Grilling Algebra (Using a Rotating Grille to interest your students in Algebra)

Spike Clancy (hclancy@dimacs.rutgers.edu)
Valetta Eshbach (eshbach@dimacs.rutgers.edu)



General Information
There are many different ways of coding a message, one simple but interesting way is to use a Rotating Grille. Girolamo Cardano, an Italian, has been credited with inventing the grille. Grilles can be made from any geometrical figure which is equilateral. This lesson will use a square grille, but please be creative.

Category
This packet of activities has been designed to interest Algebra students in cryptography while reinforcing algebraic ideas. Most activities can be used in Algebra I, II, or III.

Grade Levels
9-12 (also grades 7 & 8 with slight revisions)

Prerequisites
After building a grille, most of the activities can be carried out with no prerequisites.

Activity 1: Use a Rotating Grille to decode and code a message

Suggested Materials
Grille Key, 1 for each member of class
Graph paper, same size as grille
Pencil
Paper
A sample coded message

Procedure
1. Distribute a rotating grille (see sample) to all students along with graph paper and scissors.
2. On the graph paper outline a grid the same size as the Grille Key.
3. Mark the upper left corner.
4. Distribute a coded message to each student.
5. Students place code on grid starting in upper left corner, putting one character in each square proceeding left to right, row by row. (See sample grid with code)
6. Matching marks on upper left corner, students place Rotating Grille on the grid containing characters.
7. Copy the characters seen through the openings in the Grille again starting in upper left corner and proceeding left to right, row by row.
8. Turn the Grille 90 degrees clockwise and repeat step 7.
9. Continue turning and copying characters until the marks in upper left corner are aligned.
10. Read the message, filling in spaces between words.

Sample Coded Message:
MVREO TSARE ISRLA TLLGI YEEN SSG! SCID LEGE

Sample Decoded Message:
ROTATING GRILLES CLEVERLY HIDE MESSAGES!

Discussion Questions
1. What other shapes might be used to accomplish this task?
2. What procedure will you follow to send your friend a message?


Activity 2: Create a rotating grille

Suggested materials
1 cm graph paper
Pencil
Scissors
Transparencies showing sample grille numbered
One uncut grille
One coded grille

Procedure
1. Cut out a 6 x 6 square grid
2. Separate the grid into quadrants
3. Mark the top left corner
4. Number the 9 squares in the top left quadrant 1 through 9 working left to right and top to bottom by rows. (See sample A)
5. Rotate grid 90 degrees clockwise and repeat procedure #4. (See sample B)
6. Repeat step 4 and 5 until whole grid is filled. (See sample C and D)
7. Shade one of each number (1 through 9). (See sample E)
8. Cut out the 9 shaded squares. (See sample F)

Using the Grille
The Rotating Grille becomes the key to encoding and decoding messages. You can now exchange private messages with anyone who shares your grille. The message cannot have more characters than squares on your grid. Proceed as follows:
1. Write out the message you want to send. (ROTATING GRILLES CLEVERLY HIDE MESSAGES.
2. Place the grille on a blank grid of the same size.
3. Write one character of the message on the grid through each hole in the grille. Start in top row, work left to right, row by row. (See sample G)
4. Turn the grille 90 degrees clockwise and repeat the process of step 3. (See sample H)
5. Continue 90 degrees clockwise turns repeating the process of step 3. (See samples I, J)
6. If your message has fewer characters than there are squares, fill in dummy characters.
7. The final grid should look like sample K.

The last step is to prepare the coded message to send to your friend with the matching Grille. You want this to look like a note so no one suspects your method of encoding. Copy the characters in order starting in the top left corner and proceeding left to right, row by row. When writing them insert spaces to imply word separation. These spaces are ignored by the decoder. Sample: MRVEE ORTIA LLRSL SYAET SICNG EHSI! DGLEE

Your partner who receives the coded message will place the characters on a grid and retrieve the message by using the Rotating Grille to determine the correct order of characters. (See Activity 1 for more help)

References
Gaines, Helen Fouché, "Cryptoanalysis", Dover Publications, (c)1956
Kahn, David, "Codebreakers"