year 5 problem solving division

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Year 5 Division

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Find ways to make all the numbers 1 to 21.

3. Explore factors, multiples, primes, and composite numbers

You can help your child identify   factors   and   factor pairs   using their knowledge of times tables.

You can play a factor and multiple game with your child. Ask your child to choose an even number on a   100 square   and cross it out. You then cross out a number that is either a factor or a multiple of that number. For example:

Start at 20. A factor of 20 is 4. A multiple of 4 is 40. A multiple of 40 is 80. … and so on. The game gets harder and harder as it goes on and more numbers are crossed out!

Continue to take it in turns to choose a number that is a multiple or a factor of the number that has just been crossed out by the other player. When a player cannot cross out any numbers, that player loses the game.

Your child will also need to explain why a number is prime or   composite .  Prime numbers are numbers that only have factors of 1 and itself. They can only be divided by 1 or themselves to get a whole number. 2, 3, 5, 7, 11, and 13 are all prime numbers.  Composite numbers are numbers with more than two factors. 4, 6, 8, 9, 10, and 12 are all composite numbers.

Activity: Prime number lotto

Find out about prime numbers.

4. Explore square and cube numbers

Your child will need to use and understand square and cube numbers.

A square number is created by multiplying an integer (whole number) by itself. For example:

9 is a square number because it can be made by multiplying 3 by 3. 25 is a square number because it can be made by multiplying 5 by 5. 144 is a square number because it can be made by multiplying 12 by 12.

A square number can also be represented in the shape of a square. A square with 3cm by 3cm sides has an area of 9cm ².   Ask your child to find some objects that are all the same size, and arrange them into a square. Notice how this makes a square number.

Cube numbers are formed by multiplying an integer (whole number) by itself three times. For example:

8 is a cube number because it can be made by multiplying 2 by 2 by 2. 27 is a cube number because it can be made by multiplying 3 by 3 by 3. 125 is a cube number because it can be made by multiplying 5 by 5 by 5.

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Division Word Problem Worksheets

This page contains extensive division word problems replete with engaging scenarios that involve two-digit and three-digit dividends and single digit divisors; three-digit dividends and two-digit divisors; and advanced division worksheets (four-digit and five-digit dividends). Thumb through some of these worksheets for free!

Division Word Problems for Beginners

These printable worksheets feature simple division word problems. The divisors are in the range 2 to 9. The quotients are in the range 2 to 10. These worksheets are building blocks for children.

• Download the set

Division: Two-digit by Single-digit (without Remainder)

This set of word problems involves dividing a two-digit number by a single-digit number to arrive at a quotient. The division leaves no remainder. Answer key is included in each worksheet.

Division: Two-digit by Single-digit (with Remainder)

These word problems require the learner to divide the two-digit dividend by the single-digit divisor and write down both the quotient and the remainder. Three pdf worksheets with 15 scenarios are featured here.

Theme based Word Problems

Each worksheet has five word problems related to the given theme. Supermarket, School and Halloween party are the themes used here.

Three-digit by Single-digit Word Problems

These printable worksheets involve division word problems with three-digit dividends and single digit divisors. Apply long division method to solve each problem.

Three-digit by Two-digit Word Problems

This set of word problems will require the student to perform division operations involving three-digit numbers and two-digit numbers. Verify your answer with the answer key provided in the worksheet.

Division: Four or Five-digit by Single-digit

Interesting scenarios are presented in these advanced worksheet pdfs that involve four-digit and five-digit dividends and single digit divisors. Use long division method to find the quotient.

Related Worksheets

» Addition Word Problems

» Subtraction Word Problems

» Multiplication Word Problems

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Year 5 | Solving Division Word Problems Worksheets

In these Year 5 solving division word problems worksheets , a variety of word problems are presented, challenging learners to apply their knowledge and skills in short division to find the solutions. Each problem comes with a section for pupils to show their step-by-step working.

