## Fractions in Year 1 (age 5–6)

In Year 1, your child will learn that fractions are about sharing. They will think about fractions as a part of one object or as part of a group of objects, focusing on halves and quarters.

## What your child will learn

Take a look at the National Curriculum expectations for fractions in Year 1 (age 5–6):

## Recognise a half as 1 of 2 equal parts of something

Your child will begin to think about halving as sharing equally into two parts or groups.

First, they will try halving a shape or object into two equal parts (for example, finding half of a pizza). Then, they will learn to find half of a group of objects (for example, finding that half of 4 pizzas is 2 pizzas). They will do this by sharing the total number of objects equally between two groups, one object at a time.

Your child will also learn to halve lengths – for example, finding half the length of an 8cm ribbon.

## Recognise a quarter as 1 of 4 equal parts of something

Your child will begin to understand finding a quarter as sharing into four equal parts or groups.

First, your child will learn to find a quarter of a shape or object by splitting it into four equal parts (for example, finding a quarter of a cake). They will find a quarter by finding half and then finding half again. They will also learn to find a quarter of a quantity of objects. They will do this by sharing the total quantity of objects equally between four groups, one object at a time.

Your child will learn to find a quarter of a length (for example, finding a quarter of the length of a ribbon that measures 8cm). They could do this by first halving 8cm to get 4cm, and then halving again to get 2cm.

## How to help at home

There are lots of everyday ways you can help your child to understand fractions. Here are just a few ideas.

## 1. Fractions and food

Food always provides a fun (and tasty) opportunity to explore fractions!

When cutting a sandwich, show your child how you can cut a whole sandwich into two equal parts, and explain that we call each part a half. Point out that we could put the two halves together again to make the whole. Try halving in different ways, such as cutting a whole sandwich into two equal rectangles or triangles.

Why not show your child that we can find the quarters of a sandwich by first cutting it in half and then cutting those pieces in half again? Ask your child to find one of the equal parts and explain that this is one quarter. Try finding quarters in different ways, such as cutting a whole sandwich into four equal rectangles or triangles.

## 2. Use toys

Learn about fractions using some of your child’s favourite toys – little figures, blocks, teddies, cars, you name it.

Gather together a collection of toys. Show your child how to halve quantities by asking them to split their collection of toys into two equal groups. So, if they have ten cars, they can split them into two groups of five.

You could also show them how to find one quarter of quantities by splitting their toys into four equal groups.

## 3. Drawing shapes

Grab some paper and colouring pencils. Draw a circle, square, rectangle, and equal-sided triangle and ask them to find and colour in one half. Next, draw another shape and ask your child to colour in one quarter.

## 4. Fractions all around us

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## Fractions Word Problems- Year 1- DIFFERENTIATED

Subject: Mathematics

Age range: 5-7

Resource type: Worksheet/Activity

Last updated

22 February 2018

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## cazlydiatam

These look fantasic just wish the money ones were editable as am in Aus so needed cents :(

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## 3littlekittens

Fantastic resource and saved me so much time. Thank you :-)

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I used this with my middle group in year 1 - they loved it, great pictures and ideal for what I needed - thank you! :)

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## Unit 5: Fractions

About this unit.

In this topic, we will explore fractions conceptually and add, subtract, multiply, and divide fractions.

## Fractions intro

- Intro to fractions (Opens a modal)
- More about fractions (Opens a modal)
- Cutting shapes into equal parts (Opens a modal)
- Identifying unit fractions word problem (Opens a modal)
- Cut shapes into equal parts 7 questions Practice
- Identify unit fractions 7 questions Practice

## What fractions mean

- Recognize fractions (Opens a modal)
- Worked example: Recognizing fractions (Opens a modal)
- Recognizing fractions greater than 1 (Opens a modal)
- Identifying numerators and denominators (Opens a modal)
- Recognize fractions 7 questions Practice
- Recognize fractions greater than 1 4 questions Practice
- Understand numerators and denominators 7 questions Practice

## Fractions on the number line

- Fractions on a number line (Opens a modal)
- Fractions on number line widget (Opens a modal)
- Fractions on the number line 7 questions Practice
- Unit fractions on the number line 4 questions Practice

## Fractions and whole numbers

- Representing 1 as a fraction (Opens a modal)
- Whole numbers as fractions (Opens a modal)
- Find 1 on the number line 7 questions Practice
- Write whole numbers as fractions 7 questions Practice

## Comparing fractions

- Comparing fractions with > and < symbols (Opens a modal)
- Comparing fractions with like numerators and denominators (Opens a modal)
- Comparing fractions (Opens a modal)
- Comparing fractions 2 (unlike denominators) (Opens a modal)
- Comparing and ordering fractions (Opens a modal)
- Ordering fractions (Opens a modal)
- Compare fractions with the same numerator or denominator 7 questions Practice
- Compare fractions with different numerators and denominators 7 questions Practice
- Order fractions 4 questions Practice

## Equivalent fractions 1

- Equivalent fractions with visuals (Opens a modal)
- Equivalent fraction models (Opens a modal)
- Visualizing equivalent fractions review (Opens a modal)
- Equivalent fraction models 7 questions Practice

## Equivalent fractions on the number line

- Equivalent fraction visually (Opens a modal)
- Creating equivalent fractions (Opens a modal)
- Equivalent fractions on the number line 4 questions Practice

