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The Must-Have Math AI & Calculator for Students

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Math AI tools like Mathful are designed to assist users in solving mathematical problems, equations, and calculations. We utilize machine learning algorithms and natural language processing to interpret user input, perform complex computations, and provide step-by-step solutions.

Math AI can be beneficial for students, researchers, and professionals who require assistance in understanding and solving mathematical problems efficiently.

Can a math AI help me improve my math grades in school?

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Does this math AI tool cover all types of math?

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How long does it take for the math AI solver to get an answer to my math question?

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Can I try the Mathful AI math solver for free?

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What problems can I solve with a math calculator?

With a math calculator from Mathful, you can solve a broad spectrum of mathematical problems across various domains, including basic arithmetic operations, algebraic equations, and geometry calculations to more complex calculus derivatives and integrals.

Solve Math Problems Easily with Our Math Solver & Math Calculator

Get instant, accurate answers to your math problems and boost your math grade. Try our AI math problem solver and math calculator with steps for free today.

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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
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• Place value
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• 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
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• Intro to division
• Understand fractions
• Equivalent fractions and comparing fractions
• More with multiplication and division
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• Quadrilaterals
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• Multiply by 1-digit numbers
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• Area and perimeter
• Units of measurement
• Decimal place value
• Add decimals
• Subtract decimals
• Multi-digit multiplication and division
• Divide fractions
• Multiply decimals
• Divide decimals
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• Data and statistics
• Negative numbers: addition and subtraction
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• 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
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• Limits and continuity
• Derivatives: definition and basic rules
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• Applications of derivatives
• Analyzing functions
• Parametric equations, polar coordinates, and vector-valued functions
• Applications of integrals
• Differentiation: definition and basic derivative rules
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• Contextual applications of differentiation
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• AP Calculus AB solved free response questions from past exams
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• 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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Online Math Problem Solver

An absolutely free universal math problem solver:

Solve your math problems online. The free version gives you just answers. If you would like to see complete solutions you need to sign up for an account.

Basic Math Plan

Basic Math Solver offers you solving online fraction problems, metric conversions, power and radical problems. You can find area and volume of rectangles, circles, triangles, trapezoids, boxes, cylinders, cones, pyramids, spheres. You can simplify and evaluate expressions, factor/multiply polynomials, combine expressions.

Online Pre-Algebra(Geometry) Solver

You can solve all problems from the basic math section plus solving simple equations, inequalities and coordinate plane problems. You can also evaluate expressions, factor polynomials, combine/multiply/divide expressions.

Online Algebra Solver

Online trigonometry solver.

Solve all type of trigonometric (sin, cos, tan, sec, scs, cot) expressions, equations, inequalities. Graph trigonometric functions. Trigonometry of a right triangle.

Online Pre-calculus Solver

Include everything above plus finding limits (lim), sums, matrices.

Online Calculus Solver

Solve integral problems - definite, indefinite integrals.

Online Statistics Solver

Solve your probability, combination, permutation problems. Statistics - find median, mean (arithmetic, geometric, quadratic), mode, dispersion, mormal distributions, t-Distribution. The solver successfully do Statistical hypothesis testing

Online Chemistry Solver

You can online solve chemistry equations.

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Mathematics

Wolfram|Alpha has broad knowledge and deep computational power when it comes to math. Whether it be arithmetic, algebra, calculus, differential equations or anything in between, Wolfram|Alpha is up to the challenge. Get help with math homework, solve specific math problems or find information on mathematical subjects and topics.

Do basic arithmetic. Work with fractions, percentages and similar fundamentals. Solve place value and word problems.

Do basic arithmetic:

Do exact arithmetic with fractions:.

Compute the properties of geometric objects of various kinds in 2, 3 or higher dimensions. Explore and apply ideas from many subfields of geometry.

Compute properties of a geometric figure:

Plot a conic section and identify its type:, compute properties of a polyhedron:.

Work with various kinds of numbers. Check for membership in larger sets, such as the rationals or the transcendental numbers. Convert between bases.

Compute a decimal approximation to a specified number of digits:

Convert a decimal number to another base:.

Analyze integers; subsets of the integers, including the prime numbers; and related ideas.

