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How to Solve Math Problems

Last Updated: May 16, 2023 Fact Checked

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. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 583,722 times.

Although math problems may be solved in different ways, there is a general method of visualizing, approaching and solving math problems that may help you to solve even the most difficult problem. Using these strategies can also help you to improve your math skills overall. Keep reading to learn about some of these math problem solving strategies.

Understanding the Problem

Step 1 Identify the type of problem.

  • Draw a Venn diagram. A Venn diagram shows the relationships among the numbers in your problem. Venn diagrams can be especially helpful with word problems.
  • Draw a graph or chart.
  • Arrange the components of the problem on a line.
  • Draw simple shapes to represent more complex features of the problem.

Step 5 Look for patterns.

Developing a Plan

Step 1 Figure out what formulas you will need to solve the problem.

Solving the Problem

Step 1 Follow your plan.

Expert Q&A

Daron Cam

  • Seek help from your teacher or a math tutor if you get stuck or if you have tried multiple strategies without success. Your teacher or a math tutor may be able to easily identify what is wrong and help you to understand how to correct it. Thanks Helpful 1 Not Helpful 1
  • Keep practicing sums and diagrams. Go through the concept your class notes regularly. Write down your understanding of the methods and utilize it. Thanks Helpful 1 Not Helpful 0

the best way to solve math problems

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Do Math Proofs

  • ↑ Daron Cam. Math Tutor. Expert Interview. 29 May 2020.
  • ↑ http://www.interventioncentral.org/academic-interventions/math/math-problem-solving-combining-cognitive-metacognitive-strategies
  • ↑ http://tutorial.math.lamar.edu/Extras/StudyMath/ProblemSolving.aspx
  • ↑ https://math.berkeley.edu/~gmelvin/polya.pdf

About This Article

Daron Cam

To solve a math problem, try rewriting the problem in your own words so it's easier to solve. You can also make a drawing of the problem to help you figure out what it's asking you to do. If you're still completely stuck, try solving a different problem that's similar but easier and then use the same steps to solve the harder problem. Even if you can't figure out how to solve it, try to make an educated guess instead of leaving the question blank. To learn how to come up with a solid plan to use to help you solve a math problem, scroll down! Did this summary help you? Yes No

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

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Written by Marcus Guido

  • Teaching Strategies

“Test time. No calculators.”

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

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

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

Addition and Subtraction

1. two-step addition.

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

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

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

Broken down, the process is:

  • (393 + 7) + (89 – 7)

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

2. Two-Step Subtraction

There’s a similar method for subtraction.

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

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

Here’s the process:

  • (567 – 67) – (153 – 67)

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

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3. Subtracting from 1,000

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

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

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

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

Multiplication and Division

4. doubling and halving.

no image

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

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

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

The only prerequisite is understanding the 2 times table.

5. Multiplying by Powers of 2

This tactic is a speedy variation of doubling and halving.

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

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

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

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

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6. Multiplying by 9

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

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

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

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

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

7. Multiplying by 11

no image

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

no image

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

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

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

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

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

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

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

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

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

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

12. Calculating Percentages

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

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

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

This shortcut is a useful timesaver on tests and quizzes.

13. Balancing Averages

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

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

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

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

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

  • 10 hours – 6 hours = 4 hours

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

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Word Problems

14. identifying buzzwords.

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

This isn’t a trick. It’s a tactic.

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

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

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

15. Creating Sub-Questions

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

Each student should ask him or herself:

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

These sub-questions help students avoid overload.

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

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

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1.3: Problem Solving Strategies

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  • Michelle Manes
  • University of Hawaii

Think back to the first problem in this chapter, the ABC Problem. What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

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

How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985. [1]

George_Pólya_ca_1973.jpg

George Pólya, circa 1973

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

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

  • First, you have to understand the problem.
  • After understanding, then make a plan.
  • Carry out the plan.
  • Look back on your work. How could it be better?

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

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

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

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

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

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

Problem Solving Strategy 2 (Try Something!).

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

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

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

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem? This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

Problem Solving Strategy 3 (Draw a Picture).

Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?

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

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

Problem Solving Strategy 4 (Make Up Numbers).

Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

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

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

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

(Squares on a Chess Board)

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

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

Think / Pair / Share

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

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

Problem Solving Strategy 5 (Try a Simpler Problem).

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

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

Problem Solving Strategy 6 (Work Systematically).

If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

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

Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate).

Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

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

Problem Solving Strategy 8 (Look for and Explain Patterns).

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

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

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

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

(Broken Clock)

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

index-12_1-300x282-1.png

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

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

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

Problem Solving Strategy 9 (Find the Math, Remove the Context).

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

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

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

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

Problem Solving Strategy 10 (Check Your Assumptions).

When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

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

index-13_1-300x296.png

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

6 Tools to Help You Solve Difficult Math Problems

Math is fairly tricky, so what better way to get help than with the tech in your pocket? Here are six tools to help you solve difficult math problems.

There are tools that can help you to solve your math problems faster than you can count from one to ten. These math solvers are lifesavers for those who have mathematics anxiety, or aren't particularly strong at math.