## Equivalent fractions 2

- Equivalent fractions (Opens a modal)
- Visualizing equivalent fractions (Opens a modal)
- More on equivalent fractions (Opens a modal)
- Equivalent fractions review (Opens a modal)
- Equivalent fractions and different wholes (Opens a modal)
- Equivalent fractions (fraction models) 4 questions Practice
- Equivalent fractions 7 questions Practice
- Equivalent fractions 2 7 questions Practice
- Equivalent fractions and different wholes 7 questions Practice
- Simplify fractions 7 questions Practice

## Common denominators

- Finding common denominators (Opens a modal)
- Common denominators: 3/5 and 7/2 (Opens a modal)
- Common denominators: 1/4 and 5/6 (Opens a modal)
- Common denominators review (Opens a modal)
- Common denominators 4 questions Practice

## Decomposing fractions

- Decomposing a fraction visually (Opens a modal)
- Decomposing a mixed number (Opens a modal)
- Decomposing fractions review (Opens a modal)
- Decompose fractions 7 questions Practice

## Adding and subtracting fractions with like denominators

- Adding fractions with like denominators (Opens a modal)
- Subtracting fractions with like denominators (Opens a modal)
- Add fractions with common denominators 7 questions Practice
- Subtract fractions with common denominators 7 questions Practice

## Mixed numbers

- Writing mixed numbers as improper fractions (Opens a modal)
- Writing improper fractions as mixed numbers (Opens a modal)
- Comparing improper fractions and mixed numbers (Opens a modal)
- Mixed numbers and improper fractions review (Opens a modal)
- Mixed number or improper fraction on a number line (Opens a modal)
- Write mixed numbers and improper fractions 7 questions Practice
- Compare fractions and mixed numbers 4 questions Practice

## Adding and subtracting mixed numbers

- Intro to adding mixed numbers (Opens a modal)
- Intro to subtracting mixed numbers (Opens a modal)
- Add and subtract mixed numbers (no regrouping) 7 questions Practice
- Add and subtract mixed numbers (with regrouping) 4 questions Practice

## Visually adding and subtracting fractions with unlike denominators

- Visually adding fractions: 5/6+1/4 (Opens a modal)
- Visually subtracting fractions: 3/4-5/8 (Opens a modal)
- Visually add and subtract fractions 4 questions Practice

## Adding and subtracting fractions with unlike denominators

- Adding fractions with unlike denominators (Opens a modal)
- Subtracting fractions with unlike denominators (Opens a modal)
- Adding and subtracting 3 fractions (Opens a modal)
- Solving for the missing fraction (Opens a modal)
- Add fractions with unlike denominators 7 questions Practice
- Subtracting fractions with unlike denominators 7 questions Practice
- Add and subtract fractions 4 questions Practice

## Adding and subtracting mixed number with unlike denominators

- Adding mixed numbers: 19 3/18 + 18 2/3 (Opens a modal)
- Subtracting mixed numbers: 7 6/9 - 3 2/5 (Opens a modal)
- Adding mixed numbers with regrouping (Opens a modal)
- Subtracting mixed numbers with regrouping (unlike denominators) (Opens a modal)
- Add and subtract mixed numbers with unlike denominators (no regrouping) 4 questions Practice
- Add and subtract mixed numbers with unlike denominators (regrouping) 4 questions Practice

## Adding and subtracting fractions with unlike denominators word problems

- Adding fractions word problem: paint (Opens a modal)
- Subtracting fractions word problem: tomatoes (Opens a modal)
- Add and subtract fractions word problems 4 questions Practice

## Multiplying unit fractions and whole numbers

- Multiplying unit fractions and whole numbers (Opens a modal)
- Multiply unit fractions and whole numbers 7 questions Practice

## Multiplying whole numbers and fractions

- Multiplying fractions and whole numbers visually (Opens a modal)
- Equivalent fraction and whole number multiplication problems (Opens a modal)
- Multiply fractions and whole numbers 7 questions Practice
- Multiply fractions and whole numbers with fraction models 4 questions Practice
- Equivalent whole number and fraction multiplication expressions 7 questions Practice

## Multiplication as scaling

- Multiplication as scaling with fractions (Opens a modal)
- Fraction multiplication as scaling 7 questions Practice

## Multiplying fractions

- Intro to multiplying 2 fractions (Opens a modal)
- Multiplying 2 fractions: fraction model (Opens a modal)
- Multiplying 2 fractions: number line (Opens a modal)
- Multiplying 2 fractions: 5/6 x 2/3 (Opens a modal)
- Finding area with fractional sides 1 (Opens a modal)
- Finding area with fractional sides 2 (Opens a modal)
- Multiplying fractions review (Opens a modal)
- Multiplying fractions with visuals 4 questions Practice
- Multiplying fractions 7 questions Practice
- Area of rectangles with fraction side lengths 4 questions Practice

## Multiplying mixed numbers

- Multiplying mixed numbers (Opens a modal)
- Multiply mixed numbers 4 questions Practice

## Multiplying fractions word problems

- Multiplying fractions word problem: muffins (Opens a modal)
- Multiplying fractions word problem: laundry (Opens a modal)
- Multiplying fractions word problem: bike (Opens a modal)
- Multiplying fractions word problem: pumpkin pie (Opens a modal)
- Multiply fractions word problems 4 questions Practice

## Fractions as division

- Understanding fractions as division (Opens a modal)
- Creating a fraction through division (Opens a modal)
- Creating mixed numbers with fraction division (Opens a modal)
- Fractions as division 7 questions Practice
- Fractions as division word problems 4 questions Practice

## Dividing unit fractions by whole numbers

- No videos or articles available in this lesson
- Dividing unit fractions by whole numbers visually 4 questions Practice
- Dividing unit fractions by whole numbers 4 questions Practice

