how to solve math problems in parentheses

Choose Your Test

Sat / act prep online guides and tips, the pemdas rule: understanding order of operations.

General Education

Everyone who's taken a math class in the US has heard the acronym "PEMDAS" before. But what does it mean exactly? Here, we will explain in detail the PEMDAS meaning and how it's used before giving you some sample PEMDAS problems so you can practice what you've learned.

PEMDAS Meaning: What Does It Stand For?

PEMDAS is an acronym meant to help you remember the order of operations used to solve math problems. It's typically pronounced "pem-dass," "pem-dozz," or "pem-doss."

Here's what each letter in PEMDAS stands for:

• P arentheses
• M ultiplication and D ivision
• A ddition and S ubtraction

The order of letters shows you the order you must solve different parts of a math problem , with expressions in parentheses coming first and addition and subtraction coming last.

Many students use this mnemonic device to help them remember each letter: P lease E xcuse M y D ear A unt S ally .

In the United Kingdom and other countries, students typically learn PEMDAS as BODMAS . The BODMAS meaning is the same as the PEMDAS meaning — it just uses a couple different words. In this acronym, the B stands for "brackets" (what we in the US call parentheses) and the O stands for "orders" (or exponents). Now, how exactly do you use the PEMDAS rule? Let's take a look.

How Do You Use PEMDAS?

PEMDAS is an acronym used to remind people of the order of operations.

This means that you don't just solve math problems from left to right; rather, you solve them in a predetermined order that's given to you via the acronym PEMDAS . In other words, you'll start by simplifying any expressions in parentheses before simplifying any exponents and moving on to multiplication, etc.

But there's more to it than this. Here's exactly what PEMDAS means for solving math problems:

• Parentheses: Anything in parentheses must be simplified first
• Exponents: Anything with an exponent (or square root) must be simplified after everything in parentheses has been simplified
• Multiplication and Division: Once parentheses and exponents have been dealt with, solve any multiplication and division from left to right
• Addition and Subtraction: Once parentheses, exponents, multiplication, and division have been dealt with, solve any addition and subtraction from left to right

If any of these elements are missing (e.g., you have a math problem without exponents), you can simply skip that step and move on to the next one.

Now, let's look at a sample problem to help you understand the PEMDAS rule better:

4 (5 − 3)² − 10 ÷ 5 + 8

You might be tempted to solve this math problem left to right, but that would result in the wrong answer! So, instead, let's use PEMDAS to help us approach it the correct way.

We know that parentheses must be dealt with first. This problem has one set of parentheses: (5 − 3). Simplifying this gives us 2 , so now our equation looks like this:

4 (2)² − 10 ÷ 5 + 8

The next part of PEMDAS is exponents (and square roots). There is one exponent in this problem that squares the number 2 (i.e., what we found by simplifying the expression in the parentheses).

This gives us 2 × 2 = 4. So now our equation looks like this:

4 (4) − 10 ÷ 5 + 8    OR    4 × 4 − 10 ÷ 5 + 8

Next up is multiplication and division from left to right . Our problem contains both multiplication and division, which we'll solve from left to right (so first 4 × 4 and then 10 ÷ 5). This simplifies our equation as follows:

Finally, all we need to do now is solve the remaining addition and subtraction from left to right :

The final answer is 22. Don't believe me? Insert the whole equation into your calculator (written exactly as it is above) and you'll get the same result!

Sample Math Problems Using PEMDAS + Answers

See whether you can solve the following four problems correctly using the PEMDAS rule. We'll go over the answers after.

Sample PEMDAS Problems

11 − 8 + 5 × 6

8 ÷ 2 (2 + 2)

7 × 4 − 10 (5 − 3) ÷ 2²

√25 (4 + 2)² − 18 ÷ 3 (3 − 1) + 2³

Answer Explanations

Here, we go over each problem above and how you can use PEMDAS to get the correct answer.

#1 Answer Explanation

This math problem is a fairly straightforward example of PEMDAS that uses addition, subtraction, and multiplication only , so no having to worry about parentheses or exponents here.

We know that multiplication comes before addition and subtraction , so you'll need to start by multiplying 5 by 6 to get 30:

11 − 8 + 30

Now, we can simply work left to right on the addition and subtraction:

11 − 8 + 30 3 + 30 = 33

This brings us to the correct answer, which is 33 .

