maths problem solving sums

Get step-by-step solutions to your math problems

Try Math Solver

Get step-by-step explanations

Graph your math problems

Practice, practice, practice

Get math help in your language

Math Solver

Geogebra math solver.

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

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

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

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

Additional menu

Khan Academy Blog

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
• Integration techniques
• 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

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

Module 1: Problem Solving Strategies

• Last updated
• Save as PDF
• Page ID 10352

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.

Pólya’s 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

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

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!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (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 thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

Problem Solving in Mathematics

• Math Tutorials
• Pre Algebra & Algebra
• Exponential Decay
• Worksheets By Grade

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

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

Use Established Procedures

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

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

Look for Clue Words

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

Common clue words for addition  problems:

Common clue words for  subtraction  problems:

• How much more

Common clue words for multiplication problems:

Common clue words for division problems:

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

Read the Problem Carefully

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

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

Develop a Plan and Review Your Work

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

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

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

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

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

Tips and Hints

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

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

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

• 2nd Grade Math Word Problems
• The Horse Problem: A Math Challenge
• 2020-21 Common Application Essay Option 4—Solving a Problem
• How to Use Math Journals in Class
• The Frayer Model for Math
• Algorithms in Mathematics and Beyond
• "Grandpa's Rubik's Cube"—Sample Common Application Essay, Option #4
• Math Stumper: Use Two Squares to Make Separate Pens for Nine Pigs
• Critical Thinking Definition, Skills, and Examples
• College Interview Tips: "Tell Me About a Challenge You Overcame"
• Graphic Organizers in Math
• Christmas Word Problem Worksheets
• Solving Problems Involving Distance, Rate, and Time
• Innovative Ways to Teach Math
• Study Tips for Math Homework and Math Tests
• How to Do Algebra Word Problems

Upload a screenshot and solve any math problem instantly with MathGPT!

Drag & drop an image file here, or click to select an image.

The best way to learn math and computer science

Guided interactive problem solving that’s effective and fun. master concepts in 15 minutes a day., data analysis, computer science, programming & ai, science & engineering, join over 10 million people learning on brilliant, over 50,000 5-star reviews on ios app store and google play.

Master concepts in 15 minutes a day

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

Effective, hands-on learning

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

Learn at your level

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

Guided bite-sized lessons

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

Stay motivated

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

Guided courses for every journey

All of our courses are crafted by award-winning teachers, researchers, and professionals from MIT, Caltech, Duke, Microsoft, Google, and more.

• Foundational Math
• Software Development
• Foundational Logic
• Data Science
• High School Math
• Engineering
• Statistics and Finance

Courses in Foundational Math

• Solving Equations
• Measurement
• Mathematical Fundamentals
• Reasoning with Algebra
• Functions and Quadratics

10k+ Ratings

60k+ Ratings

We use cookies to improve your experience on Brilliant. Learn more about our cookie policy and settings .

Solver Title

Generating PDF...

• 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 Roman Numerals 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 Volume
• Pre Algebra
• One-Step Addition
• One-Step Subtraction
• One-Step Multiplication
• One-Step Division
• One-Step Decimals
• Two-Step Integers
• Two-Step Add/Subtract
• Two-Step Multiply/Divide
• Two-Step Fractions
• Two-Step Decimals
• Multi-Step Integers
• Multi-Step with Parentheses
• Multi-Step Rational
• Multi-Step Fractions
• Multi-Step Decimals
• Solve by Factoring
• Completing the Square
• Quadratic Formula
• Biquadratic
• Logarithmic
• Exponential
• 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
• Product Rule
• Quotient Rule
• Expand Power Rule
• Fraction Exponent
• Exponent Rules
• Exponential Form
• Logarithmic Form
• Absolute Value
• Rational Number
• Powers of i
• Partial Fractions
• Is Polynomial
• Leading Coefficient
• Leading Term
• Standard Form
• Complete the Square
• Synthetic Division
• Linear Factors
• Rationalize Denominator
• Rationalize Numerator
• Identify Type
• Convergence
• Interval Notation
• Pi (Product) Notation
• Boolean Algebra
• Truth Table
• Mutual Exclusive
• Cardinality
• Caretesian Product
• Age Problems
• Distance Problems
• Cost Problems
• Investment Problems
• Number Problems
• Percent Problems
• Addition/Subtraction
• Multiplication/Division
• Dice Problems
• Coin Problems
• Card Problems
• Pre Calculus
• Linear Algebra
• Trigonometry
• Conversions

Most Used Actions

Number line.