This Year 5 solving division word problems worksheet serves as an ideal practice tool for children to hone their proficiency in the short division bus stop written method while gaining a deeper understanding of how division can be effectively employed to solve real-life word problems. Engaging with division word problems requires children to analyse the information provided, identify that division is the appropriate operation to use, and apply problem-solving strategies to arrive at a solution, fostering critical thinking and analytical skills.

Our Year 5 solving division word problems worksheets are aligned with KS2 multiplication and division on the primary national curriculum. All of our Year 5 division worksheets can be combined with your ideas for primary maths activities, learning resources, teaching material, visual aids, games, differentiation, display boards, homework, and lesson plans.

You might be interested in our Year 5 Multiples Worksheets .

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Multiplication and Division KS2

This collection is one of our Primary Curriculum collections - tasks that are grouped by topic.

One Wasn't Square

Mrs Morgan, the class's teacher, pinned numbers onto the backs of three children. Use the information to find out what the three numbers were.

All the Digits

This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?

Cycling Squares

Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

Can you replace the letters with numbers? Is there only one solution in each case?

Multiplication Square Jigsaw

Can you complete this jigsaw of the multiplication square?

Shape Times Shape

These eleven shapes each stand for a different number. Can you use the number sentences to work out what they are?

What Do You Need?

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

How Do You Do It?

This group activity will encourage you to share calculation strategies and to think about which strategy might be the most efficient.

Table Patterns Go Wild!

Nearly all of us have made table patterns on hundred squares, that is 10 by 10 grids. This problem looks at the patterns on differently sized square grids.

Journeys in Numberland

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

Ordering Cards

This problem is designed to help children to learn, and to use, the two and three times tables.

Let Us Divide!

Look at different ways of dividing things. What do they mean? How might you show them in a picture, with things, with numbers and symbols?

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Sweets in a Box

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

Round and Round the Circle

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Highest and Lowest

Put operations signs between the numbers 3 4 5 6 to make the highest possible number and lowest possible number.

Zios and Zepts

On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

Abundant Numbers

48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?

Find at least one way to put in some operation signs to make these digits come to 100.

Flashing Lights

Norrie sees two lights flash at the same time, then one of them flashes every 4th second, and the other flashes every 5th second. How many times do they flash together during a whole minute?

The Moons of Vuvv

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

Mystery Matrix

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

Four Goodness Sake

Use 4 four times with simple operations so that you get the answer 12. Can you make 15, 16 and 17 too?

Multiplication Squares

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

Factor Lines

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Two Primes Make One Square

Can you make square numbers by adding two prime numbers together?

Cubes Within Cubes

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

Which Is Quicker?

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

A Square of Numbers

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

Odd Squares

Think of a number, square it and subtract your starting number. Is the number you're left with odd or even? How do the images help to explain this?

Up and Down Staircases

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

Carrying Cards

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?

An Easy Way to Multiply by 10?

Do you agree with Badger's statements? Is Badger's reasoning 'watertight'? Why or why not?

Multiples Grid

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

Factors and Multiples Game

This game can replace standard practice exercises on finding factors and multiples.

Music to My Ears

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

What's in the Box?

This big box multiplies anything that goes inside it by the same number. If you know the numbers that come out, what multiplication might be going on in the box?

Factor-multiple Chains

Can you see how these factor-multiple chains work? Find the chain which contains the smallest possible numbers. How about the largest possible numbers?

This challenge is a game for two players. Choose two of the numbers to multiply or divide, then mark your answer on the number line. Can you get four in a row?

Picture a Pyramid ...

Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

The Remainders Game

Play this game and see if you can figure out the computer's chosen number.

Which Symbol?

Choose a symbol to put into the number sentence.

Times Tables Shifts

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

Counting Cogs

Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

Light the Lights Again

Each light in this interactivity turns on according to a rule. What happens when you enter different numbers? Can you find the smallest number that lights up all four lights?

