## Squaring Magic Squares

Suzanne's Magic Squares || Multiplying Magic Squares: Contents || Exploring the Math

### Objectives: [NCTM Standards: Number and Operations, Communication, and Connections]

1. Students will work with arrays of numbers.
2. Students will calculate and compare sums of numbers in the context of magic squares.
3. Students will apply a method for multiplying two magic squares.

### Materials:

Large empty 3x3 grid
The nine cells in numerical order
Ordering the cells to make a magic square
The completed 9-cell magic square

2. Blank paper
3. Blank overhead transparency and pens
4. Rulers
5. Calculators

### Definition:

To square a number means to multiply it by itself.

[Just as we can talk about squaring a number, we can talk about squaring something with any notion of multiplication. We can square an nxn matrix by multiplying it by itself under matrix multiplication, and we can square a magic square by multiplying it by itself.]

### Considering Square A:

Remember the method we followed to multiply Square A by Square B? (To review, return to Classroom Activities.) If this method can be used to 'multiply' two magic squares, let's use it to square a magic square.

To square Square A, we'll follow the same steps, but instead of using Squares A and B, we'll use Square A twice.

### Square A           Square A

We'll again need a large grid. It will have as many cells as there are cells in Square A.

### Check for Understanding:

1. Verify that Square A is a magic square.
2. How many cells does Square A have?
3. How many cells will we need in the large grid?

### Generating the second square:

Following the multiplication method, we will fill the nine cells in the empty 3x3 grid. We will begin by adding the same number to each entry in Square A to make a new magic square.

Square A is made up of nine entries. When we add 9 to each of them, we get

We will continue this process by adding 9 to each entry to make each successive cell.

Continued next page