• \mathrm{Lauren's\:age\:is\:half\:of\:Joe's\:age.\:Emma\:is\:four\:years\:older\:than\:Joe.\:The\:sum\:of\:Lauren,\:Emma,\:and\:Joe's\:age\:is\:54.\:How\:old\:is\:Joe?}
• \mathrm{Kira\:went\:for\:a\:drive\:in\:her\:new\:car.\:She\:drove\:for\:142.5\:miles\:at\:a\:speed\:of\:57\:mph.\:For\:how\:many\:hours\:did\:she\:drive?}
• \mathrm{The\:sum\:of\:two\:numbers\:is\:249\:.\:Twice\:the\:larger\:number\:plus\:three\:times\:the\:smaller\:number\:is\:591\:.\:Find\:the\:numbers.}
• \mathrm{If\:2\:tacos\:and\:3\:drinks\:cost\:12\:and\:3\:tacos\:and\:2\:drinks\:cost\:13\:how\:much\:does\:a\:taco\:cost?}
• \mathrm{You\:deposit\:3000\:in\:an\:account\:earning\:2\%\:interest\:compounded\:monthly.\:How\:much\:will\:you\:have\:in\:the\:account\:in\:15\:years?}
• How do you solve word problems?
• To solve word problems start by reading the problem carefully and understanding what it's asking. Try underlining or highlighting key information, such as numbers and key words that indicate what operation is needed to perform. Translate the problem into mathematical expressions or equations, and use the information and equations generated to solve for the answer.
• How do you identify word problems in math?
• Word problems in math can be identified by the use of language that describes a situation or scenario. Word problems often use words and phrases which indicate that performing calculations is needed to find a solution. Additionally, word problems will often include specific information such as numbers, measurements, and units that needed to be used to solve the problem.
• Is there a calculator that can solve word problems?
• Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems.
• What is an age problem?
• An age problem is a type of word problem in math that involves calculating the age of one or more people at a specific point in time. These problems often use phrases such as 'x years ago,' 'in y years,' or 'y years later,' which indicate that the problem is related to time and age.

word-problems-calculator

• High School Math Solutions – Systems of Equations Calculator, Elimination A system of equations is a collection of two or more equations with the same set of variables. In this blog post,...

Please add a message.

Message received. Thanks for the feedback.

When it comes to math, maybe the kids aren’t the problem

The way we are teaching, testing, and requiring mathematics is damaging our society with potentially fatal results. That’s a strong statement, I know, but to get a sense of why, take a moment to think of something that you would find hard or maybe impossible to do — executing 30 pullups, playing a violin concerto, walking a tightrope over an abyss, feeding a snake, solving a complex polynomial equation. 

Now suppose our society decrees that to graduate from high school, you must be able to do that thing or be branded a failure and not earn a diploma. How are you feeling about yourself and your prospects?

Imagine trying your hardest to learn the skills, but the textbooks, online resources and sometimes even teachers confuse you, and on top of that, some exams are so badly designed they are virtually impossible to pass. 

The 2022 results from the National Assessment of Educational Progress showed only 36% of fourth graders and 26% of eighth graders were considered proficient in mathematics. The national average for high school math proficiency is just 38%.

Why are so many young people not achieving math proficiency? Consider how we define “proficiency.” 

For example, the Common Core math standards state that, among other things, high school students should “Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f (0) = f (1) = 1, f (n+1) = f (n) + f (n-1) for n ≥ 1.” And that’s just one example of many.

Suppose your child came to you with that as homework each night, possibly in tears. Now imagine that your child has special needs and deals with dyscalculia , ADD or other challenges.  

One of my sons struggles with ADD and is trying to complete an online financial algebra course offered by his school district to fulfill his mandatory “math pathway” for graduation. To help him succeed, a certified math and science tutor and I have worked with him every day through some very poorly designed online lessons provided by his district. 

