Common Core Standards: Math
2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
Some mathematical language sneaks its way out of the math classrooms and into common conversations. For example, people throw around the word, "rate," "exponentially," and "similar" all the time. Other mathematical language stays firmly inside math classrooms. We're talking words like, "integer," "polynomial," and yes, "fractional edge length."
So what exactly does fractional edge length mean, and what is this standard talking about?
In fifth grade, students were introduced to the concept of volume and the unit cube as a measure of volume. To measure the volume of a solid in cm3, for example, they'd count how many 1 cm × 1 cm × 1 cm cubes can fit inside of the solid. So far, so good.
Students also learned that to find the volume of a right rectangular prism with dimensions <em>l</em>, <em>w</em>, and <em>h</em>, the number of unit cubes they can fit inside the prism is given by the formula <em>V</em> = <em>lwh</em>. They then used this formula to find the volume of right rectangular prisms with whole-number edge lengths.
Now, we expect them to do the same thing—only this time, the edge lengths of the prism will be fractions or decimals instead of whole numbers. Psych!
For instance, how can we get the students to find and understand the volume of a rectangular prism whose dimensions are <img src="http://media1.shmoop.com/images/common-core/comcore6_math_g_latek_1.png"> foot, <img src="http://media1.shmoop.com/images/common-core/comcore6_math_g_latek_2.png"> foot, and <img src="http://media1.shmoop.com/images/common-core/comcore6_math_g_latek_3.png"> foot?
It's 1 over the least common multiple of the denominators. In our example, the unit fraction should be <img src="http://media1.shmoop.com/images/common-core/comcore6_math_g_latek_4.png">. Our unit cubes will have edge lengths of <img src="http://media1.shmoop.com/images/common-core/comcore6_math_g_latek_5.png"> foot. To figure out the volume of each of these unit cubes, we can reason that since their edge lengths are <img src="http://media1.shmoop.com/images/common-core/comcore6_math_g_latek_6.png"> foot, we can fit 6 × 6 × 6, or 216, of them into a rectangular solid whose volume is 1 ft3. This tells us that each of our unit cubes has a teeny tiny volume of <img src="http://media1.shmoop.com/images/common-core/comcore6_math_g_latek_7.png"> ft3.
We can now fill up our rectangular prism these unit cubes. We should be able to fit 24 in our prism. The volume of rectangular prism, then, is <img src="http://media1.shmoop.com/images/common-core/comcore6_math_g_latek_8.png"> ft3, or <img src="http://media1.shmoop.com/images/common-core/comcore6_math_g_latek_9.png"> ft3. Lo and behold, if we multiply our length, width, and height together, we'll get the same answer.
This way, students can understand that (gasp!) volumes of solids with fractional edge lengths work the exact same way as ones with whole-number edge lengths. And we're accomplishing this <em>visually</em> (i.e., by drawing out the fractional unit cubes) and by using ideas that students already know and feel comfortable with (i.e., the unit cube and counting cubes to find volume) to boot. Good goin', teach!
These types of activities and calculations will allow them to understand what fractional volumes and fractional unit cubes mean and why the formula <em>V</em> = <em>lwh</em> works for fractions as well as whole numbers. (Hint: because whole numbers and fractions are both, well, numbers.)
As always, we also want students to be able to apply this understanding to real-world situations. So long as students working with rectangular prisms with non-whole-number dimensions, they're good to go. Assuming, of course, that they're using the right units and interpreting their answers properly.