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Free Math Worksheets — Over 100k free practice problems on Khan Academy

Looking for free math worksheets.

You’ve found something even better!

That’s because Khan Academy has over 100,000 free practice questions. And they’re even better than traditional math worksheets – more instantaneous, more interactive, and more fun!

Just choose your grade level or topic to get access to 100% free practice questions:

Kindergarten, basic geometry, pre-algebra, algebra basics, high school geometry.

  • Trigonometry

Statistics and probability

High school statistics, ap®︎/college statistics, precalculus, differential calculus, integral calculus, ap®︎/college calculus ab, ap®︎/college calculus bc, multivariable calculus, differential equations, linear algebra.

  • Addition and subtraction
  • Place value (tens and hundreds)
  • Addition and subtraction within 20
  • Addition and subtraction within 100
  • Addition and subtraction within 1000
  • Measurement and data
  • Counting and place value
  • Measurement and geometry
  • Place value
  • Measurement, data, and geometry
  • Add and subtract within 20
  • Add and subtract within 100
  • Add and subtract within 1,000
  • Money and time
  • Measurement
  • Intro to multiplication
  • 1-digit multiplication
  • Addition, subtraction, and estimation
  • Intro to division
  • Understand fractions
  • Equivalent fractions and comparing fractions
  • More with multiplication and division
  • Arithmetic patterns and problem solving
  • Quadrilaterals
  • Represent and interpret data
  • Multiply by 1-digit numbers
  • Multiply by 2-digit numbers
  • Factors, multiples and patterns
  • Add and subtract fractions
  • Multiply fractions
  • Understand decimals
  • Plane figures
  • Measuring angles
  • Area and perimeter
  • Units of measurement
  • Decimal place value
  • Add decimals
  • Subtract decimals
  • Multi-digit multiplication and division
  • Divide fractions
  • Multiply decimals
  • Divide decimals
  • Powers of ten
  • Coordinate plane
  • Algebraic thinking
  • Converting units of measure
  • Properties of shapes
  • Ratios, rates, & percentages
  • Arithmetic operations
  • Negative numbers
  • Properties of numbers
  • Variables & expressions
  • Equations & inequalities introduction
  • Data and statistics
  • Negative numbers: addition and subtraction
  • Negative numbers: multiplication and division
  • Fractions, decimals, & percentages
  • Rates & proportional relationships
  • Expressions, equations, & inequalities
  • Numbers and operations
  • Solving equations with one unknown
  • Linear equations and functions
  • Systems of equations
  • Geometric transformations
  • Data and modeling
  • Volume and surface area
  • Pythagorean theorem
  • Transformations, congruence, and similarity
  • Arithmetic properties
  • Factors and multiples
  • Reading and interpreting data
  • Negative numbers and coordinate plane
  • Ratios, rates, proportions
  • Equations, expressions, and inequalities
  • Exponents, radicals, and scientific notation
  • Foundations
  • Algebraic expressions
  • Linear equations and inequalities
  • Graphing lines and slope
  • Expressions with exponents
  • Quadratics and polynomials
  • Equations and geometry
  • Algebra foundations
  • Solving equations & inequalities
  • Working with units
  • Linear equations & graphs
  • Forms of linear equations
  • Inequalities (systems & graphs)
  • Absolute value & piecewise functions
  • Exponents & radicals
  • Exponential growth & decay
  • Quadratics: Multiplying & factoring
  • Quadratic functions & equations
  • Irrational numbers
  • Performing transformations
  • Transformation properties and proofs
  • Right triangles & trigonometry
  • Non-right triangles & trigonometry (Advanced)
  • Analytic geometry
  • Conic sections
  • Solid geometry
  • Polynomial arithmetic
  • Complex numbers
  • Polynomial factorization
  • Polynomial division
  • Polynomial graphs
  • Rational exponents and radicals
  • Exponential models
  • Transformations of functions
  • Rational functions
  • Trigonometric functions
  • Non-right triangles & trigonometry
  • Trigonometric equations and identities
  • Analyzing categorical data
  • Displaying and comparing quantitative data
  • Summarizing quantitative data
  • Modeling data distributions
  • Exploring bivariate numerical data
  • Study design
  • Probability
  • Counting, permutations, and combinations
  • Random variables
  • Sampling distributions
  • Confidence intervals
  • Significance tests (hypothesis testing)
  • Two-sample inference for the difference between groups
  • Inference for categorical data (chi-square tests)
  • Advanced regression (inference and transforming)
  • Analysis of variance (ANOVA)
  • Scatterplots
  • Data distributions
  • Two-way tables
  • Binomial probability
  • Normal distributions
  • Displaying and describing quantitative data
  • Inference comparing two groups or populations
  • Chi-square tests for categorical data
  • More on regression
  • Prepare for the 2020 AP®︎ Statistics Exam
  • AP®︎ Statistics Standards mappings
  • Polynomials
  • Composite functions
  • Probability and combinatorics
  • Limits and continuity
  • Derivatives: definition and basic rules
  • Derivatives: chain rule and other advanced topics
  • Applications of derivatives
  • Analyzing functions
  • Parametric equations, polar coordinates, and vector-valued functions
  • Applications of integrals
  • Differentiation: definition and basic derivative rules
  • Differentiation: composite, implicit, and inverse functions
  • Contextual applications of differentiation
  • Applying derivatives to analyze functions
  • Integration and accumulation of change
  • Applications of integration
  • AP Calculus AB solved free response questions from past exams
  • AP®︎ Calculus AB Standards mappings
  • Infinite sequences and series
  • AP Calculus BC solved exams
  • AP®︎ Calculus BC Standards mappings
  • Integrals review
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  • Thinking about multivariable functions
  • Derivatives of multivariable functions
  • Applications of multivariable derivatives
  • Integrating multivariable functions
  • Green’s, Stokes’, and the divergence theorems
  • First order differential equations
  • Second order linear equations
  • Laplace transform
  • Vectors and spaces
  • Matrix transformations
  • Alternate coordinate systems (bases)

