Lesson 22
Fitting Boxes into Boxes
22.1: Determining Shipping Costs (Part 1)
Activity
In this first part of the project, students make sense of the task at hand and determine what they need to know and do to find the most economical shipping box combination. Students should recognize that they will need to:
 Find out the measurements of the jewelry boxes and shipping boxes, as well as the costs for mailing a shipping box of each size. Asking students to find out this information themselves will increase the modeling demand (MP4).
 Decide on an orientation for the jewelry boxes inside each shipping box and calculate how many jewelry boxes will fit with that particular orientation.
 Test out different orientations and how they affect the number of jewelry boxes to be fitted and the cost.
Students may want to think of a strategy for considering different configurations efficiently, rather than testing all of them, which would be time consuming and repetitive. As students make plans to try out different jewelry box orientations and associated calculations, encourage them to work systematically to minimize omissions and errors. Urge students to create drawings or models of the boxes to show the calculations they will need to make.
Consider supporting students by discussing the different orientations of jewelry boxes in a shipping box (Which orientations are possible? How much empty space would result?) and possible ways to use drawings or diagrams to show the arrangements of jewelry boxes.
Launch
Give students 1–2 minutes to read the task statement individually and to ask any clarifying questions. Demonstrate the idea of the task by putting a small box inside a larger box in different orientations. Consider displaying USPS flatrate boxes, or an image of each of the boxes.
Arrange students in groups of 4. Give students 5 minutes of quiet think time to brainstorm about what information is needed to solve this task and share to it with their group. Give another 5 minutes to plan in groups, followed by time to measure boxes or research box options and dimensions for themselves. Provide access to measuring tools.
USPS flatrate information:
 Small box: \(5\frac38\) inches by \(8\frac58\) inches by \(1\frac58\) inches. Cost: $6.80.
 Medium box 1: \(11\) inches by \(8\frac12\) inches by \( 5\frac12\) inches. Cost: $13.45.
 Medium box 2: \(11\frac78\) inches by \(3\frac38\) inches by \( 13\frac58\) inches. Cost: $13.45.
 Large box: \(12\) inches by \( 12 \) inches by \( 5\frac12\) inches. Cost: $18.75.
Design Principle(s): Support sensemaking
Student Facing
An artist makes necklaces. She packs each necklace in a small jewelry box that is \(1\frac34\) inches by \(2\frac14\) inches by \(\frac34\) inch.
A department store ordered 270 necklaces. The artist plans to ship the necklaces to the department store using flatrate shipping boxes from the post office.

Consider the problem: Which of the flatrate boxes should she use to minimize her shipping cost?
What other information would you need to be able to solve the problem?
 Discuss this information with your group. Make a plan for using this information to find the most inexpensive way to ship the jewelry boxes. Once you have agreed on a plan, write down the main steps.
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
Activity Synthesis
Reconvene as a class before continuing with the next part. Ask each group to share a couple of specific steps they have taken toward answering the question and a couple of steps they plan on taking to move forward. Highlight any ideas students might have about making the problemsolving process more efficient and systematic. If not already mentioned by students, suggest that each group divide up the calculations to be done so each person is responsible for one shipping box.
22.2: Determining Shipping Costs (Part 2)
Activity
After planning and gathering information in Part 1, students now calculate the cost of shipping jewelry boxes in each of the USPS flatrate boxes. Each member of the group will select one of the 4 flatrate shipping box sizes, decide on the best orientation for the jewelry boxes inside the shipping box, and calculate the cost to ship 270 boxes.
Notice groups using different strategies for division with fractions. Ask students to think about the different ways they have used fractions in calculations. If they are stuck, remind students that drawing the boxes, or making them from paper, might help them to visualize what calculations would be most helpful in finding a solution for the task at hand.
Launch
Keep students in the same groups of 4. Ask each group member to select a different size of shipping box so that all boxes are represented in each group. Provide access to geometry toolkits and rulers (or tape measures).
Once shipping boxes are assigned, give students quiet time to work. After most students have attempted a few box orientations and made their first calculations, ask the class to pause their work. Use MLR1 Stronger and Clearer Each Time: Students successively share their orientations and reasoning for their particular orientation, and get feedback on both language and orientation choices from their partners before the postwrite.
