Starting with the 7th grade, we ask students to develop an understand of the relationships behind linear expressions. We ask them to write linear expressions and solve linear equations from real-life situations. Whenever I did this work with my students, I found that their understanding was often very shallow and disconnected. For instance, with the table on the left, my students could easily write the relationship as 3x. The table on the right was more difficult.
I don't use a textbook for my prealgebra class. I tossed the textbook out because it finally dawned on me that the most important idea in the year was just a one-class two-page spread (back in the days of that kind of textbook). That was the lesson when students were asked to write a function for a linear table. Linear functions seemed to be the concept behind 70%+ of the book: evaluating and solving linear expressions, graphing equations, and ratios and proportions; and this seemed to me to be the essence: understanding linear functions as a pattern of steady growth or decrease. As I struggled to find a way to have students make the connection between the patterns in the numbers and the functions, I tried The Pattern and Function Connection by Fulton and Lombard, the Creative Publications' Algebraic Thinking: First Experiences, and other places where students were looking at visual patterns, making tables and writing functions. (This was long before Fawn Nguyen had created Visual Patterns).
The students could see the patterns in the visual sequences, and that made it easier to write functions from the shapes, but writing a function from a table was another matter. As I wrote above, they saw how the numbers in the right column (the range) were increasing or decreasing by a constant amount, but they couldn't understand why that appeared in the function as multiplication, not addition or subtraction. They didn't see how the function could both describe the pattern and predict other values when there were gaps in pattern, if x = 9, 32, or 105. And there was another trouble spot. When we got to graphing linear functions, the students could find the rise and the run, and they could even use slope and the y-intercept to write a function, or find those values from the table, but it seemed those connections were just correlations, and they weren't really seeing the pattern of growth behind them. They were just numbers.
That's when I figured out that the students did not see the numbers in the range as a sequence at all. They were focused on the (x,y) number pairs. Or if they did see the sequence, they had a difficult time extending it mentally, or filling in gaps. The students needed time to look at linear sequences and reason about them, find missing terms, and find connections between the generating function and the sequence. Using Dan Meyer's metaphor, if linear functions are the aspirin, then missing terms in a sequence should be the headache. I also realized that I should start with just sequences, lists not contained in a table, then introduce the concept of term numbers, then combine the term numbers and the sequence to make a horizontal table, so that the students could develop this way of seeing and understanding linear patterns and graphs as a pattern of growth in constant steps.
Proceeding in this linear fashion :), my students develop a much deeper understanding of linear relationships. I do a lot of work with situations that are linear - writing linear functions, exploring sequences, filling in tables, etc. - before we start writing linear equations and solving. You might say that gives the students a context for solving equations, but I think more importantly it gives the students a mathematical model and a tool that allows them to do independent thinking about all of the above.
I hope at a later date to post some of the materials I use with my students for exploring linear growth situations and functions. You can take a look at some of the sequence activities I use in the powerpoint I mentioned in my previous post. The powerpoint outlines a progression of increasing challenge that takes place over a couple of months for me. You can use it at any pace that works for your students. I use sequences the same way one might use Visual Patterns: present a couple of examples at the start of class once or twice a week, have a good discussion about what people did and noticed, then extend the ideas to a slightly more challenging problem the next time.