Introduction to The Toothpick Problem

The following task is designed to help you use structure to describe a pattern and find a rule that represents the relationship between the independent variable (i.e. the number of connected squares) and the dependent variable (i.e. the number of toothpicks needed to build a figure with a given number of connected squares).

Generate Solutions for The Toothpick Problem

Print out The Toothpick Problem handout

Work independently or with a partner:

Solve each part. Spend time thinking about and discussing how you can write symbolic expressions that describe the way you are seeing the arrangement of toothpicks for any given number of connected squares.
The goal here is to generate solutions with written work where you have made a clear connection between the way you see the toothpick pattern and the symbolic expressions you record.

Think and Discuss - The Toothpick Problem

After you have generated several solutions for each problem, discuss the following with your PLC:

In what ways do you see writing for structure in your solutions?
When looking at how the the number of toothpicks changed as the number of connected squares changed, did you first notice the recursive relationship or an explicit relationship? That is, did you first notice how many more toothpicks would be needed to add a square to a given figure, or did you notice how the number of toothpicks needed was based on the number of connected squares? Why did this stand out to you?