Lesson 5

The purpose of this warm-up is for students to notice that each face of a prism can be the base, which will be useful when students use a base of a prism to find the prism’s volume in a later activity. While students may notice and wonder many things about these images, the relationship between the images of the prism and the images of the rectangles are the important discussion points.

• Group size: 2
• Display the images for all to see.

• Ask students to think of at least one thing they notice and at least one thing they wonder.
• 1 minute: quiet think time
• 2 minute: partner discussion
• Share and record responses.

• “These rectangles show different faces of the prisms. Any face of a prism can be a base. We are going to learn more about this in the activities.”
• “Where do we see each base in the prism?”

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 think through all of the possible bases for each prism systematically in order to solve this problem (MP1).

• Have connecting cubes available for students who need them.

• Groups of 2

• Give students access to connecting cubes.

• 4 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who:
  • Match each prism to a rectangle that represents the base the prism is resting on.
  • Recognize that the 4 by 3 rectangle represents a base on each of the prisms.

If students do not correctly match prisms to bases, ask them to build each prism and describe how a given face does or does not match each base that is represented with rectangles A, B, and C.

• Invite previously identified students to share.
• Display student 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 by 3 base, how many layers tall would the prism be?” (6 layers)
• “We can use the word height to describe how tall a prism is.”
• “If prism 3 was resting on the 4 by 3 base, what would the height of the prism be?” (5 cubes)

The purpose of this activity is for students to describe the layered structure of rectangular prisms using the side lengths of the prism. Instead of diagrams of rectangular prisms built from cubes, students are shown a diagram of one of the bases of a prism and are asked to find the volume of the prism with different heights. Students may still use informal language, such as layers, to describe the prisms and find their volume. During the lesson synthesis, connect their informal language o the more formal math language of length, width, height, and area of the base.

• Have connecting cubes available for students who need them.

• Groups of 2
• Give students access to connecting cubes.

• 2 minutes: independent work time
• 10 minutes: partner work time
• As students work, monitor for students who:
  • describe and use layers of the prism.
  • multiply 40 by the height of the prism to determine the volume of the prisms.

• Ask previously selected students to share their solutions for the first problem.
• Display a completed table or use the one given in student solutions:

  number of layers	multiplication expression for volume	volume
  1	\(4\times10\times1\) or \(40\times1\)	40 cubes
  2	\(4\times10\times2\) or \(40\times2\)	80 cubes
  3	\(4\times10\times3\) or \(40\times3\)	120 cubes
  10	\(4\times10\times10\) or \(40\times10\)	400 cubes
  25	\(4\times10\times25\) or \(40\times25\)	1,000 cubes

• “How does the volume of the prism change in the table?” (The volume is increasing by 40 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 shows more groups of 40.)
• Display the expression \(3 \times 40\).
• “How does the expression represent the volume of the prism?” (There are 40 cubes in the base layer of the prism or 4 rows of 10 cubes. Then there are 3 of these layers.)

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

• groups of 2

• 2 minutes: quiet think time
• 4 minutes: partner work time
• Monitor for students who:
  • use informal language, such as layers.
  • use the terms length, width, height, and base in their questions.

• 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 \(3\times4\times5\) represent the prism described in the second question?” (The area of the base is \(3\times4=12\), and the height is 5, so \(3\times4\times5\) represents the product of length, width, and height.)