
Flexagon
15-30 min
Ages 8+
What Will You Learn?
Invented by Arthur H. Stone in 1939, flexagons are mathematical, multilayered objects composed of folded paper strips. They usually take the form of a square or hexagon, and there are several variations on the type of flexing and number of faces. When a flexagon is flexed, a new face is revealed. Flexing is similar to folding, but it’s not a static motion where there is a beginning and ending point; it’s a fluid and continuous forward and backward motion. For our project, we’re going to create a standard square flexagon, known as a tetraflexagon, which will reveal 2 new faces.
To Make your Flexagon
Step 1
First, cut out both of the L-shaped strips on the second page of the PDF. Note that they’re double-sided.
Step 2
Next, lay them horizontally, so that the tallest side is on your left. (The text on side B-1 will be upside down.)
Step 3
Starting from the left, fold over to the right. Repeat.
Repeat Step 3 with the second strip.



To Flex Your Flexagon
Step 4
Remember that this is a fluid, rotating motion.
Begin with face 1 (pink) facing you.
Step 5
Push both the right and left sides downward.The profile should look like a mountain peak.
Step 6
The first new face, face 2 (blue), will be revealed from the center of the flexagon; with your thumbs, open it up like a book.
Step 7
Repeat Step 2 to reveal face 3 (yellow, second new face).
Step 8
Return to face 1 by reversing the fluid movement. You will be flexing the outside edges toward the center. The profile should look like a V-shape.
NOTE: A second method of returning to the starting point is by flipping the flexagon over so that face 4 is facing you and flexing it twice. Then flip it over once more and face 1 will be facing you.



What's Next?
Try out other shapes
Try other Flexagon shapes!
https://makezine.com/2009/02/15/make-a-flexagon/
Maker Numberphile’s Forgotten Flexagon
Try Vi Hart’s Hexaflexagon
Books to read:
The Magic of Flexagons: Paper Curiosities to Cut Out & Make by David Mitchell
Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games by Martin Gardner
Flexagons Inside Out by Les Pook
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Maker Camp Project Standards
Based on NGSS (Next Generation Science Standards)
National Core Arts Standards
The National Core Arts Standards are a process that guides educators in providing a unified quality arts education for students in Pre-K through high school. These standards provide goals for Dance, Media Arts, Music, Theatre, and Visual Arts with cross-cutting anchors in Creating, Performing, Responding, and Connecting through art. The Anchor Standards include:- Generate and conceptualize artistic ideas and work.
- Organize and develop artistic ideas and work.
- Refine and complete artistic work.
- Select, analyze, and interpret artistic work for presentation.
- Develop and refine artistic techniques and work for presentation.
- Convey meaning through the presentation of artistic work.
- Perceive and analyze artistic work.
- Interpret intent and meaning in artistic work.
- Apply criteria to evaluate artistic work.
- Synthesize and relate knowledge and personal experiences to make art.
- Relate artistic ideas and works with societal, cultural, and historical context to deepen understanding.
National Core Arts Standards
The National Core Arts Standards are a process that guides educators in providing a unified quality arts education for students in Pre-K through high school. Also see Standards with cross-cutting anchors in Creating, Performing, Responding, and Connecting through art for Visual Arts.ISTE Standards (International Society for Technology in Education)
The ISTE Standards provide the competencies for learning, teaching and leading in the digital age, providing a comprehensive roadmap for the effective use of technology in schools worldwide.1.1 Empowered Learner
- Summary: Students leverage technology to take an active role in choosing, achieving, and demonstrating competency in their learning goals, informed by the learning sciences.
- 1.1.a Students articulate and set personal learning goals, develop strategies leveraging technology to achieve them and reflect on the learning process itself to improve learning outcomes.
- 1.1.b Students build networks and customize their learning environments in ways that support the learning process.
- 1.1.c Students use technology to seek feedback that informs and improves their practice and to demonstrate their learning in a variety of ways.
- 1.1.d Students understand the fundamental concepts of technology operations, demonstrate the ability to choose, use and troubleshoot current technologies and are able to transfer their knowledge to explore emerging technologies.
1.2 Digital Citizen
- Summary: Students recognize the rights, responsibilities and opportunities of living, learning and working in an interconnected digital world, and they act and model in ways that are safe, legal and ethical.
- 1.2.a Students cultivate and manage their digital identity and reputation and are aware of the permanence of their actions in the digital world.
