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.
I love fiber arts and weaving. So, I have one more weaving post for this series, but this time it’s with paper. This activity is great for all ages and can be done with ribbon, bias tape or strips of paper. I like to use origami paper strips.
The idea here is to play with repeating patterns and find where you can create secondary patterns, tessellations, and other shapes. Learners can experiment with over/under weavings and see what amazing patterns emerge. Make sure to have lots of colors, and encourage experimentation (diagonal, skipping, color patterns in warp and weft, gradients, etc.)
Math is beautiful. Math is playing with patterns and abstract thoughts. This is a wonderful activity to tickle the math parts of our brains.
Weavings above are done by my family and friends. My daughter and son really made a week of weaving papers.
Some questions to ponder:
Can you create a matrix or array that can represent your pattern?
For precalc and above – what would operations on your matrices result in if you mapped colors to numbers?
Can you create curves or other optical illusions with weaving techniques?
How can weaving relate to our numbers? (number line, even/odd, etc.)
Can you weave a function? What is the input and output?
This week we will do tessellations that fit together through translation (moving without rotation). We will look at reflection and rotation in other weeks. There are a few different ways to do this, but we will use the paper method today. I always start the class by talking about what different kinds of shapes can tessellate (triangles, trapezoids, hexagons, rectangles, etc.). We look at the tessellations around us (bricks, floor tiles, fabrics, etc.)
If you have never made a tessellation before, the easiest way is to use a rectangle sheet of paper, with a pencil, scissors and tape. Here are the instructions:
Step 1: Sketch a curve that stretches from the bottom left to the bottom right corner of your rectangle
Step 2: Cut out your curve and move it to the opposite side of your rectangle. Tape it together as perfectly as you can.
Step 3: Sketch a curve from the top left of your rectangle to the bottom left.
Step 4: Cut out the curve on the left and then tape it to the opposite side (again as perfectly as you can).
Step 5: Trace your shape on a sheet of paper and add some fun details:
This week learners can play with angles with both grand projects and smaller art projects. There are 360 degrees in a circle or 2pi radians. Learners can draw a circle and then mark every 20 degrees (or every 30 or any factor of 360).
Once the circle and tick marks are made, learners can start connecting points by skipping a set amount (skip every 5 marks). The key here is to be consistent – make sure they skip the same number of marks with each line. The lengths of the lines should be the same, so they can use that to check each line. I like to use circular protractors, but it’s not necessary.
After creating a star, or mix of polygons, learners can color them in, create a template for sewing applique, laser cut, combine them into a mobile, and more.