Frequently Asked Questions about Khan Academy and Math Worksheets

Why is khan academy even better than traditional math worksheets.

Khan Academy’s 100,000+ free practice questions give instant feedback, don’t need to be graded, and don’t require a printer.

What do Khan Academy’s interactive math worksheets look like?

Here’s an example:

What are teachers saying about Khan Academy’s interactive math worksheets?

“My students love Khan Academy because they can immediately learn from their mistakes, unlike traditional worksheets.”

Is Khan Academy free?

Khan Academy’s practice questions are 100% free—with no ads or subscriptions.

What do Khan Academy’s interactive math worksheets cover?

Our 100,000+ practice questions cover every math topic from arithmetic to calculus, as well as ELA, Science, Social Studies, and more.

Is Khan Academy a company?

Khan Academy is a nonprofit with a mission to provide a free, world-class education to anyone, anywhere.

Want to get even more out of Khan Academy?

Then be sure to check out our teacher tools . They’ll help you assign the perfect practice for each student from our full math curriculum and track your students’ progress across the year. Plus, they’re also 100% free — with no subscriptions and no ads.

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

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

Download on App Store

What can QuickMath do?

QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

  • The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
  • The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
  • The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
  • The calculus section will carry out differentiation as well as definite and indefinite integration.
  • The matrices section contains commands for the arithmetic manipulation of matrices.
  • The graphs section contains commands for plotting equations and inequalities.
  • The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

Math Topics

More solvers.

  • Add Fractions
  • Simplify Fractions

Solving Equations

What is an equation.

An equation says that two things are equal. It will have an equals sign "=" like this:

That equations says:

what is on the left (x − 2)  equals  what is on the right (4)

So an equation is like a statement " this equals that "

What is a Solution?

A Solution is a value we can put in place of a variable (such as x ) that makes the equation true .

Example: x − 2 = 4

When we put 6 in place of x we get:

which is true

So x = 6 is a solution.

How about other values for x ?

  • For x=5 we get "5−2=4" which is not true , so x=5 is not a solution .
  • For x=9 we get "9−2=4" which is not true , so x=9 is not a solution .

In this case x = 6 is the only solution.

You might like to practice solving some animated equations .

More Than One Solution

There can be more than one solution.

