Transcript Technologically Teaching Algebra to Elementary Students

Multicultural Math Fun: Learning With
Magic Squares
by
Robert Capraro, Shuhua An & Mary Margaret
Capraro
Integrating computers in the pursuit
of algebraic competence of patterns
with magic squares for elementarygrade students .
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

Constant
1
2
3
1+2+3+4+5+6+7+8+9= 45
45/#of Columns= 15
4
5
6
7
8
9
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 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.
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).
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.
Magic Square West African Method
6
The next step is to
divide the magic square
number by 3 and place
the answer in the center
of the square.
Magic Square West African Method
7
6
5
The next step is to
supply the other two
numbers on the
diagonal by that will
result in three
sequential numbers.
Magic Square West African Method
4
3
2
7
6
5
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.
Magic Square West African Method
4
3
2
7
6
5
10
9
8
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.
Magic Square West African Method
4
3 8 7
2 10 6 2 10
5 4 9
8
Now flip the numbers in
the red boxes to the
opposites ends of the white
boxes.
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 Square
1 6 5
8 4 0
3 2 7
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.
Magic Square 75
22 27 26
29 25 21
24 23 28
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.
Magic Square 108
33 38 37
+2
40+436 -4 32
-2
35 34 39
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.
Magic Squares - Chinese
Magic squares provides practice in addition and subtraction. To
construct magic squares for odd numbers squared, follow these rules.
Magic Squares - Chinese
Position the
numerals in
consecutive
order, beginning
with 1. Place
the numeral 1 in
the top center
cell.
1
Magic Squares - Chinese
Proceed diagonally
upward and to the
right from each small
square.
1
Magic Squares - Chinese
If you leave the large
square at the top, drop
to the bottom of the
column.
1
2
Magic Squares - Chinese
If you leave the large
square at the side, go
to the other end of the
row.
1
3
2
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
4
2
Magic Squares - Chinese
From 4, proceed
Diagonally upward
and to the right from
each small square.
6
1
3
4
5
2
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
4
7
5
2
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.
8
1
6
3
5
7
4
2
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
3
5
7
4
9
2
Congratulations !
You have reached the end of the
lesson on Magic Squares
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