## Dividing whole numbers by unit fractions

- Dividing whole numbers by unit fractions visually 4 questions Practice
- Dividing whole numbers by unit fractions 4 questions Practice

## Dividing fractions by fractions

- Understanding division of fractions (Opens a modal)
- Dividing fractions: 2/5 ÷ 7/3 (Opens a modal)
- Dividing fractions: 3/5 ÷ 1/2 (Opens a modal)
- Dividing fractions review (Opens a modal)
- Dividing fractions 4 questions Practice
- Divide mixed numbers 4 questions Practice

## Dividing fractions word problems

- Dividing whole numbers & fractions: t-shirts (Opens a modal)
- Dividing fractions word problems 7 questions Practice

## Free Printable Fraction Word Problems Worksheets for 1st Year

Fraction Word Problems: Discover a collection of free printable worksheets for Year 1 students, focusing on math fraction word problems. Enhance learning and build a strong foundation in mathematics.

## Explore Fraction Word Problems Worksheets by Grades

- kindergarten

## Explore Other Subject Worksheets for year 1

- Social studies
- Social emotional
- Foreign language
- Reading & Writing

## Explore printable Fraction Word Problems worksheets for 1st Year

Fraction Word Problems worksheets for Year 1 are an essential tool for teachers looking to help their students develop a strong foundation in math. These worksheets provide a variety of engaging and challenging problems that require students to apply their understanding of fractions in real-life situations. By incorporating these worksheets into their lesson plans, teachers can ensure that their students are not only learning the necessary math skills but also developing critical thinking and problem-solving abilities. With a wide range of topics covered, from simple addition and subtraction of fractions to more complex problems involving mixed numbers and improper fractions, these Fraction Word Problems worksheets for Year 1 are an invaluable resource for any teacher looking to enhance their students' math education.

Quizizz is an excellent platform that offers a variety of educational resources, including Fraction Word Problems worksheets for Year 1, to support teachers in their quest to provide the best possible math education for their students. In addition to these worksheets, Quizizz also offers interactive quizzes, engaging games, and other learning tools that can be easily integrated into the classroom. By utilizing Quizizz, teachers can not only access high-quality Fraction Word Problems worksheets for Year 1 but also track their students' progress and identify areas where additional support may be needed. This comprehensive platform is an invaluable resource for teachers looking to create a dynamic and engaging learning environment that fosters a love of math and helps students excel in their Math Word Problems.

## Fraction Worksheets

Conversion :: Addition :: Subtraction :: Multiplication :: Division

## Conversions

Fractions - addition, fractions - subtraction, fractions - multiplication, fractions - division.

## Reasoning/Problem Solving Maths Worksheets for Year 1 (age 5-6)

Reasoning about number, sorting and counting.

To build on the work done in Early Years children need to sort groups in various ways before counting the objects within the groups.

Counting a variety of things.

Sorting and counting activities.

Sorting and counting activities (3)

How many flowers, balls etc can you see in the picture?

How many boats, trains, bikes and buses can you see in the picture?

Looking for similarities in groups of sorted objects.

Explaining the criteria for sorting into groups - there may be more than one possible answer.

Guess how many objects and then count to see how accurate you are!

Sorting a variety of fruit into groups.

Sorting a variety of objects into groups.

Explaining answers: counting forwards and backwards.

## Making number sentences and using number cards

Children will be asked to find different ways of making numbers using addition and subtraction. Number cards are an excellent resource in the classroom for this type of activity.

Use number cards to find different ways of making 3.

How many ways can you make 4 using the cards shown?

How many ways can you make 5 using the numbers and signs?

Use number cards to find different ways of making 6.

How many ways can you make 8 using the numbers on the cards?

How many ways can you add 2 numbers to make 7?

Use cards to make numbers.

Use cards to make numbers, including zero.

## Reasoning about money

Using coins is a great way to explore different ways of making amounts. The later pages on shopping require logical thinking and children should be able to explain verbally or on paper how they reached their answers.

How many ways can you pay 8p?

What coins could you use to buy the ragdoll?

What shapes can you buy for 10p? Costs according to the number of sides the shape has.

Solving 2-step problems with money up to 10p.

2-step money problems.

More shopping: 2-step problems with whole pounds.

Simple reasoning with small amounts of money.

Reasoning skills with small amounts of money.

## Reasoning about addition and subtraction

Plenty of opportunity to select numbers and add and subtract to reach totals. Finding all the possible combinations can be an important part of these activities.

A first look at word problems needing addition or subtraction to solve.

More problem solving where addition or subtraction is needed.

Sharks know that there's more than one way to catch a fish. And there's usually lots of ways to make a total by adding two numbers.

'How many ways?' investigation using 7, 8 and 9.

How many different answers can you get?

Three darts to score on the board. Good adding up revision.

A little age problem.

Make subtraction sentences using the numbers around the hexagon.

Put the numbers 1, 2, 3, 4 and 5 onto the grid so that both lines add up to the same total. How many ways?

Can you put the numbers in the circles so that each side of the triangle adds up to 9?

Pick three numbers from those shown and add them. How many different additions can you find?

Completing missing symbols in number sentences.

Completing number sentences using more and less than.

Completing simple number patterns.

## More reasoning activities

More activities to challenge children’s reasoning and logical thinking. Plenty of, 'How many ways ....' activities.

Find how many cubes can be held in one hand and record data collected in a table.

How many different ways can the ice lolly be coloured, using just two colours?

How many different ways can you put the cubes in the boxes?

How many rectangles can be made?

How many ways can the 9 tiles be put together to make a square?

