Lesson 1

This warm-up prompts students to compare four images. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another. ​​​​​

Arrange students in groups of 2–4. Display the images for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as round, corners, circular, or symmetric. Also, press students on unsubstantiated claims.

If the concept of complete rotational symmetry in three dimensions does not come up during the conversation, ask students to discuss this idea. Invite them envision a vertical axis of rotation, or a line around which things rotate in three dimensions, through the center of each object. The cylindrical block, and the conical hut can rotate through any number of degrees using this axis and look identical to the original orientation. The puzzle cube must rotate a multiple of 90 degrees in order to look identical to its original positioning. If it was stacked as a cylinder, the spring toy would have an axis of rotation with symmetry for any angle of rotation. As pictured, a circle could be rotated 180 degrees to trace out the shape of the spring, but not a full 360 degrees.

In this activity, students explore solids of rotation in a hands-on way. As students visualize the connection between two- and three-dimensional objects, they are thinking abstractly and quantitatively (MP2).

Devices that can run the GeoGebra applet are needed for the digital version of this activity. If students don’t have individual access, projecting the applet would be helpful during the synthesis.

Tell students that a solid formed by rotating a two-dimensional shape using an axis of rotation is called a solid of rotation.

It will be helpful for students to practice sketching some of these three-dimensional solids on paper. Demonstrate how to sketch a sphere, a cylinder, and a cone, and ask students to practice sketching a few of each solid.

For the cylinder, draw two ellipses, one above the other. Then draw two vertical line segments connecting the left and right sides of each ellipse. For the cone, draw one ellipse and a point above it. Draw line segments connecting this point to the left and right sides of the ellipse. For the sphere, draw a circle and then draw a narrow ellipse that goes across the circle and contains the center. To show more depth, consider using dotted lines for the parts of the drawings that would be obstructed from view if you were looking at the three-dimensional solid.

In this activity, students draw the two-dimensional figure that, when rotated using the given axis of rotation, would produce the given three-dimensional solid. Monitor for students who draw a silhouette of the entire figure rather than just one half of it to highlight during discussion.

If students are using the paper and pencil version of this activity, they may struggle to draw the figure for the doughnut because it is the only solid that does not make contact with the axis of rotation. They may visualize a horizontal cross section instead of a vertical one. Ask them where the axis of rotation would be (a vertical axis through the hole in the middle of the doughnut), then to imagine slicing the doughnut in half vertically.

Ask students to share their responses and ask them to explain their reasoning. If possible, display responses that show a whole silhouette next to those that show half the silhouette. Here are some questions for discussion:

• “Which figures were easiest to draw? Which were the most difficult?” (Sample response: The barbell was difficult because the axis wasn’t horizontal or vertical.)
• “What would be the difference between rotating a two-dimensional figure that outlined the whole figure, and rotating one that outlines just half the figure?” (The half outline would need to be rotated by 360 degrees to trace out the whole solid, whereas the whole silhouette could be rotated by just 180 degrees to trace out the whole solid.)
• “Is there any two-dimensional figure that you could spin to create a rectangular prism? If so, what would it look like? If not, why not?” (There is no such figure. When a two-dimensional figure spins, it traces out circles. It can’t create a solid with a corner.)