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Common Core - State Standards Initiative

Being able to round and estimate numbers enables people to perform calculations in their head quickly and without using a calculator. Math Games motivates students to practice and hone this important skill by blending learning with play in its appealing online games!

Pupils can use our resources to practice:

• Understanding, identifying and comparing fractions of numbers and shapes
• Making equivalent fractions and reducing fractions
• Performing calculations with fractions and mixed numbers
• Converting between fractions, percents, decimals and mixed numbers

Other resources for teachers and parents include downloadable worksheets and apps, and a digital textbook. We have activities for every grade and ability level, and all of them are aligned with the Common Core State Standards for Mathematics. Choose a skill to start playing!

Fraction Talks

Description

• Author: Dan
• Posted: Mon, December 11, 2017, last modified January 28, 2020
• Topics: Common Core Math Practices , Numbers and Operations in Fraction , Operations and Algebraic Thinking
• Grades: 4 , 5 , 6 , 7 , 8 , 9

About Fraction Talks

Topics: Mental math, numerical fluency; argument & critique Materials: White board or projector Common Core: Variable, and especially MP3 and NF

This mental math routine creates powerful positive habits for students to understand and think productively about fractions.

Fraction talks developed as a way to use the Number Talk routine to target fractions. A growing body of fraction talks are available at fractiontalks.com .

I’ve created a writeup below for how to use fraction talks, as well as a collection here which are ideal for introducing fraction talks to students early in their understanding of fractions, and then continuing in slowly increasing levels of difficulty.

Why We Love Fraction Talks

Fraction Talks are a powerful way to extend the Number Talk format to the domain of fractions. They get all students involved, help them strengthen fluency, intuition, and mental math strategies, improve students’ ability to explain and critique solutions, and allow teachers a valuable window into their students’ thinking, as well as giving a framework for students to develop a more visual, conceptual framework

Download the PDF – For more information regarding this lesson, and for a collection of fraction images.

Fraction Talks Gallery

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Problem Solving Activities: 7 Strategies

• Critical Thinking

Problem solving can be a daunting aspect of effective mathematics teaching, but it does not have to be! In this post, I share seven strategic ways to integrate problem solving into your everyday math program.

In the middle of our problem solving lesson, my district math coordinator stopped by for a surprise walkthrough. 

I was so excited!

We were in the middle of what I thought was the most brilliant math lesson– teaching my students how to solve problem solving tasks using specific problem solving strategies. 

It was a proud moment for me!

Each week, I presented a new problem solving strategy and the students completed problems that emphasized the strategy. 

Genius right? 

After observing my class, my district coordinator pulled me aside to chat. I was excited to talk to her about my brilliant plan, but she told me I should provide the tasks and let my students come up with ways to solve the problems. Then, as students shared their work, I could revoice the student’s strategies and give them an official name. 

What a crushing blow! Just when I thought I did something special, I find out I did it all wrong. 

I took some time to consider her advice. Once I acknowledged she was right, I was able to make BIG changes to the way I taught problem solving in the classroom. 

When I Finally Saw the Light

To give my students an opportunity to engage in more authentic problem solving which would lead them to use a larger variety of problem solving strategies, I decided to vary the activities and the way I approached problem solving with my students. 

Problem Solving Activities

Here are seven ways to strategically reinforce problem solving skills in your classroom. 

Seasonal Problem Solving

Many teachers use word problems as problem solving tasks. Instead, try engaging your students with non-routine tasks that look like word problems but require more than the use of addition, subtraction, multiplication, and division to complete. Seasonal problem solving tasks and daily challenges are a perfect way to celebrate the season and have a little fun too!

Cooperative Problem Solving Tasks

Go cooperative! If you’ve got a few extra minutes, have students work on problem solving tasks in small groups. After working through the task, students create a poster to help explain their solution process and then post their poster around the classroom. Students then complete a gallery walk of the posters in the classroom and provide feedback via sticky notes or during a math talk session.

