multiplying fractions problem solving worksheet

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Multiplying Fractions Worksheets

Welcome to our Multiplying Fractions Worksheets page. Here you will find a range of free printable sheets and support to help your child learn to multiply fractions by integers or other fractions.

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Multiplying Fractions Online Quiz

We have a calculator to help you to multiply two fractions together.

You can multiply mixed numbers or multiply a fraction by an integer too.

All the working out is shown step-by-step so you can see how it is worked out.

• Multiply Fractions Calculator

How to Multiply Fractions Support

This support page will help you find out all about how to multiply fractions together.

There are several worked examples and a printable support sheet.

Formula for multiplying two fractions

\[{a \over b} \times {c \over d} = {ac \over bd} \]

• How do you Multiply Fractions Support

About our Multiplying Fractions Worksheets

Here you will find a selection of Fraction worksheets designed to help your child understand how to multiply a fraction by an integer or another fraction.

The sheets are carefully graded so that the easiest sheets come first, and the most difficult sheet is the last one.

The sheets have been split into two sections:

• The first section contains sheets multiplying fractions by integers
• The second section contains sheets multiplying fractions by other fractions.

Before your child tackles multiplying fractions, they should be confident with converting improper fractions to mixed numbers, and also using simplest form.

Using these sheets will help your child to:

• multiply a fraction by a whole number;
• multiply 2 fractions together;
• apply their understanding of simplest form;

Want to test yourself to see how well you have understood this skill?

• Try our NEW quick quiz at the bottom of this page.

Multiplying Fractions by an Integer Worksheets

• Multiply fractions by integer 1
• Sheet 1 Answers
• PDF version
• Multiply fractions by integer 2
• Sheet 2 Answers
• Multiply Fractions 1
• Multiply Fractions 2
• Multiplying Fractions 3
• Sheet 3 Answers
• Multiply Fractions 4
• Sheet 4 Answers
• Multiply Fractions 5
• Sheet 5 Answers

Multiplying Fractions Walkthrough Video

This short video walkthrough shows several problems from our Multiply Fractions Worksheet 2 being solved and has been produced by the West Explains Best math channel.

If you would like some support in solving the problems on these sheets, check out the video!

More Recommended Math Resources

Take a look at some more of our worksheets and resources similar to these.

Multiplying and Dividing Fractions Support Pages

Take a look at some of our other multiplying and dividing fractions support pages if you need more help with multiplying or dividing fractions.

• How to Multiply Mixed Fractions support page
• How to Divide Fractions
• How to Divide Mixed Numbers

Multiplying and Dividing Fractions Worksheets

We have more worksheets to help your child learn and practice multiplying and dividing fractions.

All the sheets are carefully graded with different levels of support.

Take a look at our full range below.

• Multiplying Mixed Fractions Worksheets
• Dividing Fractions Worksheets
• Dividing Fractions by Whole Numbers
• Dividing Mixed Fractions Worksheets
• Multiplying and Dividing Fractions (Randomly Generated)
• Add Subtract Multiply Divide Fractions (Randomly Generated)

How to Convert Improper Fractions

This support page shows how to convert improper fractions to mixed numbers, and how to convert mixed numbers to improper fractions.

You will also find printable support sheets, and several practice math worksheets for this learning fractions skill.

• Convert Improper Fractions Support

Simplifying Fractions

This is also sometimes called reducing fractions to their simplest form.

This involves dividing both the numerator and denominator by a common factor to reduce the fraction to the equivalent fraction with the smallest possible numerator and denominator.

The printable fraction page below contains more support, examples and practice about simplifying fractions.

• How to Simplify Fractions support page
• Simplify Fractions Practice Zone

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This quick quiz tests your understanding and skill at multiplying fractions.

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Fraction Multiplication Word Problems Worksheets

Our printable worksheets on multiplying fractions word problems task grade 4 through grade 7 students with reading and solving realistic scenarios by performing fraction multiplication. The problems feature both common and uncommon denominators, so the budding problem-solving stars must follow the correct procedure to obtain the products. Equipped with answer keys, these pdf resources are available in customary and metric units. Try some multiplying fractions word problems worksheets for free!

Multiplying Fractions by Whole Numbers Word Problems

Impel the eager beavers in 4th grade and 5th grade to multiply fractions by whole numbers through word problems. Kids are required to multiply proper fractions and improper fractions by whole numbers correctly.

