Investigating Linear and Nonlinear Tile Patterns

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SWBAT use multiple representations to describe linear and non-linear tile patterns and to develop method to find explicit rules for the number of tiles in a given figure.

Big Idea

How many tiles will be in the 100th figure? Students develop their own shortcuts to find explicit rules for linear and nonlinear tile patterns.


30 minutes

This warm-up gives many chances for good conversations with students and I gave them lots of time to work on it (we ended up spending almost 40 minutes!). The first three problems are really good opportunities to talk about the meaning of numbers. I found students doing all kinds of operations with numbers, and whether the operations were correct or not, I asked them: What does that number mean? I kept asking them until they produced an answer that made sense in the context of the problem, so this was a good way to bring in MP2.


Finding a method to determine the number of toothpicks in the 100th figure in sequence at left was perhaps the most challenging problem that the majority of students tackled. (The second sequence was challenging—none of my students found an explicit rule for this sequence today, but we used it as an example of a non-linear sequence.)

Different Solutions to Patchwork Problem

10 minutes

Investigation and New Learning

30 minutes

Sharing and Closing

10 minutes

Today's closing provides an opportunity for students to start looking at some of the big algebraic ideas that will be an important part of the course. The two that can be highlighted in today's lesson are (1) equivalent expressions and (2) making predictions based on patterns. 


The idea is for students to begin to connect the problems and calculations that they do throughout class to larger ideas that can be themes throughout the year. On their first reading of these big ideas, students might not have a deep understanding of the ideas or even really make sense of them, but the idea is that through repeated exposure to the same ideas students will be able to make sense of them.

You can choose whether to present one or both ideas today. I present the ideas by projecting them and reading them aloud. I ask students to choose whichever idea makes the most sense to them initially and to find some way to connect it to today's lesson. Students' connections may be initially superficial, but this will change the more they think about them. 

After students have had some time to think, I ask them to share any ideas they have. If they are unwilling to share, I start them off with a few ideas of my own and ask them to take a minute to expand those ideas. This is only a formative assessment and I use this to help me get started thinking about how to incorporate these ideas into the class. I hope that some students will write comments that I can use as quotes to give other students ideas in future lessons.