# Stuffed Bookcase

- Division
- Measurement
- Multiplication

How many books can fit in a bookcase?

Jada wants to know how many hardcover books she could fit in her bookcase. The bookcase is 4 feet wide and 9 inches deep. Each of the 3 shelves is 9 inches tall.

- How many hardcover books, like the one Jada is reading, could she fit in her bookcase?
- Can you find another way to arrange the books so more will fit?
- Which arrangement do you think is better? Why?

- How wide could Jada’s hand be? What could be the dimensions of Jada’s book?
- How could you arrange the books on one shelf?

- What if each student in your class wanted to store 6 books in a bookcase in your classroom? How many bookcases like Jada’s would your class need?
- Jada has a collection of paperback books. Most of these books are only ½ inch thick. Their front covers are about the same size as the book she’s reading now. How many more paperback books could she fit in her bookcase than hardcover books?

In this task, students will model filling a bookcase with books. Those who have been formally introduced to volume may use this understanding to help them conceptualize the situation. But students who haven’t been introduced to volume can also make sense of this task using spatial reasoning and estimations related to size. They will use their measurement understandings and the relationship between multiplication and division to model the situation. After “filling” one shelf, students may use multiplication to determine how many books will fill the entire bookcase.

Students will need to make assumptions about the size of Jada’s books. Using the image of Jada, an approximation of the size of her hand, and their understanding of the relative sizes of measurements, they can estimate the size of Jada’s book. Students will then need to consider the arrangement of books on a shelf. They might arrange books standing up with their spines out, or they might stack books on top of each other with the narrowest edge of the book facing out. They might partition the shelf into sections that match the width of one book in their chosen arrangement. If their arrangement involves stacks of books, they may multiply the number of books in one stack by the number of stacks they can fit. They may find that, while one arrangement allows for more books, it might not be the most practical arrangement.

After estimating likely measurements of Jada’s book, students may choose to model the possible book arrangements in various apps.

- In the Number Pieces app, students might use number pieces to represent the measurements of the books in inches. After building a 4-foot, or 48-inch shelf that is 9 inches tall, they may outline the space they think one book will take up and use it to either mentally or graphically determine how many books of that size will fit.
- Using the Fractions app, students might make a model of 1 shelf using what they know about the relationships between feet and inches. Dividing 1 whole into 12 equal parts, students can model each foot of the shelf in inches. They can make a similar representation of 9 inches for the height and begin drawing books based on their proposed dimensions.

If your students have learned about volume, ask: *How is this situation related to volume? Could you have used the volume of 1 book to find how many books Jada could fit in her bookcase? Will finding the volume of 1 shelf or the entire bookcase work sometimes, but not other times? Why?*

To extend students’ thinking about fractions of a foot and the operations needed to approach this situation, ask: *How do your estimates for the size of Jada’s book compare to 1 foot? Could you use fractions to help you reason in a different way? How does changing the way you put a book on a shelf change the numbers and operations you might use?*