In this lesson, students use partial quotients and long division to divide whole numbers. 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.

Students begin by recalling that division can be done in parts, by decomposing the dividend. For instance, we can calculate by calculating  , , and . These partial quotients can be recorded as a series of equations, as students may have seen in earlier grades.

Next, students analyze a method of recording division that also involves partial quotients but is arranged vertically. Students make sense of the steps for subtracting parts of the dividend below the original number and the adding of partial quotients stacked above it. The vertical calculation foreshadows the long division algorithm.

When using partial quotients to calculate quotients, all numbers and their meanings are fully and explicitly written out. For example, to find 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.

Next, students analyze a long division for . Having seen the same division calculated using partial quotients, students can better interpret what each digit represents and can focus on making sense of the structure of the algorithm. Next, students use the algorithm to perform division with whole-number dividends, divisors, and quotients.

An optional activity allows students to further practice using long division and to analyze a place-value error commonly made in long division.