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How to Solve Math Problems

Last Updated: April 15, 2024 Fact Checked

This article was co-authored by Daron Cam . Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 591,221 times.

Although math problems may be solved in different ways, there is a general method of visualizing, approaching and solving math problems that may help you to solve even the most difficult problem. Using these strategies can also help you to improve your math skills overall. Keep reading to learn about some of these math problem solving strategies.

Understanding the Problem

Step 1 Identify the type of problem.

  • Draw a Venn diagram. A Venn diagram shows the relationships among the numbers in your problem. Venn diagrams can be especially helpful with word problems.
  • Draw a graph or chart.
  • Arrange the components of the problem on a line.
  • Draw simple shapes to represent more complex features of the problem.

Step 5 Look for patterns.

Developing a Plan

Step 1 Figure out what formulas you will need to solve the problem.

Solving the Problem

Step 1 Follow your plan.

Joseph Meyer

When doing practice problems, promptly check to see if your answers are correct. Use worksheets that provide answer keys for instant feedback. Discuss answers with a classmate or find explanations online. Immediate feedback will help you correct your mistakes, avoid bad habits, and advance your learning more quickly.

Expert Q&A

Daron Cam

  • Seek help from your teacher or a math tutor if you get stuck or if you have tried multiple strategies without success. Your teacher or a math tutor may be able to easily identify what is wrong and help you to understand how to correct it. Thanks Helpful 0 Not Helpful 0
  • Keep practicing sums and diagrams. Go through the concept your class notes regularly. Write down your understanding of the methods and utilize it. Thanks Helpful 0 Not Helpful 0

how to solve math problems

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Do Math Proofs

  • ↑ Daron Cam. Math Tutor. Expert Interview. 29 May 2020.
  • ↑ http://www.interventioncentral.org/academic-interventions/math/math-problem-solving-combining-cognitive-metacognitive-strategies
  • ↑ http://tutorial.math.lamar.edu/Extras/StudyMath/ProblemSolving.aspx
  • ↑ https://math.berkeley.edu/~gmelvin/polya.pdf

About This Article

Daron Cam

To solve a math problem, try rewriting the problem in your own words so it's easier to solve. You can also make a drawing of the problem to help you figure out what it's asking you to do. If you're still completely stuck, try solving a different problem that's similar but easier and then use the same steps to solve the harder problem. Even if you can't figure out how to solve it, try to make an educated guess instead of leaving the question blank. To learn how to come up with a solid plan to use to help you solve a math problem, scroll down! Did this summary help you? Yes No

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Solving Equations

What is an equation.

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

That equations says:

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

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

What is a Solution?

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

Example: x − 2 = 4

When we put 6 in place of x we get:

which is true

So x = 6 is a solution.

How about other values for x ?

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

In this case x = 6 is the only solution.

You might like to practice solving some animated equations .

More Than One Solution

There can be more than one solution.

Example: (x−3)(x−2) = 0

When x is 3 we get:

(3−3)(3−2) = 0 × 1 = 0

And when x is 2 we get:

(2−3)(2−2) = (−1) × 0 = 0

which is also true

So the solutions are:

x = 3 , or x = 2

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

The above solution set is: {2, 3}

Solutions Everywhere!

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

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

Let's try θ = 30°:

sin(−30°) = −0.5 and

−sin(30°) = −0.5

So it is true for θ = 30°

Let's try θ = 90°:

sin(−90°) = −1 and

−sin(90°) = −1

So it is also true for θ = 90°

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

How to Solve an Equation

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

A Useful Goal

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

x = something

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

Example: Solve 3x−6 = 9

Now we have x = something ,

and a short calculation reveals that x = 5

Like a Puzzle

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

Here are some things we can do:

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

Example: Solve √(x/2) = 3

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

Special Equations

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

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

Check Your Solutions

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

How To Check

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

Example: solve for x:

2x x − 3 + 3 = 6 x − 3     (x≠3)

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

Let's multiply through by (x − 3) :

2x + 3(x−3) = 6

Bring the 6 to the left:

2x + 3(x−3) − 6 = 0

Expand and solve:

2x + 3x − 9 − 6 = 0

5x − 15 = 0

5(x − 3) = 0

Which can be solved by having x=3

Let us check x=3 using the original question:

2 × 3 3 − 3 + 3  =   6 3 − 3

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

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

x = 3 does not actually work, and so:

There is No Solution!

