A Math Forum Web Unit

Suzanne Alejandre's

Table of Contents

Objectives: NCTM Standards: Number and Operations and Connections]

1. Students will learn the definition of a magic square.
2. Students will learn the historical background of magic squares.
3. Students will experience the mathematical aspects of magic squares.

Materials:

Prepare overhead transparencies and/or handouts before presenting the activity, including:
1. Franklin's Magic Square
2. Franklin's Magic Square with Lines
3. Just the lines (for an art activity)
4. Blank paper
5. Blank overhead transparency and pens
6. Rulers

Procedure:

Display the Franklin Magic Square.

History Questions:
1. Who was Benjamin Franklin?
2. After reading about Benjamin Franklin's background, why do you think he created such an intricate magic square?
3. Was Benjamin Franklin best known as a mathematician?

Number Questions:

1. What is magic about the arrangement of the numbers in the 8x8 cell square?
2. What is the first number that Franklin used?
3. What is the last number?
4. How many numbers are there?
5. Is any number repeated?
6. What is the sum of the numbers in the1st row? 2nd row? the other rows?
7. What is the sum of the numbers in the1st column? 2nd column? the other columns?
8. If you start in the upper left hand corner and add the numbers halfway down the column, what is the sum? How does this compare to the total sum of that column?
9. What is the sum of the four corner numbers?
10. What other numerical relationships can you find?

Symmetry Questions:

1. If you vertically separate the square into two rectangles in your mind,
   are the numbers from 1 to 10 on the right or the left?
   are the numbers from 54 to 64 on the right or the left?

2. Consider the placement of 1, 2, 63, and 64. What is their sum?
3. Consider the placement of 31, 32, 33, and 34. What is their sum?
4. When you draw lines connecting the numbers in the Franklin square in order from 1 to 64, do you see a pattern?
   [Refer to Franklin Magic Square with lines.]
5. Are there symmetrical relationships?
6. Is the line design you have created an example of

   rotation symmetry?
   translation symmetry?
   reflection symmetry?
   glide reflection symmetry?
   



[Note: To discover the symmetry involved, create a transparency (trace the original) of the lines and use it to test for rotation, translation (slide), and reflection (flip).]

. .