Even if you're a math lover, these math solvers can still help you to deepen your love and appreciation for numbers. By simplifying math solutions, these tools encourage you to explore math even further.

In this article, we'll cover six such tools that can help you to solve your math problems faster.

how to use Mathway

Mathway is one of the most popular math solvers that provide students with the resources they need to understand and solve their math problems. To date, Mathway has been used to solve over 5 billion math problems.

Parents, teachers, and students use Mathway to solve different math problems in areas such as algebra, linear and quadratic equations, polynomials, inequalities, vector analysis, logarithms, matrices, and geometry, among others.

To use Mathway, simply head to the website, type your math problem in the Enter a problem area using the on-screen keyboard. Click Send and select how you want Mathway to answer. Your answer will be summarized. To view the detailed solution, click on Tap to view steps . You'll need to be signed in to view the solution steps.

Mathway is available as a web tool, as well as on Android and iOS.

2. Microsoft Math Solver

Microsoft Math Solver

Microsoft Math Solver is a free entry-level educational app developed and maintained by Microsoft. It's an intuitive tool that provides you with step-by-step solutions to your math problems. It was first released as an Edge preview feature .

Microsoft Math Solver can help you to solve math problems such as arithmetic, linear equations, simultaneous equations, quadratic equations, trigonometry, matrix, differentiation, integration, and limits.

Microsoft Math Solver covers topics in pre-algebra, algebra, trigonometry, and calculus. To try it out, head over to the website and click on the keyboard icon in the Type a math problem box. Using the on-screen keyboard, enter your math problem, then click Solve .

Related: How to Use Microsoft Edge to Solve Math Problems

It will process your math problem and give you answers. You can expand the answers to view the step-by-step solutions by clicking View solution steps .

Microsoft's Math Solver is available for the web, Android and iOS.

3. Math is Fun

Maths Is Fun

Math can be fun with the right tools and techniques. Math is Fun is a free web tool that explains math concepts in easy-to-understand language using puzzles, games, worksheets, and an illustrated dictionary suited for K-12 kids, teachers, and parents.

So, whether you're a teacher, parent of a K-12 kid, or simply a learner who wants to brush up on your math skills, Math is Fun is definitely the way to go. Here, there's something for everyone, be you in elementary school, middle school, high school, or even out of school.

The Math is Fun Forum is an active community with thousands of posts on math-related topics to support you in your journey. Here, you can even learn how to solve math problems using your computer.

Related: The Best iPhone Apps for Improving at Math

Math is Fun is a web tool that you can use on your browser without installing third-party apps, extensions, or add-ons. Go ahead and have some fun with math, using Math is Fun.

4. Zearn Math

Zearn

Zearn Math is a teacher-recommended math solver that uses a research-backed and evidence-driven approach to math education. It is designed for teachers and kids to help them to master, teach and learn K-5 mathematical concepts.

Grade 6 math is being rolled out for the 2021-22 school year, so expect to see topics like area and surface area, ratios, unit rates and percentages, dividing fractions, and arithmetic in base ten, among others.

Zearn Math is a free web-based tool that is reportedly being used by 1 in 4 elementary students across the US and is powered by Zearn, a non-profit educational organization.

5. K5 Learning

K5 Learning

K5 Learning is another valuable web-based resource focused on improving reading, science, and math skills for K-5 and even K-6 students. K5 Learning helps parents, educators, and students to excel at teaching and learning math using worksheets, flashcards, and math videos.

K5 Learning's worksheets cover a wide range of math topics from kindergarten to 6th grade. You can find worksheets for operations, fractions, decimals and percentages, measurement, geometry, exponents, proportions, and more.

K5 Learning is free and features a collection of free math worksheets that you can browse by grade and by subject. You can also access free math flashcards and inexpensive workbooks.

Related: The Best Free Online Homeschool Math Curriculums

K5 Learning works with leading teachers to improve its K-5 content. If you're one, you can either submit an article/story or create worksheets. To begin, go to the website, navigate to the footer area, and click Work with K5.

6. Symbolab

Symbolab

Last but not least is Symbolab, arguably one of the best math solvers out there. Symbolab is a computational engine that helps you to understand and solve complex college-level math problems.

Symbolab has helped students to solve over 1 billion math problems with clear step-by-step explanations and proofs. This helps not just to solve the problem at hand, but also to facilitate the learning and understanding of the underlying mathematical concepts.

To solve a problem using Symbolab, simply head to the site, enter the problem in the space provided and click Go . Symbolab covers topics in algebra, trigonometry, limits, derivatives, and integrals.

It also features different cheat sheets for easy reference. Symbolab is a web-based tool that supports up to six different languages and is also available on Android and iOS.

It's Much Easier to Solve Math Problems Than You Think

These math solvers are not meant to replace, but rather to complement your math teacher and to enhance your learning and understanding of mathematics. That being said, rather than using them to cheat, think of them as cheat sheets or shortcuts to better understanding math.

Math solvers can be handy for remote learning. It can help students get math help during the lockdown, weekends, holidays, or when their teachers are out of reach.

Learning and relearning math this way can help you to stimulate your mind and boost creativity. You can also check out our list of recommended websites where you can learn math with detailed step-by-step instructions.

