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If you REALLY like exercising your brain, figuring things 'round and 'round till you explode, then this is the page for you !

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Usable Math

(formerly 4mality), math learning through problem solving and design.

Usable Math provides interactive problem solving practice for 3rd through 6th grade students learning mathematical reasoning and computation through creative writing, NoCode slideshow design, and human-AI collaboration.

• Arrow Right MATH MODULES

Math Friends

Featuring four coaches Estella Explainer, Chef Math Bear, How-to Hound, and Visual Vicuna who offer reading, computation, strategy, and visual strategies for solving math problems.

Estella Explainer

"I help children understand the language and meaning of questions using kid-friendly vocabulary."

Chef Math Bear

"I provide computational strategies (addition, subtraction, multiplication and division) for solving problems."

How-to-Hound

" I present strategic thinking clues (rounding, estimation, elimination of wrong answers). "

Visual Vicuna

" I offer ways to see problems and their solutions using animations, pictures, charts and graphs. "

The coaches annotate hints and provide feedback to help students with various levels of knowledge solve mathematical word problems using a wide range of strategies.

Math and ISTE Standards Based

Usable Math aims to teach mathematics concepts and problem solving skills based on the Massachusetts Mathematics Curriculum Framework and the Common Core State Standards for Mathematics. Usable Math supports ISTE Standards for Students : Empowered Learner (1.1), Knowledge Constructor (1.3), and Computational Thinker (1.5).

Open Education Resource

Usable Math is an open education resource project developed in the College of Education, University of Massachusetts Amherst. Usable Math received a 2023 classroom grant from MassCUE (Massachusetts Computer Using Educators) . An initial version called 4mality was developed with funding support from the Verizon Foundation and a grant from the US Department of Education, Institute of Education (IES).

BROWSE MATH MODULES

Storywriting, history, and science modules, a sydney-the-statistician data analysis adventure, math & statistics, a tai-the-math historian time travel adventure, math & history, ai-enhanced, a sofia-the-forester adventure, math & storywriting, a jenny-the-fisher math and citizen scientist adventure, math & science, math problem-solving and design modules, area and perimeter, total problems: 6.

Total problems: 8

Multiplication and Division

Algebraic Thinking

Add & take away: how many more.

Add & Take Away: Find Totals

Geometry: Lines and Lines of Symmetry

Geometry: Maps + Grids + Ordered Pairs

Charts & graphs, total problems: 10.

Geometry: Figures, Shapes and Angles

Total problems: 7.

Total problems: 11

Place Value

Total problems: 14.

Total problems: 9

Total problems: 5

Measurement

More coming soon, ai-enhanced learning, introducing ai-enhanced math comprehension with estella explainer math bot 2, a gpt4-powered tool designed to revolutionize how elementary school students understand math word problems., free to use. a free chatgpt account is required..

Empowering Math Educators

A Reading and Language Coach: A Reading and Language Coach: Estella Bot assists teachers and tutors by providing simplified explanations of math word problems and vocabulary.

Fine-Tuning for Clarity: We have customized the bot to enhance young students' understanding of math problems through targeted fine-tuning.

Enhancing Student Understanding

Overcoming Challenges: Address the complexities of math word problems with Estella Bot’s support.

Step-by-Step Problem Solving: Inspired by George Polya’s methodology, Estella Bot helps students read and comprehend problems before solving them.

Applying the Feynman Technique: Estella Bot uses the Feynman Technique to simplify complex word problems for young learners, by breaking down complex math terms into easy-to-understand language.

Interactive Learning Partner

Natural Conversations: Engage in interactive, human-like conversations with Estella Bot to clarify and deepen math understanding.

Sample Dialogues: View examples of how Estella Bot simplifies math problems through clear and concise explanations.

Decision-Making Tool for Educators

Using the Decision Tree: Utilize the Estella Explainer AI Bot Decision Tree to guide teaching strategies and support student learning effectively (scroll down for the decision tree visual).

Best Practices for Using Estella Bot

Contextual Prompts: Provide specific educational context to receive the most accurate and helpful responses from Estella Bot.

Precision in Language: Use clear, concrete language in prompts for better bot responses.

Readability Metrics: Apply the Flesch-Kincaid Scale to ensure responses are easily understood by young learners.

Share Your Feedback

We value your input on the Estella Bot and the Usable Math platform. Please send your feedback to [email protected] , or use our contact form .

Supporting Students in Reading Math Word Problems

Challenges in Math Word Problems: Understand the common reading challenges faced by students.

Step-by-Step Problem Solving: Explore the step-by-step approach inspired by George Polya’s problem-solving techniques.

Applying the Feynman Technique: See how Estella Bot uses the Feynman Technique to simplify complex word problems for young learners.

Welcome to Usable Math. In this interactive website, you will find Learning Modules designed to develop mathematical problem solving skills among young learners in grades 3 to 6.

Our Modules explore standards-based math concepts including Fractions, Measurement, Geometry, Decimals, Money, and more. Usable Math is free to access using a computer, smartphone, or iPad.

What do we mean by Usable Math?

The word Usable can read as follows:

U Able meaning you can do math problem solving.

Us Able meaning together all of us can do math problem solving.

Usable meaning anyone is able to learn math problem solving - with practice, effort, and support.

What are the Usable Math Learning Modules?

Each learning module in Usable Math consists of a group of math word problems related to a specific mathematical concept. The problems are based on the Massachusetts Mathematics Curriculum Framework↗ as well as Common Core Standards↗ .

Each problem within a module consists of a question, three to four possible answer choices, and problem solving ideas and strategies provided by our four coaches: Estella Explainer, Chef Math Bear, How-to-Hound, and Visual Vicuna.

How are the Modules Displayed online?

Each module has been developed using Google Slides.

Click. Pause. Solve.

View each module in Slideshow.

How do teachers, students and families use each module?

We strive to make every module on Usable Math kid friendly . Clicking on a module from the selections on the Modules Homepage , each user controls what happens during the learning experience by clicking to open strategies and spending time thinking about them before answering the question. The goal is for students, by themselves, in small groups, or with a teacher, or a family member, to analyze and understand what the problem is asking them to solve before providing an answer.

A question appears without its answer choices or any problem solving strategies.

Click one time and Estella offers a problem solving strategy.

Click again and the Bear offers a different strategy.

Click again and the Hound presents a strategy.

Click again and the Vicuna has an additional strategy approach.

The next click gives the four answer choices, but not yet the correct answer.

The final click highlights the correct answer from among the answer choices.

Before going to the next problem, a motivational statement and gif appears offering encouragement to the users.

What is the purpose of the Motivational Statements between Problems?

Each motivational statement is intended to provide feedback and encouragement to students using the system. Following the insights of researchers into the use of praise and the development of growth mindsets in young learners, these motivational statements are designed to reward students’ effort, hard work, persistence, and belief in one’s self as a learner. We want youngsters to realize that they can learn anything with the right tools, the right beliefs, the right coaches, and their own work and practice.

Need more help? Or have a question?

Reach out to us and we will do our best to get back to you within 12 hours.

RESEARCH AND RESOURCES

We believe that every child deserves a strong foundation in mathematics. our platform is designed to provide engaging and effective math instruction to elementary school students, and we are proud to say that there is science behind the way we deliver this instruction..

UsableMath was formerly known as 4MALITY. As a result of our commitment to providing the best possible math instruction to elementary school students, we have rebranded our platform as UsableMath.com to better reflect our mission and approach to teaching mathematics.

Our platform is designed to provide engaging and effective math instruction to students in grades K-5, using a unique approach that emphasizes hands-on, problem-solving activities. We use interactive, multimedia elements such as videos, games, and simulations to help students understand key mathematical concepts and build a strong foundation of knowledge.

Math Coaches

The use of virtual coaches that provides students with personalized support and feedback, has become increasingly popular in the field of math education. Research has shown that learning companions can be effective in improving student engagement and motivation, as well as helping students to better understand mathematical concepts and build a stronger foundation of knowledge. UsableMath employs the concept of learning companions to help students succeed in mathematics. Our virtual math coaches serve as personal guides, providing students with individualized support and feedback as they work through mathematical concepts and problems. These coaches, or learning companions, are designed to be like friends or mentors, helping students to build their confidence, overcome challenges, and achieve their full potential.

How are we using Generative AI to enhance Usable Math Modules?

As developers of Usable Math, we are aware of both the educational potentials and complexities of Generative AI technologies. In our system, ChatGPT is used to support teachers and other adults to expand and enhance how math can be understood and taught in schools and homes. When you click on the AI icon, you are linked to a blog where we have recorded how AI proposes to solve selected math word problems found in Usable Math modules in a side-by-side view next to the hints we have authored from the perspectives of our four math coaches: Estella Explainer, Chef Math Bear, How-to-Hound, and Visual Vicuna. Our hope is that our strategies along with the AI-developed strategies will give adults more ways to inspire math learning among students.

Look for this icon for AI-enhanced guides.

Prompts for ChatGPT, BingAI and Other Generative AI Tools

Estella Explainer Prompt:

Take the personality of a math coach who provides strategies for understanding language and meaning of questions using kid-friendly vocabulary. The coach’s motto is "My job is to explain the math questions clearly so you know what you are supposed to do to solve the problem. Sometimes there are unfamiliar or confusing terms in the question. I will help you understand what they mean. The first math problem is {replace math word problem here}

Chef Math Bear Prompt:

Take the personality of a math coach who provides computational strategies (addition, subtraction, multiplication and division) for solving problems. The coach’s motto is “I am here to make sure that you know how to do the math needed to answer these questions. Sometimes you need to do addition, subtraction, multiplication or division. Some questions ask you to use fractions, decimals, large numbers, and probability. When you need ideas for what to do, I am ready.

How-to-Hound Prompt:

Take the personality of a math coach who uses strategic thinking clues (rounding, estimation, elimination of wrong answers) to solve math problems. The coach’s motto is “Answering math questions means you need a plan and my role is to help you figure out different strategies for solving problems. Sometimes you can get the correct answer by crossing out the wrong answers; other times you can round numbers up or down to make figuring a problem easier. I know other strategies as well.

Visual Vicuna Prompt:

Take the personality of a math coach who offers ways to see problems and their solutions using animations, pictures, charts and graphs. The coach’s motto is “I find math is a lot clearer when I take the numbers and words and put them into pictures and drawings or move objects around so I can see how to answer a question. When you find yourself unsure about a question, see if one of my ideas will explain what to do.

Growth Mindset Statements

As education researchers, we understand the important role that a positive attitude and motivation play in learner success. That's why we’ve integrated the use of growth mindset and motivational cues in Usable Math. After every math challenge, students receive messages that encourage them to adopt a growth mindset, reinforcing the idea that with effort and persistence, they can improve their math skills and achieve success.

A sample motivational cue from the Fractions module.

