Exposed Faces Count

Overview

In this fourth lesson of the eight-lesson unit, students apply prior learning about surface area of prisms to composite solids. They decompose prism-based objects, identify exposed and hidden faces, and calculate total exterior surface area by adding component areas and subtracting shared faces twice.

Learning intentions

Students will:

• Decompose a composite solid into rectangular prisms.
• Create a complete, labelled face inventory.
• Distinguish exposed faces from hidden or shared faces.
• Calculate exterior surface area using an efficient and justified method.
• Explain and correct common surface-area errors.

Success criteria

• I can separate a composite solid into recognizable prisms.
• I can label every face and identify whether it is exposed, shared, or hidden.
• I can calculate the area of each exposed face using correct units.
• I can explain why a shared face is subtracted twice.

Curriculum links

• Shape and Space — determine and apply surface-area strategies for three-dimensional objects.
• Shape and Space — analyse and solve problems involving rectangular prisms and composite objects.
• Mathematical processes — communicate mathematical thinking using diagrams, labels, calculations, and explanations.
• Mathematical processes — use reasoning, connections, and error analysis to verify solutions.

Lesson structure (30 minutes)

1. 0–4 min · Hook and retrieval. Display a composite solid made from two rectangular prisms using the hook and retrieval slides. Ask: “If two boxes are joined, do we still count every face?” Students sketch the object, identify a likely hidden face, and share one surface-area rule from the previous lesson.

2. 4–10 min · Model the strategy. Use the decomposition and worked-example slides to model a stepped solid with dimensions 6 cm by 4 cm by 3 cm joined to a prism measuring 4 cm by 2 cm by 2 cm. Think aloud while drawing the decomposition, naming each rectangular face, and recording a face inventory. Emphasize that the shared rectangle is counted once in each component’s surface area, so it must be subtracted twice:

• Surface area of prism A + surface area of prism B − 2(shared area). Students annotate the example and explain to a partner why the shared face is not exposed.

3. 10–13 min · Guided check. Reveal a second composite solid on the guided-check slide. Students independently decide which faces are exposed and hold up or point to their selected labels. Invite students to justify one decision; correct misconceptions before independent work.

4. 13–23 min · Error analysis and practice. Distribute the composite surface-area error-analysis worksheet. Students work in pairs, first completing a labelled face inventory for each problem, then solving and analysing two sample responses: one that double-counts the shared face and one that subtracts it only once. Partners must identify the error, calculate the corrected answer, and write a short explanation. Circulate and check that inventories are completed before calculations.

5. 23–27 min · Share and consolidate. Use the error-analysis discussion slides to display selected incorrect solutions. Students compare methods and respond to: “When is adding component surface areas useful?” and “Why is subtracting twice the shared area equivalent to removing both hidden faces?” Record the general rule on the board.

6. 27–30 min · Exit ticket. Show the final composite solid on the exit-ticket slide and ask students to complete the labelled face inventory before finding its exterior surface area. Students must show dimensions, face areas, units squared, and one sentence explaining how shared faces were handled. Collect the worksheet and exit ticket as they leave.

Resources

• the composite surface-area teaching deck
• the composite surface-area error-analysis worksheet
• Whiteboard and markers
• Calculators, if routinely used in the class
• Rulers
• Graph paper or squared paper
• Pencils and coloured pencils
• Exit-ticket response slips
• Document camera or interactive display

Assessment

• During modelling and the guided check, listen for accurate use of exposed, hidden, and shared-face language; ask students to point to evidence on the diagram.
• Review pair work for a complete face inventory, correct identification of shared faces, accurate subtraction of twice the shared area, and square units.
• Use the exit ticket to assess whether each student can independently inventory faces and justify the treatment of shared faces. Group students for the next lesson according to whether the difficulty was decomposition, area calculation, or reasoning.

Differentiation

• Provide a partially labelled diagram, colour-code each component prism, and offer the sentence frame: “The face is shared because ___, so I ___ it ___ times.”
• Allow students who need support to use a formula reference, graph paper, a calculator, and a teacher- or peer-created face inventory before completing the calculations.
• For EAL learners, display and repeatedly use the terms exposed, hidden, shared, component, and exterior, supported by labelled diagrams and gestures.
• Extend ready students by asking them to solve the same solid using both the component-surface-area method and a direct exposed-face inventory, then compare efficiency and possible sources of error.