Monday, November 24, 2014

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.

Absolute Value Inequalities & Price Is Right

I’ve been searching for a real-world scenario where absolute value is involved, specifically where there are operations outside the absolute value bars.  This year I found it!  My wife and I love playing guessing games, often going with “Price is Right” rules, where a guess over the actual amount is disqualified.  These are the standard “Price is Right” rules, with one exception.  At times, they play a game called “Cliffhanger”.  The object of the game is to guess the price of three items.  For each dollar you are off, the cliffhanger moves up the mountain.  After each item is revealed, a guess is given, the cliffhanger moves, and the actual price of the item is revealed.  If the cliffhanger moves more than 25 spaces at any point, he falls off the cliff and the contestant loses.

In class we watched this clip until right before Walter makes his guess.  I asked students to make a guess of their own, which they entered into a Google spreadsheet.  We watched Walter guess, and then watched the cliffhanger move.  We stopped the video before the actual price of the coffee maker was revealed.  I asked students “What do you know about the actual cost of coffee maker?”  Many students decided that the price must be less than $50.  I let them all mull that over for a minute or two and eventually a student said that it could have been more than $80 as well. 

We watched the end of the video, where the price of the coffee maker is revealed.  We then revisited our guesses.  I had students copy and paste the list of guesses into Excel so they could all work independently.  I asked them to create a second column in their spreadsheet which would represent the number of steps that the cliffhanger would move based on the guess.  We then generalized their process.  Some students worked through a piecewise process, treating the arithmetic differently if the guess was more or less than 45.  Others did the same arithmetic every time and “dropped the negative”.  They highlighted all of their peers that would have won the prize based on their guess for the last item.  We discussed how there were many possibilities and what this meant about the type of problem we were trying to solve.  Students made connection to inequalities and absolute value as it relates to distance. 

At this point, we graphed the guesses compared to the steps that the cliffhanger moved.  We discussed where the vertex of the graph was and why it would occur there.  We also discussed slope.  Graphically we highlighted the portion of the graph that represented winning guesses versus non-winning guesses, connecting this with the one-dimensional number line representation of the solutions to the corresponding absolute value inequality.

This time around, I focused on the inequality lx – 45l <  15, but next year, I will definitely focus on the game as a whole.  The purpose of the game is to have the cliffhanger move no more than 25 total spots.  Including this with the Walter video, we could focus on the slightly more complicated inequality of 10 + lx – 45l < 25.  This shift would help students to realize the process for when you should “split” an absolute value expression.  Mentally, the students (and I) had already had this step cemented in their heads, but bringing it to the forefront may help with some misconceptions down the line.  Reinforcing what students are already doing with the symbolic manipulation to back it up will help students develop their abstract and quantitative reasoning.