and sharpen their math skills .

Read More About Math Instruction on the DE Blog!

Related posts.

Mastering the Art of Parent-Teacher Conferences 

Fuel Curiosity with Immersive Storytelling

How to Keep Communication Alive: Teachers, Students & Caregivers 

The Five Senses: Immersion in the Classroom

The Discovery Education blog is a free resource for educators to find time-saving teaching strategies and compelling content for their daily lessons.

Full of timely tips, high-quality DE resources, and advice from our DEN community, these posts are meant to entertain and inform our users while supporting educators everywhere with new ways to engage their students in and out of the classroom.

Stay Connected

Copyright © 2024 Discovery Education. All rights reserved. TERMS OF USE PRIVACY POLICY CONTACT US COOKIE SETTINGS

To revisit this article, visit My Profile, then View saved stories .

• The Big Story
• Newsletters
• Steven Levy's Plaintext Column
• WIRED Classics from the Archive
• WIRED Insider
• WIRED Consulting

OpenAI Announces a New AI Model, Code-Named Strawberry, That Solves Difficult Problems Step by Step

OpenAI made the last big breakthrough in artificial intelligence by increasing the size of its models to dizzying proportions, when it introduced GPT-4 last year. The company today announced a new advance that signals a shift in approach—a model that can “reason” logically through many difficult problems and is significantly smarter than existing AI without a major scale-up.

The new model, dubbed OpenAI o1, can solve problems that stump existing AI models, including OpenAI’s most powerful existing model, GPT-4o . Rather than summon up an answer in one step, as a large language model normally does, it reasons through the problem, effectively thinking out loud as a person might, before arriving at the right result.

“This is what we consider the new paradigm in these models,” Mira Murati , OpenAI’s chief technology officer, tells WIRED. “It is much better at tackling very complex reasoning tasks.”

The new model was code-named Strawberry within OpenAI, and it is not a successor to GPT-4o but rather a complement to it, the company says.

Murati says that OpenAI is currently building its next master model, GPT-5, which will be considerably larger than its predecessor. But while the company still believes that scale will help wring new abilities out of AI, GPT-5 is likely to also include the reasoning technology introduced today. “There are two paradigms,” Murati says. “The scaling paradigm and this new paradigm. We expect that we will bring them together.”

LLMs typically conjure their answers from huge neural networks fed vast quantities of training data. They can exhibit remarkable linguistic and logical abilities, but traditionally struggle with surprisingly simple problems such as rudimentary math questions that involve reasoning.

Murati says OpenAI o1 uses reinforcement learning, which involves giving a model positive feedback when it gets answers right and negative feedback when it does not, in order to improve its reasoning process. “The model sharpens its thinking and fine tunes the strategies that it uses to get to the answer,” she says. Reinforcement learning has enabled computers to play games with superhuman skill and do useful tasks like designing computer chips . The technique is also a key ingredient for turning an LLM into a useful and well-behaved chatbot.

Mark Chen, vice president of research at OpenAI, demonstrated the new model to WIRED, using it to solve several problems that its prior model, GPT-4o, cannot. These included an advanced chemistry question and the following mind-bending mathematical puzzle: “A princess is as old as the prince will be when the princess is twice as old as the prince was when the princess’s age was half the sum of their present age. What is the age of the prince and princess?” (The correct answer is that the prince is 30, and the princess is 40).

“The [new] model is learning to think for itself, rather than kind of trying to imitate the way humans would think,” as a conventional LLM does, Chen says.

OpenAI says its new model performs markedly better on a number of problem sets, including ones focused on coding, math, physics, biology, and chemistry. On the American Invitational Mathematics Examination (AIME), a test for math students, GPT-4o solved on average 12 percent of the problems while o1 got 83 percent right, according to the company.

The new model is slower than GPT-4o, and OpenAI says it does not always perform better—in part because, unlike GPT-4o, it cannot search the web and it is not multimodal, meaning it cannot parse images or audio.

Improving the reasoning capabilities of LLMs has been a hot topic in research circles for some time. Indeed, rivals are pursuing similar research lines. In July, Google announced AlphaProof , a project that combines language models with reinforcement learning for solving difficult math problems.

AlphaProof was able to learn how to reason over math problems by looking at correct answers. A key challenge with broadening this kind of learning is that there are not correct answers for everything a model might encounter. Chen says OpenAI has succeeded in building a reasoning system that is much more general. “I do think we have made some breakthroughs there; I think it is part of our edge,” Chen says. “It’s actually fairly good at reasoning across all domains.”

Noah Goodman , a professor at Stanford who has published work on improving the reasoning abilities of LLMs, says the key to more generalized training may involve using a “carefully prompted language model and handcrafted data” for training. He adds that being able to consistently trade the speed of results for greater accuracy would be a “nice advance.”

Yoon Kim , an assistant professor at MIT, says how LLMs solve problems currently remains somewhat mysterious, and even if they perform step-by-step reasoning there may be key differences from human intelligence. This could be crucial as the technology becomes more widely used. “These are systems that would be potentially making decisions that affect many, many people,” he says. “The larger question is, do we need to be confident about how a computational model is arriving at the decisions?”

The technique introduced by OpenAI today also may help ensure that AI models behave well. Murati says the new model has shown itself to be better at avoiding producing unpleasant or potentially harmful output by reasoning about the outcome of its actions. “If you think about teaching children, they learn much better to align to certain norms, behaviors, and values once they can reason about why they’re doing a certain thing,” she says.