Compute a prime factorization:

Solve a diophantine equation:.

Perform numerical analysis and optimization of systems and objects, including packing and covering of objects and control systems.

Minimize or maximize a function:

Numerically integrate functions that cannot be integrated symbolically:.

Make queries about various definitions and descriptions in mathematics.

Find information about a math concept:

Get a brief definition:.

Compute properties of datasets, perform statistical inference or model data. Work with probability distributions and random variables.

Calculate basic descriptive statistics for a data set:

Find the sample size needed to estimate a binomial parameter:.

Find roots of and expand, factor or simplify mathematical expressions—everything from polynomials to fields and groups.

Solve an equation:

Factor a polynomial:, simplify an expression:.

Solve differential equations of any order. Examine solutions and plots of the solution families. Specify initial conditions to find exact solutions.

Solve a linear ordinary differential equation:

Specify initial values:, solve a nonlinear equation:.

Perform trigonometric calculations and explore properties of trigonometric functions and identities.

Compute values of trigonometric functions:

Solve a trigonometric equation:.

Explore sequences and recurrences, solve common problems in combinatorics and compute properties of graphs and lattices.

Compute a possible formula and continuation for a sequence:

Analyze a graph specified by adjacency rules:, solve a recurrence:.

Evaluate Boolean logic expressions and expressions involving sets and set operators. Solve Boolean equations. Compute truth tables. Generate Venn diagrams.

Compute a truth table:

Generate a venn diagram:.

Gather information about famous problems, conjectures, theorems and paradoxes. Learn about them and their formulators.

Get information about a mathematical conjecture:

Get historical information about a theorem:.

Compute the probabilities of certain events occurring. Compute joint, disjoint or conditional probabilities and apply them to real-world situations.

Compute the probability of a union of events:

Compute coin‐toss probabilities:.

Compute integrals, derivatives and limits as well as analyze sums, products and series.

Compute an integral:

Calculate a derivative:, solve an ordinary differential equation:.

Visualize functions, equations and inequalities. Do so in 1, 2 or 3 dimensions. Make polar and parametric plots.

Plot a function:

Plot a region satisfying multiple inequalities:.

Explore and compute properties of vectors, matrices and vector spaces.

Compute properties of a vector:

Calculate properties of a matrix:, determine whether a set of vectors is linearly independent:.

Analyze functions and expressions containing imaginary numbers or complex variables.

Compute properties of a function of a complex variable (use the variable z ):

Compute the residue of a function at a point:.

Examine the properties of mathematical functions, such as continuity, surjectivity and parity. Utilize notable special functions or number theoretic functions.

Do computations with special functions:

Do computations with number theoretic functions:, find representations for a function:.

Compute; learn about algorithms, definitions and theorems involving; or find properties of continued fractions.

Find the continued fraction representation of a number:

Find definitions of continued fraction terminology:, find continued fraction papers by author:.

Get information about math Common Core Standards for kindergarten through eighth grade.

Evaluate an expression (CCSS.Math.Content.6.EE.A.2c):

Perform multiple operations with rational numbers (ccss.math.content.7.ns.a.2c):.

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Number Line

The Algebra Calculator is a versatile online tool designed to simplify algebraic problem-solving for users of all levels. Here's how to make the most of it:

• Begin by typing your algebraic expression into the above input field, or scanning the problem with your camera.
• After entering the equation, click the 'Go' button to generate instant solutions.
• The calculator provides detailed step-by-step solutions, aiding in understanding the underlying concepts.
• -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

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Purplemath's lessons cost nothing! These lessons are free, are intended for students (rather than for instructors), and are interlinked to help you find related material. There is a "Search" box on every page to help you find whatever math content you're looking for — without encountering ads masquerading as search results. Whether you're in middle school, high school, college, or university, these free algebra lessons can help you out!

How can I use Purplemath's lessons?

Purplemath's math lessons can be used in many ways:

• Read the lesson that relates to what you're about to study in class, so you get to class at least somewhat prepared.
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Neat homework can aid your comprehension and might make your teacher like you better. Purplemath's " Homework Guidelines for Mathematics " will give you a leg up, explaining in clear terms what your math teacher is looking for. The Guidelines link to examples of common errors, and demonstrate techniques that your instructors will love! In addition, students who get in the habit of explaining themselves clearly in their homework gain greater understanding of what they're doing, and therefore tend to do much better on their tests.

