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Relate volume to multiplication and addition

Volume measures the space inside a rectangular prism and can be interpreted as the number of unit cubes it contains: multiplying length, width, and height counts equal layers of cubes. A prism can also be decomposed into smaller rectangular prisms, with total volume found by adding their volumes; this connects multiplication and addition through the distributive property and applies to whole-number and fractional measurements without double-counting shared boundaries.

Detailed Explanation: Relate volume to multiplication and addition

Volume tells how much space is inside a rectangular prism. If the prism is filled with unit cubes, its volume is the number of cubes inside:

V=length×width×height.V=\text{length}\times\text{width}\times\text{height}.

You can also split a prism into smaller rectangular prisms. Find each smaller volume, then add them. This works because multiplication distributes over addition.

Example: A rectangular prism is 88 units long, 55 units wide, and 33 units high. Find its volume by splitting it into two prisms.

  1. Split the length of 88 units into 5+35+3 units.

  2. Find the volume of the first prism:

5×5×3=75 cubic units.5\times 5\times 3=75\text{ cubic units}.
  1. Find the volume of the second prism:
3×5×3=45 cubic units.3\times 5\times 3=45\text{ cubic units}.
  1. Add the two volumes:
75+45=120 cubic units.75+45=120\text{ cubic units}.

So, the total volume is

120 cubic units.\boxed{120\text{ cubic units}}.

This matches multiplying the full dimensions at once:

8×5×3=120.8\times 5\times 3=120.

The connection is shown by

(5+3)×5×3=5×5×3+3×5×3.(5+3)\times 5\times 3 =5\times 5\times 3+3\times 5\times 3.

The shared dividing face is only a boundary, not extra space, so it is not counted twice.

Learn by doing: Relate volume to multiplication and addition

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Volume of a Rectangular Prism - Count Blocks


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