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 https://ww.youtube.com/watch?v=HjT7bSHAHAU 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.