How many ways can I join 4 triangles to make other shapes?

Make four 2-digit numbers from just two 1-digit numbers.

Put the numbers into the grid so that there is always a difference of more than one.

Find and describe routes to the bus stop.

How many numbers can you make with 6 beads on the abacus?

How many different routes are there from head to tail? Great logical thinking challenge.

Fun counting challenge which encourages logical thinking.

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## Fraction Word Problems (Difficult)

Here are some examples of more difficult fraction word problems. We will illustrate how block models (tape diagrams) can be used to help you to visualize the fraction word problems in terms of the information given and the data that needs to be found.

Related Pages Fraction Word Problems Singapore Math Lessons Fraction Problems Using Algebra Algebra Word Problems

Block modeling (also known as tape diagrams or bar models) are widely used in Singapore Math and the Common Core to help students visualize and understand math word problems.

Example: 2/9 of the people on a restaurant are adults. If there are 95 more children than adults, how many children are there in the restaurant?

Solution: Draw a diagram with 9 equal parts: 2 parts to represent the adults and 7 parts to represent the children.

5 units = 95 1 unit = 95 ÷ 5 = 19 7 units = 7 × 19 = 133

Answer: There are 133 children in the restaurant.

Example: Gary and Henry brought an equal amount of money for shopping. Gary spent $95 and Henry spent $350. After that Henry had 4/7 of what Gary had left. How much money did Gary have left after shopping?

350 – 95 = 255 3 units = 255 1 unit = 255 ÷ 3 = 85 7 units = 85 × 7 = 595

Answer: Gary has $595 after shopping.

Example: 1/9 of the shirts sold at Peter’s shop are striped. 5/8 of the remainder are printed. The rest of the shirts are plain colored shirts. If Peter’s shop has 81 plain colored shirts, how many more printed shirts than plain colored shirts does the shop have?

Solution: Draw a diagram with 9 parts. One part represents striped shirts. Out of the remaining 8 parts: 5 parts represent the printed shirts and 3 parts represent plain colored shirts.

3 units = 81 1 unit = 81 ÷ 3 = 27 Printed shirts have 2 parts more than plain shirts. 2 units = 27 × 2 = 54

Answer: Peter’s shop has 54 more printed colored shirts than plain shirts.

Solve a problem involving fractions of fractions and fractions of remaining parts

Example: 1/4 of my trail mix recipe is raisins and the rest is nuts. 3/5 of the nuts are peanuts and the rest are almonds. What fraction of my trail mix is almonds?

How to solve fraction word problem that involves addition, subtraction and multiplication using a tape diagram or block model

Example: Jenny’s mom says she has an hour before it’s bedtime. Jenny spends 3/5 of the hour texting a friend and 3/8 of the remaining time brushing her teeth and putting on her pajamas. She spends the rest of the time reading her book. How long did Jenny read?

How to solve a four step fraction word problem using tape diagrams?

Example: In an auditorium, 1/6 of the students are fifth graders, 1/3 are fourth graders, and 1/4 of the remaining students are second graders. If there are 96 students in the auditorium, how many second graders are there?

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## Fraction word problems

Read story, write fraction.

Students write fractions related to each story. The fractions may represent parts of a whole ("1/2 of the pizza had pineapple on it") or parts of a group ("1/3 of the students were boys"). Some problems include irrelevant data.

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History Daily

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

Posted: May 11, 2024 | Last updated: May 11, 2024

## Some ingenuity

Teacher Michelle Blouin Williams initiated a math competition with a bonus question tasking students to develop a new proof for the Pythagorean theorem using trigonometry, without anticipating that anyone would successfully tackle the challenge.

She said, “I was just looking for some ingenuity.”

## Williams' expectations

Calcea Johnson and Ne’Kiya Jackson exceeded Williams' expectations by successfully solving the challenge in 2023.

## St. Mary's Academy

These two teenagers, who were seniors at St. Mary's Academy in New Orleans, a renowned Catholic school for girls with impressive college acceptance and graduation rates, were featured on CBS News' "60 Minutes" to discuss their accomplishment.

## $500 reward

Initially enticed by the math competition's $500 reward, their determination to complete the task intensified as they delved into the complex bonus question.

Over a period of two months, these high school seniors dedicated themselves to completing their proof.

## Pages and pages

During an interview with "60 Minutes," CeCe Johnson, the mother of Calcea, said “It was pages and pages and pages of, like, over 20 or 30 pages for this one problem.”

## The garbage can

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.”

## Upon completion

Upon completion, the teachers at St. Mary's Academy acknowledged the significance of Calcea and Ne'Kiya's achievement and submitted their proof to the American Mathematical Society for recognition at a conference in March 2023, where the students presented their findings.

## The Pythagorean theorem

The Pythagorean theorem, at its core, states that knowing the lengths of two sides of a right triangle allows you to determine the length of the third side using the formula a² + b² = c².

While commonly attributed to the Greek mathematician Pythagoras, historical evidence suggests its existence in Babylon and Iron Age India. This theorem finds practical applications in construction, architecture, navigation, and surveying.

## A mathematical proof

A mathematical proof is a logical argument that demonstrates the truth of a mathematical theorem. American mathematician Daniel Kane likens proofs to essays, but rooted in mathematical concepts.

## Using trigonometry

As per the "60 Minutes" segment, “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.”

## Mathematician Elisha Loomis

Back in 1927, mathematician Elisha Loomis made a statement in his book "The Pythagorean Proposition." Loomis contended that a trigonometric proof of the theorem was impossible as it would create a circular argument.

