One of the realizations that I made at NCTM was that the idea of multiple representations is still new to some people and that I need to start sharing more.1
This is one of my “worksheets”2 from last fall that I made on systems of equations for my freshmen Honors Accelerated Advanced Algebra class (links: .docx or .pdf). The first problem was difficult for some of them, they didn’t like being asked to estimate the solution. Which was my whole point. It doesn’t fall on a “nice” intersection, but I wanted them to use that estimation to determine if their answer to part c was reasonable.
Even though they had all taken Algebra I in 8th grade, most of them are not used to working from multiple representations. One of my goals is for them to become comfortable in working from tables and graphs: equations they were used to (not that we’re going to stop working with equations).
The third problem asked them to solve the system from a table.
They approached parts a and c in a variety of ways . Some of them sketched a rough graph (didn’t occur to me, but okay, that works). Some of them took a more analytic approach and looked for the interval where f(x) became greater than g(x). A few got stuck. They didn’t know what to do, so they started with d and e and then worked backwards. Once others had shared their methods (yay document camera!), I heard a few Ooh, I hadn’t thought of that type comments. Which means they were listening to each other. They weren’t used to doing that at the beginning of the year either.
I’m happy to say that as the year has gone on, they have gotten much better at working with multiple representations and at explaining and justfying their work. And in realizing that I am not the mathematical authority in the classroom.
1 I think I forget that the people whose blogs I read are probably not a representative sample of math teachers.
2“Worksheet” has such a bad connotation. Just because something is on a piece of paper, does that mean it has no value? I don’t really think of this as an “activity”. It is a worksheet. One that asks them to work from multiple representations, see the connections between them, and to explain and justify. I am okay with that.
Continuities 
It’s a good worksheet, Jackie. I agree with your sentiments about how worksheets can be useful when constructed appropriately, and yours seems to do just that. I’m impressed.
Thank you Joe! (sorry your comment somehow got caught in the spam filter for a day or so, I guess my blog isn’t sure what to do with compliments 🙂)
I like to find teachers on the same page.
Just a smily story on your theme: in class this week, I don’t even remember what we were doing… finding the inverse of a function? I had kids pestering me – should we use the graph to check the algebra or the algebra to check the graph? Precious.
Jonathan
This is great. I love how it really got the kids thinking and trying different ways. I think a lot of kids lack the ability to put their thoughts into words and express it to one another and actually LISTEN to what is being said. Great job!