Week 47: Math Dance

Get up and move! This week learners can dance their favorite equations, math symbols and concepts. Whatever topic is of interest or in the process of being learned is a great one to figure out the dance moves that go with it. I recommend taking 5 to 10 of your favorite moves and making it into a mathematically choreographed dance.

Here is one approach:

  • Pile 1: Make flash cards for the vocabulary, plots, or concepts that you want to move to.
  • Pile 2: Make flash cards for the body parts that you want to move with (hands, feet, legs, arms, whole body, etc.)
  • Draw a card from each pile, turn on the music, and get some moves.
  • Note: you can play with sequences, beats, or number of repeats as well. (This works in a virtual environment as well – have a dance break!)

Here is an example generator I made:

See the Pen MathDance by Sophia (@fractalkitty) on CodePen.

To run it in a separate tab (good for screen sharing) click here.

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Week 46: Half-Square Triangles (Truchet Tiles)

If you are a quilter, then you will be a pro with this week’s activity. For the last six months my quilting mother lived with us through chemo and we watched her quilt her heart out. Now that she moved back to her home, I had to laugh because she would have been so much fun making these tiles this week – I should have done this sooner! So many traditional and historical quilts use these squares (a great segue into history – Underground Railroad, folk art, stories, family histories, etc.).

This week I encourage learners to actually sew some tiles, specifically half-square triangles. Another name for these squares is Truchet Tiles. These are tiles that are not rotationally symmetric. The tiles can create a variety of patterns and tessellations. Make some tiles and then play. (The reason I encourage sewing is that the problem-solving, process, and mistakes lead to so much learning.)

If sewing isn’t your thing, then I have a JPG file below you can print and play with. There are also tons of maker ideas for these tiles (Paint squares of cardboard, wood, felt, etc.). A piece of graph paper will also work (shade in half of the squares diagonally).

How to sew:

  • Iron the fabric that you wish to use.
  • First cut out 25 squares of Color-1 and 50 squares of a Color-2. (You may want to do 2×2″ squares for small tiles or 10×10″ for larger tiles.) You can also do 36 squares of Color-1 and 72 of Color-2. Do you see the ratio? NOTE: Cut these squares carefully (millimeters matter).
  • Next cut down the diagonals of 25 of Color-1 and 25 of Color-2 (The same amount of each color if you are doing more than 25 tiles).
  • Now, take one triangle of each color and place them face to face and sew them with a quarter inch seam (make sure your seam width is consistent for all of this work).
  • Iron the square you just made so that the seam is folded flat to one side and then place it on one of the squares of Color-2. Sew around the border to attach the half-square triangle to the square bottom.

  • Now cut the extra fabric, while squaring the tile (make sure you use a grid or corner to make sure it is square as you cut).
  • Repeat this for the other 24 tiles (or more if you so chose).

We also made draftboard versions of these tiles to play even more:

If you like what you see, please consider donating to this website.

Week 45: The Quincunx

This week learners can dive into probability through a quincunx (also known as a Bean Machine). Learners can make bean machines with building toys (Legos), pins and a cork board, or nails and wood (or other methods they devise (3d-printing, sculpture, etc)). Here is a template to use.(It’s a png and is also at the bottom of this page.) Make sure to size the printable to the size ball that you are using before you nail or pin.

What is fun about Quincunxes is that they show how possibilities play out. The idea is that every time a ball reaches a pin it has two possibilities: Left or Right. Learners can draw the possibility trees as an exercise to see how the distribution works. The middle columns have far more paths than the outer columns. The number of possible paths to a column is related to Pascal’s Triangle (or the Jia Xian triangle that was discovered much earlier).

Math is Fun has a tool to play with possibilities here.

One of my kids decided to continue to play with the pins:


Side note: I chose not to call it a Galton Board here because of the history of its inventor being a racist and coining the term for eugenics.

If you like what you see, please consider donating to this website.

Week 44: Conics, Orbits, and Projectile Motion

You don’t have to be in high school math to play with conics, orbits, and projectile motion. This week (or month) learners can play with projectile motion, orbits, and conics sections with the activities below:

1.) Slicing cones

  • Learners can mold cones with clay and slice to see the possible shapes. This will give circles, ellipses, parabolas, and hyperbolas.
  • Create a cone sculpture with an intersecting plane using paper, string, pipe cleaners, or other mediums.
  • Try John Sharp’s sliceform template.
  • Check out this Conics Geogebra tool by Irina Boyadzhiev.

2.) Observe parabolas through projectiles

  • Create a straw rocket (don’t aim it at someone) and take slow motion videos of their flight and observe the curve that is created. For a template and activity, go to NASA’s website here. Look at various launch angles and analyze the differences and similarities in curves.
  • Create a water balloon launcher with PVC pipes (or other parent/teacher approved apparatuses). Record their launches and analyze. (Water rockets are also fun.)
  • Play with a garden hose and the curves created by shooting water up into the air.
  • Play with the PhET simulator.
Rocket Launch

3.) Play with ellipses and circles through gravity

  • Use tacks and string to create ellipses (2 tacks) and circles (1 tack or compass). Create an abstract work of art with these tools.
  • Play with this gravity simulator by the NSTMF (really fun!)
  • Universe Sandbox is a program that costs money, but is excellent for playing with orbits and answering a lot of learners’ “what-if’s.” I love, love, love this tool!
  • JPL learning activities are here.
  • Use a large piece of elastic cloth and place weights in it. See if you can create orbits with marbles. We have found that a hula hoop works with swimsuit material. Here is a video as well.
  • Take time to get outside and observe the planets, comets, and astronomy that is with us every day.

There are so many other ways to play with these curves, so experiment, draw/paint, and enjoy.

