Quadratics Intro

We are starting quadratics in Algebra II.  I had groups of 4 work on a problem about putting a walkway around a pool and trying to find the maximum possible width of the walkway, given that the total area need to be less than a certain amount.

While this has been a successful task for several years, I am very curious as to why it always elicits a good deal of success.  These are some of my thoughts.

1.  There are several ways to approach a solution.  Students feel comfortable trying things out because I reinforce this idea at the beginning of the task.  I do not tell them if a method is right or wrong, but instead wait until they get an answer using their method, then ask if they can check to see if that answer makes sense.
2. The content that students are focused on is area of rectangles, which many feel very confident with.  The variable does add a wrinkle, but their confidence for areas of a rectangle is exceptionally high compared with other areas of math.
3. The problem is challenging, but not beyond their reach.  This ties in with the confidence.  As soon as they know they are finding areas of rectangles, they know that they will be able to find the solution.  Knowing that they understand the “basic math” of the situation, they stick with it, even though the content I’m focused on in writing quadratic equations.  

I was very happy with the way the conversations were turning out.  Students tried various methods.  Some started with finding an expression for the total area, then abandoned that idea to break it up into smaller rectangles, to then go back to the total area expression to check their proposed solutions.  They also were never satisfied with the guess and check method for solving their equations once they solved them. 

This lesson is also one of my go-to arguments for teaching in the style that I do.  Every group in every section of my Algebra II class was able to set up a quadratic equation that would lead them to the solution to this problem.  None of the groups was able to recognize that the way to solve the equation was to use the quadratic formula, even though the vast majority of them have heard it and used it before.  I truly believe that one of the most fundamental aspects of teaching Algebra is teaching students when to use a given tool as much as it is how to use that tool.  I think recognizing when a tool such as the quadratic formula may be useful is just as, if not more important, than working through the maze laid out by the order of operations in calculating solutions using this tool.

We left class with students wanting to find a more efficient way to solve their equations.  I told them that we’d be learning it in future classes, but did not explicitly link it to the quadratic formula just yet.  I’m hoping they’ll be able to discover that for themselves, but this was a nice activity for introducing the unit and will make a great reference point for other lessons on this topic.