Formative Assessment 103

Assessment Problem: 

Consider the sequence of W-patterns.
[Figure from Rivera/Becker reference below, p. 68.]
A. How many dots are there in pattern 6?  Explain.
B. How many dots are there in pattern 37?  Explain.
C. Find a direct (i.e., closed-form) formula for the total number of dots D in pattern n.  Explain how you obtained your answer. 
D. Zaccheus’ direct formula is as follows: D=4(n + 1) - 3.  Is his formula correct?  Why or why not?
E. A certain W-pattern has 73 dots all together.  Which pattern number is it?  Explain.

Common Responses: 

Students often get a correct formula but are not able to explain why it works.  They attempt to “deconstruct” the figure (i.e., break it down into pieces that make it up in some predictable way), but the deconstructions do not necessarily generalize in a useful way.

Mathematical Issues: 

“Deconstructing” a sequence of figures can help transform it into a numerical sequence, which might be easier to generalize. Students tend to have more difficulty with deconstructions; this exercise represents an opportunity to see where students fall on that spectrum

Patterns & Functions
Linear Patterns
Common Core Standards: 
F-LE: Linear, Quadratic, and Exponential models
Research References: 
Rivera, F., & Becker, J. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns. ZDM, 40(1), 65-82.