Magic squares provide students the opportunity to apply strategies that use the properties of addition as they add the columns, rows, and diagonals. They also provide opportunities to use the patterns they have discovered to create new magic squares.
Display the completed magic square from Question 3 in the Student Activity Book. See Figure 2. Ask students to make a new 3 + 3 magic square by adding 3 to each of the nine numbers in this puzzle: 3, 7, 11 (3 + 3, 7 + 3, 11 + 3).
- How does adding 3 to each of the numbers change the sum of the magic square? (The sum of each row, column, and diagonal changes from 21 to 30.)
- What do you notice about the pattern? (Now all of the numbers in the magic square are even numbers instead of odd numbers. The middle number is still in the center square.)
- Add 5 to each of these nine numbers to make a new magic square. How does this change the sum of the magic square? (Now the sum of the square goes up by 15, so it is 45 instead of 30.)
- How does adding 5 to each of the nine numbers change the pattern? (Now all of the numbers in the squares are odd again.)
- Why do you think a magic square is not “ruined” when you can add the same number to each of its nine numbers? (If you add the same number to each square, you are adding the same amount to each row, column, and diagonal. It stays a magic square; the sum is just bigger.)
- What will happen to the magic square if you subtract 2 from each of these nine numbers? (It will still be a magic square, but the sum will be 39 now. When you subtract 2 from each number, you are subtracting 6 from each row, column, and diagonal. The numbers in the magic square will still be odd because when you subtract two from an odd number, the answer is an odd number.)
