Magic Square Background

1. Multicultural Math Fun: Learning With Magic SquaresbyRobert Capraro, Shuhua An &Mary Margaret Capraro Integrating computers in the pursuit of algebraic competence of patterns with magic squares for elementary- grade students .

2. Magic Square Background • Magic Squares can be 2X2 and any configuration to n x n • Magic Squares all have Constant Numbers • Magic Squares have been represented in art, literature, and mathematics of many cultures • There are magic squares that are diabolical

3. Constant 1+2+3+4+5+6+7+8+9= 45 2 1 3 45/#of Columns= 15 4 5 6 9 7 8

4. Introduction • We will be addressing the topic of Algebra in grades 1-5 in accordance with the NCTM Standards through an integrated and thematic approach using technology of Power Point software to teach mathematics. • The students will be practicing addition and subtraction with patterns through concrete, pictorial, and abstract activities.

5. Magic Squares • Began with the ancient Chinese (2200 BC) who told a story about a divine tortoise named Lo who swam in the river Shu. The tortoise had a dots on his back. The pattern was a magic square that no matter how it was added horizontally, vertically, or diagonally, the sum was 15. • The Chinese left no written instructions but passed down solutions orally. The West Africans had a written method which we will look at next.

6. Magic Square West African Method The West Africans gave us a method of extending the 3 x 3 square (see red boxes that were added). 18

7. Magic Square West African Method Solve the magic square with a sum of 18. Arrange all the numbers 2 through 10; Using each number only once to make each column, row, and diagonal equal 18. 18

8. Magic Square West African Method The next step is to divide the magic square number by 3 and place the answer in the center of the square. 6 18

9. Magic Square West African Method The next step is to supply the other two numbers on the diagonal by that will result in three sequential numbers. 7 6 5 18

10. Magic Square West African Method 4 The next step is to supply the three numbers on the top diagonal that will result in three sequential numbers immediately preceding the first sequential numbers you wrote. 3 7 2 6 5 18

11. Magic Square West African Method 4 The next step is to supply the three numbers on the bottom diagonal that will result in three sequential numbers following the first sequential numbers you wrote. 3 7 2 6 10 5 9 8 18

12. Magic Square West African Method 4 Now flip the numbers in the red boxes to the opposites ends of the white boxes. 3 8 7 2 10 6 2 10 5 4 9 18 8

13. Practice on your own!! Take a piece of paper and draw a 3 x 3 square and try to do the magic square of 12 on the next slide. Read the instructions and before you click the mouse see if you can figure it out on your own.

14. Magic Square Solve the magic square with a sum of 12. Arrange all the numbers 0 through 8; Using each number only once to make each column, row, and diagonal equal 12. 1 6 5 8 4 0 3 2 7

15. Magic Square 75 Solve the magic square with a sum of 75. Arrange nine of the numbers in the range 5 through 40; Using a number only once to make each column, row, and diagonal equal 75. 22 27 26 29 25 21 24 23 28 75

16. Magic Square 108 Solve the magic square with a sum of 108. Arrange nine of the numbers in the range 2 through 70; Using a number only once to make each column, row, and diagonal equal 108. 33 38 37 -3 +2 40 36 32 +4 -4 -2 +3 35 34 39 108

17. Magic Squares - Chinese Magic squares provides practice in addition and subtraction. To construct magic squares for odd numbers squared, follow these rules.

18. Magic Squares - Chinese Position the numerals in consecutive order, beginning with 1. Place the numeral 1 in the top center cell. 1

19. Magic Squares - Chinese Proceed diagonally upward and to the right from each small square. 1

20. Magic Squares - Chinese If you leave the large square at the top, drop to the bottom of the column. 1 2

21. Magic Squares - Chinese If you leave the large square at the side, go to the other end of the row. 1 3 2

22. Magic Squares - Chinese If a number is a multiple of the number that is squared to get the total number of cells in the magic square, the next numeral is placed directly below. 1 3 2 4

23. Magic Squares - Chinese From 4, proceed Diagonally upward and to the right from each small square. 6 1 3 5 4 2

24. Magic Squares - Chinese • The square on the • top right with 6 is a special square,the next numeral • 7 is placed directly below. 6 1 3 7 5 4 2

25. Magic Squares - Chinese Proceed diagonally upward and to the right from a small Square with 7. Since you leave the large square at the side, go to the other end of the row. 1 6 8 3 7 5 2 4

26. Magic Squares - Chinese Proceed diagonally upward and to the right from a small square with 8. Since you leave the large square at the top, drop to the bottom off the column. 8 1 6 7 5 3 4 2 9

27. Congratulations ! You have reached the end of the lesson on Magic Squares

28. NCTM NCTM stands for the National Council of Teachers of Mathematics. The NCTM developed national mathematics standards that are widely accepted. In 2000, they wrote The Principles and Standards for School Mathematics. If you are on the internet click on the link below and follow it to the NCTM homepage to learn more about the professional organization. http://www.nctm.org