Follow the Numbers

What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

Curious Number

Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

Factor Track

Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules.

So It's Times!

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

Square Subtraction

Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

This Pied Piper of Hamelin

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

Multiply Multiples 1

Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?

Multiply Multiples 2

Can you work out some different ways to balance this equation?

Multiply Multiples 3

Have a go at balancing this equation. Can you find different ways of doing it?

Division Rules

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Always, Sometimes or Never? Number

Are these statements always true, sometimes true or never true?

Satisfying Four Statements

Can you find any two-digit numbers that satisfy all of these statements?

Picture Your Method

Can you match these calculation methods to their visual representations?

Compare the Calculations

Can you put these four calculations into order of difficulty? How did you decide?

Dicey Array

Watch the video of this game being played. Can you work out the rules? Which dice totals are good to get, and why?

4 by 4 Mathdokus

Can you use the clues to complete these 4 by 4 Mathematical Sudokus?

Xavi's T-shirt

How much can you read into a T-shirt?

Teens come up with answer to problem that stumped math world for centuries

A s 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.

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How two high school students solved a 2,000-year-old math puzzle

By Bill Whitaker , Aliza Chasan , Sara Kuzmarov, Mariah Campbell

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

A high school math teacher at St. Mary's Academy in New Orleans, Michelle Blouin Williams, was looking for ingenuity when she and her colleagues set a school-wide math contest with a challenging bonus question. That bonus question asked students to create a new proof for the Pythagorean Theorem, a fundamental principle of geometry, using trigonometry. The teachers weren't necessarily expecting anyone to solve it, as proofs of the Pythagorean Theorem using trigonometry were believed to be impossible for nearly 2,000 years.

But then, in December 2022, Calcea Johnson and Ne'Kiya Jackson, seniors at St. Mary's Academy, stepped up to the challenge. The $500 prize money was a motivating factor.

After months of work, they submitted their innovative proofs to their teachers. With the contest behind them, their teachers encouraged the students to present at a mathematics conference, and then to seek to publish their work. And even today, they're not done. Now in college, they've been working on further proofs of the Pythagorean Theorem and believe they have found five more proofs. Amazingly, despite their impressive achievements, they insist they're not math geniuses.

"I think that's a stretch," Calcea said.

The St. Mary's math contest

When the pair started working on the math contest they were familiar with the Pythagorean Theorem's equation: A² + B² = C², which explains that by knowing the length of two sides of a right triangle, it's possible to figure out the length of the third side.

When Calcea and Ne'Kiya set out to create a new Pythagorean Theorem proof, they didn't know that for thousands of years, one using trigonometry was thought to be impossible.  In 2009, mathematician Jason Zimba submitted one, and now Calcea and Ne'Kiya are adding to the canon.

Calcea and Ne'Kiya had studied geometry and some trigonometry when they started working on their proofs, but said they didn't feel math was easy. As the contest went on, they spent almost all their free time developing their ideas.

"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," Cal Johnson, Calcea's dad, said.

Neliska Jackson, Ne'Kiya's mother, says lightheartedly, that most of the time, her daughter's work was beyond her. 

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.

"Well, our teacher approached us and was like, 'Hey, you might be able to actually present this,'" Ne'Kiya said. "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."

Why Calcea' and Ne'kiya's work "blew up"

The reaction was insane and unexpected, Calcea said. News of their accomplishment spread around the world. The pair got a write-up in South Korea and a shoutout from former first lady Michelle Obama. They got a commendation from the governor and keys to the city of New Orleans. 

Calcea and Ne'Kiya said they think there's several reasons why people found their work so impressive. 

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

Ne'Kiya said she'd like their accomplishment to be celebrated for what it is: "a great mathematical achievement."

In spite of the community's celebration of the students' work, St. Mary's Academy president and interim principal Pamela Rogers said that with recognition came racist calls and comments. 