Example: (x−3)(x−2) = 0

When x is 3 we get:

(3−3)(3−2) = 0 × 1 = 0

And when x is 2 we get:

(2−3)(2−2) = (−1) × 0 = 0

which is also true

So the solutions are:

x = 3 , or x = 2

When we gather all solutions together it is called a Solution Set

The above solution set is: {2, 3}

Solutions Everywhere!

Some equations are true for all allowed values and are then called Identities

Example: sin(−θ) = −sin(θ) is one of the Trigonometric Identities

Let's try θ = 30°:

sin(−30°) = −0.5 and

−sin(30°) = −0.5

So it is true for θ = 30°

Let's try θ = 90°:

sin(−90°) = −1 and

−sin(90°) = −1

So it is also true for θ = 90°

Is it true for all values of θ ? Try some values for yourself!

How to Solve an Equation

There is no "one perfect way" to solve all equations.

A Useful Goal

But we often get success when our goal is to end up with:

x = something

In other words, we want to move everything except "x" (or whatever name the variable has) over to the right hand side.

Example: Solve 3x−6 = 9

Now we have x = something ,

and a short calculation reveals that x = 5

Like a Puzzle

In fact, solving an equation is just like solving a puzzle. And like puzzles, there are things we can (and cannot) do.

Here are some things we can do:

  • Add or Subtract the same value from both sides
  • Clear out any fractions by Multiplying every term by the bottom parts
  • Divide every term by the same nonzero value
  • Combine Like Terms
  • Expanding (the opposite of factoring) may also help
  • Recognizing a pattern, such as the difference of squares
  • Sometimes we can apply a function to both sides (e.g. square both sides)

Example: Solve √(x/2) = 3

And the more "tricks" and techniques you learn the better you will get.

Special Equations

There are special ways of solving some types of equations. Learn how to ...

  • solve Quadratic Equations
  • solve Radical Equations
  • solve Equations with Sine, Cosine and Tangent

Check Your Solutions

You should always check that your "solution" really is a solution.

How To Check

Take the solution(s) and put them in the original equation to see if they really work.

Example: solve for x:

2x x − 3 + 3 = 6 x − 3     (x≠3)

We have said x≠3 to avoid a division by zero.

Let's multiply through by (x − 3) :

2x + 3(x−3) = 6

Bring the 6 to the left:

2x + 3(x−3) − 6 = 0

Expand and solve:

2x + 3x − 9 − 6 = 0

5x − 15 = 0

5(x − 3) = 0

Which can be solved by having x=3

Let us check x=3 using the original question:

2 × 3 3 − 3 + 3  =   6 3 − 3

Hang On: 3 − 3 = 0 That means dividing by Zero!

And anyway, we said at the top that x≠3 , so ...

x = 3 does not actually work, and so:

There is No Solution!

That was interesting ... we thought we had found a solution, but when we looked back at the question we found it wasn't allowed!

This gives us a moral lesson:

"Solving" only gives us possible solutions, they need to be checked!

  • Note down where an expression is not defined (due to a division by zero, the square root of a negative number, or some other reason)
  • Show all the steps , so it can be checked later (by you or someone else)

Solver Title

Practice

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  • Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Mean, Median & Mode
  • Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Logical Sets Word Problems
  • Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
  • Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
  • Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
  • Linear Algebra Matrices Vectors
  • Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
  • Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
  • Physics Mechanics
  • Chemistry Chemical Reactions Chemical Properties
  • Finance Simple Interest Compound Interest Present Value Future Value
  • Economics Point of Diminishing Return
  • Conversions Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time
  • Pre Algebra
  • One-Step Addition
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  • Two-Step Integers
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  • Multi-Step with Parentheses
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  • Solve by Factoring
  • Completing the Square
  • Quadratic Formula
  • Biquadratic
  • Logarithmic
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  • Rational Roots
  • Floor/Ceiling
  • Equation Given Roots
  • Newton Raphson
  • Substitution
  • Elimination
  • Cramer's Rule
  • Gaussian Elimination
  • System of Inequalities
  • Perfect Squares
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Polynomials
  • Distributive Property
  • FOIL method
  • Perfect Cubes
  • Binomial Expansion
  • Negative Rule
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  • Exponent Rules
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  • Absolute Value
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  • Powers of i
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  • Is Polynomial
  • Leading Coefficient
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  • Standard Form
  • Complete the Square
  • Synthetic Division
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  • Rationalize Denominator
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  • Identify Type
  • Convergence
  • Interval Notation
  • Pi (Product) Notation
  • Boolean Algebra
  • Truth Table
  • Mutual Exclusive
  • Cardinality
  • Caretesian Product
  • Age Problems
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  • Cost Problems
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  • Conversions

Click to reveal more operations

Most Used Actions

Number line.