#2 Answer Explanation

If this math problem looks familiar to you, that's probably because it went viral in August 2019 due to its ambiguous setup . Many people argued over whether the correct answer was 1 or 16, but as we all know, with math there's (almost always!) only one truly correct answer.

So which is it: 1 or 16?

Let's see how PEMDAS can give us the right answer. This problem has parentheses, division, and multiplication. So we'll start by simplifying the expression in the parentheses, per PEMDAS:

While most people online agreed up until this point, many disagreed on what to do next: do you multiply 2 by 4, or divide 8 by 2?

PEMDAS can answer this question: when it comes to multiplication and division, you always work left to right. This means that you would indeed divide 8 by 2 before multiplying by 4.

It might help to look at the problem this way instead, since people tend to get tripped up on the parentheses (remember that anything next to a parenthesis is being multiplied by whatever is in the parentheses):

Now, we just solve the equation from left to right:

8 ÷ 2 × 4 4 × 4 = 16

The correct answer is 16. Anyone who argues it's 1 is definitely wrong — and clearly isn't using PEMDAS correctly!

#3 Answer Explanation

Things start to get a bit trickier now.

This math problem has parentheses, an exponent, multiplication, division, and subtraction. But don't get overwhelmed — let's work through the equation, one step at a time.

First, per the PEMDAS rule, we must simplify what's in the parentheses :

7 × 4 − 10 (2) ÷ 2²

Easy peasy, right? Next, let's simplify the exponent :

7 × 4 − 10 (2) ÷ 4

All that's left now is multiplication, division, and subtraction. Remember that with multiplication and division, we simply work from left to right:

7 × 4 − 10 (2) ÷ 4 28 − 10 (2) ÷ 4 28 − 20 ÷ 4 28 − 5

Once you've multiplied and divided, you just need to do the subtraction to solve it:

28 − 5 = 23

This gives us the correct answer of 23 .

#4 Answer Explanation

This problem might look scary, but I promise it's not! As you long as you approach it one step at a time using the PEMDAS rule , you'll be able to solve it in no time.

Right away we can see that this problem contains all components of PEMDAS : parentheses (two sets), exponents (two and a square root), multiplication, division, addition, and subtraction. But it's really no different from any other math problem we've done.

First, we must simplify what's in the two sets of parentheses:

√25 (6)² − 18 ÷ 3 (2) + 2³

Next, we must simplify all the exponents — this includes square roots, too :

5 (36) − 18 ÷ 3 (2) + 8

Now, we must do the multiplication and division from left to right:

5 (36) − 18 ÷ 3 (2) + 8 180 − 18 ÷ 3 (2) + 8 180 − 6 (2) + 8 180 − 12 + 8

Finally, we solve the remaining addition and subtraction from left to right:

180 − 12 + 8 168 + 8 = 176

This leads us to the correct answer of 176 .

What's Next?

Another math acronym you should know is SOHCAHTOA. Our expert guide tells you what the acronym SOHCAHTOAH means and how you can use it to solve problems involving triangles .

Studying for the SAT or ACT Math section? Then you'll definitely want to check out our ultimate SAT Math guide / ACT Math guide , which gives you tons of tips and strategies for this tricky section.

Interested in really big numbers? Learn what a googol and googolplex are , as well as why it's impossible to write one of these numbers out.

Hannah received her MA in Japanese Studies from the University of Michigan and holds a bachelor's degree from the University of Southern California. From 2013 to 2015, she taught English in Japan via the JET Program. She is passionate about education, writing, and travel.

Ask a Question Below

Have any questions about this article or other topics? Ask below and we'll reply!

Improve With Our Famous Guides

• For All Students

The 5 Strategies You Must Be Using to Improve 160+ SAT Points

How to Get a Perfect 1600, by a Perfect Scorer

Series: How to Get 800 on Each SAT Section:

Score 800 on SAT Math

Score 800 on SAT Reading

Score 800 on SAT Writing

Series: How to Get to 600 on Each SAT Section:

Score 600 on SAT Math

Score 600 on SAT Reading

Score 600 on SAT Writing

Free Complete Official SAT Practice Tests

What SAT Target Score Should You Be Aiming For?