We also helped him try to solve problems during a unit exam, but together, we achieved only a score of 32% on the exam. A certified math teacher, and a former statistics instructor with a Ph.D., failed a high school financial algebra exam. Why? Because the lessons were deeply flawed, the exam items were ambiguously worded, there were too many problems for the allotted time and the rules prohibited the use of resources like a financial calculator that one might enlist in real life. 

All that is not an excuse, and this column is not some reflexive “anti-testing” rant. It is, unfortunately, the truth. That leaves us asking, “If we failed the exam, how will most students, or their parents, possibly succeed?”

If you make children and their parents feel like failures, it is predictable that some are eventually going to give up on school and on themselves. Once that happens, they are far more likely to turn to behaviors that are harmful, self-destructive, and, tragically, sometimes fatal. And when significant portions of our population share those feelings, it inevitably harms our society as a whole. It is nothing less than educational and social malpractice to create unrealistic and unnecessary demands for children and then fail to provide them with the best resources to help them succeed.   

A good start would be to insist that policymakers, superintendents, school boards and school principals throughout our state attempt to learn each subject in math through the same texts or online resources students are required to use. I don’t mean just skimming a text or taking the word of someone else, I mean really digging in and trying to learn the material and take the tests firsthand. I doubt most people would accept the challenge and I am certain even fewer would pass the exams. 

That is not because the administrators and teachers aren’t smart, good people — it is because the materials and exams are so poorly designed, and the requirements are so profoundly detached from daily life. But if that is so, which the evidence suggests it is, why are we inflicting such things on our children and with what impacts?

Most Read Opinion Stories

• When it comes to math, maybe the kids aren't the problem
• School truancy crisis means more than missed classes VIEW
• Potatoes are not grains. Keep them rooted in the vegetable family
• Keep lights on as Washington transitions to clean power
• Many patients, not enough beds: Here's how to solve a care crisis

The opinions expressed in reader comments are those of the author only and do not reflect the opinions of The Seattle Times.

Help | Advanced Search

Computer Science > Computation and Language

Title: chatglm-math: improving math problem-solving in large language models with a self-critique pipeline.

Abstract: Large language models (LLMs) have shown excellent mastering of human language, but still struggle in real-world applications that require mathematical problem-solving. While many strategies and datasets to enhance LLMs' mathematics are developed, it remains a challenge to simultaneously maintain and improve both language and mathematical capabilities in deployed LLM this http URL this work, we tailor the Self-Critique pipeline, which addresses the challenge in the feedback learning stage of LLM alignment. We first train a general Math-Critique model from the LLM itself to provide feedback signals. Then, we sequentially employ rejective fine-tuning and direct preference optimization over the LLM's own generations for data collection. Based on ChatGLM3-32B, we conduct a series of experiments on both academic and our newly created challenging dataset, MathUserEval. Results show that our pipeline significantly enhances the LLM's mathematical problem-solving while still improving its language ability, outperforming LLMs that could be two times larger. Related techniques have been deployed to ChatGLM\footnote{\url{ this https URL }}, an online serving LLM. Related evaluation dataset and scripts are released at \url{ this https URL }.

Submission history

Access paper:.

• HTML (experimental)
• Other Formats

References & Citations

• Google Scholar
• Semantic Scholar

BibTeX formatted citation

Bibliographic and Citation Tools

Code, data and media associated with this article, recommenders and search tools.

• Institution

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .

share this!

April 2, 2024

This article has been reviewed according to Science X's editorial process and policies . Editors have highlighted the following attributes while ensuring the content's credibility:

fact-checked

trusted source

written by researcher(s)

How can Australia solve the math teacher shortage? It can start by training more existing teachers to teach math

by Ian Gordon, Mary P. Coupland and Merrilyn Goos, The Conversation

Imagine if you enrolled your child in swimming lessons but instead of a qualified swimming instructor, they were taught freestyle technique by a soccer coach.

Something similar is happening in classrooms around Australia every day. As part of the ongoing teacher shortage, there are significant numbers of teachers teaching " out-of-field ." This means they are teaching subjects they are not qualified to teach.

