Common Core Standards: Math
6. Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Abstract math concepts are hard to swallow—especially seventh graders—which is why a little real-world modeling can go a long, long way. Instead of finding "the internal measurements of a three-dimensional regular hexahedron with octahedral symmetry," we'd be better off with "the volume of an ice cube." Besides, hexahedrons can't chill your drink.
That's why the mantra of math teachers everywhere is Keep It Real. Or at least, it should be.
Fortunately, geometry is about as real as it gets, since literally everything in the world is made of shapes or solids. (Except for T-1000—we're still not sure what he's made of.)
At this point, your students have spent a couple years on area, volume, and surface area. There aren't any new concepts in this standard; it's just about tweaking their knowledge and giving them increasingly complex word problems to further hone their geometry chops. This time around, they'll also start to mess with composite shapes in two and three dimensions, like finding the volume of a house that's made up of a rectangular prism with a rectangular pyramid on top. Hey, it ain't much, but it's home.
We also want students to kick things into reverse and figure out the dimensions of a shape when they know its area or volume. If a cube's got a volume of 64 in3, they should be able to unpack that and realize its sides are all 4 inches long, since 4 × 4 × 4 = 64.
You could even get really wild and have them calculate the volume when they know the surface area, or vice versa. And again, it's always a good idea to couch this in real-life terms—e.g., "How much space can a present hold if it's a cube-shaped box covered in 96 in2 of wrapping paper?" (Answer: enough for a very small kitten. Just don't forget to poke air holes in the box.)
This is a great standard to mix-and-match geometry with cost-per-unit problems, too. If wallpaper costs $1.20 per square foot and Ayesha wants to cover her bedroom walls, how much will it cost if her room is a rectangular prism 8 feet tall, 12 feet long, and 10 feet wide? Students should know this means finding the surface area of the room's walls (ignoring the ceiling and floor), and then multiplying the square footage by $1.20 to get the final cost.
That's some expensive wallpaper, yo.