English teachers tell their students explicitly how to format their papers: what fonts, what page margins, what style guides, etc. Math teachers, on the other hand, frequently just complain amongst themselves in the faculty lounge about how messy their students' work is. Meanwhile, their students wonder why they've lost points on homework and tests.

Study these Guidelines , print out this " formatting " PDF, and improve your learning, retention, and test scores. Don't leave easy points on the table!

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Solvo - Math Homework Helper 4+

Problem solver & essay writer.

• #123 in Education
• 4.7 • 176 Ratings
• Offers In-App Purchases

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Solvo is your new superpower in education and beyond Maximize your academic potential with your own personal AI homework helper! Meet Solvo—an AI-powered math, chemistry, biology, physics solver & essay writer that revolutionizes the way you manage study assignments. Simply scan, type, or upload the task in front of you and let the app work its magic! Check out what Solvo can help you with: Scan & Solve • Scan and solve math problems, equations, and more Faced with a boss-level problem (e.g., you need science answers) and don’t know where to start? Snap a picture of it—you’ll see the result and in-depth solution steps. This way, you gain more insights into how to tackle certain tasks and become more confident solving them yourself next time! Math, science answers, and more—you name it, our AI homework helper helps with it in a flash. • Ace any test and quiz Our AI homework helper can answer all sorts of questions typically used in tests and quizzes, including true or false, multiple-choice, and open questions. Biology solver? Chemistry solver? It’s already in your pocket! Simply tap Text-Based Problems, snap a picture of the question, and get your answer in seconds. This feature can also help you test your knowledge and prepare for exams. Streamline Reading & Writing • Write killer essays in a breeze Have excellent ideas for your essay but find it hard to articulate them clearly? No problem—Solvo is an experienced essay writer! Simply tap Create Essay and type your subject. You can go ahead and use the output directly or to get your creative juices flowing. • Improve and reword your writing Solvo isn’t just an essay writer—it’s a great editor! Already prepared a draft of your text and need help with polishing it into something truly A grade-worthy? Just upload your writing to our AI homework helper, and the app will offer suggestions to reword and improve it. This can be a game-changer if you feel stuck with a writing assignment. • Read smarter, not harder Our AI homework helper can be a lifesaver if you need a quick overview of a book. Type the name of the book or its author, or upload the book if you've got a file, and no matter how long or complex, tap Generate Summary. Get the essentials in a breeze! Math solver, physics homework solver, essay writer, biology solver, chemistry solver—Solvo wears many hats! Yes, studies can be challenging, but with our AI homework helper, you're well-equipped to handle them! Get answers to all your problems—including tricky science answers—with prompt assistance for your tasks whenever and wherever you need it and enjoy studying with less anxiety. Be unstoppable in class with Premium! A subscription allows you to: • Remove usage limits • Get more detailed answers • Use text recognition (OCR) • Get instant responses Subscriptions are auto-billed based on the chosen plan. Privacy Policy - https://aiby.mobi/ai_study_ios/privacy Terms of Use - https://aiby.mobi/ai_study_ios/terms

Version 1.30

Get ready for the back-to-school season with Solvo! In this update: — Improved scanning and solving of visual tasks (including graphs, geometry, tables, and more) — Leave feedback after any solution, so we can continue improving task-solving — Copy and share any solution Don’t forget to send feedback to [email protected] and leave your review on the App Store! It helps us make the app even better.

Ratings and Reviews

176 Ratings

A wonderful app

Very helpful app I was needing something to help me with my daughter’s homework my daughter is horrible on taking notes in school. And if I have notes to see the task I’m good at figuring it out how to do the rest so I tried this help because I couldn’t find anything online to help me. So when this can’t up I was like why not. And I was glad I did it tells u how to solve it. And I could figure out the rest by their help. The only thing I would like if they make another app or add on this this one for younger kids. I know a couple of parents that also need help with there kid’s homework (how to help there child I don’t do it for my kids do there homework but I have to explain it to them sometimes and for that I need to refresh my mind as will) and this is a great app for that just hope they for something god younger students grades 2nd to 5th graders would help parents a lot.

