ex:

  79
+67
  16    (7 plus 9 = 16;  so we write 16)
130    (70 plus 60 = 130; so we write 130)
146           you get 146 with no carrying!

*The 16 and the 130 are called "partial sums".  They are "parts" of the total answer, and we can get them in any order.  We do not HAVE to get the 16 first.

. 79
+67
130
..16
146

Here is one where we enter partial sums in all different orders:

. 478,669
+479,949
..800,000  (hundred thousands sum)
............18   (ones sum)
..140,000   (ten-thousands sum)
....17,000    (thousands sum)
..........100    (tens sum)
.......1,500    (hundreds sum)
..958,618

You see, with partial sums, you are writing down the "parts" and therefore it does not matter that you are not working from right to left as is necessary in formal addition with carrying.

*The aspect of this addition that is most convenient is that you can work FROM LEFT TO RIGHT and stop whenever you have enough information. 

Quite frankly, the numbers on the left are far more important in an addition problem. 
They are the largest numbers and are often all you need for a good estimate.

As you work your way to the right, your estimate of the answer gets more precise until you finally have the EXACT answer.

For example, say you have $10,000 for a building project.  The items you need are priced in at:

Now on a piece of paper try adding the following.
Be sure to add them each in a different order using the partial sums.

ASSIGNMENT:

To really "own" this approach try filling  one side of a piece of notebook paper with all different kinds of addition problems.  Make them different lengths with a different number of addends each time.  Be sure to force yourself to add from left to right to get used to the idea.

top

first answer = 232
back

second answer = 2,467
back

third answer = 98,483
back

*The first rule in making math quick and easy is to change something difficult into something easy, whenever that is possible.

Now when adding two numbers, the larger a digit, the more likely it is to involve carrying.  The digit 9, when added to any digit but 0, will make carrying necessary.  On the other hand, a small digit is easy to handle, and 0 is the easiest of all.  No matter what digit you add to a 0, even a 9, no carrying is needed.

So LET'S GO AND CREATE SOME ZEROES!

Example:
Add 48 + 76. 

Let's change the 48 to 50 by adding 2 to it.   If it is 50, it will be super easy to add to anything!  I know, you are saying..."You can't do that!"

Well technically you are correct, but if I can do something to balance out this extra 2 the final answer will be the same.

To balance it out I will subtract 2 from the the answer.  The net result is 0 change because you added a number and then you subtracted the same number.

The mental benefits of this 0 change are terrific however. (this is one of those....."why didn't anyone ever show me this before"  type of things) Here let's take a look:

Instead of 48 + 76, we now have 50 + 76 which equals 126 which is 2 units too large.  So we simply subtract 2 from the 126. This balances out the problem and we are left with the correct answer.  Absolutely no carrying needed!

The algebra that verifies this is quite simple:

The sum of a and b is a + b. Suppose you add any number (n) to a and subtract that same number from the final answer.  The number a becomes a+n while the sum becomes a+n +b.  Now subtract the n we added and we get, a+n+b-n which gives us just a+b.  The n's just undo themselves.

Now try the following: write the problems and then transform them to create zeroes for easy adding.

This is a very powerful trick when adding prices at the grocery store.

YOUR TURN:
To "own" this trick you need to practice it a lot.  Soon you will find yourself adding like this naturally in mental situations.  But it DOES take practice.

You need to fill one side of a piece of notebook paper with examples of this kind of addition.  The more you do, the faster you will get!

Be sure to make up different kinds and lengths of addition problems.  Ex. whole numbers, decimals, money, etc.