Volume - Fractional Edge Lengths

🔑 Key Concepts

  • The large cube is \(6\) small cubes long, \(6\) small cubes wide, and \(6\) small cubes high.
  • Each small cube has an edge length of \(\frac{1}{2}\) inch.
  • The actual edge length of the large cube is \(6\left(\frac{1}{2}\right)=3\) inches.
  • Each horizontal layer contains \(6\times6=36\) small cubes.
  • Six layers contain \(36\times6=216\) small cubes.
  • One small cube has a volume of \(\left(\frac{1}{2}\right)^3=\frac{1}{8}\) cubic inch.
  • The total volume is \(216\left(\frac{1}{8}\right)=27\) cubic inches.
  • The same result is found using \(V=s^3=3^3=27\text{ in}^3\).

✏️ Interactive Worked Example

🧠 Math Vocabulary

  • Cube: A three-dimensional solid with six congruent square faces and equal edge lengths.
  • Edge length: The length of one edge of a solid.
  • Fractional edge length: An edge length represented by a fraction, such as \(\frac{1}{2}\) inch.
  • Congruent cubes: Cubes with the same shape and dimensions.
  • Volume: The amount of three-dimensional space contained inside a solid.
  • Cubic inch: The volume of a cube with one-inch edge lengths, written as \(\text{in}^3\).
  • Fractional cubic unit: A fraction of a full cubic unit, such as \(\frac{1}{8}\text{ in}^3\).
  • Layer: One complete horizontal level of cubes.
  • Unit cube: A cube whose edge lengths are each one unit.
  • Scale: The relationship between the dimensions of two similar measurements or models.
  • Cube of a number: A number multiplied by itself three times, such as \(3^3\).
  • Composite structure: A solid made from smaller solids joined together.

💡 Main Idea

A \(6\times6\times6\) arrangement of half-inch cubes does not create a cube with six-inch edges. Since each small edge represents only \(\frac{1}{2}\) inch, the actual edge length is \(6\left(\frac{1}{2}\right)=3\) inches. The large cube therefore has a volume of \(3^3=27\) cubic inches.

📚 What You Should Already Know

Students should know how to multiply fractions, calculate the volume of a cube, interpret cubic units, count cubes in layers, and distinguish between the number of small cubes along an edge and the actual measured edge length.

🚀 What Comes Next

Students can extend this reasoning to rectangular prisms made from cubes with other fractional edge lengths, determine missing dimensions, compare the number of cubes to the total volume, and solve packing and capacity problems involving fractional cubic units.

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