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

Looking for free math worksheets.

You’ve found something even better!

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

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

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

• Trigonometry

Statistics and probability

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

• Addition and subtraction
• Place value (tens and hundreds)
• Addition and subtraction within 20
• Addition and subtraction within 100
• Addition and subtraction within 1000
• Measurement and data
• Counting and place value
• Measurement and geometry
• Place value
• Measurement, data, and geometry
• Add and subtract within 20
• Add and subtract within 100
• Add and subtract within 1,000
• Money and time
• Measurement
• Intro to multiplication
• 1-digit multiplication
• Addition, subtraction, and estimation
• Intro to division
• Understand fractions
• Equivalent fractions and comparing fractions
• More with multiplication and division
• Arithmetic patterns and problem solving
• Quadrilaterals
• Represent and interpret data
• Multiply by 1-digit numbers
• Multiply by 2-digit numbers
• Factors, multiples and patterns
• Add and subtract fractions
• Multiply fractions
• Understand decimals
• Plane figures
• Measuring angles
• Area and perimeter
• Units of measurement
• Decimal place value
• Add decimals
• Subtract decimals
• Multi-digit multiplication and division
• Divide fractions
• Multiply decimals
• Divide decimals
• Powers of ten
• Coordinate plane
• Algebraic thinking
• Converting units of measure
• Properties of shapes
• Ratios, rates, & percentages
• Arithmetic operations
• Negative numbers
• Properties of numbers
• Variables & expressions
• Equations & inequalities introduction
• Data and statistics
• Negative numbers: addition and subtraction
• Negative numbers: multiplication and division
• Fractions, decimals, & percentages
• Rates & proportional relationships
• Expressions, equations, & inequalities
• Numbers and operations
• Solving equations with one unknown
• Linear equations and functions
• Systems of equations
• Geometric transformations
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• Pythagorean theorem
• Transformations, congruence, and similarity
• Arithmetic properties
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• Negative numbers and coordinate plane
• Ratios, rates, proportions
• Equations, expressions, and inequalities
• Exponents, radicals, and scientific notation
• Foundations
• Algebraic expressions
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• Expressions with exponents
• Quadratics and polynomials
• Equations and geometry
• Algebra foundations
• Solving equations & inequalities
• Working with units
• Linear equations & graphs
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• Inequalities (systems & graphs)
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• Irrational numbers
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• More on regression
• Prepare for the 2020 AP®︎ Statistics Exam
• AP®︎ Statistics Standards mappings
• Polynomials
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• Derivatives: definition and basic rules
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• Applications of derivatives
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• Applications of integrals
• Differentiation: definition and basic derivative rules
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• Applications of integration
• AP Calculus AB solved free response questions from past exams
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• 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.

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

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Math Worksheets

Test your math skills! Ace that test! See how far you can get! You can view them on-screen, and then print them, with or without answers.

Every worksheet has thousands of variations, so you need never run out of practice material.

Choose your Subject !

* Note: the worksheet variation number is not printed with the worksheet on purpose so others cannot simply look up the answers. If you want the answers, either bookmark the worksheet or print the answers straight away.

Also! Our forum members have put together a collection of Math Exercises .

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Free Math Worksheets

Printable math worksheets from k5 learning.

Our  free math worksheets  cover the full range of elementary school math skills from numbers and counting through fractions, decimals, word problems and more. All worksheets are printable files with answers on the 2nd page.

Math worksheets by grade:

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Sample Math Worksheet

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Free Math Worksheets by Math-Drills

Math-Drills.com includes over 70,000 free math worksheets that may be used to help students learn math. Our math worksheets are available on a broad range of topics including number sense, arithmetic, pre-algebra, geometry, measurement, money concepts and much more. There are two interactive math features: the math flash cards and dots math game.

Math-Drills.com was launched in 2005 with around 400 math worksheets. Since then, tens of thousands more math worksheets have been added. The website and content continues to be improved based on feedback and suggestions from our users and our own knowledge of effective math practices.

Most Math-Drills users are classroom teachers or parents. Classroom teachers use our math worksheets to assess student mastery of basic math facts, to give students extra math practice, to teach new math strategies, and to save precious planning time. Parents use our math worksheets to give their children extra math practice over school breaks and to enhance their math education. Home schools use our math worksheets in their programs to develop and strengthen math skills in their children.

