problem solving math 1

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

• 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

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!

• 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.

1.3: Problem Solving Strategies

• Last updated
• Save as PDF
• Page ID 9823

• Michelle Manes
• University of Hawaii

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

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

How to Solve It

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

George Pólya, circa 1973

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

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

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

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

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

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

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

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

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

Problem Solving Strategy 2 (Try Something!).

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

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

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

Think/Pair/Share

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

Problem Solving Strategy 3 (Draw a Picture).

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

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

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

Problem Solving Strategy 4 (Make Up Numbers).

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

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

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

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

(Squares on a Chess Board)

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

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

Think / Pair / Share

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

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

Problem Solving Strategy 5 (Try a Simpler Problem).

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

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

Problem Solving Strategy 6 (Work Systematically).

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

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

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

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

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

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

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

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

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

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

(Broken Clock)

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

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

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

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

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

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

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

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

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

Solver Title

Generating PDF...

• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Expanded Form 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
• Complex Form
• 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.

• -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.

algebra-calculator

• High School Math Solutions – Inequalities Calculator, Exponential Inequalities Last post, we talked about how to solve logarithmic inequalities. This post, we will learn how to solve exponential...

Please add a message.

Message received. Thanks for the feedback.

• 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

How is the problem-solving ability and scientific attitude of students in mathematics learning seen from the teacher’s perspective?

[email protected]

[email protected]

• Split-Screen
• Article contents
• Figures & tables
• Supplementary Data
• Peer Review
• Open the PDF for in another window
• Reprints and Permissions
• Cite Icon Cite
• Search Site

Nia Afrelia , Ali Mahmudi , Heri Retnawati; How is the problem-solving ability and scientific attitude of students in mathematics learning seen from the teacher’s perspective?. AIP Conf. Proc. 29 April 2024; 2622 (1): 080004. https://doi.org/10.1063/5.0133852

Download citation file:

• Ris (Zotero)
• Reference Manager

One of the main goals of learning mathematics is problem solving ability, which are important for every student to mastered. In addition, there is a domain of attitude that also supports problem-solving abilities, namely the scientific attitude. The important role of teachers is needed in training students’ problem-solving skills and scientific attitudes especially in learning mathematics. This study aims to describe how students’ mathematical problem-solving abilities and students’ scientific attitudes in the mathematics learning process based on teacher perceptions. This research is a qualitative with the type of phenomenology. Data were collected using online questionnaires and interviews with 10 mathematics teachers in Jambi province. Data analysis was carried out using the Milles & Huberman stage, which divided the steps of data analysis activities into several parts, there are data collection, data reduction, data display, and conclusion or verification. The results showed that the problem-solving ability and scientific attitude of students in learning mathematics based on the teacher’s perspective are not optimal, this happened not only because of factors from the students themselves but also exist big contribution by a teacher.

Citing articles via

Publish with us - request a quote.

Sign up for alerts

• Online ISSN 1551-7616
• Print ISSN 0094-243X
• For Researchers
• For Librarians
• For Advertisers
• Our Publishing Partners  
• Physics Today
• Conference Proceedings
• Special Topics

pubs.aip.org

• Privacy Policy
• Terms of Use

Connect with AIP Publishing

This feature is available to subscribers only.

Sign In or Create an Account

Game Central

Lisa Lightner

29 Math Problem Solving IEP Goals (Including Math Reasoning)

Posted: February 13, 2024 | Last updated: May 1, 2024

• Math problem solving is a critical skill for students with learning disabilities that requires individualized support and attention.
• Effective math problem solving IEP goals are specific, measurable, and achievable, and are developed through collaboration with parents, teachers, and other stakeholders.
• By setting realistic goals, monitoring progress, and adjusting goals as needed, educators can help students with learning disabilities build their confidence and independence in math problem solving.

When a child has math skills and can apply them to everyday life, it can be something we take for granted.

Many of us are familiar with the moaning and groaning while doing math and saying, “But I’m never going to use this in my everyday life!” But, you might!

My teen can now do a lot of math in his head. I’ve noticed it while we are out shopping and he sees something he wants. He can quickly calculate in his head if he can afford it with the money in his pocket.

Or, he can calculate in his head what the sale price will be if it is a certain percentage off the full price.