Notice and Wonder

Before beginning a problem solving task, such as a seasonal problem solving task, conduct a Notice and Wonder session. To do this, ask students what they notice about the problem. Then, ask them what they wonder about the problem. This will give students an opportunity to highlight the unique characteristics and conditions of the problem as they try to make sense of it. 

Want a better experience? Remove the stimulus, or question, and allow students to wonder about the problem. Try it! You’ll gain some great insight into how your students think about a problem.

Math Starters

Start your math block with a math starter, critical thinking activities designed to get your students thinking about math and provide opportunities to “sneak” in grade-level content and skills in a fun and engaging way. These tasks are quick, designed to take no more than five minutes, and provide a great way to turn-on your students’ brains. Read more about math starters here ! 

Create your own puzzle box! The puzzle box is a set of puzzles and math challenges I use as fast finisher tasks for my students when they finish an assignment or need an extra challenge. The box can be a file box, file crate, or even a wall chart. It includes a variety of activities so all students can find a challenge that suits their interests and ability level.

Calculators

Use calculators! For some reason, this tool is not one many students get to use frequently; however, it’s important students have a chance to practice using it in the classroom. After all, almost everyone has access to a calculator on their cell phones. There are also some standardized tests that allow students to use them, so it’s important for us to practice using calculators in the classroom. Plus, calculators can be fun learning tools all by themselves!

Three-Act Math Tasks

Use a three-act math task to engage students with a content-focused, real-world problem! These math tasks were created with math modeling in mind– students are presented with a scenario and then given clues and hints to help them solve the problem. There are several sites where you can find these awesome math tasks, including Dan Meyer’s Three-Act Math Tasks and Graham Fletcher’s 3-Acts Lessons . 

Getting the Most from Each of the Problem Solving Activities

When students participate in problem solving activities, it is important to ask guiding, not leading, questions. This provides students with the support necessary to move forward in their thinking and it provides teachers with a more in-depth understanding of student thinking. Selecting an initial question and then analyzing a student’s response tells teachers where to go next. 

Ready to jump in? Grab a free set of problem solving challenges like the ones pictured using the form below. 

Which of the problem solving activities will you try first? Respond in the comments below.

Shametria Routt Banks

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2 Responses

This is a very cool site. I hope it takes off and is well received by teachers. I work in mathematical problem solving and help prepare pre-service teachers in mathematics.

Thank you, Scott! Best wishes to you and your pre-service teachers this year!

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Introducing operations with fractions in mathematics can sometimes be confusing for students and frustrating for teachers.

Providing some context within which students can conceptualise fractions can really help solidify these abstract concepts in students’ minds, especially when adding and subtracting fractions to amounts that are greater than a whole.

In this ‘low threshold, high ceiling’ task, students take the role of the owner of a cake wholesaler, baking and supplying cakes to local café businesses. As café owners order their weekly cakes by the slice, students are required to add unit fractions together to calculate total cake orders. They then solve problems associated with subtracting fractional remainders, using equivalent fractions and converting between improper fractions and mixed numerals.

This downloadable ‘rich task’ lesson resource is designed for teachers and students in Years 5 to 7. It is mapped against Australian Curriculum (Mathematics), with an emphasis on problem solving and reasoning in operating with fractions .

The task comes with a comprehensive grading rubric to assist teachers who wish to use the project for formal assessment purposes.

This task was developed in consultation with Greta Public School (NSW), a CHOOSE MATHS Schools Outreach partner school.

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Fractions in Everyday Life - Problem Solving

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A 60 minute lesson in which students will solve simple word problems involving fractions.

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Recognise and interpret common uses of halves, quarters and eighths of shapes and collections

FractionsandDecimals - represents and models halves, quarters and eighths

Identify common uses and represent halves, quarters and eighths in relation to shapes, objects and events

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Five Simple Activities to Help Your Students Build Fraction Sense

• By Shelley Gray
• August 28, 2024
• No Comments

Do your students possess fraction sense? Fraction sense involves a deep, conceptual understanding of fractions as numbers. This goes far beyond simply shading parts of a whole or identifying numerators and denominators. It involves true, flexible understanding. 