Most Popular Free Math Worksheets this Week

Supporting Student Learning with Free Math Worksheets by Math-Drills

Math-Drills believes that education should be accessible to all children despite their socioeconomic situation or any other factors. Since it began in 2005, all the math worksheets on Math-Drills have been free-to-use with students learning math. The Math-Drills website works well on any device and worksheets can be printed or used on a screen.

Math-Drills worksheets are also used in special education, adult education, tutoring, colleges, high schools, prisons and a variety of other situations. Special education teachers especially like that we break down math skills and provide large-print options. Adult learners appreciate the simple uncluttered format that our worksheets offer. Tutors use our math worksheets to reduce their costs and focus on student learning. High school, college and university educators sometimes need remedial resources for students to allow them to continue with more advanced topics.

Students who practice their math skills with our math worksheets over school breaks keep their math skills sharp for upcoming school terms. Because we provide answer keys, students are able to self-assess and use the immediate feedback provided by an answer key to analyze and correct errors in their work. Our interactive (fillable) math worksheets allow them to fill in their answers on the screen and save or print the results.

Math-Drills.com Tour on YouTube.

Join the Math-Drills Newsletter

Join our newsletter to find out about new math worksheets and other information related to the website.

Copyright © 2005-2024 Math-Drills.com You may use the math worksheets on this website according to our Terms of Use to help students learn math.

Free Math Printable Worksheets with Answer Keys and Activities

Other free resources.

Feel free to download and enjoy these free worksheets on functions and relations. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Long Division with Remainders
• Long Division with Remainders #2 (Zeros in the Quotient)
• Long Division with 2 Digit Divisors
• Whole Number by Unit Fraction
• Equation of Circle
• Simplify Imaginary Numbers
• Adding and Subtracting Complex Numbers
• Multiplying Complex Numbers
• Dividing Complex Numbers
• Dividing Complex Number (Advanced)
• End of Unit, Review Sheet
• Distance Formula
• Simplify Rational Exponents (Algebra 2)
• Solve Equations with Rational Exponents (Algebra 2)
• Solve Equations with variables in Exponents (Algebra 2)
• Exponential Growth (no answer key on this one, sorry)
• Compound Interest Worksheet #1 (No logs)
• Compound Interest Worksheet (Logarithms required)
• Factor Trinomials Worksheet
• Factor by Grouping
• Domain and Range (Algebra 1)
• Functions vs Relations (Distinguish function from relation, state domain etc..) (Algebra 2)
• Evaluating Functions (Algebra 2)
• 1 to 1 Functions (Algebra 2)
• Composition of Functions (Algebra 2)
• Inverse Functions Worksheet (Algebra 2)
• Operations with Functions (Algebra 2)
• Functions Review Worksheet (Algebra 2)
• Logarithmic Equations
• Properties of Logarithms Worksheet
• Product Rule of Logarithms
• Power Rule of Logarithms
• Quotient Rule of Logarithms
• Solve Quadratic Equations by Factoring
• Quadratic Formula Worksheets (3 different sheets)
• Quadratic Formula Worksheet (Real solutions)
• Quadratic Formula (Complex solutions)
• Quadratic Formula (Both real and complex solutions)
• Discriminant and Nature of the Roots
• Solve Quadratic Equations by Completing the Square
• Sum and Product of Roots
• Radical Equations
• Mixed Problems on Writing Equations of Lines
• Slope Intercept Form Worksheet
• Standard Form Worksheet
• Point Slope Worksheet
• Write Equation of Line from the Slope and 1 Point
• Write Equation of Line From Two Points
• Equation of Line Parallel to Another Line and Through a Point
• Equation of Line Perpendicular to Another Line and Through a Point
• Slope of a Line
• Perpendicular Bisector of Segment
• Write Equation of Line Mixed Review
• Word Problems
• Multiplying Monomials Worksheet
• Multiplying and Dividing Monomials Sheet
• Adding and Subtracting Polynomials worksheet
• Multiplying Monomials with Polynomials Worksheet
• Multiplying Binomials Worksheet
• Multiplying Polynomials
• Simplifying Polynomials
• Factoring Trinomials
• Operations with Polynomials Worksheet
• Dividing Radicals
• Simplify Radicals Worksheet
• Adding Radicals
• Multiplying Radicals Worksheet
• Radicals Review (Mixed review worksheet on radicals and square roots)
• Rationalizing the Denominator (Algebra 2)
• Radical Equations (Algebra 2)
• Solve Systems of Equations Graphically
• Solve Systems of Equations by Elimination
• Solve by Substitution
• Solve Systems of Equations (Mixed Review)
• Activity on Systems of Equations (Create an advertisement for your favorite method to Solve Systems of Equations )
• Real World Connections (Compare cell phone plans)
• Identifying Fractions

