Math Background

Addition and Subtraction: Overview

Addition and subtraction concepts have been developed through hands-on 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.

3 + 5 = 8      and      8 5 = 3
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addends sum difference

This explains why the two operations are taught together. It is an easy way to practice and reinforce fact families.

As addition and subtraction expands to 2 digits and 3 digits, emphasize the need for proper alignment of numbers. Keeping digits in proper places helps prevent errors. Help students realize that adding or subtracting 3-digit numbers is similar to adding or subtracting 2-digit 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.

place-value chart
place-value chart

Students need to see exactly why regrouping is necessary. Imagine the sum of 135 + 278 being written as 3-10-13, 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.

place-value chart

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 “backward.” 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 — the Commutative Property and the Associative Property. The Commutative Property of Addition, which is also called the Order Property, allows the order of the addends to be changed without affecting the sum. So 3 + 5 has the same sum as 5 + 3.

number lines

Encourage students to use the Commutative Property to make addition easier. For example, 4 + 36 may be added by counting on beginning with the larger number 36 + 4. The Commutative Property justifies the switch in addends. Students will use this property again for multiplication.

The Associative Property of Addition allows addends to be grouped in different ways without affecting the sum.

 9 + (1 + 4) = 9 + 5 = 14
and (9 + 1) + 4 = 10 + 4 = 14
The sum is the same, but the addends were combined in different ways.

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). Students should recognize that 9 + (3 + 17) can be added mentally as 9 + 20, or 29. The Associative Property justifies the grouping. Students will use this property again for multiplication.


Houghton Mifflin Math Grade 3