Can you fit more boxes in a shipping container than Jane can?

Exploring some of the mathematics in packing a shipping container.

Facilitator(s): Mark
Date of Meeting: September 14, 2017
Problem: pdf

For today’s CAMI meeting, Mark was trying out a draft of a lesson that he wrote with Eric involving volume and units in a workplace context. The problem we explored involves trying to fit rectangular boxes into a shipping container. 


We started with a little estimation to warm-up, watching the following video:

We shared some noticings and wonderings, and then we took about 45 seconds to come up with our best estimate as to how many boxes of girl scout cookies were loaded into the back of this minivan.

The range was 144 to 1200, with Solange coming the closest with 840.

Then as a way to draw out background knowledge and misconceptions, we looked at the following figure and in pairs, took turns sharing observations about it:

Tell me as many things as you can about this figure

Some of the things that came out:

  • You can see it as 7 two-dimensional shapes (2 right triangles, 1 rectangle and 4 trapezoids)
  • As a three-dimensional shape is has: six sides (two square sides, 4 rectangle sides), volume (length x width x height)
  • the sides we refer to as the length , width and height might be different if we rotate the figure

Then we looked a situation involving rectangular prisms.

You are the Logistics Manager for a clothing manufacturer. You need to determine how to arrange boxes of socks so that you can fit as many boxes as you can into one shipping container. You also need to write instructions for how you want to arrange the boxes in the container so as to fit the greatest number of boxes into each shipping container. Your directions must be clear for your team of packers to follow. What is the greatest number of boxes that will fit and how should they be arranged?

The available space in the container measures 7’8” wide by 7’10” tall by 39’6” long. The boxes are all the same and measure 24” by 18” by 36”. You can arrange the boxes any way you want in the shipping container.

Here are some Support and Push Cards 
Teacher Thinking <<Spoiler Alert>>

Teachers used different methods to get to the maximum numbers of boxes if all of the boxes are oriented in the same direction. Some people figured out the 6 different ways to position the box and then tried filling the measurements of the shipping container with each. Doing we found there are two ways to fit 195 boxes into the container.

Solange’s work
The directions Solange wrote for her team of packers
Ramon drew his ideal orientation of the box in the container to fit 195 boxes

As teachers tried the different box orientations, they noticed that there was missing/unused space for each of them. Ramon (and for a while Jane) both focused on calculating which orientation results in the least unused space.

Ramon trying to minimize the remainders
Ramon and Jane discussing his method

Mark gave out a few push questions to extend the problem into using more than one positioning of the boxes.

Jeremy was able to fit 234 boxes, using 2 different orientations for the boxes.
Jane’s method for loading 240 boxes into the container, using three different orientation of boxes.

Can we fit more than 240 boxes into the shipping container? What is the maximum number of boxes and how would we know that we had the maximum?


This lesson is part of the CUNY Careerkit Project, a set of teaching resources exploring ten different industries. This problem comes from the soon to be available manufacturing careerkit.


In attendance: Solange, Jane, Deneise, Ramon, Jeremy, Eric, Mark

Programs represented: BMCC Adult Basic Education Program, PCACP, CUNY Adult Literacy/HSE PD Team

Leave a Reply

Your email address will not be published. Required fields are marked *