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The newest edition of ANN's award-winning newsletter is here! Look inside for even more math ideas you can use with your students.

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Thanks to the incredible sponsors for donating prizes. It is so nice to get goodies for our classrooms.

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Continue the conversation online. Sign up for AskANN to email with other ANN members. Sign in at ANN's web site for information on joining.

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Adult Numeracy Network - Authorization required

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These are the slides from the Main Room. The breakout room slides are provided with the classroom ready activities in the music and art columns.

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Bookmarked on Resources for 2021 ANN Institute

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Tempo Challenge Slides - from Breakout Task #1

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Tempo Challenge Teacher Notes - from Breakout Task #1

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Helpful tech tools to prepare you for the day

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Tech Tools for ANN's Institute_updated.pdf

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Reference for Breakout Task #1

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More than a tool for adding and subtracting fractions

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Polyrhythm Challenge Slides - from Breakout Task #2

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Polyrhythm Challenge Teacher Notes - from Breakout Task #2

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Prizes!! Just one more perk for participating in our Institute.

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Beauty, joy, creativity - Are these things present in your math classes? Could they be?

Mathematics, music and art are all human endeavors that help us understand the world around us and to express ourselves. Learn how the arts can deepen students' understanding of essential math concepts and help them tune-in to the math in the world around them. No music or art skills are needed to enjoy this fun-filled day of learning and growing as math practitioners. To learn more, check out our flyer: EXPERIENCING HARMONY & BEAUTY in MATHEMATICAL PATTERNS.

The two videos below offer a sneak-preview into the event.

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Beauty, joy, creativity - Are these things present in your math classes? Could they be? Mathematics, music and art are all human endeavors that help us understand the world around us and to express ourselves. Learn how the arts can deepen students' understanding of essential math concepts and help them tune-in to the math in the world around them.

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Math Manipulatives for Remote Instruction

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Math Manipulatives for Remote Instruction

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Build your own math toolbox

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This handy two-page document and companion video offers suggestions for free, downloadable resources and everyday items that can help instructors teach math remotely. Students can use these resources at home to learn and reinforce math concepts in a hands-on way even when they can't physically be in the classroom.

You may also be interested in the companion video, Remote Learning: Creating Community.

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Math Manipulatives for Remote Instruction | SABES

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This handy two-page document and companion video offers suggestions for free, downloadable resources and everyday items that can help instructors teach math remotely. Students can use these resources at home to learn and reinforce math concepts in a hands-on way even when they can’t physically be in the classroom.

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Most of us have had students who have not yet developed fluency with multiplication facts. For most of those students, the challenge is that have been taught to think of the 12 by 12 multiplication table as 144 discrete math facts to memorize.

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Unleash your creativity with the world's best virtual manipulatives!

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Push and Support cards are tools that allow teachers to dial up and dial down problems, keeping every student in a mixed-level math class engaged in high-cognitive level math tasks. This guide, written by Patricia Helmuth, explains how to use them, how to write your own, and has seven tasks with Push and Support cards already written to get you started.

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May 24, 2018 byPatricia Helmuth
This OER curriculum guide is a free resource that supports the integration of OER into teaching plans. It was made possible by funding provided by the U.S. Department of Education Office of Career, Technical and Adult Education (OCTAE) under the project titled: Power in Numbers: Advancing Math for Adult Learners, (OER Math project), managed by Luminary Labs. The opinions expressed herein do not necessarily represent the positions or policies of the U.S. Department of Education and no official endorsement by the U.S. Department of Education should be inferred.

for_oer_post.jpeg Problem Solving with Push and Support

Problem Solving In the Adult Education Classroom

Problem Solving with Push and Support

After analyzing the data collected from PIAAC, The Program for the International Assessment of Adult Competencies, Donna Curry concludes: Addressing adult learners' attitudes about math to help them build a productive disposition is possibly the greatest challenge for teachers. [1] Many of us who have taught, or who are currently teaching adults, recognize the truth in this statement. Adults many times enter our classes with a belief that they are not a "math person" and that the only way to succeed in math is to memorize procedures. Even adults who enter our classes having experienced some success in math classes in the past, almost always have at least some of those procedures bolloxed up in their memories and have minimal ability to be able to make application of procedures in messy real-world settings that require more than just surface knowledge of one procedure. In both cases, students need to develop conceptual understanding of the math behind the numbers and the symbols, and the ability to think through mathematical situations. In other words, they need to develop problem solving skills.

Students may have wide-ranging ideas on what problem solving means. This isn't a bad thing. It is valuable to draw out the thinking of our students in terms of what they are already inclined to do when faced with solving a math problem and what they intuitively know about a situation that requires mathematical thinking. Many of our adult students have gotten by in life by using unconventional means to solve math and these creative, out-of-the-box, problem solving methods are where our students are currently at, so we need to meet them there. Sometimes, we can use what they are already doing and help them to build on the foundation they have. At other times, we discover deep rooted misconceptions that need to be adjusted. Either way, our goal is to help our students to make sense of problems and persevere in solving them, MP1. [2] Our goal is not to "show" them how to solve a problem. This will not help them to pass high-stakes exams, nor will it enable them to function as numerate adults.

Collaborative Learning

Push and support cards act as a catalyst for students in their journey to construct viable arguments and critique the reasoning of others, MP3. [2] As already mentioned, our adult students usually come to our classes with some ideas (or sometimes no idea) about how to solve math problems. Creating an environment where they feel safe sharing their angst, their ideas, and their atypical problem solving strategies is vital if they are to become mathematically proficient. There are numerous ways to approach the solving of math problems and this should be encouraged and celebrated. Some of these methods may come out naturally for students as they work together, but if not, the support cards provide recommendations for ways to get started. Suggestions on the cards include, but are not limited to: drawing a picture, using manipulatives, or using a table to organize information.

As they make these connections they develop number sense and confidence that carries forward when they are faced with new and different problems to solve, because they have their knowledge of their own discoveries in problem solving and also awareness of the methods they have seen other students use and talk about. Another important aspect that comes of collaboration is that students can make sense of a math problem by talking through it. When working individually, a student may skip over or misunderstand relevant information, but when students talk about it together, they help each other to make sense of the problem.

Before Using Push and Support in Your Classroom

support_and_push_drawing_q6vTgow.jpg Examples of Push and Support Cards

One first step in understanding how your students might feel when pressed with solving a problem is to spend time solving it yourself. When doing this, if you find that you get stuck, rather than reach for the answer key that is available in the Google doc version of this curriculum guide, you might want to take a look at the push and support cards that correlate with the problem, one at a time, until you've come to a solution [3] . Even if you didn't need the support cards to solve the initial phase of the problem, think about how the support cards might be used by students to help them gain conceptual understanding. For example, some of the cards may suggest that a student draw a picture of the situation or use manipulatives to solve it. I recommend you do the same. This may be a challenge for some instructors who are very familiar with using procedures to solve math problems. What would a picture look like that models the situation? How do you anticipate that students will use the manipulatives to model the math? After this, work through the problems on the push cards; each time anticipating how students will respond to the cards.

Examples of Push and Support Cards

How to Use Push and Support in Your Classroom

Have in mind before each task how you want to scaffold the cards, moving from support to push. However, keep in mind that some cards can act as a support card or a push card depending on the student (or group). For example, it may be helpful for some students to use manipulatives or draw a picture in order to understand the mathematical situation. On the other hand, other students may understand how to manipulate numbers on paper, but be challenged when asked to draw a picture of the situation. Listed below are some simple steps to take when using the push and support cards

Allow students 3 - 5 minutes of quiet time to work individually on the initial phase of the task without giving out any cards. This quiet time is essential for some students in order for them to be able to develop their own ideas about a mathematical task. If not given the time, they may shut down in their thinking as soon as a fellow student voices an idea or problem solving strategy. You should be on the lookout for what students are writing during this quiet time and encourage them to share out during group work.

After quiet time, pair students or have them work in groups of three. Listen to what they are talking about and look at what they are writing on their paper. If they are stuck give the group one support card to share. When possible, try to hand out a card that connects to the solution pathway that they are already moving towards. Make sure that the group understands the information on the card, providing clarifying information if needed, but stop short of explaining the guidance the card is providing. [3]

Only hand out one card at a time. When the group has exhausted the sharing of their own ideas and strategies and needs more support or suggestions to move forward, give them another card to share. Meantime, if a group has finished working through the suggestion on the card and is ready to take the problem to the next level, provide them with a push card to extend their understanding to more advanced mathematics.

It is very important to keep in mind that the reason you are using push and support cards is to create a classroom environment where all students can work on the same problem, at the same time, but not necessarily all reach the same level of understanding. Some students may spend the entire time allotted for the activity working with one or two support cards, while other students may work through every push card that you have!

Note: A detailed description of how to use push and support cards is included in the write up of The Painted Cube: one of the Math Memos problems. Download and print: How to Use Push and Support in Your Classroom

Course Description

For the purposes of this curriculum guide, we will think about math problems as a situation in the real-world that involves quantities and context that students would be familiar with or a visual model that can be used by students as a bridge to gain conceptual understanding of abstract mathematical concepts.

These tasks are sometimes referred to as low-floor/high-ceiling tasks because while all students can find a way into the task (low-floor), some students will be able to extend their understanding to the more abstract level (high-ceiling). Low-Floor/High-Ceiling tasks have the following characteristics:

They have multiple starting points.

They do not have a readily apparent solution.

They can be solved using different methods.

They stimulate discussion.

They promote productive struggle.

They require students to make decisions.

These tasks, along with notice/wonder and the push and support cards, can be used with a group of heterogeneous students, but the problems contained or linked to in this guide also work very well in a one-on-one setting where a tutor is working with only one student at a time. In this case, the tutor would scaffold the push and support cards as appropriate for the learner.

A Note About Notice/Wonder

All of the tasks within or linked to in this curriculum guide have been adapted for the multi-level adult education classroom by removing any questions that appeared on the first page of the original task and replacing the question(s) with notice/wonder. Removing the original question(s) serves multiple purposes.

notice_wonder_video.JPG Ever Wonder What They'd Notice? (if only someone would ask)

For one, it acts to slow students down so that they do not immediately jump into calculations before they actually understand the information that is being presented. In addition, it gives students opportunity to express some of the math ideas that are already in their head and to verbalize something they are pondering but might not have the courage to say out loud unless everyone else is in the game. There is no pressure here to immediately come up with an answer, so students who have not yet developed confidence in their math problem-solving skills can freely participate. A more detailed, step-by-step guide on how to implement this powerful classroom strategy in your classroom can be found in OER Commons. [4]

Ever Wonder What They'd Notice? (if only someone would ask)

A well invested five minutes: To catch the excitement of how notice/wonder helps to draw out the mathematical thinking of our students, please follow this link to watch the video: Ever Wonder What They'd Notice? (if only someone would ask).

