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.

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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 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: