Pedagogical advice on number sequences

When you work with junior high students on number sequences you would often  find  that they topic are able to identify the number pattern corresponding to a number sequence, but are having difficulty explaining it in words. If for instance they consider a progression of the kind:
1   3    6   10    15 ...
they would easily  identify what's happening (going up by 2, 3, 4, 5, etc.) but
find it difficult to express the rule more formally.

The first thing you as teachers should know when working on sequences is that there are many different ways to look at them.  In this case the sequence could be described e.g. as:
the triangular numbers successive partial sums of the series 1+2+3+...
explicitly: a_n = n(n+1)/2
recursively: a₁ = 1, a_n = a_n-1 + n

In the Weblabs environment the variety of approaches is enriched by the different Toon Talk realizations of one and the same rule. So instead of forcing the students to follow your  rule behind a number sequence,  do an experiment with your class, and just let them brainstorm ways to describe the pattern. There would be many valid ways to state it, and many ways also to approach a full description starting with simple observations. It could be a useful exercise for them to come up with their own list of ways to describe it, and perhaps even a chart of different paths to the realization of what this sequence really is. You might challenge them to find as many observations as they can about the sequence, and then to decide which they would start with if they were discussing it with younger students, and which, on the other hand, give the clearest and most complete explanation of the nature of the pattern.

In the above example here is one route they might find:
We can first look for a recursive pattern, that is, a pattern in the way each term relates to the one before it. In this case, we see that the difference between successive terms increases constantly:
terms:         1   3   6   10  15
differences:    2   3   4    5

Now we might want to clarify just what we mean by "increasing"; how are the differences increasing? One way is to look at the second difference; what is the difference from one difference to the next?
We see that it is always 1:
terms:       1   3   6   10  15
differences:  2   3   4    5
second diff:    1   1    1

But that's awfully hard to grasp. We might instead look for a way to describe the differences explicitly; it can help here to write down the index of each term next to it in order to compare:
indexes:     1     2     3     4     5
terms:        1     3     6   10    15
differences:    2     3    4     5

Now we can see that the difference we add to each term to get the next is the index of the next term. That makes it a lot easier to describe the pattern: to get term N, add N to the previous term.
 
That's a perfectly good description of the pattern. But there's another direction we could go in. Looking back at the differences, we can see
terms:              1  3   6   10   15
differences:         2  3   4     5
cumulative sums:  1 + 2=3
                      1 + 2 + 3=6
                      1 + 2 + 3 + 4=10

So we've turned a recursive pattern into one that generates each term as a sum.
 
The important thing for your students will be to understand that  there are many ways
to talk about the same pattern, and that that observation itself is useful.