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Same and Different Images

Lots of images to support students in developing the habit of comparing and connecting representations. Thanks Brian Bushart for pulling them together!

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Great tasks you can do in 10-15 minutes every day...click the image and they're yours.

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Enriching Mathematics

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search

Enriching mathematics for all learners

A free-to-use collection of mathematics activities, lessons and problems designed to nurture curious, resourceful and confident learners of mathematics. Produced by The University of Cambridge.

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The Nrich Maths Project Cambridge,England. Mathematics resources for children,parents and teachers to enrich learning. Problems,children's solutions,interactivities,games,articles.

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Problem Based Curriculum Maps with Tasks

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The following Problem Based Learning (PrBL) curriculum maps are based on the Math Common Core State Standards and the associated scope and sequences. The problems and tasks have been scoured from thoughtful math bloggers who have advanced our practice by posting their materials online.

The Scope and Sequences for Algebra 1, Geometry, Algebra 2, Math 9 (Integrated), Math 10 (Integrated), and Math 11 (Integrated) are from Pearson. Other Scope and Sequences were developed by me, modeling a similar visual style.
Grade 3 CCSS PrBL Curriculum Map Grade 4 CCSS PrBL Curriculum Map Grade 5 CCSS PrBL Curriculum Map Grade 6 CCSS PrBL Curriculum Map Grade 7 CCSS PrBL Curriculum Map Grade 8 CCSS PrBL Curriculum Map Math 9 (Integrated) CCSS PrBL Curriculum Map Math 10 (Integrated) CCSS PrBL Curriculum Map Math 11 (Integrated) CCSS PrBL Curriculum Map Algebra 1 CCSS PrBL Curriculum Map Geometry CCSS PrBL Curriculum Map Algebra 2 CCSS PrBL Curriculum Map
Be sure to check back often as these curriculum maps are bound to change when new cool activities are added on math blogs worldwide. I'll be adding more and more curriculum maps in due time. Also, if you have a favorite particular task that isn't here (or just comments in general), let me know in the comments!

For more on Problem Based learning, you can check out my previous posts on the topic.

- Geoff

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Common Core Problem Based Curriculum Maps

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The following Problem Based Learning (PrBL) curriculum maps are based on the Math Common Core State Standards and the associated scope and sequences. The problems and tasks have been scoured from thoughtful math bloggers who have advanced our practice by posting their materials online. The Scope and Sequences for Algebra 1, Geometry, Algebra 2, Math...

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We think that this is a very good time to wish for peace. Above is a picture of the wreath that I made for this winter's season. Do you know this symbol? Where did it originate? Do you ever use a different symbol for peace (for instance, sideways "V"s)?

In this activity students learn about the semaphore flag symbols and how this peace symbol was first designed. They design their own symbol and try to figure out how many possibilities there are in this kind of communication.

The activity: PeaceSymbol.pdf

How should I cook my turkey? - Students judge timing, cost, tastiness, and quantity necessary as they plan for the feast. 4.MD, 5.NBT, 4.MD, 6.RP, 6.NS, 7.NS + Great video on a deep frying fire with William Shatner.

Shopping season already! - Students will be able to add some understanding and analysis to holiday shopping trips after calculating savings in dollars and percents with this timely investigation. 6.RP, 7.EE, 7.EE

Not enough mashed potatoes - Use Brian's famous mashed potatoes recipe to practice changing decimals to fractions; calculating ingredient measure for various-sized Thanksgiving servings; have students explain their reasoning; and to have students figure out how many servings 7½ pounds of potatoes would make. 5.NF, 6.RP, 7.RP

Consumers in the U.S. spend a lot of money. In this activity we look at historical data to see if there are any patterns in our spending. How does our holiday spending show up in these graphs? What accounts for spikes or drops in spending? What patterns do you notice? Which patterns will likely continue and which patterns won't? Let the class see, think, wonder and discuss in this timely activity.

Soo Yeon is a Mathematics teacher at an international school in Asia and has be an avid user of YummyMath. Formerly she was a chemical engineer and shares our intent of providing practical, relatable, and real-world contexts in teaching mathematics.

She would like to share the following with YummyMath users:

"As you know Leslie and Brian's hard work and vision of exploring math in real life contexts has been instrumental in helping students develop critical thinking, questioning skills, and a positive attitude towards Mathematics. I wholeheartedly share their vision and am dedicated to upholding their mission. Moving forward, YummyMath will continue to evolve, enhancing content and user experiences to better meet your needs both now and in the future. I understand the challenges teachers and parents face in balancing engaging activities with time management. To address this, I kindly request your input through a survey ( ) to ensure that YummyMath effectively supports your needs.

Thanks so much for your support as YummyMath starts this new chapter. "

In this activity, students work through the nomenclature to understand the possibilities of nano science and quantum dots. Questions that are answered; How many nanometers are in a millimeter? What is a semiconductor? What and why does this nano technology increase our ability to closely examine, understand and modify our tools?

The activity: Quantum-dot.pdf

On Veterans Day we honor the men and women that have served in the Armed Forces of the United States. There are now 16.2 million living veterans. We found these maps that show where veterans live in the U.S.

