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Graham Fletcher

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Modeling with mathematics is the practice of making sense of the world through a mathematical perspective. Take a moment to look around and get curious: How do you use mathematics to make decisions in your everyday life? Maybe you're deciding what to make for dinner. Does the recipe have enough servings to feed everyone or will you need to modify it, perhaps by doubling or halving the amount of each ingredient? When should you start making dinner if you'd like to eat at 6?

These questions can be viewed from a mathematical perspective. There is something to count, measure, or quantify and the answers to these questions have real and interesting implications for our lives.

Children also need opportunities to identify mathematical problems in their world, determine what information will help them solve a problem, develop mathematical models of situations, and revise their models to more closely predict real world phenomena. This is the work of modeling with mathematics, a mathematical practice identified by the Common Core State Standards as central to the work of K-12 mathematics.

Three-Act Tasks

A Three-Act Task is a lesson structure designed specifically to engage children in modeling with mathematics. The activity was created by Dan Meyer, a former high school math teacher and blogger. Teachers around the country are trying these tasks with students and sharing what they've learned. Here are just a few bloggers to check out: Graham Fletcher, Andrew Stadel, Mike Wiernicki, and Brian Bushart.

In Three-Act Tasks, students are shown an image or video that depicts an interesting situation. Examples of elementary tasks might include an image of two sisters standing side-by-side, one taller than the other, or a video of candy being poured into a jar. A good task will suggest some mathematical features or relationships that children may wonder about. After viewing the image or video, students are engaged in asking mathematical questions, identifying important information that is needed to answer those questions, constructing mathematical models of the situation, and comparing their models to the real world.

The first two videos in this series take us inside two primary classrooms where children are working on Three-Act Tasks and modeling with mathematics. As you watch Three-Act Tasks: Modeling Addition and Three-Act Tasks: Modeling Subtraction, consider:

What rich mathematical ideas do the children apply to these contextual situations?

How are Three-Act Tasks similar or different from other problem solving activities?

Two Strategies that Support Student and Teacher Learning

In Teacher Collaboration... While Teaching, we explore the ways teachers collaborate in the classroom to learn together, adapt lessons in the moment, and grow their instructional practice. The teachers use two strategies -- Teacher Time Out and Huddling -- to discuss instructional decisions together while lessons play out. These strategies are enacted by pairs of teachers working together, but can also be used by small groups of teachers who are learning about new instructional strategies or who are working together to improve their practice.

Finally, in the video Building Emotional Literacy, we get a glimpse into one small strategy that Sarah, Kristin, and their colleagues use to support students to develop emotional literacy skills and ready their bodies for learning. They link their work to a program called RULER, which involves a set of coordinated components to support students' socio-emotional learning.

Give these strategies a try and let us know in the comments below what you and your students learn.

For more on Modeling with Mathematics, visit these great resources:
Three-Act Tasks
Dan Meyer: Three-Act Tasks (grades 6-12)
Graham Fletcher: Three-Act Tasks (grades K-7)
Mike Wiernicki: Three-Act Tasks (grades 2-8)
Andrew Stadel: Three-Act Tasks (elementary-high school)
Other Resources
An entertaining 60 minute presentation by Dan Meyer: Fake-World Math: When Mathematical Modeling Goes Wrong and How to Get It Right

A brief blog post by John Pelesko with suggestions for elementary mathematical modeling: "How did the chicken cross the road?"

Have some fun with mathematical modeling! This website by Dan Meyer has interesting images and videos that you can ask mathematical questions about.
This work was made possible through support by the National Science Foundation. Elham Kazemi is a professor of mathematics education at the University of Washington. She loves to design professional learning opportunities for teachers, coaches, and principals to learn together about children's mathematical thinking. Kendra Lomax is a teacher educator at the University of Washington. She collaborates with local teachers to learn about children's mathematical thinking and designs job-embedded professional learning opportunities through projects like Teacher Education by Design. Curiosity about children's mathematical ideas is at the heart of her work.