  • -x+3\gt 2x+1
  • (x+5)(x-5)\gt 0
  • 10^{1-x}=10^4
  • \sqrt{3+x}=-2
  • 6+11x+6x^2+x^3=0
  • factor\:x^{2}-5x+6
  • simplify\:\frac{2}{3}-\frac{3}{2}+\frac{1}{4}
  • x+2y=2x-5,\:x-y=3
  • How do you solve algebraic expressions?
  • To solve an algebraic expression, simplify the expression by combining like terms, isolate the variable on one side of the equation by using inverse operations. Then, solve the equation by finding the value of the variable that makes the equation true.
  • What are the basics of algebra?
  • The basics of algebra are the commutative, associative, and distributive laws.
  • What are the 3 rules of algebra?
  • The basic rules of algebra are the commutative, associative, and distributive laws.
  • What is the golden rule of algebra?
  • The golden rule of algebra states Do unto one side of the equation what you do to others. Meaning, whatever operation is being used on one side of equation, the same will be used on the other side too.
  • What are the 5 basic laws of algebra?
  • The basic laws of algebra are the Commutative Law For Addition, Commutative Law For Multiplication, Associative Law For Addition, Associative Law For Multiplication, and the Distributive Law.

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  • Middle School Math Solutions – Inequalities Calculator Next up in our Getting Started maths solutions series is help with another middle school algebra topic - solving... Read More

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Now, we can't guarantee you'll get better grades, since the AI Math Solver is only half of the success equation, but the majority of students who use the application report full letter grade improvements in their grades. Most see improvements in their homework grades immediately.

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Our application is available 24/7 and will start working on a solutions immediately after you send it. The time it takes to solve each problem is dependant on the complexity of the problem. The application will post a step by step solution.

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Yes, simply tap or click the carmera icon next to the Solve button in the application then select the image or if you're on your phone open your camera to immediately take a picture of your math problem.

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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.

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Scientists Have Solved the 141-Year-Old ‘Reverse Sprinkler’ Problem

This brain-teaser has baffled physicists since 1883. Thanks to some innovative engineering, it finally makes sense.

water splash from sprinkler

  • For 141 years, physicists have pondered a deceptively difficult question—what would happen if a sprinkler worked in reverse?
  • Two camps formed—one arguing that a water-sucking sprinkler would have to spin clockwise, and another camp pushing for an counter-clockwise motion.
  • Scientists from New York University took care of the problem by designing just such a device, and discovered that a reverse sprinkler would spin counter-clockwise 50 times slower than if it was just a normal sprinkler.

First posed in 1883 by Austrian physicist Ernst Mach and popularized by Nobel laureate Richard Feynman, the “reverse sprinkler” question is relatively simple—if you put an s-shaped sprinkler in a tank of water , and the sprinkler sucked in water, what direction would it spin and why?

But this is where things get tricky. One camp adamantly suggests that the sucking force would pull the nozzle counter-clockwise, while others argue that inflowing water would smack the inside of the nozzle, forcing it clockwise.

“The answer is perfectly clear at first sight,” Feynman wrote in the 1985 autobiographical book, Surely You’re Joking, Mr. Feynman . “The trouble was, some guy would think it was perfectly clear one way, and another guy would think it was perfectly clear the other way.”

To finally solve this 141-year-old problem, scientists from the New York University (NYU) created custom sprinkler devices that pushed and sucked in water at controllable rates. Complete with a new kind of ultra-low-friction rotary bearing, according to the NYU press release , the sprinklers were constructed so the researchers could observe what was going on outside and inside the sprinkler. To capture the reverse sprinkler’s fluid dynamics in action, the team also added dyes and microparticles to help illuminate the waterflow via lasers . The results were published in the journal Physics Review Letters in January.