The Student Centered Classroom Routine

Some of the tasks that are recommended in this curriculum guide already contain sufficient guidance for use in the classroom as is. As for the tasks lacking this, what follows is a suggested framework. Consider how you can formatively assess your students during each step.

Give students the notice/wonder page of the activity and ask them to silently work on this for 3 - 5 minutes. This gives students time to draw up any background information they already have about the situation or visual model. Float about the room and look for some varied responses. Quietly ask a few of the students, whose thinking you would like to draw out during the whole group discussion, if they would mind sharing their thinking with the whole group. This prepares them to think about how they want to express themselves.

After quiet time, have students share out what they notice and record their responses. This is a time when some students may notice some math vocabulary or concepts that are either essential to understand the problem, or may be something you know will show up on a high-stakes exam. If a student mentions something, for example the word vertices, ask the student to elaborate: "Could you talk a little more about that?

Share out what students wonder and record their responses. If their wonderings are vague, ask them to elaborate.

Be sure to include wait time in both asking what students notice and what they wonder. This may seem a bit of an awkward silence if you are not used to wait time, but it's essential to give all students the opportunity to participate, as some students are slow to raise their hand. Did you see anything unique, funny, or witty on a student's paper that wasn't mentioned yet? Perhaps that student is ready to share out now.

Once students have shared out all of their observations and questions, the initial task can be given to students. Many times, the actual question (or a form of the question or task) that you want students to focus on, actually comes from the students, themselves, during notice/wonder. Be sure to make that connection as this invests students in the task: they own it. Give students 3 - 5 minutes of quiet time to individually work on the initial task.

As students are working through the problem, making use of the push and support cards, note the diverse problem solving strategies that students use while solving the task. Remember: not all students (or groups) may have gotten to the same level during the activity. Some students may have focused on concrete representations in the form of pictures or manipulatives, others may have used guess & check, some may have used a way to organize their information (a number line, diagram, or a function table), while others may have extended the problem to abstract procedures. Note at least three different ways students (or groups of students) worked with the problem. You may want to ask students ahead of time to share their discoveries when the whole group comes together to discuss the task.

Share out. There are different ways that you may choose to have students share out their discoveries. Below are two suggestions:

(1) Ask each group to create a poster that illustrates what they discovered and depicts the strategies they used in their exploration of the problem. The group should elect a spokesperson who will present their project to the class. The rest of the class should be encouraged to ask questions of the group when the spokesperson is finished presenting.

(2 ) Ask individual students to share out, one-at-a time, the way he or she approached the problem, sequencing the presenting of the strategies from the concrete to the more abstract. You may choose to have the student recreate the strategy on chart paper or the board, or the student may stay seated and describe the strategy, step-by-step, whilst you recreate it on the board or chart paper. As with option (1), the whole class should be asked if they have any questions for the student who shared his or her thinking.

Whichever share-out method you choose, it is best to have them displayed side-by-side so that students can compare the strategies as a whole group. How are they the same? How are they different? The point here is for students to develop flexibility when working with problems, learn new problem solving strategies, and understand the connections between the different strategies.

Download and print: The Student-Centered Classroom Routine

Tasks

Download and print all tasks: Tasks

Activity Name

CCRSAE Alignment

(Corresponds to CCSS Alignment)

The Freeway

Use units as a way to understand problems and to guide the solutions of multi-step problems. (N.Q.1)
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. (N.Q.3)

Analyze and solve linear equations and pairs of simultaneous linear equations. (8.EE.7) (8.EE.8)

Accurately Weighing Pennies*

Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations with two variables. (A.REI.6)

Saving for Mom's Present *

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (A.REI.3)

Represent and solve equations and inequalities graphically. (A.REI.10)

The Paycheck Problem*

Understand the concept of a function and use function notation. (F.IF.2)

Interpret functions that arise in applications in terms of the context. (F.IF.4)

The Painted Cube*

Distinguish between situations that can be modeled with linear functions and with exponential functions. (F.LE.1)

Reason abstractly and quantitatively (MP 2)

Look for and express regularity in repeated reasoning. (MP 8)

Bacteria Population Growth Lesson Plan*

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (F.IF.7)

Use properties of exponents to interpret expressions for exponential functions. (F.IF.8b)

*Examples of student work are available by accessing the Google Doc.

Reflection Activity

If you have been using the tasks linked to in this curriculum guide, now would be a good time to examine what some of the problems looked like before they were adapted for the multi-level adult education classroom and before the push and support cards were added to the tasks. This will give you some ideas on how you can modify the many rich tasks that already exist at OER Commons. Examine at least one of the problems listed below in its original form, designed for use in Grades 8-12, and then compare that with the repurposed version, designed for ABE/HSE adult education students. It is recommended that you complete this reflection activity before moving on to the next section.

How to Create Push and Support Cards Using OER Commons Resources

After your examination of both the original problems and the repurposed version of the of problems, you most likely have a lot of ideas now about how to create push and support cards. Keep in mind that the cards should be designed to encourage mathematical thinking as described in The Standards for Mathematical Practice. [2]

Here are a few suggestions on how to get started writing push and support cards:

Remove all questions and replace with notice/wonder

Suggest the use of manipulatives

Suggest students draw a picture to model the situation

Suggest students organize their information

Suggest students plug simpler numbers into the problem

Provide a function table with some of the information filled in

Use questions from original problem as push cards

Ask students if they see a pattern

Ask students to write a similar problem

Ask "What if?" questions

Ask "Is it possible?" questions

Provide challenges where students have to work backwards, such as solving for x

For some additional support regarding the de-evolution of a math problem, so as to include push and support, see the Math Memos write up about The Paycheck Problem.

Conclusion

While the focus of this curriculum guide has been on how to implement the strategy of push and support cards in your classroom, it is evident that there are additional factors that contribute to ensuring that these cards are used as an effective differentiation tool. The cards should be connected to an open-ended non-routine problem that all learners in a multi-level class can find a way in to, and then scaffold up to a level that is individually appropriate. Every student needs to be afforded time to allow their own thinking to emerge, and then given the opportunity to collaborate with other students, so they can hear their own voice, have their own voice, and value their own voice in the math classroom.

This means that there needs to be a relationship built between teacher and students where it is understood that the teacher is not the keeper of all knowledge. Rather, conceptual understanding of mathematics and a productive disposition that fuels problem solving happens in the students' minds and in their conversations when they question each other, explain their problem solving strategies, and rely on group conversations to confirm the solution to a math problem. Notice/wonder, high-cognitive tasks, think time, push and support cards, and collaboration are a few strategies that can be utilized to foster this kind of learning environment where all students in a multi-level classroom can become masters of their own learning.

References

[1] Curry, D. (2017). Using the PIAAC Numeracy Framework to Guide Instruction: An Introduction for Adult Educators. Retrieved May 04, 2018, from https://bit.ly/2qRLC1J. Washington, DC.

[2] Common Core State Standards Initiative, Standards for Mathematical Practice, 2018. Retrieved May 10, 2018 from http://www.corestandards.org/Math/Practice/

[3] Appleton, Eric (2017). A Strategy for Teaching in a Multi-Level Classroom: Push and Support Cards, The MATH Practitioner, Volume 22, Issue 3, Summer 2017. Retrieved May 04, 2017 from https://bit.ly/2rm6Tzi.

[4] OER Commons, Trushkowsky M. (2016) What Do You Notice? What Do You Wonder? Retrieved May 11, 2018 from https://www.oercommons.org/courses/what-do-you-notice-what-do-you-wonder

[5] OER Commons,Illustrative Mathematics, (2016). Felicia's Drive. Retrieved May 10, 2018 from https://www.oercommons.org/courses/felicia-s-drive

[6] OER Commons, Illustrative Mathematics, (2016). Fixing the Furnace. Retrieved May 10, 2018 from https://www.oercommons.org/courses/fixing-the-furnace

[7] OER Commons, Illustrative Mathematics, (2016). Accurately weighing pennies I. Retrieved May 10, 2018 from https://www.oercommons.org/courses/accurately-weighing-pennies-i

[8] OER Commons, Appleton, E. (2015). A Window into International Education and Problem Solving in the Mathematics Classroom. Retrieved May 11, 2018 from https://www.oercommons.org/courses/a-window-into-international-education-and-problem-solving-in-the-mathematics-classroom

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Push and Support Cards: A Bridge to Advanced Mathematics for the Multi-Level Adult Education Classroom Curriculum Guide

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Adult education classrooms are commonly comprised of learners who have widely disparate levels of mathematical problem-solving skills. This is true regardless of what level a student may be assessed at when entering an adult education program or what level class they are placed in.

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Chrome Music Lab is a website that makes learning music more accessible through fun, hands-on experiments.

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How were these built?

All our experiments are all built with freely accessible web technology such as Web Audio API, WebMIDI, Tone.js, and more. These tools make it easier for coders to build new interactive music experiences. You can get the open-source code to lots of these experiments here on Github.

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Music is for everyone. Play with simple experiments that let anyone, of any age, explore how music works.

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This is inside Chrome Music Lab and what we used in the Institute. Song Maker lets you make and share your own songs.

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Song Maker, an experiment in Chrome Music Lab, is a simple way for anyone to make and share a song.

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So many resources...so little time...
So, just start with one.

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Bookmarked on Resources for 2021 ANN Institute

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Desmos Activity ready for the ABE/HSE math classroom

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Think algebraically by investigating patterns found in a design of interlocking circles.

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Desmos Activity we did at the institute

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Explore these patterns with your students and watch their natural tendencies to see patterns morph into powerful algebraic thinking and reasoning. It’s an ideal routine to foster mathematical practice #7 - look for and make use of structure.

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Students will learn by interacting with mathematical representations, illustrations of the world, and their classmates. They'll represent their developing ideas with sketches, text responses, card sorts, number responses, multiple choice, and more.

Click Student Preview on any activity to experience it as a student.

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Students will learn by interacting with mathematical representations, illustrations of the world, and their classmates. They'll represent their developing ideas with sketches, text responses, card sorts, number responses, multiple choice, and more. Click Student Preview on any activity to experience it as a student.

