## Division: Overview

Students are introduced to the concepts of multiplication and division by relating them to addition and subtraction. That is, they learn that multiplication is repeated addition and division is repeated subtraction. Similarly, you can point out to students that just as addition is the inverse operation of subtraction, division is the inverse operation of multiplication. Last year, students learned to regroup to multiply and divide two- and three-digit numbers by one-digit numbers. See Regrouping to Multiply and Divide. This year they will build on that knowledge to learn the algorithm of long division.

The mathematical expression 3 x 5 represents three groups with five items in each group. To find the product, students can build a model of three groups with five items in each group as shown below.

Students can also use repeated addition to find the product. They can add 5 three times: 5 + 5 + 5 = 15.

Remember that multiplication “undoes” division and division “undoes” multiplication. In other words, since 3 x 5 = 15, then 15 ÷ 5 = 3. Division and multiplication are inverse operations. Students can use models similar to the models used in multiplication to divide. In the expression 15 ÷ 3, you begin with fifteen items and want to know how many items you can put in each of three groups. The answer, or quotient, is the number of items in each group.

Since multiplication is a form of repeated addition, division is a form of repeated subtraction. For example, 15 ÷ 5 asks you to repeatedly subtract 5 from 15 until you reach zero: 15 − 5 − 5 − 5 = 0. This process required 5 to be subtracted 3 consecutive times, so again we see that 15 ÷ 5 = 3.

The number that is divided is called the dividend and the number by which the dividend is being divided is the divisor. The answer to a division problem is the quotient.

As students master the basic division facts, they will need to learn how to divide larger dividends. Thus, it makes sense to begin with two-digit dividends divided by a one-digit divisor to introduce the long-division algorithm. Although students may know the quotient for the problem, they need to carefully learn the division algorithm, which will allow them to quickly move to larger numbers. Look at the division problem below.

It is sometimes helpful if students think of division as multiplication. The above division problem can be written as 306 ÷ 6 = ? or ? x 6 = 306. In the multiplication problem ? x 6 = 306, the number 306 is the product, 6 is a factor and ? is a missing factor. In the division problem 306 ÷ 6 = ? the number 306 can be thought of as the product, 6 as a factor, and ? as the missing factor. The missing factor is really the answer, or quotient, of a division problem. When looking at , we are trying to find what number multiplied by 6 will give us 306. Understanding this concept will help students learn the long-division algorithm.

When beginning the long-division algorithm, students need to ask questions such as “Can 3 be divided by 6?” (no) “Can 30 be divided by 6?” (yes) “What number multiplied by 6 is 30?” The answer is 5, so 5 can be written above the 0 in the tens place as shown below. Since 5 tens = 50, and 50 x 6 = 300, we take 300 away from 306, leaving 6. The process is repeated when asking “What number times 6 is less than or equal to 6?” Since the answer is 1, a 1 is written above the 6 in the ones place. Since 6 x 1 is 6, 6 is taken away from 6 leaving 0. The problem is complete since the amount left over, the remainder, is less than the divisor, 6. The complete problem is shown below.