Checker/Chessboard Square Fractions — Tic Tac Chec Game — Square Algorithm

The children had so much fun creating their own fractions using square, triangular and hexagonal paper that I wanted to continue this exploration with a view to my favorite fraction (1/2) and my passion for chess. We started with a single square with no shaded interior. Asking the children to identify the fraction that this picture represented, they had to process their dissonance before realizing that this was 0/1 (zero wholes) or just 0. Then shading in the square we had 1/1=1. Moving onto a 2x2 checker/chess board (“Board”), they easily saw 2/4=1/2; 4x4 Board with 8/16=1/2; then a 6x6 Board was a little more difficult as 36 does not roll off their tongues for 18/36; 8x8 Board (I threw a vinyl chess board into the circle) 32/64. When I drew them a quick mock-up of a 10x10 Board, they quickly said 50/100 and made a conjecture or observation that every even sided square Board would yield 1/2. 

For the 3-6th graders, I let them generate the algebraic proof of ((nxn)/2)/(nxn) = 1/2; please see the last two pages of the first pdf called Checkerboard Square Fractions.pdf. 

Then the real fun started: I drew a 3x3 Board with a corner shaded and they were able to easily see that the shaded fraction generated was 5/9; some of the children were easily able to see that 5/9 > 1/2. Some of the children immediately suggested that if we didn’t shade in a corner on this board, the fraction would be 4/9 and < 1/2. We continued on with a 5x5 Board (13/25 > 1/2), 7x7 Board (25/49 > 1/2). At this point in the lesson, it became easy for them to see the algorithm to generate the Board fraction for any n x n Board, where n is odd. They were able to make this connection since 25 is clearly half of 50 and they saw that 49+1 divided by 2 is 25.

Kindergarteners should only focus on the Boards I provided to them; some may want to color in their own Boards so I gave them the outlines; some may want to create their own from the square paper I provided.

Again, for the 3-6th graders, I let them generate the algebraic proof of (((nxn)+1)/2)/(nxn) > 1/2. The pdf will allow them to generate these fractions up to 30x30 Boards. For the even n boards, they should practice creating equivalent fractions by writing down 32/64 = 16/32 = 8/16 = 4/8 = 2/4 = 1/2. 

The children loved it when I showed them the blank square paper with 34x50 squares and 80x125 squares. They can create their own Boards and indicate the shaded fraction in simplified form.

The second half the lesson was for me to introduce them to a new game that I found called Tic Tac Chec (by Dream Green) that is played on a 4x4 Board. It only uses the pawn, rook, knight, and bishop. I found it in a store in Brookline called Eureka (worth the trip). The game is fast and teaches a very sophisticated version of chess without the millions of variables and long playing time. The object is to get four in a row and you never lose pieces. The rules are attached with a game board and black and white of the four chess pieces required to play. Of course, it is more fun if you use 3D pieces that you have at home. PLEASE PLAY THIS WITH YOUR CHILDREN AS I CAN DO MANY MATHEMATICAL CHALLENGES INVOLVING CHESS THEORY.

Finally, for the 5th-6th graders, I could not help myself in wanting to introduce them to my own algorithm for squaring numbers. It is based on algebraic proofs they won’t get until high school but I would love for them to practice this with stop watches to see how fast it is as compared to the standard algorithm for multiplication. For example, it takes me about 6 seconds to square 73  with my algorithm which 73x73=5,329; and about 10 seconds to do this with standard algorithm. For three digit squares it is about 15 seconds to 30 seconds. The worksheets allow the children to choose a 2-digit number and use a calculator to time themselves using my method and then do the same using standard algorithm. The beauty of this procedure, is that they need not check it on a calculator since they will generate the same answer if done correctly. The attached pdf goes all the way up to 6 digits but they should only do what is comfortable. Of course, some of them will be curious to move to 3 digits and beyond. If you would like any help on this, email me.

Finally, for the 5th-6th graders who worked on filling the Rhind Papyrus holes last week, I attach my answer key so they can check their answers. I excluded each row of the calculation so they would hopefully try to find their mistake (believe it or not, the mistakes usually come in the addition).