Beaded Cube Math Lesson for Kids

Several people have asked me to write a math lesson for kids using beading. The challenge is to make something mathematically interesting with bead weaving that can be completed within a single lesson, under an hour, preferably less.

So I wrote this lesson on beaded cubes (PDF). The lesson begins with some background on what a beaded cube is with lots of drawings and a photo. It uses terms like edge, face, vertex, and graph of a cube. The second section gives step-by-step instructions for how to bead weave a cube with a needle, string, and pony beads. After finishing this lesson, students will have practiced basic sewing skills like measuring thread, threading a needle, and tying square knots. The final section provides several math and spatial reasoning "challenges" to extend learners' thinking about beaded cubes.

This lesson teaches several things, especially spatial reasoning, where the learner has to move back and forth between a 2D representation on paper to the 3D model in their hands. It also teaches the importance of carefully following instructions because every step is laid out, and if you skip one of them, your work won't look like what is on the paper. The challenges encourage students to build different symmetric coloring of a cube, extend the construction to a row of cubes, and think about a minimal thread path.

I taught this lesson at the Julia Robinson Math Festival and another version at MoMath. With just 12 beads, this task is not easy, but it is enticing, approachable, and engaging for children, especially girls, and boys like it too, as do their moms and dads. It seems appropriate for children as young as fourth grade, and most of them generally need a good amount of help, but they can do it with coaching. Fifth and six graders have a bit easier time with it.  Interestingly enough, my experience in teaching this lesson to adults and children is that it is not any easier for a typical adult than it is for a typical fifth or sixth grader. In fact, I watched one man get completely lapped by his fifth grade daughter today. GO GIRL MATH POWER!

Here are the materials you need to teach a group of kids:

Pony beads
Size 18 tapestry needles, one per student
String: Cotton is good.  Something thin enough to fit through the eye of the needle, but thick enough to provide some friction.
Magnetic pin cushion: an easy way to collect the needles
Snips or scissors: I tied them to the end of some crazy yarn so they wouldn't disappear.
Beaded samples
Ruler(s)
Cups or bowls to hold beads
Hand outs: I designed the PDF file to be printed once, and then photocopied onto two sides of a sheet of paper, with one sheet per student.  Print the PDF file for best quality, not the jpgs.

I encourage you to try this lesson with your own students or children. If you have anything meaningful to contribute to making it better, please do not hesitate to send me an email or leave a comment below. If you want to see what else you can do with beaded cubes, you should search for my blog for CRAW or "cubic right angle weave" because a beaded cube forms the basic unit of one of my favorite bead weaving stitches.  Also search Planet Bead to see the many beautiful things that have been beaded with cubes.

If you think this lesson is useful, consider showing your support by perusing my Etsy shop, gwenbeads and buying yourself a little something special. You deserve it.

Thanks for looking. You're awesome. Yes, you!

Edited to add: Emilie Pritchard suggests using long plastic beads, called "spaghetti" beads, to help visualize the edges of the cube. Search the internet to find them for sale.

Free Tutorial -- A beading math lesson with David's Star for children and adults

Maybe you read my paper on using tiling theory to generate angle weaves with beads (PDF).  Then again, maybe you didn't.  One of the most elegantly simple weaves that I presented in that paper is what I called, "David's Star."  You can read about it on pages 12 and 13, and even if you don't want to read anything, you can look at the three bracelets on page 13 that use the same technique as I show here.   

I derived David's Star using mathematics, in particular the mathematics of tilings or tessellations.  Using a tiling to describe this weave, David's Star is the edge-and-cover angle weave for (6.6.6).  Let me explain that mouthful.  First we start with the regular tiling by hexagons, like what we commonly see in natural honeycombs.  This tiling is the black lines labeled (6^3) below. The standard notation for this tiling is (6.6.6), or (6^3) for short.  The 6 is because the tiles all have 6 sides (i.e., they're hexagons), and the 3 is for the 3 hexagons that meet at each vertex. 

The dark blue edge beads are the beads that you place on all of the edges of the tiling, the black line segments.  The cover beads are the beads that cover the thread between the edge beads when you sew each hexgon of beads in a loop of thread.  All of the non-dark-blue beads in the picture are cover beads.  So, David's Star is the edge-and-cover angle weave for (6.6.6).