15 Strategies to Improve Your SAT Essay

The 5 Strategies You Must Be Using to Improve 4+ ACT Points

How to Get a Perfect 36 ACT, by a Perfect Scorer

Series: How to Get 36 on Each ACT Section:

36 on ACT English

36 on ACT Math

36 on ACT Reading

36 on ACT Science

Series: How to Get to 24 on Each ACT Section:

24 on ACT English

24 on ACT Math

24 on ACT Reading

24 on ACT Science

What ACT target score should you be aiming for?

ACT Vocabulary You Must Know

ACT Writing: 15 Tips to Raise Your Essay Score

How to Get Into Harvard and the Ivy League

How to Get a Perfect 4.0 GPA

How to Write an Amazing College Essay

What Exactly Are Colleges Looking For?

Is the ACT easier than the SAT? A Comprehensive Guide

Should you retake your SAT or ACT?

When should you take the SAT or ACT?

Stay Informed

Get the latest articles and test prep tips!

Looking for Graduate School Test Prep?

Check out our top-rated graduate blogs here:

GRE Online Prep Blog

GMAT Online Prep Blog

TOEFL Online Prep Blog

Holly R. "I am absolutely overjoyed and cannot thank you enough for helping me!”

Brackets (Parentheses)

Brackets are symbols used in pairs to group things together.

When we see things inside brackets we do them first (as explained in Order of Operations ).

Example: (3 + 2) × (6 − 4)

The parentheses group 3 and 2 together, and 6 and 4 together, so they get done first:

Without the parentheses the multiplication is done first:

3 + 2 × 6 − 4 = 3 + 12 − 4 = 11 (not 10)

With more complicated grouping we can use different types of brackets :

Example: [(3 + 2) × (6 − 4) + 2] × 4

The parentheses group 3 and 2 together, and 6 and 4 together, and the square brackets tell us to do all the calculations inside them before multiplying by 4:

Curly Brackets

Curly brackets {} are used in Sets :

Example: {2, 4, 6, 8}

Is the set of even numbers from 2 to 8

Angle Brackets

Angle brackets 〈 〉 are used in Bra-Ket Notation .

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Course: 5th grade   >   Unit 13

• Constructing numerical expressions

Evaluating expressions with & without parentheses

• Evaluate expressions with parentheses
• Translating expressions with parentheses
• Translate expressions with parentheses
• Create expressions with parentheses

Want to join the conversation?

• Upvote Button navigates to signup page
• Downvote Button navigates to signup page
• Flag Button navigates to signup page

Video transcript

Parentheses, Braces, and Brackets in Math

How These Symbols Help Determine the Order of Operations

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

You'll come across many symbols in mathematics and arithmetic. In fact, the language of math is written in symbols, with some text inserted as needed for clarification. Three important—and related—symbols you'll see often in math are parentheses, brackets , and braces, which you'll encounter frequently in  prealgebra  and  algebra . That's why it's so important to understand the specific uses of these symbols in higher math.

Using Parentheses ( )

Parentheses are used to group numbers or variables, or both. When you see a math problem containing parentheses, you need to use the order of operations to solve it. For example, take the problem: 9 - 5 ÷ (8 - 3) x 2 + 6

For this problem, you must calculate the operation within the parentheses first—even if it's an operation that would normally come after the other operations in the problem. In this problem, the multiplication and division operations would normally come before subtraction (minus), however, since 8 - 3 falls within the parentheses, you'd work out this part of the problem first. Once you've taken care of the calculation that falls within the parentheses, you'd remove them. In this case (8 - 3) becomes 5, so you would solve the problem as follows:

9 - 5 ÷ (8 - 3) x 2 + 6

= 9 - 5 ÷ 5 x 2 + 6

= 9 - 1 x 2 + 6

= 9 - 2 + 6

= 7 + 6

Note that per the order of operations, you'd work what's in the parentheses first, next, calculate numbers with exponents, and then multiply and/or divide, and finally, add or subtract. Multiplication and division, as well as addition and subtraction, hold an equal place in the order of operations, so you work these from left to right.