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Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.

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Padlet of images of creatively colored interlocking circles. Add your own!

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Experiencing Harmony and Beauty in Mathematical Patterns

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Google Slides that introduce the ScrapColoring tool with 6 interlocking circles, explore line symmetry and rotational symmetry, and end with a set of symmetry challenges. Slides are ready to be used with students, complete with animations and teacher notes.

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You ought to be in pictures! Can we quote you on that? Grant us permission if so.

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, a teacher in the Twin Cities, defines math art as something which inspires curiosity about structure. If you are looking for activities to teach your students math through the making of beautiful art, check out her

Explore Celtic Knots, Sona drawings, probability designs, Islamic Geometry, Hitomezashi stitching, Hilbert curves, Mondrian puzzles, spirals, and more. Some are advanced, but most are accessible to all.

A good place to get started is to scroll through the Math Art Challenge page linked above. Annie included a picture of each project in the list, so it's easier to find one to try. You can also browse the hashtag on Twitter to see what others have done.

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a teacher in the Twin Cities, defines math art as something which inspires curiosity about structure. If you are looking for activities to teach your students math through the making of beautiful art, check out her Explore Celtic Knots, Sona drawings, probability designs, Islamic Geometry, Hitomezashi stitching, Hilbert curves, Mondrian puzzles, spirals, and more.

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Both Math & Art activities were inspired by this pattern, introduced by Sophie Griffith at a meeting of the NYC Community of Adult Math Instructors (CAMI). Sophie's activity uses the pattern of circles and sentence-starters to support learners in posing their own mathematical questions.

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Community of Adult Math Instructors (CAMI)

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Facilitator(s): Sophie Grifith
Date of Meeting: November 13th
Problem: pdf

In CAMI Meetings and in class with students, we often want a prompt to get students generating their own mathematical questions to answer, rather than giving them a predetermined math problem that everyone needs to solve. In the November evening meeting, we started off by considering some prompts and sentence starters to get students asking questions that will lead to math explorations.

Here are a sentence-starters that we came up with:

And a few questions we can ask to get students thinking mathematically:

After that, Sophie showed us a drawing of some circles.

What do you notice? What do you wonder?

When we looked at these circles, we had a *lot* of ideas and questions. In the exploration that followed, we found that one of the biggest challenges was drawing new images, especially with more than four circles. Some of us made pencil/paper drawings, and some of us tried to use online software. Sarah proposed making some pre-existing images on Geogebra or Desmos so that students could play with the images and explore math questions without needing to struggle with making the images themselves.

Check out some of the images we came up with!

Jamboard: (includes photos of pen-and-paper drawings) https://jamboard.google.com/d/1hWOdKlHIPZ-J42ENTMsIkSBaW-zJidgd2x1sJYHUYxI/viewer

Geogebra: (includes images that one group made during the meeting)
https://www.geogebra.org/classic/dnxmszv3
Categories Posted on Visual Patterns

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Facilitator(s): Sophie Grifith Date of Meeting: November 13th Problem: pdf In CAMI Meetings and in class with students, we often want a prompt to get students generating their own mathematical questions to answer, rather than giving them a predetermined math problem that everyone needs to solve.

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Annie Perkins, a teacher in the Twin Cities, defines math art as something which inspires curiosity about structure. If you are looking for more activities to teach your students math through the making of beautiful art, check out her Math Art Challenges. Explore Celtic Knots, Sona drawings, probability designs, Islamic Geometry, Hitomezashi stitching, Hilbert curves, Mondrian puzzles, spirals, and more. Some are advanced, but most are accessible to all.

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Hey folk! Here are links to all of the #MathArtChallenge…challenges. :) Please feel free to share widely. This dropbox folder has lots of videos/pictures, too. Also, if you have an idea for a…

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Clarissa Grandi hosts Artful Maths where she shares Math Art lessons with full instructions to explore geometry and draw curves of pursuit, impossible objects, mandalas, Celtic knotwork, and more!

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On this page you will find a selection of fully-resourced lessons with a 'mathematical art' theme. The activities in these lessons will help to reinforce many important mathematical skills such as accurate measuring and constructing, visual problem solving and strategic forward planning.

Each resource is accompanied by brief instructions and photos of students' work. If you have any questions about these resources, please do not hesitate to contact me.

I hope you have found something here to try out with your students. I would love to see photos of your students' work if you do. If you post anything on social media, perhaps use the hashtag #mathsart. And do watch this space for new ideas - I have plans for a 'constructing a mandala' lesson, and one on Islamic tiling patterns too... oh, and maybe one on fractals... So much mathematical art, so little time!

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Fully resourced lessons with a mathematical art theme.

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Your feedback is appreciated!

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Teaching & Learning Institute Evaluation 2021

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Responses are anonymous and will be used to improve future institutes and training events.

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Paula Beardell Krieg is an artist who makes works with paper, teaches book arts, and thinks a lot about math. Her website is filled with activities for students to explore paper folding, lines, fractions, patterns, symmetry, rigid transformations and familiar shapes (like triangles, hexagons and pentagons) and unfamiliar shapes (like kaliedocycles and hexaflexagons).

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Symmetry challenges and investigations, as well as tools including coloring 6, 7, 8, 9, 10, 11, and 12 interlocking circles with the ScrapColoring tool.

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Demos Graphing Calculator: Interlocking Circles w/Slider

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Demos Graphing Calculator: Interlocking Circles w/Slider

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Butour Ngale - African polyrhythm demonstration

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Butour Ngale - African polyrhythm demonstration

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The video we watched during the institute. Watch and listen for the two different beats

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Butour Ngale - African polyrhythm demonstration

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Counting the same rhythm in “threes” and “twos”. Nice (short video) and visual of the “sixths” at the bottom, very similar to our fraction strip activity

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Free tutorial showing you how to count a 3/2 polyrhythm. Ideal free guitar lesson for fingerpicking guitarists to improve their rhythm

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44th Anniversary of the Birth of Hip Hop

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44th Anniversary of the Birth of Hip Hop

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Google Doodle from August 11, 2017. This highly interactive tool allows you to play and mix two classic beats and play around with the BPM - either matching them or speeding them up/slowing them down.

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This is the tool we used to create our coloring pages for the institute.
You can upload pictures to turn into coloring pages.

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Convert your Drawings and Photos to Coloring Pages

New! You can turn into a coloring page any drawing, image or photo (that you found on the Internet or that you have drawn and scanned). The coloring page you create can then be colored online with the colorful gradients and patterns of ScrapColoring!

All you need to do is to upload your image with the form below, and it will be automatically and almost instantly into shapes that can be colored.

Notes:

Converting the pixels from the picture into shapes is a process called vectorisation. Results are generally not perfect. They vary greatly from the images used as input. Shapes with sharp strokes, filled with flat colors (or black and white) work best. Photographs with a lot of details, many colors and/or fuzzy strokes are more difficult. You will often get surprising but always interesting results! You can also try with already colored drawings, posters etc.

The optimal size of the input image is 600x600 pixels. If you send an image with a different size, it will be automatically cropped and resized.

We hope you will have much fun using our coloring page generator as wed had creating it and testing it! Please tell us what you think and show us your creations!

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ScrapColoring - Convert your Drawings and Photos to Coloring Pages

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Free online coloring pages for kids with a rich variety of colorful patterns, gradients, fabrics, papers and textures for hours of fun and creativity.

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The Magic and Logic of Color: How Josef Albers Revolutionized Visual Culture and the Art of Seeing

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The Magic and Logic of Color: How Josef Albers Revolutionized Visual Culture and the Art of Seeing

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"Hundreds of people can talk, for one who can think," John Ruskin wrote, "but thousands of people can think, for one who can see." "We see, but we do not see: we use our eyes, but our gaze is glancing, frivolously considering its object," Alexandra Horowitz lamented in her sublime meditation on looking. Hardly anyone has accomplished more in revolutionizing the art of seeing than German-born American artist, poet, printmaker, and educator Josef Albers (March 19, 1888-March 25, 1976), as celebrated for his iconic abstract paintings as he was for his vibrant wit and spellbinding presence as a classroom performer. In 1963, he launched into the world what would become the most influential exploration of the art, science, psychology, practical application, and magic of color - an experiment, radical and brave at the time, seeking to cultivate a new way of studying and understanding color through experience and trial-and-error rather than through didactic, theoretical dogma. Half a century later, Interaction of Color (public library), with its illuminating visual exercises and mind-bending optical illusions, remains an indispensable blueprint to the art of seeing.

Albers, who headed the legendary Black Mountain College that shaped such luminaries as Zen composer John Cage and reconstructionist Ruth Asawa, lays out the book's beautifully fulfilled and timeless promise in the original introduction:

In visual perception a color is almost never seen as it really is - as it physically is. This fact makes color the most relative medium in art.

In order to use color effectively it is necessary to recognize that color deceives continually. To this end, the beginning is not a study of color systems.

First, it should be learned that one and the same color evokes innumerable readings. Instead of mechanically applying or merely implying laws and rules of color harmony, distinct color effects are produced-through recognition of the interaction of color-by making, for instance, two very different colors look alike, or nearly alike.

Albers defied the standard academic approach of "theory and practice," focusing instead on "development of observation and articulation," with an emphasis not only on seeing color, but also feeling the relationships between colors. He writes:

[ Interaction of Color] reverses this order and places practice before theory, which after all, is the conclusion of practice. ... Just as the knowledge of acoustics does not make one musical - neither on the productive nor on the appreciative side - so no color system by itself can develop one's sensitivity for color. This is parallel to the recognition that no theory of composition by itself leads to the production of music, or of art.

Practical exercises demonstrate through color deception (illusion) the relativity and instability of color. And experience teaches that in visual perception there is a discrepancy between physical fact and psychic effect. What counts here - first and last - is not so-called knowledge of so-called facts, but vision - seeing. Seeing here implies Schauen (as in Weltanschauung) and is coupled with fantasy, with imagination.

To mark the book's fiftieth anniversary, Debbie Millman, who is herself a master of color, sits down to discuss Albers's far-reaching legacy and his fundamental contributions to our everyday understanding of color with Brenda Danilowitz, Chief Curator at the Josef and Anni Albers Foundation, and Philip Tiongson, who designed the magnificent iPad app accompanying the new edition of the book (an app so exceptional, in fact, that Millman rightly calls it "the example the world has been waiting for in order to begin to understand how it's possible that books will never, ever go away"). Here are some of the highlights from an altogether fascinating conversation.

