# Mountain Climber Problem: A Nice Follow up to Graphing Stories

I have a new job this year: teaching Algebra 1 and Algebra 2 at an Expeditionary Learning High School.  So far, it is an excellent place; strong and present leadership, dedicated colleagues, and a mission of rigor, relationships, and relevance.  I am very lucky.

Like many Algebra teachers tuned-in to blogland, My students and I worked on some graphing stories to begin the year.  We started with Dan’s Site, and graphed some of the actions (stories) included in the video work there.  Students then created their own actions and short videos (we have 1-1 iPads here this year! Hurray!).  I hope to produce some of these videos with accompanying graphs, and publish them here.  It would be great to have them for next year’s kids to use as well.

After our video analysis, we turned to some work from The Language of Functions and Graphs, an excellent (if somewhat dated) text.  Though the problems are not visually seductive and compelling for students in the way that the graphing videos grab their attention, they are very provocative in terms of the math thinking they demand – and like it or not, we live in a world where students need to practice with traditional looking materials so they’re not caught off guard when taking standardized tests or in their next math class where the teacher may have a more traditional approach.  I found an electronic version of this text HERE, and modified some of the activities – for example, I removed the text but used the graphic from page 42 and asked students to write a “story” depicted by the action in this graph.

The Mountain Climber

As a follow up to this graphing work, I asked them to grapple with a problem involving a mountain climber.  My teaching partner reminded me of this problem, but I’m sure I’ve seen it before.  If you know where it came from, please let me know so I can give credit where it’s due.  Here is the problem:  A climber leaves base camp at 6AM one day, climbs up to the peak and arrives at 6PM.  The next day, she leaves the peak at 6 AM, and begins to climb down.  At a certain point on the trail, she notices that she was at exactly the same spot at exactly the same time on the previous day on the way up.  The question is: What are the chances of this happening?  I asked for initial guesses, and there was unanimous agreement that this would be highly unlikely.  Students talked about this for 5 minutes or so, and as no one was graphing, I suggested that creating a graph might be one way to examine this problem.  After another few minutes, I strongly suggested graphing.  It is so difficult for students to connect yesterday’s work with today’s sometimes – I need to work on this!

I love this problem because the answer becomes totally clear when you make a time vs. elevation graph – and the answer violates nearly everyone’s expectations and leads to a surprise!  Many students got stuck in their initial guess, and even when we went over together what the intersection of the two lines implied, they tried desperately to draw a version of the graph where the two lines didn’t intersect.  When they figured out that even skydiving down wouldn’t work, some resorted to teleportation.

As a nice reminder, the whole Junior class hiked up a mountain to end the week, and at least 6 students brought the problem up to me during our hike.  It was cool that this one stuck with them.

UPDATE: I added some of the materials I used below in case anyone would like them.