Errors in fraction calculations can often be traced back to poor fraction sense. For example, think of a student who might add three-fourths and three-fourths and get six-eighths. 

This student is likely getting mixed up with a rule that he learned for adding the numerators only when adding two fractions with the same denominator.

But we don’t want our students to rely on a meaningless rule or procedure. We want them to actually understand fractions. 

So how do we ensure that our students develop this fraction sense rather than having to rely on meaningless rules? 

Here are five simple activities you can incorporate into your classroom to help your students build their fraction sense.

Use Manipulatives and Visual Models

Fractions are an abstract concept for many kids, especially since they may not have much real-life experience with fractions yet. But concrete manipulatives can be a game-changer, making fractions easy to visualize and understand. 

I love the circular foam fraction pieces for students to explore concepts like fraction addition/subtraction and equivalent fractions. They also make improper and mixed fractions simple to visualize.

Another amazing manipulative for fractions is Cuisinaire rods. Look at the examples below for just a couple ways these can be used.

Another visual that I love for fractions is bar models. Bar models can be used to make simple calculations more conceptual, or as an effective problem-solving tool. See this post on Instagram for more information on how to use them.

Paper Folding

Simple paper-folding activities can take fraction concepts from abstract to concrete for your students. There is some prep involved to get these paper pieces ready for students, but the actual act of your students creating the fraction pieces will be well worth it.

To prepare for this activity each student will need several different square pieces of paper. I like to have different colors. In this picture, I used black as one whole. Then students can fold the blue piece in half. Each of these pieces represents one-half. The pink piece was folded in half, and then in half again. These represent one-fourth. 

The act of folding and cutting is powerful, because it helps reinforce that one-fourth is half of one-half, one-eighth is half of one-fourth, etc. 

BUT! Here’s the important part. You may be tempted to have students label the pieces as 1, 1/2, 1/4, etc. However, using them unlabelled will allow for far more deep thinking. For example, consider how these tasks might look:

• If the black piece represents 4, what does the blue piece represent? How about the pink piece?
•  Show one-half in three different ways.
• How does the orange piece relate to the black piece?

Conduct Whole Class or Small Group Fraction Talks

Simply talking about fractions can have a profound impact on students’ understanding. This can be done in a whole class or small group setting and requires no prep or special materials. 

For example, you could simply draw an image on the board, like the one here, and ask students what they notice about the drawing.

If students are stuck for ideas, prompts like these could help get the conversation started:

• What equations could we write to represent this drawing?
• How are these circles the same and different?

If you own my Math Conversations number talk resources, you’ll also find some great fraction talk slides in there.

Think of all the possiblities that students might come up with for this question. Maybe it represents 8 groups of one-fourth.

Maybe it represents two wholes and 8 groups of three-fourths.

When we start looking at how students might group the circles to add them together, there are loads of different options.

Count By Fractions

Just like we can count by 1s, 2s, 5s, etc, we can count by fractions! This helps students see fractions as numbers. 

I’ve made a few videos that you can use in your classroom to help your students make connections between the fractions and the visual representations.  See all available videos here.

Make Fractions Meaningful and Relevant to Real Life

There is no better way to encourage true understanding than to relate fractions to how they are actually used in real life. Just think of all the amazing fraction skills we learn by baking! Here are just a few other examples of real-life experiences we can use to help our students understand fractions:

• Sports statistics
• Discounts at a store or restaurant
• Time (half of an hour, etc.)

Or see all real-life math projects here.

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6 Free Math Performance Tasks for Grades K-5

Innovative performance tasks to differentiate and extend math learning for all students!

Marshall Cavendish’s Primary Mathematics program empowers all students to develop critical-thinking skills, master concepts, and build the confidence needed to set the stage for long-term math achievement.