On how the brain's conditioning to notice only what it expects cheats us of the richness of seeing:

Albers believed that in normal seeing, we use our eyes so much because the world is controlled by our vision, but we become so accustomed to it that we take things for granted. And when he talked about visual perception, he meant something much more profound than just the way we look at the world - he would stop and look at the world, look at the smallest object, smallest event, and see through it in a deep kind of way. ... He would see magic, he would see something deeper. And he believed that the majority of people just missed the true reality - it was available for everyone to see, but nobody was looking. And that was where his notion of "to open eyes" really comes from.

On Albers's unconventional approach as an art educator and the mesmerism he had over his students:

The one word that to Josef Albers was absolute anathema was "self-expression." He said you do not express yourself - you have to learn, you have to have these skills, and then you create something.

Fittingly, one of Albers's most memorable quotes:

Easy to know that diamonds are precious. Good to learn that rubies have depth. But more to see that pebbles are miraculous.

On how Albers embodied the aphorism that "the art of teaching is the art of assisting discovery" and challenged his Black Mountain College students to experiment with materials in a way that counters the assumptions of perceptual reality:

He believed in experiential teaching - not in putting out a rule and teaching students how to execute that rule. He believed in discovery in the classroom, and that is why his classes were always new and different.

On Albers's intention with building not a theoretical treatise but a practical toolkit for understanding color:

Albers was not interested in creating a treatise on color. He was not giving you rules about color - he was giving you tools to unlock what he considered the magic of color.

Hear the full interview below, and subscribe to the indispensable Design Matters on iTunes or SoundCloud:

Interaction of Color (public library) is an essential piece of visual literacy, exploring such fascinating subjects and phenomena as color recollection and visual memory, the relativity of color, transparence and space-illusion, temperature and humidity in color, and the afterimage effect. Complement it with Goethe on the psychology of color and emotion and The Black Book of Colors.

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The Magic and Logic of Color: How Josef Albers Revolutionized Visual Culture and the Art of Seeing

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"Hundreds of people can talk, for one who can think," John Ruskin wrote, "but thousands of people can think, for one who can see." "We see, but we do not see: we use our eyes, but our gaze is glancing, frivolously considering its object," Alexandra Horowitz lamented in her sublime meditation on looking.

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Given what's happening in my community right now, I felt it was necessary to highlight some Black brilliance today.

The Challenge: (Re)create a Sona drawing. I did a couple hours of research yesterday (which is totally insufficient to fully understand it), but what I can tell you is that these drawings originate with the Chokwe people in southwestern Africa, specifically Angola and the southern part of the Democratic Republic of the Congo. The drawings are told in conjunction with a story, and the goal of most is to draw them in as unbroken a line. Please check out some of the resources below, and if you have others to add, I'd love to see them and link them here.

Materials Needed: grid, pencil, paper, perhaps graphing software.
Math concepts you could explore with this challenge: angles, counting, geometry, graph theory, proportions/ratios, slope, symmetry, vertices/intersections

As you can see, I made a number of mistakes in doing these. I wanted to leave those mistakes in because I think it's important, as a teacher, to model making mistakes, and also because I want you to appreciate the challenge of some of these drawings. To draw in a wholly unbroken line is not always an obvious task.

All of the drawings I did here were highlighted in the articles I found - not yet confident enough to create my own images, although this is absolutely a thing I'll return to when I'm done.
Mathematical_Experiments_with_African_Sona_Designs Depending on how you use this activity, you may engage with different mathematical standards. I've listed possible connected math content above. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

1.) Make sense of problems and persevere in solving them. Can you create a description to guide a complete circuit throughout any given drawing? How do you know when to turn around, make a 180 degree turn?

8.) Look for and express regularity in repeated reasoning. What can you say about the total number of lines needed to complete an nxm grid?

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Given what's happening in my community right now, I felt it was necessary to highlight some Black brilliance today. The Challenge: (Re)create a Sona drawing. I did a couple hours of research yesterday (which is totally insufficient to fully understand it), but what I can tell you is that these drawings originate with the Chokwe...

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Two teams of teenagers use algebra to match the tempo of an electronic drum track to the tempo of an instrumental sample created by DobleFlo.

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Manny Dominguez and Luis Lopez of DobleFlo talk about how their duo got started, how they use math in producing hip-hop music, and set up a music-related algebra challenge.

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In the K-8 Art Studio with Anita Sagastegui

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Description of the Unit - Students will explore Josef Albers' series Homage to the Square, using his artwork and our discussion as a guide to creating their own series of squares, concentrating on contrast and depth.

Activity statement - Josef Albers' 25 year-long series Homage to the Square, begun in 1950, explores the optical effects created by superimposing colored squares over one another. The paintings are deceptively simple compositions of straight-from-the-tube oil paint applied with a palette knife on primed white Masonite panels (so cool!). According to the Guggenheim's website, "the optical effects Albers created-shimmering color contrasts and the illusion of receding and advancing planes-were meant not so much to deceive the eye as to challenge the viewer's faculties of visual reception. This shift in emphasis from perception willed by the artist to reception engineered by the viewer is the philosophical root of the Homage to the Square series. Albers tried to teach the mechanics of vision and show even the uninformed viewer how to see," ( Guggenheim).

Study for Homage to the square, 1956

"With Rays" - 1959, from Albers' Homage to the Square series

Students eagerly respond to this visual challenge. As simple as the images may seem, right away students pick up on the use of color to create the illusion of these advancing and receding planes. Students considered which colors created the most contrast and depth (analogous or complimentary?) and contemplated how placement of the squares, as well as their diminishing size, enhances the illusion of three dimensions. As one student noted, the squares "were like an echo, but with color."

Besides thoughtfully arranging their own squares, this project is a wonderful way for students to develop their fine motor skills, as cutting straight lines into a square is not the easiest on little 1 st grade hands. Some students request to use rulers, which is fine, as they still have to cut that line they drew, but it's not required of they don't want to.

Goals - Students should...

Understand:

Complimentary (contrasting) colors on the color wheel

Understand analogous colors on the color wheel

How color can contribute to optical illusions, such as the illusion of depth

How placement and size of squares contributes to the illusion of depth?

Know:

What is meant by an optical illusion

What is meant by advancing/receding planes

Be able to:

Use complimentary and analogous colors in a thoughtful arrangement of squares
Resources and materials -

Examples of Albers' paintings from Homage to the Square

Multicolored construction paper

Scissors

Rulers

Glue

Questions -

What do you notice about (chosen Albers' example) painting?

What can you tell me about the colors in (chosen Albers' example) painting? Are these analogous colors or complimentary? Which ones are analogous? Which ones are complimentary?

What does it feel like to you to look at the squares? Are the squares moving away from you, or towards you?

Are the squares in the center of the painting?

How do you think Albers created this painting?

Evaluation - Did students:

Understand the meaning of analogous colors and complimentary colors?

Understand how color arrangement can create a three-dimensional illusion?

Understand how the placement and size of squares creates the illusion of depth?

Thoughtfully choose colors and arrange their own squares to attempt the sense of depth?

Informal:

Group discussions

Oral and responses to essential questions

At least one finished piece

1st grade student examples:

1st grade student work - Homage to the Square

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1st Grade - Josef Albers' Homage to the Square

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Description of the Unit - Students will explore Josef Albers' series Homage to the Square, using his artwork and our discussion as a guide to creating their own series of squares, concentrating on contrast and depth. Activity statement - Josef Albers' 25 year-long series Homage to the Square, begun in 1950, explores the optical effects created by superimposing colored squares over one another.

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Polyrhythms: Interactive Pattern Generator 

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Polyrhythms: Interactive Pattern Generator 

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Tip : click the blue dots to load their associated settings.
● 1-2-5, 1-2-7 and 1-2-9 are all good battle noises. Keep the 1 stronger than the other two though.

● 5:9 feels really hasty - really fun rhythmic effect :)

● 2:3:5 sounds surprisingly simple but is still really dreamy.

● 1:3:9 is basically the rhythm cubed, so it creates a very interesting auditory effect.

● 4:6:9 is where its at (its two different 2:3 polyrhythms "stacked")

● 3-5-7 sounds like something falling down the stairs.

● Love this pattern generator, this setting gives a complex and uplifting rhythm.

● Came here for relaxing, but instead I am dancing.

● Really great interface for experimenting with polyrhythms! Wish I could control accents on the individual drums, as sometimes it is confusing where the 1 is! Thanks for this :-)

● This one is the longest one you can do. It only repeats ever 210 beats.

● 1/2/9. This one sounds like something really important is about to happen in a movie.

● 2-3-4-5-6 Best one!

● 1/4/5/8 sounds awesome.

● Wow! This is very cool. Some of the more complex polyrhythms sound like tabla patterns.

● Great tool to learn more complex rhythms.

● The page is great and really easy to use, would recommend to anyone trying to gain a better understanding of rhythm!

● I'm practicing some Chopin these days, and he puts a 7:6 in a lick, and an extended implied 8:6 in another piece. So this metronome is awesome for getting that engrained in me!

● This is a wonderful tool, thanks for it! I would love a way to change which beats are accented for each number, i.e. you could click on 8 and a popup would appear that allowed you to choose to accent say beats 1 and 5 or 1, 4 and 7. Keep up the good work!

● This one sounds cool to me.

● Totaly honest here, 2 and 5 at max volume, 4 and 10 at about 75% volume. This sounds amazing.

● Lmao the person who said they have a smart child when they literally just put up all the sliders except one.

● I think I'll try to use this rhythm to make a song some time. I really like how simple this website is to use!

● This seems like a war preparation.

● I have a smart child and she made this! It's a 2\3\4\5\6\7\8!

● My only complaint is that I cant make the tempo go extremely high or extremely low but other than that this is perfection.

● 2-4-6-8 Which website do we appreciate? myNoise! myNoise! Haha.

● Also useful for practicing on the piano. Like a tier up from a normal metronome.

● Keeps me at a steady pace.

● 1 2 4 5 10

● I really like 3/7/10, Really unique feel.

● This generator can also be used for tuplets.

● The sound resembles popcorn popping.

● This is literally rain, try it lol!

● 2-5-10 straight up Crash Bandicoot Vibes

● This is very useful for understanding and practicing polyrhythms. The African Trance generator is lovely, but definitely not the same at all; it is possible, with some fiddling, to run them both concurrently, giving a much more drum-based trance experience.