In the problem above, after taking care of the subtraction in the parentheses, you need to first divide 5 by 5, yielding 1; then multiply 1 by 2, yielding 2; then subtract 2 from 9, yielding 7; and then add 7 and 6, yielding a final answer of 13.

Parentheses Can Also Mean Multiplication

In the problem: 3(2 + 5), the parentheses tell you to multiply. However, you wouldn't multiply until you complete the operation inside the parentheses—2 + 5—so you would solve the problem as follows:

3(2 + 5)

Examples of Brackets [ ]

Brackets are used after the parentheses to group numbers and variables as well. Typically, you'd use the parentheses first, then brackets, followed by braces. Here is an example of a problem using brackets:

 4 - 3[4 - 2(6 - 3)] ÷ 3

= 4 - 3[4 - 2(3)] ÷ 3 (Do the operation in the parentheses first; leave the parentheses.)

= 4 - 3[4 - 6] ÷ 3 (Do the operation in the brackets.)

= 4 - 3[-2] ÷ 3 (The bracket informs you to multiply the number within, which is -3 x -2.)

= 4 + 6 ÷ 3

= 4 + 2

Examples of Braces { }

Braces are also used to group numbers and variables. This example problem uses parentheses, brackets, and braces. Parentheses inside other parentheses (or brackets and braces) are also referred to as " nested parentheses ." Remember, when you have parentheses inside brackets and braces, or nested parentheses, always work from the inside out:

 2{1 + [4(2 + 1) + 3]}

= 2{1 + [4(3) + 3]}

= 2{1 + [12 + 3]}

= 2{1 + [15]}

= 2{16}

Notes About Parentheses, Brackets, and Braces

Parentheses, brackets, and braces are sometimes referred to as "round," "square," and "curly" brackets, respectively. Braces are also used in sets, as in:

{2, 3, 6, 8, 10...}

When working with nested parentheses, the order will always be parentheses, brackets, braces, as follows:

• Use BEDMAS to Remember the Order of Operations
• How to Use Parentheses in Writing
• The Associative and Commutative Properties
• The Rules of Using Positive and Negative Integers
• Order of Operations Worksheets
• Probabilities for Rolling Three Dice
• Popular Math Terms and Definitions
• The Associative Property in Math
• What You Need to Know About Consecutive Numbers
• How to Use Brackets Correctly in Writing
• Understanding Equivalent Equations in Algebra
• Computations With Fractions
• Multiply Fractions With Common Denominators Worksheets
• Fraction Tests and Worksheets
• Arithmetic and Geometric Sequences
• Simplifying Expressions With the Distributive Property Law

Video Crash Courses

Junior Math

Math Essentials

Tutor-on-Demand

Encyclopedia

Digital Tools

Equations with Fractions and Parentheses

Here, you’ll learn how to solve equations that contain fractions . The only major difference from what you’ve done so far is that you must find a common denominator and multiply all the terms by it. This will allow you to cancel out the denominators and isolate the variable. The whole point of this method is to get rid of the fractions, which simplifies the calculations!

Solving Equations with Fractions and Parentheses

Solve the equation x 3 + 1 = 4 − 2 x 3

• HW Guidelines
• Study Skills Quiz
• Find Local Tutors
• Demo MathHelp.com
• Join MathHelp.com

Select a Course Below

• ACCUPLACER Math
• Math Placement Test
• PRAXIS Math
• + more tests
• 5th Grade Math
• 6th Grade Math
• Pre-Algebra
• College Pre-Algebra
• Introductory Algebra
• Intermediate Algebra
• College Algebra

Simplifying Nested Parentheses

Intro Nested Solving v. Simplifying

Parentheses inside of other parentheses are called "nested" parentheses. The process of simplification works the same way as in the simpler examples on the previous page, but we do need to be a little more careful as we work our way through the grouping symbols.

And, by "working our way through", I mean "work our way out from the inside, simplifying as we go".

Content Continues Below

MathHelp.com

Simplify 3[2 − 1(3 2 − 2 3 )]

The innermost parenthetical is the exponent stuff inside the parentheses. I'll start by simplifying this portion of the expression:

3[2 − 1(3 2 − 2 3 )]

3[2 − 1(9 − 8)]

3[2 − 1(1)]

Simplifying inside the square brackets comes next.