• Download the set

Multiplying Fractions by Cross-Cancelling Word Problems

Say a whole-hearted yes to our no-prep printable worksheets on fraction multiplication that have you fully covered! Interpret the word problems and multiply fractions with a special emphasis on cross cancelling.

Multiplying Mixed Numbers Word Problems

Multiplying mixed numbers may initially feel incomprehensible, but practice makes the pain melt away. Let grade 6 and grade 7 kids convert the mixed numbers into fractions and work out the products.

Multiplying Mixed Numbers and Fractions Word Problems

Let not the challenges of obtaining solutions for problems befall you! Our pdf worksheets on multiplying fractions word problems, equipped with answer key for a quick self-validation, help ease into the process.

Themed Fraction Multiplication Word Problems

Revive 5th grade, 6th grade, and 7th grade students' problem-solving fortunes with our themed word problems, featuring a good mix of like fractions, unlike fractions, whole numbers, and mixed numbers!

Related Worksheets

» Multiplying Fractions on a Number Line

» Multiplying Fractions by Whole Numbers

» Multiplying Fractions with Cross Cancelling

» Multiplying Mixed Numbers

» Fraction Word Problems

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Multiplying Fractions

Students will find the products of fraction pairs in these worksheets. This page is divided into four subheaders: multiplying fractions, multiplying fractions by whole numbers, multiplying mixed numbers, and mixed skills.

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Multiplying Fractions by Whole Numbers

Mixed skills: multiplying fractions, whole numbers, and mixed numbers, multiplying mixed numbers.

Practice dividing fractions and mixed numbers with these printable pages. Many worksheets include illustrations and models, as well as word problems.

Worksheets for teaching basic fraction recognition skills and fraction concepts, as well as operations with fractions.

Add fractions with same and different denominators; Also add mixed numbers

You may also want to check out these worksheets on reciprocal fractions.

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Multiplying Fractions Worksheets

When multiplying fractions, students need to do cross cancellations, if possible. Then they can give the answer in mixed or proper fractions.

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• 6th Grade Math
• Multiplying fractions

6th Grade multiplying fractions worksheets with answers

Hello! Are you ready to engage in another enjoyable math adventure? Then join us today, and let’s discuss multiplying fractions skills. In this article, we’ll share an outstanding collection of 6th Grade multiplying fractions worksheets with answers covering fun and engaging activities and exercises for multiplying fractions.

Learn how to multiply fractions like a pro in 6th Grade with engaging printable worksheets

In 6th grade, understanding and multiplying fractions is a crucial skill that sets the foundation for advanced math concepts. But fear not because we have the solution to make learning fractions a breeze. Our aim for introducing Mathskills4kids’ engaging and printable worksheets is to help your child in 6 th Grade learn how to multiply fractions like a pro .

These worksheets are carefully crafted to ensure your child will grasp multiplying fractions and find joy. With colorful visuals, interactive exercises, and step-by-step explanations, our worksheets provide a hands-on learning experience that will keep your child engaged and motivated.

Whether your child is a visual learner or prefers a more hands-on approach, our worksheets cater to all learning styles. So, say goodbye to those confusing fractions and hello to confidence in math. Get ready to multiply fractions easily with our captivating printable worksheets for 6th-grade students!

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Start practice on Sixth Grade here

Why 6th-grade students need to learn how to multiply fractions.

Do you know why 6th-grade students need to learn how to multiply fractions ? Maybe no. This is because, aside from being an essential skill in 6 th Grade, the fractions multiplication concept helps students understand the concept of fractions as parts of a whole and how to combine fractions with different denominators.

Multiplying fractions also prepare 6 th graders for more advanced topics in math, such as ratios, proportions, decimals, and algebra. By learning how to multiply fractions, 6th-grade students can solve real-world problems involving fractions, such as finding the area of a rectangle, scaling a recipe, or dividing a pizza.

Benefits of using Mathskills4kids’ printable worksheets for fractions multiplication practice

Mathskills4kids’ Printable worksheets are among the best ways to help 6th-grade students practice fractions multiplication . Printable worksheets offer many benefits , such as:

• They provide a variety of problems with different levels of difficulty and formats, such as word problems, equations, and diagrams.
• They allow students to work at their own pace and check their answers with the answer key.
• They can be used individually or in groups in class or at home.
• They can be customized to suit the needs and interests of each student by choosing the topics, themes, and number of problems.
• They provide an answer sheet guide for quick and easy corrections.
• They can be used to review, reinforce, or extend the concepts taught in the lesson.

