Last year, I taught my kids to graph parabolas, and to solve all those drop a ball off of a balcony problems.
And they did horrible at it come test time1.
I don't have a complete answer to the word problems yet, but they knocked the graphing questions out of the park. The trick is that, on the test, you either get one graph and four choices of equations to match it to, or one equation and four graphs.
This is a series of worksheets, designed to take up just part of a class period every day for two weeks. I'd hand them out in the last 8 or so minutes of class, and make them tickets out. I'd also insist that students fill in the relevant information that let them distinguish between their choices.
The early worksheets are simple, requiring only happy/sad and y intercept to differentiate the choices. Later worksheets require more knowledge.
It turns out that the incremental requirements between these sheets was perfect for my students - everyone could get the first couple, and it introduced new ideas just quickly enough that they could adopt them and use them effectively.
The win was that they could use the knowledge from these worksheets not just to ace the test, but to quickly be able to predict what a graph for an equation would look like, or what they should expect when finding an equation for a graph.
1 Our district does quarterly tests that tend to be pretty good predictors of how the kids will do on the CST. I find it useful to have that feedback, since I have no idea what the CST questions are, much less how the kids fail at them. (back)
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I love these for several reasons:
* Each individual worksheet is simple, has straightforward directions, and is meaningful, and as a bonus, doesn't take much time to give or grade-- this factor alone is a strength in a worksheet/instructional activity.
* Repeated, these worksheets reinforce for students the most important characteristics of a parabola to pay attention to and also build their confidence in their ability to solve these problems (because they've done it so many times already)-- the repetition and consistency makes these even more powerful.
* As you mentioned, the worksheets build in complexity and level of skill required, so students aren't caught off guard by trickier ones but rather are able to practice up to them, and the work never gets redundant because there's always something slightly new and challenging-- so this exercise is a teaching and concept-building activity in addition to just practice.
* Also as you mentioned, students gradually develop an intuition for working with parabolas-- and what more could we ask?
Thanks for sharing. I'm inspired :)
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