One of the subjects where out-of-field teaching is particularly common is math.

A 2021 report on Australia's teaching workforce found that 40% of those teaching high school mathematics are out-of-field (English and science were 28% and 29%, respectively).

Another 2021 study of students in Year 8 found they were more likely to be taught by teachers who had specialist training in both math and math education if they went to a school in an affluent area rather than a disadvantaged one (54% compared with 31%).

Our new report looks at how we can fix this situation by training more existing teachers in math education.

Why is this a problem?

Mathematics is one of the key parts of school education. But we are seeing worrying signs students are not receiving the math education they need.

The 2021 study of Year 8 students showed those taught by teachers with a university degree majoring in math had markedly higher results, compared with those taught by out-of-field teachers.

We also know math skills are desperately needed in the broader workforce. The burgeoning worlds of big data and artificial intelligence rely on mathematical and statistical thinking, formulae and algorithms. Math has also been identified as a national skill shortages priority area .

What do we do about this?

There have been repeated efforts to address teacher shortages, including trying to retain existing mathematics teachers, having specialist teachers teaching across multiple schools and higher salaries . There is also a push to train more teachers from scratch, which of course will take many years to implement.

There is one strategy, however, that has not yet been given much attention by policy makers : upgrading current teachers' math and statistics knowledge and their skills in how to teach these subjects.

They already have training and expertise in how to teach and a commitment to the profession. Specific training in math will mean they can move from being out-of-field to "in-field".

How to give teachers this training

A new report commissioned by mathematics and statistics organizations in Australia (including the Australian Mathematical Sciences Institute) looks at what is currently available in Australia to train teachers in math.

It identified 12 different courses to give existing teachers math teaching skills. They varied in terms of location, duration (from six months to 18 months full-time) and aims.

For example, some were only targeted at teachers who want to teach math in the junior and middle years of high school. Some taught university-level math and others taught school-level math. Some had government funding support; others could cost students more than A$37,000.

Overall, we found the current system is confusing for teachers to navigate. There are complex differences between states about what qualifies a teacher to be "in-field" for a subject area.

In the current incentive environment, we found these courses cater to a very small number of teachers. For example, in 2024 in New South Wales this year there are only about 50 government-sponsored places available.

This is not adequate. Pre-COVID, it was estimated we were losing more than 1,000 equivalent full-time math teachers per year to attrition and retirement and new graduates were at best in the low hundreds.

But we don't know exactly how many extra teachers need to be trained in math. One of the key recommendations of the report is for accurate national data of every teacher 's content specializations.

We need a national approach

The report also recommends a national strategy to train more existing teachers to be math teachers. This would replace the current piecemeal approach.

It would involve a standard training regime across Australia with government and school-system incentives for people to take up extra training in math.

There is international evidence to show a major upskilling program like this could work.

In Ireland, where the same problem was identified, the government funds a scheme run by a group of universities. Since 2012, teachers have been able to get a formal qualification (a professional diploma). Between 2009 and 2018 the percentage of out-of-field math teaching in Ireland dropped from 48% to 25%.

To develop a similar scheme here in Australia, we would need coordination between federal and state governments and universities. Based on the Irish experience, it would also require several million dollars in funding.

But with students receiving crucial math lessons every day by teachers who are not trained to teach math, the need is urgent.

The report mentioned in this article was commissioned by the Australian Mathematical Sciences Institute, the Australian Mathematical Society, the Statistical Society of Australia, the Mathematics Education Research Group of Australasia and the Actuaries Institute.

Provided by The Conversation

Explore further

Feedback to editors

'Swallowed,' torn up or live on: How Earth will fare when the sun dies

12 minutes ago

Research team exerts electrical control over polaritons, hybridized light-matter particles, at room temperature

19 minutes ago

A natural touch for coastal defense: Hybrid solutions may offer more benefits in lower-risk areas

4 hours ago

Public transit agencies may need to adapt to the rise of remote work, says new study

Broken record: March is 10th straight month to be hottest on record, scientists say

5 hours ago

Early medieval money mystery solved

14 hours ago

Finding new chemistry to capture double the carbon

17 hours ago

Americans are bad at recognizing conspiracy theories when they believe they're true, says study

18 hours ago

A total solar eclipse races across North America as clouds part along totality

New statistical-modeling workflow may help advance drug discovery and synthetic chemistry

Relevant physicsforums posts, how to pick some numbers out of 13 integers, by a 4 digits code.