How to effectively use Mathskills4kids’ printable worksheets for multiplying fractions

To make the most of Mathskills4kids’ printable worksheets for multiplying fractions , here are some tips:

• Before giving the worksheets to the students, review the steps and rules for multiplying fractions, such as finding the common denominator, simplifying the fractions, and cross-canceling.
• Explain the purpose and objectives of each worksheet, and give clear instructions on how to complete it.
• Monitor the student's progress and provide feedback and guidance as needed.
• Encourage the students to show their work and explain their reasoning for each problem.
• Review the worksheets with the whole class and discuss the solutions and strategies used by the students.

Tips for teaching multiplication of fractions

Teaching multiplication of fractions can be challenging but also rewarding. Here are some tips to make it easier and more effective:

• Start with simple fractions with common denominators that can be easily multiplied mentally or with a calculator.
• Use concrete examples and contexts that relate to the students' lives and interests, such as sports, music, or food.
• Use multiple representations and methods to show how to multiply fractions, such as pictures, diagrams, number lines, area models, or algorithms.
• Connect multiplying fractions to other math concepts and operations that the students already know, such as adding fractions, dividing fractions, or multiplying whole numbers.
• Scaffold the learning process by gradually increasing the complexity and variety of the problems and fractions.

Common mistakes to avoid when multiplying fractions

When multiplying fractions, 6th-grade students may make mistakes that can affect their accuracy and understanding. Some of the common mistakes to avoid when multiplying fractions are:

• Forgetting to find the common denominator before multiplying fractions with different denominators.
• Forgetting to simplify or reduce the fractions before or after multiplying them.
• Forgetting to cross-cancel or cross-multiply when multiplying fractions in their lowest terms.
• Confusing the order of operations when multiplying fractions with mixed numbers or other operations.
• Confusing multiplying fractions with adding or subtracting fractions.

Teachers should emphasize the importance of following the steps and rules for multiplying fractions correctly to avoid these mistakes. They should also model how to check the answers for reasonableness and accuracy.

Multiply fractions: Meaning and example

Fractions are parts of a whole, and multiplying them means finding how many parts of one fraction are in another. For example, if you have 2/3 of a pizza and want to share it with 4 friends, how much pizza does each friend get? To find out, multiply 2/3 by 1/4, i.e., the fraction representing one friend's share.

To multiply fractions, you need to follow these steps:

• Multiply the numerators (the top numbers) of both fractions. In our example, that would be 2 x 1 = 2.
• Multiply the denominators (the bottom numbers) of both fractions. In our example, that would be 3 x 4 = 12.
• Simplify the resulting fraction by dividing the numerator and the denominator by their greatest common factor (GCF). The GCF is the largest number that can divide both numbers evenly. In our example, the GCF of 2 and 12 is 2, so we divide both by 2 and get 1/6.

The final answer is 1/6, which means each friend gets one-sixth of the pizza. You can check your answer by adding all the fractions and seeing if they equal the original fraction. In our example, if we add up 1/6 four times, we get 4/6, equivalent to 2/3.

Engaging activities and exercises for multiplying fractions

Besides Mathskills4kids’ 6th Grade multiplying fractions worksheets with answers , there are many other fun and engaging activities and exercises that can help 6th-grade students practice multiplying fractions, such as:

• Playing games that involve multiplying fractions, such as bingo, dominoes, or card games.
• Using manipulatives representing fractions, such as fraction strips, fraction circles, or fraction bars.
• Creating posters or presentations illustrating fractions multiplication using visual models or real-life examples.
• Making crafts or recipes that require multiplying fractions , such as paper snowflakes, origami animals, or cookies.
• Taking quizzes or tests that assess the student's understanding and skills in multiplying fractions.

More fun and engaging activities and exercises to improve 6 th grader’s multiplying fractions skills are:

To play, one person draws a fraction card and reads it aloud. Then, everyone multiplies the fraction by another fraction of their choice and looks for the answer on their bingo card. The first person to get five answers in a row, column, or diagonal wins.

To play, shuffle the dominoes and deal seven to each player. The rest of the dominoes form the draw pile. The first player places a domino on the table. Then, each player takes turns placing a domino next to an existing one so that the fraction product on one end matches the fraction on the other.

If a player cannot place a domino, they will draw one from the pile. The game ends when one player runs out of dominoes or when no one can place a domino. The player with the least number of dominoes left wins.