● Try with Canyon Drone. It is beautiful. It has taken me to my tribe again..

● Excellent for picturing signal drums in a story! Recommending especially: 2/3, 2/5, 3/4, 4/6, 6/9

● This sound like a tribal chant!

● Reminds me of knights galloping into battle, with the epic drummers close behind.

● 3 5 7

● 2-3-4-6

● Bring them up one at a time and it sorta sounds like some sort of build up..

● Sounds jazzy and feels like pop at the same time.

● This sounds like a bunch of professional soldiers ordered to skip-march! Sounds a little silly and serious at the same time.

● These sounds vibrate along my head in a good way :D

● Strong 1 and lower with 7 and 8: seems like the time slows down...

● When I use 3 and 5, what does it sound like? (hint: think of the sounds you remembered as a 90's kid)

● Pair this with Huu Chant and Duduk Song. Makes it sound like an intimidating video game soundtrack.

● This and Implanted Memories together is great for focusing at work! Implanted Memories is calming, and the drums fit in so perfect as a "driving" energy!

● 2 3 5 6 and 9 I found interesting

● Very interesting beat, easy to play over once you get 3 over 4 down.

● 5 - 6 - 9

● 2 on 3 on 5 on 7. Chaotic and solid at the same time.

● This 5 on 7 pattern has a surprisingly unified, easy-to-follow rhythm. Together, they hit almost all of the sixteenth notes!

● 4, 5, 10 sounds super dope.

● Try 1, 3, 4, 6 and 8. Sounds good!

● 1, 3, 4, 6, sounds nice tbh.

● I use this to practice my drumming. It is like having your drum circle.

● 4/5 thing :)

● This one is extremely meditative for me-it fills up my whole head and is very relaxing.

● This one seems simple, but look closer.

● Playing around with the sliders for the 6 7's and 9's I came up with this Driving Rhythm, sounds like your in a Fast paced Race against time. Maybe a good workout tone to try...

● 1 4 and 6 sounds like Carol of the Bells :D

● Great when paired w/ the fireplace. Sounds like they're dancing and beating drums around a fire pit.

● 1, 3, and 4 -- sounds a little like 'Money' by Pink Floyd lol.

● Came here because Adam Neely, stayed here because of half an hour of uninterrupted fun.

● 7:5 sounds nice.

● Try 4, 5, 10.

● 1, 3, 5 and 7 go really well together. Must try!

● I like altering the beats until it works for me, catching enough of my 'distraction radar' to stop me form actually being distracted!

● Rush! Three, five and their multiples. This is exciting!

● Decapod Gallop: Frequencies increasing arithmetically by three, gives rise to a galloping 1:4:7:10 polyrhythm that is quite mesmerising. For a less insistent sound, lower the first slider a bit. Helps me focus :-)

● Simple yet addictive. Goes well with pads or melodies.

● This pattern sounds like a 2 against 4 against 8 rhythm, but it has subtleties to make it sound nice. Using this Rhythm with Marimbas helps me focus.

● Listen closely. There's a lot of really cool subtleties going on in this 5-against-2.

● This is ceremonial.

● Three drums, beating in 3, 5 and 7, will keep your hands, head and feet moving independently but together - a lovely way to loosen up :-)

● War Drums!

● It's like happy walking to me.

● When first heard it sounds like a 1 2 & 3 rhythm, but listen closely and there's a whole host of other stuff going on (because of drums 4, 5, 7). Really encourages you to focus!

● This one sounds almost Native American.

● When I need to focus and do really detailed work, I add this to a more "chill" soundscape. It adds some forward motion and changes the mood from "mellow and relaxed" to "do all the things without getting frantic."

● I combine it with Bilateral Harmonics for work and it's very brain stimulating :)

● This is what we used to call 'a long walk'.

● Sounds intense. I'm not sure what situation it fits in, but it's definitely not calm.

● It's like the perfect rhythm for a hunt or chase scene.

● Really useful for song maker. Helps visualize poly-rhythms. This is defs something I am using on my next album.

● This one sounds like an epic movie.

● Twisted dance--helps me to do maths and dance.

● This feels like a chase scene on a nature documentary and it's awesome!

● ← How 'bout this?

● This is one of my all-time faves to combine with any others. Thank you so much for making these available! I will donate again soon. :) Cheers!

● I think this is my favorite, just put in on in the default setting at work or home, keeps me focussed but in a flow. Absolutely gorgeous with singing bowls, sliders on animate.

● Sounds interesting and different from what I usually listened to.

● It feels like I'm running away from something in a suspense-action film.

● This sounds intense!

● Breathe slowly & deeply, while listening.

● This sounds so intense, like running from a wild animal!

● When I have a migraine and regular music feels too strong and high pitched, this quick and simple beat seems to really help relax my mind.

● ← These settings remind me of the march of an army. It feels very active.

● I used one of the presets mentioned here and combined it with Canyon. While Canyon gives peace and cuts out external noise African rhythm doesn't allow me to space out. Perfect for work and helps to avoid overwhelm when I have a pile of things to do! Thanks a lot.

● Wow. I'm sick at home but listening to this with Distant Thunder makes me feel like I could do anything.

● I can hear the 3, 5, or 7 subdivision in this setting. It mainly sounds like a groove in 5 to my ear.

● I play this one with Didgeridrone when I need to actively focus on something for an extended time. Highly recommend!

● 1 2 3 and Azerbaijan + Australia from Planet earth give perfect sound kit for a tribal ritual.

● I feel this one makes me happy, and in general sounds real good :)

● This is a great tool to help practice with playing polyrhythms! 5 4 3 has been giving me a lot of trouble, but now with this, I can hear and play the rhythms clearly!

● Shamanic ritual & war dance vibes. Badass groove for fighting spirits.

● Powerful and strangely erotic.

● I use this one when playing Chivalry on Steam... This beat keeps me amped.

● Animated, this swings through several beautiful, driving rhythms alternating with calm meditative ones. I've been using it in conjunction with RPG Elements. It is truly amazing how much the texture changes as the five rhythms ebb and flow. I have been using it as a background for writing, for working out, and for meditating, and somehow works well for all those disparate things.

● Reading the end of David Lindsay's A Voyage to Arcturus, I thought how fitting it would be if I listened to 4 and 2 on this generator to mimic Surtur's drumlike beat. It added a pounding severity to the experience of that imaginative and revelatory ending.

● Very nice beat going on here. Sounds like an fight or chase scene.

● 1+2+9 with the Folk Tradition Flutes, particularly with the FT set to vary over time, it just transports me to another place entirely! -E-

● A lot of the recommended settings talk about battle music, if you want to hear real battle music use this and do the heavieness setting on Electric Guitar Mashup. Add Siren song and turn up slider 7 to add some epic fantasy vocals!

● I really like this tribal beat...

● 1+2+4+5+10 with a little 8 is great! I've probably spent too much time here... back to work! Thanks.

● I can use this for compositional ideas!

● ← I just like the way this set talks like it needs 90 seconds to tell the story.

● This setting combined with Isochronic Brainwave Generator set to 'relaxed' creates a fantastic platform for out of the box percussion practice - I play and teach percussion and hand drumming and the many colliding rhythms paradoxically sets me free of rhythmical constraints and enables a really liberated practice - Like a rhythmical meditation with the brain and hands set free to wander and create.

● 1-(2)-3-(4)-(6) is really nice and interesting :)

● Oh my goodness, this is PERFECT for my trance journey and meditation needs! I must share this! -E-

● Haha, this is a fun one! Feels like going on a dangerous hunt with this preset...

● This blends with all other sound generators nicely. What a way to induce trance! Absolutely love it.

● Wow I love this one. <3. I suggest 1, 2, 3, 4, 6, and 8 all together at the same volume.

● This sounds almost like a bodhran doing a 4/4.

● So here I am, studying for my accounting final... at the same time here I am facing my own mind in glorious battle. That should keep the ADD busy while I study... Paired with Tibetan Sprit (Low in Altitude).

● Playing this with the Shepard rising tone is a great way to build up tension, infinitely...

● Basically, the African Trance is the perfect backbeat for any of the other sounds on this site. It makes any other soundscape ten times more intense and powerful, instantly.

● Now this is a BEAT so energetic and exciting; my BRAIN is DANCING!

● Intense enough to keep you calm, yet energetic and focused.

● I can't believe how amazingly it works together with circular breeze. Because both are very different soundscapes, but amazingly they help me focus in way that I have never been able to do before. Its insanely amazing.

● This rhthym sounds like war has begun. Helps me be more focused.

● Play this with Himalayan Voices, Tibetan Choir, and Temple Bells, and you will forget where you are.

● Very energetic for me, keeps focus.

● Makes me feel that I am escaping a village and into the jungle!

● The fight has begun.

● Sounds like a hunt is on.

● Polyrhythm with Didgeridrone on animate creates a energetic combination!

● Most of the time when I'm trying to concentrate on math, noise generators of any kind drive me crazy. But this one is really helpful; something about the intensity keeps up with my mood when I feel frustrated, and keeping it on animate keeps it from getting to repetitive--there's always a slightly different rhythm rising to the top. Thank you very much for this!

● 10/10, animate, plus the Tibetan Choir drone generator.. A feast for the ears! Very inspiring when I need something dynamic and primal to fuel me.

● I want this to be played at my funeral so no one has the chance to be sad.

● This is really intense! LOVE IT.

● Takes me back to my days of djembe drumming in a circle. There was always a certain point in drumming where you almost left your sense of a body behind as you became ingrained into the rhythm. The tricky part was that as soon you became aware of this, the rhythm completely left you! Djembe drumming is so much more than just hitting a drum.

● It sounds like a tribal festival and it's super relaxing! Love it!

● I personally used the African Trance along side Indian Drone and just set both of them to animate. I really suggest trying to put two Noise Machines together. It's a lot of fun!

● The heart of battle against an admirable foe of logic and ever-changing specifications. This is my war song: keyboard warriors unite! Pair this with Digeridrone and Tibetan Choir for the ultimate in ambiance.

● Makes me feel like I'm about to ride into battle...

● I really like this on animate. The inconsistency helps with my ADHD during really stressful days and gives me something to focus on.

● A variation on the 1-3-6-9 pattern below. While this helps me focus, it's also hard to stop tapping along with my pen/pencil to add additional rhythms over the top.

● Hypnotizing!

