🔑 Key Concepts
- One foot is equal to \(12\) inches.
- A cubic foot measures \(1\text{ ft}\times1\text{ ft}\times1\text{ ft}\).
- After converting each edge to inches, one cubic foot measures \(12\text{ in}\times12\text{ in}\times12\text{ in}\).
- The base contains \(12\times12=144\) square inches.
- Each one-inch-high layer contains \(144\) cubic inches.
- Twelve layers contain \(144\times12=1{,}728\) cubic inches.
- Therefore, \(1\text{ ft}^3=1{,}728\text{ in}^3\).
✏️ Interactive Worked Example
🧠 Math Vocabulary
- Cubic foot: The volume of a cube with edge lengths of one foot, written as \(\text{ft}^3\).
- Cubic inch: The volume of a cube with edge lengths of one inch, written as \(\text{in}^3\).
- Linear unit: A unit used to measure one-dimensional length, such as an inch or foot.
- Square unit: A unit used to measure two-dimensional area.
- Cubic unit: A unit used to measure three-dimensional volume.
- Unit cube: A cube whose length, width, and height are each one unit.
- Volume: The amount of three-dimensional space contained inside a solid.
- Base area: The area of a selected base of a prism, represented by \(B\).
- Layer: One complete level of unit cubes covering the base.
- Conversion factor: A ratio used to rename a measurement in a different unit without changing its value.
- Equivalent measurements: Measurements that represent the same quantity using different units.
- Cube of a number: The product of a number multiplied by itself three times, such as \(12^3\).
- Capacity: The amount a three-dimensional container can hold.
💡 Main Idea
One cubic foot is a cube measuring \(12\) inches in every dimension. Its base contains \(12\times12=144\) square inches, so each one-inch-high layer contains \(144\) cubic inches. Stacking twelve layers gives \(144\times12=1{,}728\) cubic inches.
📚 What You Should Already Know
Students should know that one foot equals twelve inches, how to multiply whole numbers, how to calculate the area of a rectangle, and how square units differ from cubic units.
🚀 What Comes Next
Students can extend this reasoning to convert between other cubic measurements, compare the capacities of different containers, solve multistep volume problems, and explain why cubic conversion factors are the cubes of their corresponding linear conversion factors.
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