Source: Do my math homework
Early Algebra
I’ve been in a number of junior classrooms lately looking at how we use pattern rules for growing patterns. Visual representations can significantly help with this, and the transition to an algebraic representation.
There’s usually a disconnect between how we use pattern rules in late primary / early junior, and the transition to algebraic representations later on.
Consider this number sequence: 5, 7, 9, …
In grade 3 we tend to accept “Start with 5 and add 2 each time”
But the algebraic representation is 3 + 2n
The difference is that algebraically, we add something to each stage including the first stage.
I’ve always thought about the idea that we seem to change the idea of the pattern rule over the grades without being explicit about it to the students. So should we just start with the pattern rule that correlates to the algebraic representation? In words, should it be something more like “Start at 1, and add 2 to every term (including the first one)”?
This is easier for students to see pictorially or with manipulatives.
Visually

In words Start with 3 and add 2 each time

Algebraically 3 + 2n 
Number Sequence 5, 7, 9, … 
To test this out I visited a grade 4 class who had been exploring growing patterns (but not visually).
When asked to draw their own pattern for Start with 1 and add 2 each time, most drew pattern similar to  
But a couple drew 
Over the course of a 20 minute discussion and students defending their point of view, the entire class decided the second representation actually depicted the given pattern rule better.