Question

why is the product of the number of pieces and the area of each piece always the same? explain how you know.

Explanation:

When a figure (e.g., a shape or a region) is divided into pieces, the total area of the figure is equal to the sum of the areas of all the individual pieces. If we let \( n \) be the number of pieces and \( A_{piece} \) be the area of each piece, the total area \( A_{total} \) can be expressed as \( A_{total}=n\times A_{piece} \) (since the sum of \( n \) equal - sized pieces' areas is \( n \) times the area of one piece). Since the original figure's total area does not change when we divide it (we are just partitioning it, not adding or removing area), the product \( n\times A_{piece} \) (which is the total area) must always be the same, no matter how many pieces we have or what the area of each individual piece is (as long as we are partitioning the same original figure).

Answer:

The product of the number of pieces and the area of each piece is always the same because it represents the total area of the original figure, which remains constant regardless of how it is divided.