# The Box Factory

## Extending Multiplication with the Array

By Catherine Twomey Fosnot, Miki Jensen

The Box Factory: Extending Multiplication with the Array is one of five units in the Contexts for Learning Mathematics’ Investigating Multiplication and Division (3–5)

The focus of this unit is the deepening and extending of students’ understanding of multiplication, specifically the associative and commutative properties and their use with computation, systematic factoring, and the extension of students’ understanding of two-dimensional rectangular arrays to three-dimensional arrays within rectangular prisms.

The unit includes a series of
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The Context for Learning Mathematics Series

#### Full Description

The Box Factory: Extending Multiplication with the Array is one of five units in the Contexts for Learning Mathematics’ Investigating Multiplication and Division (3–5)

The focus of this unit is the deepening and extending of students’ understanding of multiplication, specifically the associative and commutative properties and their use with computation, systematic factoring, and the extension of students’ understanding of two-dimensional rectangular arrays to three-dimensional arrays within rectangular prisms.

The unit includes a series of investigations based on the context of a cardboard box factory. Initially students design a variety of boxes (rectangular prisms) that hold 24 items arranged in rows, columns, and layers. The questions posed in the first investigation (how many box arrangements are possible, and how do we know for certain that we have found all the possibilities) give students an opportunity to explore the associative and commutative properties, factor pairs, doubling and halving strategies, and systematic ways of organizing their work to determine all possible cases. In the second and third investigations, students analyze the amount and cost of the cardboard needed for their boxes, deepening their understanding of the associative property, examining congruency vs. equivalency, and exploring the relationship of surface area to the shape of the box. Subsequent investigations involve using two different cubic boxes as units of measurement, and determining the volume of a shipping box that measures 4 feet by 6 feet by 4 feet. By the end of the unit, formulas for surface area and volume of rectangular prisms are the focus.

Several minilessons for multiplication are included in the unit as well—these are structures as strings of related problems explicitly designed to guide learners toward computational fluency with whole-number multiplication, by focusing on factors and efficient grouping.