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.
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!
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).
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:
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.
Games don’t have to be complicated to require some good thinking skills. We all learn tic-tac-toe when we are younger. We soon learn how to always come to a stalemate with an equal opponent. Once you get the strategy, it can get a little boring… But what if we add a layer (or two) of moves. What does this do to the strategy of the game? Is it so easy to predict your opponents next move?
This week I recommend learners create a tic-tac-toe board that has tic-tac-toe boards in each square. Here is a video of how to play:
My learners contemplated:
How many times can you nest the game before it’s too complex?
With each layer added, how much longer and more difficult would it be? (how many moves are there?)
How is this like a fractal?
Could you keep a game going with one move a day for how many days with 2-nested?, 3-nested, 4?
What does the game tree look like?
How many ways can you play tic-tac-toe vs ultimate tic-tac-toe?(think combinations). What is the combinatorics calculation look like for this?
If you are wondering how I was able to do this in the time of Covid… I use a digital white board and label squares so it is easy to say the next move. You can also use a shared google drawing or a google spreadsheet to play (here is one for you.)
Another blog (Games for Young Minds) that does a great post on this game is here. Math with Bad Drawings also has a great post here. As you can see, this is a fun game with us mathy folks everywhere.
Let’s get out our pencils, isometric paper, and thinking caps this week! Isometric drawings are often used in engineering and design as a way to display 3D ideas. They can also be used to create optical illusions and escheresque works of art.
To start, print some isometric paper, or set your digital drawing program to isometric drawing guides. Start by drawing simple objects, like a cube, and play with shading.
Once comfortable with basics, start making skeletons for shapes, linking sides that don’t make physical sense, and thinking about objects that would allow you to go up and down at the same time. Below are some examples and videos to play with:
Sometimes the simplest things have wonder hidden within. This week, learners can play with the angles of polygons. How many degrees are in a triangle? In a quadrilateral? In a hexagon? Is there a pattern?
Here is a warm-up activity:
Draw a triangle (any triangle), and cut it out.
Next, rip the corners off:
Now, here is the fun part… put the pointy angles together. What do you get? Try it with lots of triangles and see if you always get a straight line. Rather than lecturing or telling learners that triangles have 180 degrees (or pi radians), let them discover. They can even create art ( I like to make my angles into perspective path doodles.)
Now do the same with a four sided shape. What do you notice? Is it the same for all the ones you can create?
Now do the same with 5, 6 or more sided shapes. There is a rule to be found. Try to discover it if you don’t know. I will put the rule at the bottom of this post.
I did this twice last week with virtual classrooms through the Covid-19 isolation. Students from kindergarten to middle-school ate it up. We used it as a warm up activity (10-15 minutes) prior to doing some loop-doodle math and/or other activities.
The rule for simple polygons is that for n sides there are 180(n-2) degrees. Or you add 180 degrees every time you add a side.
I am trying to tell my 15yr old daughter that an elective high school credit in Graph Theory would be fun next year. Of course I do this as subtly as possible – I start drawing coloring sheets for this post on my iPad and then carefully shade them in. All three of my children slowly sneak up behind me and breath in my ear.
“You know that it will never take more than four colors” I state.
“Really?” I hear my oldest daughter say with a sense of wonder in her voice. “Can I make one?” she asks reaching for my device.
She takes over the iPad. I go for a run. I clean up a bit. She is still designing, thinking, coloring. A wave of gratitude flows over me “Thank God that coloring isn’t just for kindergarten.” We are so blessed to have the abundance and time to be able to color, play, and contemplate.
She finishes her design. “It looks like the beautiful cobbles on our Oregon beaches.” I think, then say.
“That’s what I was going for.” She says. Then gets up and goes back to her school work.
This week I challenge learners to play with coloring sheets. Make your own. Share them. Color them. Contemplate them. Can you restrict the coloring to four colors? It may take some problem solving for more complex sheets.
In graph theory, there is the study of graphs that are made up of nodes (vertices) that are connected with lines (edges). Create a graph for one of your coloring sheets, where the regions are nodes and lines connect the regions that touch.
You could also create a graph with nodes and edges and then the coloring sheet to go with it.
Below are a couple examples (some blank for you, my readers, to use):
This week learners can brainstorm game ideas and test them out with family and friends. Games can be prototyped with paper, clay, cardboard, maker equipment, and/or craft supplies. When I do this with classes, we often play or analyze games that we love prior to designing our own. This allows learners to incorporate aspects that work and exclude things they don’t like. Some rankings from students have been on ease of setup, how long it plays, how long it takes to learn, balance of strategy/chance, fun factor, and uniqueness.
Once learners are ready for their own game design, you can encourage them with the following prompts:
Does your game have a theme or story?
(Sometimes a theme or story can engage different sets of users.)
Is your game competitive or collaborative?
Do you want to work together or separate?
Is your game going to be more strategy or chance?
How can you add elements of strategy and/or chance?
What does the set-up look like for the game?
Does it take a long time, or is it easy?
How does a player take a turn?
What is the algorithm for turns?
What is the goal of the game/how do you win?
How many players can play without making it take to long?
Is there a way to change how the game plays each time?
How can you add variety to game play?
Items that learners may want to include in their game: pieces, board, box, instructions, dice, cards, tokens, etc.
Apollonian Gaskets are a creative way to play with circles, fractals, and mindfulness in math. Students can cut out circles and place them within circles or practice their drafting skills with a compass and ruler. The idea is to draw a large circle and then fit smaller and smaller circles inside as closely as possible (tangent circles). For a great resource on gaskets, click here. For the theorem behind them click here (Descartes’ Theorem). This is a great time to review circle geometry. For middle and high school students, learning Descartes’ Theorem can be a fun. I have found that learners’ desire to be exact in their art has lead them to want to learn the math. However, these sketches need not be perfect; just have fun.
Step 1 – Draw a circle with a compass. Mark the center and sketch a diameter line. It is nice to use pencil and ink for these steps to be able to erase some lines and ink in your circles.
Step 2 – Add marks on the diameter that divide it into fouths. Draw two circles that have a diameter of half of your original circle by placing your compass center at the 1/4th and 3/4ths marks.
Step 3 – Draws circles that fit in the largest two spaces, (1/3 of the radius of the original). Once you set your compass to 1/3 the size of the original circle, zero in on the center of the next circle by using the perpendicular line to the original diameter and moving the compass around until it only touches the outer circle and two inner circles.
Step 4 – If you want to be exact, then you need to use Descartes’ theorem to calculate the size of each circle to proceed. You can also use a circle template or stencil set to eyeball tangent circles. The idea is to continue to fill in the spaces with more and more tangent circles. The sketch below is just approximated for a “quick sketch” and not done with the precision of Descartes’ Theorem.
Before writing this entry, I played with cutting an Apollonian Gasket from acrylic. Here is my design for a math toy. My files are on Etsy.