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“Teachers should teach math in a way that encourages students to engage in sense-making and not merely to memorize or internalize exactly what the teacher says or does,” says Jon R. Star.

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One way is the wrong way to do math. Here’s the right way.

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Research by Ed School psychologist reinforces case for stressing multiple problem-solving paths over memorization

There’s never just one way to solve a math problem, says Jon R. Star , a psychologist and professor of education at the Harvard Graduate School of Education. With researchers from Vanderbilt University, Star found that teaching students multiple ways to solve math problems instead of using a single method improves teaching and learning. In an interview with the Gazette, Star, a former math teacher, outlined the research and explained how anyone, with the right instruction, can develop a knack for numbers.

Jon R. Star

GAZETTE: What is the most common misconception about math learning?

STAR: That you’re either a math person or you’re not a math person — that some people are just born with math smarts, and they can do math, and other people are just not, and there’s not much you can do about it.

GAZETTE: What does science say about the process of learning math?

STAR: One thing we know from psychology about the learning process is that the act of reaching into your brain, grabbing some knowledge, pulling it out, chewing on it, talking about it, and putting it back helps you learn. Psychologists call this elaborative encoding. The more times you can do that process — putting knowledge in, getting it out, elaborating on it, putting it back in — the more you will have learned, remembered, and understood the material. We’re trying to get math teachers to help students engage in that process of elaborative encoding.

GAZETTE: How did you learn math yourself?

STAR: Learning math should involve some sense-making. It’s necessary that we listen to what our teacher tells us about the math and try to make sense of it in our minds. Math learning is not about pouring the words directly from the teacher’s mouth into the students’ ears and brains. That’s not the way it works. I think that’s how I learned math. But that’s not how I hope students learn math and that’s not how I hope teachers think about the teaching of math. Teachers should teach math in a way that encourages students to engage in sense-making and not merely to memorize or internalize exactly what the teacher says or does.

GAZETTE: Tell us about the teaching method described in the research.

STAR: One of the strategies that some teachers may use when teaching math is to show students how to solve problems and expect that the student is going to end up using the same method that the teacher showed. But there are many ways to solve math problems; there’s never just one way.

The strategy we developed asks that teachers compare two ways for solving a problem, side by side, and that they follow an instructional routine to lead a discussion to help students understand the difference between the two methods. That discussion is really the heart of this routine because it is fundamentally about sharing reasoning: Teachers ask students to explain why a strategy works, and students must dig into their heads and try to say what they understand. And listening to other people’s reasoning reinforces the process of learning.

GAZETTE: Why is this strategy an improvement over just learning a single method?

STAR: We think that learning multiple strategies for solving problems deepens students’ understanding of the content. There is a direct benefit to learning through comparing multiple methods, but there are also other types of benefits to students’ motivation. In this process, students come to see math a little differently — not just as a set of problems, each of which has exactly one way to solve it that you must memorize, but rather, as a terrain where there are always decisions to be made and multiple strategies that one might need to justify or debate. Because that is what math is.

For teachers, this can also be empowering because they are interested in increasing their students’ understanding, and we’ve given them a set of tools that can help them do that and potentially make the class more interesting as well. It’s important to note, too, that this approach is not something that we invented. In this case, what we’re asking teachers to do is something that they do a little bit of already. Every high school math teacher, for certain topics, is teaching students multiple strategies. It’s built into the curriculum. All that we’re saying is, first, you should do it more because it’s a good thing, and second, when you do it, this is a certain way that we found to be especially effective, both in terms of the visual materials and the pedagogy. It’s not a big stretch for most teachers. Conversations around ways to teach math for the past 30 or 40 years, and perhaps longer, have been emphasizing the use of multiple strategies.

GAZETTE: What are the potential challenges for math teachers to put this in practice?

STAR: If we want teachers to introduce students to multiple ways to solve problems, we must recognize that that is a lot of information for students and teachers. There is a concern that there could be information overload, and that’s very legitimate. Also, a well-intentioned teacher might take our strategy too far. A teacher might say something like, “Well, if comparing two strategies is good, then why don’t I compare three or four or five?” Not that that’s impossible to do well. But the visual materials you would have to design to help students manage that information overload are quite challenging. We don’t recommend that.

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10 Strategies for Problem Solving in Math

Created: December 25, 2023

Last updated: January 6, 2024

strategies for problem solving in math

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

What Are Problem Solving Strategies in Math?

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

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

Strategies for Problem-solving in Math

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

Understand the Problem

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

Math for Kids

Guess and Check

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

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

Work It Out

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

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

Work Backwards

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

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

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

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

Find a Pattern

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

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

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

Draw a Picture or Diagram

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

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

Trial and Error Method

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

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

Review Answers with Peers

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

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

Check out the Printable Math Worksheets for Your Kids!

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

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

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Introduction to Algebra

Algebra is great fun - you get to solve puzzles!

What is the missing number?

OK, the answer is 6, right? Because 6 − 2 = 4 . Easy stuff.

Well, in Algebra we don't use blank boxes, we use a letter (usually an x or y, but any letter is fine). So we write:

It is really that simple. The letter (in this case an x) just means "we don't know this yet", and is often called the unknown or the variable .