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

This gives us a moral lesson:

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

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

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The algebra section of QuickMath allows you to manipulate mathematical expressions in all sorts of useful ways. At the moment, QuickMath can expand, factor or simplify virtually any expression, cancel common factors within fractions, split fractions up into smaller ('partial') fractions and join two or more fractions together into a single fraction. More specialized commands are on the way.

What is algebra?

Algebra is the branch of elementary mathematics which uses symbols to stand for unknown quantities. In a more basic sense, it consists of solving equations or manipulating expressions which contain symbols (usually letters, like x, y or z) as well as numbers and functions. Although solving equations is really a part of algebra, it is such a big area that it has its own section in QuickMath.

This part of QuickMath deals only with algebraic expressions. These are mathematical statements which contain letters, numbers and functions, but no equals signs. Here are a few examples of simple algebraic expressions :

The expand command is used mainly to rewrite polynomials with all brackets and whole number powers multiplied out and all like terms collected together. In the advanced section, you also have the option of expanding trigonometric functions, expanding modulo any integer and leaving certain parts of the expression untouched whilst expanding the rest.

Go to the Expand page

The factor command will try to rewrite an expression as a product of smaller expressions. It takes care of such things as taking out common factors, factoring by pairs, quadratic trinomials, differences of two squares, sums and differences of two cubes, and a whole lot more. The advanced section includes options for factoring trigonometric functions, factoring modulo any integer, factoring over the field of Gaussian integers (just the thing for those tricky sums of squares), and even extending the field over which factoring occurs with your own custom extensions.

Go to the Factor page

Simplifying is perhaps the most difficult of all the commands to describe. The way simplification is performed in QuickMath involves looking at many different combinations of transformations of an expression and choosing the one which has the smallest number of parts. Amongst other things, the Simplify command will take care of canceling common factors from the top and bottom of a fraction and collecting like terms. The advanced options allow you to simplify trigonometric functions or to instruct QuickMath to try harder to find a simplified expression.

Go to the Simplify page

The cancel command allows you to cancel out common factors in the denominator and numerator of any fraction appearing in an expression. This command works by canceling the greatest common divisor of the denominator and numerator.

Go to the Cancel page

Partial Fractions

The partial fractions command allows you to split a rational function into a sum or difference of fractions. A rational function is simply a quotient of two polynomials. Any rational function can be written as a sum of fractions, where the denominators of the fractions are powers of the factors of the denominator of the original expression. This command is especially useful if you need to integrate a rational function. By splitting it into partial fractions first, the integration can often be made much simpler.

Go to the Partial Fractions page

Join Fractions

The join fractions command essentially does the reverse of the partial fractions command. It will rewrite a number of fractions which are added or subtracted as a single fraction. The denominator of this single fraction will usually be the lowest common multiple of the denominators of all the fractions being added or subtracted. Any common factors in the numerator and denominator of the answer will automatically be cancelled out.

Go to the Join Fractions page

Introduction to Algebraic Functions

The notion of correspondence is encountered frequently in everyday life. For example, to each book in a library there corresponds the number of pages in the book. As another example, to each human being there corresponds a birth date. To cite a third example, if the temperature of the air is recorded throughout a day, then at each instant of time there is a corresponding temperature.

The examples of correspondences we have given involve two sets X and Y. In our first example, X denotes the set of books in a library and Y the set of positive integers. For each book x in X there corresponds a positive integer y, namely the number of pages in the book. In the second example, if we let X denote the set of all human beings and Y the set of all possible dates, then to each person x in X there corresponds a birth date y.

We sometimes represent correspondences by diagrams of the type shown in Figure 1.17, where the sets X and Y are represented by points within regions in a plane. The curved arrow indicates that the element y of Y corresponds to the element x of X. We have pictured X and Y as different sets. However, X and Y may have elements in common. As a matter of fact, we often have X = Y.

how to solve math problems

A function f from a set X to a set Y is a correspondence that assigns to each element x of X a unique element y of Y. The element y is called the image of x under f and is denoted by f(x). The set X is called the domain of the function. The range of the function consists of all images of elements of X.

