Composite Object Surface Area

Overview

Students learn to find the total exposed surface area of a composite object by decomposing it into familiar solids and excluding faces hidden where components join. This lesson builds on calculating the surface area of rectangular prisms and cubes and prepares students to design a themed composite object in the final unit project.

Learning intentions

Students will:

• Decompose a composite object into familiar rectangular prisms or cubes.
• Identify faces that become hidden when components are joined.
• Calculate total exposed surface area using appropriate dimensions, units, and operations.
• Explain why joined faces are subtracted twice.

Success criteria

• I can split a composite object into familiar components.
• I can mark the faces that are hidden at a join.
• I can calculate each component’s surface area and combine the results accurately.
• I can explain why each shared face is removed twice.

Curriculum links

• Shape and Space — determine and apply surface-area strategies for three-dimensional objects.
• Measurement — select and use appropriate units and dimensions in real-life problems.
• Mathematical reasoning and problem solving — decompose, represent, calculate, verify, and explain a solution.
• Communication and connections — use diagrams, mathematical language, and models to communicate thinking.

Lesson structure (30 minutes)

1. 0–3 min · Warm-up: What disappears? Teacher displays two joined blocks using the hook and warm-up slides and asks, “Which faces disappear when two blocks are joined?” Students observe a physical model or diagram, turn and talk, then point to faces that are no longer exposed. Emphasize that hidden faces are still part of each original component but cannot be seen on the finished object.

2. 3–10 min · Mini-lesson: Build the method. Teacher uses the composite-surface-area method slides to model the sequence: split the object into familiar components; calculate each component’s surface area; identify the joined faces; subtract hidden faces; label the final answer in square units. Record this general rule:

Total exposed surface area = sum of component surface areas − hidden joined faces.

Explain that if two equal rectangular faces meet, both copies were included in the separate surface-area totals, so both must be subtracted. Use the student note: DOODLE 🧱➕🧱: “Count what you can see!” Students copy the rule, annotate a simple diagram, and identify which measurements belong to each component. Address common mistakes: adding volumes, counting internal faces, mixing dimensions, or omitting square units.

3. 10–18 min · Guided practice: Worked example. Teacher displays the example in the guided-example slides and solves it with student input:

A 4 × 3 × 2 cm rectangular prism is joined to a 3 × 3 × 3 cm cube along a 3 × 2 cm rectangular face.

• Prism surface area: (2(4×3 + 4×2 + 3×2) = 52\text{ cm}^2)
• Cube surface area: (6(3×3) = 54\text{ cm}^2)
• Shared face: (3×2 = 6\text{ cm}^2)
• Total exposed surface area: (52 + 54 - 2(6) = 94\text{ cm}^2)

Students calculate alongside the teacher, then justify why (2(6)), rather than (6), is subtracted. Check understanding by asking students to hold up one finger for “subtract once” or two fingers for “subtract twice,” followed by a brief explanation.

4. 18–26 min · Collaborative task: Worksheet F. Teacher places students in groups of three or four and distributes Worksheet F — Composite Objects. Assign roles such as reader/organizer, model builder, calculator/checker, and reporter. Students use linking cubes or the provided diagrams to decompose each object, list the components, mark or shade hidden faces, calculate component surface areas, and determine total exposed surface area. Circulate and ask: “Which faces are included in both component totals?” and “How do you know no hidden join has been counted?” Each group must be ready to point to one excluded face.

5. 26–28 min · Formative check and project briefing. Teacher asks each group to point to one face excluded from its calculation and explain why it is hidden. Use the project-briefing slides to introduce the next design task: students will create a themed composite object from two to four paper-card components, such as a robot, tiny building, animal, or game piece. Students record one possible theme and identify the familiar solids they might combine.

6. 28–30 min · Closure: Exit ticket. Teacher displays the final prompt on the closure and exit-ticket slide: “Why subtract joined faces twice?” Students write: “Each component’s surface area counted the shared face once, but it is not exposed in the composite.” Collect responses and note whether students can distinguish exposed from internal faces.

Resources

• the Composite Objects teaching deck
• Worksheet F — Composite Objects
• Linking cubes or rectangular-prism and cube models
• Coloured pencils or highlighters for hidden faces
• Calculators
• Rulers and board space for worked calculations
• Paper-card components or sample design materials for the project briefing

Assessment

• During the warm-up and guided example, listen for accurate identification of faces that disappear and ask students to justify subtracting both copies of a shared face.
• During group work, check diagrams, component lists, dimensions, operations, units, and whether internal faces have been excluded.
• Use the exit-ticket explanation to identify students who still subtract a shared face only once or confuse surface area with volume.

Differentiation

• Support students with physical blocks, colour-coded hidden faces, a surface-area formula reference, and a partially completed decomposition diagram.
• Provide sentence starters: “The shared face is ___ because…”, “I subtract it twice because…”, and “The final exposed surface area is…”.
• Pair students strategically and allow verbal explanations, labelled diagrams, or calculator use while maintaining the requirement to show the method.
• Extend confident students by asking them to create two valid decompositions of the same composite object and compare whether both methods produce the same total surface area.