Trigonomnetry

• Law of Sines and Cosines Worksheet (This sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well as on using both formulas to solve for a single triangle's side or angle)
• Law of Sines
• Ambiguous Case of the Law of Sines
• Law of Cosines
• Vector Worksheet
• Sine, Cosine, Tangent, to Find Side Length
• Sine, Cosine, Tangent Chart
• Inverse Trig Functions
• Real World Applications of SOHCATOA
• Mixed Review
• Unit Circle Worksheet
• Graphing Sine and Cosine Worksheet
• Sine Cosine Graphs with Vertical Translations
• Sine, Cosine, Tangent Graphs with Phase Shifts
• Sine, Cosine, Tangent Graphs with Change in Period, Amplitude and Phase Shifts (All Translations)
• Tangent Equation, Graph Worksheet
• Graphing Sine, Cosine, Tangent with Change in Period
• Cumulative, Summative Worksheet on Periodic Trig Functions - period, amplitude, phase shift, radians, degrees,unit circle
• Ratio and Proportion
• Similar Polygons
• Area of Triangle
• Interior Angles of Polygons
• Exterior Angles of Polygons

• Identifying Fractions Worksheet
• Associated Powerpoint
• Simplify Fractions Worksheet (Regular Difficulty)
• Associated PowerPoint
• Simplify Fractions Worksheet (Challenging Difficulty level for advanced learners)
• System of Linear Equations Worksheet
• System of Linear Equations - Real World Application
• Compositions of Reflections. Reflections Over Intersecting Lines as Rotations

All of these worksheets and activities are available for free so long as they are used solely for educational, noncommercial purposes and are not distributed outside of a specific teacher's classroom.

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

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Math Games offers online games and printable worksheets to make learning math fun. Kids from pre-K to 8th grade can practice math skills recommended by the Common Core State Standards in exciting game formats. Never associated learning algebra with rescuing animals or destroying zombies? Time to think again!

Kids learn better when they're having fun . They also learn better when they get to practice new skills repeatedly . Math Games lets them do both - in school or at home .

Teachers and parents can create custom assignments that assess or review particular math skills. Activities are tailored so pupils work at appropriate grade levels . Worksheets can be downloaded and printed for classroom use , or activities can be completed and automatically graded online .

Best of all? It's completely free! Click on any of our games above to get started.

Get updates on what we do by following us on Twitter at @mathgames . Send us your comments, queries or suggestions here .

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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.

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Stanford Online

Introduction to mathematical thinking.

HSTAR-Y0001

Stanford Graduate School of Education

The goal of the course is to help you develop a valuable mental ability – a powerful way of thinking that our ancestors have developed over three thousand years.

Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today's world. This course helps to develop that crucial way of thinking.

The course is offered in two versions. The eight-week-long Basic Course is designed for people who want to develop or improve mathematics-based, analytic thinking for professional or general life purposes. The ten-week-long Extended Course is aimed primarily at first-year students at college or university who are thinking of majoring in mathematics or a mathematically-dependent subject, or high school seniors who have such a college career in mind. The final two weeks are more intensive and require more mathematical background than the Basic Course. There is no need to make a formal election between the two. Simply skip or drop out of the final two weeks if you decide you want to complete only the Basic Course.

Subtitles for all video lectures available in: Portuguese (provided by  The Lemann Foundation ), English

Course Syllabus

Instructor's welcome and introduction

•  Introductory material
•  Analysis of language – the logical combinators
•  Analysis of language – implication
•  Analysis of language – equivalence
•  Analysis of language – quantifiers
•  Working with quantifiers
•  Proofs
•  Proofs involving quantifiers
•  Elements of number theory
• Beginning real analysis

Recommended Background

High school mathematics. Specific requirements are familiarity with elementary symbolic algebra, the concept of a number system (in particular, the characteristics of, and distinctions between, the natural numbers, the integers, the rational numbers, and the real numbers), and some elementary set theory (including inequalities and intervals of the real line). Students whose familiarity with these topics is somewhat rusty typically find that with a little extra effort they can pick up what is required along the way. The only heavy use of these topics is in the (optional) final two weeks of the Extended Course.

