#Illustrative Mathematics

Task

Fred has some colored kitchen floor tiles and wants to choose a pattern using them to make a border around white tiles. He generates patterns by starting with a row of four white tiles. He surrounds these four tiles with a border of colored tiles (Border 1). The design continues as shown below:

Fred writes the expression $4(b-1)+10\,$ for the number of tiles in each border, where $b$ is the border number, $b \ge 1$.

1. Explain why Fred's expression is correct.
2. Emma wants to start with five tiles in a row. She reasons, “Fred started with four tiles and his expression was $4(b-1)+10$. So if I start with five tiles, the expression will be $5(b-1)+10$. Is Emma’s statement correct? Explain your reasoning.
3. If Emma starts with a row of $n$ tiles, what should the expression be?

IM Commentary

The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning.

Analyzing and generalizing geometric patterns such as the one in this task may be familiar to students from work in previous grades, so part a may be a review of that process. It requires students to make use of the structure in the expression (MP7), to notice and express the regularity in the repeated geometric construction (MP8), and to explain and justify the reasoning of others (MP3). Part b requires a deeper analysis of the expression, identifying the referents for its various parts (MP7). Students may still need guidance in writing the formula for part c since it introduces a second variable.

This task includes an experimental GeoGebra worksheet, with the intent that instructors might use it to more interactively demonstrate the relevant content material. The file should be considered a draft version, and feedback on it in the comment section is highly encouraged, both in terms of suggestions for improvement and for ideas on using it effectively. The file can be run via the free online application GeoGebra. or run locally if GeoGebra has been installed on a computer.