9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
If your students start to find these geometry topics a bit two-dimensional—well, they might be onto something. Most of these geometry concepts are in two dimensions. It's simpler, clearer—but, alas!—boring-er. You know what'll really get their adrenaline pumping? Let's go 3D.
Yeah, we're talking about volume.
Students should understand that volume is a measure of three-dimensional space. Like area, but with an extra dimension added in. While you could always find the volume by counting how many little cubes you can fit into a figure, there's an easier way. Plus, those little cubes get to be a drag when you have to carry them around everywhere.
Instead, your students can make use of the volume formulas. They should already know how to calculate the volumes of simpler three-dimensional figures, like prisms and pyramids. Might be time to round off the corners and get to know cones, cylinders, and spheres.
Students should know not only the volume formulas of cylinders, cones, and spheres (V = πr2h, V = ⅓πr2h, and V = 4⁄3πr3, where r is the radius and h is the height), but also have a basic understanding of where they come from. It might help to compare the volume formulas of prisms and cylinders, looking for similarities and differences. (Spoiler alert: they're really the same formula!)
Once students have a solid understanding of how to use the formulas and which dimension to plug in where, they can work on applying these formulas to real-life scenarios where the dimensions aren't as explicitly stated. Also, combine these formulas with other geometric concepts and formulas that the students should already know.
Don't forget to tell your students about the importance of units and how to convert between them! Volume is always in units cubed because we're dealing with three dimensions—so the conversions are also cubed, too. It might seem like a minor thing, but it's likely to trip them up from time to time, especially if they're too caught up in playing the "Which Formula Do I Use?" game (which is rarely as fun as it sounds).
- SAT Math 4.2 Geometry and Measurement
- Volumen de un Cubo
- Volumes of Pyramids and Cones old
- SAT Math 1.1 Geometry and Measurement