High School: Geometry
Modeling with Geometry HSG-MG.A.2
2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
How many monkeys can fit in a barrel? How many slams are there in an old screen door? How many licks does it take to get to the Tootsie Roll center of a Tootsie Pop?
If your students are packed into your classroom like sardines, they may begin to question the school board's answer to, "How many students can we squeeze into a classroom?" But they won't make much ground complaining about the lack of elbowroom. Instead, they can reflect on the population density of their state compared with the neighbors, the salinity of the aquarium in the library, or how nutrient-dense the soil is in the bean field across the street.
Students should understand that density is a ratio of mass (or heat or people or things) to area or volume. The real world boasts many situations modeled by density concepts, but understanding doesn't come from chucking objects into water to see whether they sink or float. Instead, students can take a look at all the different examples of density all around us.
- Ohio asks its mail carriers to help count cottontail rabbits in order to determine the population density of that species of wildlife throughout the state. No lie.
- The USDA provides reports about the national yield of bushels of corn harvested per acre planted.
- Geologists record the quantity of volcanic eruptions (or other rude awakenings) that occurs in the Ring of Fire. Er, uh…we meant the Pacific Ring of Fire.
Like the professor trying to inspire his students to kick back and enjoy a few cold, uh, Yoohoo's with friends, students should try to come up with their own examples of how many things can fit into different spaces and the implications of those densities.
Students must also be able to use density to calculate other quantities related to it and interpret these answers in terms of their contexts. Using the right units, among other things, is good way to check that an answer makes sense.
If we're looking for mass, our answer should be in units of grams or pounds. If we're looking for the number of licks it takes to get to the center of a Tootsie Pop, we should have units of licks per Tootsie Pop. What is that number? The world may never know.