Lesson 11

In previous grades, students learned that the volume of a prism with whole-number edge lengths is the product of the edge lengths. Now they consider the volume of a prism with dimensions \(1 \frac 12\) inch by 2 inches by \(2 \frac 12\) inches. They picture it as being packed with cubes whose edge length is \(\frac 12\) inch, making it a prism that is 3 cubes by 4 cubes by 5 cubes, for a total of 60 cubes, because \(3 \boldcdot 4 \boldcdot 5 = 60\). At the same time, they see that each of these \(\frac12\)-inch cubes has a volume of \(\frac18\) cubic inches, because we can fit 8 of them into a unit cube. They conclude that the volume of the prism is \(60 \boldcdot \frac18 = 7 \frac12\) cubic inches.

By repeating this reasoning and generalizing (MP8), students see that the volume of a rectangular prism with fractional edge lengths can also be found by multiplying its edge lengths directly (e.g., \(\left(1 \frac12 \right) \boldcdot 2 \boldcdot \left( 2\frac12 \right) = 7 \frac12\)). They use this understanding to find the volume of rectangular prisms given the edge lengths, and to find unknown edge lengths given the volume and other edge lengths.

Problems about rectangles and triangles in the previous lesson involved three quantities: length, width, and area; or base, height, and area. Problems in this lesson involve four quantities: length, width, height, and volume. So finding an unknown quantity might involve an extra step, for example, multiplying two known lengths first and then dividing the volume by this product, or dividing the volume twice, once by each known length.

In tackling problems with increasing complexity and less scaffolding, students must make sense of problems and persevere in solving them (MP1).