## Stuart Anderson

Stuart Anderson, a retired mathematics professor from Texas A&M University-Commerce, mentioned to Scientific American, “A lot of the basic trig ‘identities’ are nothing more than Pythagoras’ theorem.”

## Trigonometric functions

Loomis contended that attempting to use trigonometric functions to prove the Pythagorean theorem would result in a circular reasoning loop, as the functions themselves are built upon the theorem. This, according to Loomis, would constitute a fundamental mathematical mistake.

## The law of sines

As reported by Scientific American, the teenagers challenged this notion during their presentation in 2023 and stated 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.”

## Final peer review

Calcea and Ne'Kiya are now part of a select few who have achieved a similar milestone, such as mathematician Jason Zimba, who developed a new proof in 2009. They submitted their proof for the final peer review earlier this year and are actively working on crafting additional proofs.

## In recognition

In recognition of their accomplishment, the teenagers were honored with the keys to the city of New Orleans and received commendations from the governor of Louisiana, along with other public acknowledgments.

## Widespread recognition

Despite the widespread recognition of their accomplishment which “blew up,” as Ne'Kiya expressed it, the two students maintain their humility and even chuckled at being labeled as geniuses.

## Surprised and skeptical

Upon the revelation of their achievement, there were individuals who appeared surprised and skeptical, dismissing the news as untrue, as mentioned by St. Mary's president Pamela Rogers during the interview.

## African Americans

“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.”

## The significant reaction

When questioned by interviewer Bill Whitaker about their thoughts on the significant reaction, 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.”

## Great mathematical achievement

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

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## A Plan to Remake the Middle East

While talks for a cease-fire between israel and hamas continue, another set of negotiations is happening behind the scenes..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From New York Times, I’m Michael Barbaro. This is The Daily.

[MUSIC CONTINUES]

Today, if and when Israel and Hamas reach a deal for a ceasefire fire, the United States will immediately turn to a different set of negotiations over a grand diplomatic bargain that it believes could rebuild Gaza and remake the Middle East. My colleague Michael Crowley has been reporting on that plan and explains why those involved in it believe they have so little time left to get it done.

It’s Wednesday, May 8.

Michael, I want to start with what feels like a pretty dizzying set of developments in this conflict over the past few days. Just walk us through them?

Well, over the weekend, there was an intense round of negotiations in an effort, backed by the United States, to reach a ceasefire in the Gaza war.

The latest ceasefire proposal would reportedly see as many as 33 Israeli hostages released in exchange for potentially hundreds of Palestinian prisoners.

US officials were very eager to get this deal.

Pressure for a ceasefire has been building ahead of a threatened Israeli assault on Rafah.

Because Israel has been threatening a military offensive in the Southern Palestinian city of Rafah, where a huge number of people are crowded.

Fleeing the violence to the North. And now they’re packed into Rafah. Exposed and vulnerable, they need to be protected.

And the US says it would be a humanitarian catastrophe on top of the emergency that’s already underway.

Breaking news this hour — very important breaking news. An official Hamas source has told The BBC that it does accept a proposal for a ceasefire deal in Gaza.

And for a few hours on Monday, it looked like there might have been a major breakthrough when Hamas put out a statement saying that it had accepted a negotiating proposal.

Israeli Prime Minister Benjamin Netanyahu says the ceasefire proposal does not meet his country’s requirements. But Netanyahu says he will send a delegation of mediators to continue those talks. Now, the terms —

But those hopes were dashed pretty quickly when the Israelis took a look at what Hamas was saying and said that it was not a proposal that they had agreed to. It had been modified.

And overnight —

Israeli troops stormed into Rafah. Video showing tanks crashing over a sign at the entrance of the city.

— the Israelis launched a partial invasion of Rafah.

It says Hamas used the area to launch a deadly attack on Israeli troops over the weekend.

And they have now secured a border crossing at the Southern end of Gaza and are conducting targeted strikes. This is not yet the full scale invasion that President Biden has adamantly warned Israel against undertaking, but it is an escalation by Israel.

So while all that drama might suggest that these talks are in big trouble, these talks are very much still alive and ongoing and there is still a possibility of a ceasefire deal.

And the reason that’s so important is not just to stop the fighting in Gaza and relieve the suffering there, but a ceasefire also opens the door to a grand diplomatic bargain, one that involves Israel and its Arab neighbors and the Palestinians, and would have very far-reaching implications.

And what is that grand bargain. Describe what you’re talking about?

Well, it’s incredibly ambitious. It would reshape Israel’s relationship with its Arab neighbors, principally Saudi Arabia. But it’s important to understand that this is a vision that has actually been around since well before October 7. This was a diplomatic project that President Biden had been investing in and negotiating actually in a very real and tangible way long before the Hamas attacks and the Gaza war.

And President Biden was looking to build on something that President Trump had done, which was a series of agreements that the Trump administration struck in which Israel and some of its Arab neighbors agreed to have normal diplomatic relations for the first time.

Right, they’re called the Abraham Accords.

That’s right. And, you know, Biden doesn’t like a lot of things, most things that Trump did. But he actually likes this, because the idea is that they contribute to stability and economic integration in the Middle East, the US likes Israel having friends and likes having a tight-knit alliance against Iran.

President Biden agrees with the Saudis and with the Israelis, that Iran is really the top threat to everybody here. So, how can you build on this? How can you expand it? Well, the next and biggest step would be normalizing relations between Israel and Saudi Arabia.

And the Saudis have made clear that they want to do this and that they’re ready to do this. They weren’t ready to do it in the Trump years. But Mohammed bin Salman, the Crown Prince of Saudi Arabia, has made clear he wants to do it now.

