In this activity, each student determines the number of gift boxes that can fit in a particular shipping box and finds how many shipping boxes are needed to contain 270 necklaces in gift boxes.

Along the way, students consider how the arrangement of the gift boxes inside each shipping box affects the number of gift boxes that can fit. Encourage students to create drawings of the boxes to show the calculations that they need to make and to visualize various constraints.

Students also have an opportunity to make assumptions as they model with mathematics (MP4). For instance, they may assume that only the same type or size of shipping boxes could be used, that some room for protective padding is needed, and so on.

In the Activity Synthesis, students are prompted to explain their reasoning and listen to the reasoning of others (MP3). They discuss their decisions, evaluate the accuracy of their calculations, and revise their work based on feedback.

If additional modeling demand is desired, consider asking students to research information on sizes and costs of shipping boxes or to measure the shipping boxes rather than providing this information for them.

Focus this time on small-group discussions. After students have made their first round of calculations, ask them to partner with a fellow group member and take turns explaining their work. Explain that their job is to check each other’s reasoning, give feedback to one another, correct any errors, and revise their work based on the feedback.

Display questions such as the following. Ask students to use them to guide their discussion.

• “How many different ways can the gift boxes fit into this shipping box?”
• “For this box, does one orientation create less or more wasted space than others? Why?”
• “Can you use diagrams to show and compare the unused spaces in different arrangements?”
• “Is there a way to fit more gift boxes into this shipping box? How?”
• “Is it possible to fit 270 gift boxes in fewer shipping boxes of this size? How?”

If time permits, facilitate a second round of small-group discussions. This time, ask students to discuss their work with a classmate from another group who worked on the same shipping box as they did and to revise their thinking as needed.

In this activity, students calculate the costs of shipping the 270 necklaces in different-size boxes, decide on the least expensive option, and discuss their conclusions. Students also reflect on their reasoning and problem-solving process.

As students work, monitor for groups who arrive at different conclusions on the number of shipping boxes needed or the most economical options. Consider asking them to share during class discussion.

If time allows, select 4–5 groups to present their findings about the 4–5 shipping boxes (with each group presenting one option).

Otherwise, invite students to share their conclusions on the most economical shipping option for the artist. Display or record their reasoning for all to see.

To involve more students in the conversation, consider asking:

• “Did anyone arrive at the same conclusion but for a different reason?”
• “Did anyone arrive at a different conclusion?”
• “Do you agree or disagree? Why?”
• “Is any group rethinking their original conclusion after hearing another group’s reasoning? Which parts of their reasoning led you to reconsider your answer?”