Earlier, we introduced the notation f(x) for the element of Y which corresponds to x. This is usually read "f of x." We also call f(x) the value of f at x. In terms of the pictorial representation given earlier, we may now sketch a diagram as in Figure 1.18. The curved arrows indicate that the elements f(x), f(w), f(z), and f(a) of Y correspond to the elements x, y, z and a of X. Let us repeat the important fact that to each x in X there is assigned precisely one image f(x) in Y; however, different elements of X such as w and z in Figure 1.18 may have the same image in Y.

how to solve math problems

Solution As in Example 1, finding images under f is simply a matter of substituting the appropriate number for x in the expression for f(x). Thus:

how to solve math problems

Many formulas which occur in mathematics and the sciences determine functions. As an illustration, the formula A = pi*r 2 for the area A of a circle of radius r assigns to each positive real number r a unique value of A. This determines a function f, where f(r) = pi*r 2 , and we may write A= f(r). The letter r, which represents an arbitrary number from the domain off, is often called an independent variable. The letter A, which represents a number from the range off, is called a dependent variable, since its value depends on the number assigned tor. When two variables r and A are related in this manner, it is customary to use the phrase A is a function of r. To cite another example, if an automobile travels at a uniform rate of 50 miles per hour, then the distance d (miles) traveled in time t (hours) is given by d = 50t and hence the distance d is a function of time t.

We have seen that different elements in the domain of a function may have the same image. If images are always different, then, as in the next definition, the function is called one-to-one.  

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Problem Solver Subjects

Our math problem solver that lets you input a wide variety of math math problems and it will provide a step by step answer. This math solver excels at math word problems as well as a wide range of math subjects.

  • Math Word Problems
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  • Linear Algebra

Here are example math problems within each subject that can be input into the calculator and solved. This list is constanstly growing as functionality is added to the calculator.

Basic Math Solutions

Below are examples of basic math problems that can be solved.

  • Long Arithmetic
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  • Factors, Fractions, and Exponents
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Math Word Problem Solutions

Math word problems require interpreting what is being asked and simplifying that into a basic math equation. Once you have the equation you can then enter that into the problem solver as a basic math or algebra question to be correctly solved. Below are math word problem examples and their simplified forms.

Word Problem: Rachel has 17 apples. She gives some to Sarah. Sarah now has 8 apples. How many apples did Rachel give her?

Simplified Equation: 17 - x = 8

Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b)

Simplified Equation: {r = d + 12, d = b + 6, r = 2 �� b}

Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left?

Simplified: 40 - 10 - 5

Pre-Algebra Solutions

Below are examples of Pre-Algebra math problems that can be solved.

  • Variables, Expressions, and Integers
  • Simplifying and Evaluating Expressions
  • Solving Equations
  • Multi-Step Equations and Inequalities
  • Ratios, Proportions, and Percents
  • Linear Equations and Inequalities

Algebra Solutions

Below are examples of Algebra math problems that can be solved.

  • Algebra Concepts and Expressions
  • Points, Lines, and Line Segments
  • Simplifying Polynomials
  • Factoring Polynomials
  • Linear Equations
  • Absolute Value Expressions and Equations
  • Radical Expressions and Equations
  • Systems of Equations
  • Quadratic Equations
  • Inequalities
  • Complex Numbers and Vector Analysis
  • Logarithmic Expressions and Equations
  • Exponential Expressions and Equations
  • Conic Sections
  • Vector Spaces
  • 3d Coordinate System
  • Eigenvalues and Eigenvectors
  • Linear Transformations
  • Number Sets
  • Analytic Geometry

Trigonometry Solutions

Below are examples of Trigonometry math problems that can be solved.

  • Algebra Concepts and Expressions Review
  • Right Triangle Trigonometry
  • Radian Measure and Circular Functions
  • Graphing Trigonometric Functions
  • Simplifying Trigonometric Expressions
  • Verifying Trigonometric Identities
  • Solving Trigonometric Equations
  • Complex Numbers
  • Analytic Geometry in Polar Coordinates
  • Exponential and Logarithmic Functions
  • Vector Arithmetic

Precalculus Solutions

Below are examples of Precalculus math problems that can be solved.

  • Operations on Functions
  • Rational Expressions and Equations
  • Polynomial and Rational Functions
  • Analytic Trigonometry
  • Sequences and Series
  • Analytic Geometry in Rectangular Coordinates
  • Limits and an Introduction to Calculus

Calculus Solutions

Below are examples of Calculus math problems that can be solved.