Collaborative Problem Solving

We believe in the power of collaboration and teamwork when it comes to learning mathematics. Our platform creates a learning climate that promotes collaborative problem solving, providing students with opportunities to work together and explore mathematical concepts in a supportive and inclusive environment. Whether you are a student, teacher, or parent, we invite you to explore our platform and experience the science behind the way we deliver math instruction to elementary school students. Read more about our work on the Journal of STEM Education↗

Papers, Presentations and Blogs

UsableMath GenAI Prompts: Learn Math with Our Tailor-Made Prompts for ChatGPT, Claude, and other GenAI tools. Usable Math Blog. https://blog.usablemath.org/usablemath-genai-prompts .

Maloy, R. W. & Gattupalli, S. (2024). Prompt Literacy. EdTechnica: The Open Encyclopedia of Educational Technology . https://edtechbooks.org/encyclopedia/prompt_literacy

Gattupalli, S., & Maloy, R. W. (2024). On Human-Centered AI in Education. https://doi.org/10.7275/KXAP-FN13

Gattupalli, S., Edwards, S.A, Maloy, R. W., & Rancourt, M. (2023, October). Designing for Learning: Key Decisions for an Open Online Math Tutor for Elementary Students. Digital Experiences in Mathematics Education . https://doi.org/10.1007/s40751-023-00128-3 .

Gattupalli, S., Maloy, R.W., Edwards, S.A. & Gearty, A. (2023, August 23). Prompt Literacy for STEM Educators: Enhance Your Teaching and Learning with Generative AI. Berkshire Resources for Learning and Innovation (BRLI) Teaching with Technology Conference, Pittsfield, MA. ScholarWorks@UMass.

Blending Gardens and Geometry: Socio-cultural Approaches in Math Ed. Usable Math Blog. https://blog.usablemath.org/blending-gardens-and-geometry-socio-cultural-approaches-in-math-education .

Maloy, R. W., Gattupalli, S., & Edwards, S. A. (2023). Developing Usable Math Online Tutor for Elementary Math Learners with NoCode Tools . Scholarworks@UMass.

Gattupalli, S., Maloy, R. W., & Edwards, S. A. (2023). Prompt Literacy: A Pivotal Educational Skill in the Age of AI . Scholarworks@UMass.

Gattupalli, S., Maloy, R. W., & Edwards, S. (2023). Comparing Teacher-Written and AI-Generated Math Problem Solving Strategies for Elementary School Students: Implications for Classroom Learning . https://doi.org/10.7275/8sgx-xj08

Making Math Usable for Young Learners . Guest post on Rachelle Dené Poth's EdTech blog Learning as I go: Experiences, Reflections, Lessons Learned . January, 2023.

Math Learning Digital Choice Board (2020) . ScholarWorks, University of Massachusetts Amherst.

Maloy, R.W., Razzaq, L., & Edwards, S.A. (2014). Learning by Choosing: Fourth Graders Use of an Online Multimedia Tutoring System for Math Problem Solving . Journal of Interactive Learning Research , 25(1), 51-64.

Razzaq, L., Maloy, R. W., Edwards, S. A., Arroyo, I., & Woolf, B.P. (2011). “4MALITY: Coaching Students with Different Problem Solving Strategies Using an Online Tutoring System” (p. 359-364). In J. A. Konstan, Ricardo Conejo, Jose L, Marzo & Nuria Oliver, User Modeling, Adaptation and Personalization: 19th International Conference, UMAP 2011, Girona, Spain, July 11-15 Proceedings . Berlin: Springer Verlag.

Maloy, R.W., Edwards,S. A. & Anderson G. (2010, January-June). “Teaching Math Problem Solving Using a Web-based Tutoring System, Learning Games, and Students’ Writing .” Journal of STEM Education: Innovations and Research, 11 (1&2).

Edwards, S. A., Maloy, R.W., & Anderson G. (2010, February). “Classroom Characters Coach Students to Success.” Teaching Children Mathematics, 16 (6), 342-349.

Edwards, S. A., Maloy, R. W., & Anderson G. (2009, Summer). “Reading Coaching of Math Word Problems.” Literacy Coaching Clearinghouse . http://www.literacycoachingonline.org/briefs.html .

MEET OUR TEAM

Sharon Edwards , Ph.D.

Teacher Education & Curriculum Studies

College of Education, University of Massachusetts Amherst

Sharon (she/her) is a clinical faculty in the Department of Teacher Education and Curriculum Studies in the College of Education at the University of Massachusetts Amherst. Sharon is the big brains behind the development of Usable Math online math tutor.

Email : sae at umass dot edu

Robert Maloy , Ph.D.

Elementary Math and History

Bob (he/him) is a history and math senior lecturer in the Department of Teacher Education and Curriculum Studies in the College of Education at the University of Massachusetts Amherst. Bob is the creative math content creator and storytelling artist behind Usable Math.

Email : rwm at umass dot edu

Sai Gattupalli

Math, Science & Learning Technologies (MSLT)

Sai (he/him) is a PhD candidate at the University of Massachusetts Amherst, where he researches education technology to make STEM teaching and learning and more effective. Sai is passionate about understanding learner culture to create effective learning experiences. Email : sgattupalli at umass dot edu Website : gattupalli.com

Marguerite Rancourt

Lead Teacher, Discovery School at Four Corners

Greenfield, Massachusetts

Marguerite (she/her) teaches fourth grade at the Discovery School in addition to serving as Lead Teacher for the school. She has created and taught professional development workshop for other elementary school teachers. In 2018, she received the Pioneer Valley Excellence in Teaching Award. Students in her class have been contributing to the design of system throughout the 2022-2023 school year.

Aubrey Coyne

Math Content Designer and Reviewer

College of Education, Commonwealth Honors College, University of Massachusetts Amherst.

Aubrey Coyne (she/her) is a sophomore at the University of Massachusetts Amherst. She is a math tutor and is studying to be an elementary teacher. Aubrey is passionate about finding ways to make learning accessible and enjoyable for all students.

Graduate Student, Math and Digital Media Research Assistant

Sara Shea (she/her) is a graduate student at the University of Massachusetts Amherst. She is currently part of the university’s Collaborative Teacher Education Pathway program, working towards earning her master’s degree in elementary education.

Katie Allan

Math and Digital Media Research Assistant

Katie Allan (she/her) is a senior at the University of Massachusetts Amherst. She is a math major with a concentration in education and passionate about math education.

SUGGESTIONS AND FEEDBACK

We welcome ideas from teachers, students, and families about the usable math system..

Please complete our UsableMath Module Review and Feedback↗ form.

Your responses will help us to improve how the system works instructionally and technically. Let us know any additional thoughts about the problems, characters, hints, gifs, mindset statements and more.

Your message has been received. We will get back to you shortly. The average response time is approximately 6 hours.

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Problem Solving Games, Videos and Worksheets

Word Problems Games for Kids

Let's bring math concepts to life with our word problems games! These interactive word problem games improve kids' critical thinking and problem-solving skills in a fun and engaging way. Learn to solve word problems involving four arithmetic operations, multi-step word problems, and word problems o ... Read more n fractions and money concepts. Parents and teachers love how these games help kids connect math with real-world scenarios through playful exercises. Happy solving! Start now for free!

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Addition Word Problems Games

Solve Word Problems with Add to Scenarios Game

Shine bright in the math world by learning how to solve word problems with "Add to" scenarios.

Adding Within 5 by Making a Model Game

Add more arrows to your child’s math quiver by adding within 5 by making a model.

Word Problems to Add Multiples of 10 Game

Learn to solve word problems to add multiples of 10.

Adding One by Making a Model Game

Treat yourself to an immersive learning experience with our 'Adding One by Making a Model' game.

Subtraction Word Problems Games

Solve Subtraction Scenarios Game

Apply your knowledge to solve subtraction scenarios.

Solve Scenarios with 'Difference Unknown' Game

Shine bright in the math world by learning how to solve scenarios with 'Difference Unknown'.

Solving Word Problems Game

Enjoy the marvel of math-multiverse by exploring how to solve word problems.

Solve Difference Unknown Scenarios Game

Kids must solve difference unknown scenarios to practice subtraction.

Multiplication Word Problems Games

Solve Word Problems on Decimal Multiplication Game

Kids must solve word problems on decimal multiplication to practice decimals.

Division Word Problems Games

Word Problems on Dividing Decimals by Decimals Game

Enjoy the marvel of math-multiverse by solving word problems on dividing decimals by decimals.

Word Problems on Dividing Decimals Game

Enjoy the marvel of mathematics by exploring how to solve word problems on dividing decimals.

Fraction Word Problems Games

Solve the Word Problems on Fraction Subtraction Game

Add more arrows to your child’s math quiver by solving word problems on fraction subtraction.

Solve the Word Problems on Fraction Addition Game

Have your own math-themed party by learning how to solve the word problems on fraction addition.

Word Problems on Addition of Fractions Game

Use your fraction skills to solve word problems on addition of fractions.

Solve Word Problems on Fraction-Whole Number Multiplication Game

Apply your knowledge to solve word problems on fraction-whole number multiplication.

Money Word Problems Games

Add the Same Type of Coins and Compare Game

Enjoy the marvel of mathematics by exploring how to add the same type of coins and compare them.

Find the Remaining Amount of Money Game

Solve math problems to find the remaining amount of money.

Add Coins of Different Types and Compare Game

Kids must add coins of different types and compare them to practice counting money.

Find the Change Game

Use your counting money skills to find the change.

All Word Problems Games

Solve the Word Problems Related to Multiplication Game

Unearth the wisdom of mathematics by learning to solve word problems related to multiplication.

Word Problems on How many Tens Game

Learn to solve world problems on 'How many Tens' with this game.

• Word Problems: Subtracting One Game

Sharpen your math skills with the 'Word Problems: Subtracting One' game.

Solve Word Problems on Division Game

Learn to solve math problems by solving word problems on division.

Model and Add (Within 10) Game

Unearth the wisdom of mathematics by learning how to model and add (within 10).

Word Problems: Subtracting Within 10 Game

Enjoy the marvel of math-multiverse by practicing to solve word problems on subtracting within 10.

Solve Word Problems using Division Game

Apply your knowledge to solve word problems using division.

Solve 'Add To' Scenarios Game

Add more arrows to your child’s math quiver by solving 'Add To' scenarios.

Solve 'Take Apart' Scenarios Game

Take the pressure off by simplifying subtraction by solving 'Take Apart' scenarios.

• Solve Comparison Word Problems Game

Unearth the wisdom of mathematics by learning how to solve comparison word problems.

Solve 'Put Together' Scenarios Game

Shine bright in the math world by learning how to solve 'Put Together' scenarios.

Subtraction Scenario Game

Take a look at subtraction scenarios with this game.

Find the Number of Groups Game

Find the number of groups to practice division.

Word Problems on Subtraction of Fractions Game

Have your own math-themed party by learning how to solve word problems on subtraction of fractions.

Find the Amount that was Paid Game

Take a look at how to find the amount that was paid with this counting money game.

Solve 'Add To' Word Problems Game

Unearth the wisdom of mathematics by learning how to solve 'Add To' word problems.

Take Away Scenario Game

Use your subtraction skills to solve 'Take Away' scenarios.

Solve Word Problems Related to Division Game

Add more arrows to your child’s math quiver by solving word problems related to division.