I like this bead weave for several reasons.  First, it is very simple to stitch and works up relatively quickly.  Therefore, I think it's a good pattern for beginners who want to make a wide flat bracelet.  Second, the arrangement of beads allows for lots of different and beautiful ways to color the beads. Finally, the beads fit together really well: David's Star doesn't show thread or bead holes.  Oh, and one more thing, you can weave it in any direction.

I drew this picture today in preparation for an invited plenary talk and workshop I'll be giving in Washington State in April at a math conference.  For a moment, I considered writing up a complete tutorial and putting it in my Etsy shop.  Then I reconsidered.  For something this simple and basic to the art and math of beadweaving, I think this information should be generally available for free to those who are interested.  So here it is.  Also, a lot of people over the years have suggested that we should use beading more to teach mathematics, and I think that this particular weave is a nice choice for a math lesson. I made the beadwork in the photos here with pony beads and fishing line, and without a needle.  Beading the patch above is a good lesson in visualization.   Using pony beads and fishing line or stretchy thread makes it suitable for children and adults alike. Thin yarn with a size 10 tapestry needle is also a good choice. The friction on the yarn helps hold the beads together.

After trying this pattern, you can ask lots of extension questions in both math and art.  For example, ask, "What does an edge only weave of (6.6.6) look like?"  (Answer: Hexagon angle weave.) Or you could ask, "Draw the tilings (4^4) and (3^6)." "Can you draw the edge-and-cover weaves for these tilings?" and, "Can you bead weave them?"  The answers to these questions are explored in the paper I linked to above.

If you want to play with the art, draw a picture of an interesting coloring for David's Star and then weave it.  If you want a real challenge, (1) pick your own tiling, (2) draw an edge-and-cover weave for it, and then (3) bead it.

If you made it this far, please remember, I make most of my living selling my tutorials and other artwork.  So if you liked this little free-bee, and try it yourself or with a kid, maybe you'll be so pleased that you'll want to hop over to my website or Etsy shop and show your appreciation by buying something. It's like a buy-one-get-one-free, but in the opposite order.  If you've already purchased something, then consider this a thank you gift for supporting my work as an artist and teacher.  Without you, I couldn't afford the time to write this blog every week, and I'd have to get a normal job.  In any case, I hope you enjoyed this little mathematical beading lesson.  Thanks for looking.

Math Anxiety Camp: An emotional art piece

For the last few years, I have been attending a festival in the middle of the Nevada desert called the Burning Man Art Festival, where tens of thousands of artists and revelers join together each summer to celebrate art while camping in some of the harshest conditions in the US.  One of the peculiar features of Burning Man is that groups of people band together into "theme camps" which offer free gifts in the form of art, music, food, drinks, experiences, and other life pleasures and fancies to almost anybody who stops by.  Many of the gifts come with the touch of a prankster, designed to shock and amuse. Last year, we created the idea of Math Anxiety Camp, where we would give people math problems in an attempt to provoke math anxiety.  Burning Man has a culture of gift giving, and we joked, "Math anxiety: It's our gift."

because math matters

Great art provokes emotions.  This is a reason why music is so popular and powerful.  Music provokes intense emotions in listeners.  Have you ever cried to a love song? I have.  Now, I have never seen a piece of artwork that was designed specifically to provoke math anxiety.  So I created Math Anxiety Camp with the help of my camp mates.  For Burning Man 2013, we wrote a short book of 38 math problems, and I designed a sticker so we could hand out awards to those who achieved math anxiety.

Math Anxiety Camp Achievement Award

My campmates and I drilled unsuspecting Burning Man attendees (i.e., burners) with our math problems in the hopes of provoking the emotion of math anxiety for the sake of art.  We did all of the things you're not supposed to do as a good math teacher, like telling our examinees, "You should already know this," "You should have learned this last year," and "Work faster! Faster, faster, FASTER!"  When participants got wrong answers, we made loud buzzer noises.  The purpose was not to focus on the math, but to focus on experiencing and emphasizing the emotion of math anxiety.