  • Evaluating Limits
  • Derivatives
  • Applications of Differentiation
  • Applications of Integration
  • Techniques of Integration
  • Parametric Equations and Polar Coordinates
  • Differential Equations

Statistics Solutions

Below are examples of Statistics problems that can be solved.

  • Algebra Review
  • Average Descriptive Statistics
  • Dispersion Statistics
  • Probability
  • Probability Distributions
  • Frequency Distribution
  • Normal Distributions
  • t-Distributions
  • Hypothesis Testing
  • Estimation and Sample Size
  • Correlation and Regression

Finite Math Solutions

Below are examples of Finite Math problems that can be solved.

  • Polynomials and Expressions
  • Equations and Inequalities
  • Linear Functions and Points
  • Systems of Linear Equations
  • Mathematics of Finance
  • Statistical Distributions

Linear Algebra Solutions

Below are examples of Linear Algebra math problems that can be solved.

  • Introduction to Matrices
  • Linear Independence and Combinations

Chemistry Solutions

Below are examples of Chemistry problems that can be solved.

  • Unit Conversion
  • Atomic Structure
  • Molecules and Compounds
  • Chemical Equations and Reactions
  • Behavior of Gases
  • Solutions and Concentrations

Physics Solutions

Below are examples of Physics math problems that can be solved.

  • Static Equilibrium
  • Dynamic Equilibrium
  • Kinematics Equations
  • Electricity
  • Thermodymanics

Geometry Graphing Solutions

Below are examples of Geometry and graphing math problems that can be solved.

  • Step By Step Graphing
  • Linear Equations and Functions
  • Polar Equations

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Android's Circle to Search can now help students solve math and physics homework

It can show step-by-step instructions on how to solve a range of math and physics problems..

Google has introduced another capability for its Circle to Search feature at the company's annual I/O developer conference, and it's something that could help students better understand potentially difficult class topics . The feature will now be able to show them step-by-step instructions for a "range of physics and math word problems." They just have to activate the feature by long-pressing the home button or navigation bar and then circling the problem that's got them stumped, though some math problems will require users to be signed up for Google's experimental Search Labs feature.

The company says Circle to Search's new capability was made possible by its new family of AI models called LearnLM that was specifically created and fine-tuned for learning. It's also planning to make adjustments to this particular capability and to roll out an upgraded version later this year that could solve even more complex problems "involving symbolic formulas, diagrams, graphs and more." Google launched Circle to Search earlier this year at a Samsung Unpacked event, because the feature was initially available on Galaxy 24, as well as on Pixel 8 devices. It's now also out for the Galaxy S23, Galaxy S22, Z Fold, Z Flip, Pixel 6 and Pixel 7 devices, and it'll likely make its way to more hardware in the future.

In addition to the new Circle to Search capability, Google has also revealed that devices that can support the Gemini for Android chatbot assistant will now be able to bring it up as an overlay on top of the application that's currently open. Users can then drag and drop images straight from the overlay into apps like Gmail, for instance, or use the overlay to look up information without having to swipe away from whatever they're doing. They can tap "Ask this video" to find specific information within a YouTube video that's open, and if they have access to Gemini Advanced, they can use the "Ask this PDF" option to find information from within lengthy documents.

Google is also rolling out multimodal capabilities to Nano, the smallest model in the Gemini family that can process information on-device. The updated Gemini Nano, which will be able to process sights, sounds and spoken language, is coming to Google's TalkBack screen reader later this year. Gemini Nano will enable TalkBack to describe images onscreen more quickly and even without an internet connection. Finally, Google is currently testing a Gemini Nano feature that can alert users while a call is ongoing if it detects common conversation patterns associated with scams. Users will be alerted, for instance, if they're talking to someone asking them for their PINs or passwords or to someone asking them to buy gift cards.

Catch up on all the news from Google I/O 2024 right here !

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Google’s Circle to Search will help you with your math homework

Circle to search on android can now help students learn to solve mathematical word problems thanks to google’s new learnlm model..

By Umar Shakir , a news writer fond of the electric vehicle lifestyle and things that plug in via USB-C. He spent over 15 years in IT support before joining The Verge.

Share this story

A math word problem that says a car takes 8 seconds to accelerate from 0 meters per second to 24 meters per second. Calculate the acceleration of the car. There’s a drawn circle around the word problem. A metallic blue android bot is in the corner with a Gemini star above it.

