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.
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:
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.
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.
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.
I love to incorporate drawing skills into math education. This week I encourage learners to start seeing birds (or other animals/people) as shapes. Heads are circles, torsos are ellipses, beaks are triangles, wings are long ellipses…
Sketching is a skill. A skill is something that you can master in time (think growth mindset). This week I challenge learners to start a daily doodle routine. Just doodle something (anything) for 1-2 minutes a day.
Here is an example of the activity. I will use a hummingbird as guidance, but please feel free to pick any object/bird/animal. I tried to do this as a quick sketch example:
Some of the concepts and discussions around sketching can include proportions, ratios, what shapes fit best, etc. I encourage learners to research and dive deeper into sketching skills and drills. I truly believe that art and spatial awareness can be beautifully integrated into learning math.
Additional activity: For high school students in Algebra 2 or higher, they can use Geogebra to sketch the shapes for an animal. How do you plot a circle? an ellipse? triangles? etc. Desmos can be used at a precalculus and calculus level.
This week the Spiral of Theodorus can be used to enhance understanding of the pythagorean theorem, right triangles, pi, and more. The spiral goes by many names (square root, Pythagorean, or Einstein Spiral) and approximates the Archimedean Spiral.
1.) Create a right isosceles triangle where the sides that are the same measure 1 unit (I used inches).
2.) Add a 1 unit line segment perpendicular to the hypotenuse of your first triangle and then connect it to create another right triangle.
3.) Add another 1 unit line segment to the hypotenuse created in step 3 and connect it to the center to create another triangle.
4.) Repeat step 3 as many times as you wish to expand your spiral.
5.) When your done, you can transform your work into a fun sketch:
Possible reflection/discussion questions:
How does the Pythagorean Theorem apply here? What pattern do the hypotenuses make?
Can you create a function that would reflect the rate of growth for the hypotenuses?
There are so many ways to say “right angle” – can you say it three other ways?
Can you create an algorithm for making these spirals?