A good way to assess if your  basic  school background is adequate (even if currently rusty) is to glance at the topics in the book  Adding It Up: Helping Children Learn Mathematics  (free download), published by the US National Academies Press in 2001. Though aimed at K-8 mathematics teachers and teacher educators, it provides an excellent coverage of what constitutes a good basic mathematics education for life in the Twenty-First Century (which was the National Academies' aim in producing it).

Dr Keith Devlin, Co-founder and Executive Director H-STAR Institute

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Top-Rated Tips From Experts: How to Write a Maths Assignment

Mathematics Mastery: Trust Our Experts for Impeccable Assignment Writing, Ensuring Precision and Academic Excellence.

Writing a mathematics assignment is rewarding yet challenging, requiring precision, clarity, and a deep understanding of mathematical concepts. Navigating this academic challenge successfully can be exhaustive for students. Under this circumstance, consider the insights and tips our expert professionals share. These tips enhance your approach to creating high-quality and accurate math assignments.

From effective problem-solving strategies to the importance of clear communication, these expert-recommended guidelines offer a comprehensive framework for achieving excellence in math homework. Whether you're a student looking to improve your assignment skills or an educator guiding others through the process, incorporating these tips can elevate the quality and impact of your math assignments.

Mastering Math Assignments: Strategies for Success

Embracing the world of mathematics can be enjoyable and intellectually stimulating when approached with the right mindset. Here are some valuable tips to enhance your ability to solve math assignments efficiently.

Eliminate Distractions: Dedicate focused time to complete your math assignments, as they often involve multi-step problem-solving. Minimize distractions by putting away your phone and notifying others of your study plans. Seeking assistance from math assignment help services can provide expert guidance and support you require for math assignments.

Utilize Mathematical Tricks: Learn and apply quick mathematical tricks from your professors for calculations. You can also opt for assistance from a math assignment helper to ace different techniques. Familiarity with these tricks can significantly expedite the problem-solving of math writing assignment.

Thoroughly Read and Follow Instructions: Carefully read the instructions provided by your instructor before diving into the assignment. Use a notepad to jot down essential information and create an outline for a better understanding. Clearly articulate your problem-solving process, ensuring you address all aspects mentioned in the instructions.

Start with Challenging Problems: Prioritize your math problems based on difficulty and importance. Tackle the most challenging problems first and gradually work your way down. This approach ensures efficient utilization of your energy and time.

Take Regular, Short Breaks: Maintain concentration by taking short breaks to alleviate stress and prevent burnout. Breaks of 15 to 20 minutes allow you to recharge without losing focus on your assignments.

Study with a Friend: Collaborating with a study companion can provide motivation and a fresh perspective. Explaining concepts to each other enhances understanding and promotes effective learning.

Seek Expert Assistance: Don't hesitate to ask for help from experts at any stage of your assignment. Technology facilitates instant access to professionals who can assist with complex problems, ensuring academic success.

Practice Regularly: Building a strong foundation in mathematics requires consistent practice. Regular practice eases complex computations and explores various methods. It also helps you to analyze shortcuts for rapid problem-solving.

Common Worries About Math Assignments:

Many students experience common worries about math assignments. They often feel anxious about understanding complex concepts and solving difficult problems. Time management is another concern, as math assignments can be time-consuming and require focused attention. The fear of making mistakes and the pressure to achieve high grades can also contribute to stress. Additionally, some students worry about not having adequate resources or support to complete their assignments successfully. These concerns can impact their confidence and performance, but with the right strategies and help, such as tutoring and practice, students can overcome these challenges and excel in their math assignments.

Understanding why students may struggle with math assignments is essential. Common reasons include:

Problems in Math Concept Understanding

Difficulty in understanding math arises from a combination of factors.

Some learners find the abstract nature of mathematical concepts challenging. Others face problems due to confusion from previously perceived knowledge. Most students acknowledged they often find it hard to visualize and relate math problems to real-world scenarios. Foundational knowledge gaps hinder comprehension. Effective math understanding requires a supportive learning environment, including several techniques that we have explained in the visual. These steps emphasize building a solid foundation for understanding and solving mathematical problems.

