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:
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!
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:
I love tessellating! This week we are playing with hexagons. Learners can either draw or cut various hexagonal designs with various colors. It is fun to see what secondary patterns can occur. Do other polygons tessellate?
This is a wonderful activity to practice mindfulness and presence as you play with these shapes. It is an opportune project for students to learn single pointed mindedness.
Use lots of colors – tissue paper is also fun. I didn’t post many pictures, because I don’t like to give all the patterns away – I really love the discovery that happens with this project. A paper cutter can come in handy here if you have a large class.
Here is a template to cut from. It is also a good time to break out protractors and comapasses and play with 120 degrees (the outer angle of a regular hexagon).
In our home, we have been playing with laser cut hexagons this week; and if you are a hobbyist with a laser and interested in a set, then I have SVG files here.