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Unit 08 · Area Volume and Surface AreaLesson

Day 114Packing Fractional Unit Cubes

LearnClassworkAnswer keyPDF packet ↗
Standards 6.G.A.2
LEARNING TARGET

I can connect fractional unit cubes with a prism volume formula.

01 · LEARN

Build the idea

For cubes with edge 1/n, each cube has volume 1/n^3 cubic unit. Divide each prism edge by 1/n to count cubes along it. Multiply the counts, then multiply by the volume of one small cube.

WORKED EXAMPLE 1

Fill a prism 5/2 × 2 × 1 units with cubes of edge 1/2 unit. How many cubes fit? Use their volumes to verify lwh.

40 cubes; each 1/8 cubic unit; prism 5 cubic units

Counts along edges: 5, 4, 2. Product 40; 40 × 1/8 = 5, also lwh.

WORKED EXAMPLE 2

Fill a prism 5/3 × 7/3 × 5/3 units with cubes of edge 1/3 unit. How many cubes fit? Use their volumes to verify lwh.

175 cubes; each 1/27 cubic unit; prism 175/27 cubic units

Counts along edges: 5, 7, 5. Product 175; 175 × 1/27 = 175/27, also lwh.

02 · PRACTICE

Your turn

Work in order. Start with the foundations, then build toward explanation and application.

  1. Fill a prism 3 × 7/2 × 3/2 units with cubes of edge 1/2 unit. How many cubes fit? Use their volumes to verify lwh.

  2. Fill a prism 4 × 1 × 5/2 units with cubes of edge 1/2 unit. How many cubes fit? Use their volumes to verify lwh.

  3. Fill a prism 4 × 3/2 × 3/2 units with cubes of edge 1/2 unit. How many cubes fit? Use their volumes to verify lwh.

  4. Fill a prism 4 × 7/2 × 1 units with cubes of edge 1/2 unit. How many cubes fit? Use their volumes to verify lwh.

  5. Fill a prism 5/2 × 5/2 × 3/2 units with cubes of edge 1/2 unit. How many cubes fit? Use their volumes to verify lwh.

  6. Fill a prism 1 × 3 × 5/2 units with cubes of edge 1/2 unit. How many cubes fit? Use their volumes to verify lwh.

  7. Fill a prism 2/3 × 2/3 × 2 units with cubes of edge 1/3 unit. How many cubes fit? Use their volumes to verify lwh.

  8. Fill a prism 7/3 × 1 × 5/3 units with cubes of edge 1/3 unit. How many cubes fit? Use their volumes to verify lwh.

  9. Fill a prism 4/3 × 2/3 × 2/3 units with cubes of edge 1/3 unit. How many cubes fit? Use their volumes to verify lwh.

  10. Fill a prism 2/3 × 2 × 4/3 units with cubes of edge 1/3 unit. How many cubes fit? Use their volumes to verify lwh.

03 · CHECK

Answers and reasoning

Reveal the teacher answer key
  1. 126 cubes; each 1/8 cubic unit; prism 63/4 cubic units

    Counts along edges: 6, 7, 3. Product 126; 126 × 1/8 = 63/4, also lwh.

  2. 80 cubes; each 1/8 cubic unit; prism 10 cubic units

    Counts along edges: 8, 2, 5. Product 80; 80 × 1/8 = 10, also lwh.

  3. 72 cubes; each 1/8 cubic unit; prism 9 cubic units

    Counts along edges: 8, 3, 3. Product 72; 72 × 1/8 = 9, also lwh.

  4. 112 cubes; each 1/8 cubic unit; prism 14 cubic units

    Counts along edges: 8, 7, 2. Product 112; 112 × 1/8 = 14, also lwh.

  5. 75 cubes; each 1/8 cubic unit; prism 75/8 cubic units

    Counts along edges: 5, 5, 3. Product 75; 75 × 1/8 = 75/8, also lwh.

  6. 60 cubes; each 1/8 cubic unit; prism 15/2 cubic units

    Counts along edges: 2, 6, 5. Product 60; 60 × 1/8 = 15/2, also lwh.

  7. 24 cubes; each 1/27 cubic unit; prism 8/9 cubic units

    Counts along edges: 2, 2, 6. Product 24; 24 × 1/27 = 8/9, also lwh.

  8. 105 cubes; each 1/27 cubic unit; prism 35/9 cubic units

    Counts along edges: 7, 3, 5. Product 105; 105 × 1/27 = 35/9, also lwh.

  9. 16 cubes; each 1/27 cubic unit; prism 16/27 cubic units

    Counts along edges: 4, 2, 2. Product 16; 16 × 1/27 = 16/27, also lwh.

  10. 48 cubes; each 1/27 cubic unit; prism 16/9 cubic units

    Counts along edges: 2, 6, 4. Product 48; 48 × 1/27 = 16/9, also lwh.

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