Recently, he asked for a subscription to Spotify. I made him an offer to gather and take out the trash weekly in exchange for this. He quickly calculated how much he was being “paid” to take out the trash every week and decided that it was worth a Spotify subscription.

Math Problem Solving IEP Goals

In every-day conversations, math enters, and we don’t even realize it. Just the other day, I said to him, “Well, I heard that about 70% of the population has the COVID vaccine.” Without math skills and being able to visualize and apply them, this sentence has no meaning.

Even if you say something as mundane as “But you only ate half of your dinner!” you must have visualization and math skills for this phrase to have any impact.

Math problem solving is a crucial skill that students need to develop to succeed academically and in their future careers. Or, at least, support them if they are struggling to learn them. For students with learning disabilities, math problem solving can be a particularly challenging area that requires individualized support and attention.

This is where Individualized Education Program (IEP) goals come in.

IEP goals are specific, measurable objectives that are designed to help students with learning disabilities achieve academic success.

In the context of math problem solving, IEP goals can focus on a range of skills, including understanding mathematical concepts, applying problem-solving strategies, and monitoring progress. By setting realistic and achievable goals, educators can help students with learning disabilities build their confidence and independence in math problem solving.

To create effective math problem solving IEP goals, teachers must collaborate with the IEP team to identify the student’s strengths and areas of need, assess their current level of mathematical understanding, and develop a plan for achieving their goals.

Teachers can help students with learning disabilities develop the skills they need to succeed in math and beyond. And parents can better understand what a good math IEP goal looks like.

Understanding IEP Goals in Math Problem Solving

Individualized Education Programs (IEPs) are designed to help students with disabilities receive the support they need to succeed in school. IEP goals in math problem solving are specific objectives that are tailored to meet the needs of each individual student.

These goals are designed to help students develop the skills they need to solve math problems and succeed in math class.

IEP goals in math problem solving can cover a wide range of skills, including:

• Understanding math concepts
• Solving math problems
• Using math tools and technology
• Applying math skills to real-world situations

When developing IEP goals in math problem solving, it is important to consider the individual needs of each student. IEP Goals should be specific, measurable, achievable, relevant, and time-bound (SMART) .

This means that they should be clear and concise, and should include specific details about what the student is expected to achieve and when they are expected to achieve it.

To help ensure that IEP goals in math problem solving are effective, it is important to involve parents, teachers, and other members of the student’s IEP team in the goal-setting process.

This can help to ensure that goals are realistic and achievable, and that they are tailored to meet the individual needs of each student.

IEP goals in math problem solving are an important tool for helping students with disabilities succeed in math class.

By setting clear, specific goals that are tailored to meet the individual needs of each student, educators can help to ensure that students are able to develop the skills they need to succeed in math and beyond.

Involve the student in goal-setting

Involving the student in the goal-setting process can help increase motivation and ownership. Students can provide input on their strengths and weaknesses , suggest goals that are meaningful to them, and track their progress towards achieving their goals.

Teachers can use a variety of strategies, such as student-led conferences, goal-setting worksheets, and progress monitoring tools , to involve students in the goal-setting process.

By following these tips, teachers can develop realistic and effective IEP goals for math problem solving that help students achieve success and build confidence in their math abilities.

Collaboration with Parents and Teachers

Collaboration between parents and teachers is essential for the success of students with math problem-solving IEP goals. Parents can provide valuable information about their child’s strengths and weaknesses , which can help teachers create more effective IEP goals.

Teachers, in turn, can provide parents with feedback on their child’s progress and offer suggestions for how they can support their child’s learning at home.

One way to facilitate collaboration is to hold regular meetings between parents and teachers. These meetings can be used to discuss the student’s progress, set goals, and identify areas where additional support may be needed.

It’s important that both parties come prepared to these meetings with specific examples of what’s been working and what hasn’t, as well as any questions or concerns they may have.

Another way to foster collaboration is to provide parents with resources and strategies they can use to support their child’s learning at home. This might include providing access to online math resources, suggesting math games and activities, or offering tips on how to help their child with homework.

Finally, it’s important to keep lines of communication open between parents and teachers throughout the school year. This can be done through regular progress reports, email updates, or phone calls.

By working together, parents and teachers can help ensure that students with math problem-solving IEP goals are receiving the support they need to succeed.