Challenges Due to Teaching

Students face diverse study challenges due to teaching methodologies. One of them is the lack of effective communication between educators and students. If teachers fail to articulate concepts clearly, students may struggle to grasp subject materials. Inadequate teacher-student interaction can hinder the resolution of doubts or concept clarification. Outdated teaching methods fail to engage students, resulting in a lack of interest and motivation to study. Effective teaching involves adapting to the diverse needs of students. It fosters a positive and interactive study environment for a better learning experience.

Inconsistency in Practice:

Inconsistency in practice poses a significant challenge in various contexts, ranging from skill development to routine tasks. Irregular practice can impede progress. Inconsistency in practicing new skills or habits can hinder the cultivation of expertise or the formation of positive routines. The key to overcoming this challenge is establishing and adhering to a consistent, structured practice regimen. By doing so, individuals can enhance their learning, skill acquisition, and personal growth, fostering a more sustainable and effective path toward mastery and success.

Quick Tips for Improved Math Skills

Improve your math skills with these quick tips: practice regularly, review and understand basic concepts, seek help when needed, use online resources, and solve various problem types. Stay organized, work on time management, and maintain a positive attitude towards learning. Consistency and persistence are key to mastering math.

Enhance your arithmetic abilities and speed up problem-solving with these additional tips:

Master the Basics:

Proficiency in fundamental operations is crucial for tackling more advanced topics. It ensures a solid foundation by mastering the core principles of arithmetic.

Understand Concepts, Don't Memorize:

Focus on understanding the underlying concepts rather than memorizing procedures. It enables flexible problem-solving and a deeper grasp of mathematical principles.

Use Online Resources:

Explore online platforms offering interactive math exercises and tutorials. Math assignment help websites provide valuable resources for offering academic assistance.

Break Down Complex Problems:

Break down complex problems into smaller, manageable steps. This approach makes problem-solving more approachable and helps prevent feeling overwhelmed.

Teach Others:

Teaching a concept to someone else is an effective way to solidify your understanding. This process forces you to articulate ideas and reinforces your knowledge.

Seek Help When Needed:

Don't hesitate to ask for help when you encounter difficulties. Teachers, classmates, or online communities can provide valuable insights and clarification.

Review and Reflect:

Regularly review previously learned material and reflect on your progress. Identify areas of weakness and focus on reinforcing those concepts.

Utilize Educational Apps and Games:

Explore educational apps and games that make learning math enjoyable. These approaches can add an element of fun to your practice sessions.

Assignments are integral to academic growth, and mastering math is a journey that requires dedication and effective time management. Seek assistance when needed, and you'll develop strong math skills with consistent practice.

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Math Assignment

 What is an annotated bibliography?

Assignments in math courses are usually a list of problems to be solved. Each question may be of various difficulties and types, which are marked and returned to the student to be used as a source of feedback. Math assignments are designed to provide opportunities for 'doing math' and to consolidate students’ understanding of the content. The questions often come from the most recent week of learned material, but some questions may require students to synthesize concepts from further back. This is because learning math is cumulative by nature; you continuously build upon what you've learned before. Because math assignments are typically used as a checkpoint for understanding, they tend to weigh less than other assessments in terms of grade, therefore, it's important to treat them as learning opportunities instead of a tool for maximizing course grades.

Click on the Timeline for a visual representation of the timeline. Click on the Checklist for a document containing the checklist items for a math assignment.

According to your start and end dates ( 2024-08-23 to 2024-08-28 ), you have 5 days to finish your assignment.

Step 1: Understand the questions Complete by Fri Aug 23, 2024

Read through the assignment once it’s released and understand the questions. Read the questions early to prime your brain; too many students start too late.

• For some courses, assignments come out before or as content is covered, so having the questions in your mind as you see the content can help prime you to actively learn the content and will also help you know where to look for relevant material in the lecture notes and/or videos. Active learning of content means you are asking yourself questions as you are learning, actively monitoring your understanding of the topic, and anticipating next steps.
• Read all the questions to estimate the range of tested material, make note of anything you don’t understand. Understanding the question correctly reduces the chance of getting stuck.

The hardest thing being with a mathematician is that they always have problems.

– Tendai Chitewere

Step 2: Gather materials and review Complete by Sat Aug 24, 2024

• Gather required materials (lecture notes and/or videos, textbook sections).
• Study the material (definitions, concepts, solved problems, etc.) until you are familiar with it. Identify connections across concepts/topics and practice additional problems. Regularly reviewing your material will also help you to find relevant information quickly if you get stuck on a question.

