Addition and SubtractionAddition and subtraction concepts have been developed through handson experiences with countable objects or place value blocks. Addition is the "putting together" of two groups of objects and finding how many in all. Subtraction tells "how many are left" or "how many more or less." We say addition and subtraction are inverse operations because one operation can "undo" the other operation. Adding 3 and 5 to get 8 is the opposite of 8 minus 5, leaving 3.
This explains why the two operations are taught together. It is an easy way to practice and reinforce fact families. See Grade 1: Addition and Subtraction. As addition and subtraction expands to 2 and 3digits, emphasize the need for proper alignment of numbers. Keeping digits in proper places helps prevent errors. Help students realize that adding or subtracting 3digit numbers is similar to adding or subtracting 2digit numbers. Using a place value chart offers a sure way of getting digits in the correct places. For example, to solve an addition problem, model the addends in the place value chart, regroup the blocks as needed, and then write the answer in standard form. See Grade 2: Place Value to 1,000. The following example shows 135 + 278.
Children need to see exactly why regrouping is necessary. Imagine the sum of 135 + 278 being written as 31013, where 3 represents the sum of the hundreds, 10 represents the sum of the tens, and 13 represents the sum of the ones! Instead, when a column has more than 10 blocks, such as in the ones column and tens column in the example above, 10 of the blocks in each column are changed to a ten block or hundred block and placed in the column on the left. This is known as regrouping.
The sum of 413 is now shown. A thorough mastery of regrouping is essential because it is also used in other operations. In subtraction, regrouping goes "backwards." This means 1 ten becomes 10 ones or 1 hundred becomes 10 tens. This gives enough blocks to subtract. You can strengthen addition skills by investigating two special properties. Students may be familiar with the Order Property. It is also called the Commutative Property of Addition because it allows the order of the addends to be changed without affecting the sum. So, 3 + 5 has the same sum as 5 + 3.
Subtraction, however, is not commutative. When the order of the numbers in a subtraction problem are changed, the answer is different. For example 8 – 5 is not the same difference as 5  8.
Encourage children to use the Commutative Property to make addition easier. For example, 4 + 36 may be added by counting on when using 36 + 4. The Commutative Property justifies the switch in addends. Children will use this property again for multiplication. See Multiplication Facts.
The Associative Property of Addition allows addends to be grouped in different ways without affecting the sum.
The Associative Property can also make addition easier. For example, 9 + 3 + 17 may be found as (9 + 3) + 17 or as 9 + (3 + 17). Children should recognize that 9 + (3 + 17) can be added mentally as 9 + 20, or 29. The Associative Property justifies the grouping. Children will use this property again. See Grade 3: Multiplication. 