Design Principle(s): Optimize output (for explanation)
Supports accessibility for: Organization; Attention
Student Facing
Work with your group to find the best plan for shipping the boxes of necklaces. Each member of your group should select a different type of flatrate shipping box and answer the following questions. Recall that each jewelry box is \(1\frac34\) inches by \(2\frac14\) inches by \(\frac34\) inch, and that there are 270 jewelry boxes to be shipped.
For each type of flatrate shipping box:
 Find how many jewelry boxes can fit into the box. Explain or show how the jewelry boxes can be packed in the shipping box. Draw a sketch to show your thinking, if needed.
 Calculate the total cost of shipping all 270 jewelry boxes in shipping boxes of that type. Show your reasoning and organize your work so it can be followed by others.
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
Activity Synthesis
Smallgroup and wholeclass reflections will occur in the next activity.
22.3: Determining Shipping Costs (Part 3)
Activity
In the final phase of the shipping project, students present, reflect on, and revise their work within their small group. They discuss their decisions, evaluate the accuracy of their calculations, and then revise them as needed. In groups, they discuss which shipping box size, or combination of sizes, will be the most economical for shipping 270 jewelry boxes.
Launch
Keep students in the same groups of 4. Give them 10–12 minutes to share each member’s work in small groups and make revisions as needed. Use MLR 2 (Collect and Display) and refer to select student language while engaging in the whole group discussion. Display questions such as the following. Ask students to use them to guide their discussion.
 How many different ways can the jewelry boxes fit into each shipping box?
 How does the orientation of the jewelry boxes affect how they fit within the shipping boxes?
 Do some shipping boxes have more wasted space than others? Why?
 Can you use diagrams to show and compare the unused spaces in different configurations?
 Are there ways to reduce the amount of wasted space when shipping exactly 270 jewelry boxes?
 How does the orientation of the jewelry boxes affect the cost of shipping with each shipping box?
 Is there a way to increase the number of jewelry boxes that will fit into a shipping box? How?
Once each group member has had a chance to share individual work and before discussing this problem as a whole class, give students 4–5 minutes to decide on the best (least expensive) option for shipping 270 jewelry boxes and write down ideas for explaining their strategies.
Supports accessibility for: Language; Organization
Student Facing
 Share and discuss your work with the other members of your group. Your teacher will display questions to guide your discussion. Note the feedback from your group so you can use it to revise your work.
 Using the feedback from your group, revise your work to improve its correctness, clarity, and accuracy. Correct any errors. You may also want to add notes or diagrams, or remove unnecessary information.
 Which shipping boxes should the artist use? As a group, decide which boxes you recommend for shipping 270 jewelry boxes. Be prepared to share your reasoning.
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
Activity Synthesis
After small groups have reached an agreement on shipping box recommendations, discuss as a class so students can see a variety of strategies for orientation and calculation. Depending on the time available, students could just stand and share, groups could create a gallery walk, or each group could present more formally. Select one group to present their findings about each shipping box. Select an additional 1–2 groups to share their recommended shipping box size, or combination of sizes, needed to ship all 270 jewelry boxes.
To help students tie everything together, consider discussing the following questions:
 “How did the choice of jewelry box orientation affect how many would fit into each shipping box?”
 “How did the quantity of jewelry boxes (270) affect the choice of shipping box size?”
 “How did you calculate how many jewelry boxes would fit in a box? Did you multiply the lengths of the jewelry boxes or divide the lengths of the shipping boxes?”
 “Did the size of fractions affect how you performed division? What methods did you use to divide?”
 “How did you confirm or check your calculations?”
 “If you had a chance to solve a similar problem, what might you do differently to improve the efficiency or accuracy of your work?”
Design Principle(s): Support sensemaking; Maximize metaawareness
Lesson Synthesis
Lesson Synthesis
The bulk of students’ reflection about the mathematics of the unit should happen in the last task of the lesson. To wrap up this culminating lesson, consider highlighting instances of mathematical modeling in the lesson by asking questions such as:
 “When did you have to make assumptions to make the problem solving possible or more manageable? What assumptions did you make?” (Possible responses: When deciding which shipping boxes to use, I assumed that only one type of shipping box would be used. When deciding how many jewelry boxes would fit into a shipping box, I assumed that no bubble wraps or other packing materials were needed. When deciding how to pack the jewelry boxes, I assumed that using the same orientation within a box was preferable.)
 “Was there any missing information you had to find out before you could proceed?”
 “Were there times when you had to change course or strategy because the approach you had chosen was not productive?”