Step3: Solve questions and get help Complete by Tue Aug 27, 2024

Part a: attempt to solve each question.

• Rule of thumb for time allocation: one hard question may take significantly longer than all easy questions combined. Consider how long you’ve taken to solve all the easy questions. It may take at least twice as long to solve one hard question. Try all easy questions as soon as possible to gauge how long the rest could take.
• Try giving yourself a break for a day before attempting them again to allow time for your mind to continue working subconsciously.
• Still stuck? Check out Math problems: What to do when you're stuck for more strategies and tips.
• Keep track of unsolved questions and what stages you were stuck at. Don’t throw away any work in progress.Your instructor or TA can provide better help if you can narrow down on the issue.

The only way to learn mathematics is to do mathematics.

– Paul Halmos

Mathematics is not a deductive science -- that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.

– Paul Halmos  

Part B: Get help on unsolved questions

• The sooner you ask for support, the more time you and the instructor/TA will have for follow-up conversations.
• Focus on asking questions to help you close the gap between your understanding, and the understanding you need to complete the question.  Questions such as “Can I get a hint on #4?” or “What is the answer?” are not typically helpful in increasing your problem-solving abilities.
• For questions you were challenged by, but were able to solve after receiving help from the instructor or TA, make sure that you fully understand how to come up with the solution in case a similar question is asked on a test.
• Many students find it helpful to study together. Make sure you understand what your instructor’s expectations are around group work (if you’re not sure, ask!). Be mindful about how much you post about an assignment solution on the course’s discussion board and if you’re in doubt, consider posting privately. Familiarize yourself with the academic integrity standards expected in your courses.

The goal of seeking help is to try to find out how to approach questions that you are stuck on. From this extra help you still may not see the complete picture of how to solve the problem. To see this may require that you go back to Part A , now with the instructor/TA’s suggestions in mind.

I have not failed. I've just found 10,000 ways that won't work.

– Thomas Alva Edison

Step 4: Check your solutions Complete by Wed Aug 28, 2024

• It’s important that you spend some time away from your solutions before you check them. For problems you did solve, try to figure out if you can check your own answer like you would have to on a test (i.e. don’t look up answers in a book or online).
• You could ask the instructor/TA for guidance, but avoid questions such as “Did I do this right?” Instead, ask yourself whether you are asking the right questions about your solution. 
• Remember not to focus only on the final answer, but also on how you communicated your answer. Ask yourself if a typical student in the class could follow along with what you’ve written. 
• Remember to cite any external sources that you used in your work, including work completed with others (if this was permitted).

Step 5: Review Marked Assignments Complete by Wed Aug 28, 2024

After receiving your marked assignment check over incorrect questions as well as those you got right. Review posted solutions and read them critically. Often, there are different approaches to a problem and you may learn about some of these new approaches through this review.

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What are the rules of assignment?

Recently I've been reviewing some of the basics of algebra, and one thing that is taught rather flimsily are the rules of when we want to 'assign' to a symbol. Is there a formal language that allows us to define an assignment, for example we often see 'if $x=...$ or 'when $x=...$ '. The first question that a student might have is 'WHEN precisely is $x=3$ ' I understand in 'model theory' there exists an idea of 'assignment', but it does not seem naturally possible to link it to the simpler concept, Perhaps here we find the origins of this concept, but I cannot see it? Likewise, can we use the original 'variable' name? For example 'under the assignment $x=3$ , we know that $x^2=9$ ' in this case we are using the variable $x$ but under a particular assignment. If we can do this, is there any difference between a 'general' expression for the variable $x$ and an expression under an 'assignment'.

• algebra-precalculus
• $\begingroup$ Logical rules for equality. $\endgroup$ –  Mauro ALLEGRANZA Commented Jun 5, 2022 at 17:30
• 2 $\begingroup$ This question is a great example of the extremely blurry line between mathematics and philosophy. $\endgroup$ –  Silvio Mayolo Commented Jun 6, 2022 at 2:51
• $\begingroup$ I think this is a great question, and one that is often confusing to students who get stuck in early levels of mathematics. The meaning of the equal sign changes subtly depending on context, and it can mean very different things depending on that context. $\endgroup$ –  Amos Joshua Commented Jun 6, 2022 at 4:51

4 Answers 4

The problem here is that in actual practice, we are mixing both mathematical language and natural language, and it's not that the mathematical language or the natural language by themselves are tricky, but rather it's the mixing of the two that cause these concepts to become blurred.

