In this lesson, students visualize solids of rotation created from rotating two-dimensional figures using an axis of rotation. These visualization skills will be useful in later lessons when students need to analyze two-dimensional figures that are cross sections of three-dimensional figures. For example, sometimes it is necessary to consider vertical cross sections of cones or pyramids to solve problems involving volume. In future courses, these visualization skills will be applicable when using calculus to compute volumes or when using linear algebra in three or more dimensions.
As students describe characteristics of solids, they have the opportunity to use language precisely (MP6). When students visualize the relationship between two- and three-dimensional objects, they are thinking abstractly (MP2).
- Determine the two-dimensional shape that when rotated using a given axis generates a particular three-dimensional solid.
- Identify and describe (using words and other representations) the three-dimensional solid created by rotating a two-dimensional figure using a linear axis.
- Let’s rotate two-dimensional shapes to make three-dimensional shapes.
Devices are required for the digital versions of the activity Axis of Rotation and From Three Dimensions to Two. If using the digital versions, acquire devices that can run the GeoGebra applet, one for every 2-3 students.
If using the paper and pencil version of the Axis of Rotation activity, cut out a set of shapes from card stock for each group of 3–4 students, with enough for at least 1 shape per student. Shapes can include rectangles, circles, and various kinds of triangles.
If desired, find some honeycomb paper decorations to display to students to help them visualize solids of rotation in the Axis of Rotation activity.
- I can draw the two-dimensional shape that creates a particular three-dimensional solid when rotated using a given axis.
- I can identify the three-dimensional solid created by rotating a two-dimensional figure using a linear axis.
axis of rotation
A line about which a two-dimensional figure is rotated to produce a three-dimensional figure, called a solid of rotation. The dashed line is the axis of rotation for the solid of rotation formed by rotating the green triangle.
solid of rotation
A three-dimensional figure formed by rotating a two-dimensional figure using a line called the axis of rotation.
The axis of rotation is the dashed line. The green triangle is rotated about the axis of rotation line to form a solid of rotation.