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Fitting Boxes into Boxes

Lesson # 17

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2019 Open Up Resources |

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Let’s use what we learned about fractions to find shipping costs.

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Today’s Goals

I can use multiplication and division of fractions to reason about real-world volume problems.

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Determining Shipping Costs Part 1

Activity 1

MLR6: Three Reads

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 flat-rate 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 flat-rate information:

Small box: 5 ⅜ inches by 8 ⅝ inches by 1 ⅝Cost: $6.80.
Medium box 1: 11 inches by 8 ½ inches by 5 ½ inches. Cost: $13.45.
Medium box 2: 11 ⅞ inches by 3 ⅜ inches by 13 ⅝Cost: $13.45.
Large box: 12inches by 12inches by 5 ½ inches. Cost: $18.75.

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1st Read: Shared Reading�What is this situation about?

2nd Read: Team Reading�What is countable or measureable in this story?

3rd Read: Team Reading�How might you begin to solve this task?

Start working on your own. �Then we’ll discuss your thinking as a class!

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USPS Flat Rate Info

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Determining Shipping Costs (Part 2)

Activity 2

MLR1: Stronger & Clearer Each Time
MLR7: Compare & Connect

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Determining Shipping Costs (Part 3)

Activity 3

MLR8: Discussion Supports

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.

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Let’s Finish This Up

Share and discuss your work with the other members of your group.I 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.

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Questions to Discuss:

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?

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Let’s Talk About It

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?

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?”

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When did you have to make assumptions to make the problem solving process more manageable? What assumptions did you make?
Was there any missing information you had to find out before you could proceed?
Were there times you had to change course or strategy because the course you had chosen was not productive?

Lesson Synthesis

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Today’s Goals

I can use multiplication and division of fractions to reason about real-world volume problems.