For week 52 of 52 Weeks of Hands-On Math, I couldn’t resist a play on words: Math rocks! This week I encourage learners to share their math in the neighborhood. Create sidewalk chalk art with Fibonacci hopscotch, paint a math rock garden, make a math obstacle course by your home, or have a piece of outside artwork to share.
Our family created a math rock garden. Rocks are often painted, traded, and shared in our area. We may still do another epic hopscotch course before the summer is over. Math rocks, so let’s share it with our world.
Acrylic non-toxic paint pens were used with these rocks, but we have used watercolors, markers, and ink on rocks in the past.
I love thinking of mirror images when I am block printing. I will never forget the time I printed SPARK backwards on accident for a summer art camp and my kids laughed at the reverse phonics. This week I encourage learners to take a math concept, tessellation, or shape and create a print.
Ways to create plates for printing:
For younger learners, foam boards easily take impressions.
For those that are semi-comfortable carving, potatoes, apples, or rubber blocks can provide semi-soft mediums to carve.
For those that are skilled with sharp objects, wood or lino-blocks may be preferred.
When I teach I say these words at least a few dozen times:
Do not ever force a blade.
Do not cut towards yourself or others.
Keep your tools sharp and cleaned.
Be in control.
If you are going to carve on a block, foam, or rubber sheet:
Sketch a design on paper with a pencil. (Keep in mind the size of your carving surface.)
Transfer the design by rubbing the pencil onto the carving surface.
Carve your design. (You can either carve in, tracing your lines, or around them.)
Roll ink on the block with a brayer.
Place a sheet of paper (or fabric) on the block and burnish (or rub) it with a flat surface to make sure that it makes contact with the block.
Peel the paper/fabric off. (This can take a few tries to get it right.)
For this post I carved a circle composition with the Fibonacci sequence in mind. I think I run faster with math shirts, so I printed one as well:
One of my children jumped in and we did a Sierpinski potato triangle. To use potatoes: Draw a sketch on the potato, cut out the design, and then treat it like a stamp.
This week I encourage learners to play with their animation skills. Take a math concept, problem, or design and play with ideas to animate it. Start simple to warm up and then build on the ideas. Flip books are fun. I recommend using thinner paper that can be seen through so it is easier to draw on top of the previous frame. Try the paper out before spending the time to draw. Flip books are an analogue GIF.
Here are the basic steps to drawing a flip book:
Decide on your math topic to animate. (Fractals, projectiles, growth, decay, etc.)
Draw the starting frame:
Draw the next frame by tracing and/or using the previous sheet as a reference:
Repeat the process of making new frames by referencing the previous:
Once your frames are complete – flip it!
There are other ways to animate as well. There are apps such as iMotion to make stop animation films. I also enjoy using Procreate and Looom to create animated GIFs. Learners may prefer to use technology for their animations. I plan to cover this in some STEAM posts at some point. Please share your animations!
Pendulums are wonderful physics toys that are great for exploring periodic functions. For week 49, I encourage learners to get out some string, weights, and stop watches.
Here are some ideas for playing with math and pendulums:
Start with a string and a weight to observe basic pendulum motion. Nuts, bolts, tennis balls, and other objects can work as weights.
Record how long it takes for the pendulum to return to its start (this is called the period).
Change the length of the string to see if it changes the period.
Change the mass of the weight to see if it changes the period.
For older learners, you may want to try to find the relationship between time, period and length.
With all ages, it can be fun to look at trigonometric functions with this activity.
Another activity is to create a wave pendulum. This can be done with a broomstick and weights, or other construction toys. I like to bring in a maker box and materials and have learners devise a way to make a wave pendulum. Once you have one working, then see if learners can explain why it’s called a wave pendulum. For those that like to code, look at creating a wave pendulum . My code is here and running below. At the bottom of this post I have a portable laser cut pendulum I designed.
Another activity is a painting pendulum. Hang a cup full of paint over a canvas and push it into a circular motion and watch the curves that are created as the pendulum is dampened. These can be a lot of fun with large tripods in bigger spaces.
Finally, I love to create chaotic pendulums because they are such a contrast to the previous activities. To create a chaotic pendulum, you can use building toys to create a tripod with a pendulum and then put a magnet in the weight. Put other magnets just beyond the reach of the pendulum and kick it off. The motion observed will be nothing like the periodic and predictable motion of the previous exercises. This is a great opportunity to talk about dynamic systems, chaos, and unpredictable behaviors.
Another topic that can be discussed is how pendulums dampen and why they do. What would happen in a vacuum?
Pipe cleaners have so many uses and one of the best ways to use them is to make bubbles. This week I encourage learners to build mathematical structures with pipe cleaners, straws, string, or other waterproof toys to create beautiful structures. I used Zometools in some of my classes as well, and they were a big hit. If this is being done inside, then a fan can be a great tool with a small tub of soap. I did this in my Village Home classes from a few years ago and it was great for 5yrs to 16yrs to adults. Diluted dish-soap worked well for us, but some folks have special formulas for bubble solutions to make them last longer.
