Formative Assessment 119

Assessment Problem: 

Square Tiles Task
Tiles are arranged to form pictures like the ones below. Find a formula that enables you to calculate the number of square tiles in Picture “n.” How did you obtain your formula?
(needs image)
A.     If the solution has been obtained numerically, respond to the following question: Is there a way to explain your formula from the figures?
B.     How many square tiles will there be in Picture 75? Explain.
C.     Can you think of another way of finding a direct formula?
D.    Two sixth graders came up with the following two formulas:
Kevin’s direct formula is: T=(n x 2)+(n x 2)+ 1, where n means Picture number and T means total number of squares. Is his formula correct? Why or why not?
Melanie’s direct formula is: T=(n x 2)+ 1+(n x 2)+ 1-1, where n and T mean the same thing as in (D) above. Is her formula correct? Why or why not?
Which formula is correct: Kevin’s formula, Melanie’s formula, or your formula? Explain.

Patterns & Functions
Common Core Standards: 
Research References: 
Becker, J., & Rivera, F. (2006). Establishing and justifying algebraic generalization at the sixth grade level. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 465-472. Prague: PME.