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.
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