Area Word Problems on Product of Fractions Game

Learn to solve area word problems on product of fractions.

Word Problems on Adding Money Game

Shine bright in the math world by learning how to solve word problems on adding money.

Solve 'Put Together' Word Problems Game

Let your child see the world through math-colored shades by solving 'Put Together' word problems!

Word Problems on Subtraction Game

Add more arrows to your child’s math quiver by solving word problems on subtraction.

Find the Dividend Game

Enter the madness of math-multiverse by exploring how to find the dividend.

Solve Word Problems on Adding Money Game

Enjoy the marvel of mathematics by exploring how to solve word problems on adding money.

Solve Count On Scenarios Game

Begin the exciting journey of becoming a math wizard by learning how to solve count on scenarios.

Word Problems on Take Away Game

Kids must solve word problems on 'Take Away' to play.

Subtracting Money Game

Have your own math-themed party by learning how to subtracting money.

Represent 'Add To' Scenarios Game

Learn to solve math problems through our 'Represent 'Add To' Scenarios' game.

Solve Word Problems to Find the Difference Game

Kids must solve word problems to find the difference to practice subtraction.

Word Problems on Subtracting Money Game

Take a deep dive into the world of math by solving word problems on subtracting money.

Represent 'Put Together' Scenarios Game

Have your own math-themed party by learning how to represent 'Put Together' scenarios.

Solve 'Change Unknown' Scenarios Game

Begin the exciting journey of becoming a math wizard by solving 'Change Unknown' scenarios.

• Solve Word Problems on Subtracting Money Game

Have your own math-themed party by learning how to solve word problems on subtracting money.

Addition Word Problems on Finding the Total Game

Learn to solve addition word problems on finding the total.

What Are Word Problems in Math?

Word problems in math are mathematical exercises presented in the form of real-life scenarios or situations described using words. These problems require students to analyze the information given, extract relevant mathematical details, and formulate equations or calculations to solve them. 

Word problems help bridge the gap between abstract mathematical concepts and practical applications, enhancing problem-solving skills and critical thinking. They are commonly used to assess students' ability to apply mathematical knowledge to real-world situations and develop their proficiency in translating verbal information into mathematical expressions.

SplashLearn addresses a common challenge in learning word problems - understanding the language used in questions. Children often struggle to grasp the question's meaning, making problem-solving difficult. SplashLearn's games tackle this issue by enhancing comprehension of descriptive language and boosting quick problem-solving skills. 

The games adopt a visual-centric, interactive approach to learning, making it engaging and enjoyable for kids. This innovative approach is revolutionizing the learning experience for children, making learning fascinating and leaving them eager for more.

Types of Word Problems

Addition and subtraction word problems.

Addition and subtraction word problems involve real-world scenarios and test students' ability to translate textual information into mathematical expressions, apply addition and subtraction skills, and arrive at accurate solutions.

Here's a simple example of an addition and subtraction word problem:

Example : Emma had 8 cookies, and her friend gave her 5 more cookies. Later, she shared 3 cookies with her brother. How many cookies does Emma have now?

Emma's friend gave her 5 cookies, so she has 8 + 5 = 13 cookies.

After sharing 3 cookies, she has 13 - 3 = 10 cookies left.

Answer : Emma has 10 cookies now.

Ready to tackle word problems in a fun and engaging way? Dive into our interactive addition and subtraction word problems games to boost your problem-solving skills while having a blast! Get started now and conquer math challenges with confidence.

Multiplication and Division Word Problems

Multiplication and division word problems involve real-life scenarios where multiplication and division operations are applied to solve practical situations. These problems require students to decipher information from the text, use multiplication and division skills, and arrive at accurate solutions. Solving these word problems sharpens critical thinking, mathematical reasoning, and problem-solving abilities while connecting math to everyday contexts.

Example : Lisa wants to buy 4 packs of stickers, with each pack containing 6 stickers. How many stickers will she have in total?

Solution : To find the total number of stickers, we multiply the number of packs (4) by the number of stickers in each pack (6):

Total stickers = 4 × 6 = 24 stickers.

Answer : Lisa will have 24 stickers in total.

Apply your multiplication and division skills to solve word problems! Practice with the help of our exciting multiplication and division word problems games . Master word problems while having fun! 

Fractions Word Problems

Fractions word problems present real-world scenarios where fractions are used to solve practical situations. These problems challenge students to interpret textual information, apply fraction skills, and find solutions accurately. Solving fractions word problems cultivates critical thinking, mathematical reasoning, and problem-solving capabilities while linking math to everyday scenarios.

Example : Sarah ate 3 out of 4 slices of pizza. What fraction of the pizza did she eat?

Solution : Sarah ate 3 slices out of 4, which can be represented as \(\frac{3}{4}\).

Answer : Sarah ate \(\frac{3}{4}\) of the pizza.

Master fractions through enjoyable Fraction Word Problems Games ! Strengthen your fractions skills with fun activities!

Money Word Problems

Money word problems involve real-life situations where currency values, coins, paper bills are utilized to solve practical scenarios. These problems challenge students to comprehend textual information, apply money-related skills, and arrive at accurate solutions. Solving money word problems fosters critical thinking, financial literacy, and problem-solving abilities while connecting math to everyday financial contexts.

Example : Mark bought a toy for $12 and a book for $7. How much money did he spend in total?

Solution : To find the total cost, we add the prices of the toy and the book:

Total cost = $12 + $7 = $19.

Answer : Mark spent a total of $19.

Boost your money-related skills through engaging Money Word Problem Games ! 

Grade-wise Progression of Types and Complexity of Word Problems  

What Are the Real-world Advantages of Solving Word Problems for Kids?

• Everyday Application : Solving word problems helps kids calculate costs while shopping (finding total cost), measuring ingredients for recipes, and determining travel distances, making math useful in daily scenarios.
• Language Comprehension : Word problems enhance language skills as kids decode and understand problem statements, improving their ability to interpret information accurately.
• Critical Thinking : When solving a word problem like, kids must analyze, subtract, and divide to make decisions, honing their critical thinking skills.
• Strategic Problem-Solving Skills : When figuring out the approach to solve a problem, kids learn to divide and think strategically to distribute resources evenly.

Importance of Introducing Word Problems to Kids through Games

• Engagement : Games make word problems relatable and exciting, capturing kids' attention and motivating them to practice problem-solving skills in a fun way.
• Contextual Learning : Interactive game scenarios mirror real-life situations, helping kids grasp the practical relevance of math concepts embedded in word problems.
• Skill Transfer : Math games facilitate the transfer of problem-solving skills learned in virtual settings to real-world situations, improving kids' ability to handle challenges effectively.
• Confidence Building : Success in solving game-based word problems boosts kids' confidence in their mathematical abilities, encouraging them to tackle more complex problems with enthusiasm.

Mastering Word Problems: Math Word Problem Games vs. Traditional Methods

• Engaging Learning Environment : Online games create an interactive and immersive learning experience, keeping kids engaged and motivated to tackle word problems compared to traditional static worksheets .
• Immediate Feedback : Games provide instant feedback on answers, enabling kids to understand mistakes and correct them immediately, leading to faster learning and improvement.
• Adaptive Learning : Online word problem games adapt to a child's skill level, gradually increasing complexity as they progress, ensuring personalized learning and preventing frustration.
• Real-world Application : Game-based word problems simulate real-life scenarios, helping kids understand the practical relevance of math concepts and encouraging them to apply problem-solving skills in everyday situations.

What Are 5 Fun Word Problem Games and Activities for Kids?

 Here are 5 fun word problem games for kids by SplashLearn:

• Solve ‘Put Together’ Scenarios Game
• Calculate the Total Change You Will Get Back Game

How can kids practice word problems?

• Worksheets : Solve math-oriented word problems through structured learning resources such as printable Word Problems Worksheets , reinforcing critical thinking and math application.
• Online Games : Engage in interactive word problem games that make learning enjoyable, enhancing problem-solving abilities in a dynamic setting.
• Daily Activities : Connect math to real-life scenarios, like measuring ingredients while cooking or calculating distances during travel, embedding problem-solving skills in everyday experiences.

How can I teach word problems effectively to kids?

• Real-life Scenarios : Use relatable situations, like sharing cookies or shopping, to frame word problems, making them relevant and easier for kids to grasp.
• Visual Aids : Utilize drawings, objects, or diagrams to help kids visualize the problems, aiding comprehension and making abstract concepts more concrete.
• Step-by-Step Approach : Break down complex problems into smaller steps, guiding kids through understanding, solving, and verifying their answers systematically.
• Interactive Practice : Engage kids in hands-on activities and games that involve solving word problems, fostering active learning and skill application.

What are some specific skills that children can develop through Word Problem games?

• Word problem games develop skills like logical thinking, problem-solving, understanding patterns, and using math in everyday situations, building a strong foundation for practical math skills.
• Word problem games help kids think carefully to solve challenges. They also show how math is useful in real life by turning everyday situations into math problems.

Your one stop solution for all grade learning needs.

Word Problems or Problem Solving Math Games and Worksheets

Related Pages More Math Games Math Worksheets

Our directory of Word Problems Math Games or Problem Solving Math Games available on the Internet - games that teach, build or strengthen your math problem solving skills. We categorize and review the games listed here to help you find the math games you are looking for. We have word problem games for kids and word problem games that can be solved using block models, bar models or tape diagrams. We have added more free word problem games that can be played on PCs, Tablets, iPads and Mobiles.

Word Problem Solving Games for Kids

Word problem math games using modeling, algebra worksheets.

Evaluate Expressions by Substitution Simplify Expressions by Combining Like Terms Simplify Expressions using Distributive Property Simplify Expressions using Distributive Property & Combine Like Terms

Solving Equations Solving One-Step Equations Solving Two-Step Equations Solving Equations by Combining Like Terms Solving Equations with the Variable on Both Sides Solving Equations by Distributive Property Solving Multi-Step Equations Solving Multi-Step Equations Solving Word Problems using Algebra

Solving System of Equations by Substitution Solving System of Equations by Addition Solving System of Equationsby Subtraction Solving System of Equations by Addition or Subtraction Solving System of Equations by Multiplication and then Addition or Subtraction

Number Word Problems Worksheets SAT Test Prep SAT practice problems with step-by-step video solutions.

Disaster Math

Word problems about different disasters Disaster Master Build a Kit

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Play these fun Maths Games for 5-7 year olds

Choose a category:, problem solving games.

Bead Numbers - Place Value

Bead Numbers is a place value investigation involving a tens and ones abacus. The game provides a good context for encouraging learners to think systematically.

Shape Patterns

Shape Patterns is a sequencing game where children from 3 to 8 years of age need to complete the pattern of different coloured 2D shapes. Three levels of difficulty.

Bobbie Bear

How many different outfits can Bobbie Bear wear in each game given the options for shirts and trousers. Can you find a way to predict how many before trying all the combinations?

Thinking of a Number

Guess the number by revealing the clues on the clouds one by one. Children will need knowledge of rounding, odd and even and tens and ones.