As the sprinkler sucked in water, the NYU scientists discovered that it created a kind of “inside-out rocket ,” which very slowly rotated the sprinkler counter-clockwise.

When the sprinkler works as intended (a.k.a. spraying water) it effectively acts like a mini-rocket, with the water acting as the propellant that spins the sprinkler. When flowing in reverse, however, these jets meet inside the sprinkler chamber—but don’t smash head-on due to the flow caused by the sprinkler’s curved arms. This slight misalignment causes the sprinkler to slowly rotate in reverse, about 50 times slower than when the sprinkler is operating normally.

“Our study solves the problem by combining precision lab experiments with mathematical modeling that explains how a reverse sprinkler operates,” NYU associate professor and senior author Leif Ristroph said in a press release. “We found that the reverse sprinkler spins in the ‘reverse’ or opposite direction when taking in water as it does when ejecting it, and the cause is subtle and surprising.”

Ristroph and his team didn’t undertake this 141-year-old problem just for laughs. Understanding these fluid dynamics could help us better understand the sustainable sources of energy flowing around us, such as “wind in our atmosphere as well as waves and currents in our oceans and rivers,” Ristroph said.

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Darren lives in Portland, has a cat, and writes/edits about sci-fi and how our world works. You can find his previous stuff at Gizmodo and Paste if you look hard enough. 

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Easy Finger Math Tricks to Help Kids Solve Problems

While using your fingers isn't the fastest way to recall a multiplication fact while doing a problem, finger math tricks can help kids figure out how to answer the problem at hand — and as they work on their math, they will eventually learn all the facts by repetition.

Note that before your child can understand other finger tricks, they must be able to count by 2s, 5s, and 10s and multiply by 2s, 3s, and 4s.

Quick Finger Math Tricks for Threes and Fours

The tricks for multiplying by threes and fours are really a matter of counting out the answer on your fingers. As your children count out the answer repeatedly, they'll memorize it and then be able to move on to larger numbers.

Multiplying by Three

Did you realize that all of your fingers have three segments? Therefore, you can figure out anything from 3 x 1 to 3 x 10 by counting the segments on each finger. To start:

  • Hold up the number of fingers you're going to multiply by 3. For example, if the problem is 3 x 4 — hold up four fingers.
  • Count each segment on each finger you're holding up, and you should come up with 12 — which is the correct answer.

Multiplying by Four

Multiplying by four is the same as multiplying by two — twice. To start:

  • Hold up the number of fingers to correspond with the number you are multiplying by four. For example, if you are multiplying 4 x 6 — hold up six fingers.
  • Count each finger by two, moving from left to right. Then count each finger again, continuing to count by twos, until you've counted every finger twice.
Helpful Hack To keep track of the fingers you've counted twice, sometimes it's easier to put your finger down as you count the first time, and back up as you count the second time.

Finger Math Tricks for Multiplying by 6, 7, 8, and 9

While numbers one through five are easy for most kids to remember, six and up often pose a problem. This handy trick will make it a little easier to work those problems out.

Multiplying 6, 7, 8, and 9 by Hand

To begin, assign each finger a number. For example, your thumbs represent 6, your index fingers each represent 7, etc. This will remain the same throughout the finger math hack.

Your left hand will represent the first number that you are multiplying and your right hand will represent the second number you are multiplying. In this example, we are multiplying 7 x 8. 

To Determine the Part of Your Answer:

  • On your left hand, put down the finger that represents the number you are multiplying as well as any fingers whose number value is less than this figure. In this example, you are multiplying 7 x 8, so the left hand will represent 7. You will drop your index finger (number 7) and your thumb (number 6).
  • Similarly, the right hand will represent eight, so you will drop down your middle finger (number 8), your index finger (number 7), and your thumb (number 6).
  • Now, just multiply the fingers that are still pointed upwards. In this case, you will have three fingers on your left hand and two on your right, so you will multiply 3 x 2 to get 6. This is the first part of your answer!

To Determine the Second Part of Your Answer:

  • Keeping your fingers in the same positions, count how many fingers are folded down. In the 7 x 8 example, you should have five fingers folded. 
  • You will count each of these in quantities of ten. So, 10, 20, 30, 40, 50.
  • 50 is your answer.

