Describing Relationships Among Numbers in a Table

When we talk about relationships between numbers on a table, we really go two different directions with it. First, we dipped our toes into algebraic thinking by discussing variables and algebraic expressions. We used variables to look at different tables and either determine the rule of the table, or use the rule to fill in missing information in the table.  

Secondly, we discussed different ways we can describe a table. Developmentally, this is harder for a 4th grader than you might think. We began by looking at the relationship between the numbers 1 and 5. I asked the kids to think of a real-life relationship between 1 and 5. Someone said hands and fingers. We started discussing how many fingers are on 1 hand, 3 hands, 6 hands, etc. I asked them how we might organize our information so that we could see the relationship more clearly. We decided to make a table. Once our table was made, we looked at our relationship and came up with ways to describe it. The hardest part about this is that the kids tend to want to describe it in a way that is opposite of what is actually happening. For example, one of the ways our table was described was, "The number of hands is the number of fingers multiplied by 5." This is opposite of what is really there. This is a very common mistake for 4th graders! It led to a great discussion, though, about looking at what is reasonable. Could we ever have more hands than fingers? Not unless we'd had some very unfortunate circumstances. ;)