Building Understanding of Sequence Notation

Sequences and series are never a particularly hard topic for my Algebra 2 kids- conceptually, at least. They get it. They've been looking for patterns since they were tiny little humans and it's a fun puzzle for many of them to do so in class. What they struggle with consistently is the notation we demand of them at this level of mathematics. Some mastered it in Algebra 1, but many just show up and pray they do it right this time or memorize what their teacher told them to do instead of trying to battle the sense-making involved to understand and be able to apply this knowledge. This activity is one I use to combat that and it builds to have students generate their own equation for recursive sequences. 

Recursion Notation Challenge

Phase I:
We start by doing a sorting activity, where students are given the cards at right and asked to order them. This is an easy enough task- definitely low floor for some of my weaker students. This is a springboard for discussing the "why" and asking questions like:

• Which comes directly after n? How do you know?
  
• Which comes directly before n? How do you know?
  
• If you have n-3, which would come 2 after that? 

You can really extend it as far as you want to go here. 

Phase II:
From there, we look at the actual expressions we use for terms. 

 Using what we previously did, ordering these isn't usually much of a challenge. From there, students work in their small groups to extend:

Phase III:

Here is where the wrap up discussion as a class occurs and we begin to test our understanding of the notation. In small groups, we first just examine the differences between position in the sequence and actual value of the term:

Then, we start translating from words into notation. 

I've always heard it said that students are more receptive to an idea if they think it came from themselves or another student....this activity has been a huge help in clarifying misconceptions with the help of other students, not just a teacher re-explaining it the same way for the 400th time. 

If you have anything else that you love for teaching recursion, sequences, or series, please share!!