Number Trains

Make a train by sequencing the numbers on their carriages. The numbers are represented in a range of formats such as words, numerals, dice dots or counting frames. The levels progress in difficulty. Level 5 has sequences of twos, fives and tens.

*BRAND NEW* Self-paced Introduction to Counting & Probability.   Enroll   today!

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I really enjoyed taking my first AoPS class. I always thought that online classes weren't very good. I was mistaken after I took this class. Not only did Introduction to Geometry help me become a better math student, it also helped me to think more abstractly and look for multiple roads to solve a problem. I enjoy the way the classroom is run and the pace was very nice. The class wasn’t boring at all. This class challenged me to become a better thinker and was way more challenging than regular school courses. I am also very glad that AoPS also included a little bit of a social aspect to math with the message boards. I made a lot of friends through this course. In school, I am always made fun of for being ahead of all of the other students. On AoPS I actually feel accepted for once because I can interact with kids that are more advanced than I am and also I learn from them. I will definitely take another AoPS class.

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Discover Frequently Asked Math Questions and Their Answers

• How do you write the quadratic function #y=x^2+14x+11# in vertex form?
• How many points does #y=-2x^2+x-3# have in common with the vertex and where is the vertex in relation to the x axis?
• How do you solve #4x^4 - 16x^2 + 15 = 0#?
• How do you solve #2x^2+3x-2=0#?
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Introducing Qwen2-Math

GITHUB HUGGING FACE MODELSCOPE DISCORD

🚨 This model mainly supports English. We will release bilingual (English and Chinese) math models soon.

Introduction #

Over the past year, we have dedicated significant effort to researching and enhancing the reasoning capabilities of large language models, with a particular focus on their ability to solve arithmetic and mathematical problems. Today, we are delighted to introduce a series of math-specific large language models of our Qwen2 series, Qwen2-Math and Qwen2-Math-Instruct-1.5B/7B/72B. Qwen2-Math is a series of specialized math language models built upon the Qwen2 LLMs, which significantly outperforms the mathematical capabilities of open-source models and even closed-source models (e.g., GPT-4o). We hope that Qwen2-Math can contribute to the community for solving complex mathematical problems.

We evaluate our math-specific models on a series of math benchmarks. The results below demonstrate that our largest math-specific model Qwen2-Math-72B-Instruct outperforms the state-of-the-art models, including GPT-4o, Claude-3.5-Sonnet, Gemini-1.5-Pro, and Llama-3.1-405B.

Qwen2-Math: Base Models #

The base models of Qwen2-Math are initialized with Qwen2-1.5B/7B/72B, and then pretrained on a meticulously designed Mathematics-specific Corpus. This corpus contains large-scale high-quality mathematical web texts, books, codes, exam questions, and mathematical pre-training data synthesized by Qwen2.

We evaluate our Qwen2-Math base models on three widely used English math benchmarks GSM8K, Math, and MMLU-STEM. In addition, we also evaluate three Chinese math benchmarks CMATH, GaoKao Math Cloze, and GaoKao Math QA. All evaluations are tested with few-shot chain-of-thought prompting.

Qwen2-Math-Instruct: Instruction-Tuned Models #

We first trained a math-specific reward model based on Qwen2-Math-72B. We then combined this dense reward signal with a binary signal indicating whether the model answered correctly. This combined signal is used as supervision for constructing the SFT data through Rejection Sampling and also in the reinforcement learning with Group Relative Policy Optimization (GRPO) after SFT.

We evaluate Qwen2-Math-Instruct on mathematical benchmarks in both English and Chinese. In addition to the widely-used benchmarks, such as GSM8K and Math, we also involve more exams that are much challenging to fully inspect the capabilities of Qwen2-Math-Instruct, such as OlympiadBench, CollegeMath, GaoKao, AIME2024, and AMC2023. For Chinese mathematical benchmarks, we use CMATH, Gaokao (Chinese college entrance examination 2024), and CN Middle School 24 (China High School Entrance Examination 2024).

We report greedy , Maj@8 and RM@8 performance on all benchmarks in the zero-shot setting, except for the multi-choice benchmarks (including MMLU STEM and multiple-choice problems in GaoKao and CN Middle School 24) with a 5-shot setting. Qwen2-Math-Instruct achieves the best performance among models of the same size, with RM@8 outperforming Maj@8, particularly in the 1.5B and 7B models. This demonstrates the effectiveness of our math reward model.

In more complex mathematical competition evaluations such as AIME 2024 and AMC 2023, Qwen2-Math-Instruct also performs well across various settings, including Greedy, Maj@64, RM@64, and RM@256.

Case Study #

Here we list some test cases, which include some IMO math problems. From the experimental results and case study, we find that Qwen2-Math is capable of solving simple math competition problems. Feel free to click the expandable blocks to check the cases!

All the solutions are generated by our model without modification. Please note that we do not guarantee the correctness of the claims in the process.

What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with

$$ x^3_1+x^3_2+,\ldots,+x^3_t=2002^{2002}, $$

Qwen2-Math-72B-Instruct Solution

To determine the smallest positive integer $ t $ such that there exist integers $ x_1, x_2, \ldots, x_t $ satisfying the equation

$$ x_1^3 + x_2^3 + \cdots + x_t^3 = 2002^{2002}, $$

we start by analyzing the properties of cubes and their sums. We need to find the smallest $ t $ for which the sum of $ t $ cubes equals $ 2002^{2002} $.

First, we note that $ 2002 $ can be factored into prime factors:

$$ 2002 = 2 \times 7 \times 11 \times 13. $$

$$ 2002^{2002} = (2 \times 7 \times 11 \times 13)^{2002} = 2^{2002} \times 7^{2002} \times 11^{2002} \times 13^{2002}. $$

Next, we consider the properties of cubes modulo 9. Cubes modulo 9 can only be congruent to $ 0, 1, $ or $ -1 $. This is because:

$$ \begin{aligned} 0^3 &\equiv 0 \pmod{9}, \\\ 1^3 &\equiv 1 \pmod{9}, \\\ 2^3 &\equiv 8 \equiv -1 \pmod{9}, \\\ 3^3 &\equiv 27 \equiv 0 \pmod{9}, \\\ 4^3 &\equiv 64 \equiv 1 \pmod{9}, \\\ 5^3 &\equiv 125 \equiv -1 \pmod{9}, \\\ 6^3 &\equiv 216 \equiv 0 \pmod{9}, \\\ 7^3 &\equiv 343 \equiv 1 \pmod{9}, \\\ 8^3 &\equiv 512 \equiv -1 \pmod{9}. \end{aligned} $$

Now, we need to determine $ 2002^{2002} \mod 9 $. First, we find $ 2002 \mod 9 $:

$$ 2002 \div 9 = 222 \text{ remainder } 4 \quad \Rightarrow \quad 2002 \equiv 4 \pmod{9}. $$

$$ 2002^{2002} \equiv 4^{2002} \pmod{9}. $$

Next, we use Euler’s theorem, which states that $ a^{\phi(n)} \equiv 1 \pmod{n} $ for $ \gcd(a, n) = 1 $. Here, $ \phi(9) = 6 $, so:

$$ 4^6 \equiv 1 \pmod{9}. $$

We can reduce the exponent $ 2002 \mod 6 $:

$$ 2002 \div 6 = 333 \text{ remainder } 4 \quad \Rightarrow \quad 2002 \equiv 4 \pmod{6}. $$

$$ 4^{2002} \equiv 4^4 \pmod{9}. $$

$$ 4^4 = 256. $$

Finally, we find $ 256 \mod 9 $:

$$ 256 \div 9 = 28 \text{ remainder } 4 \quad \Rightarrow \quad 256 \equiv 4 \pmod{9}. $$

$$ 2002^{2002} \equiv 4 \pmod{9}. $$

We need to find the smallest $ t $ such that the sum of $ t $ cubes is congruent to 4 modulo 9. The possible sums of cubes modulo 9 are:

$$ \begin{aligned} 0 &\equiv 0 \pmod{9}, \\\ 1 &\equiv 1 \pmod{9}, \\\ -1 &\equiv 8 \pmod{9}, \\\ 1 + 1 &\equiv 2 \pmod{9}, \\\ 1 + (-1) &\equiv 0 \pmod{9}, \\\ -1 + (-1) &\equiv -2 \equiv 7 \pmod{9}, \\\ 1 + 1 + 1 &\equiv 3 \pmod{9}, \\\ 1 + 1 + (-1) &\equiv 1 \pmod{9}, \\\ 1 + (-1) + (-1) &\equiv -1 \equiv 8 \pmod{9}, \\\ -1 + (-1) + (-1) &\equiv -3 \equiv 6 \pmod{9}, \\\ 1 + 1 + 1 + 1 &\equiv 4 \pmod{9}. \end{aligned} $$

We see that the sum of four cubes can be congruent to 4 modulo 9. Therefore, the smallest $ t $ is 4.

Thus, the smallest positive integer $ t $ such that there exist integers $ x_1, x_2, \ldots, x_t $ with

$$ x_1^3 + x_2^3 + \cdots + x_t^3 = 2002^{2002} $$

$$ \boxed{4}. $$

For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $ 0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?

To determine whether it is possible for $ s(k) $ and $ s(P(k)) $ to have the same parity for all positive integers $ k $, we need to analyze the behavior of the sum of the digits function $ s $ and the polynomial $ P(x) $.

First, recall that the sum of the digits of a number $ k $ modulo 2 is equivalent to $ k \mod 2 $. This is because the sum of the digits of $ k $ in base 10 is congruent to $ k \mod 9 $, and since $ 9 \equiv 1 \mod 2 $, the sum of the digits modulo 2 is the same as $ k \mod 2 $.

Given the polynomial $ P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 $, we need to determine the parity of $ P(k) $ for any positive integer $ k $. Specifically, we need to check if $ P(k) \mod 2 $ is the same as $ k \mod 2 $.

Let’s consider the polynomial modulo 2:

$$ P(x) \equiv x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \mod 2. $$

Since $ a_i $ are positive integers, $ a_i \mod 2 $ is either 0 or 1. Let $ b_i = a_i \mod 2 $. Then we can rewrite the polynomial modulo 2 as:

$$ P(x) \equiv x^n + b_{n-1}x^{n-1} + \cdots + b_1x + b_0 \mod 2. $$

Now, we need to check the parity of $ P(k) $ for $ k \equiv 0 \mod 2 $ and $ k \equiv 1 \mod 2 $.

Case 1: $ k \equiv 0 \mod 2 $

$$ P(0) \equiv b_0 \mod 2. $$

For $ s(P(0)) $ to have the same parity as $ s(0) $, $ b_0 $ must be 0. Therefore, $ a_0 $ must be even.

Case 2: $ k \equiv 1 \mod 2 $

$$ P(1) \equiv 1^n + b_{n-1} \cdot 1^{n-1} + \cdots + b_1 \cdot 1 + b_0 \equiv 1 + b_{n-1} + \cdots + b_1 + b_0 \mod 2. $$

For $ s(P(1)) $ to have the same parity as $ s(1) $, $ 1 + b_{n-1} + \cdots + b_1 + b_0 $ must be odd. Since $ 1 $ is odd, the sum $ b_{n-1} + \cdots + b_1 + b_0 $ must be even.