3[2 − 1]

There's nothing more that can be done with this, so my answer is:

Advertisement

You may be wondering why I didn't put "equals" signs between each of the lines above. While it would have been legitimate (mathematically) to do so, I've seen many students get confused about when the "equals" signs "matter", and when they don't.

To be clear, one must have "equals" signs when one is working with equations (that is, when one is working with "(one expression) equals (another expression)", and one has to solve for the value of a variable). One must have "equals" signs for working with equations. One does not have have to use them for simplifying stand-alone expressions. In hopes of avoiding some confusion, I try not to use "equals" signs when only simplifying. (More on this topic later .)

Simplify 4[ x + 3(2 x + 1)]

With nested parentheses, the safest plan is to work from the inside out. So I'll take the 3 through the inner parentheses first, before I even think about dealing with the 4 and the square brackets on the outside. I'll also simplify as much as I can, as I go along. I will write each step out completely as I go. I start with the original exercise, and then take the 3 through the innermost parenthetical:

4[ x + 3 (2 x + 1)]

4[ x + 3 (2 x ) + 3 (1)]

4[ x + 6 x + 3]

Now I can combine like terms (being the two variable-containing terms) to get:

With the insides of the square brackets simplified, I can now start moving the 4 through the brackets:

4 [7 x + 3]

4 [7 x ] + 4 [3]

I can't combine these two terms, so I'm done.

By the way, there is no particular significance to the square brackets ("[" and "]") versus the parentheses versus the curly braces ("{" and "}"). Using the different grouping symbols is just a nice way of helping the user keep track of the different pairs of symbols. This is similar to working in certain spreadsheets, where the pairs of parentheses in a highlighted formula will be color-coded, as you can see below:

This color-coding helps you see which ")" character (that is, with which "close-paren") goes with which "(" character (that is, with which "open-paren") in a formula. The different types of grouping symbols in mathematics serve the same purpose as do the colored parentheses in the spreadsheet.

FYI: The traditional sequence of grouping symbols, working from the inside out, is "parentheses", then "square brackets", and then "curly braces"; then you repeat the sequence, as necessary. But this is not, to my knowledge, a rule; it's just a common convention.

Simplify −5{3 − 2[1 − 4(3 − 2 2 )]}

Yeesh! Negatives, much?

I'll start from the innermost grouping symbols, being the parentheses.

−5{3 − 2[1 − 4(3 − 2 2 )]}

−5{3 − 2[1 − 4(3 − 4)]}

−5{3 − 2[1 − 4(−1)]}

Now I can multiply by the 4 in front of the parentheses, and simplify inside the square brackets:

−5{3 − 2[1 − (−4)]}

−5{3 − 2[1 + 4]}

−5{3 − 2[5]}

And now I can simplify inside the curly braces, and then take the 5 through the braces to complete the simplification:

−5{3 − 10}

−5{−7}

This expression happened not to have any variables, so I was able to simplify all the way down to a simple number, which is my hand-in answer (though I'll leave off the "plus" in front of the number, because it is neither necessary nor customary).

Yes, you should expect at least one problem like this, with all the "minus" signs, on the next test. Don't try to do too many steps at once, when you're simplifying messes like this. Take your time, and write out however many steps you need.

Simplify 9 − 3[ x − (3 x + 2)] + 4

I won't do anything with the "9 −" or the "+ 4" until I simplify what's inside the brackets and the parentheses. I'll work from the inside out, inserting the "understood" 1 where I find it helpful:

9 − 3[ x − (3 x + 2)] + 4

9 − 3[ x − 1 (3 x + 2)] + 4

9 − 3[ x − 1 (3 x ) − 1 (2)] + 4

9 − 3[ x − 3 x − 2] + 4

9 − 3[−2 x − 2] + 4

9 − 3[−2 x ] − 3[−2] + 4

9 + 6 x + 6 + 4

I can't combine these terms, so there's nothing left to simplify. My answer is:

It is not required that you write out this many (or this few) steps. You should be careful to do one step at a time, though, writing things out completely and simplifying as you go. You should do as many steps as you need in order to consistently arrive at the correct answer.

