In this lesson, students use what they learned about volume and about operations with fractions to solve an optimization problem. Given several shipping boxes with fractional edge lengths, students determine the most efficient and economical way to pack and ship 270 necklaces, each placed in a smaller gift box.

In the first activity, students make sense of the context and problem, outline what they will need to know and do to answer the question, and map out a plan.

Next, students use available measurements to determine the number of gift boxes that can fit each shipping box and the number of shipping boxes needed. This investigation involves experimenting with different orientations of the gift boxes to optimize the space in a shipping box. (The quantities in this lesson are based on the flat-rate shipping options of the United States Postal Service (USPS), but other sizes and rates can be used as well. )

Finally, students use what they find about the capacity of each shipping box and the necessary number of boxes to calculate costs, compare them, and identify the least expensive option.

Throughout the lesson, students have opportunities to explain their thinking, listen to feedback from their peers, and critique the reasoning of others (MP3). To model a situation mathematically, they also need to make assumptions about relevant conditions or constraints (MP4).

To complete all activities would likely take more than a typical class meeting. The “How Many Shipping Boxes?” activity allows students to consolidate key concepts from this unit and is therefore prioritized. The last activity, which prompts students to calculate shipping costs, summarize their findings, and make a recommendation, is marked as optional.