To Determine Your Final Answer:

  • Add your two numbers together. In this example, you would add 6 + 50, which gives you 56!

Another Finger Math Trick Just for Nine

There is a trick that works separately, just for multiplying by the number nine.

  • To start, hold up all ten fingers, with your palms facing you.
  • Assign each finger a number, starting with your left-hand thumb and ending with your right-hand thumb. The left-hand thumb will be one, the left-hand index finger will be two, and so on until you reach the number 10 for your right-hand thumb.
  • To tackle a problem, put down the corresponding finger of the number you're multiplying by nine. For example, if you are multiplying 9 x 8, you'd put down the eighth finger (which will be on your right hand).
  • Count all the fingers to the left of the finger you have folded down. This will give you 7. This is the first digit of your answer.
  • Count all the fingers to the right of the finger you have folded down. This will give you 2. This is the second digit of your answer.
  • Put the numbers together! Your answer is 72.

Finger Multiplication Tricks Can Make Math Easy and Fun

While the hope is that your kids will eventually memorize their multiplication charts , using some quick hand tricks for multiplication and letting them count things out on their fingers is not a bad way to learn. It keeps frustration at bay since the answer is always a fingertip away, and the repetition of having to figure it out will help cement those facts into their brains.

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February 5, 2024

How String Theory Solved Math’s Monstrous Moonshine Problem

A concept from theoretical physics helped confirm the strange connection between two completely different areas of mathematics

By Manon Bischoff

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After the star-studded mystery thriller The Number 23 debuted in cinemas in 2007, many people became convinced that they were seeing the eponymous number everywhere. I was in school at that time, and some of my classmates would shudder whenever the number 23 appeared in any context. Other people became fascinated by this form of numerology because as soon as you pay more attention to a certain thing—including a number— you get the feeling that you see it too often to be purely coincidence.

For a long time, people assumed that the late mathematician John McKay might have fallen victim to this same phenomenon, known as the “frequency illusion,” or the Baader-Meinhof phenomenon . In McKay’s case, the number that captured his imagination was 196,884.

It doesn’t seem too surprising that a two-digit number such as 23 might come up repeatedly. But would a six-digit figure do so? McKay came across this number by chance in 1978 when he was looking through a paper in a mathematical field that was not his specialty. He was working in geometry and was studying the symmetry of figures. That day, however, he was looking at results from number theory , which deals with the properties of integers such as prime numbers . He came across a sequence of numbers that started with the value 196,884.

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This figure sounded familiar to McKay. He had previously worked on a mathematical structure—still hypothetical at the time— known as the monster . This strange algebraic structure was intended to describe the symmetries of a geometric object that lives in 196,883 dimensions (only one fewer than the number 196,884). And because a one-dimensional point fulfills every symmetry anyway, the monster can also describe its symmetrical properties. So McKay found the number 196,884 again in an extraordinary way. He added the first two dimensions in which mathematicians believed the monster’s symmetry applied: 196,883 + 1 = 196,884.

Does that sound far-fetched? Others thought so, too. Experts paid little attention to McKay’s result. After all, a structure such as the monster contains a number of numbers, as does the consequence from number theory that McKay had associated with it. “If you have a whole lot of numbers, then a few of them are going to be roughly the same as each other just by coincidence,” said mathematician Richard Borcherds, who has made major contributions to the field, in an explanatory YouTube video .

But McKay couldn’t shake the feeling that the two extremely different mathematical fields of geometry and number theory could be connected. He even reportedly wore T-shirts with the inscription “196,883 + 1 = 196,884” at conferences.

Complete Madness or a Stroke of Genius?

A short time later, mathematician John Thompson realized that there might be something to McKay’s suspicions after all. He succeeded in linking the next higher dimension, in which an object follows the symmetries of the monster, with the next member of the mysterious sequence of numbers from number theory. The dimension is 21,296,876. The values differ—but if you add up all the monster dimensions as before (1 + 196,883 + 21,296,876), the result is 21,493,760.

That was surprising because, as you may recall, when McKay first spotted 196,884, he was looking at a special sequence in number theory. The second number in that sequence is 21,493,760—Thompson’s result. In other words, it began to seem that there really could be a link between two seemingly unrelated areas of mathematics.