Those who are tardy don't get a fruit cup (Thank you Kimberly Laabs)

Vi Hart and I wrote the book of the math problems together with several features in mind.

Vi and me sitting on Bat Country

All of our math problems were designed to be actual math problems that have at least one right answer (some have more).  Topics included arithmetic, combinatorics, geometry, calculus, and logic. Problems ranged in difficulty from trivially easy (e.g., "Name a number that is 3.") to tricky (e.g., "Name a triangle with two right angles.") but all were chosen to be simple enough that most of our subjects would at least understand the question, even if they couldn't solve the problem.   Some problems were designed to be funny.  We included several classic, well studied math problems that are known to confuse people.  Most of the problems have multiple choice answers, and the distracters (incorrect wrong answers) were designed to be funny or deliberately confusing or deceiving.  We included lots of "All of the above" and "None of the above" options because of their cringe value. We added scenarios relevant to the art festival, and where the characters were in mortal danger.

My campmates eagerly distributed math problems, books, and awards throughout the festival.  I was pleasantly surprised at how many people engaged in the project.  We handed out nearly 500 achievement award stickers and almost 20 math books to specific people who wanted to own a copy.  I listened to and heard about several people who read the entire book, thoroughly musing over each the 38 problems.  Some were math teachers, married to math teachers, physicists, geologists, engineers, and others who just enjoy the satisfaction of solving a good math problem.  Here is a PDF copy of the 2013 Math Anxiety Math Book if you'd like to take a look.  This version is edited to make it more suitable for a general audience.

Ethan Port brought a whole suitcase of math books, and Paul McGlaughlin painted us a sign for the front of our camp so that passers by would know we were there and what we had to offer.  As one young man biked past, we overheard, "Math Anxiety Camp!  That's TERRIFYING!"  So of course, we invited him in to share our problems and win an achievement award.

Camp Sign
Photo by Daniel Thornton

When people asked us about our sign, they often confused us with "Math Camp at BRC", a different theme camp that touts themselves as "a safe place for mathematics." We were the opposite: an unsafe place for mathematics.  We were there to give problems, not solve them. "Our problems are your problems," and, "We have so many problems, we'll give you some!"  I saw people tense up instantly when they heard these statements.  Although we gave them lots of problems to solve, and many people successfully solved them, my favorite part always was watching their anxiety transition into laughter when they were presented with anxiety achievement awards.

Paul painted half of the sign with chalkboard paint so we could write a new problem each day. We left chalk by the board, and many of the problems were solved by the next morning.

Question of the Day
Photo by Daniel Thornton

In addition to working independently, Math Anxiety Camp also joined forces with Camp UFOm and the Civil Defense Camp.  UFOm provided an "interblastive foam experience" where participants performed foam art.  UFOm was so popular, that the Civil Defense crew was enlisted to conduct drills on the revelers in an attempt to slow down the line of entry into UFOm and deter all but the most dedicated from entering. 

Civil Defense Bunker, Tent and Ropes Course

Burners were subjected to drills including physical exams such as a rope course, running laps, push-ups, jumping jacks, rolling in the dust, and games of duck-duck-goose.  There were also oral exams on outdated American civil defense literature from the 1940s, and people were drilled with math problems from the Math Anxiety Camp Math Book.

Civil Defense Drills: Photo by Ben Harper

A little background on the Math Anxiety Camp project:  I am a former teacher of mathematics, and the idea of provoking math anxiety on purpose is simply ridiculous to me.  I spent many years of my life trying just about anything to minimize, or at least reduce math anxiety in my students because students who are too anxious don't perform well in school.  Math anxiety makes people hate math and avoid it.  Wanting people to love math and engage, I read many papers on math anxiety, attended lectures on the subject, wrote worksheets and led discussions in my classrooms that were designed to reduce my students' anxiety.  It seemed to be an ever-present problem in my classrooms of college students, many of whom had learned to fear math from a very young age, typically spawned from negative interaction with their teachers and parents.  I was often surprised at how quickly some of my students were to state their disdain for mathematics publicly, even though they were studying for professions that would require them to do mathematics regularly, like engineering or teaching children.