The central bone of contention, as you're observing, is what exactly is the following hybrid of mathematical and natural language (English), in this case:

Let $x = 5$ .

What does it say about $=$ ? Does it mean $=$ is simply a verb indicating that something holds true, or is it something else? What I would suggest is that the above piece of language is formally ambiguous as to its interpretation, but both interpretations can be carried out in a consistent manner, and both of them have been suggested separately in the two answers given here.

The first is that this is introducing a hypothesis . The variable $x$ never actually gets a value. Instead, what is going on here is we are really saying "Suppose it is true that $x = 5$ ", and for sake of brevity, we use the term "let". That is, we are building what in formal language would look something like

$$(x = 5) \rightarrow \cdots$$

The second is that this statement, by its enunciation, actually does cause the fact that $x = 5$ to become true. That is, "let" expands to "I declare hereby that $x = 5$ ." This is the assignment version. Linguistically, this type of morsel is called in some circles a " performative ": it is a bit of language that causes its own truth simply by virtue of putting it forward. (This is similar to, but not quite the same as, the more commonly-known imperative : an imperative is a command given from one person to another. They both "do something", but the imperative requires further action to cause change, the performative does that all by itself.)

However, there then becomes the question of when and where the assignment ceases being valid, as you point out, since clearly it is not the case that from here on forever and everywhere $x = 5$ must hold otherwise we'd have a heck of a time trying to do maths and everyone would be fighting over what " $x$ ", a very common symbol, must mean! :) Hence if you take the assignment picture, you must thus also take the idea of scope . Unfortunately, while there is a semi-common symbol, $:=$ , for assignment, there is not a maths-only symbol for a scope. Thus, scopes are still implicit, not explicit.

Insofar as the "rules of assignment", it's basically this:

• Declare a scope. (No notation!)
• Make the assignments within that scope.
• Choose this as the scope to do some argumentation, then do the argumentation. In that argumentation, the given assignments are the meanings of the variables.

Outside the scope, the variables' meanings are once more undetermined (or not fully determined). That, I believe, answers your last question: the statement's meaning changes depending on what scope it is enclosed in, but scopes are not written explicitly in mathematical papers. Instead, you have to mentally track them and it's something you just pick up with experience.

ADD: I see some comments that suggest you may be asking for, say, properties or axioms that assignment follows. I'd suggest that there's pretty much only one: once $a := b$ has been enunciated, then $a = b$ necessarily evaluates to "true".

Of course, the relationship between these two approaches is, of course, that once you make the assignment, it becomes a true hypothesis, and thus we can try reasoning forward from it. However, which approach you choose to formulate maths is not something with a single answer. It's a choice. In the strong computer-coding affinity this has, it's basically the difference between functional and imperative programming. In functional programming, everything is in terms of "this relates to that". In imperative programming, you more explicitly issue commands to the computer that make the computer do stuff, like set variables to certain values. Of course, since the computer is ultimately doing stuff, the functional program must somehow become translated to imperative, but it is not written like it is imperative.

That said, I'd still argue in the end that ultimately you can't get away from the assignment: after all, even if we take the hypothetical approach, for the result to have any use, somewhere $x$ ultimately has to get assigned. The trick, then, is that that would happen in a particular application. Like if you're calculating a geometric quantity, your variable assignments happen - again implicitly - as part of the specific problem in question.

I would claim that, indeed, in the practice of mathematics, the sign $=$ is used in (at least) three different ways. The tradition is to pretend that there is only one use, that of an assertion of a relationship, such as $2=1+1$ .

However, in practice, there is also assignment , as the question observes. So, for example, if we're doing Peano arithmetic, we might define $2$ by "let $2=1+1$ ".

Also, there are queries or tests, which do not assert a relationship, and do not assign anything. As in "if $x>y$ , then...". Yes, we can say that the smaller logical piece " $x>y$ " expresses a possible relationship, but it does not assert it. Depending on one's formal logical framework, this may be an empty distinction. However, in dealing with informal logic and basic mathematics, this conceptually distinct.

Some of this is blurred by what are, in my opinion, "fake implications", where one is looking at cases, or making certain assumptions, but expresses it as "if... then...", where, in fact, the "if" component is not in doubt. E.g., "if $x=2$ , then $x+x=4$ ". My quibble could be expressed as, "well, how do we test whether or not $x=2$ ?". It makes much more practical sense to say "for $x=2$ ..." or "when $x=2$ ..." or, again, an assignment: "let $x=2$ . Then $x+x=4$ ."

