Illusory shapes, which impel more than one perspective, help students understand alternative ways of approaching shapes and problem solving. Lock Haven University professor Donald E. Simanek provides examples for students like isometric drawings, the tribar and Schuster’s conundrum and explains that the illusions are more math than art. As an exercise, you can reproduce illusory images and use them for discussion by showing one point of view and then another. Then, ask the class to help deconstruct the picture. With isometric drawings, point out how some use negative space, for example, when three-sided cubes are carefully positioned to create a six-sided star.
Much of M.C. Escher’s artwork was based on grids. On a large sheet of graph paper, draw some basic interlocking shapes, such as sketching two diamonds out of some of the grid squares. Label horizontal and vertical axes on the graph paper and number the important points in the shape. This introduces students to simple coordinate geometry by having them locate the points, listing the horizontal and then vertical axis numbers. Work with positive numbers only, so that your axes resemble an “L.” You can also explore how quilting, needlepoint, and embroidery like Japanese sashiko use isometric patterns as further engagement activities.
String art combines lessons on patterns, geometrical shapes, measuring and arithmetic with the fine-motor skill of basic hand sewing. You will need card stock, sharp sewing needles, masking tape and enough scissors for the class. An alternative to needles and thread is colored pencils or crayons. You can either prepare the card stock yourself by drawing large circles or have your students do so with a stencil. With the needles, and knowing the circumference of the circle, instruct the children to create 12 evenly spaced holes on their circles. Show them how to thread their needles, and with the first stitch, use masking tape to hold it on the back of the card stock. Have some students stitch every other hole with one color and follow up with a different color every two holes. Think of the 12 points like a clock and ask students, with 12 o’clock equaling 0, to represent their patterns using numbers. For example, every other point would be 0, 2, 4, 6, 8 and 10, back to 0. Using the thread adds texture, but pencil works as well. You can also have students practice by using paper first and numbering the points on their circles. For further enrichment, try a string art pattern using a Fibonacci series such as 0, 1, 1, 2, 3, 5 and 8. With modulo-12 arithmetic, students can continue the pattern until they stitch or draw the same lines.
On a keyboard, label a chromatic scale with key names and numbers, starting with middle C at 0 and B above middle C as 11. Students will not need to know the names of notes or to read music to do this activity. Explore some interval patterns with students by playing the two notes and identifying the number of those notes. For a major third starting on C, the notes would be “0” and “4.” Have the students use the piano or create a keyboard chart to count the number of steps that made that major third. Knowing it takes four steps to make a major third, have the students calculate the major third on a different part of the keyboard. They will quickly notice that the keyboard has more than 12 notes. Let them know that they can use the same numbers for any octave. You can show them a modulo-12 application, however, if you allow them to count the C above middle C as 12, C-sharp as 13, D as 14 and so forth. A major third starting on A, for example would go from 9 to 13. To figure out what 13 means, subtract 12 from that number and you would get 1, which is C-sharp.