● This is great. :D

● Dunno why but makes me think of 'Carol of the Bells'

● 10/10 would listen

● Inspiring and majestic

● African Transe drums over Three Friends of Winter somehow reminds me of Firefly's soundtrack - dynamic yet subtle, good for a long day of coding.

● I like how this sounds. I've got it on a Fibonacci sequence of 1, 2, 3, 5, and 8... then animate it! It drowns out the distracting conversations and keeps me from falling asleep as I work :-)

● Jovial but unhurried, drums to add a little dance to your day!

● This is so simple but so gorgeous to listen to.

● This what I listen to to get me in the mood for basketball

● Reminiscent of Halo's combat vibes

● Simply beautiful, it calms me so much when I'm having a stressful day.

● I love to layer this with quiet brook in water stream - makes me think of horses galloping alongside a rushing river.

● I like putting this and Winter Walk (using only the stream and birds) and writing books.

● This polyrhythm is fun because it mixes two different 3-over-2 beats. If you concentrate, you can hear either the slower (6-over-4) or the faster (9-over-6) beat. The challenge is: Can you hear both of them, at the same time?

● &karr; This is what I expected to hear when I clicked on the war drum preset

● This one is my favorite

● ← 2-4-5-10 has a nice groove

● ← This is great; 2 on 3 with a syncopating backbeat.

● This with the binaural beat generator (lethargic 1 hz) setting with volume cranked up to max on the beat is great

● My favorite "incessant and compelling" beat in order to focus - Layering "African Trance" in 2 to 3 separate tabs. Works well (for me) using the same 'Groovy' preset in each tab and setting each tab at a different volume. You can play with the synchronization by restarting one of the players. Note: using Chrome.

● Just the default pattern along with the Canyon Drone and Wind Noise makes me feel like I should be sitting out on the edge of the Grand Canyon watching the clouds roll by.

● Maasai beat that's use in mating ceremonies

● Like fireworks almost.

● This is probably not in the spirit of the generator - it's a 2-4-8 rhythm combination - but I find it grounding and steadying, and it helps put the chaos in my head in order.

● It's like an endless loop out of something you could hear on Battlestar Galactica. Perfect for chilling out or just on the background while playing games or somesuch, gets you in the zone.

● This arrangement sounds amazing! It feels like the pounding of someones footsteps running from something or someone. I can practically see the way the camera zooms up close to the characters face as he bolts through the forest floor.

● African death-robot march :D

● I love this generator! It allows me to create such a vast spectrum of music! This one sounds like hundreds of people slowly marching to battle, or maybe panning around a treacherous terrain that the characters have to cross.

● I think this beat sounds familiar, like a sort of hunting sound.

● You enter the tower and the ages old stone construct smash's threw the wall.. with eyes of flame and claws of gold. Fight for your life.

● Variations of rhythm.

● This sound is catchy.

● Very spirited serious drumming.

● I totally feel the heat of Africa... This is amazing...

● Wow. Just... wow. Catchy and fun

● This really helps bring me back to the present from bad memories and the rhythm keeps me present.

● Epic hunt-like beat - enjoy :P

● Raining drums! Relaxing and exciting at the same time

● ← party crasher because at some point it sounds like it gets dark

● I find this noise particularly relaxing. The interactions between the sounds are so interesting, yet calming.

● Reminding of a horse or tribal judgement. Good for trance too.

● This is a trance beat i did not found on the other presets. Greetings from Brazil. Cool noise generators by the way. I really enjoy this experience.

● Very interesting beat, I like it alot! Very lovely work you've done with this.

● Really great beats! Soothing, simple, but interesting enough to keep my mind occupied, or relaxed, or whatever I need.

● While the settings produce a relatively simple sound, there's something profound about the interaction of the beats -- like the feeling of a storm brewing, or something big about to happen. For so simple a setting, it's remarkably intense.

● Concentration

● A fast, continuous yet somewhat complex beat... instantly brings me into the shoes of a running hunter. Relaxing in a way. -R.A

● Totally reminded me of LOZ!

● When a noise like that start in a video game, you know you're screwd...

● Bongo!

● This setting makes me feel like I running form something really bad. In other words i feel epic

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Polyrhythm Pattern Generator * Design Your Own

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Tip : click the blue dots to load their associated settings. ● 1-2-5, 1-2-7 and 1-2-9 are all good battle noises. Keep the 1 stronger than the other two though. ● 5:9 feels really hasty - really fun rhythmic effect :) ● 2:3:5 sounds surprisingly simple but is still really dreamy.

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Thanks to Katherine Seaton for sharing this idea!

The Challenge: Using grid paper, assign each row/column a 0 or a 1. Then "stitch" both ways. You could assign the 0s and 1s as you prefer, or with a coin, or you could code something in binary!

Materials Needed: Grid/dot paper (You can print some or make some without too much trouble), or if you have the stitching materials...
Math concepts you could explore with this challenge: binary numbers, randomness, probability, symmetry

Yes, this is based on stitch patterns made with running stitch i.e. alternating between front and back of fabric. Either the first stitch is present (1) or absent (0) on the surface.
- Katherine Seaton (@maths_kath) March 29, 2020

Here's a brilliant long form tutorial from Neil Butler, if that helps!

And here is a brilliant way for you to adapt this to teach experimental probability:
Depending on how you use this activity, you may engage with different mathematical standards. I've listed possible connected math content above. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!
4.) Model with mathematics. Discuss designs like those shared by Constance Rojas-Molina above. What do you notice about the relative structure and symmetries provided by Rojas-Molina? What does "random" really mean?

7.) Look for and make use of structure. Notice how many "layers" are created by each design. Is there a way to increase the number of layers? How do the areas of each layer compare? How many colors are needed to color each design and why?

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#MathArtChallenge Day 14: Hitomezashi stitching (Suggested by Katherine Seaton)

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Several examples of Hitomezashi stitiching. Some on paper, some on cloth Thanks to Katherine Seaton for sharing this idea! The Challenge: Using grid paper, assign each row/column a 0 or a 1. Then "stitch" both ways. You could assign the 0s and 1s as you prefer, or with a coin, or you could code something...

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4th Grade – Intro to Orphism with Sonia and Robert Delaunay

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4th Grade – Intro to Orphism with Sonia and Robert Delaunay

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In the K-8 Art Studio with Anita Sagastegui

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Description of the Unit - Students will discover Orphism through the work of Sonia and Robert Delaunay, creating their own Orphism-style design, focusing specifically on the dynamics of color.

Activity statement - Artists Sonia & Robert Delaunay pioneered a form of abstract expressionism, Orphism, that invited the viewer to be drawn in and affected by the intensity and relationship of color-daring you to feel as moved by an abstract work of art as you would feel by a representational one (in which the subject itself affected you). This worked by very deliberately playing with the dynamic between the composition of shapes on a format and the colors within those shapes. Sonia Delaunay carried this idea beyond painting into textile design, innovating a new approach to fashion and decor. Sonia and Robert were both inspired by one another's work and one could say their art was a sort of call and response riffing between the two artists.

Sonia & Robert Delaunay

Through the Delaunay's paintings, students will explore the expresiveness of color, in the same way Robert Delaunay "was particularly captivated by how the interaction of various colors generated impressions of movement and depth without allusion to nature" ( https://www.theartstory.org/artist/delaunay-robert/) How can this be done? How can such compositions inspire feelings of surprise and joy? As Sonia Delaunay said, "color is the skin of the world... (and color) brings me joy."

Electric Prism, 1914, Sonia Delaunay

Color Explosion, 1939, Robert Delaunay

Rhythm in Colour, 1959, Sonia Delaunay

Goals - Students should...

Understand:

The concept of abstract art

What makes for a dynamic, active abstract composition when using a variety of shapes

Know:

Who the Delaunay's were, and the basic characteristics of the Orphism movement

Be able to:

Identify primary and secondary colors

Match complimentary (opposite) primary and secondary colors

Objectives - Students will: Create one Orphism-inspired design in oil pastels
Resources and materials -

Examples of work by Sonia & Robert Delaunay

White Bristol paper or tagboard in a square aspect (I like 12" x 12")

Multicolored oil pastels

Rulers

Circle stencil in various sizes

Questions -

(While observing the Delaunay Orphism paintings): What do you notice?

What colors can you identify?

What do you notice about which colors are next to other colors?

What kinds of shapes do you notice?

What feelings do you notice experiencing as you look at this painting? Why do you think that is? Describe...

Do you recognize any images from real life?

What colors would you combine in your own painting to give an excited feeling to it? Why?

Evaluation - Did students:

Design a piece using a thoughtful, deliberate relationship between colors?

Create a compelling dynamic between circles and angular geometric shapes?

Informal:

Student questions

Group discussions

Oral responses to essential questions

Elaboration and risk-taking

One finished composition

4th grade art - Orphism-style oil pastel illustrations

4th grade art - Two Orphism-style oil pastel illustrations

4th grade art - Two Orphism-style oil pastel illustrations

4th grade art - Two Orphism-style oil pastel illustrations

4th grade art - An Orphism-inspired oil pastel illustration

4th grade art - An Orphism-inspired oil pastel illustration

4th grade art - An Orphism-inspired oil pastel illustration. It's interesting to note that sometimes it's hard for students to go fully abstract and not do some form of representation, as in this sun setting over an ocean.

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4th Grade - Intro to Orphism with Sonia and Robert Delaunay

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Description of the Unit - Students will discover Orphism through the work of Sonia and Robert Delaunay, creating their own Orphism-style design, focusing specifically on the dynamics of color.

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Check out the Beat Sequencer in GarageBand to explore fractions!

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What do fractions sound like?  Check out the Beat Sequencer in GarageBand to explore!  #EveryoneCanCreate #AppleEDUchat @AppleEDU

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What do fractions sound like?  Check out the Beat Sequencer in GarageBand to explore!  #EveryoneCanCreate #AppleEDUchat @AppleEDU https://t.co/EEMyXQPOyK

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What do fractions sound like? Check out the Beat Sequencer in GarageBand to explore! #EveryoneCanCreate #AppleEDUchat @AppleEDU pic.twitter.com/EEMyXQPOyK
— Mary Kemper (@MrsKemper) September 17, 2018

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What do fractions sound like?  Check out the Beat Sequencer in GarageBand to explore!  #EveryoneCanCreate #AppleEDUchat @AppleEDU https://t.co/EEMyXQPOyK

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What do fractions sound like?  Check out the Beat Sequencer in GarageBand to explore!  #EveryoneCanCreate #AppleEDUchat @AppleEDU https://t.co/EEMyXQPOyK

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What do fractions sound like?  Check out the Beat Sequencer in GarageBand to explore!  #EveryoneCanCreate #AppleEDUchat @AppleEDU

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Community of Adult Math Instructors (CAMI)

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We learned how to draw interlocking knots and investigated mathematical questions related to counting features of the knots.