More for You

Another drugstore chain files for Chapter 11 bankruptcy

Ranking the 21 'American Idol' winners

Here’s What the US Minimum Wage Was the Year You Were Born

25 Healthier Fast Food Swaps That Won’t Disappoint

I was fired from a new job in less than a week after I started. It taught me not every opportunity is a good opportunity.

Kids who do these 6 things have 'high emotional intelligence,' says parenting expert who studied over 200 children

What Do All the Heart Emojis Mean? A Guide To Using the Symbols of Love

Do I have to pay off my spouse's debts when they die? Here's what you're responsible for and what you aren't after a loved one's death

Lost Planet Theia Is Hidden Inside the Earth, New Study Says

Map reveals best places to live in the US if nuclear war breaks out

18 ‘Normal’ Things From the ’80s and ’90s That Are Considered Luxuries Now

Experts Say These Are The 5 Worst Foods For Your Cholesterol

Charles Barkley Names 76ers Fatal Flaw Following Game 6 Loss

The top country music star from every year since 1970

A NATO country says it could join Ukraine's war with Russia if 2 conditions are met

Spacecraft spots "spiders" scattered across surface of Mars

Employers Are Avoiding Hiring Gen Z Workers- Here's Why

Taylor Swift Claims Record Top 14 Spots on Billboard Hot 100

I Am Doing a PhD at 16—My Mother's Death Is the Reason

11 Facts You Should Know About Hard-Boiled Eggs

Math Calculator

Enter the expression you want to evaluate.

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

• a special character: @$#!%*?&

Learning to sample initial solution for solving 0–1 discrete optimization problem by local search

• AI Methods for Optimization Problems
• Published: 29 April 2024

Cite this article

• Xin Liu 1 ,
• Jianyong Sun 1 &
• Zongben Xu 1  

Local search methods are convenient alternatives for solving discrete optimization problems (DOPs). These easy-to-implement methods are able to find approximate optimal solutions within a tolerable time limit. It is known that the quality of the initial solution greatly affects the quality of the approximated solution found by a local search method. In this paper, we propose to take the initial solution as a random variable and learn its preferable probability distribution. The aim is to sample a good initial solution from the learned distribution so that the local search can find a high-quality solution. We develop two different deep network models to deal with DOPs established on set (the knapsack problem) and graph (the maximum clique problem), respectively. The deep neural network learns the representation of an optimization problem instance and transforms the representation to its probability vector. Experimental results show that given the initial solution sampled from the learned probability distribution, a local search method can acquire much better approximate solutions than the randomly-sampled initial solution on the synthesized knapsack instances and the Erdős-Rényi random graph instances. Furthermore, with sampled initial solutions, a classical genetic algorithm can achieve better solutions than a random initialized population in solving the maximum clique problems on DIMACS instances. Particularly, we emphasize that the developed models can generalize in dimensions and across graphs with various densities, which is an important advantage on generalizing deep-learning-based optimization algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price includes VAT (Russian Federation)

Instant access to the full article PDF.

Rent this article via DeepDyve

Institutional subscriptions

Abdel-Basset M, Abdel-Fatah L, Sangaiah A K. Metaheuristic algorithms: A comprehensive review. In: Sangaiah A K, Zhang Z, Sheng M, eds. Computational Intelligence for Multimedia Big Data on the Cloud with Engineering Applications. New York: Academic Press, 2018, 185–231

Chapter   Google Scholar  

Ansótegui C, Heymann, B, Pon J, et al. Hyper-reactive tabu search for MaxSAT. In: Proceedings of the International Conference on Learning and Intelligent Optimization. Berlin-Heidelberg: Springer, 2019, 309–325

Bahdanau D, Cho K, Bengio Y. Neural machine translation by jointly learning to align and translate. In: Proceedings of the 3rd International Conference on Learning Representations. New Orleans: OpenReview.net, 2015, 1–15

Google Scholar  

Bello I, Pham H, Le Q V, et al. Neural combinatorial optimization with reinforcement learning. In: Proceedings of the 5th International Conference on Learning Representations. New Orleans: OpenReview.net, 2017, 1–15

Bengio Y, Lodi A, Prouvost A. Machine learning for combinatorial optimization: A methodological tour d’horizon. Euro J Oper Res, 2021, 290: 405–421

Article   MathSciNet   Google Scholar  

Bertsekas D P. Dynamic Programming and Optimal Control: Volume I. Cambridge: Athena Scientific, 2012

