This Warm-up prompts students to compare four open-top boxes with specific dimensions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the boxes in comparison to one another.

Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure that the reasons given are correct.

• “How do you know ?”
• “What do you mean by ?”
• “Can you say that in another way?”

Math Community
At the end of the Warm-up, display the Math Community Chart. Tell students that norms are expectations that help everyone in the room feel safe, comfortable, and productive doing math together. Using the Math Community Chart, offer an example of how the “Doing Math” actions can be used to create norms. For example, “In the last exercise, many of you said that our math community sounds like ‘sharing ideas.’ A norm that supports that is ‘We listen as others share their ideas.’ For a teacher norm, ‘questioning vs telling’ is very important to me, so a norm to support that is ‘Ask questions first to make sure I understand how someone is thinking.’”

Invite students to reflect on both individual and group actions. Ask, “As we work together in our mathematical community, what norms, or expectations, should we keep in mind?” Give 1–2 minutes of quiet think time and then invite as many students as time allows to share either their own norm suggestion or to “+1” another student’s suggestion. Record student thinking in the student and teacher “Norms” sections on the Math Community Chart. 

Conclude the discussion by telling students that what they made today is only a first draft of math community norms and that they can suggest other additions during the Cool-down. Throughout the year, students will revise, add, or remove norms based on those that are and are not supporting the community. 

This activity offers a hands-on introduction to the mathematical work of modeling the volume of a box using a polynomial function. Students do not need to develop an equation with variables for the volume of the box or identify the greatest possible volume at this time, as that is the focus of the following activity.

Monitor for students using logical reasoning to figure out the volume of their boxes instead of, or in addition to, measuring directly to share during the whole-class discussion.

In this activity students write an expression that models the volume of a box as a function of the side length  of the square cutouts. Then students use graphing technology to graph the function. The graphing technology allows students to estimate the value of that will produce the box with the largest volume, and to establish a reasonable domain for the function by considering the context.

Students interpret their expression in context (MP2). Of particular interest, values of greater than 5.5 inches result in positive volumes even though cutouts of this size do not make sense when constructing a box from an 8.5-inch by 11-inch sheet of paper. It is not necessary to rewrite the expression for in standard form. Students will practice rewriting equations for polynomials in different forms in future lessons.