Struggling to find math resources that resonate with your students and align with your curriculum goals? Discover how Marshall Cavendish’s Primary Mathematics can transform your math instruction. Try out these free performance tasks with the corresponding teacher guides to experience how students can demonstrate mastery through real-life application of mathematics. The newly launched curriculum is a print/digital core math program for elementary students. Get the performance tasks for grades K-5 and see just how many ways these resources will enhance your students’ math skills.

Let’s explore the performance task for each grade level topic below.

Kindergarten: Cultivate a love for numbers with Ande’s Nature Walk

A captivating performance task designed to immerse young learners in the fundamentals of composing and decomposing numbers up to 10. Through this real-world adventure, students are not just learning, they are connecting, exploring, and finding joy in the world of mathematics.

1st Grade: Explore addition and subtraction with Balloon Adventures

Math becomes an adventure in this performance task filled with narrative-rich word problems in addition and subtraction. Join characters Aki, Brooke, and Isabel as they navigate through balloon-related challenges, turning abstract mathematical concepts into relatable and engaging experiences.

2nd Grade: Enhance problem-solving skills with Shopping Math Adventures

A thoughtfully crafted performance task focusing on the essential skills of addition and subtraction using bar models. In this real-world shopping scenario, students employ bar models as a visual tool to solve both two-step and one-step word problems, strengthening their understanding and application of mathematical concepts.

3rd Grade: Discovering the World of Multiplication and Division

Embark on a mathematical adventure with your students as they delve into the complexities of multiplication and division with numbers 6 through 9. This performance task masterfully balances previous knowledge and introduces new challenges, guiding students through a progressive journey of understanding and skill acquisition.

4th Grade: The Mysteries of Multiplication and Division

This task is a journey of numerical exploration, building on previous knowledge while introducing new challenges and concepts related to numbers 6 through 9.

5th Grade: Mastering fractions through pizza-purchase scenarios

Engage students in a captivating and relatable performance task centered around adding and subtracting fractions in real-life pizza-purchase scenarios. This task boosts critical thinking and problem-solving skills while solidifying students’ understanding of fractions.

What sets Primary Mathematics apart?

The newly launched Primary Mathematics is a comprehensive K-5 math curriculum that uniquely combines U.S. standards with the renowned Singapore Math pedagogy. This program is meticulously designed to cater to every student, promoting equity and differentiation in the classroom. Here’s why it stands out:

• Print/Digital Format: Integrated print/digital resources and interactive tools provide detailed teacher support to plan, develop, and deliver each lesson with confidence and effectiveness.
• Singapore Math Pedagogy: This renowned approach ensures a deep understanding and the development of strong critical thinking and problem-solving skills through the use of rich tasks, visual models, productive struggle, and mathematical discourse. 
• Equity and Differentiation: Every student is given the opportunity to succeed, regardless of their starting point, with just-in-time differentiation resources for each lesson that help teachers meet students’ needs whether reteaching or extending math concepts.
• Performance Tasks Aligned With Chapter Goals: Practical application and explanation of mathematical knowledge reinforce learning objectives.
• Progression to Mastery: Skill integration from previous chapters and even grade levels builds a comprehensive understanding. Mastery is achieved through intentional sequencing and variation of concepts developing from the basic to the complex. Students learn to think mathematically, use the depth of knowledge from previous lessons, and apply the concepts and strategies in new ways.
• Student and Teacher Rubrics: Tools for self-reflection, assessment, and targeted feedback ensure personalized learning.
• Point-of-Use Teaching Support: Teacher resources provide support throughout the learning process with question prompts, lesson and chapter planning, teacher tips, and differentiation tools to address the needs of all learners.

Embrace the transformative power of Primary Mathematics and join a community of educators dedicated to nurturing mathematical excellence.

Download these free math performance tasks today to get a sneak peek at the engaging resources primary mathematics offers., you might also like.