Google is enhancing Android’s Circle to Search — the feature that lets you literally circle something on your Android phone’s screen to search it on Google — with a new ability to generate instructions on how to solve school math and physics problems .

Using an Android phone or tablet, students can now use Circle to Search to get AI assistance on mathematical word problems from their homework. The feature will help unpack the problem and list what the student needs to do to get the correct answer. According to Google, it won’t actually do the homework for you — only help you approach the problem.

Over the past year, the use of AI tools like ChatGPT has become a hot topic in the field of education, with plenty of concern over how students can and will use it to get work done quickly. Google, however, is explicitly positioning this as a feature to support education, potentially walking around some of the concerns about AI doing all of the work for students.

Later this year, Circle to Search will also gain the ability to solve complex math equations that involve formulas, diagrams, graphs, and more. Google is using LearnLM, its new AI model that’s fine-tuned for learning, to make the new Circle to Search abilities work.

Circle to Search first launched on  Samsung’s Galaxy S24 series in January and then on the Pixel 8 and 8 Pro later the same month. It’s one of the star new features of Android, and although iOS users can’t yet circle their math homework for help, anything is possible .

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Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson and Calcea Johnson

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Michelle Blouin Williams

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Bill Whitaker with Calcea Johnson and Ne'Kiya Jackson

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

St. Mary's Academy president and interim principal Pamela Rogers

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Bill Whitaker and Gloria Ladson-Billings

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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how to solve math problems

Android's Circle to Search can now help students solve math and physics homework

G oogle has introduced another capability for its Circle to Search feature at the company's annual I/O developer conference, and it's something that could help students better understand potentially difficult class topics . The feature will now be able to show them step-by-step instructions for a "range of physics and math word problems." They just have to activate the feature by long-pressing the home button or navigation bar and then circling the problem that's got them stumped, though some math problems will require users to be signed up for Google's experimental Search Labs feature.

The company says Circle to Search's new capability was made possible by its new family of AI models called LearnLM that was specifically created and fine-tuned for learning. It's also planning to make adjustments to this particular capability and to roll out an upgraded version later this year that could solve even more complex problems "involving symbolic formulas, diagrams, graphs and more." Google launched Circle to Search earlier this year at a Samsung Unpacked event, because the feature was initially available on Galaxy 24, as well as on Pixel 8 devices. It's now also out for the Galaxy S23, Galaxy S22, Z Fold, Z Flip, Pixel 6 and Pixel 7 devices, and it'll likely make its way to more hardware in the future. 

In addition to the new Circle to Search capability, Google has also revealed that devices that can support the Gemini for Android chatbot assistant will now be able to bring it up as an overlay on top of the application that's currently open. Users can then drag and drop images straight from the overlay into apps like Gmail, for instance, or use the overlay to look up information without having to swipe away from whatever they're doing. They can tap "Ask this video" to find specific information within a YouTube video that's open, and if they have access to Gemini Advanced, they can use the "Ask this PDF" option to find information from within lengthy documents. 

Google is also rolling out multimodal capabilities to Nano, the smallest model in the Gemini family that can process information on-device. The updated Gemini Nano, which will be able to process sights, sounds and spoken language, is coming to Google's TalkBack screen reader later this year. Gemini Nano will enable TalkBack to describe images onscreen more quickly and even without an internet connection. Finally, Google is currently testing a Gemini Nano feature that can alert users while a call is ongoing if it detects common conversation patterns associated with scams. Users will be alerted, for instance, if they're talking to someone asking them for their PINs or passwords or to someone asking them to buy gift cards. 

Catch up on all the news from Google I/O 2024 right here !

Android's Circle to Search can now help students solve math and physics homework

ChatGPT’s Next Magic Trick Is Singing and Solving Math Problems With Your Phone Camera

GPT-4o responds as fast as humans.

Open AI's GPT-4o being demoed.

ChatGPT 4 may still be relatively new, but OpenAI is already iterating with an upgrade that can respond as quickly as humans do in normal conversation. The company showed off GPT-4o in a live demo , showing off its ability to use your phone’s camera to solve math equations and deliver a much more conversational voice assistant experience.

While we only have the event demo to go off of, GPT-4o looks impressive. It doesn’t even have to wait for you to finish your request and can roll with interruptions mid-prompt, bringing one step closer to living out Her in real life.

