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Concepts of addition problems

Here's a sample addition problem of 2 2-digit numbers:
  

    36

+ 48

  

 
Referring to that problem above, on a piece of paper, I would make 6 and 8 dots for the place value of ones and 3 and 4 vertical lines for the place value of tens.  This would help any kid understand the structure of each number so that he or she could learn how to add them.  To prove that the digits with the lowest place value of ones have to be added first, I would show the kid that this is done by first adding the one-digit numbers, 6 and 8 in the problem above, of the smallest pieces which are dots.  Then, if the sum is at least 10, this will verify that 10 dots are put together to make a vertical line for an extra ten called a carry that will be added to the original group of tens and thereby regroup; nevertheless, the number of remaining dots is kept and is therefore the final digit in the ones place of the sum.  Next, with the final group of vertical lines representing tens, I would ask the kid to count all those symbols to see whether or not he/she will have to yield another carry.  This would teach him/her how to add two numbers by adding each two digits sharing the same place value in the order of place values from right to left.