Great App / One Major Issue

I love this app. Its saved me multiple times on upcoming tests, and the great thing about it is it thoroughly goes through the topic step-by-step making sure you understand how the AI got to the solution. All that to be said, I really wish there was a feature to edit the text that was scanned in the picture. I think its already an intended design because theres text displayed saying if you’d made typos heres the time to fix it, but it doesn't work. Tapping on the screen doesn't do anything. You can copy and paste the text but theres no way to edit it where the users keyboard opens. This is a 10/10 if I could edit the prompt.

Great academic support

Solvo has truly been a remarkable discovery for me as a busy working mom. My son has been facing difficulties with certain subjects in school, and finding the time and energy to assist him with homework has been a challenge for me. Since we found Solvo, everything has changed for the better. My son doesn’t give up on his assignments when they’re difficult. Solvo gives me peace of mind. I know my son receives the help he needs. The app has empowered my son to become more independent in tackling his academic challenges without unnecessary stress. I highly recommend Solvo to all working parents who want to actively support their children’s education. It’s an invention that has made a significant difference in our lives.

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Solving Problems with the Unknown in any position

Beginning as early as 1st grade, students should have experience solving addition and subtraction problems with the unknown in any position . Traditionally, we have tended to focus on result unknown problems, such as 3 + 2 = ⬜. But students also need to be able to solve problems such as 3 + ⬜ = 5. So exactly how do we go about building that understanding?

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

number bonds

Flexibility with numbers begins in Kindergarten when students learn all the combinations for the numbers through 10. These combinations are often referred to as number bonds . Here you see a common way to show number bond relationships. In this case, you see all the ways to make 5.

You may notice that number bonds look a lot like fact families, and they are similar. The difference is how we approach building an understanding of the relationship between the numbers. When we taught fact families, it was typically done as a rote skill. Students knew that they needed to have two addition equations and two subtraction equations using the same numbers, yet they didn’t have an understanding of what the equations represented. We knew that because we would often see unreasonable results, like 2 – 5 = 3.

Students in Kindergarten need to first work with number bonds in a totally concrete way. You can find lots of different games, many with free downloads, in this post .

We can use number bond cards to help students understand the part/whole relationship of the numbers that make up a number bond. We want to start out with result unknown cards and provide students the concrete support of counters.

Here we see that they start out by putting teddy bear counters on the two parts that are known—3 bears on one part and 2 on the other. Next, students move the bears representing the two parts to the unknown whole section, finding that the whole is 5.

After lots of practice with result unknown , students can move to working with the cards with the unknown as one of the parts. Keep in mind that this won’t happen at the same time for all children. Differentiation is critical. We begin by placing the 5 bears on the whole. Next, we move 2 of the bears to the part we know. Finally the remaining 3 bears are moved to the unknown part. Be sure to provide plenty of guided practice during small group instruction before asking students to work with part unknown cards independently.

3 reads protocol

Using word problems makes abstract concepts more concrete because they put the numbers in a familiar context. However, we need to make sure that we help students develop reading comprehension skills to allow them to understand what the numbers represent in the context of the story. Enter a powerful strategy called the 3 Read Protocol.

The Three Reads Protocol, not surprisingly, involves reading a word problem three times, with each read having a different purpose.

But here are some things that might surprise you.

The problem in Read 1 has NO numbers and NO question. When you take out the numbers, students have to focus on the words. This helps them learn to make mental pictures of what’s taking place, which helps them understand what math to do.

Read 2 provides the numbers, but still does not have a question. Now the focus shifts to the numbers and what they represent in the story.

Finally, in Read 3, students come up with questions that could complete the word problem.

While this sample problem is a result unknown problem (but could also be a comparison problem, right?), you would gradually introduce stories that have the unknown in other positions.

For more information on 3 Reads and how to incorporate it into your instructional routine, check out this post .

