Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
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.
Launch
Arrange students in groups of 2–4. Display the images for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three figures that go together and can explain why. Next, tell students to share their response with their group and then together to find as many sets of three as they can.
Activity
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Student Task Statement
Which three go together? Why do they go together?
A
B
C
D
Activity Synthesis
Invite each group to share one reason why a particular set of three goes together. 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 three go together, attend to students’ explanations, and ensure the reasons given are correct.
During the discussion, ask students to explain the meaning of any geometric terminology they use, such as “round,” “corners,” “circular,” or “symmetric,” and to clarify their reasoning as needed. Consider asking:
“How do you know _____?”
“What do you mean by _____?”
“Can you say that in another way?”
Focus students’ attention on the concept of complete rotational symmetry in three dimensions. If it does not come up during the discussion, introduce the idea. Invite students to 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 were 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 rotated a full 360 degrees.
In this lesson, students made solids of rotation, and identified two-dimensional figures that create particular solids when rotated, using a given axis of rotation. Here are some questions for discussion:
“What is the difference between a rotation in two dimensions and a rotation in three dimensions?” (In two dimensions, points rotate in a circle around a central point. In three dimensions, points rotate in a circle around an axis in a plane perpendicular to the axis.)
“Where do you see solids of rotation in the world?” (Sample response: The axle of a car is the axis of rotation for the tires. Earth spins on an axis of rotation. Wet clay spins on a board in pottery making. Basketballs spin around an axis as they travel toward the hoop.)
To help students consider the effects of the axis of rotation, display these images. Ask students to sketch and describe the different solids that would result from rotating the same two-dimensional figure using each of the given axes of rotation.
Student Lesson Summary
Imagine a coin spinning on its edge. If the coin moves fast enough and stays vertical it would look like a sphere.
This triangular flag is made of metal. What shape will it make if the pole is spun quickly?
When the flag rotates around the pole, we can see a cone.
In both pictures, the object spins around a line called the axis of rotation, and that spinning creates a solid of rotation. When the flag spins around a different axis of rotation, it would make a different solid of rotation. This is what it looks like when the flag is rotated using a different axis of rotation.
A machine called a lathe cuts away at a rotating block of material. For example, a lathe could be used to make a decorative sculpture shaped like the figure on the right from a block of wood. The image on the left shows the cross-sectional shape that would remain after the lathe carved away part of the spinning block of wood. If we rotate this two-dimensional shape using the vertical axis shown, it produces the sculpture shape.
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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).
In the digital version of the activity, students use an applet to check their predictions for the solids that result from rotating figures around different axes. The applet allows students to observe solids get traced out and to see the solids from different perspectives. Use the digital version if students would benefit from using a dynamic visual to see the solids traced out by a shape.
If students don’t have individual access, projecting the applet would be helpful during the Activity Synthesis.
Launch
Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 3–4. Distribute a different shape to each student in a group. Ask students to look at the figures they received and to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their group the things they notice and wonder. (There are different shapes. Some are curved. They all have one straight edge. What are we doing with them? Why these shapes?)
Explain that students will tape an edge of their shape to a pencil, and then quickly roll the pencil between their fingers. The pencil will act as an axis of rotation. Consider using honeycomb paper decorations, such as the one in these images, to demonstrate this concept.
Activity
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Student Task Statement
Your teacher will give you a shape. Tape one side of the shape to a pencil.
Spin the pencil between your hands. What solid is traced out as you rotate the shape? Draw the solid.
Predict what solids will be formed by the shapes of other members of your group. Confirm by asking them to rotate their shapes.
Activity Synthesis
The goal of this discussion is to make sure students understand the two-dimensional figures that are visible on a solid of rotation. Tell students that a solid formed by rotating a two-dimensional shape using an axis of rotation is called a solid of rotation.
Display the applet and check only box 1. Drag the slider to 360 degrees and explain that the triangle has now completed a full rotation around the y-axis, the axis of rotation for this solid. It will be helpful for students to practice sketching some of these three-dimensional solids on paper. Rotate the solid and ask students to identify the shapes they see (circle, ellipse, right triangle, acute triangle).
Demonstrate how to sketch a cone, a cylinder, and a sphere, and ask students to practice sketching a few of each solid.
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 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 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.
Representation: Access for Perception. Provide access to physical three-dimensional models of spheres, cylinders, and cones for students to view or manipulate. Ask students to identify correspondences between the three-dimensional models and their sketches. Supports accessibility for: Visual-Spatial Processing
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Activity Narrative
The purpose of this activity is for students to practice visualizing the two-dimensional figure that was rotated to make a three-dimensional shape.
In this activity, students draw the two-dimensional figure that, when rotated around 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.
In the digital version of the activity, students use an applet to create their own three-dimensional solids. The applet allows students to view a created solid from different perspectives before drawing its original two-dimensional figure. If time allows, the digital version may be preferable because students create the three-dimensional shapes themselves.
Launch
Engagement: Internalize Self-Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to choose and draw two-dimensional figures for three out of the four three-dimensional solids. Supports accessibility for: Organization, Attention
Activity
None
Student Task Statement
Draw the two-dimensional shape that, when rotated around the given axis of rotation, produces each solid of rotation. Ignore any non-symmetric aspects of the solid.
Activity Synthesis
The goal is to make sure students understand that solids of rotation are created by rotating a two-dimensional figure 360 degrees around an axis of rotation. 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.)
MLR8 Discussion Supports. For each response that is shared, invite students to turn to a partner and restate what they heard, using precise mathematical language. Advances: Listening, Speaking
Standards Alignment
Building On
Addressing
HSG-GMD.B.4
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
If students are using the paper-and-pencil version of this activity, they may struggle to draw the figure for the bagel 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 bagel), then to imagine slicing the bagel in half vertically.