From the above cases, we have:

• $ a_0 $ must be even.
• The sum $ b_{n-1} + \cdots + b_1 + b_0 $ must be even.

However, if $ a_0 $ is even, then $ b_0 = 0 $. This means the sum $ b_{n-1} + \cdots + b_1 $ must be even. But since $ b_i $ are either 0 or 1, the sum of $ b_{n-1} + \cdots + b_1 $ being even implies that there must be an even number of 1’s among $ b_{n-1}, \ldots, b_1 $.

Now, consider the polynomial $ P(x) = x^n + 1 $. For this polynomial:

• $ a_0 = 1 $ (odd), which contradicts our requirement that $ a_0 $ must be even.
• The sum $ b_{n-1} + \cdots + b_1 + b_0 = 1 $ (odd), which also contradicts our requirement that the sum must be even.

Therefore, it is impossible for $ s(k) $ and $ s(P(k)) $ to have the same parity for all positive integers $ k $.

Thus, the final answer is:

$$ \boxed{\text{No}} $$

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for each $x \in \mathbb{R}^+$, there is exactly one $y \in \mathbb{R}^+$ satisfying

$$ xf(y)+yf(x) \leq 2 $$

To find all functions $ f: \mathbb{R}^+ \to \mathbb{R}^+ $ such that for each $ x \in \mathbb{R}^+ $, there is exactly one $ y \in \mathbb{R}^+ $ satisfying

$$ xf(y) + yf(x) \leq 2, $$

we will proceed step by step.

Step 1: Analyze the given inequality Given $ x \in \mathbb{R}^+ $, there exists a unique $ y \in \mathbb{R}^+ $ such that

$$ xf(y) + yf(x) \leq 2. $$

We need to explore the implications of this condition.

Step 2: Consider the case $ x = y $ Substitute $ y = x $ into the inequality:

$$ xf(x) + xf(x) = 2xf(x) \leq 2. $$

This simplifies to:

$$ xf(x) \leq 1. $$

Thus, for all $ x \in \mathbb{R}^+ $,

$$ f(x) \leq \frac{1}{x}. $$

Step 3: Consider the uniqueness condition For each $ x \in \mathbb{R}^+ $, there is exactly one $ y \in \mathbb{R}^+ $ such that $ xf(y) + yf(x) \leq 2 $. We need to ensure that this $ y $ is unique.

Step 4: Test the function $ f(x) = \frac{1}{x} $ Consider the function $ f(x) = \frac{1}{x} $. Substitute $ f(x) = \frac{1}{x} $ and $ f(y) = \frac{1}{y} $ into the inequality:

$$ x \cdot \frac{1}{y} + y \cdot \frac{1}{x} = \frac{x}{y} + \frac{y}{x}. $$

$$ \frac{x}{y} + \frac{y}{x} \leq 2. $$

Using the AM-GM inequality, we know:

$$ \frac{x}{y} + \frac{y}{x} \geq 2. $$

Equality holds if and only if $ \frac{x}{y} = 1 $, which implies $ x = y $.

Thus, for $ f(x) = \frac{1}{x} $, the inequality $ xf(y) + yf(x) \leq 2 $ holds if and only if $ x = y $. This satisfies the condition that there is exactly one $ y $ for each $ x $.

Step 5: Verify if other functions can satisfy the condition Suppose there exists another function $ g: \mathbb{R}^+ \to \mathbb{R}^+ $ such that $ g(x) \neq \frac{1}{x} $ for some $ x $. Then, for some $ x $, $ g(x) < \frac{1}{x} $ or $ g(x) > \frac{1}{x} $.

• If $ g(x) < \frac{1}{x} $, then $ xg(y) + yg(x) $ might not satisfy the inequality for all $ y $.
• If $ g(x) > \frac{1}{x} $, then $ xg(y) + yg(x) $ might exceed 2 for some $ y $.

Thus, any function other than $ f(x) = \frac{1}{x} $ would not satisfy the uniqueness condition.

Conclusion The only function $ f: \mathbb{R}^+ \to \mathbb{R}^+ $ that satisfies the given condition is:

$$ \boxed{f(x) = \frac{1}{x}}. $$

Let $Z$ be the set of all integers. Find all the function $f: Z \to Z$ such that

$$ f(4x+3y)=f(3x+y)+f(x+2y) $$

For all integers $x,y$

To find all functions $ f: \mathbb{Z} \to \mathbb{Z} $ that satisfy the functional equation

$$ f(4x + 3y) = f(3x + y) + f(x + 2y) $$

for all integers $ x $ and $ y $, we will proceed step by step.

Step 1: Initial Substitution

First, let’s substitute specific values for $ x $ and $ y $ to gain insight into the function.

Substitution 1: $ x = 0 $

$$ f(4 \cdot 0 + 3y) = f(3 \cdot 0 + y) + f(0 + 2y) $$

$$ f(3y) = f(y) + f(2y) $$

Substitution 2: $ y = 0 $

$$ f(4x + 3 \cdot 0) = f(3x + 0) + f(x + 2 \cdot 0) $$

$$ f(4x) = f(3x) + f(x) $$

Step 2: Exploring Further Substitutions

We now have two equations:

• $ f(3y) = f(y) + f(2y) $
• $ f(4x) = f(3x) + f(x) $

Substitution 3: $ y = 1 $ in the first equation

$$ f(3 \cdot 1) = f(1) + f(2 \cdot 1) $$

$$ f(3) = f(1) + f(2) $$

Substitution 4: $ x = 1 $ in the second equation

$$ f(4 \cdot 1) = f(3 \cdot 1) + f(1) $$

$$ f(4) = f(3) + f(1) $$

Step 3: Using the Equations to Find a Pattern

Substitute $ f(3) $ from the first equation into the second equation:

$$ f(4) = (f(1) + f(2)) + f(1) $$

$$ f(4) = 2f(1) + f(2) $$

Step 4: Generalizing the Function

We hypothesize that $ f $ might be a linear function. Let’s assume $ f(x) = kx $ for some integer $ k $.

Verification Substitute $ f(x) = kx $ into the original functional equation:

$$ f(4x + 3y) = k(4x + 3y) $$

$$ f(3x + y) = k(3x + y) $$

$$ f(x + 2y) = k(x + 2y) $$

The original equation becomes:

$$ k(4x + 3y) = k(3x + y) + k(x + 2y) $$

$$ 4kx + 3ky = 3kx + ky + kx + 2ky $$

$$ 4kx + 3ky = 4kx + 3ky $$

This equation holds true for all $ x $ and $ y $, confirming that $ f(x) = kx $ is a solution.

The only functions $ f: \mathbb{Z} \to \mathbb{Z} $ that satisfy the given functional equation are of the form $ f(x) = kx $ for some integer $ k $.

$$ \boxed{f(x) = kx \text{ for some integer } k} $$

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$ f(f(x)+y)+xf(y)=f(xy+y)+f(x) $$

Qwen2-Math-72B-Instruct Solution:

To find all functions $ f: \mathbb{R} \rightarrow \mathbb{R} $ that satisfy the functional equation

$$ f(f(x) + y) + x f(y) = f(xy + y) + f(x), $$

First, let’s substitute $ y = 0 $ into the equation:

$$ f(f(x) + 0) + x f(0) = f(x \cdot 0 + 0) + f(x). $$

$$ f(f(x)) + x f(0) = f(0) + f(x). $$

Rearranging terms, we get:

$$ f(f(x)) = f(x) + f(0) - x f(0). $$

Step 2: Analyzing the Simplified Equation

Let $ c = f(0) $. The equation becomes:

$$ f(f(x)) = f(x) + c - x c. $$

Step 3: Substituting $ x = 0 $

Next, substitute $ x = 0 $ into the original equation:

$$ f(f(0) + y) + 0 \cdot f(y) = f(0 \cdot y + y) + f(0). $$

$$ f(c + y) = f(y) + c. $$

Step 4: Substituting $ y = -c $

Substitute $ y = -c $ into the equation $ f(c + y) = f(y) + c $:

$$ f(c - c) = f(-c) + c. $$

$$ f(0) = f(-c) + c. $$

Since $ f(0) = c $, we have:

$$ c = f(-c) + c. $$

Subtracting $ c $ from both sides, we get:

$$ f(-c) = 0. $$

Step 5: Substituting $ y = -f(x) $

Substitute $ y = -f(x) $ into the original equation:

$$ f(f(x) - f(x)) + x f(-f(x)) = f(x(-f(x)) - f(x)) + f(x). $$

$$ f(0) + x f(-f(x)) = f(-x f(x) - f(x)) + f(x). $$

$$ c + x f(-f(x)) = f(-x f(x) - f(x)) + f(x). $$

Step 6: Considering $ f(x) = 0 $

Assume $ f(x) = 0 $ for all $ x $. Substitute $ f(x) = 0 $ into the original equation:

$$ f(0 + y) + x f(y) = f(xy + y) + f(x). $$

$$ f(y) + 0 = f(y) + 0, $$

which is always true.

Step 7: Verifying $ f(x) = x $

Assume $ f(x) = x $. Substitute $ f(x) = x $ into the original equation:

$$ f(x + y) + x f(y) = f(xy + y) + f(x). $$

$$ x + y + xy = xy + y + x, $$

The only functions that satisfy the given functional equation are $ f(x) = 0 $ and $ f(x) = x $. Therefore, the solutions are:

$$ \boxed{f(x) = 0 \text{ or } f(x) = x}. $$

A long time ago, there existed Martians with $ 3$ different colours: red, green and blue. As Mars was devastated by an intergalactic war, only $ 2$ Martians of each colours survived. In order to reconstruct the Martian population, they decided to use a machine that transforms two Martians of distinct colours into four Martians of colour different to the two initial ones. For example, if a red Martian and a blue Martian use the machine, they’ll be transformed into four green Martians.

a) Is it possible that, after using that machine finitely many times, we have $ 2022 $ red Martians, $ 2022$ green Martians and $ 2022$ blue Martians?

b) Is it possible that, after using that machine finitely many times, we have $ 2021$ red Martians, $ 2022$ green Martians and $ 2023$ blue Martians?

Let’s analyze the problem step by step.

We start with 2 red, 2 green, and 2 blue Martians. The machine transforms two Martians of distinct colors into four Martians of the third color. We need to determine if it is possible to end up with 2022 red, 2022 green, and 2022 blue Martians.

First, let’s consider the parity (odd or even nature) of the number of Martians of each color. Initially, we have:

• 2 red (even)
• 2 green (even)
• 2 blue (even)

Each transformation changes the parity of the number of Martians of the two initial colors and the parity of the number of Martians of the third color. Specifically:

• If we transform 1 red and 1 green Martian, we get 4 blue Martians. This changes the parity of red and green Martians from even to odd, and the parity of blue Martians from even to odd.
• If we transform 1 red and 1 blue Martian, we get 4 green Martians. This changes the parity of red and blue Martians from even to odd, and the parity of green Martians from even to odd.
• If we transform 1 green and 1 blue Martian, we get 4 red Martians. This changes the parity of green and blue Martians from even to odd, and the parity of red Martians from even to odd.

