Lesson 11

The purpose of this warm-up is to review important characteristics of prisms, pyramids, and polyhedra. Students should be able to interpret the two-dimensional pictures and three-dimensional objects, understanding that the dotted lines indicate hidden lines and identify all of the parts of the polyhedra. 

Students may think that a pyramid must have its apex over the center of the base. Students may think that a prism (or pyramid) must have base at the “bottom”.

Ask students to describe each shape. Record and display their responses for all to see. After each student shares, ask the group if they have anything to add before moving on to the next shape. 

If not mentioned by students, explain: 

• A prism is a polyhedron with two identical polygon bases, connected by rectangles.
• A pyramid is a polyhedron with one polygon base, and all other faces are triangles meeting at a point.

The goal of this activity is to help visualize cross sections of a three-dimensional object. One way to do this is to cut a solid object and use one or both of the pieces to stamp the resulting cross section onto paper. This helps students see the two-dimensional shape that results from cutting a three-dimensional object. During the launch of this activity, students see a demonstration of cutting a fruit or vegetable and are asked to describe the shape of the cross section. Students are then asked to describe the shape of a cross section of a three-dimensional object given to them in the task statement.

As students work on the task, monitor for students who can describe the two-dimensional shape produced from each cross section described.

Some students may struggle to visualize slicing the solids that are shown. It may be helpful to use a three-dimensional model of the rectangular prism and rectangular pyramid to demonstrate where the cut is happening in each question. Building the solids out of salt dough and slicing them with dental floss is another option.

Select previously identified students to describe the shapes of cross sections of the objects. Consider asking some of the following questions:

• “How do the cross sections in the different objects compare to one another?” (One is a scaled copy of the other.)
• “How do the cross sections in each object compare to its own base?” (In the cube, the cross section is the same as the base, in the pyramid, the cross section is a scaled copy of the base.)

Explain to students that in the next activity they will get another chance to determine shapes of different cross sections.

In this activity, students practice visualizing cross sections in a more abstract way by looking at images of a solid object that has been cut by a plane and matching those images to the shapes created by the cuts. The cuts made in this activity vary from the previous activity in that the cuts are not all parallel to the base of the three-dimensional object. 

As students work on the task, monitor for groups of students who use different reasons to sort their cards.

Arrange students into groups of 3. Supply each group with cards cut from the blackline master. Tell students that these cards have different things in common so different groups of students might have different reasons for grouping certain images together. Give students 2–3 minutes of quiet work time followed by a whole-class discussion.

Select previously identified groups to share their groupings and reasons for grouping them that way.

If not mentioned by students, explain that there are a few ways to sort the cards:

• Based on the solid object that is being cut. (rectangular prism, triangular prism, square-based pyramid, triangle based pyramid)
• Based on the cross section made by the cuts. (parallel to the base, perpendicular to the base, oblique to the base)
• Based on the shape of cross section. Note that there could be two or three groups for these cards: triangles, and quadrilaterals or triangles, rectangles, and trapezoids.

Explain to students that it is possible to create other cross section shapes by cutting these objects in other ways.

In this activity, students are given pictures and descriptions of planes cutting prisms and pyramids. Students are asked to draw cross sections freehand but this is not a skill that is required in order for students to be able to describe two-dimensional shapes created from cross sections, which is why this is an optional activity. Some pictures are of a moving plane. Students describe how the cross section changes as the plane moves.

If students have access to the digital activity there are applets to explore the cross sections. 

Adapted from applets created in GeoGebra by Anthony C.M. OR.

Select students to share their drawings and descriptions. Consider asking some of the following questions:

• “How did you figure out the shape of the cross section?”
• “What helped you visualize the shape?”
• “Were any of the shapes you drew here similar to the shapes you described in the previous activity?”