At this point the math community began to get curious. Maybe McKay was right after all—even if that sounded totally absurd. What could this strange structure, which described symmetries of unimaginable objects and had not even been fully constructed, have to do with number theory?

By 1979 evidence was mounting that other numbers and dimensions seemed to follow this unexpected pattern. Mathematicians John Conway and Simon Norton finally published a paper entitled “Monstrous Moonshine,” in which they set out the conjecture of a connection between geometry and number theory. “They called it moonshine because it appeared so far-fetched,” said number theorist Don Zagier of the Max Planck Institute for Mathematics in Bonn, Germany, to Quanta Magazine in 2015.

And indeed, there was likely very little hope of ever proving this moonshine conjecture. Quite apart from the fact that there was no indication that the two distant mathematical areas were connected, it was not even completely clear whether the monster really existed.

The Monster in the Moonlight

The monster was a theoretical prediction of group theory, an area of geometry that deals with the symmetrical properties of objects. In the 1970s mathematicians began to create a kind of periodic table of groups: they wanted to find the “atoms” of finite symmetries. According to this way of thinking, every finite group can be represented by a combination of these atoms. After decades of research, the geometers finally seemed to have reached their goal. Unlike the chemical elements, there are an infinite number of “finite simple groups,” but almost all can be divided into 18 categories, the arrangement of which is reminiscent of the periodic table. In addition, the experts came across a total of 26 outsiders that do not fit into these 18 classes.

Triangles labeled to show varied symmetries.

The first of these outliers was the “monster,” which mathematicians Bernd Fischer and Robert Griess predicted in 1973. The name comes from the sheer size of this group: it contains more than 8 x 10 53 symmetries. For comparison, the symmetry group of a 20-sided “D20” die ( an icosahedron ) contains 60 symmetries, meaning 60 possible transformations (rotations or reflections) can be carried out without changing the orientation of the D20.

Colorful circles connected by lines represent groups of symmetries.

Because of its sheer size, the monster presented mathematicians with massive challenges. “Most people thought it was going to be hopeless to construct it since much, much, much smaller groups required computer constructions at that time,” explained Borcherds in his YouTube video. Meanwhile even powerful computers struggle with a structure consisting of 8 x 10 53 elements.

Yet this pessimistic forecast ultimately proved wrong. In 1980 Griess constructed the monster and thus proved its existence —without the help of computers.

A Sine Function on Steroids

Number theory is mostly about integers, which seems quite simple at first glance. But to investigate the relationships between them, experts resort to complicated concepts, such as so-called modular forms. These are functions f ( z ) that are extremely symmetrical. As with the sine function, you only need to know a specific section of a modular form to know what it looks like everywhere else.

“Modular forms are something like trigonometric functions, but on steroids,” mathematician Ken Ono told Quanta Magazine .

Colorful arcs represent moduli space

Nevertheless, they play an extremely important role in mathematics. Andrew Wiles of the University of Oxford used them, for example, to prove Fermat’s theorem, and Maryna Viazovska of the Swiss Federal Institute of Technology in Lausanne used them to find the densest sphere-packing arrangement in eight spatial dimensions . Because modular forms are so complicated, however, they are often approximated by an infinitely long polynomial, such as:

f (q ) = ( 1 ⁄ q ) + 744 + 19,688 q + 21,493,760 q 2 + 864,299,970 q 3 + …

The prefactors in front of the variable q form a number sequence with interesting properties from a number-theoretical perspective. McKay associated this sequence of numbers with the monster.

A Surprising Link

Borcherds first heard about the moonshine conjecture in the 1980s. “I was just completely blown away by this,” he recalled in an interview with YouTuber Curt Jaimungal . Borcherds was sitting in one of Conway’s lectures at the time and learned that number theory and group theory could be mysteriously connected. The subject never let go of him. He began to search for the suspected connection until he found it. In 1992 he published his groundbreaking result , for which he received a Fields Medal, one of the highest awards in mathematics, six years later. His conclusion: a highly speculative area of physics, string theory, could provide the missing piece of the puzzle between the monster and the sequence of numbers.

