Unit Plan 16 (Grade 7 Math): Slicing Solids—Cross-Sections
7th graders explore cross-sections of right rectangular prisms and pyramids by slicing with planes in different orientations. They predict, construct, and justify resulting 2D shapes—rectangles, parallelograms, triangles, or trapezoids—using precise geometric reasoning and vocabulary.
Focus: Describe two-dimensional figures resulting from slicing right rectangular prisms/pyramids.
Grade Level: 7
Subject Area: Mathematics (Geometry • Spatial Reasoning)
Total Unit Duration: 5 sessions (one week), 45–60 minutes per session
I. Introduction
Students investigate how planes slice three-dimensional solids and what 2D shapes appear as the cross-sections. Using hands-on models and precise language, they describe cross-sections of right rectangular prisms and right rectangular pyramids for planes that are parallel, perpendicular, or oblique to faces/bases. Emphasis: naming the resulting polygons, identifying when shapes are similar to the base, and justifying claims with geometric reasoning (edges/angles parallelism, triangle inequality isn’t needed here, but parallel and perpendicular relationships are).
Essential Questions
- How does the orientation of a slicing plane (parallel, perpendicular, oblique) determine the 2D cross-section you see?
- When is a cross-section congruent or similar to the base, and why?
- What clear evidence (edges cut, faces intersected, parallel lines) supports your description of the cross-section?
II. Objectives and Standards
Learning Objectives — Students will be able to…
- Use precise vocabulary (plane, cross-section, parallel, perpendicular, oblique) to describe slices of right rectangular prisms and pyramids.
- Predict and justify the shape of a cross-section given the plane’s orientation (for example, rectangles, squares, parallelograms in prisms; rectangles similar to the base, triangles, trapezoids in pyramids).
- Explain why a plane parallel to a base of a prism yields a congruent rectangle, and why a plane parallel to a pyramid’s base yields a similar rectangle (scaled).
- Construct or model physical/drawn cross-sections and label key features (edges intersected, angle relationships).
- Communicate reasoning clearly with diagrams and written justifications.
Standards Alignment — CCSS Grade 7
- 7.G.3: Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
- Mathematical Practices emphasized: MP.1 (make sense), MP.3 (justify), MP.5 (use tools strategically), MP.6 (precision), MP.7 (look for structure).
Success Criteria — Student Language
- I can identify the plane’s orientation and name the 2D shape that appears.
- I can explain why a parallel slice to a prism’s base gives a congruent rectangle, and to a pyramid’s base gives a similar rectangle.
- I can recognize when a prism slice makes a parallelogram (oblique to faces) or when a pyramid slice makes a triangle/trapezoid.
- I can label a diagram that proves my description (which faces/edges the plane crosses).
- I can write a brief justification connecting orientation to the resulting polygon.