Bron C, Kerbosch J. Algorithm 457: Finding all cliques of an undirected graph. Commun ACM, 1973, 16: 575–577

Article   Google Scholar  

Bruna J, Zaremba W, Szlam A, et al. Spectral networks and locally connected networks on graphs. In: Proceedings of the 2nd International Conference on Learning Representations. New Orleans: OpenReview.net, 2014, 1–14

Cacchiani V, Iori M, Locatelli A, et al. Knapsack problems - an overview of recent advances. part I: Single knapsack problems. Comput Oper Res, 2022, 143: 105692

Chen Z, Liu J, Wang X, et al. On representing linear programs by graph neural networks. In: Proceedings of the 11th International Conference on Learning Representations. New Orleans: OpenReview.net, 2023, 1–29

Cobos C, Dulcey H, Ortega J, et al. A binary fisherman search procedure for the 0/1 knapsack problem. In: Proceedings of the Advances in Artificial Intelligence. Berlin-Heidelberg: Springer, 2016, 447–457

Cortez P. Local Search. Cham: Springerg, 2021

Book   Google Scholar  

Crama Y, Kolen A W J, Pesch E J. Local Search in Combinatorial Optimization. Berlin-Heidelberg: Springer, 1995

Csirik J, Frenk J, Labbé M, et al. Heuristics for the 0–1 min-knapsack problem. Acta Cybernet, 1991, 10: 15–20

MathSciNet   Google Scholar  

Dai H, Khalil E B, Zhang Y, et al. Learning combinatorial optimization algorithms over graphs. In: Proceedings of the 31st International Conference on Neural Information Processing Systems. San Francisco: Curran Associates, 2017, 6351–6361

Defferrard M, Bresson X, Vandergheynst P. Convolutional neural networks on graphs with fast localized spectral filtering. In: Proceedings of the 30th International Conference on Neural Information Processing Systems. San Francisco: Curran Associates, 2016, 3844–3852

Doerr B, Neumann F. A survey on recent progress in the theory of evolutionary algorithms for discrete optimization. ACM Trans Evo Learn Optim, 2021, 1: 1–43

Erdős P, Rényi A. On the Evolution of Random Graphs. Princeton: Princeton Univ Press, 2006

Ezugwu A E, Shukla A K, Nath R, et al. Metaheuristics: A comprehensive overview and classification along with bibliometric analysis. Art Intell Rev, 2021, 54: 4237–4316

Gasse M, Chetelat D, Ferroni N, et al. Exact combinatorial optimization with graph convolutional neural networks. In: Proceedings of the 33rd International Conference on Neural Information Processing Systems. San Francisco: Curran Associates, 2019, 15554–15566

Glover F. Future paths for integer programming and links to artificial intelligence. Comput Oper Res, 1986, 13: 533–549

Goodfellow I, Bengio Y, Courville A. Optimization for Training Deep Models. Cambridge: MIT Press, 2016

Haghir Chehreghani M. Half a decade of graph convolutional networks. Nat Mach Intell, 2021, 4: 192–193

Hansen P, Mladenovic N. Variable Neighborhood Search. Cham: Springer, 2018

Hottung A, Tanaka S, Tierney K. Deep learning assisted heuristic tree search for the container pre-marshalling problem. Comput Oper Res, 2020, 113: 104781

Hussein A, Gaber M M, Elyan E, et al. Imitation learning: A survey of learning methods. ACM Comput Sur, 2017, 50: 1–35

Jiang H, Li C-M, Manyà F. Combining efficient preprocessing and incremental MaxSAT reasoning for maxclique in large graphs. In: Proceedings of the Twenty-Second European Conference on Artificial Intelligence. Palo Alto: AAAI Press, 2016, 939–947

Johnson D J, Trick M A. Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge. Providence: Amer Math Soce, 1996

Karimi-Mamaghan M, Mohammadi M, Meyer P, et al. Machine learning at the service of meta-heuristics for solving combinatorial optimization problems: A state-of-the-art. Euro J Oper Res, 2022, 296: 393–422

Khalil E B, Dilkina B, Nemhauser G L, et al. Learning to run heuristics in tree search. In: Proceedings of the 26th International Joint Conference on Artificial Intelligence. Palo Alto: AAAI Press, 2017, 659–666