String theory attempts to unite the four fundamental forces of physics (electromagnetism, strong and weak nuclear forces and gravity). Instead of relying on particles or waves to make up the basic building blocks of the universe, as in conventional theories, string theory involves one-dimensional structures: tiny threads vibrate like the strings of an instrument and thus generate the familiar particles and interactions that we perceive in the universe.

Borcherds knew that string theory was based on many mathematical principles related to symmetries. As it turns out, moduli also play a role. When the tiny threads are closed and move through spacetime in a wobbly manner, their track forms a two-dimensional tube. This structure has the same symmetry as modular shapes—regardless of how the thread oscillates.

The type of string theory that Borcherds investigated can only be mathematically formulated in 25 spatial dimensions. Because our world consists of only three visible spatial dimensions, however, string theorists assume that the remaining 22 dimensions are rolled up into tiny spheres or doughnut-shaped tori. But the physics depends on their exact shape: a string theory in which the dimensions are rolled up as cylinders provides different predictions than one in which they form a sphere. In order to describe the particles and their interactions in a way that fits our world, physicists have to find the right “compactification” in their calculations.

Borcherds rolled up 24 dimensions into a 24-dimensional doughnut surface and discovered that the associated string theory had the symmetry of the monster. The fact that only one free spatial dimension remained did not bother him. After all, he was interested in the mathematical properties of the model and not in a physical theory that describes our world.

In this constructed world, the threads swing along the 24-dimensional doughnut. The dimensions of the monster count all the ways in which a thread can vibrate at a certain energy. So at the lowest energy, it only vibrates in one way; at the next highest energy, there are already 196,883 different possibilities. And the trace that the thread leaves behind has the symmetry of a modular shape.

Borcherds had thus proven the connection between the monster group and a modular form. And it was not to remain the only such case: in the meantime, mathematicians have been able to connect other finite groups with other modular forms —and there, too, string theory provides the link. So even if it turns out that the speculative theory is not suitable for describing our universe, it can still help us discover completely new mathematical worlds.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

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  6. Solve

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  11. List of unsolved problems in mathematics

    Lists of unsolved problems in mathematics Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions.

  12. DoYourMath.com

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  13. MathGPT

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  15. 10 Hard Math Problems That May Never Be Solved

    One of the greatest unsolved mysteries in math is also very easy to write. Goldbach's Conjecture is, "Every even number (greater than two) is the sum of two primes.". You check this in your ...

  16. The Top Unsolved Questions in Mathematics Remain Mostly Mysterious

    August 2021 Issue The Sciences Twenty-one years ago this week, mathematicians released a list of the top seven unsolved problems in the field. Answering them would offer major new insights in...

  17. 6 Deceptively Simple Maths Problems That No One Can Solve

    The Collatz conjecture is one of the most famous unsolved mathematical problems, because it's so simple, you can explain it to a primary-school-aged kid, and they'll probably be intrigued enough to try and find the answer for themselves. So here's how it goes: pick a number, any number. If it's even, divide it by 2.

  18. AI Math Problem Solver

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  19. 10 Best strategies for solving math word problems in 2024

    2. Identify Key Information and Variables. 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.

  20. 7 of the hardest math problem in the world yet to be solved

    Math problems like the Poincaré conjecture and Fermat's last theorem took centuries to solve. However, others like the Riemann hypothesis and Goldbach's conjecture still haunt mathematicians...

  21. Scientists Have Solved the 141-Year-Old 'Reverse Sprinkler' Problem

    To finally solve this 141-year-old problem, ... 10 of the Hardest Math Problems Ever Solved. Solution to Riddle of the Week #12: Licking Frogs. Advertisement - Continue Reading Below.

  22. Solve

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  23. Equation Solver

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  24. Easy Finger Math Tricks to Help Kids Solve Problems

    Finger Math Tricks for Multiplying by 6, 7, 8, and 9 While numbers one through five are easy for most kids to remember, six and up often pose a problem. This handy trick will make it a little ...

  25. How String Theory Solved Math's Monstrous Moonshine Problem

    Some of the red, green and blue sporadic groups are related to one another. The white sporadic groups are considered outsiders. Credit: Drschawrz/Wikimedia Commons (CC BY-SA 3.0) Because of its ...

  26. Mathway

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