"[People said] 'they could not have done it. African Americans don't have the brains to do it.' Of course, we sheltered our girls from that," Rogers said. "But we absolutely did not expect it to come in the volume that it came."

Rogers said too often society has a vision of who can be successful.

"To some people, it is not always an African American female," Rogers said. "And to us, it's always an African American female."

Success at St. Marys 

St. Mary's, a private Catholic elementary and high school, was started for young Black women just after the Civil War. Ne'Kiya and Calcea follow a long line of barrier-breaking graduates. Leah Chase , the late queen of Creole cuisine, was an alum. So was Michelle Woodfork, the first African American female New Orleans police chief, and Dana Douglas, a judge for the Fifth Circuit Court of Appeals. 

Math teacher Michelle Blouin Williams, who initiated the math contest, said Calcea and Ne'Kiya are typical St. Mary's students. She said if they're "unicorns," then every student who's matriculated through the school is a "beautiful, Black unicorn."

Students hear that message from the moment they walk in the door, Rogers said. 

"We believe all students can succeed, all students can learn," the principal said. "It does not matter the environment that you live in."

About half the students at St. Mary's get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. There's no test to get in, but expectations are high and rules are strict: cellphones are not allowed and modest skirts and hair in its natural color are required. 

Students said they appreciate the rules and rigor.

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

What's next for Ne'Kiya and Calcea

Last year when Ne'Kiya and Calcea graduated, all their classmates were accepted into college and received scholarship offers. The school has had a 100% graduation rate and a 100% college acceptance rate for 17 years, according to Rogers.

Ne'Kiya got a full ride in the pharmacy department at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University. Neither one is pursuing a career in math, though Calcea said she may minor in math.

"People might expect too much out of me if I become a mathematician," Ne'Kiya said wryly. 

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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The Protesters and the President

Over the past week, thousands of students protesting the war in gaza have been arrested..

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Over the past week, students at dozens of universities held demonstrations, set up encampments and, at times, seized academic buildings. In response, administrators at many of those colleges decided to crack down and called in the local police to detain and arrest demonstrators.

As of Thursday, the police had arrested 2,000 people across more than 40 campuses, a situation so startling that President Biden could no longer ignore it.

Jonathan Wolfe, who has been covering the student protests for The Times, and Peter Baker, the chief White House correspondent, discuss the history-making week.

On today’s episode

Jonathan Wolfe , a senior staff editor on the newsletters team at The New York Times.

Peter Baker , the chief White House correspondent for The New York Times covering President Biden and his administration.

Background reading

As crews cleared the remnants of an encampment at U.C.L.A., students and faculty members wondered how the university could have handled protests over the war in Gaza so badly .

Biden denounced violence on campus , breaking his silence after a rash of arrests.

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After years of delay, Boeing to try again with Starliner space capsule

Monday’s planned launch comes as the aerospace giant is dealing with fallout from problems in its airline division.

Before a door-size panel blew out of a Boeing 737 Max , leaving a gaping hole in the side of an Alaska Airlines aircraft shortly after takeoff; before whistleblowers came forward to say they were threatened for bringing up safety issues at the company; and before the Justice Department opened a criminal investigation into the blowout incident, Boeing was struggling with another set of issues, on another high-profile vehicle.

Its Starliner spacecraft, designed to fly astronauts to orbit under a $4.2 billion contract from NASA, had suffered a series of problems that put its launch with astronauts years behind schedule. Its onboard computer had failed during its first test flight. A second test flight was scrubbed after valves in the vehicle’s service module stuck and wouldn’t operate. Then, after the craft finally flew a test mission successfully without anyone on board, Boeing discovered that tape used as insulation on wiring inside the capsule was flammable and would need to be removed. The parachute system also had problems, which forced the company to redesign and strengthen a link between the parachutes and the spacecraft.

