This activity aims to give students a concrete context for thinking about division. Students estimate how many of each given object are needed to make a stack of a particular height. To do so, they use what they know about the sizes of familiar objects (boxes, egg cartons, notebooks, and coins) and their intuition that it takes more of a thinner object and fewer of a thicker object to reach the same height. Later, they will use this idea to think about division more generally.

We often refer to certain objects (such as coins and books) as having a thickness rather than a height. Clarify that in these examples "thickness" and “height” refer to the same dimension.

As students discuss in groups, monitor for those who can:

• Explain clearly the relationship between the height of the object being stacked and the height of a stack.
• Explain clearly why the situation can be represented with a division expression.

Students engage in abstract and quantitative reasoning (MP2) as they represent a situation mathematically and interpret mathematical expressions in context.

If students are unsure how to make estimates for the first set of questions, ask them which object we would need the most of and the least of to reach 1 foot. Next, prompt them to reason about how many of each object it would take to reach 1 foot before doing the same for a height of 5 feet.

In this activity, students continue to explore the relationship between dividends, divisors, and quotients.

First, they study two sets of division expressions and arrange them in the order of their values—from the largest to smallest. The first set of expressions has the same dividend (800). The second set has the same divisor (25).

Next, students estimate the values of another set of expressions and classify the quotients as being close to 0, close to 1, or much greater than 1.

As students work, monitor how they think about placing the expressions. Select students or groups to explain their rationale later.

Also notice the expressions that students find difficult to put in order. Expressions with fractional or decimal divisors, or expressions in which the dividends and divisors are very close to each other (such as and ), may be particularly challenging. This is an opportunity for students to make use of the structure that relates the three quantities in a division (MP7): to see that as the divisor gets larger, even if only by a very small amount, the quotient necessarily gets smaller.

In Are You Ready for More? students have the option to work in groups of 2. Instead of writing a list of expressions, partners may take turns writing expressions with values that are increasingly closer to 1 without equaling 1.

Direct students’ attention to the reference created using Collect and Display. Ask students or groups to share how they ordered the two sets of expressions in the first question. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

Discuss division expressions that students found hard to place (such as those involving decimal, fractional, or very large divisors), and select other students to share how they made their decisions.

At this point, students may find it helpful to have more concise terms to describe the numbers in a division expression.

Introduce (or remind students of) the terms “dividend” for the number being divided and “divisor” for the number used to divide. Display a division equation labeled with these terms so students can refer to it as needed:

Next, discuss ways to reason about the size of quotients relative to 0 and 1. Encourage students to use the new terms as they respond to the following questions:

• “How did you decide that a quotient is close to 0?” (The divisor is much larger than the dividend.)
• “How did you decide that a quotient is close to 1?” (If the divisor is close to the dividend, then it would be closer to 1.)
• “Is there a way to tell if a quotient is less than 1 or more than 1?” (If the divisor is smaller than the dividend, then the quotient is more than 1. Otherwise, it is less than 1.)
• “Suppose a divisor is less than the dividend. How can we tell if the quotient is just a little larger than 1 or much larger than 1?” (If the dividend and divisor are very far apart in size, then the quotient is much larger than 1.)