Fitting Boxes into Boxes

15 min

Teacher Prep
Setup
1–2 minutes to read and clarify task statement. Students in groups of 4 with access to paper, rulers, tape-measures, USPS flat-rate boxes (optional). 5 minutes to determine what information is needed to solve this problem. 5 minutes to plan in groups, followed by time to measure boxes or research box dimensions.

Narrative

This activity introduces the context and constraints of a shipping problem. It invites students to make sense of the situation (packing 270 necklaces in boxes) and determine the information they need to solve the problem (finding the least expensive way to ship them).

To find the most economical shipping box combination, students will need to:

  • Find out the measurements of the gift boxes and shipping boxes, as well as the costs for mailing a shipping box of each size.
  • Decide on an orientation for the gift boxes inside each shipping box and calculate how many gift boxes will fit with that particular orientation.
  • Test out different orientations and how they affect the number of gift boxes to be fitted and the cost.

As they think about necessary information and steps to solve a real-world problem, students engage in aspects of modeling (MP4).

This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.

Launch

Use Three Reads to support reading comprehension and sense-making about this problem. Display only the two paragraphs, without revealing the last sentence (“She wants to know . . .”) or the questions.

  • For the first read, read the problem aloud then ask, “What is this situation about?” (An artist is packing necklaces in individual boxes to ship to a store.) Listen for and clarify any questions about the context, such as “flat rate” or “shipping rates.”
  • After the second read, ask students to list any quantities that can be counted or measured. (The number of necklaces, the edge lengths of each gift box)
  • After the third read, reveal the statement: “She wants to know which boxes to use to minimize her shipping cost” and ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referring to quantities from the second read. (Find out the size and cost of each shipping box. Figure out how many gift boxes can fit in it. Think about different ways to arrange the gift boxes.)

Arrange students in groups of 4. Give students 2 minutes of quiet think time to brainstorm the information needed to solve this problem. Then invite students to share their ideas, and record them for all to see.

Consider displaying some flat-rate boxes from the United States Postal Service (USPS) or an image of each of the boxes. Demonstrate the idea of the task by putting a small box inside a larger box in different orientations. Tell students that the postal service offers shipping boxes in a few standard sizes and charges a fix rated for shipping a box of each size.
Next, give students 4–5 minutes to work with their group and to think about how to effectively find the best shipping option.

Student Task

An artist makes necklaces. She packs each necklace in a small gift box that is 1341\frac34 inches by 2142\frac14 inches by 34\frac34 inch.

A department store ordered 270 necklaces. The artist plans to ship the necklaces to the store using flat-rate shipping boxes from the post office. She wants to know which boxes to use to minimize her shipping cost.

  1. What information would she need to find out?

  2. How would you use this information to find the most inexpensive way to ship the necklaces? With your group, make a plan and write down the main steps.

Sample Response

Sample response:

  1. Information needed:
    • The size and cost of each shipping box
    • The measurements of each shipping box
    • Whether there are any rules about how heavy each box can be or how the gift boxes need to be packed in a shipping box
    • Whether the artist has a budget for shipping
  2. A plan for solving the problem:
    • Decide which group member or members will work with which shipping box.
    • Find out how many gift boxes fit into each shipping box.
    • Find out how many and what combination of shipping boxes we need.
    • Calculate the total cost for shipping the gift boxes.
    • Compare the costs.
Activity Synthesis (Teacher Notes)

The purpose of this discussion is to elicit ideas students have for managing the problem-solving process. Ask each group to share a couple of specific steps that they could take to answer the question. If not mentioned by students, suggest that each group split up the calculations to be done so each person is responsible for finding the cost associated with one shipping box.

Standards
Addressing
  • 6.G.2·Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
  • 6.G.A.2·Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas <span class="math">\(V = l w h\)</span> and <span class="math">\(V = b h\)</span> to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
  • 6.NS.1·Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. <em>For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?</em>
  • 6.NS.A.1·Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. <span>For example, create a story context for <span class="math">\((2/3) \div (3/4)\)</span> and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that <span class="math">\((2/3) \div (3/4) = 8/9\)</span> because <span class="math">\(3/4\)</span> of <span class="math">\(8/9\)</span> is <span class="math">\(2/3\)</span>. (In general, <span class="math">\((a/b) \div (c/d) = ad/bc\)</span>.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? </span>

30 min