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Teachers: Resources for Middle Grades (6-8)

The North Carolina Collaborative for Mathematics Learning (NC 2 ML) aims to support NC math educators in implementing the revised mathematics content standards in ways that align with what we know from research on students’ mathematical thinking, mathematics teaching, and teacher learning. To do so, we bring together mathematics educators to co-design research-based resources and professional learning opportunities.

6-8 Resources Home

First Week Problem Solving Tasks

The Instructional Frameworks at each grade level recommend spending the first week of school doing general, high cognitive demand tasks with students in order to establish strong communication practices (SMP 3). Students can be enculturated into the discourse, listening and writing practices essential for strong mathematical reasoning while working these problems.

Additional Supporting Articles

Herbel-Eisenmenn, B. & Breyfogle, M. (2005). Questioning our patterns of questioning. Mathematics Teaching in the Middle School, 10(9), 484-489.

Stephan, M. (2014). Establishing standards for mathematical practice. Mathematics Teaching in the Middle School, 19(9), 532-538.

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Rich Problems – Part 1

Rich problems – part 1, by marvin cohen and karen rothschild.

One of the underlying beliefs that guides Math for All is that in order to learn mathematics well, students must engage with rich problems. Rich problems allow ALL students, with a variety of neurodevelopmental strengths and challenges, to engage in mathematical reasoning and become flexible and creative thinkers about mathematical ideas. In this Math for All Updates, we review what rich problems are, why they are important, and where to find some ready to use. In a later Math for All Updates we will discuss how to create your own rich problems customized for your curriculum.

What are Rich Problems?

At Math for All, we believe that all rich problems provide:

• opportunities to engage the problem solver in thinking about mathematical ideas in a variety of non-routine ways.
• an appropriate level of productive struggle.
• an opportunity for students to communicate their thinking about mathematical ideas.

Rich problems increase both the problem solver’s reasoning skills and the depth of their mathematical understanding. Rich problems are rich because they are not amenable to the application of a known algorithm, but require non-routine use of the student’s knowledge, skills, and ingenuity. They usually offer multiple entry pathways and methods of representation. This provides students with diverse abilities and challenges the opportunity to create solution strategies that leverage their particular strengths.

Rich problems usually have one or more of the following characteristics:

• Several correct answers. For example, “Find four numbers whose sum is 20.”
• A single answer but with many pathways to a solution. For example, “There are 10 animals in the barnyard, some chickens, some pigs. Altogether there are 24 legs. How many of the animals are chickens and how many are pigs?”
• A level of complexity that may require an entire class period or more to solve.
• An opportunity to look for patterns and make connections to previous problems, other students’ strategies, and other areas of mathematics. For example, see the staircase problem below.
• A “low floor and high ceiling,” meaning both that all your students will be able to engage with the mathematics of the problem in some way, and that the problem has sufficient complexity to challenge all your students. NRICH summarizes this approach as “everyone can get started, and everyone can get stuck” (2013). For example, a problem could have a variety of questions related to the following sequence, such as: How many squares are in the next staircase? How many in the 20th staircase? What is the rule for finding the number of squares in any staircase?

• An expectation that the student be able to communicate their ideas and defend their approach.
• An opportunity for students to choose from a range of tools and strategies to solve the problem based on their own neurodevelopmental strengths.
• An opportunity to learn some new mathematics (a mathematical residue) through working on the problem.
• An opportunity to practice routine skills in the service of engaging with a complex problem.
• An opportunity for a teacher to deepen their understanding of their students as learners and to build new lessons based on what students know, their developmental level, and their neurodevelopmental strengths and challenges.

Why Rich Problems?