What do you notice and what do you wonder when seeing these maps?

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We think that this is a very good time to wish for peace. Above is a picture of the wreath that I made for this winter's season. Do you know this symbol? Where did it originate? Do you ever use a different symbol for peace (for instance, sideways "V"s)?

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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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Grades K-12

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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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Tasks by Grade Level/Domains

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Providing instructional and assessment tasks, lesson plans, and other resources for teachers, assessment writers, and curriculum developers since 2011.

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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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Jo Boaler

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North Carolina Dept. of Public Instruction (K-2)

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North Carolina Dept. of Public Instruction (K-2)

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Gr. K-2 Assessment Tasks

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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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North Carolina Dept. of Public Instruction (3-5)

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North Carolina Dept. of Public Instruction (3-5)

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Gr. 3-5 Assessment Tasks

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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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Past Problem Sets

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Problem Set #76 Problem Set #75 Problem Set #74 Problem Set #73 Problem Set #72 Problem Set #71 Problem Set #70 Problem Set #69 Problem Set #68 Problem Set #67 Problem Set #66 Problem Set #65 Problem Set #64 Problem Set #63 Problem Set #62 Problem Set #61 Problem Set #60 Problem Set #59 Problem...

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Low Floor, High Ceiling

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Performance Assessment Tasks

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Performance Assessment Tasks | Inside Mathematics

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These tasks are grade-level formative performance assessment tasks with accompanying scoring rubrics and discussion of student work samples. They are aligned to the Common Core State Standards for Mathematics. You may download and use these tasks for professional development purposes without modifyi…

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Problem of the Day and Tasks

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Please Note: Balanced Assessment printed materials are available from this site, except for those indicated below. Please use the order form and follow the ordering directions carefully as they have changed. Current prices can be found on our order form.

Please also note that the Balanced Assessment Primary & Elementary Tasks have been published by Corwin Press. The Balanced Assessment Transition & Middle School Tasks have been published by Teachers' College Press. These tasks may still be viewed in.pdf format on this website but they may not be copied or printed.

About the program

From 1993 to 2003, the Balanced Assessment in Mathematics Program existed at the Harvard Graduate School of Education. The project group developed a large collection of innovative mathematics assessment tasks for grades K to 12, and trained teachers to use these assessments in their classrooms.

The library of over 300 mathematics assessment tasks developed during the project remains freely available through this web site. Teachers may use these materials in their own classrooms at no cost.

This site also offers reports on our approach to mathematics assessment and our distinctive objects x actions scoring system. A more detailed overview of the project is also available.

One of the most satisfying results of this ten-year project was the enthusiastic response of teachers to our assessment tasks and publications. We hope that you will continue to find the Balanced Assessment tasks to be a valuable resource.

Ordering Balanced Assessment publications

It is possible to purchase printed Balanced Assessment publications using our order form.

Our library of assessment tasks

If you are looking for a specific task, you can find it by name or by number.

If you're just getting to know our materials, we'd like to suggest a few of our favorite tasks:

Our approach to assessment and scoring

These reports and materials were developed while the Balanced Assessment program operated at the Harvard Graduate School of Education, between 1993 and 2003.

Assessing Mathematical Understanding and Skills Effectively (AMUSE)
Also available as a Word document.
This 1995 report describes our project group's approach to mathematics assessment.

Mathematics Content and Process Scoring System (MCAPS)
Also available as a Word document.
This report describes our objects x actions approach to scoring assessment tasks.

MCAPS Scoring Tool
Requires Microsoft Excel or compatible spreadsheet.
This spreadsheet implementation of the MCAPS scoring system enables teachers to efficiently use the system in their classes.

Judah L. Schwartz,
Principal Investigator/Director
Joan Kenney,
Project Coordinator/Director
Susan Ilias
Kevin Kelly
Teresa Sienkiewicz
Yesha Sivan
Victor Steinbok
Amy Warren
Michal Yerushalmy

For information on using BA tasks in the classroom, professional development for teachers and administrators, and issues regarding orders, please contact:
Joan Kenney
Joan_Kenney@post.harvard.edu

For technical web site issues, contact:
dan@mathshell.org

The materials on this web site were developed under National Science Foundation grants MDR-9252902 and ESI-9726403 (subcontracts from the University of California at Berkeley and Michigan State University), with additional funding from the American Academy of Arts and Sciences, the Boston Public Schools, the Cambridge Public Schools, and the Noyce Foundation..

All documents are Copyright © 1995-2003, President and Fellows of Harvard College.With the exceptions listed above in the heading to this page you are hereby granted the right to duplicate or print these materials for use in your classroom or school. Otherwise, these materials are protected by copyright.

This web site was originally designed by Kevin Kelly.

Last Update: 10/17/2015
Copyright © 2020, The Concord Consortium. All rights reserved.

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Please Note: Balanced Assessment printed materials are available from this site, except for those indicated below. Please use the order form and follow the ordering directions carefully as they have changed. Current prices can be found on our order form. Please also note that the Balanced Assessment Primary & Elementary Tasks have been published by Corwin Press.