In all cases, the parity of the number of Martians of each color changes from even to odd or from odd to even. Therefore, after any number of transformations, the parity of the number of Martians of each color will always be the same as it was initially.

Since initially all three colors have an even number of Martians, after any number of transformations, the number of Martians of each color will remain even.

Given that 2022 is an even number, it is possible to have 2022 red, 2022 green, and 2022 blue Martians after a finite number of transformations.

Thus, the answer to part (a) is:

$$ \boxed{\text{Yes}} $$

We start again with 2 red, 2 green, and 2 blue Martians. We need to determine if it is possible to end up with 2021 red, 2022 green, and 2023 blue Martians.

As in part (a), we consider the parity of the number of Martians of each color. Initially, we have:

Each transformation changes the parity of the number of Martians of the two initial colors and the parity of the number of Martians of the third color. Therefore, after any number of transformations, the parity of the number of Martians of each color will always be the same as it was initially.

Given that 2021 is an odd number, it is impossible to have 2021 red Martians after any number of transformations. Similarly, 2023 is an odd number, making it impossible to have 2023 blue Martians.

Thus, the answer to part (b) is:

Joe and Penny play a game. Initially there are $ 5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $ k$-th move, any number of stones between $ 1$ and $ k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?

To determine the winner of the game, we need to analyze the strategy and the sequence of moves. Let’s break down the problem step by step.

Understanding the Game Rules:

• Initially, there are 5000 stones.
• Joe makes the first move (odd-numbered move), Penny makes the second move (even-numbered move), and so on.
• On the $ k $-th move, a player can remove between 1 and $ k $ stones.
• The player who removes the last stone wins.

Key Insight:

• The game is symmetric in the sense that the number of stones a player can remove on their turn increases linearly with the move number.
• The total number of moves in the game is equal to the number of stones initially in the pile, which is 5000.

Total Number of Moves:

• Since there are 5000 stones, there will be 5000 moves in total.
• Joe makes the odd-numbered moves: 1st, 3rd, 5th, …, 4999th.
• Penny makes the even-numbered moves: 2nd, 4th, 6th, …, 5000th.

Determining the Winner:

• The last move (5000th move) is an even-numbered move.
• Therefore, Penny will make the last move.

Conclusion:

• Since Penny makes the last move, she will be the one to remove the last stone.
• Hence, Penny wins the game if both players play perfectly.

$$ \boxed{\text{Penny}} $$

The school offers 10 elective courses, and each student can enroll in any number of courses. The director selects $k$ students, where although each student’s combination of courses is different, any two students have at least one course in common. At this point, it is found that any student outside these $k$ students cannot be classmates with these $k$ students regardless of how they enroll (having one course in common is enough to be classmates). Find the maximum number of $k$.

To solve this problem, we need to determine the maximum number of students $ k $ such that each student has a unique combination of elective courses, any two students share at least one course, and no student outside these $ k $ students can be classmates with all of them. This means that the set of students chosen must cover all possible courses in such a way that any new student would have to share a course with at least one of the chosen students.

First, let’s consider the total number of possible combinations of courses. Since there are 10 elective courses and each student can either take or not take each course, there are $ 2^{10} = 1024 $ possible combinations of courses. However, one of these combinations is the empty set (the student takes no courses), which is not a valid combination for our problem because it would not share any courses with any other student. Therefore, there are $ 1024 - 1 = 1023 $ non-empty combinations of courses.

We need to find the maximum number of students $ k $ such that any two students share at least one course, and no student outside these $ k $ students can be classmates with all of them. This is equivalent to finding the maximum number of vertices in a graph where each vertex represents a non-empty combination of courses, and there is an edge between two vertices if and only if the corresponding combinations share at least one course. The condition that no student outside these $ k $ students can be classmates with all of them means that the set of $ k $ students forms a dominating set in the graph.

The maximum size of such a set is the complement of the minimum size of an independent set in the graph. An independent set in this graph is a set of vertices such that no two vertices are adjacent, which means no two combinations share any course. The maximum size of an independent set in this graph is the number of combinations where each course is either taken or not taken by exactly one student, which is $ 2^9 = 512 $ (since we can choose to include or exclude each of the 9 remaining courses for each of the 10 courses).

Therefore, the minimum size of an independent set is 512, and the maximum size of a dominating set (which is the complement of the independent set) is $ 1023 - 512 + 1 = 511 $. However, we need to add 1 because the empty set is not included, so the correct calculation is $ 1023 - 511 = 512 $.

Thus, the maximum number of students $ k $ is $\boxed{512}$.

Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.

To determine whether it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, and $IE$ to all have integer lengths in triangle $ABC$ with $\angle A = 90^\circ$, we start by analyzing the given conditions and properties of the triangle.

• $\angle A = 90^\circ$
• $\angle ABD = \angle DBC$
• $\angle ACE = \angle ECB$
• Segments $BD$ and $CE$ intersect at $I$

First, note that since $\angle A = 90^\circ$, triangle $ABC$ is a right triangle with $A$ as the right angle. Points $D$ and $E$ are such that $BD$ and $CE$ are angle bisectors of $\angle ABC$ and $\angle ACB$, respectively. Therefore, $I$ is the incenter of triangle $ABC$.

Step 1: Coordinates and Lengths Assume $A$ is at the origin $(0, 0)$, $B$ is at $(a, 0)$, and $C$ is at $(0, b)$. The lengths of the sides are:

• $BC = \sqrt{a^2 + b^2}$

Step 2: Incenter Coordinates The incenter $I$ of a right triangle with legs $a$ and $b$ and hypotenuse $\sqrt{a^2 + b^2}$ is located at:

$$ I = \left( \frac{a}{2}, \frac{b}{2} \right) $$

Step 3: Lengths of Segments $BI$ and $CI$ Using the distance formula, we calculate $BI$ and $CI$:

$$ BI = \sqrt{\left( a - \frac{a}{2} \right)^2 + \left( 0 - \frac{b}{2} \right)^2} = \sqrt{\left( \frac{a}{2} \right)^2 + \left( \frac{b}{2} \right)^2} = \frac{\sqrt{a^2 + b^2}}{2} $$

$$ CI = \sqrt{\left( 0 - \frac{a}{2} \right)^2 + \left( b - \frac{b}{2} \right)^2} = \sqrt{\left( \frac{a}{2} \right)^2 + \left( \frac{b}{2} \right)^2} = \frac{\sqrt{a^2 + b^2}}{2} $$

Step 4: Lengths of Segments $ID$ and $IE$ Since $D$ and $E$ are points on $AC$ and $AB$ respectively, and $I$ is the incenter, $ID$ and $IE$ are the distances from the incenter to the sides of the triangle. For a right triangle, these distances are given by:

$$ ID = IE = r = \frac{a + b - \sqrt{a^2 + b^2}}{2} $$

Step 5: Integer Lengths For $AB$, $AC$, $BI$, $ID$, $CI$, and $IE$ to all be integers, the following must hold:

• $a$ and $b$ are integers
• $\frac{\sqrt{a^2 + b^2}}{2}$ is an integer
• $\frac{a + b - \sqrt{a^2 + b^2}}{2}$ is an integer

Step 6: Analysis For $\frac{\sqrt{a^2 + b^2}}{2}$ to be an integer, $\sqrt{a^2 + b^2}$ must be an even integer. Let $\sqrt{a^2 + b^2} = 2k$ for some integer $k$. Then:

$$ a^2 + b^2 = 4k^2 $$

For $\frac{a + b - \sqrt{a^2 + b^2}}{2}$ to be an integer, $a + b - 2k$ must be even. Since $a$ and $b$ are integers, $a + b$ must be even.

Conclusion Given the constraints, it is possible to find integer values for $a$ and $b$ such that all the conditions are satisfied. For example, if $a = 3$ and $b = 4$, then:

• $BI = CI = \frac{5}{2} = 2.5$ (not an integer)

Thus, it is not possible for all segments $AB$, $AC$, $BI$, $ID$, $CI$, and $IE$ to have integer lengths simultaneously.

Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle ${\angle{CGH}}$.

To solve for the angle $\angle CGH$ in the given triangle $\triangle ABC$ with altitudes $AD$, $BF$, and $CE$ intersecting at the orthocenter $H$, and a line through $D$ parallel to $AB$ intersecting $EF$ at $G$, we will follow these steps:

Identify Key Points and Properties:

• $D$ is the foot of the altitude from $A$ to $BC$.
• $E$ is the foot of the altitude from $B$ to $AC$.
• $F$ is the foot of the altitude from $C$ to $AB$.
• $H$ is the orthocenter of $\triangle ABC$.
• $G$ is the intersection of the line through $D$ parallel to $AB$ and the line $EF$.

Use the Property of the Orthocenter:

• The orthocenter $H$ lies on the altitudes $AD$, $BF$, and $CE$.
• The points $D$, $E$, and $F$ are collinear with $H$ on the respective altitudes.

Analyze the Parallel Line:

• Since $DG \parallel AB$, $\angle ADG = \angle DAB$ (corresponding angles).

Consider the Cyclic Quadrilateral:

• The quadrilateral $AEHF$ is cyclic because $\angle AEH = \angle AFH = 90^\circ$.
• Therefore, $\angle EHF = 180^\circ - \angle EAF$.

Use the Parallel Line Property Again:

• Since $DG \parallel AB$, $\angle DGE = \angle BAE$ (alterFnate interior angles).

Relate Angles in Cyclic Quadrilateral:

• In $\triangle AEF$, $\angle EAF = \angle EHF$ because they are opposite angles in the cyclic quadrilateral $AEHF$.

Determine $\angle CGH$:

• Since $G$ lies on $EF$ and $DG \parallel AB$, $\angle DGE = \angle BAE$.
• $\angle BAE = \angle EAF$ (as established).
• Therefore, $\angle DGE = \angle EAF$.

Final Angle Calculation:

• Since $\angle EAF = \angle EHF$ and $\angle EHF = 90^\circ$ (as $H$ is the orthocenter and $E$ and $F$ are feet of perpendiculars), we have: $$ \angle CGH = \angle EHF = 90^\circ. $$

Thus, the angle $\angle CGH$ is $\boxed{90^\circ}$.

Decontamination #

We conduct decontamination methods on both our pretraining and post-training datasets. Specifically, for pretraining data, we target on math datasets, including GSM8K, MATH, and remove samples that have significant overlaps with the test sets. We use exact match to remove the identical samples and further apply 13-gram deduplication (with a condition that the ratio of longest common sequence should be larger than 0.6) to remove more samples that might cause contamination. For post-training data, we remove more postitive contaminated samples that have overlaps with GSM8K, MATH, Aqua, SAT Math, OlympiadBench, College Math, AIME24, AMC23, etc. with the same filtering method.

This time, we’re releasing a new model series focused on mathematical capabilities, Qwen2-Math, built upon the Qwen2 foundation. Our flagship model, Qwen2-Math-72B-Instruct, outperforms proprietary models such as GPT-4o and Claude 3.5 in math-related tasks. Given the current limitation of English-only support, we plan to release bilingual models that support both English and Chinese shortly, with the development of multilingual models also in the pipeline. Moreover, we will continue to enhance our models’ ability to solve complex and challenging mathematical problems.