My experiences as a math teacher showed me just how common math anxiety really is and how intensely some people suffer from it.  Some people will go to great lengths to avoid math at all costs just to avoid the anxiety that goes with it, and this makes me sad.  But outside of the context of teaching, it seems that math anxiety is an emotion that is rarely discussed in depth, especially in the art world.  Creating math anxiety in a novel context devoid of high-stakes consequences seemed like a good way, a safe way for people to confront their negative emotions about math.

Math has been a theme for me and my camp mates already at Burning Man.  This year, Math Anxiety Camp was also an art support theme camp, building Bat Country, a Sierpinski tetrahedron jungle gym.  Here you can see Bat Country this year on the night of the Man burn.

Bat Country
Photo by Daniel Thornton

 Here is Bat Country with the Rainbow Bridge art car.

Bat Country and the Rainbow Bridge

Significantly, we were not able to elicit math anxiety in all of our subjects.  Many participants easily and eagerly solved our math problems without anxiety.  It's not terribly surprising that burners were quick to engage in the idea of Math Anxiety Camp.  My sense is that burners are more mathematically literate than the average American population, which probably correlates with the maker attitude of the festival participants.  In addition to Math Camp at BRC, there is a thriving tradition of beautiful mathematical art at Burning Man.  My favorite returning piece this year is the honorarium art project Zonotopia and The Quasicrystalline Conjunction by Rob Bell

Zonotopia and The Quasicrystalline Conjunction

The mathematics behind these "pavilions" is polar rhombizonahedra, one of my all time favorite mathematical structures.  I've been watching this series of inhabitable structures evolve, changing from year to year, but still maintaining its same aesthetic and mathematical essense.  This portion below was the new addition to the set for 2013.  The panels have a lot more details than most of the older forms made in earlier years. Beautiful. 

My favorite new piece of mathematical art this year is the honorarium art project, The Penrose Triangle by Blake Courter and Blake Courtney. 

The Penrose Triangle

 This triangle looks very different from different perspectives.

The Penrose Triangle

Unfortunately, I missed a shot of the triangle in perfect perspective where all of the lines look straight, but this one is pretty close.

The Penrose Triangle

Fortunately, I climbed up to the top of this triangle and successfully climbed back down without killing myself. Burning Man always has a plethora of climbable objects, and I love to watch the acrobats and other "monkeys" climb and hang off these piece.  However, I rarely climb anything at Burning Man, which is a little ironic since I brought my own jungle gym (see Bat Country above).  Although I don't generally suffer from math anxiety, I do suffer from a fear of heights (or high anxiety), but I don't really want to talk about that emotion.

Pavel Curtis sent me this link.

Starburst Necklace (Water) Part 3

I finished my Starburst Galaxy necklace in this color scheme inspired by water. 

In the Starburst Galaxy class students will learn to make a whole galaxy of beaded stars.  I wanted students to make a nice sample of the different sizes, so the kits we are assembling will make the 5 different stars shown here.   (These 5 stars are just a few of the 20 or more possibilities you can make with this technique.)    The kits include enough materials to make 9 different stars:  1 large 8-star and 2 each of the smaller stars.   With 9 stars, you can make earrings, pendants, a bracelet or a necklace, or some combination of these.  We thought it would be the most fun and informative for you if the kits make a bunch of different stars, and then let to you decide how to finish your stars into jewelry.  I finished mine into a necklace with some twisted cubic right angle weave in the back.

I'm going to make two more beaded star kits for this class.  For the "Fire" and "Gothic" Starburst kits, we just ordered a HUGE pile of Swarovski crystals yesterday morning.  Once the crystals arrive, I'll start beading some sample, and show you photos as I take them.  Thanks for looking.

Desiging Starburst Galaxy Kits Water

I'm working on classroom kits for the Starburst Galaxy class that Florence Turnour and I will be teaching at the Bead & Button Show in June 2013.    The class will teach you to make stars in four different sizes, each with any numbers of points.  The five stars here are just a few of the 20 or more possibilities.

These stars are links, meaning that they are components that you can connect together with jump rings.  You can also dangle wired drops from their points.  After I finish weaving the rest of the set, I'll link them together to make a large necklace and show you. 

We now have patterns and kits available for the Beaded Starburst Galaxy so you can learn to make them yourself.