And when we solve it we write:

Why Use a Letter?

So x is simply better than having an empty box. We aren't trying to make words with it!

And it doesn't have to be x , it could be y or w ... or any letter or symbol we like.

How to Solve

Algebra is just like a puzzle where we start with something like "x − 2 = 4" and we want to end up with something like "x = 6".

But instead of saying " obviously x=6", use this neat step-by-step approach:

  • Work out what to remove to get "x = ..."
  • Remove it by doing the opposite (adding is the opposite of subtracting)
  • Do that to both sides

Here is an example:

To remove it, do the opposite , in this case add 2

Do it to both sides

Which is ...

Why did we add 2 to both sides?

To "keep the balance"....

Just remember this:

See this in action at the Algebra Balance Animation .

Another Puzzle

What we want is an answer like "x = ...", but the +5 is in the way of that! We can cancel out the +5 with a −5 (because 5−5=0)

Have a Try Yourself

Now practice on this Simple Algebra Worksheet and then check your answers. Try to use the steps we have shown you here, rather than just guessing!

Try the questions below, then read Introduction to Algebra - Multiplication

Mathway: Math Problem Solver 4+

Homework scanner & calculator, chegg, inc..

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Mathway is the world’s smartest math calculator for algebra, graphing, calculus and more! Mathway gives you unlimited access to math solutions that can help you understand complex concepts. Simply point your camera and snap a photo or type your math homework question for step-by-step answers. If a premium subscription option is selected: • Payment will be charged to iTunes Account at confirmation of purchase • Subscription automatically renews unless auto-renew is turned off at least 24-hours before the end of the current period • Account will be charged for renewal within 24-hours prior to the end of the current period, at the same monthly or annual rate selected at the beginning of the subscription • Subscriptions may be managed by the user and auto-renewal may be turned off by going to the user's Account Settings after purchase Terms of Use: https://www.mathway.com/terms Privacy Policy: https://www.mathway.com/privacy

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This update includes bug fixes and performance improvements. Powered by Chegg, Mathway is your one-stop shop for homework help. Is your equation too complex to type? Save time with our snap and solve feature! Just take a photo and instantly see the solution. Our interactive calculator also allows you to select your class subject (now including physics!) to generate all the functions you need to solve that equation. Need more than an answer? Upgrade your subscription and get unlimited access to detailed equation breakdowns, helping you learn how to solve it one step at a time.

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Recommend!!!!!!?

Once I was doing my homework I couldn’t figure it out I asked my mom to help me and I showed her and she fell asleep so I kind of started panicking Went downstairs to the table I started struggling and thinking and thinking so I went to the App Store looked and looked and looked and looked left to right for helping your homeworkIs the best I really do recommend it it is the best it will save your life and when I mean save your life I mean really save your life nobody really understands when they say that they saved their lifeThey never understand how much that person means they don’t even know how serious the persons like saying I really do recommend this app messed up in the world five star review amazing helpful save your life get it get it now get it now it’s the best app ever students allowed teacher teachers not allowed can students really need this app so don’t get mad at them if they have this app it’s the best time with them so hope you get it

All it does is give the answer

Seriously saying, this app is literally just useless unless if you know the answer to math problems and want to check your answers. Otherwise, paying wouldn’t be a consideration for me. For those who are on a tight budget, and have a strong desire to reach their future career, I’d say paying wouldn’t be worth it, especially if you are just doing a living only as a student with no career and if you do not have enough financial support. I’ve lived a long way with this app studying all day and all night and yet there is high demand for paying just to get the work shown to get the idea why a question is answered in a certain way. This is why nowadays I read reviews from various books titled for math and read books that give a full explanation of how an answer is earned. As a life lesson it’s just good to say the fact that spending is not worth it for this app since there are various places and apps to ask questions related to learning math (at least in my area). This app should be like a substitute teacher or tutor for those who are having trouble, but the fact that there is such huge fee just ruins the reason why I should install this app. Teachers ask for showing work, and yet I’m stuck, having no idea what to do to get the right answers. This also goes to understanding how an answer is caught as well.

It works but the full version hasn’t for me

I have been using this a lot this semester so I decided to pay for the monthly subscription then I could cancel the membership at the end of the semester, but it doesn’t even work. The subscription shows in my subscriptions list in my settings but I still can’t view the steps, and when I attempt to see how the problem is solved it brings me to the menu to subscribe to the paid part of the app. Honestly though, other than that. This app works pretty great, solves problems the right way and you can select the method of how you want the problem solved. You can change what kind of math you are trying to solve, and the answers are pretty spot on. The predictive text is even really good on this, it’ll help solve word problems and you kind of don’t em have to type anything in sometimes. However, the keyboard is very buggy and you can’t type very fast. It’s also kind of hard to navigate as there are like 4 different keyboard menus to choose from, I always forget where to find things and it kind of takes a while to load the text sometimes. But it does solve the problems.

Developer Response ,

Sorry to hear of the issues you're having - please select 'restore subscription' from the settings of your Mathway account. This should turn the steps "ON". For further assistance you can reach Mathway customer support by emailing: [email protected].

