Consider the types of cups to make available to students and how the cups can be manipulated For example, paper cups can be cut apart and unrolled, but plastic cups can’t. If materials are limited and need to be reused in multiple class periods, consider providing an intact cup and a cup that has been cut apart so that students can examine both.

After students complete the task, either invite them to share how they knew what shape the cup would trace out, or conduct a Gallery Walk so that they can see and comment on each other’s arguments.

Then invite students to share their responses to the last question: How did they make a cup that would trace out the largest possible area? If no student mentions it, ask what would happen if the sides of the cup weren’t slanted at all. (In theory, the cup would roll forever and trace out an infinite area, but in reality, it would run into an obstacle eventually and stop.)

Technology can be helpful when students experiment with different possible cup shapes. If technology is available, the Geogebra applet 'Rolling Cups' is available here: https://www.geogebra.org/m/kr5zfsdc

Consider the types of cups to make available to students and how the cups can be manipulated. For example, paper cups can be cut apart and unrolled, but plastic cups can’t. If materials are limited and need to be reused in multiple class periods, consider providing an intact cup and a cup that has been cut apart so that students can examine both.

The questions in Part 2 give away the shape that the cup traces out, so hand out Part 2 after the discussion of Part 1. As long as all students have developed some ideas about the shape, Part 1 can be discussed before all students have finished their explanations. For the discussion, ask students to share what shape they think the cup traces out and why they think so. Students do not need to have fully developed proofs, but take the opportunity to press them to clarify their terminology and justify their assumptions.

After students complete the task, invite them to share their responses to the last question: How did they make a cup that would trace out a larger area? (The bottom radius is closer to the top radius so that the radius of the ring is larger, or the cup is taller so that the ring is thicker.) Then ask how they would make a shape that traces out the largest possible area. If no student mentions it, ask what would happen if the sides of the cup weren’t slanted at all. (In theory, the cup would roll forever and trace out an infinite area, but in reality, it would run into an obstacle eventually and stop.)

Technology can be helpful when students experiment with different possible cup shapes. If technology is available, the Geogebra applet ‘Rolling Cups’ is available here: https://www.geogebra.org/m/kr5zfsdc