Parallel Cross-sections of Three-dimensional Objects

Learning goal

Students visualise and construct slices parallel to a base, identify when cross-sections remain congruent and compare prisms, cylinders, pyramids and cones.

Success criteria

• I can represent or identify the concept.
• I can explain the underlying relationship.
• I can select an appropriate strategy or feature.
• I can apply it in a new context.
• I can justify and verify the response.

Teaching routine

1. Represent
2. Reason
3. Calculate
4. Interpret
5. Verify

Slice a prism parallel to its base

Use the visual model first. Ask students to identify the quantities, structure or conditions before calculating or explaining.

Compare constant and changing cross-sections

Connect the central relationship to a new context, then verify the conclusion with a second representation, estimate, inverse operation or reasonableness check.

AC9M6SP01: compare the parallel cross-sections of objects and recognise their relationships to right prisms

• E1: using objects made of foam or polystyrene, slice along different cross-sections, and record the different shapes of faces that result; comparing cross-sections of different objects
• E2: using different pieces of fruit, slicing across different cross-sections, drawing the cross-section; reporting back to the class the results of the investigation
• E3: observing and drawing the shapes resulting from different ways of slicing through familiar objects; for example, slicing carrots at different angles or cutting through playdough models of objects; using playdough models, fruit or similar to establish which objects can be cut in such a way that the cross-section will always be the same shape
• E4: understanding that right prisms are objects where parallel cross-sections to the base of the prism are the same shape and size
• E5: connecting different right prisms to the shape of their parallel cross-sections, such as a triangular prism which can be described as a stack of the same sized triangles, and a cube or square prism, which can be described as a stack of the same sized squares
• E6: investigating the design of First Nations Australians’ dwellings, exploring the relationship between the cross-sections and the dwellings’ construction

Use the central and application models above to connect each elaboration to the same underlying concept.

Check understanding

• Predict a prism slice.
• Compare prism/pyramid.
• Identify a cone section.
• Explain parallel.
• Describe a non-parallel cut.

Evidence of mastery

• Represent or identify the concept
• Explain the underlying relationship
• Select an appropriate strategy or feature
• Apply it in a new context
• Justify and verify the response

Decision: continue when students can explain the model, apply it to a new example and justify their check. Otherwise return to the central model and reduce the numerical or representational load.

Australian Curriculum v9.0

AC9M6SP01: recognise and describe the parallel cross-sections of objects and recognise their relationships to the object’s base and shape

The Australian Curriculum code and wording are exact. International teachers can use the underlying mathematical concept while matching the lesson to their local grade or year outcomes.

• Australian Curriculum Version 9.0
• NSW Curriculum
• Victorian Curriculum F–10