If you like what you see, please consider donating to this website.

Week 43: Circles and Art

Circles are so much fun! This week I encourage learners to get out their compasses or a circle to trace and start making patterns on paper. Patterns with circles can start simple, but can also get really complex. You can combine your compass with a straight edge and get amazing patterns and tiles. Try intersecting circles and then placing your compass at intersections and adding more circles. Shading with markers, ink, or colored pencils can make beautiful stained glass-like mosaics.

Here are some ideas to play with circles and art:

  • The Metropolitan Museum of art has a great activity here to take a look at Islamic art and geometry.
  • Cut out some of your patterns to make a puzzle.
  • Use tissue paper and make a see-through design on wax paper.
  • Go big with string and chalk outside for your designs!
  • Dip the top of a can or cup in paint and use it to create circle patterns on boards or fabric.
  • Try playing with the negative space within the art – see how it changes the tiles and overall appearance.
  • Use a digital drawing or design app and play with color pallets and design.
  • Try adding details and embellish.
  • Play with the Girih app (costs money) from the Apple app store.
  • Play with an app: https://girihdesigner.com/.

I made tiles for my kids to play with and we’ve been having a blast:

If you like what you see, please consider donating to this website.

Week 37: Cantor Set Kirigami

For this week’s activity, learners can play with Cantor Set Kirigami. The Cantor Set is created by drawing a line. Next, remove the middle third of that line (this will create 2 lines). For each of the two lines just created, remove the middle third (this will create 4 lines). Continue with this process until the lines are too thin to work with.

Menger Lux Blox

Some of the fun characteristics to notice is the pattern of the line lengths (1, 1/3, 1/9, 1/27,…), the number of lines generated with each iteration (1, 2, 4, 8, 16, …), the fact that this set is infinite, yet not countable and that it gets smaller and smaller with each iteration.

I created a fun Kirigami Cantor Set and have the template below with a video how-to. Enjoy!

Learners can also draw Cantor Set cities, roads, abstract art, and find many ways to represent this simple fractal. The Menger Sponge is one form of a Cantor set in 3d.

If you need a jpg. of the sheet:

Week 36: Golden Angle Scavenger Hunt and Drawing Phi-Nominal Phi-lowers

The Golden ratio appears in nature all around us. Flowers and other botanicals often grow at an optimal (Golden) angle of about 137.5 degrees. For the 52-weeks of math activity, I encourage learners to seek out the Golden angle on a scavenger hunt. Take pictures or sketch in a nature journal the pinecones, flowers, and other botanicals that grow in Fibonacci/Golden Ratio spirals. Count the petals, trace the spirals, and collage the scavenger hunt together. Nature is one of the best ways to explore math.

Additionally, I created a Golden Angle grid paper for learners to sketch their own “Phinominal Phi-lowers.” Feel free to print it and play with the spirals and dots. Sometimes seeing flowers, pinecones and succulents can provide inspiration for unique flowers.

For a digital Phi playground and some more background information on Phi (click here).

Golden Angle Grid Paper

I’m Attracted to Attractors

So many plots and mathematical musings throughout my life have brought on a sense of artistic beauty and awe within my being. In the windowless halls of engineering firms I have smiled at harmonics, or in a homeschooling room squealed in glee when I stumbled upon Pisano periods by trying to play Fibonacci on the piano. Lately I have been playing with attractors. These dynamic systems make me stay up late fiddling with their metamorphic and chaotic beauty.

I came from a Matlab world and have had to teach myself some more cost efficient means of play with javascript and python. The code below is just one of my playgrounds. I don’t know if there is a name for this attractor (please let me know if you know its name). Enjoy:


With functions you have inputs (x) and outputs (y) that can be plotted on a plane (x,y). With the images below, the x and y values are computed using an initial value of (1,1) and then the next (x,y) is computed using the previous coordinate’s values. The equations are shown below:

x = sin(a1 * oldx) * cos(a1 * oldy) – sin(a2 * oldx);
y = cos(a3 * oldx) – cos(a3 * oldx) * sin(a4 * oldy);

Here is a gallery of some of my outputs:

For the images above, I calculated x and y using a1, a2, a3 and a4 coefficients and the previous x and y values (oldx, oldy). The initial point was (1,1). In the code below, there are only 300,000 points (compared to millions in higher res images). You can play with the values of a1, a2, a3, and a4.

See the Pen webcode by Sophia (@fractalkitty) on CodePen.

I like the p5.js editor. Click here to play. I would say that fiddling with this is a great idea for an “Hour of Code.”

If you like to play with sheets or excel, which is not near as pretty, I made a sheet for you here. This is also handy if you want to see the array of values for (x,y).

Week 35: Yarn-it-up Hyperbolic Space

This week let’s play with yarn! We are going to play with hyperbolic space. You will need some yarn and a crochet hook. You don’t need to know how to crochet, but you will need a little patience and a lot of desire to play. These don’t have to be perfect, and “mistakes” just add to their beauty. There is a great TED Talk on crochet coral that is a great intro into this activity as well (click here), or just watch the videos I put together below. I thought about drawing hyperbolic space as an activity, but decided that having the tactile fluffy math in hands would be much more exciting this week:

Week 34: Kirigami

I love paper cutting, so last week I did kirigami with some of my classes. What was so fun about this activity is the amount of play and discovery that happened with two simple supplies (paper and scissors).

Below are the videos I recorded for my classes to be able to go back and work at their own pace. These videos are just a starting place. There are so many methods for folding, cutting, and scoring that can be discovered and explored. My son made dioramas of forests and landscapes that fold with his creations. If you like pop-up books, this is a great place to start.