Spirolaterals are beautiful, and making them feels like a meditation. You can start on square grid graph paper, and then experiment with other paper types. It is a good idea to do most of your math work on graph paper, and to keep a good supply at home, but here is a good printables site for custom graph paper. Use paper with larger cells with younger kids! Some spirolateral artists, such as Robert Krawczuk, use complex grids and even three dimensions for their spirolateral creations.
Square grid spirolaterals, from Wolfram's Math World
3d, 45-degree turn spirolateral on a cone, by Robert Krawczuk
It is hard to believe, but I could not find any spirolateral movies on Youtube. I think we will make it this week's project to produce one.
Spirolaterals are based on patterns and turns. Multiplication is a pattern, so a spirolateral can depict multiplication tables. Times two tables are a good easy start. An order three spirolateral needs three numbers, so you can use 2, 4, 6. Trace a line two units long, turn right, trace four units, turn right, trace six units, turn right, trace two units... and so on. It is a bit harder than it sounds. I turn my maps driving, and I have to physically turn my paper drawing spirolaterals, or go very slowly. But drawing them feels so good! It probably does something between brain hemispheres, because counting is sequential and turning is visual-spatial.
Keep going until you see a pattern. Enjoy it for a while, maybe color it. And now for a little surprise! Take any other three consequtive numbers from any times table, say, 6, 9, 12. Make an order three spirolateral out of them. What do you see now? Why?! (Hint: this part has to do with proportions. You will notice it, visually, when you draw it.)
You can also make spirolaterals of order four, five, six, or even infinite if you use an infinite pattern for your steps instead of repeating a cycle. You can use patterns other than multiplication tables for your spirolateral art. For example, try powers, doubling and doubling again: 2, 4, 8... Another thing to change is the angle of your turn. On square paper, you always turn in right angles. Experiment with hex or triangle paper to turn by different angles. Share your pictures!