They will need a set of fraction puzzle pieces to be made or bought online. To do this activity, choose a puzzle and separate the pieces. Then, solve each fraction multiplication problem on the pieces and find the matching piece with the answer. Fit the pieces together until the picture is complete.

Bonus: Resources for boosting multiplying fractions skills for 6th-graders

If you want to help your 6th-grade students master multiplying fractions, you can also use some online resources that offer interactive games, videos, quizzes, and more. Here are some of the best websites that you can use to supplement Mathskills4kids’ 6th Grade multiplying fractions worksheets with answers :

• Math Playground : This website has a variety of fun and challenging games that let students practice multiplying fractions in different contexts, such as pizza, candy, and fractions of fractions. Students can also watch videos explaining multiplying fractions concepts and strategies. https://www.mathplayground.com/index_fractions.html .
• Khan Academy : This website has a comprehensive course on multiplying fractions that covers everything from the basics to the advanced topics, such as mixed numbers, word problems, and cross-canceling. Students can watch videos, do exercises, and take quizzes to check their understanding and progress. https://www.khanacademy.org/math/arithmetic/fraction-arithmetic/arith-review-multiply-fractions .
• IXL : This website has many practice questions on multiplying fractions aligned with the Common Core standards. Students can choose from different difficulty levels and get instant feedback and explanations. They can also earn awards and certificates for their achievements. https://www.ixl.com/math/grade-6/multiply-fractions .
• Math Games : This website has many engaging and interactive games that let students practice multiplying fractions in a fun way. Students can play against the computer or with other players online. They can also customize their avatars and earn coins and trophies. https://www.mathgames.com/skill/6.42-multiply-fractions .

Thank you for sharing the links of MathSkills4Kids.com with your loved ones. Your choice is greatly appreciated.

Multiplying fractions is an essential skill that 6th-grade students need to learn and master. It helps them develop their mathematical thinking and problem-solving abilities.

By using Mathskills4kids’ 6th Grade multiplying fractions worksheets with answers and other activities and exercises, teachers can help their students practice and reinforce their multiplying fractions skills.

Also, with fun tips and strategies, teachers can make their teaching more effective and engaging. Moreover, with the suggested online resources in this article, teachers can provide more opportunities and challenges for their students and encourage their 6th-grade them to excel in multiplying fractions and enjoy math.

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2-digit by 2-digit Multiplication with Grid Support Including Regrouping (A)

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Summer Math Workbook For Grade 4 With Addition, Subtraction, Multiplication, Division, Word Problems, Fractions: Daily Math Practice Workbook For 4th ... For Grade 4 | Math Worksheets For Grade 4 Paperback – May 21, 2024

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How to Teach Long Division to Kids in 6 Easy Steps

15 Famous Mathematicians in History That Kids Should Know

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Step 1: Start with Division Vocabulary

Step 2: introduce long division layout, step 3: demonstrate with an examples, step 4: practice without bringing down zeros, step 5: teach bringing down zeros, step 6: explain problems with remainders.

Long division can be challenging for kids because it involves many steps and new vocabulary, making it easy to lose track of the process. This complexity often confuses students, especially when bringing down digits or handling remainders. Teaching long division step by step helps kids understand it better. This guide on “ how to teach long division to kids ” will provide clear steps to simplify learning for kids, making it less overwhelming.

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What Should My Child Know Before Learning Long Division?

Before diving into teaching long division, children must have a solid foundation in several key areas. These pre-requisites ensure that they are prepared for the complexities of long division:

• Understanding Multiplication: Kids should be comfortable with multiplication facts up to 12. This knowledge is fundamental because long division requires frequent multiplication to check and subtract products during the division steps.

Here are some fun activities to brush up your child’s understanding of multiplication facts:

• Basic Division Concepts: Ensure children understand the concept of division as sharing or grouping equally. They should be able to perform simple division tasks without remainders, using numbers that are easy to handle.

Here are some fun activities to brush up your child’s understanding of division facts:

• Familiarity with Subtraction: Subtraction is a significant part of the long division process, used to find the difference after multiplying the divisor. Children should be confident in subtracting large numbers, as they will need to do this repeatedly in each step of long division.

Assess how well your kids are versed with subtraction with these fun subtraction games :

• Number Sense: Good number sense , or an understanding of how numbers work and relate to each other, helps children anticipate the results of multiplication and division, making the process of long division smoother and more intuitive.

