Drawing with perspective is a wonderful way to play with ratios, similar triangles, transformations, and more in math. This week I encourage learners to try to draw with various perspectives. This is a fun activity for all ages and the math can either stay simple or dive into transformation matrices.
Here are some activities to start:
One point perspective: Start with a horizon and one vanishing point. Rectangular prisms are easy objects to start with.
Two point perspective: Start with a horizon and two vanishing points. Again, rectangular prisms are easy objects to start with.
Three point perspective: With three points, sketch a rectangular prism.
Distortion (try sketching through a lens, glass, or glass sphere).
After you play with some of the basic sketches:
Look at shading and details. (Draw a city, room, fantasy land, or scene. Where is the light source? How can you show it through shading?)
Draw shadows with perspective.
Try a reoccurring object (power line poles, people, light posts) to see how it appears further away, yet the same height. (Look at ratios and similar triangles here.)
Try to grid a square tile floor with perspective (hint: look at the diagonals for the squares).
Sketch perspective using Geogebra and look at relationships in distances (some activities are here).
Birdwatching and math go hand-in-hand. There are statistics on populations, migrations, observations, and so much more. One of the ways to get in touch with nature is to become aware of the birds that frequent your home and walkabouts.
Birdwatching gets us to access so many areas of our mathematical minds. We see a size, color, shape, or feather and we start to put the puzzle together to identify what species. Some may use their ears and hear the minor differences in calls and trills to know what is around them and what to look for. This problem-solving activity is so mathematical in nature.
This week, I encourage learners to bring their senses to their windows, backyards, and trails to put the pieces together to identify the flighty, feathery friends that we have all around us. I recommend having a bird book (The Sibley Guide to Birds, 2nd Edition – please get it from a local store), and also using Cornell’s Lab of Ornithology. The Cornell lab will let you track your birds, see what’s in your area, and let you contribute to the birding community.
Here are some of the math activities that you can add to your bird watching:
Do a frequency chart for times of day vs. number of birds observed. (This would be many days or months of data that you could keep at a window or table at home.)
Try to find the rhythm of some of the bird calls that you hear and keep a journal.
Read up on your favorite birds, their stats, and behaviors and see if you can observe what you read (ex: Crows have a long childhood that may contribute to their intelligence. We have been observing baby crows play with sticks in our bird bath and try to use them as tools).
Be a scientist and record observations. Learners may find that they observe something new and interesting to share.
Draw/sketch the geometries of the various shapes and sizes of birds. You could do this in the different categories of birds (raptors, thrushes, shorebirds, etc.).
Map the migration patterns of birds that you see. (ex: We love seeing our Townsend Warblers come in and we love to think about where they are off to when they leave.)
My daughter had the wonderful idea of plotting, sketching, and observing the flight patterns of birds (rollercoaster, gliding, soaring, diving, etc.). This is a great art project idea.
In math we find balance and equilibrium. We balance equations. We keep balance by using properties of identity (multiply by 1 or add zero), Properties of Equality (mirroring operations), and by using the many other ways to manipulate and play with structures in math.
This week’s math is about equilibrium through building tensegrities (see photo above). Buckminster Fuller coined the term “Tensegrity” by combining “tensional” and “integrity.” He described the structure as, “Islands of compression in an ocean of tension.”
For this week’s activity, grab building toys, hot glue and sticks, or straws and string to create tensegrities. Learners can work on the simple design in the photo above, or on polyhedrons, bridges, sculptures, and more. Some challenging shapes would be polyhedral structures or towers. The goal is to create a structure that uses the tension of strings and the weight of the objects in positions that reach equilibrium.
For high school or middle school students, force diagrams may be a fun activity as well. Think about the moments and forces that balance in each structure created. There is a great “Beyond the Brick” video here. For strings at angles, there are some great trigonometry applications to play with.
A box of toothpicks can lead to an afternoon of entertainment. This week learners can play with the toothpick sequence. The sequence produces really interesting geometries and lines as it grows. I recommend watching Numberphile’s Youtube video on this sequence here. There is also OEIS’ website that allows for play with variations and many iterations. Grab a box of toothpicks and let’s begin:
Start by placing a single toothpick:
And then place toothpicks centered at each end:
And then place toothpicks centered at each end again:
Repeat this process at the ends that are available:
I also made a GIF in Procreate (stop animation is a wonderful way to play with all sorts of math):
Allow for play with the toothpicks to see what other mathematical patterns and tessellations are created.
Another option is to use graph paper to draw this sequence. Have fun!
I love playing with knots. Last year I designed a Knotty Math toy with wooden tiles. It is part of a series of toys I have been working on that help create single pointed mindfulness with math. These are for kids and adults alike. I think sand, clay, tiles, and tessellations can all be instruments for this meditatively, mindful, mathematical state.
This week learners can print paper versions of these tiles and see what amazing designs they can come up with. As a challenge, learners can try to create some of the mathematical knots in knot theory with various numbers of crossings (see comic further down).
Note: The design purposely uses only two tiles. I like limiting the tile types (no single straight tiles) to prompt more problem solving thought and as a reminder that less is more.
Here is the printable:
Here is a Knotty Kitty – See if you can make an unknot (0_1), trefoil (3_1), or others on this diagram (you may need to print more tiles).
Some more examples (I have tons more, but don’t want to spoil the fun of your discovery). You can create tons of links, knots or tiled art – enjoy!
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.