Account for every part of a joined solid

diagnostic. A 2 cm × 2 cm × 2 cm prism and a 2 cm × 2 cm × 3 cm prism meet along one 2 cm × 2 cm face, with no overlapping interiors. What is their combined volume? Explain whether the shared face removes any cubic centimetres.

Answer: 20 cubic cm: 8 + 12. The shared face has area 4 square cm and no thickness; it is not a duplicated 4-cubic-cm region.

Teacher response: This distinguishes adding disjoint volumes from subtracting a contact-face area as though it were volume.

model. A base prism is 6 cm long, 3 cm wide and 2 cm high. A top prism is 2 cm long, 3 cm wide and 2 cm high; it sits above the leftmost 2 cm of the base, sharing a 2 cm × 3 cm face. Their interiors do not overlap. What is the volume of the stepped solid?

1. Sketch the long base and place the shorter prism above its left end. Colour the base and top differently. Mark their shared face and state that no cube-sized region lies inside both parts.
2. Record the base's three dimensions in one row: 6 × 3 × 2 = 36 cubic cm. Record the top's own dimensions in a second row: 2 × 3 × 2 = 12 cubic cm. Its 2 cm height is additional to the base's height at that end.
3. Add the volumes, 36 + 12 = 48 cubic cm. The shared 2 cm × 3 cm face is a boundary with area 6 square cm; it has no thickness and does not duplicate a volume to subtract.
4. Check the coverage of the record: the base row covers the whole lower prism and the top row covers only the raised prism. The right-hand space above the base is empty. The outer 6 cm by 3 cm by 4 cm dimensions describe a containing box, not a solid prism filled everywhere.

Worked example: 48 cubic cm. The base contributes 6 × 3 × 2 = 36 cubic cm and the top contributes 2 × 3 × 2 = 12 cubic cm. Their non-overlapping interiors cover the solid once, so add 36 + 12.

p1. Support: prisms measuring 3 cm × 2 cm × 1 cm and 2 cm × 2 cm × 1 cm join end-to-end along 2 cm × 1 cm faces, with no overlapping interiors. Enter the total volume in cubic cm and give one record row for each part.

Answer: 10

Teacher response: The parts occupy 6 and 4 cubic cm, so together they occupy 10 cubic cm. Their contact face does not duplicate any filled region.

p2. Core: a base is 5 cm × 4 cm × 1 cm. A 2 cm × 4 cm × 2 cm prism sits above its leftmost 2 cm, with no overlapping interiors. Enter the total volume in cubic cm. State the total height at the raised end and explain which height belongs in the top part's calculation.

Answer: 36

Teacher response: The base contributes 20 cubic cm and the top contributes 16 cubic cm, giving 36. The raised end is 3 cm high, but the top part alone is 2 cm high; using 3 for that part would count some base volume again.

p3. Challenge: a plan labels region A as a 4 cm × 2 cm × 1 cm prism and region B as a 3 cm × 1 cm × 1 cm prism. These regions overlap in a 2 cm × 1 cm × 1 cm prism. B's remaining 1 cm × 1 cm × 1 cm part extends beyond A. Enter the actual solid's volume in cubic cm, then redraw a partition using all of A and only B's extension.

Answer: 9

Teacher response: A contributes 8 cubic cm. Only B's 1-cubic-cm extension is new, so the disjoint partition gives 8 + 1 = 9. Adding the original 8 and 3 would count the 2-cubic-cm overlap twice. This overlap is a filled region, unlike a contact face.

e1. Independent check: a base is 4 cm × 3 cm × 2 cm. A 2 cm × 3 cm × 1 cm top prism sits above its leftmost 2 cm, with no overlapping interiors. Enter the total volume in cubic cm. Sketch and label the two parts, give both part volumes and explain why their shared face is not subtracted.

Answer: 30

Look for:

• Identifies rectangular-prism parts whose interiors do not overlap and whose union covers the whole solid.
• Uses each part's own three dimensions and gives volume in cubic cm.
• Adds occupied volumes once each and explains that a shared face has area rather than filled volume.
• Distinguishes an empty region in a containing box from an occupied part and can repair an overlapping partition.

Teacher response: The base occupies 24 cubic cm and the top occupies 6 cubic cm, giving 30. The shared 2 cm by 3 cm face has area but no thickness, so it is not a duplicated filled region.

transfer. For the worked stepped solid, a learner multiplies its outer length 6 cm, width 3 cm and maximum height 4 cm to obtain 72 cubic cm. Identify the exact empty rectangular prism included in that calculation and use it to check the 48-cubic-cm answer.

Answer: Above the rightmost 4 cm of the base there is an empty region 4 cm long, 3 cm wide and 2 cm high. Its volume is 24 cubic cm. The containing box holds 72 cubic cm, and removing that empty region leaves 72 − 24 = 48 cubic cm, agreeing with the sum of the two occupied parts.

teach-1. 5 minutes: Give the diagnostic, including its explanation about the contact face. Use the support or ready route. Ask learners to distinguish the units for a face area from the units for a filled volume.

teach-2. 10 minutes: Sketch and colour the worked stepped solid. One partner records the base, the other records the top. Check each dimension against the description, add the rows and explain which regions are occupied or empty. Discuss why contact changes the outline without removing either part's volume.

teach-3. 15 minutes: Support learners complete p1 and the 6-plus-8 recheck. Ready learners use p2 to distinguish a top block's own height from the total height. Pairs examine p3's overlapping region labels, redraw them as two disjoint parts and write an addition that counts every occupied region once.

teach-4. 10 minutes: Allow four minutes for e1 independently with a coloured partition and volume record. Use three minutes to check cubic units and disjoint coverage, then three minutes for the transfer's containing-box check. Return double-counted work to the partition; ask successful learners to identify and calculate the exact empty prism.

misconception. Multiply the three outer dimensions to obtain 72 cubic cm, or subtract the shared-face area of 6 from the sum of the part volumes.

Teacher response: For 72, have the learner point to the empty region that the containing box includes. For subtraction of 6, ask what unit the shared face measures and whether it has thickness. Add volumes only after the coloured parts cover each occupied region once.

support. Use one colour per prism and write its dimensions in a separate row before multiplying. Point to each coloured interior while adding. Draw the common face as a line in the side view and label it a boundary, then recheck a 6-plus-8 cubic-cm example.

extend. Repartition the worked solid vertically: the left prism is 2 cm × 3 cm × 4 cm and the right prism is 4 cm × 3 cm × 2 cm. Verify 24 + 24 = 48 cubic cm and explain why this different pair of disjoint parts covers the same solid.

support-route. If the learner subtracts the shared-face area or counts only one prism, colour the two prism interiors separately and list 8 and 12 cubic cm in separate rows. Point to the contact face as a boundary, then use p1. Recheck two prisms of volumes 6 and 8 cubic cm meeting only at a face: the total is 14 cubic cm.

ready-route. If the learner gives 20 cubic cm and explains that the interiors do not overlap, move to the stepped solid. Ask for one row per prism and a statement that the parts cover the solid exactly once.