Now, a decade after NASA awarded Boeing a contract to fly astronauts to the International Space Station, Boeing will finally attempt to fly its Starliner spacecraft with people onboard. If all goes to plan, at 10:34 p.m. on Monday, the company is set to fly a pair of veteran astronauts, Sunita Williams and Barry “Butch” Wilmore, on a mission that will be one of the most significant tests for Boeing’s space division — and for NASA — in years.

The flight is intended to see how the spacecraft performs in space with a crew onboard. If all goes well, the spacecraft will catch up with the space station — which travels at 17,500 mph — about a day after lifting off. Along the way, the crew members will test manually flying the spacecraft before it docks autonomously with the station. NASA and Boeing will also be eager to see how the spacecraft’s heat shield and parachutes work as it brings Williams and Wilmore back to Earth after about eight days.

NASA officials express confidence in Boeing and say the company has gone to extraordinary lengths to ensure that the mission will be successful. They are eager to have another spacecraft, in addition to the one SpaceX flies, that can ferry astronauts to the station. “I can say with confidence that the teams have absolutely done their due diligence,” James Free, NASA’s associate administrator, said at a briefing last week.

“I see a total focus on this mission and making it successful from the way that we always have done human spaceflight in my time at NASA,” said Steve Stich, who oversees NASA’s commercial crew program. “It’s really just taking it step by step, methodically getting to flight.”

Still, launching humans to orbit is extremely difficult and dangerous, especially on a new vehicle that has a history of trouble and has never flown people before.

“The first crewed flight of a new spacecraft is an absolutely critical milestone‚” Free said. “The lives of our crew members, Suni Williams and Butch Wilmore, are at stake. We don’t take that lightly at all. The most important thing we can do is protect those two people, as well as our crew currently on board the space station.”

Safety, he said, “has and always will be our primary core value at NASA.”

The mission comes at a crucial time for Boeing. After the panel blew off the Alaska Airlines flight, the company announced it would reshuffle its leadership and replace CEO David Calhoun . Calhoun took over the company after another disaster — the crashes of two 737 Max airplanes in 2018 and 2019 that killed 346 people.

Boeing’s commercial aviation division and its space program are managed separately. But both have faced similar quality-control issues.

After the failure of Starliner’s first test flight, managers at Boeing acknowledged they had cut short a test that would have caught the software problem . That caused NASA to offer an unusually harsh assessment of its partner, saying that “there were numerous instances where the Boeing software quality processes either should have or could have uncovered the defects.”

Mark Nappi, Boeing’s Starliner program manager, told reporters the past week that the problems with the company’s airline division had not added to pressures in the mission.

“We signed up to go do this, and we’re going to go do it and be successful at it,” he said. “So I don’t think of it in terms of what’s important for Boeing, as much as I think of it in terms of what’s important for this program, to follow through with the commitments that we made to our customer.”

Boeing’s woes have been amplified by the fact that SpaceX has been flying astronauts on its spacecraft for four years , even though the companies received their contracts at the same time and SpaceX was paid $2.6 billion, far less than Boeing . Since then, SpaceX has flown multiple missions for NASA, as well as taken private astronauts to the station and to orbit. It also has received an extension on its contract with NASA to fly astronaut missions.

Not only has Boeing yet to fly people on Starliner but the delays have meant about $1.4 billion in cost overruns. In an interview last year, John Shannon, vice president of Boeing Exploration Systems, said the future of Starliner was uncertain . “Probably the biggest challenge I have is defining how do I make this into a positive business case, given the market conditions as we see them right now,” he said then.

Asked about the future of Starliner recently, Nappi was noncommittal. He said the company would fulfill its contract with NASA, flying a handful of missions, which would take the company to the end of the decade.

“So we’ve got plenty of time to think about what’s after that,” he said. “And we will do that. But right now, the focus is on CFT”— or the crewed flight test.

