# Lesson 18

Using Long Division

### Lesson Narrative

Prior to grade 6, students reasoned about division of whole numbers and decimals to the hundredths in different ways. In this first lesson on division, they revisit two methods for finding quotients of whole numbers without remainder: using base-ten diagrams and using partial quotients. Reviewing these strategies reinforces students’ understanding of the underlying principles of base-ten division—which are based on the structure of place value, the properties of operations, and the relationship between multiplication and division—and paves the way for understanding the long division algorithm. Here, partial quotients are presented as vertical calculations, which also foreshadows long division.

Then this lesson introduces students to long division. Students see that in long division the meaning of each digit is intimately tied to its place value, and that it is an efficient way to find quotients. In the partial quotients method, all numbers and their meaning are fully and explicitly written out. For example, to find $$657 \div 3$$ we write that there are at least 3 groups of 200, record a subtraction of 600, and show a difference of 57. In long division, instead of writing out all the digits, we rely on the position of any digit—of the quotient, of the number being subtracted, or of a difference—to convey its meaning, which simplifies the calculation.

In addition to making sense of long division and using it to calculate quotients, students also analyze some place-value errors commonly made in long division (MP3).

### Learning Goals

Teacher Facing

• Divide whole numbers that result in a whole-number quotient, and explain the reasoning (using words and other representations).
• Interpret the long division method, and compare and contrast it (orally) with other methods for computing the quotient of whole numbers.
• Recognize and explain (orally) that long division is an efficient strategy for dividing numbers, especially with multi-digit dividends.

### Student Facing

Let’s divide whole numbers.

### Required Preparation

Some students might find it helpful to use graph paper to help them align the digits as they divide using long division and the partial quotients method. Consider having graph paper accessible throughout the lesson.

### Student Facing

• I can use long division to find a quotient of two whole numbers when the quotient is a whole number.

Building On

### Glossary Entries

• long division

Long division is an algorithm for finding the quotient of two numbers expressed in decimal form. It works by building up the quotient one digit at a time, from left to right. Each time you get a new digit, you multiply the divisor by the corresponding base ten value and subtract that from the dividend.

Using long division we see that $$513 \div 4 = 128 \frac14$$. We can also write this as $$513 = 128 \times 4 + 1$$.

$$\displaystyle \require{enclose} \begin{array}{r} 128 \\[-3pt] 4 \enclose{longdiv}{513}\kern-.2ex \\[-3pt] \underline{4{00}} \\[-3pt] 113 \\[-3pt] \underline{\phantom{0}8{0}} \\[-3pt] \phantom{0}33 \\[-3pt] \underline{\phantom{0}32} \\[-3pt] \phantom{00}1 \end{array}$$