- 1.2.b Students engage in positive, safe, legal and ethical behavior when using technology, including social interactions online or when using networked devices.
- 1.2.c Students demonstrate an understanding of and respect for the rights and obligations of using and sharing intellectual property.
- 1.2.d Students manage their personal data to maintain digital privacy and security and are aware of data-collection technology used to track their navigation online.
1.3 Knowledge Constructor
- Summary: Students critically curate a variety of resources using digital tools to construct knowledge, produce creative artifacts and make meaningful learning experiences for themselves and others.
- 1.3.a Students plan and employ effective research strategies to locate information and other resources for their intellectual or creative pursuits.
- 1.3.b Students evaluate the accuracy, perspective, credibility and relevance of information, media, data or other resources.
- 1.3.c Students curate information from digital resources using a variety of tools and methods to create collections of artifacts that demonstrate meaningful connections or conclusions.
- 1.3.d Students build knowledge by actively exploring real-world issues and problems, developing ideas and theories and pursuing answers and solutions.
1.4 Innovative Designer
- Summary: Students use a variety of technologies within a design process to identify and solve problems by creating new, useful or imaginative solutions.
- 1.4.a Students know and use a deliberate design process for generating ideas, testing theories, creating innovative artifacts or solving authentic problems.
- 1.4.b Students select and use digital tools to plan and manage a design process that considers design constraints and calculated risks.
- 1.4.c Students develop, test and refine prototypes as part of a cyclical design process.
- 1.4.d Students exhibit a tolerance for ambiguity, perseverance and the capacity to work with open-ended problems.
1.5 Computational Thinker
- Summary: Students develop and employ strategies for understanding and solving problems in ways that leverage the power of technological methods to develop and test solutions.
- 1.5.a Students formulate problem definitions suited for technology-assisted methods such as data analysis, abstract models and algorithmic thinking in exploring and finding solutions.
- 1.5.b Students collect data or identify relevant data sets, use digital tools to analyze them, and represent data in various ways to facilitate problem-solving and decision-making.
- 1.5.c Students break problems into component parts, extract key information, and develop descriptive models to understand complex systems or facilitate problem-solving.
- 1.5.d Students understand how automation works and use algorithmic thinking to develop a sequence of steps to create and test automated solutions.
NGSS MS.Engineering Design
The Next Generation Science Standards (NGSS) are K–12 science content standards.- MS-ETS1-1. Define the criteria and constraints of a design problem with sufficient precision to ensure a successful solution, taking into account relevant scientific principles and potential impacts on people and the natural environment that may limit possible solutions.
- MS-ETS1-2. Evaluate competing design solutions using a systematic process to determine how well they meet the criteria and constraints of the problem.
- MS-ETS1-3. Analyze data from tests to determine similarities and differences among several design solutions to identify the best characteristics of each that can be combined into a new solution to better meet the criteria for success.
- MS-ETS1-4. Develop a model to generate data for iterative testing and modification of a proposed object, tool, or process such that an optimal design can be achieved.
NGSS HS.Engineering Design
The Next Generation Science Standards (NGSS) are K–12 science content standards.- HS-ETS1-1. Analyze a major global challenge to specify qualitative and quantitative criteria and constraints for solutions that account for societal needs and wants.
- HS-ETS1-2. Design a solution to a complex real-world problem by breaking it down into smaller, more manageable problems that can be solved through engineering.
- HS-ETS1-3. Evaluate a solution to a complex real-world problem based on prioritized criteria and trade-offs that account for a range of constraints, including cost, safety, reliability, and aesthetics as well as possible social, cultural, and environmental impacts.
- HS-ETS1-4. Use a computer simulation to model the impact of proposed solutions to a complex real-world problem with numerous criteria and constraints on interactions within and between systems relevant to the problem.
NGSS 3-5.Engineering Design
The Next Generation Science Standards (NGSS) are K–12 science content standards.- 3-5-ETS1-1. Define a simple design problem reflecting a need or a want that includes specified criteria for success and constraints on materials, time, or cost.
- 3-5-ETS1-2. Generate and compare multiple possible solutions to a problem based on how well each is likely to meet the criteria and constraints of the problem.
- 3-5-ETS1-3. Plan and carry out fair tests in which variables are controlled and failure points are considered to identify aspects of a model or prototype that can be improved.