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Maths Tricks

Maths tricks are the ways to solve complex mathematical problems easily and quickly.  Mathematics is not only limited to learning from textbooks, there are different learning styles that make mathematics easier. Simple Maths magic tricks help us with fast calculations and improve our mathematical skills.  For example, the multiplication tricks will help students to learn maths tables and quick multiplication.

Maths is not easy for some students. The Maths tricks are not only helpful for school-going kids but also supports you to manage time in final exams as well as in the comp etitive exam and solve the Maths questions with accuracy. These tricks are very helpful for Class 5, 6, 7, 8, 9 and 10 students.

Learning Quick Maths Tricks

Mathematical tricks are the best way to make Maths a fun subject. Therefore, learning simple Maths tricks will help the students to gain their confidence and enhance problem-solving skills. With these learning skills, they can achieve a big success in the competitive exams and upcoming future.    

16 Maths Tricks (With Examples)

Imagine how mathematics would be easy and interesting when you have the ability to calculate the problems in a matter of seconds using some Maths tricks. There are different kinds of arithmetic operations  like addition, subtraction, division, multiplication, squaring, roots, powers, logarithms, divisions, etc. Here are some of the best tricks, which will help students to perform arithmetic calculations easily.

1. Maths Tricks for Addition

With the help of basic principles of tens and unit places, the addition of two-digit numbers is performed by

  • Take 43 + 34
  • Split the second number into tens and unit places. 34 = 30 + 4
  • Finish the ten’s addition. 43 + 30 = 73
  • Finally, add the remaining unit place digit. 73 + 4 = 77.

2. Maths Tricks for Subtraction

Here is an example that requires a lot of borrowing

  • Consider two numbers say 1000 and 676
  • Subtract 1 from both the numbers; we get 999 and 675
  • Then subtract 675 from 999, we get 324
  • So, 1000 – 676 = 324.

 3.  Quick Multiplication Tricks by Breaking Down Numbers

  • Let’s try the numbers 24 and 16
  • First split the number 24, which gives 4 x 6
  • Then multiply 6 with 16, we get 96
  • Finally multiply the number, 96 x 4 = 384
  • So, the multiplication of two numbers 24 x 16 gives the solution 384.

4. Multiplied By 15

  • Consider the multiplication of two numbers say 56 and 15
  • Now add zero at the end of the first number, it becomes 560.
  • Divide that number by 2; we get 560/2 = 280
  • Add the resultant number with 560, so 560 + 280 = 840.
  • So the answer for 56 and 15 is 840.

5. Multiplication of Two-Digit Numbers

If anyone of the given numbers is an even number, then follow the steps to solve

  • Consider an example, 18 x 37
  • Here 18 is an even number, then divide the first number in half, so that 18/2 = 9
  • Then double the second number. 37 x 2 =74
  • Finally, multiply the resultant numbers. It becomes 74 x 9 = 666

6. Maths Division Tricks

The numbers that can be evenly divided by certain numbers are:

  • If a number is an even number and ends in 0, 2, 4, 6 or 8, it is divided by 2.
  • A number is divisible by 3 if the sum of the digits is divisible by 3. Consider the number 12 = 1 + 3 and 3 is divisible by 3.
  • A number is divisible by 4 if the last two digits are divisible by 4. Example: 9312. Here the last two digits are 12, and 12 is divisible by 4.
  • If the last digit is 0 or 5, it is divisible by 5
  • If a number is divisible by 2 and 3, then it is divisible by 6, since 6 is the product of 2 and 3.
  • If the number is divisible by 8, the last three digits of the numbers are divisible by 8.
  • If a number is divisible by 9, the sum of the digits is divided by 9. Let us consider the example, 4518 = 4 + 5 + 1 + 8 = 18, which is divisible by 9.
  • If the final digit of the number is 0, it is divisible by 10.

7. Maths Trick to Find Percentage

Let us take; we have to find the percentage of the number 5% of 475, follow the steps.

  • For the given number, move the decimal point over by one place. 475 becomes 47.5
  • Then divide the number 47.5 by 2, we get 23.75.
  • 23.75 is the solution to the given problem.

8. Maths Magic tricks to Calculate Squares ending with digit 5

  • Let’s consider the number 75 to find its square.
  • Start writing the answer of last two digits number that is 25 because any number that ends with 5 is 25
  • Take the first digit of the number 75. That is 7 and take the number that follows 7 is 8.
  • Now, multiply 7 and 8, we get the number 56.
  • Finally, write the number 56 in the prefix and combine with 25 what we already wrote.
  • So, the answer is 5625.
  • Squares Ending in 5:  n5 = n(n + 1)5 2 = n(n + 1)25 , where n is the first digit.
  • Example: Let’s consider the number 75 to find its square. Here n = 7,

So, 75 = 7(7 + 1)25 = (7 x 8) 25  = 5625.

9. Tricks to Multiply by 2 and 4

When a number is multiplied by 2 or 4, then the last digit of the resulting value will be an even number always.

  • 19 x 2 = 38
  • 19 x 4 = 76

10. Multiplication by 5

When a number is multiplied by 5, then the resulting value will either end with 0 or 5.

