The key to exponential growth
Learn the difference between growth by a constant multiplicative factor and growth by a constant additive factor (F-LE.A.1$^\star$).
In this section, students are introduced the underlying growth law for an exponential function, namely that the output changes by a constant multiplicative factor for a constant additive change in the input variable. They gain a quantitative sense of the difference this makes through an application to population growth.
WHAT: Students are presented with a Mythbusters video showing an attempt to fold a large piece of paper in half more than seven times. They are prompted to notice how quickly the height of the folded paper increases, and then asked how many folds it would take (given a large enough sheet of paper) to reach the moon. The core idea is that each fold of the piece of paper doubles the height of the stack. Combined with an estimate of the original thickness of the paper and the distance to the moon, this is enough information to deduce the minimum number of folds to get there.
WHY: This is a very open-ended task designed for students to develop some of the basic ideas surrounding exponential growth. A teacher may opt to give values for the thickness of the paper and distance to the moon; however, requiring students to decide which values are important and research them could tap into more parts of the modeling cycle (MP4). Results of calculations could prompt a discussion of reasonable levels of numerical accuracy (MP6).
WHAT: Students are presented with a poster claiming that one female cat can produce 2,000 descendants in 18 months, along with certain facts about gestation and litter size. They construct a model to decide whether or not the claim is plausible, making assumptions about the process (how many kittens are male/female in a litter? do they always have 4 - 6 kittens per litter? etc.) (MP4). They write an explanation of their conclusion about the claim (MP3).
WHY: Some students may model the situation with a linear model, forgetting that each kitten can have kittens of their own. A whole class discussion of the various approaches can lead to an appreciation of the power of a growth law in which each successive quantity is obtained from the previous one by multiplication, not addition. The activity not only serves as a good introduction to the basic growth law of exponential functions, but also to the comparison of linear and exponential functions (F-LE.A.1$^\star$).