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Cuboid Volume by Unit Cubes: Analyze Surface Area Change

Cuboid Volume by Unit Cubes: Analyze Surface Area Change
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2023-04-05 15:32
Solid GeometryGeometric ConstructionSlider Animation
In collections Quader und Würfel - Volumen
This GeoGebra lesson demonstrates how to find the volume of a cuboid by decomposing it into suitable unit cubes, and then explores what happens to surface area when edge lengths change. Key ideas include cuboid (rectangular prism) volume, verifying volume via counting unit cubes, applying and explaining the volume formula, and investigating the effect of length variations on surface area. Construction and interaction are organized by building multiple “box” solids and reference points, then using geometry-programming commands to drive labels and dynamic steps: Midpoint (compute key points), Shuffle (vary selections), Join (combine elements), If (switch logic based on conditions), Polygon (create auxiliary outlines), Textfield (let students input parameters), Element (reference existing objects), Append/Remove (manage dynamic creation/deletion), and Segment (show measuring edges). With objects such as Box1–Box6 and MPBox1–MPBox6 plus layout helpers (LayoutHelper/LayoutList/FIxedPos), the model supports guided exploration for grades 5–6 and early solid-geometry lessons.

GeoGebra Commands & Parameters

  1. Midpoint(Corner[Box1, 2], Corner[Box1, 1]) → MPBox1
  2. Midpoint(Corner[Box4, 1], Corner[Box4, 2]) → MPBox4
  3. Midpoint(Corner[Box3, 1], Corner[Box3, 2]) → MPBox3
  4. Midpoint(Corner[Box2, 2], Corner[Box2, 1]) → MPBox2
  5. Shuffle(1…5) → LayoutHelper, {4, 5, 3, 1, 2}
  6. Join({{4}, LayoutHelper \ {4}}) → LayoutList
  7. If(layout ≟ 1, MPBox2, If[layout ≟ 2, MPBox3, If[layout ≟ 3 ∨ layout ≟ 4 ∨ layout ≟ 5, MPBox4]]) → MP1
  8. Midpoint(Corner[Box5, 1], Corner[Box5, 2]) → MPBox5
  9. If(layout ≟ 1, MPBox4, If[layout ≟ 2, MPBox5, If[layout ≟ 3, MPBox2, If[layout ≟ 4, MPBox2, If[layout ≟ 5, MPBox1]]]]) → MP3
  10. Midpoint(Corner[Box6, 1], Corner[Box6, 2]) → MPBox6
  11. If(layout ≟ 1, MPBox3, If[layout ≟ 2, MPBox2, If[layout ≟ 3, MPBox6, If[layout ≟ 4, MPBox5, If[layout ≟ 5, MPBox2]]]]) → MP2
  12. Polygon(M, I, O, N) → VRegal, m, i, o, n
  13. If(layout ≟ 1, 40, If[layout ≟ 2, 15, If[layout ≟ 3, 50, If[layout ≟ 4, 3, If[layout ≟ 5, 30]]]]) → B1E1
  14. If(layout ≟ 1, 40, If[layout ≟ 2, 10, If[layout ≟ 3, 50, If[layout ≟ 4, 4, If[layout ≟ 5, 40]]]]) → B1E2
  15. If(layout ≟ 1, 40, If[layout ≟ 2, 10, If[layout ≟ 3, 20, If[layout ≟ 4, 7, If[layout ≟ 5, 45]]]]) → B1E3
  16. If(layout ≟ 1, 30, If[layout ≟ 2, 10, If[layout ≟ 3, 30, If[layout ≟ 4, 5, If[layout ≟ 5, 30]]]]) → B2E1
  17. If(layout ≟ 1, 40, If[layout ≟ 2, 10, If[layout ≟ 3, 50, If[layout ≟ 4, 6, If[layout ≟ 5, 30]]]]) → B2E2
  18. If(layout ≟ 1, 20, If[layout ≟ 2, 10, If[layout ≟ 3, 20, If[layout ≟ 4, 9, If[layout ≟ 5, 30]]]]) → B2E3
  19. If(layout ≟ 1, 14, If[layout ≟ 2, 10, If[layout ≟ 3, 30, If[layout ≟ 4, 4, If[layout ≟ 5, 60]]]]) → B3E1
  20. If(layout ≟ 1, 60, If[layout ≟ 2, 10, If[layout ≟ 3, 30, If[layout ≟ 4, 4, If[layout ≟ 5, 30]]]]) → B3E2
  21. If(layout ≟ 1, 40, If[layout ≟ 2, 30, If[layout ≟ 3, 30, If[layout ≟ 4, 4, If[layout ≟ 5, 18]]]]) → B3E3
  22. Textfield(Input1) → InputBox1
  23. Textfield(Input2) → InputBox2
  24. Polygon(PtopLeftCornerFeedback, PtopRightCornerFeedback, PbottomRightCornerFeedback, PbottomLeftCornerFeedback) → rectangleCorrect, ptopleftcornerfeedback, ptoprightcornerfeedback, pbottomrightcornerfeedback, pbottomleftcornerfeedback
