The purpose of this activity is for students to recognize that a base of a prism is a two-dimensional rectangle and any face of a prism can be a base. Students may start with a possible rectangular base and try to visualize which face of a given prism matches the base or they may start with the prism, study the faces, and try to find an appropriate base to match. In either case, they need to persevere and systematically think through all possible bases for each prism in order to solve this problem (MP1).

If students consider only one face of each prism when they match prisms to bases, consider asking,

• “How did you decide to match each prism with a rectangle that represents its base?”
• Invite students to build each prism. “Where do you see this base in one of the prisms?”

• Invite previously identified students to share.
• Display students' work that shows it is possible for Rectangle B to be a base for any of the prisms.
• “Where do we see this rectangle as a base in each prism?” (If we rotated each prism to be resting on the face that is facing us, each prism would be resting on Base B.)
• “If Prism 2 was resting on the 4-unit-by-3-unit base, the prism would be how many layers tall?” (6 layers)
• “We can use the word ‘height’ to describe how tall a prism is.”
• “If Prism 3 was resting on the 4-unit-by-3-unit base, what would be the height of the prism?” (5 cubes)

The purpose of this activity is for students to describe the layered structure of a rectangular prism, using the side lengths of the prism. Instead of a diagram of a rectangular prism built from cubes, students are shown a diagram of one of the bases of a prism and are asked to find the volumes of the prism with different heights. Students still may use informal language, such as “layer,” to describe the prism and find the volumes. During the Lesson Synthesis, connect students’ informal language to the more formal math language of “length,” “width,” “height,” and “area of the base.”

• Ask previously selected students to share their solutions for the first problem.
• Display a completed table, such as the table shown in the Student Responses.
• “How does the volume of the prism change in the table?” (The volume is increasing by 40 unit cubes for each layer added to the prism.)
• “How is the change in volume represented by the multiplication expressions in each row?” (The expressions in each row show more groups of 40.)
• Display the expression: 
• “How does the expression represent the volume of the prism?” (There are 40 unit cubes in the base layer of the prism or 4 rows of 10 unit cubes. Then there are 3 of these layers.)

This activity provides students an opportunity to interpret a calculation in the context of a situation (MP2) when a scenario is given with solutions to unknown questions. Students have to interpret equations and ask the questions, the answers to which are given. Numbers are chosen specifically to prompt students to consider the structure of rectangular prisms.

• Ask previously selected students to share their solutions.
• Connect the informal language to the math terms “length,” “width,” “height,” and “area of a base.”
• “How does the expression  represent the prism described in the fourth question?” (The area of the base is or 12 square units, and the height is 5 units, so represents the product of the length, the width, and the height.)