Explanation:
 Observing the change or vaiation of two related
quantities
 Covariation approach builds an understanding of
functions through observation of a pattern (or
operation) of change in each variable
Characteristics:
 Looks for generalizations in a table of values
 Focus is on the input/output, unit of change
 Can often lead to a correspondence view—
algebraic expression to describe the relationship
Covariation
Example:
1
4
9
?
1
2
3
4
A student using the covaiational approach would observe
that the 4th square would have 16 because the 1st in the
pattern is 1, the 2nd is 4 (added 3), the 3rd is 9 (added 5), so
the 4th would be 16 because 7 would be added.
Explanation:
 Developing a relationship that can be expressed
algebraically
 Allows a student to develop an algebraic relation
from their understanding of the pattern (or
operation). Then, use appropriate values from the
domain to create additional range values
Common issues:
1. Students may not always see
the patterns we expect them to see. They may need
additional time/reflection to revise their insights.
2. Students (especially younger ones) often form algebraic
relationships that are non-standard. Instead of y=mx+b,
they may tend to develop y=b+xm, indicating a starting
value, b, and a unit increase/decrease of m, where x
indicates the number of unit changes.
Characteristics:
 Focuses on creating an algebraic expression as a
mapping rule from one set to another
Correspondence
Example:
1
4
9
?
1
2
3
4
A student reasoning with a correspondence approach
would observe that 12 = 1, 22 = 4, 32=9, so, the 4th square
would contain 42 or 16 squares.
Common issues:
1. Can quickly be removed from a
context, one of the most important uses of functions (to
quantify and describe real-world situations)
2. Students often have trouble re-contextualizing to the
problem or real-world setting.