Part/whole models

Another tool that can be used to help students understand that the unknown can be in any position is a part/whole diagram. To illustrate, let’s revisit the word problem from the last section, but now let’s make at a part unknown problem.

Now let’s listen in on what it would sound like to incorporate part/whole thinking.

TEACHER: [displays the word problem and a blank part/whole diagram] Let’s read this problem together and decide how each number fits into our part/whole diagram. First we’ll read the whole problem and talk about what’s happening in this story. Then we’ll read each sentence and add the numbers to our diagram. [teacher and students read the story]

TEACHER: Who is this story about? [Juliet and her grandmother] What’s happening in the story? [Juliet is saving money for a video game. She already has some money. Her grandmother gives her money for her birthday.]

TEACHER: Okay, let’s go back to the first sentence:  Juliet has saved $15 for a video game.  Is the $15 she had already saved the whole, her total money, or is it part of her money. [part of her money] What part is it? [the part she had already saved] Great! Let’s add that to our part/whole diagram and label it money she had . Sound good?

TEACHER: Next sentence:  Her grandmother gave her some  [teacher shrugs her shoulders when she says some ] money for her birthday. Huh, do we know how much her grandmother gave her? [no] So that is our unknown in this problem! Is the money her grandmother gave her part of her money or all of the money, the total? [part] Can we label that part  money her grandmother gave her?  If we don’t have a number for the money her grandmother gave her, what should we put in that part of our diagram? [a question mark]

TEACHER: Next sentence:  Now she has $33. Is that all of her money, the whole, or one of the parts? [all of her money] Why don’t we label the whole All of her money  and add the $33 to our diagram.

TEACHER: It seem like we’re getting really close to solving this problem! Let’s read the question and make sure it matches where we placed the unknown in our diagram:  How much money did her grandmother give her? Does that match our diagram? [yes] Yes, because we have our question mark in the part labeled money her grandmother gave her .

TEACHER: Great work! Now work with your partner to solve the problem.

Let me add an important note—I used a missing part problem for this example. Keep in mind that your students would have been using the part/whole diagram and this process for talking through the problem on less complicated, result unknown problems extensively before moving on to unknown parts.

Another note—be careful of calling this a subtraction problem. Yes, most students will use subtraction to solve the problem, but they could also use a counting up strategy ( 18 and 2 more is 20. It’s 10 more to 30, and another 3 to 33. So her grandmother gave her $15 ). Allow for flexible strategies in solving ALL problems!

I hope that gives you some fresh ideas for tackling this tricky concept!

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Problem of the Week

The Problem of the Week is designed to provide students with an ongoing opportunity to solve mathematical problems. Each week, problems from various areas of mathematics will be posted here and e-mailed to educators for use with their students from Grades 3  to 12. The problems vary in difficulty and cover a wide range of mathematical concepts, making it an excellent tool for both learning and enrichment. 

Problem of the Week has wrapped up for the 2023-2024 school year. Below you can find all problems and solutions that were sent. Problem of the Week will resume in September 2024.

Below you will find the weekly problem across five grade levels.

• Grades 11/12

Gathering Treasure

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Sign up to receive Problem of the Week straight in your inbox. New problems are sent every Thursday, and their solutions are sent one week later. 

Problems and Solutions from 2024/2025 

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Problem and Solution Archive from 2023/2024, 2022/2023

Visit the archive page to access links to booklets containing all the problems and solutions from the last two years. The problems are organized into themes, grouping problems into various areas of the curriculum.  

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Problems are organized into five themes: Algebra (A), Computational Thinking (C), Data Management (D), Geometry and Measurement (G), Number Sense (N). A problem may have more than one theme.

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Game-Based Learning

How to transform lessons into adventures, by abbie misha     aug 28, 2024.

Image Credit: Prodigy Education

Imagine a classroom with students deeply engaged in their lessons, eagerly solving math problems and exploring English skills through fun digital games. Their excitement is evident as they face challenges and earn rewards, turning traditional learning into an adventure. Game-based learning not only motivates students through elements like challenges, rewards and narratives but also fosters critical skills such as problem-solving, strategic thinking and collaboration.