All adults need mathematical understanding to solve problems in their daily lives. Most adults use calculators and computers to perform routine computation beyond what they can do mentally. They must, however, understand enough mathematics to know what to enter into the machines and how to evaluate what comes out. Our personal financial situations are deeply affected by our understanding of pricing schemes for the things we buy, the mortgages we hold, and fees we pay. As citizens, understanding mathematics can help us evaluate government policies, understand political polls, and make decisions. Building and designing our homes, and scaling up recipes for crowds also require math. Now especially, mathematical understanding is crucial for making sense of policies related to the pandemic. Decisions about shutdowns, medical treatments, and vaccines are all grounded in mathematics. For all these reasons, it is important students develop their capacities to reason about mathematics. Research has demonstrated that experience with rich problems improves children’s mathematical reasoning (Hattie, Fisher, & Frey, 2017).

Where to Find Rich Problems

Several types of rich problems are available online, ready to use or adapt. The sites below are some of many places where rich problems can be found:

• Which One Doesn’t Belong – These problems consist of squares divided into 4 quadrants with numbers, shapes, or graphs. In every problem there is at least one way that each of the quadrants “doesn’t belong.” Thus, any quadrant can be argued to be different from the others.
• “Open Middle” Problems – These are problems with a single answer but with many ways to reach the answer. They are organized by both topic and grade level.
• NRICH Maths – This is a multifaceted site from the University of Cambridge in Great Britain. It has both articles and ready-made problems. The site includes  problems for grades 1–5 (scroll down to the “Collections” section) and problems for younger children . We encourage you to explore NRICH more fully as well. There are many informative articles and discussions on the site.
• Rich tasks from Virginia – These are tasks published by the Virginia Department of education. They come with complete lesson plans as well as example anticipated student responses.
• Rich tasks from Georgia – This site contains a complete framework of tasks designed to address all standards at all grades. They include 3-Act Tasks , YouCubed Tasks , and many other tasks that are open ended or feature an open middle approach.

The problems can be used “as is” or adapted to the specific neurodevelopmental strengths and challenges of your students. Carefully adapted, they can engage ALL your students in thinking about mathematical ideas in a variety of ways, thereby not only increasing their skills but also their abilities to think flexibly and deeply.

Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics, grades K-12: What works best to optimize student learning. Thousand Oaks, CA: Corwin Mathematics.

NRICH Team. (2013). Low Threshold High Ceiling – an Introduction . Cambridge University, United Kingdom: NRICH Maths.

The contents of this blog post were developed under a grant from the Department of Education. However, those contents do not necessarily represent the policy of the Department of Education, and you should not assume endorsement by the Federal Government.

Math for All is a professional development program that brings general and special education teachers together to enhance their skills in planning and adapting mathematics lessons to ensure that all students achieve high-quality learning outcomes in mathematics.

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ChatGPT as a Math Questioner? Evaluating ChatGPT on Generating Pre-university Math Question

Mathematical questioning is crucial for assessing students problem-solving skills. Since manually creating such questions requires substantial effort, automatic methods have been explored. Existing state-of-the-art models rely on fine-tuning strategies and struggle to generate questions that heavily involve multiple steps of logical and arithmetic reasoning. Meanwhile, large language models(LLMs) such as ChatGPT have excelled in many NLP tasks involving logical and arithmetic reasoning. Nonetheless, their applications in generating educational questions are underutilized, especially in the field of mathematics. To bridge this gap, we take the first step to conduct an in-depth analysis of ChatGPT in generating pre-university math questions. Our analysis is categorized into two main settings: context-aware and context-unaware. In the context-aware setting, we evaluate ChatGPT on existing math question-answering benchmarks covering elementary, secondary, and ternary classes. In the context-unaware setting, we evaluate ChatGPT in generating math questions for each lesson from pre-university math curriculums that we crawl. Our crawling results in TopicMath, a comprehensive and novel collection of pre-university math curriculums collected from 121 math topics and 428 lessons from elementary, secondary, and tertiary classes. Through this analysis, we aim to provide insight into the potential of ChatGPT as a math questioner.

Nhat M. Hoang

Research intern (jul ‘24).

My name is Nhat, research assistant at NTU Nail Lab, Singapore. I’m interested in generative AI, multimodal learning, and large language models.