In December 2019, Starliner launched successfully to orbit, but the spacecraft ran into problems shortly after it separated from the launch vehicle. Engineers on the ground were able to diagnose the problem — the onboard computer was 11 hours off, so the spacecraft had started executing commands for an entirely different part of the flight.

Engineers also soon discovered a second software problem, which could have caused the service module to crash into the crew capsule during separation. The issues were so severe that NASA officials said the spacecraft could have been lost because of either of them, threatening the lives of astronauts, had any been on board. The flight never reached the space station, and after it returned to Earth, Boeing said it would attempt a second flight test after reviewing 1 million lines of code in the spacecraft.

The second attempt to fly the capsule without anyone onboard came about a year and a half later, during the summer of 2021. By then, Boeing said it had fixed the software issues. But this time several of the valves in the spacecraft’s service module were corroded shut, and Boeing was unable to fix them while the craft was mounted on top of the rocket at the launchpad.

Taking it down led to another delay.

Boeing finally completed the uncrewed test flight in May 2022. But a little over a year later, as it was gearing up to fly astronauts, it announced the problem with the parachute system and the flammable tape.

Part of the purpose of the crewed test flight is to see how the vehicle reacts in the vacuum of space, and NASA officials said they anticipate that the astronauts and ground controllers will have to react to unexpected issues. That, they said, is how they make the vehicle safer and more reliable.

“Let me just remind everybody again: This is a new spacecraft. I’ll also remind you this is a test flight,” Free said. “We certainly have some unknowns in this mission, things we expect to learn. This being a test mission, we may encounter things we don’t expect. But our job now is to remain vigilant and keep looking for issues.”

Williams, who was assigned years ago to the commercial crew program and has been waiting to fly since then, said she’s convinced the teams are ready.

“We do anticipate everything’s going to go as planned,” she said in an interview. “But if it doesn’t, we’ll take a moment and analyze it and talk about it, and we’ll be okay. So our confidence in the mission is high.”

An earlier version of this story gave an incorrect year for when Boeing completed its uncrewed Starliner test flight. It was 2022. This version has been corrected.

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Division - Divide the Number Line with Remainder Worksheets Math Problems

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Division - Divide the Number Line with Remainder Worksheets Math Problems Empower your students with the essential skills needed to master division with our comprehensive Division Worksheets. Designed to provide structured practice and reinforcement of division concepts, these worksheets cover a wide range of topics, from basic division facts to long division and problem-solving. With engaging activities and diverse problem sets, students will develop fluency, accuracy, and confidence in their division abilities. Conceptual Understanding: Our worksheets begin with clear explanations and examples of division concepts, ensuring that students understand the principles behind division operations. From understanding division as sharing and repeated subtraction to learning division strategies, students build a solid foundation in division. Variety of Problems: With a diverse range of problems covering single-digit division, multi-digit division, long division, division with remainders, and word problems, our worksheets offer comprehensive coverage of division skills. Students encounter different problem types and scenarios, allowing for targeted practice and skill development. Step-by-Step Guidance: Each worksheet provides step-by-step guidance on how to solve division problems, including strategies for long division and problem-solving techniques. Students follow structured processes to divide numbers, check their solutions, and interpret division results, fostering independence and problem-solving skills. Interactive Activities: Engaging activities such as fill-in-the-blank exercises, matching problems, and real-world scenarios make learning division interactive and enjoyable. Students actively participate in the learning process, making connections between mathematical concepts and everyday situations. Assessment Tools: Answer keys are included for easy assessment and self-checking. Teachers and students can quickly verify the correctness of solutions, identify areas for improvement, and track progress. Aligned with Standards: Our worksheets are aligned with common core and state standards, ensuring that they complement existing curriculum frameworks and learning objectives. Worksheets are made in 8.5” x 11” Standard Letter Size. This resource is helpful in students’ assessment, Independent Studies, group activities, practice and homework. This product is available in PDF format and ready to print as well.

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