Dividing by 11 -- Short Division Method

All Even Digit Palindromes Are Evenly Divisible by 11. 

To prove that this statement is true, you will see the steps below to use my Short Divsion Method. After you get comfortable with the example, start to write down even digit palindromes and divide them by 11 to see that there is no remainder at the end.

This method teaches division but requires the student to be thinking of the first nine multiples of the divisor (in this case 11), and the art of subtraction between the multiple and the number into which you are dividing 11. if the student is looking to subtract 66 from 72, they should be thinking that it takes 4 more to get from 66 to 70 and 2 more to get from 70 to 72; essentially 4+2 or 6 is the difference. I call this method of subtraction "addition to and from multiples of 10."

The second pdf attachment shows the student an example of dividing two into any number using the short division method.

Step 1: Choose an even digit palindrome like 44, 2772, 341143, 91788719, etc.

Step 2: Set up the division problem as set forth in the attached pdf. We are using the example, 2,772 divided by 11.

Step 3: We look at the first two digits of the palindrome and ask how many times does 11 divide into 27. Since the multiples of 11 we need to focus on are 11,22,33,44,55,66,77,88,and 99. The beauty of 11 is that it divides into 22 two times, into 77 seven times, etc.  So here, we know that the first quotient (answer to a division problem) is 2.

Step 4: We put the quotient of 2 above the ones column of the number 27 so the 2 goes above the 7.

Step 5: Then we ask what is the difference between 27 and 22; this number 5 is the remainder. We put the remainder of 5 to the upper left corner of the next digit in our palindrome of 7.

Step 6: Now, we are asking how many times does 11 divide into 57. Since we know that 55 is the fifth multiple of 11 our quotient is 5 and goes above the next 7.

Step 7: Repeat step 5 asking what is the difference between 57 and 55 and, of course, we get a difference of 2. Again, we place this remainder of 2 next to the upper left corner of the next digit in our palindrome of 2. 

Step 8: Then, we repeat step 6 by asking how many times does 11 divide into 22. Since we know that 22 is the second multiple of 11, the quotient is 2 and their is no remainder. We place the 2 above the number two in the palindrome. and we have our final answer of 252. So, 2,772 is divisible by 11; 11 x 252 = 2,772.

Special Note: If you are analyzing how many times 11 is divisible into a number like 66 with no remainder, and the 66 is in the middle of your palindrome, when you look at the next digit in the palindrome, you will see that 11 will not go into that one digit number any number of times, so the quotient placed above is 0. Let's say that the one digit number was 4, you would put the quotient of zero above the 4 and place a remainder of 4 in the upper left hand corner of the next digit in the palindrome, and continue with step 6.