About Clock Arithmetic Modular arithmetic is a fundamental topic in algebra with applications ranging from coding to crystal symmetries. The RSA public key code and some error detection schemes in binary coding are based on principles of modular arithmetic. Modular arithmetic is closely connected conceptually with rotation symmetry. In the 1960's modular arithmetic was part of the elementary curriculum, part of the trend known then as "New Math". The basic ideas behind "clock arithmetic" are easily accessible to children and are a still a popular enrichment topic for the elementary classroom. The Display Please note that you will need to attach pairs of pages in the middle panel of the trifold. The printing runs across two 8 1/2"x11" sheets. The Activities Number Bracelet The number bracelet activity comes from two sources: Marilyn Burns' book About Teaching Mathematics, a K-8 Resource, and Susan Addington's webpage The Number Bracelets Game. Materials needed: 6 mm by 9 mm pony beads (about 25 for each child) and string. Pascal's Triangle The patterns formed by modular equivalents of the numbers in Pascal's triangle are fascinating. The larger the triangle, the more apparent the pattern. I found that it was easier to generate Pascal's triangle working always in the modulus, because the numbers stay much smaller. In mod 4, the largest number is 3. I also found that the patterns and the ideas behind them were more apparent if I constructed Pascal's triangle from scratch working in the modulus, with 2 + 2 = 0, 3 + 2 = 1, etc. I have provided blank triangles and numbered triangles so that kids doing this activity can choose to build the triangle by adding in the modulus of their choice or they can choose to calculate equivalents in their modulus for each of the numbers in the numbered triangle. A supplemental activity would be to build a wall-sized Pascal's triangle out of colored paper squares, using a modulus other than 4 (because that is on the display board). The person setting up the display should start the first few rows of the triangle, taping paper squares high up on the wall. Each child visiting the display would add a square, using modular arithmetic to figure out which color to place where. The resulting triangle should look something like the one on the display board, except that the pattern would be different if a different modulus is used. Materials needed: copies of Pascal's triangle to color, crayons, squares of colored paper (if the wall sized-triangle activity is offered). How to construct Pascal's triangle: In ordinary arithmetic (not modular) Pascal's triangle is constructed by adding adjacent pairs of numbers in a row to produce the next row. The sum goes on the next row in between the numbers in the row above. The source for this idea is a website with an interactive version of this activity. Mathematics for the Liberal Arts Student by Fred Richman, Carol L. Walker, Robert J. Wisner, and James W. Brewer. Modular Calculators The modular calculators need to be printed on cardstock or heavy paper, cut out, and assembled with a paper fastener. The smaller polygon goes on top of the larger polygon with the centers lined up. It might be a good idea to protect the surface by lamination, contact paper, or acrylic spray. The calculators should help kids understand how modular arithmetic works and make the connection with clocks and possibly also (subconsciously) with rotation symmetry. Children can use them as an aid in figuring out the modular arithmetic to color Pascal's triangle and to assemble the bead bracelet. Mancala Mancala is a game with great mathematical potential. It is easy to play and hard to play well. Interesting mathematics such as the figurate numbers turn up in the game's strategy. To find out more about mancala and mathematics, read Board Games Round the World by Robbie Bell and Michael Cornelius, and "Seeding Ethnomathematics with Oware" by Arthur Powell and Oshon L. Temple in Teaching Children Mathematics, Feb. 2001, Vol 7, No. 6. Variations of this game are played all over Southeast Asia and Africa. The variation given in this activity is the one which seems to be most common in the United States: these rules are found in the Family 7 Mancala set found in stores in the US and on several websites. I have chosen this version for that reason and because these rules seem easiest to learn. This website, among several others, offers the rules given here (Egyptian version): Game Cabinet: Mancala Rules. The connection between mancala and modular arithmetic is that it is useful to know where you sow your last seed. You can think of this as a mod 12 problem where you always subtract an extra seed as you pass the mancala, or a mod 13 problem where you re-number the holes on the other side of the board. The answer to the problem on the board is 3 + 15 mod 12 = 6; subtract 1 for the mancala to get 5. The last seed goes into hole number 5 and the blue player captures a quantity of seeds from the opposite hole. Materials needed: A mancala board and playing pieces. I use an egg carton, a couple of coffee jar lids, and beans. It would be a good idea to have several mancala sets available in case several pairs of children wanted to play at the same time. Written May 7, 2002 by Janet A. Dixon Revised 1/10/06