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Harris Chooses Walz

A guide to the career, politics and sudden stardom of gov. tim walz of minnesota, now vice president kamala harris’s running mate..

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.

Hey, it’s Michael. Before we get started, I want to tell you about another show made by “The New York Times” that pairs perfectly with “The Daily.” It’s called “The Headlines.” It’s a show hosted by my colleague, Tracy Mumford, that quickly catches you up on the day’s top stories and features insights from “The Times” reporters who are covering them, all in about 10 minutes or less.

So if you like “The Daily”— and if you’re listening, I have to assume you do — I hope that means you’re going to “The Headlines” as well. You can now find “The Headlines” wherever you get your podcasts. So find it, subscribe to it, and thank you. And now, here’s today’s “Daily.”

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

[MUSIC PLAYING]

Today, the story of how a little known Midwestern governor became Kamala Harris’s choice for a running mate. My colleague Ernesto Londoño walks us through the career, politics, and sudden stardom of Governor Tim Walz of Minnesota.

It’s Wednesday, August 7.

Ernesto, over the past few days, we watched Vice President Harris bring the final three contenders for her running mate to her house in Washington, DC, for a set of in-person interviews. And then we watched as she seemed to narrow her pool of choices down to a final two — the governor of Pennsylvania, Josh Shapiro, and the governor of Minnesota, Tim Walz. And now, of course, we know that she has made her choice. What has she told us about her campaign strategy, the way she views this race, in ultimately choosing Tim Walz?

Michael, I think what the choice tells us is that Kamala Harris was drawn to two qualities that Governor Walz brings to the table. And what’s interesting is they may seem to be in tension. For starters, here’s the ultimate everyday man, somebody who grew up in a small town in Middle America, served in the National Guard, was a high school teacher, a football coach, very plain-spoken, goes to campaign events wearing T-shirts and baseball caps, is a gun owner and very proud about it. He sort of embodies the Midwest.

And she clearly thinks that that is going to bring the kind of moderate, white, working class voters that the campaign needs in swing states to come to them, to make this feel like a balanced ticket and something that will give her enough of the crucial votes to defeat Donald Trump in the fall.

On the other hand, as governor, he passed a slew of pretty progressive legislation in the past couple of years, everything from abortion rights to gun control. So these things are likely to appeal to bread and butter Democrats.

But the question is, when voters have examined these two facets of Tim Walz, may it bring them enough enthusiasm from the base and enough undecided voters that the campaign desperately needs, or at some point, do these two aspects of him start canceling each other out?

Right. In short, you’re saying Harris is betting on a dual appeal from Walz to two essential constituencies, but the risk is that the appeal to one of them is just much, much greater than to the other.

Right. You could definitely see a scenario where voters, once they’ve examined Tim Walz’s story and legacy, may conclude that both of these candidates are quite liberal.

OK, so tell us the story of Tim Walz, a story that I think a lot of us don’t know because we really don’t know Walz all that well, and how he has come to embody these two qualities and that tension that you just described.

Michael, the origin story of Tim Walz’s political career is quite fascinating.

He and his wife were teachers in a small city south of Minneapolis. And in 2004, when George W. Bush was running for re-election, Walz took a group of his students to a political rally in his hometown. They wanted to just see the president make his case. And a strange scuffle happened when they were trying to get in.

Well, one of the kids had a John Kerry sticker on his wallet. And this is where the individual says, well, you’re not going to be allowed to enter. You’ve been deemed a threat.

Apparently, one of the students had a sticker for Bush’s rival, John Kerry, on his wallet. And security officials at the rally didn’t want to let them in.

And I said, oh, it’s OK. They’re with me. And who are you? And I said, I’m Tim Walz. I’m their teacher here, and showed them my ID. And they said, well, you two have been deemed a threat to the president. And I said, well, that’s not true. And it kind of escalated.

And this really ticked off Tim Walz. He was really upset. There was a fight and a confrontation at the rally.

At this point in time, I’m kind of nervous. I’m getting arrested. So I’m like saying, well, I’m Teacher of the Year in Mankato. And they didn’t care about that. And it was kind of a sad epiphany moment, how it felt for people to be looked right through by people. These people didn’t see me. And this is happening.

And ultimately, he sort of walks away from this moment feeling really sick of the Bush administration, the politics of the day. And he turns around and volunteers for the Kerry campaign.

And then the more interested he becomes in politics in this era, he starts looking around his congressional district, and there’s a Republican who’s held the seat for many, many years. This was a largely rural district in southern Minnesota. And there’s no reason to believe that a newcomer to politics, somebody without a donor base, could make a run for this seat and win.

But Walz signs up for this weekend boot camp, where expert campaigners train newcomers who want to run for office. And he gets really enthused by the idea that he can pull it off. So he starts raising money with the support of an army of students who become so thrilled and energized by the prospect that their nerdy and kind geography teacher is making this uphill bid for a congressional race.

So his campaign staff is basically his former students.

That’s right. And he proves to be a formidable candidate. He draws a lot of attention to his experience in the classroom and as a coach.

When I coached football, these stands held about 3,000 people. That’s a lot. It’s also the number of American soldiers who have died fighting in Iraq.

He’s a very strong advocate for pulling out of the war in Iraq.

Serving right now are kids that I taught, coached, and trained to be soldiers. They deserve a plan for Iraq to govern itself, so they can come home.

And one thing that happens in the campaign that is really surprising to people is he comes out as being in favor of same-sex marriage. Now, it’s useful to remember that this is 2006, when the vast majority of Democrats, Democrats running for most elected office, were not ready to come out in favor of same-sex marriage.

And here’s a guy who’s new to politics, who’s trying to unseat a Republican who’s held on to his seat for more than 12 years, taking what appeared to be a reckless position on something. And when he was asked about it at the time, Tim Walz told a supporter, this just happens to be what I believe in. And I’d rather lose a race that I’ve ran being true and consistent to my values than try to run as somebody I’m not.

And of course, he wins.

Yes. To everybody’s surprise, he pulled it off.

So from the get-go, he shows a kind of maverick, “politics be damned” quality, taking stands that he knows may be unpopular among the voters he’s trying to win over. But he’s got some innate political gifts that are all making it work.

Yeah, I think that first campaign showed us that Tim Walz had real political chops. He was a very effective campaigner. And people really liked him. When he was knocking on doors, when he was introducing himself to voters, they saw him as somebody who was very genuine and who was admirable.

So once he gets elected in this conservative leaning district in Minnesota, what does he actually do in Congress?

In Congress, he develops a reputation for being somebody who can work across the aisle. And this is a period where Democrats and Republicans were deeply polarized over the Iraq War. He spends a lot of his time lobbying to expand benefits for veterans, so it’s easier for them to go to college after their service, and also becomes a leading voice in the quest to repeal Don’t Ask, Don’t Tell, the policy that prohibited openly gay servicemen from serving in uniform.

And he remained really popular. He easily won re-election five times. The last time he runs for his seat happens to be 2016, when President Trump wins his district by about 15 points.

And still, voters kept Tim Walz in office.

I think it’s important to note what you just said. Walz is distinguishing himself as a Democrat who can take some pretty progressive positions, as he did in that first campaign on gay rights, as he did with Don’t Ask, Don’t Tell, and keep winning in very Trump-friendly districts of his state.

That’s right. And as he’s serving his sixth term in office, he sets his sights on the governor’s mansion and decides to run for office in 2018. He wins that race easily. And early on, during his time as governor, the eyes of the world are on Minnesota after a police officer kills George Floyd. And what we see is massive looting and protests in Minneapolis.

Right, and remind us how Governor Walz handles that violence, those protests.

Yeah, I think that’s a crucial chapter in Tim Walz’s political career and one that will come under scrutiny in the days ahead.

After George Floyd was killed on a Monday —

People are upset, and they’re tired. And being Black in Minnesota already has a stigma and a mark on your back.

— protests took root in Minneapolis.

Y’all want to sit out here and shoot off your rubber bullets and tear gas.

And they got progressively larger and more violent.

There comes a point where the mayor and the police chief in Minneapolis plead for help. They ask the governor to send in the National Guard. And crucially, that request was not immediately heeded.

This is the third precinct here. There are fires burning to the left of it at the —

And at the height of the crisis, a police precinct building was abandoned.

There’s someone climbing up the wall right now, kicking the window in, trying to climb up the wall.

Because city officials grew concerned that protesters were about to overrun it and may attack the cops inside their own turf.

[EXPLOSIONS]

And the building is set on fire.

Right, a very memorable image. I can recall it happening in real-time.

Yeah, and in the days that followed, I think there were a lot of questions of why the governor didn’t send in troops earlier and whether a more muscular, decisive response could have averted some of the destruction that spread through the city.

And how does Walz end up explaining his decision not to send in the National Guard more quickly?

The governor and his administration have said that they were really, really dealing with an unprecedented challenge. And I think there was a concern that sending in troops into this really, really tense situation could have done more to escalate rather than pacify things on the street.

But in the weeks and months that followed, there were a lot of questions about Governor Walz’s leadership. And there were critics who said, during what may have been the most challenging week of his life, we saw a governor who was indecisive and who waited too long to send in resources that ultimately allowed the city to get to a semblance of order.

Right, and it feels like this is a moment that will almost assuredly be used against him by Donald Trump and JD Vance, the Republican ticket, which has made law and order so central to their message in this campaign.

Yeah, absolutely. And here in Minnesota, that was certainly a liability for him when he ran for re-election in 2022. But voters kept him in office, and he won that race handily. And not only did he win, but Democrats managed to flip the Senate and have full control of the legislature on his watch.

And that sets in motion one of the most productive legislative sessions in Minnesota history, where Tim Walz and his allies in the House and the Senate managed to pass a trove of really progressive legislation, oftentimes on a party vote.

Tell us about some of that legislation.

Well, Minnesota becomes the first state in the wake of the Supreme Court ending the constitutional right to abortion to actually codify this right under state statute. And they did a lot more stuff. They had a huge budget surplus, and they used that, for instance, to fund meals for all school children.

They managed to pass a couple of gun control laws that were very contentious. They gave the right to undocumented immigrants to get driver’s licenses. They legalized recreational marijuana. And finally, the governor takes a pretty bold stance on this issue of gender affirming care for transgender kids and teenagers, and says that Minnesota will be a safe haven for people who want that health care.

So, Ernesto, so how should we think about that blitz of legislation and the largely progressive tone of it, given the way that Walz had campaigned and succeeded up to that moment as somebody with such broad appeal across the political spectrum?

When the governor was asked whether this had been too much too quickly in terms of progressive legislation, his answer was that these were broadly popular policies, that these are issues Democrats had campaigned on. And here, Democrats had a window of opportunity where they were in control of the governor’s mansion and control of the House, the Senate, and that when you have political capital, you spend it.