Oren Etzioni , a professor emeritus at the University of Washington and a prominent AI expert, says it’s “essential to enable LLMs to engage in multi-step problem solving, use tools, and solve complex problems.” He adds, “Pure scale up will not deliver this.” Etzioni says, however, that there are further challenges ahead. “Even if reasoning were solved, we would still have the challenge of hallucination and factuality.”

OpenAI’s Chen says that the new reasoning approach developed by the company shows that advancing AI need not cost ungodly amounts of compute power. “One of the exciting things about the paradigm is we believe that it’ll allow us to ship intelligence cheaper,” he says, “and I think that really is the core mission of our company.”

You Might Also Like …

In your inbox: The best and weirdest stories from WIRED’s archive

Elon Musk is a national security risk

Interview: Meredith Whittaker is out to prove capitalism wrong

How do you solve a problem like Polestar ?

Event: Join us for The Big Interview on December 3 in San Francisco

• a. Send us an email
• b. Anonymous form
• Buyer's Guide
• Upcoming Products
• Tips / Contact Us
• Podcast Instagram Facebook Twitter Mastodon YouTube Notifications RSS Newsletter

iOS 18: Use Math Notes in the Calculator App

In iOS 18, Apple has added a powerful new feature to your iPhone's Calculator app: Math Notes. This integration between Calculator and Notes offers a versatile tool for all your calculation needs. It's particularly handy for splitting bills, calculating group expenses, or working through more complex mathematical problems.

Here's how you can start using Math Notes in the Calculator app in iOS 18:

• Open the Calculator app on your iPhone.
• Tap the calculator symbol at the bottom left of the screen.

• Tap the new note symbol in the bottom right corner.

You're not limited to accessing Math Notes through the Calculator app – you can also use the feature directly within the Notes app using any new or existing note.

If you need to switch back to the standard calculator while using Math Notes, simply tap the calculator icon again and choose either Basic or Scientific mode.

You can in fact get Math Notes results almost anywhere in the operating system. If you type an equation into search, for example, you'll get a result, and the same goes for apps like Messages.

Math Notes on iPad

In ‌iPadOS 18‌, Math Notes works in the same way, but there is an added bonus - you can use an Apple Pencil. Once you start a Math Note from the Calculator app or the Notes app, you can write your equations by hand and have them solved in the exact same way.

Get weekly top MacRumors stories in your inbox.

Popular Stories

iOS 18 Available Now With These 8 New Features For Your iPhone

Here's When iOS 18 Rolls Out Today in Every Time Zone

M4 Macs, New iPad Mini, and iPad 11 Expected at Upcoming Apple Event

Apple Leaks New Mac Mini With 5 USB-C Ports

iPhone 16 Pro Demand Has Been Lower Than Expected, Analyst Says

10 Reasons to Wait for Next Year's iPhone 17

Next article.

Our comprehensive guide highlighting every major new addition in iOS 18, plus how-tos that walk you through using the new features.

iOS 18 brings major changes to the Messages app, with Apple adding several fun features.

The Photos app redesign has been one of the most controversial changes that Apple made in iOS 18.

iOS 18 is an update designed for the iPhone, but it adds several new features to the AirPods

Updates to Mail, Photos, Messages, and more, with Apple Intelligence features throughout.

iPhone Mirroring, Safari improvements, new Passwords app, and Apple Intelligence features throughout.

Apple Intelligence features, Smart Script handwriting refinement, Calculator app, and more.

New hand gestures, rearrange apps in Home View, larger Mac Virtual Display, and more.

Other Stories

14 hours ago by Joe Rossignol

19 hours ago by Tim Hardwick

21 hours ago by Tim Hardwick

21 hours ago by Joe Rossignol

22 hours ago by Tim Hardwick

Advertisement

Supported by

OpenAI Unveils New ChatGPT That Can Reason Through Math and Science

Driven by new technology called OpenAI o1, the chatbot can test various strategies and try to identify mistakes as it tackles complex tasks.

• Share full article

By Cade Metz

Reporting from San Francisco

Online chatbots like ChatGPT from OpenAI and Gemini from Google sometimes struggle with simple math problems . The computer code they generate is often buggy and incomplete. From time to time, they even make stuff up .

On Thursday, OpenAI unveiled a new version of ChatGPT that could alleviate these flaws. The company said the chatbot, underpinned by new artificial intelligence technology called OpenAI o1, could “reason” through tasks involving math, coding and science.

“With previous models like ChatGPT, you ask them a question and they immediately start responding,” said Jakub Pachocki, OpenAI’s chief scientist. “This model can take its time. It can think through the problem — in English — and try to break it down and look for angles in an effort to provide the best answer.”

In a demonstration for The New York Times, Dr. Pachocki and Szymon Sidor, an OpenAI technical fellow, showed the chatbot solving an acrostic, a kind of word puzzle that is significantly more complex than an ordinary crossword puzzle. The chatbot also answered a Ph.D.-level chemistry question and diagnosed an illness based on a detailed report about a patient’s symptoms and history.

The new technology is part of a wider effort to build A.I. that can reason through complex tasks. Companies like Google and Meta are building similar technologies, while Microsoft and its subsidiary GitHub are working to incorporate OpenAI’s new system into their products.

We are having trouble retrieving the article content.

Please enable JavaScript in your browser settings.

Thank you for your patience while we verify access. If you are in Reader mode please exit and  log into  your Times account, or  subscribe  for all of The Times.

Thank you for your patience while we verify access.

Already a subscriber?  Log in .

Want all of The Times?  Subscribe .