So this kind of triangular deal began to take shape before October 7, in which the US, Israel, and Saudi Arabia would enter this three way agreement in which everyone would get something that they wanted.

And just walk through what each side gets in this pre-October 7th version of these negotiations?

So for Israel, you get normalized ties with its most important Arab neighbor and really the country that sets the tone for the whole Muslim world, which is Saudi Arabia of course. It makes Israel feel safer and more secure. Again, it helps to build this alliance against Iran, which Israel considers its greatest threat, and it comes with benefits like economic ties and travel and tourism. And Prime Minister Benjamin Netanyahu has been very open, at least before October 7th, that this was his highest diplomatic and foreign policy priority.

For the Saudis, the rationale is similar when it comes to Israel. They think that it will bring stability. They like having a more explicitly close ally against Iran. There are economic and cultural benefits. Saudi Arabia is opening itself up in general, encouraging more tourism.

But I think that what’s most important to the Crown Prince, Mohammed bin Salman, is what he can get from the United States. And what he has been asking for are a couple of essential things. One is a security agreement whose details have always been a little bit vague, but I think essentially come down to reliable arms supplies from the United States that are not going to be cut off or paused on a whim, as he felt happened when President Biden stopped arms deliveries in 2021 because of how Saudi was conducting its war in Yemen. The Saudis were furious about that.

Saudi Arabia also wants to start a domestic nuclear power program. They are planning for a very long-term future, possibly a post-oil future. And they need help getting a nuclear program off the ground.

And they want that from the US?

And they want that from the US.

Now, those are big asks from the us. But from the perspective of President Biden, there are some really enticing things about this possible agreement. One is that it will hopefully produce more stability in the region. Again, the US likes having a tight-knit alliance against Iran.

The US also wants to have a strong relationship with Saudi Arabia. You know, despite the anger at Mohammed bin Salman over the murder of the Saudi dissident Jamal Khashoggi, the Biden administration recognizes that given the Saudis control over global oil production and their strategic importance in the Middle East, they need to have a good relationship with them. And the administration has been worried about the influence of China in the region and with the Saudis in particular.

So this is an opportunity for the US to draw the Saudis closer. Whatever our moral qualms might be about bin Salman and the Saudi government, this is an opportunity to bring the Saudis closer, which is something the Biden administration sees as a strategic benefit.

All three of these countries — big, disparate countries that normally don’t see eye-to-eye, this was a win-win-win on a military, economic, and strategic front.

That’s right. But there was one important actor in the region that did not see itself as winning, and that was the Palestinians.

[MUSIC PLAYING]

First, it’s important to understand that the Palestinians have always expected that the Arab countries in the Middle East would insist that Israel recognize a Palestinian state before those countries were willing to essentially make total peace and have normal relations with Israel.

So when the Abraham Accords happened in the Trump administration, the Palestinians felt like they’d been thrown under the bus because the Abraham Accords gave them virtually nothing. But the Palestinians did still hold out hope that Saudi Arabia would be their savior. And for years, Saudi Arabia has said that Israel must give the Palestinians a state if there’s going to be a normal relationship between Israel and Saudi Arabia.

Now the Palestinians see the Saudis in discussions with the US and Israel about a normalization agreement, and there appears to be very little on offer for the Palestinians. And they are feeling like they’re going to be left out in the cold here.

Right. And in the minds of the Palestinians, having already been essentially sold out by all their other Arab neighbors, the prospect that Saudi Arabia, of all countries, the most important Muslim Arab country in the region, would sell them out, had to be extremely painful.

It was a nightmare scenario for them. And in the minds of many analysts and US officials, this was a factor, one of many, in Hamas’s decision to stage the October 7th attacks.

Hamas, like other Palestinian leaders, was seeing the prospect that the Middle East was moving on and essentially, in their view, giving up on the Palestinian cause, and that Israel would be able to have friendly, normal relations with Arab countries around the region, and that it could continue with hardline policies toward the Palestinians and a refusal, as Prime Minister Benjamin Netanyahu has said publicly, to accept a Palestinian state.

Right. So Michael, once Hamas carries out the October 7th attacks in an effort to destroy a status quo that it thinks is leaving them less and less relevant, more and more hopeless, including potentially this prospect that Saudi Arabia is going to normalize relations with Israel, what happens to these pre-October 7th negotiations between the US, Saudi Arabia, and Israel?

Well, I think there was a snap assumption that these talks were dead and buried. That they couldn’t possibly survive a cataclysm like this.

But then something surprising happened. It became clear that all the parties were still determined to pull-off the normalization.

And most surprisingly of all, perhaps, was the continued eagerness of Saudi Arabia, which publicly was professing outrage over the Israeli response to the Hamas attacks, but privately was still very much engaged in these conversations and trying to move them forward.

And in fact, what has happened is that the scope of this effort has grown substantially. October 7th didn’t kill these talks. It actually made them bigger, more complicated, and some people would argue, more important than ever.

We’ll be right back.

Michael, walk us through what exactly happens to these three-way negotiations after October 7th that ends up making them, as you just said, more complicated and more important than ever?

Well, it’s more important than ever because of the incredible need in Gaza. And it’s going to take a deal like this and the approval of Saudi Arabia to unlock the kind of massive reconstruction project required to essentially rebuild Gaza from the rubble. Saudi Arabia and its Arab friends are also going to be instrumental in figuring out how Gaza is governed, and they might even provide troops to help secure it. None of those things are going to happen without a deal like this.

Fascinating.