  25. Shuffle(Sequence[Random[0, 10], i, 1, 100]) → randomNosForX, {6, 3, 4, 1, 3, 8, 0, 8, 3, 0, 7, 7, 8, 10, 4, 7, 9, 1, 1, 8, 4, 7, 4, 1, 9, 1, 0, 1, 7, 8, 0, 3, 9, 5, 8, 9, 3, 10, 5, 3, 1, 1, 3, 2, 8, 8, 2, 5, 4, 9, 9, 2, 5, 2, 3, 7, 6, 6, 7, 6, 3, 2, 9, 1, 3, 8, 0, 2, 9, 0, 3, 5, 4, 3, 0, 3, 6, 9, 2, 1, 0, 1, 2, 0, 1, 5, 3, 4, 9, 9, 6, 8, 7, 9, 7, 7, 5, 1, 9, 4}
  26. Element(randomNosForX, Mod[questionCount, 100] + 1) → numX
  27. Shuffle(Sequence[Random[0, 10], i, 1, 100]) → randomNosForY, {5, 0, 5, 0, 2, 9, 10, 9, 6, 2, 0, 10, 10, 9, 7, 4, 0, 10, 10, 9, 10, 3, 1, 7, 6, 1, 6, 4, 3, 9, 7, 9, 4, 3, 2, 8, 10, 8, 9, 1, 8, 4, 4, 7, 7, 9, 6, 5, 0, 8, 4, 7, 2, 3, 3, 4, 1, 4, 1, 9, 10, 6, 1, 4, 6, 4, 2, 2, 6, 0, 7, 8, 2, 10, 10, 7, 6, 5, 4, 4, 4, 7, 1, 6, 9, 0, 3, 3, 2, 6, 8, 4, 5, 4, 6, 7, 7, 7, 6, 1}
  28. Element(randomNosForY, Mod[questionCount, 100] + 1) → numY
  29. Append(true, Sequence[If[randomNosForX(i) ≟ randomNosForX(i - 1), false, true], i, 2, 100]) → stayX
  30. Append(true, Sequence[If[randomNosForY(i) ≟ randomNosForY(i - 1), false, true], i, 2, 100]) → stayY
  31. Remove(Sequence[If[stayX(i), randomNosForX(i), 0], i, 1, 100], Sequence[0, i, 1, 100]) → randomNosForXwithoutRepeat
  32. Remove(Sequence[If[stayY(i), randomNosForY(i), 0], i, 1, 100], Sequence[0, i, 1, 100]) → randomNosForYwithoutRepeat
  33. Element(randomNosForXwithoutRepeat, Mod[questionCount, Length[randomNosForXwithoutRepeat]] + 1) → numXwithoutRepeat
  34. Element(randomNosForYwithoutRepeat, Mod[questionCount, Length[randomNosForYwithoutRepeat]] + 1) → numYwithoutRepeat
  35. If(layout ≟ 1, 30, If[layout ≟ 2, 10, If[layout ≟ 3, 30, If[layout ≟ 4, 5, If[layout ≟ 5, 30]]]]) → B2E1_{1}
  36. If(layout ≟ 1, 60, If[layout ≟ 2, 10, If[layout ≟ 3, 30, If[layout ≟ 4, 4, If[layout ≟ 5, 30]]]]) → B3E2_{1}
  37. If(layout ≟ 1, 40, If[layout ≟ 2, 50, If[layout ≟ 3, 30, If[layout ≟ 4, 4, If[layout ≟ 5, 18]]]]) → B3E3_{1}
  38. Textfield(Input3) → InputBox3
  39. Segment(MP1, J) → f
  40. Segment(MP2, J_{1}) → f_{1}
  41. Segment(MP3, J_{2}) → f_{2}
  42. Segment((4, 2), (4, 4)) → vRegal1
  43. Segment((7, 4), (7, 2)) → vRegal2
  44. Polygon(M, M_2, L_2, I) → VRegal1, g, h, j, k
  45. Polygon(L_2, M_2, O_2, N_2) → VRegal2, l, p, q, r
  46. Polygon(N_2, O_2, N, O) → VRegal3, s, t, a, b
  47. Polygon(PtopLeftCornerFeedback_{1}, PtopRightCornerFeedback_{1}, PbottomRightCornerFeedback_{1}, PbottomLeftCornerFeedback_{1}) → rectangleCorrect_{1}, ptopleftcornerfeedback_{1}, ptoprightcornerfeedback_{1}, pbottomrightcornerfeedback_{1}, pbottomleftcornerfeedback_{1}
  48. Polygon(PtopLeftCornerFeedback_{2}, PtopRightCornerFeedback_{2}, PbottomRightCornerFeedback_{2}, PbottomLeftCornerFeedback_{2}) → rectangleCorrect_{2}, ptopleftcornerfeedback_{2}, ptoprightcornerfeedback_{2}, pbottomrightcornerfeedback_{2}, pbottomleftcornerfeedback_{2}
  49. If(Step ≟ 2, 2, Step ≟ 3, 1) → attemptCountHelper

AI Drawing Prompt Reference

Draw a cuboid volume analysis graph, including six boxes and six midpoint points, the relationship and size change between boxes, the change of volume and surface area, including some auxiliary lines and labels, the graph contains some random numbers and dynamic elements, the graph should be clear, accurate, and easy to understand, including some interactive elements, allowing users to input parameters and view results, the graph contains some auxiliary lines and labels to help understand and analyze.
The prompt above was compiled from the original drawing commands and is for reference only.