Ensuring kids are proficient in these areas makes teaching long division more effective, setting them up for success.

How to Teach Long Division in 6 Steps

Before learning the steps of long division, kids need to understand the key terms used throughout the process. Knowing these will help them follow long division strategies more easily.

• Dividend: This is the number being divided in the problem. In the long division layout, the dividend is written inside the long division bracket (often called the “house”).
• Divisor: The number dividing the dividend is called the divisor . The divisor is placed outside the long division bracket.
• Quotient: The quotient is the answer or the result of the division problem. This is written above the bracket, aligned with the digits of the dividend being divided.
• Remainder: If the division isn’t exact, there’s an amount left over called the remainder . The remainder is written as “R” followed by the remaining value after the division process.

Setting up a long division problem correctly is an essential skill for kids to learn. Here’s how to structure it visually:

• Write the Dividend Inside: Start by placing the dividend (the number being divided) inside the long division bracket, often called the “house.” This makes it clear that this number is the one being split into smaller groups.
• Place the Divisor Outside: Write the divisor (the number doing the dividing) outside, to the left of the bracket. This shows which number will be used repeatedly to divide the dividend into smaller groups.
• Draw Lines for the Quotient: The space above the bracket is where the quotient (the result of the division) will be written. Each digit of the quotient will align above the digit or digits of the dividend that are being divided.

1: Set Up the Problem

Write 1564 under the division bar and 27 outside the division bar.

• Look at the first two digits of the dividend (15). Since 15 is less than 27, we need to include the next digit, making it 156.
• Now, determine how many times 27 can fit into 156. 27 goes into 156 a total of 5 times because 27 multiplied by 6 (the next higher number) would exceed 156.

3. Multiply

• Multiply the quotient digit found (5) by the divisor (27), which equals 135.
• Write 135 under 156.

4. Subtract

• Subtract 135 from 156, which equals 21.
• Write 21 below 135.

5. Bring Down

Bring down the next digit of the dividend (the 4 in 1564), making the new number 214.

6. Repeat the Process

• Determine how many times 27 can fit into 214. It fits 7 times (since 27 multiplied by 8 exceeds 214).
• Multiply 27 by 7 (the next quotient digit) to get 189.
• Write 189 under 214, and subtract to find the difference: 214 – 189 = 25.

7. Remainder

• Since there are no more digits to bring down, 25 is the remainder.
• The complete quotient is 57 with a remainder of 25.

After understanding the layout, kids can apply what they’ve learned by playing these games in, each focusing on different levels of difficulty: 

• Dividing 2-Digit Numbers by 1-Digit

• Dividing 3-Digit Numbers by 1-Digit

• Dividing 4-Digit Numbers by 1-Digit

• Divide 4-Digit By 2-Digit Numbers

To build a strong foundation in long division, it’s crucial for children to practice problems where they bring down each digit consecutively without the complication of zeros . This helps them focus on mastering the basic steps of division:

Sequential Bringing Down: Start with simpler numbers where each digit in the dividend needs to be brought down one at a time. This practice solidifies their understanding of the division process step by step.

Example Problems:

• 21 ÷ 3: Begin by dividing the first digit. Since 2 divided by 3 doesn’t work, we use 21. 21 divided by 3 equals 7, so 7 is written above the division bar.
• 154 ÷ 7: Start with the first digit. 1 divided by 7 is not possible, so consider the first two digits, 15. 15 divided by 7 equals 2. Write 2 above the division bar, multiply 2 by 7 (which is 14), subtract 14 from 15 to get 1, then bring down the next digit to make it 14. Continue this way.

Handling zeros can be tricky, but it’s one of the key ways to teach long division. Here’s how to approach problems involving zeros:

Example: 120 ÷ 4

• Divide: Start by dividing the first non-zero digit. Since 1 is smaller than 4, look at the first two digits, 12. Divide 12 by 4 to get 3.
• Write the Quotient: Write 3 above the long division bracket.
• Multiply and Subtract: Multiply 3 by 4 (giving 12) and subtract this from 12, leaving 0.
• Bring Down Zeros: Bring down the next digit, which is 0, making it 0. Divide this 0 by 4 to get 0. Write 0 next to the 3 in the quotient.
• Multiply and Subtract: Multiply 0 by 4 to get 0 and subtract, leaving no remainder.