But when you start listening to Republicans in Minnesota, they say, here’s a guy who campaigned on this mantra of “One Minnesota.” That was his campaign slogan. And he sort of came into office with this promise that he would govern in a bipartisan way, reach across the aisle.

But when they had all the votes they needed to pass their policies, Republicans felt that Walz was not bothering to bring them into the fold and to pass legislation that was going to be palatable to conservatives in the state. So I think people who once regarded him as a moderate now start seeing him as somebody who, when he had the power, acted in ways that were really progressive and liberal.

So at the height of his power, Governor Walz emerges as somebody who, when given a shot at getting done what he really wants to get done with a Democratic legislature, is a pretty progressive leader, even at the risk of being somewhat at odds with his earlier image as more moderate, because in his mind, enough people in the state are behind these policies.

Yeah, and I think he assumed that he had banked enough goodwill and that people across the state liked him enough to tolerate policies they may have disagreed with. And I think it’s safe to say, among the people who cover him here regularly, there was never any real hint that Tim Walz was eyeing a run for higher office. He’s not somebody who has written the kind of political memoir that oftentimes serves as a case of what you would bring to a national ticket or to the White House. And he seems pretty happy with a state job.

So it was a huge surprise when Tim Walz starts going viral through a string of cable news appearances right after President Biden drops out of the race, and the Democrats are scrambling to put Harris at the top of the ticket. And what becomes clear is that Walz is very forcefully auditioning for the role of vice president, and Vice President. Harris starts taking him very seriously.

We’ll be right back.

So, Ernesto, tell us about this cable news audition that Governor Walz undertakes over the past few weeks and how, ultimately, it seemed to help him land this job of being Harris’s running mate.

I think Walz does something really interesting, and that is that he says that Democrats shouldn’t be talking about Trump and Vance as existential threats. He kind of makes the case that Democrats have been in this state of fear and paralysis for too long, and that it’s not serving them well. So the word he latches onto is “weird.”

Well, it’s true. These guys are just weird.

It is. It is.

And they’re running for he-man women hater’s club or something. That’s what they go at. That’s not what people are interested in.

And I think one other thing we see in Walz is somebody who’s putting himself out there as a foil to JD Vance.

That angst that JD Vance talks about in “Hillbilly Elegy,” none of my hillbilly cousins went to Yale, and none of them went on to be venture capitalists or whatever. It’s not —

I think the case he’s making is that Tim Walz is a more authentic embodiment of small town values.

What I know is, is that people like JD Vance know nothing about small town America. My town had 400 people in it, 24 kids in my graduating class. 12 were cousins. And he gets it all wrong. It’s not about hate.

And behind the scenes, people from Tim Walz’s days on Capitol Hill start calling everybody they know in the Harris campaign and the Harris orbit and saying, here’s a guy who has executive experience as governor, but also somebody who has a really impressive record from his time on Capitol Hill and somebody who could be an asset in helping a Harris administration pass tough legislation. So you should take a hard look at this guy.

Which is, of course, exactly what Harris ends up doing. And I want to talk for a moment about how Harris announces Walz as her running mate on Tuesday morning. She did it in an Instagram message. And it felt like the way she did it very much embraced this idea that you raised earlier, Ernesto, that Walz contains these two appeals, one to the Democratic base, one to the white working class.

Harris specifically cites the work that Walz did with Republicans on infrastructure and then cites his work on gun control. She mentions that he was a football coach and the founder of the high school Gay Straight Alliance. She’s straddling these two versions of Walz.

But I want to linger on the idea for a moment of Walz’s vulnerabilities, because once he becomes Harris’s running mate, Harris and Walz are going to lose a fair amount of control over how they present him to the country, because he’s going to become the subject of very fierce attacks from the Republicans in this race. So talk about that for just a moment.

Yeah, I mean, it’s important to keep in mind that Governor Walz has never endured the scrutiny of a presidential race. So the questions he’s going to be asked and the way his record is going to be looked at is going to be different and sharper. I think the Harris campaign is billing him as, first and foremost, a fighter for the middle class. And I think that certainly will have some appeal.

But I think in coming days, there’s going to be a lot of attention drawn to parts of his record that may be unpopular with many voters. For instance, giving undocumented immigrants driver’s licenses, which Governor Walz championed. It’s likely to provide fodder for an attack ad.

The very dramatic footage of Minneapolis burning in 2020 is also something that I think people will be drawn to. And there’s going to be interest in reexamining what the governor did and what he could have done differently to avert the chaos.

And on Tuesday, we saw that the Trump campaign wasted no time in trying to define Tim Walz as soft on crime, permissive on immigration policy. And they also made clear they wanted to relitigate the era of George Floyd’s killing. And specifically, they want to try to tie him to the effort at the time to defund the police, which is a movement that Walz personally never endorsed.

So the Republican attack here will be pretty simple. Walz is liberal. Harris is liberal. So, in their efforts to speak to especially white working class and rural voters in swing states, the Trump campaign is going to say this is not the ticket for that group of voters. This is the ticket of burning police precincts and gun control. And of course, that may not be fair, but that’s very likely going to be the message over the next couple of months.

Right. I think there’s going to be effort to portray him as a radical liberal who has used his small town roots to put on this sort of veneer of being a moderate and a really sort of understanding and being part of the segments of the electorate that I think are critical in this election.

I want to speak for just a moment about the person Harris did not pick when she chose Walz because many Democrats had felt that Walz was a potentially too liberal seeming running mate for a candidate, Kamala Harris, who herself comes from a blue state and is caricatured by the Republicans as liberal herself.

And the person she didn’t choose was Governor Josh Shapiro of Pennsylvania, who was seen as having a huge appeal in that particular key swing state, but also presented risks of his own of alienating parts of the Democratic base with his well-documented support for Israel and his criticism of campus protesters. How should we think about the fact that, ultimately, Harris chose Walz over Shapiro?

Yeah, I think in the final stretch of this campaign to be the vice presidential pick, we started seeing a lot of acrimony in pockets of the Democratic base, drawing attention to the fact that Governor Shapiro could be divisive on Gaza, which has really sort of split the party in recent months.

So I think at the end of the day, they made a calculation that Tim Walz would be more of a unifying figure and would be somebody who would inspire and energize enough pockets of the electorate that they need, particularly in the Midwest, to make him the stronger and more exciting pick and somebody who wouldn’t force them to go back to defending and relitigating the Biden administration’s record on Israel and on the war in Gaza.

Right, and then, on Tuesday night, we got our first glimpse of Harris and Walz together on stage for the first time at a campaign rally. I’m curious, what struck you about their debut together.

Good evening, Philadelphia.

I think everybody was watching the opening scene of this rally to see what the chemistry between these two people was going to be like. And they both seemed giddy. They were literally, at times, bouncing with enthusiasm.

Since the day that I announced my candidacy, I set out to find a partner who can help build this brighter future.

So Pennsylvania, I’m here today because I found such a leader.

Governor Tim Walz of the great state of Minnesota.

They soon got down to business. And that business was how to define Tim Waltz for voters who don’t know him well.

To those who know him best, Tim is more than a governor.

And right off the bat, we saw that Kamala Harris really highlighted a lot of pieces of his pre-political career.

To his former high school football players, he was Coach.

She repeatedly called him Coach Walz, Mr. Walz, evoking his time in the classroom, and even used his military title from his days in the Army.

To his fellow veterans, he is Sergeant Major Walz.

And then when it came time for Tim Walz to introduce himself on this massive stage —

Welcome the next vice president of the United States, Tim Walz.

— he drew a lot of attention to his small town roots.

I was born in West Point, Nebraska. I lived in Butte, a small town of 400.

He said something that he said repeatedly recently in campaign appearances, which is —

In Minnesota, we respect our neighbors and their personal choices that they make. Even if we wouldn’t make the same choice for ourselves, there’s a golden rule — mind your own damn business.

The golden rule of small towns is you mind your own damn business, which is something he said in the context of his argument that Republicans have been limiting, rather than expanding, people’s rights. But he also drew attention to the fact that he’s a gun owner.

By the way, as you heard, I was one of the best shots in Congress. But in Minnesota, we believe in the Second Amendment, but we also believe in common sense gun violence laws.

And then when it came time to draw a sharp contrast with their opponents, Tim Walz said, these guys are phonies.

Donald Trump is not fighting for you or your family. He never sat at that kitchen table like the one I grew up at, wondering how we were going to pay the bills. He sat at his country club in Mar-a-Lago, wondering how he can cut taxes for his rich friends.

He said it’s actually people like me and Kamala Harris who come from humble origins and showed what is possible in America when you hail from a working class background, and you seize opportunities that were available to you.

Thank you, Philadelphia. Thank you, Vice President. God bless America.

So when it comes to this question of Walz’s dual identities and dual appeals, what did we learn on day one of this new Democratic ticket, do you think?

I think the campaign is trying to convey that these two facets of Tim Walz’s life are not mutually exclusive, that they don’t need to be in tension. They don’t cancel each other out. They’re both part of Tim Walz’s story. And I think that’s how they’re going to present him from now until Election Day.

Ernesto, thank you very much. We appreciate it.

It’s my pleasure, Michael.

Here’s what else you need to know today. On Tuesday, Hamas said that Yahya Sinwar, one of the masterminds behind the deadly October 7 attacks on Israel, had consolidated his power over the entire organization. Until now, Sinwar had held the title of Hamas’s leader in Gaza. But with the assassination of Hamas’s top political leader by Israel last week, Hamas said that Sinwar would take on that title as well. Sinwar remains a major target of Israel and is believed to have been hiding in tunnels underneath Gaza since October 7.

And the US Department of Justice has charged a Pakistani man with ties to Iran with trying to hire a hitman to assassinate political figures in the United States. The man recently traveled to the US and was arrested in New York last month. American authorities believe that his potential targets likely included former President Trump.

Today’s episode was produced by Alex Stern, Eric Krupke, and Olivia Natt. It was edited by Lisa Chow and Patricia Willens, contains original music by Pat McCusker and Marion Lozano, and was engineered by Alyssa Moxley. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Nick Pittman and Minnesota Public Radio.

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

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Earlier this summer, few Democrats could have identified Gov. Tim Walz of Minnesota.

But, in a matter of weeks, Mr. Walz has garnered an enthusiastic following in his party, particularly among the liberals who cheer on his progressive policies. On Tuesday, Vice President Kamala Harris named him as her running mate. Ernesto Londoño, who reports for The Times from Minnesota, walks us through Mr. Walz’s career, politics and sudden stardom.

On today’s episode

Ernesto Londoño , a reporter for The Times based in Minnesota, covering news in the Midwest.

Background reading

Who is Tim Walz , Kamala Harris’s running mate?

Mr. Walz has faced criticism for his response to the George Floyd protests.

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, Michael 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, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, 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, Maddy Masiello, Isabella Anderson, Nina Lassam and Nick Pitman.

An earlier version of this episode misstated the subject that Walz’s wife taught. She taught English, not Social Studies.

How we handle corrections

Ernesto Londoño is a Times reporter based in Minnesota, covering news in the Midwest and drug use and counternarcotics policy. More about Ernesto Londoño

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