Simplify 5 + 2{[3 + (2 x − 1) + x ] − 2}

I'll work carefully from the inside out, starting with the parentheses in the middle, then moving outward to the brackets, and then on to the braces.

5 + 2{ [3 + (2 x − 1) + x ] − 2}

5 + 2{ [3 + 2 x − 1 + x ] − 2}

5 + 2{ [2 x + x + 3 − 1] − 2}

5 + 2{ [3 x + 2] − 2}

5 + 2{3 x + 2 − 2}

5 + 2{3 x }

This answer is mathematically correct, but I'll rearrange the terms into descending order, for custom's sake.

URL: https://www.purplemath.com/modules/simparen2.htm

You can use the Mathway widget below to practice simplifying with parentheses. Try the entered exercise, or type in your own exercise. Then click the button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)

Please accept "preferences" cookies in order to enable this widget.

(Click "Tap to view steps" to be taken directly to the Mathway site for a paid upgrade.)

Page 1 Page 2 Page 3

Standardized Test Prep

College math, homeschool math, share this page.

• Terms of Use
• About Purplemath
• About the Author
• Tutoring from PM
• Advertising
• Linking to PM
• Site licencing

Visit Our Profiles

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

What can QuickMath do?

QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

• The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
• The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
• The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
• The calculus section will carry out differentiation as well as definite and indefinite integration.
• The matrices section contains commands for the arithmetic manipulation of matrices.
• The graphs section contains commands for plotting equations and inequalities.
• The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

Math Topics

More solvers.

• Add Fractions
• Simplify Fractions

Watch CBS News

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: 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.

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?

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.

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?

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 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.

More from CBS News

Israel preparing for Rafah invasion in Gaza amid increasing tension with U.S.

60 Minutes Archive: Coverage of North Korea

How much does it cost to file for bankruptcy?

As a Social Security cut looms, should seniors buy long-term care insurance now?

Game Central

Similar problems from web search.

New Orleans teens solve 2,000-year-old math problem

W hen she started a math contest with a bonus question challenging students to create a new proof for the Pythagorean theorem using trigonometry, teacher Michelle Blouin Williams didn’t expect anyone to complete the task.

“I was just looking for some ingenuity,” she said, per CBS News .

Calcea Johnson and Ne’Kiya Jackson, however, blew Williams’ expectations out of the water by figuring it out in 2023. The teens were seniors at St. Mary’s Academy in New Orleans, a prestigious Catholic school for girls which has maintained a 100% acceptance rate to colleges and 100% graduation rate for 17 years, CBS News reported.

They appeared in an episode of CBS News’ “60 Minutes ” on Sunday to talk about their achievement.

How did the teens find the answer?

While they were motivated initially by the math competition’s $500 prize, an internal drive to finish what they started manifested when they reached the tricky bonus question. For two months, the high school seniors worked tirelessly to finish their proof.

CeCe Johnson, Calcea’s mother, told “60 Minutes,” “It was pages and pages and pages of, like, over 20 or 30 pages for this one problem.”

Her father, Cal Johnson, added, “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.”

When they finished, teachers at St. Mary’s recognized the importance of their work and submitted their proof to the American Mathematical Society for recognition at a conference in March 2023, where the students presented their work.

What is the Pythagorean theorem and what’s a proof?

In essence, the mathematical theorem states that knowing the lengths of two sides of a right triangle enables you to figure out the length of the third using this formula: a² + b² = c².

It’s associated with Greek mathematician Pythagoras, but evidence suggests it was known earlier, in Babylon and Iron Age India, per Britannica . Its practical uses include construction and architecture, two-dimensional navigation, and surveying.

A mathematical proof is exactly what it sounds like: reasoning that proves a mathematical theorem is true. American mathematician Daniel Kane explains proofs as being like essays, but using math.

Why is Calcea Johnson and Ne’Kiya Jackson’s work significant?

According to the “ 60 Minutes ” episode, “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.”

In 1927, mathematician Elisha Loomis said as much in his book, “ The Pythagorean Proposition .” Loomis argued that there could be no trigonometric proof of the theorem because it would be circular.

Stuart Anderson, a professor emeritus of mathematics at Texas A&M University–Commerce, told Scientific American , “A lot of the basic trig ‘identities’ are nothing more than Pythagoras’ theorem.”