Recently, EdSurge spoke to Dr. Josh Prieur about how to maximize the impact of game-based learning in classrooms. Prieur began his career as a high school English teacher in Palm Beach County, Florida, before transitioning to a school-level administrator role for six and a half years, during which he earned his doctorate in educational leadership. He later moved into digital game-based learning and currently serves as the director of educator enablement at Prodigy Education , partnering with schools and districts to enhance the use of game-based learning tools.

EdSurge: How do you define game-based learning?

Prieur: Game-based learning encompasses any game used for an educational purpose, from traditional physical games to those on computers. Not only are games a great way to embed practice into the classroom, but they can also help students find a state of flow where they are fully immersed in something they enjoy. Psychologists have posited that you enter a state of flow when you're deeply engaged . Game-based learning is a phenomenal way to get students into a state of flow because they then become deeply engaged, not only in the game but also in the learning.

I often hear from teachers who use Prodigy about how impactful games can be in their classrooms. Students using Prodigy Math or Prodigy English enjoy the storyline and get excited about the learning content. We want them to have fun while they're learning, and that's what makes effective game-based learning truly special.

How can game-based learning be integrated into classrooms and beyond?

The key is intentionality . At its core, Prodigy offers adaptive, standards-aligned content, which is fantastic. If students pick it up and use it for fun and learning, that's great. Many parents discover Prodigy and share it with their kids, and the kids enjoy playing it. Teachers can align the in-game educational content with classroom lessons in under 30 seconds, making the practice more impactful. This can seamlessly blend into activities like station rotation, math workshops, exit tickets, technology time or even whole-group instruction. There are many natural ways to incorporate games into the classroom.

What data privacy considerations should be addressed when adopting new edtech?

Privacy and security have been at the top of educators' minds, and rightfully so. I look for tools that integrate with popular cloud-based productivity suites, education-focused single sign-on platforms and digital learning ecosystems. On the cybersecurity front, I encourage educators to look for products aligned with the NIST Cybersecurity Framework . Additionally, look for products from companies that are part of the Student Data Privacy Consortium ; they've standardized the data privacy agreement process, which is fantastic. I also recommend looking for tools certified by 1EdTech and that have signed the TrustEd Apps Pledge . Additionally, Prodigy has achieved a 93 percent pass rating by Common Sense Privacy thanks to our stringent approach to student data privacy.

How can game-based learning tools be customized to meet the diverse needs of different classrooms and individual students?

Games should have multiple supports built right into them to assist students with varying needs. For example, having a read-out-loud functionality is something that I see as critical. We know it's helpful for emerging readers and English language learners to have that auditory element to go along with the reading. Having hints is also really important because it provides students with that extra scaffolding when they need it. Hints allow us to give content to students that is appropriately leveled for them — it's just hard enough. It keeps them in their Zone of Proximal Development . I also consider readability in games. For example, look for tools that use Lexend fonts , which may help improve readability for those with dyslexia.

Digital Promise offers a certification called Learner Variability , which Prodigy has earned. We keep learner support top-of-mind, and we're really proud of the fact that we take the time to make sure that Prodigy is accessible. It's something that teachers should look for as they evaluate tools for their classrooms.

To learn more about Prodigy, please visit www.prodigygame.com .

This article was sponsored by Prodigy Education and produced by the Solutions Studio team.

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A framework for solving parabolic partial differential equations

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Computer graphics and geometry processing research provide the tools needed to simulate physical phenomena like fire and flames, aiding the creation of visual effects in video games and movies as well as the fabrication of complex geometric shapes using tools like 3D printing.

Under the hood, mathematical problems called partial differential equations (PDEs) model these natural processes. Among the many PDEs used in physics and computer graphics, a class called second-order parabolic PDEs explain how phenomena can become smooth over time. The most famous example in this class is the heat equation, which predicts how heat diffuses along a surface or in a volume over time.

Researchers in geometry processing have designed numerous algorithms to solve these problems on curved surfaces, but their methods often apply only to linear problems or to a single PDE. A more general approach by researchers from MIT’s Computer Science and Artificial Intelligence Laboratory (CSAIL) tackles a general class of these potentially nonlinear problems.  In a paper recently published in the Transactions on Graphics journal and presented at the SIGGRAPH conference, they describe an algorithm that solves different nonlinear parabolic PDEs on triangle meshes by splitting them into three simpler equations that can be solved with techniques graphics researchers already have in their software toolkit. This framework can help better analyze shapes and model complex dynamical processes.

