Long-Term Project: Algebraic Magic Squares

These web-based activities were developed, during summer sessions from 1997 through 1999, by students at Mainland High School, located in Daytona Beach, FL. As part of Mainland's CCTT grant, the students were given permission by the book's publisher to convert the ancillary materials of the County's adopted Algebra 1 textbook (Glencoe Algebra 1 — ISBN: 9780028253282) into html format.

Long-Term Project
Algebraic Magic Squares

A magic square is a square array of numbers arranged so that the sum of each row, column, and main diagonal is equal to the same "magic-square" constant. In this project you will investigate some properties of magic squares and construct magic squares by using algebraic expressions. Then you will learn another way to arrange algebraic expressions so that they have "magical" properties.

1. Add the numbers in each row, column, and diagonal of this square array of numbers. How are the sums related? __________________________________________________ _________________________________________________ _________________________________________________
2. This array has six broken diagonals: (8, 3, 9, 14); (10, 6, 7, 11); (10, 13, 7, 4); (8, 12, 9, 5); (12, 2, 5, 15); and (12, 2, 5, 15). Find the sum of the numbers in each broken diagonal. How are the sums related? __________________________________________________ _________________________________________________
3. A magic square is said to be panmagic if the sum of each broken diagonal is also equal to the magic square constant. Is the square array in Exercise 1 a panmagic square? Explain. __________________________________________________ __________________________________________________
4. Rotate the square array from Exercise 1 clockwise 90 degrees by filling in this square array. Describe any "magical" qualities of the new square array. __________________________________________________ __________________________________________________
5. Now reflect the square array from Exercise 4 about the vertical axis by filling in this square array. Describe any "magical" qualities of this square array. __________________________________________________ __________________________________________________
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