For algebra homework emergency press play

You might not know this about me, but I used to be a math teacher, a math professor, actually.  I taught math in the American public education system for 15 years.  I still often think of myself as a math teacher, even though I don't work in a classroom anymore.  It's part of me, my identity.  My friends tease me about being a math nerd, which I kind of like.  I have always enjoyed sharing math ideas with other people, and I've been known to do it in the strangest of times and places.  That's why when I was asked this week to make some Doceri videos that address the Common Core State Standards in middle school mathematics, I was happy to comply.  

The first video is about Paulo who peels potatoes. If Paulo can peel 3 potatoes each minute, how long will it take Paulo to peel 20 potatoes? 

What if 5 potatoes are already peeled for him? Then how long will it take?

This video uses ratios to solve the first problem, and a linear relationship in the second problem.  Ratios are taught in the upper elementary grades, and linear relationships are taught in seventh grade. This content address Common Core State Standard CCSS.Math.Content.7.EE.B.4b. 

Here is a lesson on how to prove the quadratic formula that addresses an eighth grade standard.  This is likely the most sophisticated concept that is commonly addressed in middle school algebra.  I can't tell you how many times I proved this for my students.  Many.  And I never taught a full course on 8th grade math.  Even still, this proof came up a lot.   This content addresses the Common Core State Standard CCSS.Math.Content.HSA-REI.B.4a.

There's something quite satisfying about making a video that answers a question that I have answered so many times before.  Thanks for looking.

I made these videos with Doceri software on my iPad. 

My second Color Medallion Pendant

Since my first Color Medallion was so bright, I decided to use more subdued colors for my second one.  Here it is, in blues with a touch of purple.

Florence and I will teach this class together at the Bead & Button Show 2013. Now that the show is over, we have patterns and kits available.

More Starburst Galaxy for Bead & Button 2013

I'm still working on designs for the Starburst Galaxy class that Florence Turnour and I will be teaching at the Bead & Button Show in June 2013.  On line registration begins January 8.  That's tomorrow.  With more than 20 different stars to make in this "galaxy" I'm still making new ones.  The pattern shows stars in four different sizes.  This is the largest 3-pointed star.

The night before last, I made one, and I didn't know what to do with it.   So I made another one. Then I made a couple more and linked them all together. I think it's going to be a bracelet. It's about an inch wide. I needed to find a clasp. So today I visited two different bead shops and I couldn't find anything quite right, so I came home and searched in my sewing box instead.

Now, I've had these hook and eye clasps since my Nana passed away about 20 years ago, when I inherited her sewing box. You can see she paid 19 cents for the whole card of them, and they're still in great condition. Thanks Nana. They didn't rust, just like the card promised. I am reminded that seed bead weaving is a form of needlecraft, and sometimes, a clasp can be readily found in sewing notions rather than in jewelry findings.  I think I'm happy with this solution.  It's not the fanciest clasp, but it's inconspicuous, and I think it will be secure and reasonably easy to take on and off.  I want the focus on the stars, not the clasp. 

I also used a hook and eye clasp on this Starburst Galaxy necklace.  It shows 6-pointed stars in all four sizes.

Here's another necklace with a few different stars.  This piece shows how the stars are links that you can link together with wire wrapped loops.

Here's a medium 8-pointed star from the galaxy.  I like this as a simple pendant.

And here's a large one.  See, I just added a jump ring and cord, and voila!  It's done. 

Florence and I will also be teaching a class together that she designed, called the Color Medallion B130596- Sun. June 2 • 9:00am-5:30pm.  Florence made this pendant.  Isn't it pretty?   I haven't seen this particular pendant yet, but I saw one in another color scheme, and WOW, it is dazzling in person. 

Here's another post I wrote about our Bead & Button classes if you want to see more.  Oh yes, there's more!  There's always more...  Seriously, I haven't beaded all of the stars yet...

Starburst Galaxy Kits Bead & Button 2013

I'm working on designing kits for the Starburst Galaxy class that Florence Turnour and I will be teaching at the Bead & Button Show in June 2013.