Even Faster Response Times

According to OpenAI, the GPT-4o model can respond as fast as 232 milliseconds to audio inputs. More realistically, it averages around 320 milliseconds to respond, which OpenAI said is similar to how fast humans respond in conversation.

On top of the speed, GPT-4o can handle interruptions and any adjustment requests. As seen in the bedtime story demo, GPT-4o immediately stopped talking when interrupted and quickly handled requests like adding more dramatic inflections, narrating in a robot voice, and even singing the entire prompt out loud. If that demo doesn’t convince you, two GPT-4o models improvising a song together should.

GPT-4o isn’t just more responsive to voice, it can also see better. The new vision features allow it to see through your device’s camera and understand things like handwritten math equations or messages . It’s eerie how genuinely touched GPT-4o sounds when it sees and understands a message that says “I Heart ChatGPT.” Even more impressive, GPT-4o can handle coding tasks and live translations between two people. This should feel way more natural than Google Translate when you’re trying to have a conversation in a foreign country.

Available for Free

OpenAI said the text and image capabilities for GPT-4o roll out today, but the voice feature will be coming to alpha within ChatGPT Plus in the coming weeks. Once it’s fully ready, the upgraded ChatGPT model will be available to all users, subscribers or otherwise. However, if you pay $20 per month for ChatGPT Plus , you’ll get five times the message limits of GPT-4o compared to the free version.

Anytime a large language model gets such an impressive update, we have to consider the potential for misuse . Considering how smoothly the live demo for solving the equation went, it looks like an even better way to help students get out of their math homework. However, OpenAI said that GPT-4o was built with new safety systems to offer guardrails on voice outputs. We’ll have to wait and see if these guardrails are enough.

how to solve math problems

Experience Google AI in even more ways on Android

May 14, 2024

[[read-time]] min read

By building AI right into the Android operating system, we're reimagining how you can interact with your phone.

NZ9_3301-Edit-2 (1)

  • Bullet points
  • Circle to Search gets smarter, helping students solve physics and math problems directly from their phones and tablets.
  • Gemini on Android improves context understanding, allowing users to drag and drop generated images and ask questions about videos and PDFs.
  • Gemini Nano with Multimodality coming to Pixel, bringing multimodal capabilities for richer image descriptions and scam alerts during phone calls.
  • Android 15 and ecosystem updates coming tomorrow.
  • Basic explainer

Google is making Android phones smarter with AI.

Circle to Search can now help students with homework.

Gemini, a new AI assistant, can understand what's on your screen and help you do things.

Android phones will soon be able to alert you to suspected scams during phone calls.

Explore other styles:

An Android mascot stands in front of screens displaying Circle to Search for homework help, scam detection alert, TalkBack, and Gemini’s overlay with a button that says "Ask this PDF."

We’re at a once-in-a-generation moment where the latest advancements in AI are reinventing what phones can do. With Google AI at the core of Android’s operating system, the billions of people who use Android can now interact with their devices in entirely new ways.

Today, we’re sharing updates that let you experience Google AI on Android.

Circle to Search can now help students with homework

With Circle to Search built directly into the user experience, you can search anything you see on your phone using a simple gesture — without having to stop what you’re doing or switch to a different app. Since launching at Samsung Unpacked , we’ve added new capabilities to Circle to Search, like full-screen translation , and we’ve expanded availability to more Pixel and Samsung devices.

Starting today, Circle to Search can now help students with homework, giving them a deeper understanding, not just an answer — directly from their phones and tablets. When students circle a prompt they’re stuck on, they’ll get step-by-step instructions to solve a range of physics and math 1 word problems without leaving their digital info sheet or syllabus. Later this year, Circle to Search will be able to help solve even more complex problems involving symbolic formulas, diagrams, graphs and more. This is all possible due to our LearnLM effort to enhance our models and products for learning.

Circle to Search is already available on more than 100 million devices today. With plans to bring the experience to more devices, we’re on track to double that by the end of the year.

Gemini will get even better at understanding context to assist you in getting things done

Gemini on Android is a new kind of assistant that uses generative AI to help you be more creative and productive. This experience, which is integrated into Android, is getting even better at understanding the context of what’s on your screen and what app you’re using.

Soon, you’ll be able to bring up Gemini's overlay on top of the app you're in to easily use Gemini in more ways. For example, you can drag and drop generated images into Gmail, Google Messages and other places, or tap “Ask this video” to find specific information in a YouTube video. If you have Gemini Advanced, you’ll also have the option to “Ask this PDF” to quickly get answers without having to scroll through multiple pages. This update will roll out to hundreds of millions of devices over the next few months.