In some cases, division doesn’t result in an even quotient. This means there’s an amount leftover, called a remainder. Here’s how to handle it:

Example: 29 ÷ 4

• Divide: Start by dividing the first digit, 2, by 4. Since 2 is smaller than 4, consider the first two digits, 29.
• Find the Quotient: Divide 29 by 4 to get 7. Write 7 above the long division bracket.
• Multiply and Subtract: Multiply 7 by 4, which gives 28. Subtract 28 from 29, leaving 1.
• Write the Remainder: Since there’s nothing left to bring down, write “R1” (remainder of 1) next to the 7.

Knowing how to recognize and handle remainders is one of the essential long division tricks that help kids understand why not all division problems divide evenly. 

5 Fun Ways to Teach Long Division

Practicing long division is crucial for reinforcing the steps and strategies. Here are five effective ways to help kids master this skill:

• Online Long Division Games 

Use interactive online games that challenge students to solve division problems step-by-step. These games make learning fun while providing instant feedback to help them understand mistakes.

• Long Division Worksheets

Provide worksheets that start with simpler problems and progressively increase in complexity. Use colorful designs and real-world scenarios to keep kids engaged.

Create or find flashcards that present division problems on one side and answers on the other. This simple tool is a quick way to reinforce division facts and promote speedy calculations.

• Real-Life Problems

Apply division concepts to real-world situations, like dividing treats among friends or organizing items into equal groups. This practical approach makes long division relatable.

• Group Activities

Collaborative exercises where kids work together to solve division problems encourage discussion and peer learning. This can help clarify challenging steps and foster teamwork.

Learning long division can be challenging, but kids can grasp the concept effectively with clear steps, consistent practice, and engaging resources like games and worksheets. With these strategies on how to teach long division, children will build confidence and become proficient problem-solvers at their own pace.

Frequently Asked Questions (FAQs)

What is an easy way to learn division.

An easy way to learn division is by starting with simple problems and using visual aids like counters or drawings to illustrate the process. Gradually move to more complex problems as confidence builds.

What are some tips for teaching long division to struggling students?

Break down the steps into smaller, manageable parts, and offer lots of practice with engaging resources like games and worksheets. Provide positive reinforcement to encourage their progress and address specific challenges with patience.

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The Algebra Problem: How Middle School Math Became a National Flashpoint

Top students can benefit greatly by being offered the subject early. But many districts offer few Black and Latino eighth graders a chance to study it.

By Troy Closson

From suburbs in the Northeast to major cities on the West Coast, a surprising subject is prompting ballot measures, lawsuits and bitter fights among parents: algebra.

Students have been required for decades to learn to solve for the variable x, and to find the slope of a line. Most complete the course in their first year of high school. But top-achievers are sometimes allowed to enroll earlier, typically in eighth grade.

The dual pathways inspire some of the most fiery debates over equity and academic opportunity in American education.

Do bias and inequality keep Black and Latino children off the fast track? Should middle schools eliminate algebra to level the playing field? What if standout pupils lose the chance to challenge themselves?

The questions are so fraught because algebra functions as a crucial crossroads in the education system. Students who fail it are far less likely to graduate. Those who take it early can take calculus by 12th grade, giving them a potential edge when applying to elite universities and lifting them toward society’s most high-status and lucrative professions.

But racial and economic gaps in math achievement are wide in the United States, and grew wider during the pandemic. In some states, nearly four in five poor children do not meet math standards.

To close those gaps, New York City’s previous mayor, Bill de Blasio, adopted a goal embraced by many districts elsewhere. Every middle school would offer algebra, and principals could opt to enroll all of their eighth graders in the class. San Francisco took an opposite approach: If some children could not reach algebra by middle school, no one would be allowed to take it.

The central mission in both cities was to help disadvantaged students. But solving the algebra dilemma can be more complex than solving the quadratic formula.

New York’s dream of “algebra for all” was never fully realized, and Mayor Eric Adams’s administration changed the goal to improving outcomes for ninth graders taking algebra. In San Francisco, dismantling middle-school algebra did little to end racial inequities among students in advanced math classes. After a huge public outcry, the district decided to reverse course.

“You wouldn’t think that there could be a more boring topic in the world,” said Thurston Domina, a professor at the University of North Carolina. “And yet, it’s this place of incredibly high passions.”

“Things run hot,” he said.

In some cities, disputes over algebra have been so intense that parents have sued school districts, protested outside mayors’ offices and campaigned for the ouster of school board members.

