multicultural math fun: learning with magic squares by robert capraro, shuhua an & mary margaret...
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Multicultural Math Fun: Learning With Multicultural Math Fun: Learning With Magic SquaresMagic Squares
byby
Robert Capraro, Shuhua An &Robert Capraro, Shuhua An & Mary Margaret Mary Margaret CapraroCapraro
Integrating computers in the pursuit of algebraic competence of patterns with magic squares for elementary-
grade students .
Magic Square BackgroundMagic 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
ConstantConstant
8
1
6
3
5
7
4
2
9
1+2+3+4+5+6+7+8+9= 45
45/#of Columns= 15
Introduction 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.
Magic SquaresMagic 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.
The West Africans gave us a method of extending the 3 x 3 square (see red boxes that were added).
Magic Square Magic Square West African MethodWest African Method
Magic Square Magic Square West African MethodWest 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.
Magic Square Magic Square West African MethodWest 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
Magic Square Magic Square West African MethodWest African Method
The next step is to supply the other two numbers on the diagonal by that will result in three sequential numbers.6
5
7
Magic Square Magic Square West African MethodWest African Method
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.
65
73
2
4
Magic Square Magic Square West African MethodWest African Method
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.
65
73
2
4
89
10
Magic Square Magic Square West African MethodWest African Method
Now flip the numbers in the red boxes to the opposites ends of the white boxes.
5
6
9
73 8
210
4
2
4
10
8
Practice on your own!!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.
Magic SquareMagic SquareSolve 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.
3
4
7
51 6
08
2
Magic Square 75Magic Square 75Solve 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.
24
25
28
2622 27
2129
23
Magic Square 108Magic Square 108Solve 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.
35
36
39
3733 38
3240
34-2
+2
-4+4
-3
+3
Magic Squares - Chinese Magic Squares - Chinese Magic squares provides practice in addition and subtraction. To Magic squares provides practice in addition and subtraction. To
construct magic squares for odd numbers squared, follow these rules.construct magic squares for odd numbers squared, follow these rules.
Magic Squares - ChineseMagic Squares - Chinese
Position the numerals in consecutive order, beginning with 1. Place the numeral 1 in the top center cell.
1
Magic Squares - ChineseMagic Squares - ChineseProceed diagonally
upward and to the right from each small square. 1
Magic Squares - ChineseMagic Squares - Chinese
If you leave the large square at the top, drop to the bottom of the column.
1
2
Magic Squares - ChineseMagic Squares - Chinese
If you leave the large square at the side, go to the other end of the row.
1
2
3
Magic Squares - ChineseMagic 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
2
3
4
Magic Squares - ChineseMagic Squares - Chinese
From 4, proceed
Diagonally upward
and to the right from
each small square.
1
2
3
4
5
6
The square on the
top right with 6 is a special square, the next numeral
7 is placed directly below.
Magic Squares - Chinese
1
2
3
4
5
6
7
Magic Squares - ChineseMagic 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.
8 1 6
3
4
5 7
2
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.
Magic Squares - Chinese
8 1 6
3 5 7
4 29
Congratulations !Congratulations !
You have reached the end of the lesson on Magic Squares
NCTMNCTMNCTM 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