But this is all much more complicated now because the price for a deal like this has gone up.

And by price, you mean?

What Israel would have to give up. [MUSIC PLAYING]

From Saudi Arabia’s perspective, you have an Arab population that is furious at Israel. It now feels like a really hard time to do a normalization deal with the Israelis. It was never going to be easy, but this is about as bad a time to do it as there has been in a generation at least. And I think that President Biden and the people around him understand that the status quo between Israel and the Palestinians is intolerable and it is going to lead to chaos and violence indefinitely.

So now you have two of the three parties to this agreement, the Saudis and the Americans, basically asking a new price after October 7th, and saying to the Israelis, if we’re going to do this deal, it has to not only do something for the Palestinians, it has to do something really big. You have to commit to the creation of a Palestinian state. Now, I’ll be specific and say that what you hear the Secretary of State, Antony Blinken, say is that the agreement has to include an irreversible time-bound path to a Palestinian state.

We don’t know exactly what that looks like, but it’s some kind of a firm commitment, the likes of which the world and certainly the Israelis have not made before.

Something that was very much not present in the pre-October 7th vision of this negotiation. So much so that, as we just talked about, the Palestinians were left feeling completely out in the cold and furious at it.

That’s right. There was no sign that people were thinking that ambitiously about the Palestinians in this deal before October 7th. And the Palestinians certainly felt like they weren’t going to get much out of it. And that has completely changed now.

So, Michael, once this big new dimension after October 7th, which is the insistence by Saudi Arabia and the US that there be a Palestinian state or a path to a Palestinian state, what is the reaction specifically from Israel, which is, of course, the third major party to this entire conversation?

Well, Israel, or at least its political leadership, hates it. You know, this is just an extremely tough sell in Israel. It would have been a tough sell before October 7th. It’s even harder now.

Prime Minister Benjamin Netanyahu is completely unrepentantly open in saying that there’s not going to be a Palestinian state on his watch. He won’t accept it. He says that it’s a strategic risk to his country. He says that it would, in effect, reward Hamas.

His argument is that terrorism has forced a conversation about statehood onto the table that wasn’t there before October 7th. Sure, it’s always in the background. It’s a perennial issue in global affairs, but it was not something certainly that the US and Israel’s Arab neighbors were actively pushing. Netanyahu also has — you know, he governs with the support of very right-wing members of a political coalition that he has cobbled together. And that coalition is quite likely to fall apart if he does embrace a Palestinian state or a path to a Palestinian state.

Now, he might be able to cobble together some sort of alternative, but it creates a political crisis for him.

And finally, you know, I think in any conversation about Israel, it’s worth bearing in mind something you hear from senior US officials these days, which is that although there is often finger pointing at Netanyahu and a desire to blame Netanyahu as this obstructionist who won’t agree to deals, what they say is Netanyahu is largely reflecting his population and the political establishment of his country, not just the right-wingers in his coalition who are clearly extremist.

But actually the prevailing views of the Israeli public. And the Israeli public and their political leaders across the spectrum right now with few exceptions, are not interested in talking about a Palestinian state when there are still dozens and dozens of Israeli hostages in tunnels beneath Gaza.

So it very much looks like this giant agreement that once seemed doable before October 7th might be more important to everyone involved than ever, given that it’s a plan for rebuilding Gaza and potentially preventing future October 7th’s from happening, but because of this higher price that Israel would have to pay, which is the acceptance of a Palestinian state, it seems from everything you’re saying, that this is more and more out of reach than ever before and hard to imagine happening in the immediate future. So if the people negotiating it are being honest, Michael, are they ready to acknowledge that it doesn’t look like this is going to happen?

Well, not quite yet. As time goes by, they certainly say it’s getting harder and harder, but they’re still trying, and they still think there’s a chance. But both the Saudis and the Biden administration understand that there’s very little time left to do this.

Well, what do you mean there’s very little time left? It would seem like time might benefit this negotiation in that it might give Israel distance from October 7th to think potentially differently about a Palestinian state?

Potentially. But Saudi Arabia wants to get this deal done in the Biden administration because Mohammed bin Salman has concluded this has to be done under a Democratic president.

Because Democrats in Congress are going to be very reluctant to approve a security agreement between the United States and Saudi Arabia.

It’s important to understand that if there is a security agreement, that’s something Congress is going to have to approve. And you’re just not going to get enough Democrats in Congress to support a deal with Saudi Arabia, who a lot of Democrats don’t like to begin with, because they see them as human rights abusers.

But if a Democratic president is asking them to do it, they’re much more likely to go along.

Right. So Saudi Arabia fears that if Biden loses and Trump is president, that those same Democrats would balk at this deal in a way that they wouldn’t if it were being negotiated under President Biden?

Exactly. Now, from President Biden’s perspective, politically, think about a president who’s running for re-election, who is presiding right now over chaos in the Middle East, who doesn’t seem to have good answers for the Israeli-Palestinian question, this is an opportunity for President Biden to deliver what could be at least what he would present as a diplomatic masterstroke that does multiple things at once, including creating a new pathway for Israel and the Palestinians to coexist, to break through the logjam, even as he is also improving Israel’s relations with Saudi Arabia.

So Biden and the Crown Prince hope that they can somehow persuade Bibi Netanyahu that in spite of all the reasons that he thinks this is a terrible idea, that this is a bet worth taking on Israel’s and the region’s long-term security and future?

