Add volumes of non-overlapping rectangular prisms

Learning goal: Decompose a solid into non-overlapping right rectangular prisms and add their volumes. Explain why the parts cover the entire solid exactly once. A shared face does not add thickness or extra volume.

Scope: This lesson teaches 5.MD.5c and applies the formulas of 5.MD.5b to each part. Packing single prisms with unit cubes and the associative-property link (5.MD.5a) are taught in the companion lesson on unit cubes and prism volume.

Build a low block measuring four units long, three units wide and two units high. Its volume is 4 × 3 × 2 = 24 cubic units. Place a smaller block on top of its left half. The upper block is two units long, three units wide and one unit high, so its volume is 2 × 3 × 1 = 6 cubic units. The completed stepped solid contains 24 + 6 = 30 cubic units. With V = B × h, the low block has base area 4 × 3 = 12 square units and height 2, so 12 × 2 = 24; the upper block has base area 2 × 3 = 6 and height 1, so 6 × 1 = 6.

The blocks touch along a face but do not occupy the same space. Every unit cube belongs either to the low block or to the upper block. That is why adding the volumes counts the full solid exactly once. If one described part included the upper space already counted in another part, simply adding would count some space twice.

A layer model makes the structure clear. The bottom two layers each contain twelve cube positions. The third layer contains only six positions over the left half. The total is 12 + 12 + 6 = 30. The missing six positions above the right half are empty space, not hidden cubes. A rectangular outline around the whole shape does not turn that empty region into part of the solid.

A different decomposition gives the same result. Cut the stepped solid vertically between the left and right halves. The left prism is 2 × 3 × 3 with volume 18. The right prism is 2 × 3 × 2 with volume 12. Adding gives 18 + 12 = 30. Both decompositions are valid because their parts have no shared interior and together fill the original solid.

You can also imagine the surrounding full prism with dimensions 4 × 3 × 3. It would contain 36 cubic units. The missing upper-right part measures 2 × 3 × 1, or 6 cubic units. Subtracting that missing space gives 36 − 6 = 30. This is a check, not permission to count empty space as though it were present.

Keep dimensions attached to the correct part. The height three applies only on the taller left side; the entire four-by-three base is not three units high. Multiplying the largest length, width and height without accounting for the step would overcount. Similarly, using the smaller height for the whole footprint would omit the six upper cubes.

The context determines what volume describes. A solid model built from cubes has material volume thirty cubic units. A hollow storage container with the same outside dimensions would not necessarily hold thirty cubic units because walls, gaps and interior dimensions matter. This lesson's models are solid prisms with stated dimensions; use an appropriate model before applying a real-world formula.

For another composite solid, join prisms of dimensions 3 × 2 × 4 and 2 × 2 × 4 along a full two-by-four face. Their volumes are 24 and 16, and together they make a 5 × 2 × 4 prism with volume 40. The shared face contributes no duplicate cubic layer, so addition agrees with the single-prism product.

Worked decomposition: The stepped model's horizontal split gives 24 + 6, while its vertical split gives 18 + 12. Name the dimensions of all four parts and explain why each pair covers the same thirty cube positions.

Worked error: A student gives 36 because the maximum dimensions are 4, 3 and 3. Draw the upper-right empty region and count its six missing positions. The correction 36 − 6 = 30 identifies exactly what the first method counted incorrectly.

Worked real-world problem: A fictional solid concrete platform has a lower block 10 feet long, 4 feet wide and 2 feet high. A step block 10 feet long, 2 feet wide and 1 foot high rests on top of it. The two blocks do not overlap, so the concrete needed is 10 × 4 × 2 + 10 × 2 × 1 = 80 + 20 = 100 cubic feet. This is an invented arithmetic example, not a building plan.

Guided investigation: Build a stepped solid from cubes, sketch its layers and separate it into two rectangular prisms. Reassemble it, then choose a different separation. Check that the total stays constant.

Independent performance: Design a composite solid from two non-overlapping prisms, label all dimensions and compute each part's volume. Explain how you know no space is omitted or counted twice. Have a partner verify by layer counting or another decomposition. Retain the model observations and calculations; a formula alone does not establish a valid decomposition.