  • 11 x 5 = 55
  • 121 x 5 = 605

11. Multiplication by 10

When a number is multiplied by 10, then the resulting value ends with 0 always.

  • 5 x 10 = 50
  • 10 x 10 = 100
  • 11 x 10 = 110
  • 17 x 10 = 170
  • 211 x 10 = 2110

12. Tricks to Memorise Table of 9

It is easy to remember the table of 9. Just we need to focus on the pattern.

09, 18, 27, 36, 45, 54, 63, 72, 81, 90

We can see the numbers at the ten’s place are increasing by 1, and the numbers at the unit place are decreasing by 1.

13. Trick for Squaring a Two-digit Number

Given number: 57 

Here, Tens digit = 5 Unit digit = 7

7 + 57 = 64

64 × 5 = 320

3200 + 49 = 3249

14. Trick for Unit Digit of a Cube Number

Below are the numbers which have the same unit digit for their cubes.

This will help you in finding the cube root of a number (which is a definite cube). 

15. Trick for Cube Root of a Number

Let’s consider a number: 117649

To find the cube root of this number, we need to divide the number into parts.

Take the three rightmost digits as one part and the remaining digits as another part.

Part 1: 649

Here, the unit digit is 9, and we know the cube root of this number will contain 9 as its unit digit.

Part 2: 117

Identify the cube of a number which is smaller than 117.

4 3 = 64 and 5 3 = 125

Thus, the cube root of 117649 is 49.

Note: This trick will work for finding the cube root of definite cube numbers.

16. Trick for Multiplication Table of 15

Here is the trick for writing the 15 times table quickly.

Maths is a fun subject. Adding tricks to this subject will make it more interesting. Students will be able to solve all the complex problems using these Maths magic tricks. These tricks also help students to improve their problem-solving skills and boost their confidence.

Maths Tricks Practice Questions

Find more maths tricks questions for practice here.

  • A merchant can place 8 large boxes or 10 small boxes into a carton for shipping. In one shipment, he sent a total of 96 boxes. If there are more large boxes than small boxes, how many cartons did he ship? (A) 11 (B) 10  (C) 12   (D) 15 (E) 17
  • 339% of 803 + 77.8% of 1107 = ? (A) 3175 (B) 3320 (C) 3580 (D) 3710 (E) 3950
  • 78.54 ÷ 0.03 + 22.8 ÷ 0.8 – 1470 × 1.25 = ? (A) 809 (B) 807.5 (C) 805 (D) 802.5 (E) 801
  • The cost of 8 dozen eggs is Rs. 256. Which calculation is needed to find the cost of 9 eggs? (A) (9 × 256) × (8 ÷ 12) (B) (12 × 256) ÷ (8 × 9) (C) (8 × 256) ÷ (9 × 12) (D) (9 × 256) × (8 × 12) (E) (9 × 256) ÷ (8 × 12)

Maths Tricks Related Articles

Frequently asked questions on maths tricks, what is the use of maths tricks, what are multiplication tricks, how to add in a faster way, how to multiply quickly, how to find if a number is divisible by 9, how to find if a number is divisible by 13, how to multiply 5 with an even number.

Assume that, 5 should be multiplied by 48. Follow the below steps to find the product value. Step 1: Divide the given even number by 2. Hence, 48 divided by 2 is 24. Step 2: Add zero to the result, obtained in step 1 to get the product value. So, we get 240. Therefore, the product of 5 and 48 is 240.

How to multiply 5 with an odd number?

Let us consider an example, 5 × 25. Here, 25 is an odd number. Go through the below steps to find the product. Step 1: Subtract 1 from the odd number (i.e. the number that is being multiplied by 5). So, 25 – 1 = 24. Step 2: Half the number that is obtained in step 1. Hence, 24/2 = 12. Step 3: Keep the number 5 in unit digits and append the result obtained in step 2. Hence, we get 125. So, we get 5 × 25 is 125.

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Research shows the best ways to learn math.

New Stanford paper says speed drills and timed testing in math can be damaging for students. (Cherries/Shutterstock)

Students learn math best when they approach the subject as something they enjoy. Speed pressure, timed testing and blind memorization pose high hurdles in the pursuit of math, according to Jo Boaler, professor of mathematics education  at Stanford Graduate School of Education and lead author on a new working paper called "Fluency Without Fear."

"There is a common and damaging misconception in mathematics – the idea that strong math students are fast math students," said Boaler, also cofounder of YouCubed at Stanford, which aims to inspire and empower math educators by making accessible in the most practical way the latest research on math learning.

Fortunately, said Boaler , the new national curriculum standards known as the Common Core Standards for K-12 schools de-emphasize the rote memorization of math facts. Maths facts are fundamental assumptions about math, such as the times tables (2 x 2 = 4), for example. Still, the expectation of rote memorization continues in classrooms and households across the United States.

While research shows that knowledge of math facts is important, Boaler said the best way for students to know math facts is by using them regularly and developing understanding of numerical relations. Memorization, speed and test pressure can be damaging, she added.

Number sense is critical

On the other hand, people with "number sense" are those who can use numbers flexibly, she said. For example, when asked to solve the problem of 7 x 8, someone with number sense may have memorized 56, but they would also be able to use a strategy such as working out 10 x 7 and subtracting two 7s (70-14).