That’s right. Now, no one has explained very clearly exactly how this is going to work, and it’s probably going to require artful diplomacy, possibly even a scenario where the Israelis would agree to something that maybe means one thing to them and means something else to other people. But Biden officials refuse to say that it’s hopeless and they refuse to essentially take Netanyahu’s preliminary no’s for an answer. And they still see some way that they can thread this incredibly narrow needle.

Michael, I’m curious about a constituency that we haven’t been talking about because they’re not at the table in these discussions that we are talking about here. And that would be Hamas. How does Hamas feel about the prospect of such a deal like this ever taking shape. Do they see it as any kind of a victory and vindication for what they did on October 7th?

So it’s hard to know exactly what Hamas’s leadership is thinking. I think they can feel two things. I think they can feel on the one hand, that they have established themselves as the champions of the Palestinian people who struck a blow against Israel and against a diplomatic process that was potentially going to leave the Palestinians out in the cold.

At the same time, Hamas has no interest in the kind of two-state solution that the US is trying to promote. They think Israel should be destroyed. They think the Palestinian state should cover the entire geography of what is now Israel, and they want to lead a state like that. And that’s not something that the US, Saudi Arabia, or anyone else is going to tolerate.

So what Hamas wants is to fight, to be the leader of the Palestinian people, and to destroy Israel. And they’re not interested in any sort of a peace process or statehood process.

It seems very clear from everything you’ve said here that neither Israel nor Hamas is ready to have the conversation about a grand bargain diplomatic program. And I wonder if that inevitably has any bearing on the ceasefire negotiations that are going on right now between the two of them that are supposed to bring this conflict to some sort of an end, even if it’s just temporary?

Because if, as you said, Michael, a ceasefire opens the door to this larger diplomatic solution, and these two players don’t necessarily want that larger diplomatic solution, doesn’t that inevitably impact their enthusiasm for even reaching a ceasefire?

Well, it certainly doesn’t help. You know, this is such a hellish problem. And of course, you first have the question of whether Israel and Hamas can make a deal on these immediate issues, including the hostages, Palestinian prisoners, and what the Israeli military is going to do, how long a ceasefire might last.

But on top of that, you have these much bigger diplomatic questions that are looming over them. And it’s not clear that either side is ready to turn and face those bigger questions.

So while for the Biden administration and for Saudi Arabia, this is a way out of this crisis, these larger diplomatic solutions, it’s not clear that it’s a conversation that the two parties that are actually at war here are prepared to start having.

Well, Michael, thank you very much. We appreciate it.

On Tuesday afternoon, under intense pressure from the US, delegations from Israel and Hamas arrived in Cairo to resume negotiations over a potential ceasefire. But in a statement, Israel’s Prime Minister Benjamin Netanyahu made clear that even with the talks underway, his government would, quote, “continue to wage war against Hamas.”

Here’s what else you need to know today. In a dramatic day of testimony, Stormy Daniels offered explicit details about an alleged sexual encounter with Donald Trump that ultimately led to the hush money payment at the center of his trial. Daniels testified that Trump answered the door in pajamas, that he told her not to worry that he was married, and that he did not use a condom when they had sex.

That prompted lawyers for Trump to seek a mistrial based on what they called prejudicial testimony. But the judge in the case rejected that request. And,

We’ve seen a ferocious surge of anti-Semitism in America and around the world.

In a speech on Tuesday honoring victims of the Holocaust, President Biden condemned what he said was the alarming rise of anti-Semitism in the United States after the October 7th attacks on Israel. And he expressed worry that too many Americans were already forgetting the horrors of that attack.

The Jewish community, I want you to know I see your fear, your hurt, and your pain. Let me reassure you, as your president, you’re not alone. You belong. You always have and you always will.

Today’s episode was produced by Nina Feldman, Clare Toeniskoetter, and Rikki Novetsky. It was edited by Liz O. Baylen, contains original music by Marion Lozano, Elisheba Ittoop, and Dan Powell, and was engineered by Alyssa Moxley. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for The Daily. I’m Michael Barbaro. See you tomorrow.

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Hosted by Michael Barbaro

Featuring Michael Crowley

Produced by Nina Feldman , Clare Toeniskoetter and Rikki Novetsky

Edited by Liz O. Baylen

Original music by Marion Lozano , Elisheba Ittoop and Dan Powell

Engineered by Alyssa Moxley

## Listen and follow The Daily Apple Podcasts | Spotify | Amazon Music | YouTube

If and when Israel and Hamas reach a deal for a cease-fire, the United States will immediately turn to a different set of negotiations over a grand diplomatic bargain that it believes could rebuild Gaza and remake the Middle East.

Michael Crowley, who covers the State Department and U.S. foreign policy for The Times, explains why those involved in this plan believe they have so little time left to get it done.

## On today’s episode

Michael Crowley , a reporter covering the State Department and U.S. foreign policy for The New York Times.

## Background reading :

Talks on a cease-fire in the Gaza war are once again at an uncertain stage .

Here’s how the push for a deal between Israel and Saudi Arabia looked before Oct. 7 .

From early in the war, President Biden has said that a lasting resolution requires a “real” Palestinian state .

Here’s what Israeli officials are discussing about postwar Gaza.

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Dan Farrell, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Renan Borelli, Maddy Masiello, Isabella Anderson and Nina Lassam.

Michael Crowley covers the State Department and U.S. foreign policy for The Times. He has reported from nearly three dozen countries and often travels with the secretary of state. More about Michael Crowley

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Hosted by Michael Barbaro. Featuring Michael Crowley. Produced by Nina Feldman , Clare Toeniskoetter and Rikki Novetsky. Edited by Liz O. Baylen. Original music by Marion Lozano , Elisheba Ittoop ...