And we’ll continue to improve Gemini to give you more dynamic suggestions related to what’s on your screen.

Phone frame showing an overlay with text that answers a question about a pickleball video. Below the text shows the video’s title card that says Learn Pickleball in 3 Minutes. Under the overlay is part of the video playing on YouTube of a pickleball court.

Full multimodal capabilities coming to Gemini Nano

Android is the first mobile operating system that includes a built-in, on-device foundation model. With Gemini Nano, we’re able to bring experiences to you quickly and keep your information completely private to you. Starting with Pixel later this year, we’ll be introducing our latest model, Gemini Nano with Multimodality. This means your phone will not just be able to process text input but also understand more information in context like sights, sounds and spoken language.

Clearer descriptions with TalkBack

Later this year, Gemini Nano’s multimodal capabilities are coming to TalkBack, helping people who experience blindness or low vision get richer and clearer descriptions of what’s happening in an image. On average, TalkBack users come across 90 unlabeled images per day. This update will help fill in missing information — whether it’s more details about what’s in a photo that family or friends sent or the style and cut of clothes when shopping online. Since Gemini Nano is on-device, these descriptions happen quickly and even work when there's no network connection.

Receive alerts for suspected scams during phone calls

According to a recent report , in a 12-month period, people lost more than $1 trillion to fraud. We’re testing a new feature that uses Gemini Nano to provide real-time alerts during a call if it detects conversation patterns commonly associated with scams. For example, you would receive an alert if a “bank representative” asks you to urgently transfer funds, make a payment with a gift card or requests personal information like card PINs or passwords, which are uncommon bank requests . This protection all happens on-device, so your conversation stays private to you. We’ll share more about this opt-in feature later this year.

Red box showing the scam detection alert with text that reads “Likely Scam, Banks will never ask you to move your money to keep it safe.” Below is the option to ‘Dismiss & Continue” or “End Call”.

More to come on Android

We’re just getting started with how on-device AI can change what your phone can do, and we’ll continue building Google AI into every part of the smartphone experience with Pixel, Samsung and more. If you’re a developer, check out the Android Developers blog to learn how you can build with our latest AI models and tools, like Gemini Nano and Gemini in Android Studio.

Stay tuned tomorrow for more of our latest Android 15 and ecosystem updates.

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Available for some math word problems when opted into Search Labs.

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  17. How to Solve Math Problems Faster: 15 Techniques to Show Students

    Here are 15 techniques to show students, helping them solve math problems faster: Addition and Subtraction. 1. Two-Step Addition. Many students struggle when learning to add integers of three digits or higher together, but changing the process's steps can make it easier.

  18. Equation Solver

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  19. Algebra

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  21. Algebra Calculator

    Polynomial. In mathematics, a polynomial is a mathematical expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x² − 4x + 7. An example with three indeterminates ...

  22. Math Problem Solver

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  23. Android's Circle to Search can now help students solve math ...

    Android's Circle to Search can now help students solve math and physics homework. It can show step-by-step instructions on how to solve a range of math and physics problems. Google has introduced ...

  24. Google's Circle to Search will help you with your math homework

    Circle to Search on Android can now help students learn to solve mathematical word problems thanks to Google's new LearnLM model. By Umar Shakir, a news writer fond of the electric vehicle ...

  25. Teens come up with trigonometry proof for Pythagorean Theorem, a

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  26. Solve

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  27. Google I/O 2024

    Google has recently upgraded its Android feature, Circle to Search, with fresh capabilities aimed at assisting school students with math and physics problems. The announcement comes at the latest ...

  28. Android's Circle to Search can now help students solve math and ...

    The feature will now be able to show them step-by-step instructions for a "range of physics and math word problems." They just have to activate the feature by long-pressing the home button or ...

  29. OpenAI's GPT-4o Can Sing and Solve Math Problems With Your ...

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  30. I/O 2024: New ways to experience Google AI on Android

    When students circle a prompt they're stuck on, they'll get step-by-step instructions to solve a range of physics and math 1 word problems without leaving their digital info sheet or syllabus. Later this year, Circle to Search will be able to help solve even more complex problems involving symbolic formulas, diagrams, graphs and more.