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Modeling with mathematics is the practice of making sense of the world through a mathematical perspective. Take a moment to look around and get curious: How do you use mathematics to make decisions in your everyday life? Maybe you're deciding what to make for dinner.

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Planning Document

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Grades K-12

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Illustrative Mathematics provides resources and support for giving their students an enduring understanding of mathematics.

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Open Middle - Challenging math problems worth solving

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Open Middle - Challenging math problems worth solving

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Open Middle - Challenging math problems worth solving

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Open Middle offers interactive math problems designed to foster critical thinking. Explore unique challenges that boost problem-solving skills and creativity.

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K-2 Formative Instructional and Assessment Tasks 

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K-2 Formative Instructional and Assessment Tasks 

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North Carolina Department of Public Instruction

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K-2 Formative Instructional and Assessment Tasks - home

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In response to North Carolina legislative and State Board requirements, the North Carolina Department of Public Instruction provides Local Education Agencies with state-developed assessments to be implemented for Kindergarten, First and Second Grades. These assessments include documented, on-going individualized assessments throughout the year and a summative evaluation at the end of the year.

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Teaching Children Mathematics

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Jo Boaler

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(Grades 6-12)

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Our summative tasks are divided into three classes - Novice, Apprentice and Expert. This is not a measurement of the difficulty of the content, but indicates the extent to which they require students to autonomously apply their mathematical knowledge: Novice tasks explicitly test particular items of content knowledge, Expert tasks are far less structured and require strategic problem-solving skills in addition to content knowledge. A "balanced diet" of Novice, Apprentice and Expert tasks is needed to properly assess the Mathematical Practices.

Each task includes a scoring rubric, a set of pre-scored sample student work, and the same set of work without the scores. Scoring practice for this type of task can be an effective Professional Development activity.

A selection of test forms using these styles of task is also available.

Choosing Tasks

There are several ways of navigating:

1. Use the menu on the left to browse by grade and topic area, or search for a key phrase.

2. Go to the Standards tab to find tasks linked to a particular content standard or practice.

3. The complete set of tasks is listed below in alphabetic order.

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Our summative tasks are divided into three classes - Novice, Apprentice and Expert. This is not a measurement of the difficulty of the content, but indicates the extent to which they require students to autonomously apply their mathematical knowledge: Novice tasks explicitly test particular items of content knowledge, Expert tasks are far less structured and require strategic problem-solving skills in addition to content knowledge.

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3 Act Tasks | Under the Dome  (Grades 2-8)

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3 Act Tasks | Under the Dome  (Grades 2-8)

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Mike Wiernicki

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Dan Meyer introduced the idea of 3-Act Math Tasks with his TED Talk: Math Class Needs a Makeover.

Below is a table of the 3-Act Tasks that I've created. Those marked with a * are newly added to this website as of 8/21/2023.

5.NBT.7, 5.MD.5

Multiplying Decimals, Volume

How much money is that?

Card House
3.OA.3, 3.OA.7, 3.NBT.3, 5.NBT.7, 8.G.7

Multiply Whole Numbers
Multiply Decimals
Pythagorean Theorem

How tall will the card house be when finished? How many cards/decks were used to build the card house? How many card triangles were used to build the card house?

For more tasks to try, check out the work offered by these amazing mathematics educators:

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Dan Meyer introduced the idea of 3-Act Math Tasks with his TED Talk: Math Class Needs a Makeover. Below is a table of the 3-Act Tasks that I've created. Those marked with a * are newly added to this website as of 8/21/2023. Task Name with LinkStandard(s)Big Idea(s)Main Question/WonderSweet Tart Hearts4.NF.4, 3.NF.1Multiplying Fractions and Whole...

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Robert Kaplinsky

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My growing list of free real world problem-based lessons for grades 2 through high school. Includes links to download multimedia and teacher guide.

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Dan Meyer

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Graham Fletcher

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Graham Fletcher