Kipf T N, Welling M. Semi-supervised classification with graph convolutional networks. In: Proceedings of the 5th International Conference on Learning Representations. New Orleans: OpenReview.net, 2017, 1–14

Kirkpatrick S, Gelatt C D, Vecchi M P. Optimization by simulated annealing. Science, 1983, 220: 671–680

Kool W, van Hoof H, Welling M. Attention, learn to solve routing problems! In: Proceedings of the 7th International Conference on Learning Representations. New Orleans: OpenReview.net, 2019, 1–25

Korte B, Vygen J. Combinatorial Optimization. Berlin-Heidelberg: Springer, 2018

Kruber M, Lübbecke M E, Parmentier A. Learning when to use a decomposition. In: Proceedings of the Integration of AI and OR Techniques in Constraint Programming. Berlin-Heidelberg: Springer, 2017, 202–210

Land A H, Doig A G. An automatic method of solving discrete programming problems. Econometrica, 1960, 28: 497–520

Lemos H, Prates M, Avelar P, et al. Graph colouring meets deep learning: Effective graph neural network models for combinatorial problems. In: Proceedings of the 2019 IEEE 31st International Conference on Tools with Artificial Intelligence (ICTAI). San Francisco: IEEE, 2019, 879–885

Li X, Olafsson S. Discovering dispatching rules using data mining. J Schedul, 2005, 8: 515–527

Li Z, Chen Q, Koltun V. Combinatorial optimization with graph convolutional networks and guided tree search. In: Proceedings of the 32nd International Conference on Neural Information Processing Systems. San Francisco: Curran Associates, 2018, 537–546

Liu X, Sun J, Zhang Q, et al. Learning to learn evolutionary algorithm: A learnable differential evolution. IEEE Trans Emerg Top Comput Intell, 2023, 7: 1605–1620

Lodi A, Zarpellon G. On learning and branching: A survey. TOP, 2017, 25: 207–236

Louis S, McDonnell J. Learning with case-injected genetic algorithms. IEEE Trans Evol Comput, 2004, 8: 316–328

Mahmood R, Babier A, McNiven A, et al. Automated treatment planning in radiation therapy using generative adversarial networks. In: Proceedings of the 3rd Machine Learning for Healthcare Conference. Ann Arbor: PMLR, 2018, 484–499

Marchiori E. A simple heuristic based genetic algorithm for the maximum clique problem. In: Proceedings of the 1998 ACM Symposium on Applied Computing. New York: Association for Computing Machinery, 1998, 366–373

Marchiori E. Genetic, iterated and multistart local search for the maximum clique problem. In: Proceedings of the Applications of Evolutionary Computing. Berlin-Heidelberg: Springer, 2002, 112–121

Martí R, Reinelt G. Exact and Heuristic Methods in Combinatorial Optimization. Berlin: Springer, 2022

Mencarelli L, D’Ambrosio C, Di Zio A, et al. Heuristics for the general multiple non-linear knapsack problem. Electron Note Discret Math, 2016, 55: 69–72

Milan A, Rezatofighi S H, Garg R, et al. Data-driven approximations to np-hard problems. In: Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence. Palo Alto: AAAI Press, 2017, 1453–1459

Nasiri M M, Salesi S, Rahbari A, et al. A data mining approach for population-based methods to solve the JSSP. Soft Comput, 2019, 23: 11107–11122

Nazari M, Oroojlooy A, Takáč M, et al. Reinforcement learning for solving the vehicle routing problem. In: Proceedings of the 32nd International Conference on Neural Information Processing Systems. San Francisco: Curran Associates, 2018, 9861–9871

Peres F, Castelli M. Combinatorial optimization problems and metaheuristics: Review, challenges, design, and development. Appl Sci, 2021, 11: 1–39

Pisinger D. Where are the hard knapsack problems? Comput Oper Res, 2005, 32: 2271–2284

Rossi R A, Gleich D F, Gebremedhin A H. Parallel maximum clique algorithms with applications to network analysis. SIAM J Sci Comput, 2015, 37: C589–C616

San Segundo P, Lopez A, Pardalos P M. A new exact maximum clique algorithm for large and massive sparse graphs. Comput Oper Res, 2016, 66: 81–94

Selsam D, Lamm M, Buünz B, et al. Learning a SAT solver from single-bit supervision. In: Proceedings of the 7th International Conference on Learning Representations. New Orleans: OpenReview.net, 2019, 1–11