We started by looking at a few Celtic knots and thinking about what we noticed and wondered.

Maybe you'll take a few notes before you move to the next page...

Some of what we noticed:

There are multiple passages

No exit, no entrance

The rectangular red knot is one loop; the square knot is four loops

Both knots are hand-drawn

Some of our questions:

What is the application of this graphic?

How can these knots be categorized?

Could they created with real objects (weaving, for example)?

What is the significance of the color?

What is the history of Celtic knots?

Is there a function that describes these knots?

What can be counted in these images?

We then identified the following features that can counted in the knot images:

countable features of the Celtic knot

# of loops in each image (1 in the rectangular knot above, 4 in the square red knot)

# of "crosses" (17 in the red knot)

# of inside "triangles"-these are the holes inside the drawing and aren't exactly triangular- (10 in the red knot)

# of diamonds (8 in the red knot)

# of holes-inside "triangles" + diamonds- (18 in the red knot)

# of outside "triangles" (10 in the red knot)

# of "bumps" along the outside (10 in the red knot)-There are 4 pointy bumps on the corners and 6 rounded bumps in between.

At this point, a conjecture was shared:

Rectangular knots are made with a single loop. Square knots are made with multiple loops (the number is the same as the number of bumps along the top)

And a related question was asked: Is it possible to draw a rectangular knot with multiple loops?

After this, we watched a couple short videos demonstrating how to draw the Celtic knot, then tried drawing them ourselves. After 20 minutes or so of drawing, we came back together to discuss some of the features of the Celtic knot and take stock. After reviewing the features we had identified, we talked about how we could categorize these knots. How could we refer to them so that we each know which one we're talking about. One way is to refer to the number of loops. The knot below is a 5 "looper" because it requires 5 different loops or pieces of string.

We then noticed we could categorize the knots by the number of bumps along the edge. The red knot below is a 4×3 single strand knot.

It seems that there is a lot left to explore in these Celtic knots. Maybe we'll return to them in another meeting or continue the conversation online.

After some #noticewonder, the biggest questions for me:
How can I draw these?
How could I categorize these?
What can be counted? pic.twitter.com/uuLV0ATVAa
- Mark Trushkowsky (@mtrushkowsky) March 20, 2019

Speaking of which, check out this cool online tool that automatically creates Celtic knots. Thank you to @mathrecreation for sharing!

In Attendance: Eric, Hector, Mark, Tom, and Yi

Programs Represented: CUNY Adult Literacy PD team, CUNY Start, LAGCC Bridge program, LAGCC ALC
Respectfully submitted by Eric.

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We learned how to draw interlocking knots and investigated mathematical questions related to counting features of the knots. We started by looking at a few Celtic knots and thinking about what we noticed and wondered. Maybe you'll take a few notes before you move to the next page...

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Math and Music Using Beat Sequencer in GarageBand

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Math and Music Using Beat Sequencer in GarageBand

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Everyone Can Create with Math and Music (Teachers)

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Create powerful spreadsheets, share them with a link, and collaborate with others in real time.

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Set a base color and this tool calculates different harmonious colors (complementary, monochromatic, analogous, split complementary, triadic, and tetradic)

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1. Pick a Color.

Click the first field to open the color picker and slide to select your desired hue (such as red, violet, or blue). Drag the dot right and left to adjust saturation or up and down to adjust value.

If you know the hexadecimal, RGB, or CMYK values for your base color enter them in the fields. Click plus to add up to three base colors

2. Choose a Harmony.

Choose one of six color combinations to work with your starting color. Click the color harmony symbol to complete the color scheme.

3. See Results.

The colors making up your harmony will display in the color calculator swatches and on the interactive color wheel.

Tweak or explore these choices by selecting and comparing different harmonies, viewing the same harmony with different colors, adjusting saturation or value, or adding additional input colors. Clear All to start over.

Like what you're seeing? Create Color Scheme to see a color report-and save the hexadecimal, RGB, and CMYK colors for your Web or print projects.

See your swatch applied to design samples. Print the page, save it as a PDF, share it with friends and family. It's a colorful world.

What is the Color Wheel?

The color wheel is a chart representing the relationships between colors. Based on a circle showing the colors of the spectrum originally fashioned by Sir Isaac Newton in 1666, the colour wheel he created serves many purposes today. Painters use it to identify colors to mix and designers use it to choose colors that go well together.

The classic color wheel shows hues arranged in a circle, connected by lines or shapes. The colors include primary colors (red, yellow, and blue), secondary colors (orange, green, and violet), and tertiary colors (yellow green, blue green, blue violet, red violet, red orange, and yellow orange).Secondary colors are created by mixing primary colors. For example, mixing red and yellow creates orange; mixing yellow and blue creates green.

NOD Newsletter

Want to see more design resources, tips, and news sent to your inbox? Subscribe to our Notes on Design Newsletter, a Sessions College blog dedicated to art and design.

Developing a Color Scheme

Designers often start a project by developing a color scheme: a set of colors that will work well together for the client or task at hand. Though you'll sometimes start from scratch, generally you'll begin with one or two base colors around with other colors will be built.

Color Harmony Basics How to choose colors that really work? Use the color wheel (or our color calculator) to help you identify harmonious color combinations. The following color harmonies are based on geometric relationships on the color wheel. For this reason, we can represent them as shapes. Rotate these shapes around the central point of the color wheel to create limitless color combinations.

Choosing the Right Colors Choosing the right colors for you requires a lot of creativity and experimentation. Bear in mind that color is very psychological and different color harmonies produce different effects. For example, analogous colors are similar in hue, creating a smooth transition from one color to the next. Complementary colors are opposite to each other on the color wheel, so they create a strong contrast. Monochromatic color schemes can be subtle and sophisticated.

Experiment with different harmonies to achieve the desired mood or effect.

Adjusting Color Value and Saturation Once you've selected your colors, you may wish to adjust the value of a specific color or colors-how light or dark the color is. Or you may wish to adjust its saturation, how rich it is. Each hue on the online color wheel has a different inherent value. Yellow, for example, is lighter than blue.

To increase contrast in your color scheme, you may need to adjust the value of a specific color-by making a yellow darker or lighter, say. Or perhaps adjust the saturation to vary the intensity. You can do both using the color calculator.

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Color Wheel - Color Calculator | Sessions College

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Use the color wheel to find the perfect color scheme with the Sessions College color calculator. Endless free color combinations, working from any base color.

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From Pythagoras' observations of the fundamental mathematical relationship between vibrating strings and harmony to the digitized musical world we enjoy today, The Majesty of Music and Mathematics with the Santa Fe Symphony and the Santa Fe Institute will explore the remarkable interweaving of the languages of music and mathematics.

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Familiar songs in different time signatures

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I have 117 time signatures examples in songs from various genres that are not in 4/4 time. The examples I have are in 2/2, 2/4, 3/4, 6/8, 12/8, as well as odd signatures and polyrhythms. I'm up to 120 song examples and growing.

If you are a musician like me, then you probably find 4/4 time easy to play and write as it is the most common time signature in music. The 4/4 beat is so prevalent that it's hard to feel any other rhythm.

That can be a problem if you need to play or write music in other time signatures. What you need is to hear familiar songs you like that are in time signatures other than 4/4 time.

Examples of various time signatures

There are a number of time signatures other than 4/4 time, also known as common time, that are also fairly common in music. They are:

2/2 time or cut time
2/4 time (the marching time)
3/4 time (the rhythm of waltzes)
6/8 time
12/8 time

They are the most common after 4/4 time. There are also what is known as odd time signatures examples of which are 7/4 and 11/8. Then there are time signatures known as polyrhythms which are songs that have more than one time signature.

Almost 2/3rds of the songs listed below are from the rock, or a closely related genre, with another 25% from country and folk. The other 9% or so of songs are from bluegrass, heavy metal, blues, and Christmas & children's songs.

Focus on the songs you recognize and like. It's much easier to write in time signatures other than 4/4 if you can "feel" the beats in songs you already know.

35 song examples in cut time, or 2/2 time signature

Songs with 2 beats per measure lasting for a 1/2 note each is called either cut time or 2/2 time. You may also see the term "ala breve" in reference to 2/2 time. Cut time is represented by the letter "C" with a vertical slash thru it. Here is the symbol which is more accurate than the graphic I made above:

Listed below are 19 rock songs, 11 country, 3 bluegrass, 1 folk and 1 Christmas song in 2/2 time. I have songs from Silverman's Folk Encyclopedia listed for a lot of songs because most of them are traditional and the songwriter is unknown.

Albert Lee: Bullish Boogie, Country Boy

Beatles: All Together Now, Help, Magical Mystery Tour. Paperback Writer

Bob Dylan: Blowin' in the Wind, Subterranean Homesick Blues

Buck Owens: I've Got a Tiger by the Tail

Earl Scruggs: Foggy Mountain Breakdown

Grateful Dead: Dark Star, Fire on the Mountain, Friend of the Devil, It Must Have Been the Roses, Jack Straw, Ramble on Rose, Stella Blue, Tennessee Jed

Guy Clark: Heartbroke

Hank Garland: Sugarfoot Rag

Hank Snow: Golden Rocket

James Pierpoint: Jingle Bells

Jimmy Bryant: Stratosphere Boogie

Johnny Cash: Folsom Prison Blues

Led Zeppelin: Your Time Is Gonna Come

Little Feat: Dixie Chicken, Rocket in My Pocket, Spanish Moon, Willin'

Merle Travis: Smoke, Smoke, Smoke (That Cigarette)

Ricky Skaggs: Life's Too Long (To Live Like This)

Silverman's Folk Song Encyclopedia Vol II: Don't Let Your Deal Go Down, Mountain Dew, Roll in My Sweet Baby's Arms

Simon & Garfunkel: Mrs. Robinson

11 song examples with a 2/4 time signature

Eight of the songs in 2/4 time are folk or traditional with 2 rock and 1 Christmas song.