"They would not have to rely on a distant memory," Boaler wrote in the paper.

In fact, in one research project the investigators found that the high-achieving students actually used number sense, rather than rote memory, and the low-achieving students did not.

The conclusion was that the low achievers are often low achievers not because they know less but because they don't use numbers flexibly.

"They have been set on the wrong path, often from an early age, of trying to memorize methods instead of interacting with numbers flexibly," she wrote. Number sense is the foundation for all higher-level mathematics, she noted.

Role of the brain

Boaler said that some students will be slower when memorizing, but still possess exceptional mathematics potential.

"Math facts are a very small part of mathematics, but unfortunately students who don't memorize math facts well often come to believe that they can never be successful with math and turn away from the subject," she said.

Prior research found that students who memorized more easily were not higher achieving – in fact, they did not have what the researchers described as more "math ability" or higher IQ scores. Using an MRI scanner, the only brain differences the researchers found were in a brain region called the hippocampus, which is the area in the brain responsible for memorizing facts – the working memory section.

But according to Boaler, when students are stressed – such as when they are solving math questions under time pressure – the working memory becomes blocked and the students cannot as easily recall the math facts they had previously studied. This particularly occurs among higher achieving students and female students, she said.

Some estimates suggest that at least a third of students experience extreme stress or "math anxiety" when they take a timed test, no matter their level of achievement. "When we put students through this anxiety-provoking experience, we lose students from mathematics," she said.

Math treated differently

Boaler contrasts the common approach to teaching math with that of teaching English. In English, a student reads and understands novels or poetry, without needing to memorize the meanings of words through testing. They learn words by using them in many different situations – talking, reading and writing.

"No English student would say or think that learning about English is about the fast memorization and fast recall of words," she added.

Strategies, activities

In the paper, coauthored by Cathy Williams, cofounder of YouCubed, and Amanda Confer, a Stanford graduate student in education, the scholars provide activities for teachers and parents that help students learn math facts at the same time as developing number sense. These include number talks, addition and multiplication activities, and math cards.

Importantly, Boaler said, these activities include a focus on the visual representation of number facts. When students connect visual and symbolic representations of numbers, they are using different pathways in the brain, which deepens their learning, as shown by recent brain research.

"Math fluency" is often misinterpreted, with an over-emphasis on speed and memorization, she said. "I work with a lot of mathematicians, and one thing I notice about them is that they are not particularly fast with numbers; in fact some of them are rather slow. This is not a bad thing; they are slow because they think deeply and carefully about mathematics."

She quotes the famous French mathematician, Laurent Schwartz. He wrote in his autobiography that he often felt stupid in school, as he was one of the slowest math thinkers in class.

Math anxiety and fear play a big role in students dropping out of mathematics, said Boaler.

"When we emphasize memorization and testing in the name of fluency we are harming children, we are risking the future of our ever-quantitative society and we are threatening the discipline of mathematics," she said. "We have the research knowledge we need to change this and to enable all children to be powerful mathematics learners. Now is the time to use it."

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11 Helpful Math Tricks You Need to Know

I magine being able to slowly rewire your brain to become more logical and more efficient at solving problems. Many people are surprised to learn this occurs when they study mathematics. Math tends to be one of those subjects that people either love or hate.

Regardless of your level of enthusiasm, there are key math tips that can help you navigate difficult problems more fluidly. Let’s explore some of the most notable math tricks to keep in mind.

1. Multiplying by 6

This is one of the best ways to improve your fast math skills. When you multiply six by an even number, the answer you get will always end in the same digit as the number you multiply it by. For instance, imagine you multiply six by four. The answer is 24.

Multiplying six by two, four, six, and eight produces similar results. Keep this tip in mind so you can streamline your workflow. The less often you have to use a calculator, the better.

2. Multiplying 5 Times Any Number

If you’re multiplying five by an even number, take the number you’re multiplying five by and halve it. If you’re multiplying five by four, you’d reduce four to two. You then add zero to that number.

In the above example, you’d get 20. If you’re multiplying five by an odd number, you subtract one from that number. Multiplying five by five would require you to briefly multiply five by four.

You then halve this number and add five as the last digit. You’d get 25 by using this method for the aforementioned problem.

3. Squaring Numbers Ending in Five

This one’s a bit more specific than the others on this list, but it’s still fairly useful. Let’s consider 35 in this example. First, multiply the first digit by itself.

You then add one to this number. Afterwards, put 25 at the end. This would give you 12 and 25. Combine these two numbers to get 1,225.

In the situations where this tip is applicable, you’ll find it saves you a substantial amount of time. Numbers that end in five are some of the most common you’ll encounter.

4. Subtracting from 1,000

Dealing with the number 1,000 is something most people do daily. This is true when calculating expenses, paying bills, etc. When subtracting a number from 1,000, subtract each number from nine except for the last digit. Instead, subtract the last number from 10.

For example, assume you subtract 365 from 1,000. You would subtract three from nine, subtract six from nine, and then subtract five from 10. The answer would be 635. Try it out with different numbers to see for yourself!

