2. Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Representative IMP Year 2 Lesson:

Building the Best Fence (Do Bees Build It Best?), 328

In previous activities, students investigated which dimensions for a rectangular corral yielded the most area. They also investigated whether or not regular polygons had more area than non-regular polygons with the same number of sides. In Building the Best Fence, students build upon their prior work to generalize a formula to find the regular polygon that has the most area given the constraint of limited perimeter

Representative IMP Year 2 Lesson:

Blue Book (Small World, Isn’t It?), 445

High school students often misunderstand depreciation, because they may not have experienced it in their everyday lives. The problem asks students to build a set of data regarding the depreciation of a car over a set number of years. After examining their data, students are asked to generalize a formula to calculate how much a car has depreciated t years after its purchase.