Teaching math in middle school is a challenge for educators in part because that is when the material becomes more complex, with students moving from multiplication tables to equations and abstract concepts. Students who have not mastered the basic skills can quickly become lost, and it can be difficult for them to catch up.

Many school districts have traditionally responded to divergent achievement levels by simply separating children into distinct pathways, placing some in general math classes while offering others algebra as an accelerated option. Such sorting, known as tracking, appeals to parents who want their children to reach advanced math as quickly as possible.

But tracking has cast an uncomfortable spotlight on inequality. Around a quarter of all students in the United States take algebra in middle school. But only about 12 percent of Black and Latino eighth graders do, compared with roughly 24 percent of white pupils, a federal report found .

“That’s why middle school math is this flashpoint,” said Joshua Goodman, an associate professor of education and economics at Boston University. “It’s the first moment where you potentially make it very obvious and explicit that there are knowledge gaps opening up.”

In the decades-long war over math, San Francisco has emerged as a prominent battleground.

California once required that all eighth graders take algebra. But lower-performing middle school students often struggle when forced to enroll in the class, research shows. San Francisco later stopped offering the class in eighth grade. But the ban did little to close achievement gaps in more advanced math classes, recent research has found.

As the pendulum swung, the only constant was anger. Leading Bay Area academics disparaged one another’s research . A group of parents even sued the district last spring. “Denying students the opportunity to skip ahead in math when their intellectual ability clearly allows for it greatly harms their potential for future achievement,” their lawsuit said.

The city is now back to where it began: Middle school algebra — for some, not necessarily for all — will return in August. The experience underscored how every approach carries risks.

“Schools really don’t know what to do,” said Jon R. Star, an educational psychologist at Harvard who has studied algebra education. “And it’s just leading to a lot of tension.”

In Cambridge, Mass., the school district phased out middle school algebra before the pandemic. But some argued that the move had backfired: Families who could afford to simply paid for their children to take accelerated math outside of school.

“It’s the worst of all possible worlds for equity,” Jacob Barandes, a Cambridge parent, said at a school board meeting.

Elsewhere, many students lack options to take the class early: One of Philadelphia’s most prestigious high schools requires students to pass algebra before enrolling, preventing many low-income children from applying because they attend middle schools that do not offer the class.

In New York, Mr. de Blasio sought to tackle the disparities when he announced a plan in 2015 to offer algebra — but not require it — in all of the city’s middle schools. More than 15,000 eighth graders did not have the class at their schools at the time.

Since then, the number of middle schools that offer algebra has risen to about 80 percent from 60 percent. But white and Asian American students still pass state algebra tests at higher rates than their peers.

The city’s current schools chancellor, David Banks, also shifted the system’s algebra focus to high schools, requiring the same ninth-grade curriculum at many schools in a move that has won both support and backlash from educators.

And some New York City families are still worried about middle school. A group of parent leaders in Manhattan recently asked the district to create more accelerated math options before high school, saying that many young students must seek out higher-level instruction outside the public school system.

In a vast district like New York — where some schools are filled with children from well-off families and others mainly educate homeless children — the challenge in math education can be that “incredible diversity,” said Pedro A. Noguera, the dean of the University of Southern California’s Rossier School of Education.

“You have some kids who are ready for algebra in fourth grade, and they should not be denied it,” Mr. Noguera said. “Others are still struggling with arithmetic in high school, and they need support.”

Many schools are unequipped to teach children with disparate math skills in a single classroom. Some educators lack the training they need to help students who have fallen behind, while also challenging those working at grade level or beyond.

Some schools have tried to find ways to tackle the issue on their own. KIPP charter schools in New York have added an additional half-hour of math time to many students’ schedules, to give children more time for practice and support so they can be ready for algebra by eighth grade.

At Middle School 50 in Brooklyn, where all eighth graders take algebra, teachers rewrote lesson plans for sixth- and seventh-grade students to lay the groundwork for the class.

The school’s principal, Ben Honoroff, said he expected that some students would have to retake the class in high school. But after starting a small algebra pilot program a few years ago, he came to believe that exposing children early could benefit everyone — as long as students came into it well prepared.

Looking around at the students who were not enrolling in the class, Mr. Honoroff said, “we asked, ‘Are there other kids that would excel in this?’”

“The answer was 100 percent, yes,” he added. “That was not something that I could live with.”

Troy Closson reports on K-12 schools in New York City for The Times. More about Troy Closson

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