So, because trigonometric functions are based upon the Pythagorean theorem, using them reflexively to prove the theorem would be akin to going in circles and a fundamental mathematical error, Loomis argued.

According to Scientific American , the teens refuted this in their presentation in 2023 and said that “a trigonometric identity called the law of sines didn’t depend on the Pythagorean theorem and that they could use it to prove the theorem.”

Calcea and Ne’Kiya have joined an extremely small group who’ve accomplished the same feat, including mathematician Jason Zimba, who successfully created a new proof in 2009. The two submitted their proof for final peer review this spring and continue to work on creating more proofs.

How did the world respond to their accomplishment?

The teens were given the keys to the city of New Orleans and a commendation from the governor of Louisiana, among other public recognitions.

While their achievement “blew up,” as Ne’Kiya described it, the two students remain humble, and laughed at being called geniuses.

When news of their accomplishment broke, some people seemed to be shocked and dismissed the news as fake, St. Mary’s president Pamela Rogers said in the interview .

“They were saying, ‘Oh, they could not have done it. African Americans don’t have the brains to do it.’ ... 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.”

When interviewer Bill Whitaker asked why they thought there’d been such a response, Ne’Kiya said, “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.”

“I’d like to actually be celebrated for what it is. Like, it’s a great mathematical achievement,” she continued.

Math Games for 3rd Grade 2024 4+

Educational learning for kids, smart kidz club inc., designed for ipad.

• 4.0 • 1 Rating
• Offers In-App Purchases

Screenshots

Description.

Smart Kidz Club Grade 3 Math App is curriculum-aligned for 8-9-year-olds to practice and master all the required grade 3 math skills. This app follows the US Common Core State Standards and consists of comprehensive math skills essential for GRADE 3 school students. Resources consist of activity books with math problems. Each Math Activity that requires complex solving, consists of a SCRATCH PAD option where the child can freely write on the screen (overlay) to solve the problems in writing if they so desire. Math activities cover all the required grade 3 skills such as decimal numbers, equations, variables, measurement, geometry, understanding fractions including equivalent fractions, operations with fractions, logical reasoning, numbers up to thousands, place value, estimation, rounding, probability, statistics, Roman Numerals, data, graphs, properties and importance of parenthesis, time, and addition, subtraction, multiplication, and division expertise concepts. Math Activities consist of 2 modes, practice and test. Each of these modes comes with a choice of the number of questions/problems that the child wishes to solve— 1) Practice Mode: The Practice Mode allows the child to solve each problem before advancing to the next problem while prompting them to try again if they choose the wrong answer. The Practice Mode is not scored. At the end of the practice mode, the child can try again with a new set of problems. 2) Test Mode: The Test Mode is scored and does not prompt the child during their attempt to solve the problem. At the end of the Test Mode, the child can try again with a set of new math problems or review their answers with detailed feedback and explanation. This app supports maths learning in homes by helping kids in grade 3 with their mathematical understanding and mastering math skills without any harmful distractions or ads.

Version 1.6

Performance Improvements

Ratings and Reviews

App privacy.

The developer, Smart Kidz Club Inc. , indicated that the app’s privacy practices may include handling of data as described below. For more information, see the developer’s privacy policy .

Data Not Collected

The developer does not collect any data from this app.

Privacy practices may vary based on, for example, the features you use or your age. Learn More

Information

• Math Practice for Grade 3:1W AED 7.99
• Lifetime access to all content AED 39.99
• Math Practice for Grade 3:3M AED 14.99
• Math Practice for Grade 3:1M AED 12.99
• Math Practice for Grade 3:6M AED 19.99
• App Support
• Privacy Policy

More By This Developer

Kids Books Reading, Math Games

Read-Along Books, Reading App

ABC Flash Cards Phonics Games

Fun Maths Games for Children

Phonics Fun: Learn & Read UK

Year 1 Reading Adventure 2024

You Might Also Like

Learn Math 3rd Grade

Math Ace 3rd Grade

Learn Spelling 3rd Grade

Mental Math: Grade 3 Adventure

Math Kid Grade 3

3rd Grade Math Tutor