Sergienko I V, Shylo V P. Problems of discrete optimization: Challenges and main approaches to solve them. Cybernet Syst Anal, 2006, 42: 465–482

Stüutzle T, Ruiz R. Iterated Local Search. New York-Berlin: Springer, 2018

Sun J, Liu X, Back T, et al. Learning adaptive differential evolution algorithm from optimization experiences by policy gradient. IEEE Trans Evo Comput, 2021, 25: 666–680

Sutton R S, Barto A G. Reinforcement Learning: An Introduction. Cambridge: MIT press, 2018

Tahami H, Fakhravar H. A literature review on combining heuristics and exact algorithms in combinatorial optimization. Euro J Inform Tech Comput Sci, 2022, 2: 6–12

Talbi E-G. Machine learning into metaheuristics: A survey and taxonomy. ACM Comput Surv, 2021, 54: 1–32

Toth P. Dynamic programming algorithms for the zero-one knapsack problem. Computing, 1980, 25: 29–45

Toyer S, Trevizan F W, Thiébaux S, et al. Action schema networks: Generalised policies with deep learning. In: Proceedings of the 32nd AAAI Conference on Artificial Intelligence. Palo Alto: AAAI Press, 2018, 6294–6301

Vaswani A, Shazeer N, Parmar N, et al. Attention is all you need. In: Proceedings of the 31st International Conference on Neural Information Processing Systems. New York: Association for Computing Machinery, 2017, 6000–6010

Vince A. A framework for the greedy algorithm. Discrete Appl Math, 2002, 121: 247–260

Vinyals O, Fortunato M, Jaitly N. Pointer networks. In: Proceedings of the 28th International Conference on Neural Information Processing Systems. New York: Association for Computing Machinery, 2015, 2692–2700

Wu Q, Hao J-K. A review on algorithms for maximum clique problems. Euro J Oper Res, 2015, 242: 693–709

Yang X-S. Nature-Inspired Optimization Algorithms. New York: Academic Press, 2021

Zaheer M, Kottur S, Ravanbakhsh S, et al. Deep sets. In: Proceedings of the 31st International Conference on Neural Information Processing Systems. New York: Association for Computing Machinery, 2017, 3391–3401

Zhou J, Cui G, Hu S, et al. Graph neural networks: A review of methods and applications. AI Open, 2020, 1: 57–81

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11991023 and 62076197) and Key Research and Development Project of Shaanxi Province (Grant No. 2022GXLH-01-15).

Author information

Authors and affiliations.

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

Xin Liu, Jianyong Sun & Zongben Xu

You can also search for this author in PubMed   Google Scholar

Corresponding author

Correspondence to Jianyong Sun .

Rights and permissions

Reprints and permissions

About this article

Liu, X., Sun, J. & Xu, Z. Learning to sample initial solution for solving 0–1 discrete optimization problem by local search. Sci. China Math. (2024). https://doi.org/10.1007/s11425-023-2290-y

Download citation

Received : 01 July 2023

Accepted : 01 April 2024

Published : 29 April 2024

DOI : https://doi.org/10.1007/s11425-023-2290-y

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• discrete optimization
• deep learning
• graph convolutional network
• local search
• Find a journal
• Publish with us
• Track your research

Help | Advanced Search

Computer Science > Artificial Intelligence

Title: gold: geometry problem solver with natural language description.

Abstract: Addressing the challenge of automated geometry math problem-solving in artificial intelligence (AI) involves understanding multi-modal information and mathematics. Current methods struggle with accurately interpreting geometry diagrams, which hinders effective problem-solving. To tackle this issue, we present the Geometry problem sOlver with natural Language Description (GOLD) model. GOLD enhances the extraction of geometric relations by separately processing symbols and geometric primitives within the diagram. Subsequently, it converts the extracted relations into natural language descriptions, efficiently utilizing large language models to solve geometry math problems. Experiments show that the GOLD model outperforms the Geoformer model, the previous best method on the UniGeo dataset, by achieving accuracy improvements of 12.7% and 42.1% in calculation and proving subsets. Additionally, it surpasses the former best model on the PGPS9K and Geometry3K datasets, PGPSNet, by obtaining accuracy enhancements of 1.8% and 3.2%, respectively.

Submission history

Access paper:.

• 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 .