In previous grades, students learned the formula for the volume of a cylinder. In this task, students compare and contrast methods for finding volumes of cylinders and prisms. Monitor for students who use vocabulary such as diameter and radius, and who use expressions like , , and .

The goal is to make sure students understand what the formula means in terms of the two figures. In the discussion, be sure the expressions and come up. Ask students if they’ve seen the expression , and to describe how this relates to . Make sure they can describe what the formula means in words—to find the volume of a prism or cylinder, we multiply the area of the solid’s base by the height of the solid.

In previous grades, students learned how to calculate the volume of a cylinder. In this activity, they revisit that process. Students are prompted to think about volume in 1-unit layers. This will be helpful in upcoming lessons when students learn about Cavalieri’s Principle, or the idea that if two solids have equal-area cross-sections at all heights, then the solids have equal volumes. Cavalieri’s Principle will be used to derive a formula for the volume of a pyramid.

Note that in this activity, “volume of a container” is used as shorthand for “the volume of the region enclosed by the container.”

Monitor for students who use these approaches:

• Calculate the volume of the water in the prism and then work backward to find the height of the water in the cylinder.
• Use deductive logic (for example, reasoning that because the two bases have equal area, the height of the water must be equal in both solids).

Plan to have students present in this order to build from the familiar method toward the reasoning needed in subsequent lessons.

Students reason abstractly and quantitatively (MP2) when they compare the prism and cylinder volumes.

In this activity, students combine their experience with solids of rotation and their understanding of cylinder volume.

No specific directions are given in regard to specifying how students should express their volume answer. Monitor for students who:

• Leave their answers in terms of .
• Find an approximate decimal answer.

This is the first time Math Language Routine 7: Compare and Connect is suggested in this course. In this routine, students are given a problem that can be approached using multiple strategies or representations, and they record their method for all to see. They then compare and identify correspondences across strategies by means of a teacher-led gallery walk with commentary or teacher think-aloud (such as “I notice . I wonder ”). A typical discussion question is “What is the same and what is different?” prompting students to compare their own strategy to the others. The purpose of this routine is to allow students to make sense of mathematical strategies and, through constructive conversations, develop awareness of the language used as they compare, contrast, and connect other ways of thinking to their own.