# Toothpick Transformations

Many of us looked at the growing square toothpick pattern last year.  (See link to the lesson on PBS)

toothpick transformations  We begin with making the first square with 4 toothpicks.  Then we make a 2x2  second square with 12 toothpicks and so on.  We ask students to find a rule to find the number of toothpicks for the nth square.

Many students were able to recognize the pattern as quadratic, but had difficulty finding algebraic models for the relationship between the square and the total number of toothpicks.

In the first attachment below, I have color coded some of the possible solution strategies for this task.

The second attachment is a possible plan for the launch of the task.

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Gerri, your modifications make this very user friendly.  However, one thing I struggle with is the recursive rules and getting the students from there to the explicit equations.  Part of this is because I don't think I validate the recursive "thinkers" in our classroom discussions enough, and from what I can tell this is true in the earlier grades, too.  Any tips about how to address this issue?  Also, when should the quadratic regression with the calculator be shown or used in this activity?  I'm thinking that maybe at the end of the discussion as a tool for checking their thinking or is that too restrictive with the technology?

HI Janet :)

Recursive rules are a struggle for many.  In this quadratic relationship the recursive rule maps nicely onto the geometric pattern, but there isn't a great connection to the explicit rule.  It is something that makes quadratics unique from exponential and linear functions where the connection from recursive to explicit is so strong.  At the same time though, the recursive pattern in the table is easy for students to comprehend.

We did an interesting study of nearly 1000 pieces of student work for a problem that asked students to describe both a linear and quadratic pattern of change.  An overwhelming majority of the papers exhibited recursive reasoning.  It made me stop and think about students seemingly inate ability to think recursively and how we might tap into this ability to build a deeper understanding of explicit rules.  I personally love to color code patterns of change in a table and rewrite dependent values of a function in expanded form and simplified form to explicate a pattern.  For example:

0             4                     =  4 + 3(0)

1             7  = 4 + 3         =  4 + 3(1)

2           10  = 4 + 3 + 3   =  4 + 3(2)

3           13  = 4 + 3+3+3 =  4 + 3(3)                Next = Now +3      y=4 when x=1

Then we have a discussion about how the repeated addition in the table and recursive rule connect to the explicit rule.  Similar beauty with exponential functions :)

As for the quadratic regression, I don't think waiting until the end of the discussion is too restrictive with the technology.  If the goal of the lesson was more about statistical modeling, I might go there sooner, but not in this case.