You seem to be thinking about mathematical variables like computer language variables. There are no scope rules. If I say "let $x=3$ " I mean just that. It is expected that what I say next will expose a reason for the statement.

In mathematical writing, there is no difference between a declaration and an assignment because the symbols are not names for boxes or references to boxes.

In fact, you probably wouldn't see a statement like $$\text{let }x=3$$ . You would more likely see something like $$\text{let }x\in\mathbb R\text{ be a real number}$$ and later see a statement like $$\text{if }x=3\text{ then ...}$$

• 1 $\begingroup$ True, there are few scope "rules" in most mathematics, but the concept of "scope" does play a large role, whether explicitly acknowledged (or codified) or not, and in particular when the ubiquitous "$x$" gets re-used endlessly. :) $\endgroup$ –  paul garrett Commented Jun 5, 2022 at 18:09
• 2 $\begingroup$ @paulgarrett I have never seen $x$ get re-used in a mathematical proof. Re-using variables is a thing of computer programming. It isn't done in mathematical writing. $\endgroup$ –  John Douma Commented Jun 5, 2022 at 18:11
• 1 $\begingroup$ I disagree. In fact, I often see (and frequently write) "let $x=3\in\mathbb R$"... and perhaps later in the same not-so-short proof, "now let $x>3$". $\endgroup$ –  paul garrett Commented Jun 5, 2022 at 18:17
• 1 $\begingroup$ @paulgarrett would we not think that we are talking about different things that could or could not be true depending on the value of $x$ rather than 're-using' $x$? As we will replace every instance of $x$ when substituting which would not be the case if we were re-using $x$ for something else., $\endgroup$ –  user1047679 Commented Jun 5, 2022 at 21:14
• 1 $\begingroup$ @user37577, yes, there is indeed an issue of "scope", although not usually recognized in mathematics, so that we know "which $x$" we're referring-to, and so on. An extreme case in practical basic math is "the variable of integration", where we write $\int f(x)\,dx$. In reality, there is no need to name the argument to the function $f$ in order to integrate it, etc. But it is very traditional to write this kind of thing. And then there is a traditional confusion about whether the output of that integration "is a function of $x$" (usually it is construed so!), or ... what? :) $\endgroup$ –  paul garrett Commented Jun 5, 2022 at 21:25

You want to consider asking on the math educators site if you're specifically interested in how to teach students about $=$ or answer their questions about what it is.

Talking about assigning to a symbol in math is borrowing a metaphor from programming. That's not completely true; let $x$ $=$ something constructions are very old but I think that thinking of them as an assignment rather than a hypothesis is a programming metaphor. You can get a lot of mileage out of this metaphor, but it is not the only way to think about what $=$ is doing and it might be misleading, since $1 + 1 = 2$ is not an assignment. It is possible to lean into the programming metaphor further and use a dedicated symbol like $ := $ for assigning to variables, but that might be counterproductive.

In model theory / the semantics of first-order logic, an assignment has a specific meaning. It is a mapping from free variables to their denotations.

You need this map in order to define the notion of truth in a model.

For example, here is one such rule, giving the meaning of the existential quantifier.

$$ M, \vec{v} \models \exists x \mathop. \varphi(x, \vec{v}) \;\; \textit{if and only if}\;\; \text{there exists a variable assignment $\vec{u}$ extending $\vec{v}$ such that $M, \vec{u} \models \varphi(\vec{u})$} $$

A variable assignment $\vec{u}$ extending $\vec{v}$ means that the variables assigned any denotation at all by $\vec{u}$ are a superset of the ones assigned a denotation by $\vec{v}$ and that $\vec{u}$ and $\vec{v}$ agree on their common subset.

This is kind of unsatisfying though and doesn't really make the intuitive concept of hypothesizing a value for a variable clearer .

The semantics of equality are similarly unsatisfying and appeal to an intuitive notion of equality that already exists in the background.

$$ M, \vec{v} \models t_1(\vec{v}) = t_2(\vec{v}) \;\;\textit{if and only if}\;\; \text{the denotation of $t_1(\vec{v})$ is equal to the denotation of $t_2(\vec{v})$ in $M$} $$

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