Magically Squaring Numbers

My college math students lack confidence (I classify them as mathphobics.); so, I like to show them math "tricks" which they can use to impress their peers.  I encourage them to know their squares through 25. (Yes, I know they can use a calculator, but the mind is so much quicker!)  When we get to solving equations using the Pythagorean Theorem, I introduce this trick. Please note: For the trick to work, it must be a two digit number that ends in 5.
Suppose we have 35².  (This means will be making a square.)

• First, look at the number in the hundred’s place. In this case, it is the “3”. 
•  Next think of the number that comes directly after 3. That would be “4”. 
•  Now, in your head, multiply 3 × 4. The answer is 12. 
• Finally, multiply 5 × 5 which is 25. 
•  Place 12 in front of 25 to get the answer. Thirty-five squared is 1,255.
• 3 × 4 = 12      5 × 5 = 25       
• The answer is 1,225.

This means that we can build a square that is 35 by 35, and it will contain 1,225 squares or have an area of 1225 squares.

Now let's try 65². 

• One more than 6 is 7; so, 6 x 7 is 42. 
• Place 42 in front of 25 (5 x 5) and so 65 squared is 4,225.
• 6 × 7 = 42      5 × 5 = 25      
• The answer is 4,225.

How about finding the square root? We begin by looking at the numbers in the thousands and hundreds place. In the answer of 1,225, we would use the 12. Think of the factors of 12 that are consecutive numbers. In this case, they would be 3 and 4. Use the smaller of the two which, in this case, is 3. Now place a five after it. You now know the square root of 1,225 is 35.

Thirty-five represents the length of one of the sides of a square that contains 1,225 squares.

Now, try some numbers on your own. When you get comfortable with the "trick", try it with your students. They will find out that math can be magical!