
I start every semester with the discussion of patterns. Patterns are relatable to teenagers, variables, not so much. It also gives students the opportunity to be successful. They have been pattern-finders all their lives whether they know it or not. They use patterns to simplify everything they do, create schemas about the world around them, and anticipate their own futures. Most students can discover some sort of pattern given any set of numbers or objects. They are very creative when they feel any solution is acceptable as long as they defend it. This is an easy environment to foster with problems about patterns.
In my beginning pattern finding exercises I choose sets with multiple patterns and encourage students to find as many as they can. In order to formalize our description of numbers we see in these patterns, we talk about number categories. We start trying to predict a couple steps into the future. When we start trying to predict further, they start using variables without even knowing it.
The problem above is a good starter question because there are multiple patterns present at different ability levels. The students who need to draw every stage could feasibly do it for 50, and although no one wants to – a few probably will. Students who know the suit will want to know how students found the value of the card as well. We can class discuss and compare different methods. If we need extra examples, I keep a deck of cards in the room and can easily put another one under the document camera.
Most students are familiar with a deck of cards and can be encouraged to create their own pattern that incorporates both suit and value of the card. For higher learners, I can discuss the unique characteristics for the set of numbers included in a deck of cards; and which properties might apply to this set of numbers.