Try to create cubes, pyramids, octahedrons, dodecahedrons, cylinders, and two dimensional portals for bubbles. What is so cool about bubbles is that they can fill in the sides/faces for the skeletons that are created, and yet curved bubbles emerge when they exit the structure. Things to discuss would be volumes, vertices, faces, paths, hypercubes, ellipsoids, air currents vs bubble size, etc..
Get up and move! This week learners can dance their favorite equations, math symbols and concepts. Whatever topic is of interest or in the process of being learned is a great one to figure out the dance moves that go with it. I recommend taking 5 to 10 of your favorite moves and making it into a mathematically choreographed dance.
Here is one approach:
Pile 1: Make flash cards for the vocabulary, plots, or concepts that you want to move to.
Pile 2: Make flash cards for the body parts that you want to move with (hands, feet, legs, arms, whole body, etc.)
Draw a card from each pile, turn on the music, and get some moves.
Note: you can play with sequences, beats, or number of repeats as well. (This works in a virtual environment as well – have a dance break!)
If you are a quilter, then you will be a pro with this week’s activity. For the last six months my quilting mother lived with us through chemo and we watched her quilt her heart out. Now that she moved back to her home, I had to laugh because she would have been so much fun making these tiles this week – I should have done this sooner! So many traditional and historical quilts use these squares (a great segue into history – Underground Railroad, folk art, stories, family histories, etc.).
This week I encourage learners to actually sew some tiles, specifically half-square triangles. Another name for these squares is Truchet Tiles. These are tiles that are not rotationally symmetric. The tiles can create a variety of patterns and tessellations. Make some tiles and then play. (The reason I encourage sewing is that the problem-solving, process, and mistakes lead to so much learning.)
If sewing isn’t your thing, then I have a JPG file below you can print and play with. There are also tons of maker ideas for these tiles (Paint squares of cardboard, wood, felt, etc.). A piece of graph paper will also work (shade in half of the squares diagonally).
How to sew:
Iron the fabric that you wish to use.
First cut out 25 squares of Color-1 and 50 squares of a Color-2. (You may want to do 2×2″ squares for small tiles or 10×10″ for larger tiles.) You can also do 36 squares of Color-1 and 72 of Color-2. Do you see the ratio? NOTE: Cut these squares carefully (millimeters matter).
Next cut down the diagonals of 25 of Color-1 and 25 of Color-2 (The same amount of each color if you are doing more than 25 tiles).
Now, take one triangle of each color and place them face to face and sew them with a quarter inch seam (make sure your seam width is consistent for all of this work).
Iron the square you just made so that the seam is folded flat to one side and then place it on one of the squares of Color-2. Sew around the border to attach the half-square triangle to the square bottom.
Now cut the extra fabric, while squaring the tile (make sure you use a grid or corner to make sure it is square as you cut).
Repeat this for the other 24 tiles (or more if you so chose).
We also made draftboard versions of these tiles to play even more:
This week learners can dive into probability through a quincunx (also known as a Bean Machine). Learners can make bean machines with building toys (Legos), pins and a cork board, or nails and wood (or other methods they devise (3d-printing, sculpture, etc)). Here is a template to use.(It’s a png and is also at the bottom of this page.) Make sure to size the printable to the size ball that you are using before you nail or pin.
What is fun about Quincunxes is that they show how possibilities play out. The idea is that every time a ball reaches a pin it has two possibilities: Left or Right. Learners can draw the possibility trees as an exercise to see how the distribution works. The middle columns have far more paths than the outer columns. The number of possible paths to a column is related to Pascal’s Triangle (or the Jia Xian triangle that was discovered much earlier).
You don’t have to be in high school math to play with conics, orbits, and projectile motion. This week (or month) learners can play with projectile motion, orbits, and conics sections with the activities below:
1.) Slicing cones
Learners can mold cones with clay and slice to see the possible shapes. This will give circles, ellipses, parabolas, and hyperbolas.
Create a cone sculpture with an intersecting plane using paper, string, pipe cleaners, or other mediums.
Create a straw rocket (don’t aim it at someone) and take slow motion videos of their flight and observe the curve that is created. For a template and activity, go to NASA’s website here. Look at various launch angles and analyze the differences and similarities in curves.
Create a water balloon launcher with PVC pipes (or other parent/teacher approved apparatuses). Record their launches and analyze. (Water rockets are also fun.)
Play with a garden hose and the curves created by shooting water up into the air.
Circles are so much fun! This week I encourage learners to get out their compasses or a circle to trace and start making patterns on paper. Patterns with circles can start simple, but can also get really complex. You can combine your compass with a straight edge and get amazing patterns and tiles. Try intersecting circles and then placing your compass at intersections and adding more circles. Shading with markers, ink, or colored pencils can make beautiful stained glass-like mosaics.
Here are some ideas to play with circles and art:
The Metropolitan Museum of art has a great activity here to take a look at Islamic art and geometry.
Cut out some of your patterns to make a puzzle.
Use tissue paper and make a see-through design on wax paper.
Go big with string and chalk outside for your designs!
Dip the top of a can or cup in paint and use it to create circle patterns on boards or fabric.
Try playing with the negative space within the art – see how it changes the tiles and overall appearance.
Use a digital drawing or design app and play with color pallets and design.
Try adding details and embellish.
Play with the Girih app (costs money) from the Apple app store.