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.

Student Graphing StoryLike 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!

Mountain Climber GraphI 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.

Graphing Stories 2-Create your own

Graphing Story Entrance Ticket

Graphing Story Exit Ticket

14 thoughts on “Mountain Climber Problem: A Nice Follow up to Graphing Stories

  1. Dan Meyer

    Love the counter-intuitive moment at the end. A sense of surprise in math class is a rare and pretty wonderful thing.

    Can you let me know the technology workflow you find for this right here?

    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.

    1. Nat Post author

      The surprise is a wonderful thing, and all too rare. You gotta love the double exclamation point in the picture.

      As for the tech flow, students used the built in camera on their iPad to shoot a 15 second action, and create an accompanying graph on paper. They took a photo of the paper, and we projected their video for the class to graph. My school uses ebackpack, which I’m just trying out for the first time, and for this test assignment, some students submitted the files electronically. So far, ebackpack is nice because it labels the work with student names and puts all the submissions for one assignment into a single folder. I used dropbox for this kind of thing last year, but it was an organizational challenge, especially when students forgot to label files properly (every time). So far, ebackpack is much better.

      Sadly, I’ll likely be the one to do the video production, as it can be hard to find class time for this. I may try to enlist a student if I can find one who knows iMovie or Final Cut, or maybe we can make time right before a holiday for some production work. But even without the slow-mo clip, a video action and a still picture of a graph made by kids they likely know from last year will be great to use next time.

      I’ll update the post to include the sheet I made for students to create their own actions, and I’ll let you know when I get some student graphs up.

      Thanks for checking in!

  2. Pingback: dy/dan » Blog Archive » Great Classroom Action

  3. Jennifer Fairbanks

    Perfect timing. I just did thisisindexed today with index card graphs and needed a connector class for tomorrow’s class. I have been to before but forgot about it. I love the exit/entrance cards too. Thanks!

    1. Nat Post author

      This is a perfect index card problem. So glad that you can use the idea and some of the materials. Thanks for your comment.

    1. Nat Post author

      Ahhh… of course this would be a Gardner problem. Thanks so much for the reference. I’ll have to get a copy of that book.

  4. Mark D. James

    Thanks for sharing this lesson. I can’t wait to use it next week as a follow-up to graphing stories and an introduction to solving systems. I think it will be a great way to present the “big idea” (which often goes unremembered by my students) that the solution to a system of equations is the ordered pair where they intersect, and that the ordered pair is the SAME for both equations.

  5. Joel Patterson

    Nice one! I had not realized you could use this for Algebra. I use this story of the monk in the Limits and Continuity chapter–because both traveling up and down are continuous functions. IF you know the 2 endpoints (top & bottom of hill) AND you know the function is continuous (it’s not a quantum monk) THEN the Intermediate Value Theorem tells you the function hits every value between the endpoint values. The IVT sets the stage by emphasizing how important it is to consider endpoints (and you do that for the Extreme Value Theorem and the Fundamental Theorem of Calculus).
    This climbing up and down a mountain problem offers a lot.

  6. Nat Post author

    Love this connection – but even assuming that your monk is not a quantum monk, can he be differentiated?

    The monk problem does provide a lovely low entry point, but you’ve raised the ceiling considerably with this take. Thanks for stopping in and for sharing.

  7. Bum

    What is the name of the theorem that you use in the mountain climber problem?

    Tanck you!! :D Your work are amazing!

    Please excuse my limited English

  8. Jessica

    Just wanted to let you know that we presented this to a room full of math teachers and had a fantastic response! We gave them some time to think about it on their own and then had them discuss with other people around them. Lots of really great discussion! What a great problem for multiple entry points… it challenges students and teachers alike!

    1. Nat Post author

      So cool that you used this with a group of teachers. I found that almost everyone’s intuition steers them wrong on this one. Thanks for the report!


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