“We provide a recipe: If you want to numerically solve a second-order parabolic PDE, you can follow a set of three steps,” says lead author Leticia Mattos Da Silva SM ’23, an MIT PhD student in electrical engineering and computer science (EECS) and CSAIL affiliate. “For each of the steps in this approach, you’re solving a simpler problem using simpler tools from geometry processing, but at the end, you get a solution to the more challenging second-order parabolic PDE.” To accomplish this, Da Silva and her coauthors used Strang splitting, a technique that allows geometry processing researchers to break the PDE down into problems they know how to solve efficiently.

First, their algorithm advances a solution forward in time by solving the heat equation (also called the “diffusion equation”), which models how heat from a source spreads over a shape. Picture using a blow torch to warm up a metal plate — this equation describes how heat from that spot would diffuse over it. 
This step can be completed easily with linear algebra.

Now, imagine that the parabolic PDE has additional nonlinear behaviors that are not described by the spread of heat. This is where the second step of the algorithm comes in: it accounts for the nonlinear piece by solving a Hamilton-Jacobi (HJ) equation, a first-order nonlinear PDE.  While generic HJ equations can be hard to solve, Mattos Da Silva and coauthors prove that their splitting method applied to many important PDEs yields an HJ equation that can be solved via convex optimization algorithms. Convex optimization is a standard tool for which researchers in geometry processing already have efficient and reliable software. In the final step, the algorithm advances a solution forward in time using the heat equation again to advance the more complex second-order parabolic PDE forward in time.


Among other applications, the framework could help simulate fire and flames more efficiently. “There’s a huge pipeline that creates a video with flames being simulated, but at the heart of it is a PDE solver,” says Mattos Da Silva. For these pipelines, an essential step is solving the G-equation, a nonlinear parabolic PDE that models the front propagation of the flame and can be solved using the researchers’ framework.

The team’s algorithm can also solve the diffusion equation in the logarithmic domain, where it becomes nonlinear. Senior author Justin Solomon, associate professor of EECS and leader of the CSAIL Geometric Data Processing Group, previously developed a state-of-the-art technique for optimal transport that requires taking the logarithm of the result of heat diffusion. Mattos Da Silva’s framework provided more reliable computations by doing diffusion directly in the logarithmic domain. This enabled a more stable way to, for example, find a geometric notion of average among distributions on surface meshes like a model of a koala. Even though their framework focuses on general, nonlinear problems, it can also be used to solve linear PDE. For instance, the method solves the Fokker-Planck equation, where heat diffuses in a linear way, but there are additional terms that drift in the same direction heat is spreading. In a straightforward application, the approach modeled how swirls would evolve over the surface of a triangulated sphere. The result resembles purple-and-brown latte art.

The researchers note that this project is a starting point for tackling the nonlinearity in other PDEs that appear in graphics and geometry processing head-on. For example, they focused on static surfaces but would like to apply their work to moving ones, too. Moreover, their framework solves problems involving a single parabolic PDE, but the team would also like to tackle problems involving coupled parabolic PDE. These types of problems arise in biology and chemistry, where the equation describing the evolution of each agent in a mixture, for example, is linked to the others’ equations.

Mattos Da Silva and Solomon wrote the paper with Oded Stein, assistant professor at the University of Southern California’s Viterbi School of Engineering. Their work was supported, in part, by an MIT Schwarzman College of Computing Fellowship funded by Google, a MathWorks Fellowship, the Swiss National Science Foundation, the U.S. Army Research Office, the U.S. Air Force Office of Scientific Research, the U.S. National Science Foundation, MIT-IBM Watson AI Lab, the Toyota-CSAIL Joint Research Center, Adobe Systems, and Google Research.

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• Leticia Mattos Da Silva
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• Computer Science and Artificial Intelligence Laboratory (CSAIL)
• MIT-IBM Watson AI Lab
• Department of Electrical Engineering and Computer Science

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