2 hours ago

Calculating the inverse of a function involving the error function

Apr 5, 2024

Reprise: Calculate the distance between two points without using a coordinate system

Apr 4, 2024

Non-sinusoidal waveform model

Apr 3, 2024

I've been trying to understand the proof for the binomial theorem

Formal definition of multiplication for real and complex numbers.

Mar 29, 2024

More from General Math

Related Stories

Studies recommend increased research into achievement, engagement to raise student math scores

Feb 15, 2024

AI can teach math teachers how to improve student skills

Dec 8, 2023

First study on teacher effectiveness for students with and without disabilities

Mar 25, 2024

Students taught by highly qualified teachers more likely to obtain bachelor's degree

May 22, 2018

Former math teacher explains why some students are 'good' at math and others lag behind

Nov 3, 2022

Why teachers are letting students solve math problems in lots of different ways

Feb 1, 2023

Recommended for you

Earth, the sun and a bike wheel: Why your high-school textbook was wrong about the shape of Earth's orbit

20 hours ago

Touchibo, a robot that fosters inclusion in education through touch

More than money, family and community bonds prep teens for college success: Study

A periodic table of primes: Research team claims that prime numbers can be predicted

Research reveals significant effects of onscreen instructors during video classes in aiding student learning

Prestigious journals make it hard for scientists who don't speak English to get published, study finds

Mar 23, 2024

Let us know if there is a problem with our content

Use this form if you have come across a typo, inaccuracy or would like to send an edit request for the content on this page. For general inquiries, please use our contact form . For general feedback, use the public comments section below (please adhere to guidelines ).

Please select the most appropriate category to facilitate processing of your request

Thank you for taking time to provide your feedback to the editors.

Your feedback is important to us. However, we do not guarantee individual replies due to the high volume of messages.

E-mail the story

Your email address is used only to let the recipient know who sent the email. Neither your address nor the recipient's address will be used for any other purpose. The information you enter will appear in your e-mail message and is not retained by Phys.org in any form.

Newsletter sign up

Get weekly and/or daily updates delivered to your inbox. You can unsubscribe at any time and we'll never share your details to third parties.

More information Privacy policy

Donate and enjoy an ad-free experience

We keep our content available to everyone. Consider supporting Science X's mission by getting a premium account.

E-mail newsletter

• For a new problem, you will need to begin a new live expert session.
• You can contact support with any questions regarding your current subscription.
• You will be able to enter math problems once our session is over.
• I am only able to help with one math problem per session. Which problem would you like to work on?
• Does that make sense?
• I am currently working on this problem.
• Are you still there?
• It appears we may have a connection issue. I will end the session - please reconnect if you still need assistance.
• Let me take a look...
• Can you please send an image of the problem you are seeing in your book or homework?
• If you click on "Tap to view steps..." you will see the steps are now numbered. Which step # do you have a question on?
• Please make sure you are in the correct subject. To change subjects, please exit out of this live expert session and select the appropriate subject from the menu located in the upper left corner of the Mathway screen.
• What are you trying to do with this input?
• While we cover a very wide range of problems, we are currently unable to assist with this specific problem. I spoke with my team and we will make note of this for future training. Is there a different problem you would like further assistance with?
• Mathway currently does not support this subject. We are more than happy to answer any math specific question you may have about this problem.
• Mathway currently does not support Ask an Expert Live in Chemistry. If this is what you were looking for, please contact support.
• Mathway currently only computes linear regressions.
• We are here to assist you with your math questions. You will need to get assistance from your school if you are having problems entering the answers into your online assignment.
• Have a great day!
• Hope that helps!
• You're welcome!
• Per our terms of use, Mathway's live experts will not knowingly provide solutions to students while they are taking a test or quiz.

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

• a special character: @$#!%*?&