Led Zeppelin: Bron-Y-Aur Stomp

Silverman's Folk Song Encyclopedia Vol I: Peggy-O, The Gallow's Pole, Joy to the World

Silverman's Folk Song Encyclopedia Vol II: Little Birdie, Man of Constant Sorrow, Samson, Goin' Down the Road Feelin' Bad, I Know You Rider

Simon & Garfunkel: Kathy's Song

Steve Earle: Goodbye

27 song examples with a 3/4 time signature

The 3/4 songs below are mostly rock tunes (17 of them). One of the two heavy metal songs in this article is below, as well as the only blues song I could find not in 4/4. The rest of the songs are country and Christmas songs, as well as the Star-Spangled Banner.

Beatles: Lucy in the Sky with Diamonds

Billy Joel: Piano Man

Bob Dylan: A Hard Rain's a Gonna Fall, All I Really Want to Do, Mr. Bojangles, My Back Pages, The Times They Are A-Changing

Eric Clapton: River of Tears

Grateful Dead: To Lay Me Down

Hank Williams: Alone and Forsaken, I'm So Lonesome I Could Cry

James Taylor: Sweet Baby James

Led Zeppelin: Since I've Been Loving You

Merle Haggard: You Take Me For Granted

Rodney Crowell: The Last Waltz

Silverman's Folk Song Encyclopedia Vol I: Amazing Grace, Greensleeves, We Wish You a Merry Christmas, and Silent Night, Holy Night,

Silverman's Folk Song Encyclopedia Vol II: In the Pines I, The Star-Spangled Banner

Simon & Garfunkel: America, Scarborough Fair

Steve Earle: The Mountain

Travis Tritt: Here's a Quarter (Call Someone Who Cares)

Unknown: Happy Birthday

Willie Nelson: Somebody Pick Up My Pieces

11 (+5 nursery) songs with a 6/8 time signature

I was hard-pressed to find songs in 6/8 time so I added some children's \ nursery songs. The nursery songs are lame, but they are recognizable so it may help if you intend to write a song in this time signature.

Animals: House of the Rising Sun

Beatles: You've Got to Hide Your Love Away

Bob Dylan: Rainy Day Women #12 and #35

Queen: We Are the Champions

David Gray: This Year's Love

R.E.M.: Everybody Hurts

Mazzie Star: Fade Into You

Metallica: Nothing Else Matter

Seal: Kiss from a Rose

Red Hot Chili Peppers: Breaking the Girl

Don Henley: Please Come Home for Christmas

Unknown: The Farmer in the Dell, Hickory, Dickory, Dock, Humpty Dumpty, The Mulberry Bush, Jack and Jill.

8 song examples with a 12/8 time signature

This is another time signature where I could not find a lot of examples. Also, Whippin' Post is the only song that is in two different lists in this article, though I only counted it once.

Allman Brothers: Whippin' Post

Beatles: Norwegian Wood, Oh Darling!

Grateful Dead: Cryptical Envelopment, Truckin'

Led Zeppelin: Dazed and Confused

Leonard Cohen: Hallelujah

Stevie Ray Vaughan: Texas Flood

17 song examples with unusual time

The Grateful Dead are some talented musicians. They know their stuff and write in a variety of time signatures including the 5 songs below.

I'm including the song Whippin' Post because the introduction is in a different time signature than the rest of the song.

Below are eight songs in 7/4 time, five in 5/4, two in 9/8, and one each in 10/4 and 11/8 time.

Grateful Dead: Playing in the Band in 10/4, The Eleven in 11/8, Estimated Prophet, Lazy Lightning & Supplication in 7/4

Iron Maiden: Number of the Beast (first part) in 5/4

Jethro Tull: Living in the Past in 5/4

Jimi Hendrix: Manic Depression in 9/8 (most people label it as 3/4)

Lalo Schifrin: Mission Impossible Theme in 5/4

Led Zeppelin: Four Sticks & The Crunge in 5/4

Peter Gabriel: Solsbury Hill in 7/4

Pink Floyd: Money in 7/4

Seal: Dreaming in Metaphors in 7/4

Soundgarden: Spoonman in 7/4

Sting: I Was Brought to My Senses in 7/4, Big Lie Small World in 9/8

11 song examples with polyrhythms

I'm not going to go into detail of when and where the different time signatures occur in each song. It varies from song to song and it would take a lot of text to cover all the changes. Get yourself a copy of the sheet music if you are interested in writing songs with polyrhythms.

And look at the Grateful Dead in this list as well!

Grateful Dead

Blues for Allah: 4/4, 3/4 & 2/4
Cream Puff War: 4/4, 3/4 & 2/4
Pride of Cucamonga: 4/4 & 6/4
Unbroken Chain: there are many time signature changes
Uncle John's Band: 4/4, 3/4 & 7/4
Viola Lee Blues: 4/4 & 2/4
St. Stephen: 4/4, 3/4 & 2/4
Terrapin Station: 2/4, 3/4, and 4/4

Happiness is a Warm Gun: 4/4, 2/4, 5/4, 3/8, 9/8, 10/8
Strawberry Fields Forever: end of chorus is 2/4, 3/4, 4/4
Here Comes The Sun: bridge varies between 6/8, 5/8, 4/4, 7/8

Allman Brothers, Whippin' Post: the intro is in 11/8, rest is n 12/8

Notes on Silverman's Folk Encyclopedia

I was not able to find folk songs examples online with time signatures not in 4/4 time, so I pulled out my 2 folk encyclopedia books. Here are some stats from those books:

There are 1000+ songs between volumes I and II.

About 50% had songs that were in 4/4 time.

That left about 50% in other time signatures.

Out of the 523 songs in other time signatures, there were 102 songs in 2/2 time, 159 in 2/4, 148 in 3/4, and 88 in 6/8

There were also 28 songs either with polyrhythm or odd time signatures. For the odd time signatures, I saw examples of 3/2, 6/4, 7/8, 2/8, 3/8, 9/8 and one song without any time signature but with the notes "play freely" (Free Time).

You may want to get those books for the wealth of folk songs in them. But if you are only going to get one of the books, get Volume II since it has a bluegrass section and other songs that you will recognize.

Final Thoughts

It's not easy when you first write a song in a time signature other than 4/4 time. Use some of your favorite songs above to give you an idea of the feel of these other time signatures.

Some of the songs above are in my AAA Song Form article, so take a look at that for details on those examples. Also, take a look at the 2 Wikipedia articles on Time Signatures and Unusual Time Signatures for more examples and theory on rhythm in music.

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Time Signatures Examples: 117 Songs Not In 4/4 Time

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I have 117 time signatures examples in songs from various genres that are not in 4/4 time. The examples I have are in 2/2, 2/4, 3/4, 6/8, 12/8, as well as odd signatures and polyrhythms. I'm up to 120 song examples and growing.

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This interactive rhythm player allows you to explore polyrhythms in salsa music with the core salsa percussion instruments (clave, conga, cow bell, timbale, and guiro).

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Introduction

Salsa music has a complex rhythm. Understanding the rhythm can help you to find the 1-beat or find the 2-beat so that you know when to start dancing. On this page I introduce the most important salsa percussion instruments and explain their common rhythms. You can listen to each instrument as you read about it or combine them in the interactive rhythm player at the end.

Not necessarily all these instruments will be used in salsa music and even when they are, the rhythm that each instrument plays can vary throughout the music - some coming in during the chorus and others in the verse. Sometimes an instrument will play a variation on the basic rhythm presented here.

Clave

Clave (pronounced "clah-vay") is both a rhythm and an instrument. The "son clave" rhythm is the fundamental rhythm in salsa music and has two main variations: the "3-2 clave" shown below, and the "2-3 clave" which has the two bars in the reverse order. The choice of 2-3 or 3-2 depends on the music's melody. It may start or stop during the music but will not change from one form to the other in a single piece. The "3-2" variation is used in the player below.

Conga

The conga is a tall drum that is played with both hands. It may be played as part of a set of congas of different sizes. The basic salsa conga rhythm is called the "tumbaó" ( demonstrated by Nate Torres in this video). It is quite complex, but the loudest notes are the ones to watch for: the "boom-boom" open tones that occur on beats 4 and 4½ and again on beats 8 and 8½, and the "slap" on beats 2 and 6. These accented notes are played by the "Conga (basic)" rhythm in the player below.

The full tumbaó rhythm can also be played ("complex"). The notation used is pairs of letters: "L" or "R" for left or right hand and then "H" for heel, "T" for toe or tip, "S" for slap and "O" for open.

Cow Bell

The cow bell (or "campana" or "bongo bell") is played by the bongo player during the louder parts of salsa music (the "montuno" or "mambo" sections). In the player below there is a basic rhythm which plays on just beats 1, 3, 5 and 7. These are the "strong" or "fundamental" beats of salsa. A more complex rhythm is also included in the player which adds some extra beats to the basic rhythm. The extra beats are different in the first and second bars and so give a clue about which bar is playing even when the clave has stopped.

Timbale

In the verses and softer sections of the music a rhythm is often played with a drum stick on the side of a drum such as the timbale or tom. This rhythm is called the "cáscara" rhythm (meaning "shell" as it is played on the shell of the timbale).

Güiro

This is the one that makes that scraping sound. Like the cow bell, the güiro accents the 1, 3, 5 and 7 fundamental beats of salsa with long notes. Between these long notes it plays two short notes.

Player

In this rhythm player you can listen to the main salsa percussion instruments together or one by one by clicking on the instrument names to turn them off and on. I've put a dark green square in the beats where an instrument plays loudly and a light green square in the beats where it plays softly. The lines in grey are not playing.

I've also shown the men's steps for three different styles of salsa: "LA", "New York" and "Palladium". The notation I have used is pairs of letters: first "L" or "R" for left or right foot, then "F", "S" or "B" for forward, on the spot or backward. I have also put a slash ("/") for when the break is, i.e when you change direction. For LA and Palladium start with your right foot in front of your left, but for New York style start with your left foot in front. There's no way you can learn to dance salsa from reading about it, but the steps are there to let you see how the steps and the percussion instruments interact.

Notes

The audio functions on this page require JavaScript to be enabled. It is known to work well in the latest Chrome, Firefox, Safari and Edge web browsers, including mobile versions (which implement the Web Audio API and can be downloaded here). It cannot work in Internet Explorer.

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Salsa music has a complex rhythm. Understanding the rhythm can help you to find the 1-beat or find the 2-beat so that you know when to start dancing. On this page I introduce the most important salsa percussion instruments and explain their common rhythms. You can listen to each instrument as you read about it or combine them in the interactive rhythm player at the end.

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Polyrhythms with Snarky Puppy and the NPR Music Tiny Desk Concert

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