Formative Assessment 108

Assessment Problem: 

A. The first two entries in a table of numbers are given on the right. Write five more numbers in the table to create a linear relationship. Write five more numbers in the table to create an exponential relationship. Write five more numbers in the table to create a quadratic relationship.
B. Explain how you generated the numbers.
C. Write an equation for each relationship and explain how you got your equation.

Common Responses: 

1, 4, 7, 10, 13, 16, 19; 1, 4, 9, 16, 25, 36, 49 and 1, 4, 6, 7, 6, 4, 1; 1, 4, 16, 64, 256, 1024, 4096. 

Mathematical Issues: 

This problem was used as a professional development tool for teachers. Some believed that the quadratic pattern was unique because the other two were. On the other hand, some believed that any pattern that rose and then fell symmetrically would be a quadratic pattern; e.g., 1, 4, 6, 7, 7, 6, 4, 1. Discussions among the teachers about student solutions were very productive.

Domain: 
Patterns & Functions
Theme: 
Functions
Common Core Standards: 
A-CED: Create equations that describe numbers or relationships
Research References: 
Herbel-Eisenmann, B., & Phillips, E. (2008). Analyzing students' work: A context for connecting and extending algebraic knowledge for teaching. In C. Greenes & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics (pp. 295-311). Reston, VA: National Council of Teachers of Mathematics.