5. Multiplying by 9

Need to quickly multiply a number by nine? Let’s assume you’re multiplying nine by three. Subtract one from the number you’re multiplying by nine. For context, you’d subtract one from three to get two.

Then, subtract that number from nine. In this case, subtracting two from nine will yield seven. Combine these two numbers to get the result.

To clarify, multiplying nine by three would get you 27.

6. Multiplying Numbers Ending in Zero

This is a great way to make large numbers seem much more manageable. It simply involves multiplying the other digits of the number and adding extra zeros at the end. Let’s consider 200 multiplied by 400.

You’d multiply two by four and then add the total number of zeros in the original problem at the end. When working with 200 x 400, there are four zeros total. If you follow these instructions correctly in this problem, you’d get 80,000.

7. Division Tips

You can make division much easier for yourself when you quickly understand when a number can be evenly divided by certain other numbers. If a number ends in zero, you can divide it by 10. When dividing large numbers, add up the digits together.

If they’re evenly divisible by nine, then so is the original number. If the last three digits of a number are evenly divisible by eight, you can divide the original number by eight.

The same is true if the number ends with three zeros. Add up the digits of an even number you’re working with. If this result is divisible by three, then you can divide the original by three evenly.

To check if a number is divisible by five, it simply needs to end in a zero or a five. Numbers that end in two zeros or two digits that are divisible by four are also divisible by four.

You can assess if a number is divisible by three if you add up the number’s digits and they’re also divisible by three. Numbers that end in zero, two, four, six, or eight are divisible by two.

9. Finding Percentages

Many people hate working with percentages, but there are ways to streamline this process. Imagine you need to find 5% of 235.

Move the decimal one place to the left, which would give us 23.5 in this case. You can then divide this number by two to get the answer to the original problem.

10. Complex Multiplication

You can quickly multiply large numbers by using this technique. However, it only works if one of the numbers is even. Halve the even number and double the other number. When multiplying 20 x 135, this would briefly create 10 x 270, which in turn yields 2,700.

11. Finding Least Common Multiples

Using the least common multiple solving techniques is essential for keeping your workflow quick and free of interruptions. First, find the greatest common factor (GCF). This is the largest factor that all numbers share.

So, the GCF of 18 and 30 is six. Divide either number by the GCF. Multiply the answer by the remaining number. For instance, you could use 18 divided by six.

This will yield three, which you would multiply by 30 to get 90.

Keep These Math Tricks in Mind

The math tricks in this guide will ensure you overcome issues you would’ve otherwise struggled with. Whether you learn math as part of your job or curriculum, you can take your performance to the next level.

Looking for other useful lifestyle information that can help you out in the future? Our blog has plenty of articles like this one. Be sure to check them out today!

This article is published by NYTech in collaboration with Syndication Cloud.

11 Helpful Math Tricks You Need to Know

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Solving word problem chart

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.

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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:

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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:

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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:

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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:

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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

A word problem game

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:

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2. Practice Regularly with Diverse Problems

Word problem worksheet

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:

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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.

the best way to solve math problems

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the best way to solve math problems

Introduction:

Math is an essential skill for everyone, but it can be challenging at times, leading to rapid errors. One of the best strategies for becoming adept at solving math problems is to check your work carefully, ensuring accuracy and building confidence. In this article, we will explore three simple ways to check your math problems easily and effectively.

1. Backward-Checking:

One of the most straightforward methods for checking your work in a math problem involves working backward from the answer you have found. This method is particularly efficient when applied to algebraic equations and word problems.

To work backward, simply start with your computed solution and employ the inverse process that you utilized in solving the problem. For example, if the original question asked you to multiply two numbers together, divide your answer by one of those numbers to see if it equates to the other factor. If the process leads you back to the original equation, it’s likely that you’ve found the correct answer.

2. Estimation:

Estimation is another helpful technique for verifying whether your solution is plausible or not. This method involves approximating what a reasonable answer would be by using rounded or simplified values in the problem.

To use estimation, round each number in your problem according to its most significant digit, then perform calculations using these simplified numbers to obtain a rough estimate of the solution. Then, compare this estimated answer with your actual result. If both are close enough, chances are you have a correct solution.

For example, let’s say you are asked to multiply 87 by 12:

– Round 87 to 90 and 12 to 10

– Multiply 90 by 10, resulting in: 900

Now compare this value (900) with the precise answer (1044); they are close enough so that you can reasonably conclude that your actual solution is likely correct.

3. Cross-Checking:

Cross-checking is yet another technique used to verify your math problem solutions. It involves comparing different solutions to the same problem acquired using alternate methods. If both methods produce the same result, it’s a clear indicator that your answer is accurate.

For example, suppose you have a system with two linear equations:

– Method 1: Solve the system using substitution

– Method 2: Solve the system using elimination

If both methods produce the same solution, it’s safe to say that you have reached a correct answer.

Conclusion:

Checking your work while solving math problems is a fundamental skill that helps ensure accuracy and deepens your understanding of the subject. Practicing these three techniques – backward-checking, estimation, and cross-checking – will make verifying your math problems much more manageable and enable you to become more confident in your mathematical abilities.

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