So far, the pattern is 20 pages and counting. So far, the pattern is 20 pages and counting.  I am pretty sure you can make at least two dozen different stars with the techniques we describe in the pattern, even though  we've only made about a third of that number so far.  I am designing the pattern so that you'll learn to make the whole set, all two dozen, maybe more: large stars, medium, small, and mini stars, with any number of points from 3 to 8 or maybe more.  There's so many different stars, all made with the same techniques, we haven't even tried them all yet.  The written pattern will hold your hand through several different sizes, and then show you lots of photos with some text and charts so you can work several more, and fiddle your way through the rest of the set without too much trouble.

Certainly, the kits won't make every possible star, so I'm designing smaller kits in coordinated colors so students can purchase more than one kit if they want to make something big.  Or they can purchase one kit in different colors that just make a single pendant or a pair of earrings. Here is my first color scheme: white opalite crystal and silver.  All of the faceted beads and gems in there are Swarovski crystal. Get these under the right lights, and there are rainbows everywhere!  It's a pity it's so hard to capture that in a photo. 

Here is a sample earring.  I poled a bunch of friends on Facebook and we decided that the kits will NOT include the red heart drops.  Apparently, sometimes "more" is actually "too much." But I like the photo because it shows how you can easily link the stars together with jump rings.

Here's info on the class in case you're thinking of signing up.
Starburst Galaxy 130411- Tue. June 4 - 9:00am-5:30pm

Florence and I will also be teaching a class together that she designed, called the Color Medallion B130596- Sun. June 2 • 9:00am-5:30pm.  I've been bugging her for months to write the pattern so I can make one for myself.  This pendant is big, dazzling and super groovy in real life.  Florence, hurry up! I want one!!!

Bead & Button 2013 Starburst Galaxy

We just got word... Florence Turnour and I will be teaching two classes together at the Bead & Button Show in June 2013.  Here are a couple photos from the workshop I designed.  First is an 8-pointed star. 

And here is a necklace that I made from 6-pointed stars in four sizes.

I call the workshop the "Starburst Galaxy Made With Right Angle Weave."  I call it a galaxy because you can make more than 20 different stars in different shapes and sizes, all with one combination of techniques.    I'll show you more photos of the different stars you can make in the coming months as I assemble kits. 

Here are the class details. B130411- Tue. June 4 - 9:00am-5:30pm (with 1.5 hr. break) - DC Show Floor 22.  Registration doesn't begin until January, I think.  I'll give you more info as I get it.

Bead & Button Show 2012

We made it to the Bead & Button Show this year, and I thought I'd share some photos with you.  Here are Florence Turnour and me a "Meet the Teachers" right before it started at our sales table.  This was our first show, so you can see how excited we were.

And here we are with Cindy Holsclaw who was at the table right next to us.

Florence and I successfully taught two classes, and we have leftover kits available for sale on our website.  This is one of the kits for the Rivoli Urchin Necklace.

This is one of the kits for the Infinity Ubercube & Other Beaded Cubes.  Check out the links to see all of the kit colors we have available.  Thanks for looking.

Infinity Ubercube and Other Beaded Cubes

Florence Turnour and I will be teaching two classes on the Infinity Ubercube at the Bead & Button Show 2012.  Lately, I've been working diligently on our kit offerings. The kits each make 11 beaded beads.  This is the complete green kit. (Click on the photos to see them larger).

To make some jewelry, our students will each receive a "Findings Packet" including things like wire, ear wires, crimps, ribbon, etc.  We will not be including larger beads, however, because we want to encourage them to be creative with their beaded beads and make their own jewelry their own.  For example, I added a few extra beads from my own stash including hand made borosilicate glass, a glass button, dichroic glass and a big glossy prehnite.  With these, I made a pair of earrings and a simply strung necklace.

Here is a close-up of the pair of earrings I made with the mini cube beaded beads.

These are the 11 beaded beads in the pink and purple kit,

and this is the necklace I made with the beaded beads. I've had those purple glass roundels for years, and they finally found a home.

Florence designed this kit in juicy fall colors.  Those purple drops just want to be plucked!  

I'm looking forward to seeing what she and all of our students make with their kits. You can now purchase the pattern here: http://www.beadinfinitum.com/Kits/index.html#Ubercube Thanks for looking!