Working the steps of a rule or procedure backward
b. Finding input from output, or initial conditions from a solution
Building rules to represent functions
a. Organizing information in ways useful for uncovering patterns and the rules that
define the patterns
b. Noticing a rule at work and trying to predict how it works
c. Looking for repeating chucks in information that reveal how a pattern works
d. Describing the steps of a rule without using specific inputs
e. Wondering what different information about a situation or problem may be given
by different representations, then trying the different representations
f. Describing change in a process or relationship
g. Justifying why a rule works for “any number”
Abstracting from computations
a. Looking for shortcuts in computation, based on an understanding of how
operations work
b. Thinking about calculations independently of the particular numbers used
c. Going beyond a few examples to create generalized expressions, describe sets of
numbers, or either state or conjecture the conditions under which particular
mathematical statements are valid
d. Recognizing equivalence between expressions
e. Expressing generalizations about operations symbolically
f. Using generalizations about operations to justify computational shortcuts