This lesson invites students to investigate the relationships between the numbers in a division situation. Students learn that reasoning about the relative sizes of the divisor and the dividend can tell us about the size of the quotient.

Students begin exploring these relationships in concrete situations. They estimate how many thinner and thicker objects are needed to make a stack of a given height, and they then describe these relationships in terms of division. For example, . As they interpret division situations and equations that represent them, students practice reasoning abstractly and quantitatively (MP2).

Then students think about the relationships more generally. They reason about the values of division expressions with the same dividend but different-size divisors (such as , , and ), as well as those with different dividends but the same divisor (such as , , and ). Along the way, students have opportunities to look for structure (MP7).

Students observe that dividing by a number that is much smaller than the dividend results in a quotient that is larger than 1, that dividing by a number that is much larger than the dividend gives a quotient that is close to 0, and that dividing by a number that is close to the dividend results in a quotient that is close to 1.