Taxi Fares: MAPs Activity- Cutting out what you don’t need

I really enjoy using MAPs tasks in class, but with my recent “cutting out what you don’t need” outlook, I am finding myself looking twice even at this resource.

Consider the following graph:

Taxi Cab Graph

What do you notice?  What patterns do you see?

Depending on the level of student, you can get a variety of answers.  It has a positive correlation, or that it’s an increasing function that is not continuous.  You can get responses about statistics: the range, min/mas, mode, median, etc.  All of those observations are great for what the original worksheet has students do.  I have even considered cutting out the last sentence in the directions, hiding the axis labels- having students provide them as well.

One question a student asked: “Why would he graph his fares?”  A good question- where would this problem be placed in Dan Meyer’s (Real World,Fake Work) quadrant?
I answered: “That is a good question,  I think he did that so he could look for a pattern.”
Student: “Do I need to use the graph?”
Me: “How do you want to look at it?”
Student: “Didn’t it say he wrote down the times and distances?  I’ll use his list.”  Well, that is a problem since I didn’t create one and part of why I am having students work on this problem is to gather that information from graphs.
Me: “I don’t have it with me, is there a way you can re-create his list?”
Student: “Yes, I can use the graph.”
The student then starts working to make his list.

My point is: students can connect a lot of background experience to the graph, and it does the class a world of good to talk about them.  After you get those ideas floating around in their heads you hit em with the real math:

David needs to make $30 per hour.  Should he charge by hour or mile?  How much should he charge?  Explain your reasoning.

I can walk around and look at student work, asking questions here and there to understand their approach and thinking.  Students would solve this question in whatever manner makes sense to them.

This is what the problem actually looks like from MAPs:

Taxi Problem

I tried this with one class, and they blindly followed the directions for 1 without considering why 4 hours was chosen.  The bad thing about that is that they assumed that the work they did for 1, they were supposed to extend to 2.  They came up with 14min 40 sec for a 4mile trip, which they then approximated to 4 trips per hour, and 16 miles total.  They then got an answer of $1.88, which was close.  They did all of this work without thinking about what or why they used 4 mile trips, so when the second handout came- they were lost because it did not resemble their work.  We then backtracked and re-examined their work (I knew this was coming so I didn’t intervene right away), and they did the process using all of David’s trips.  Many students struggled with this, they did not have the academic stamina to redo a problem that was solved (in their minds).  To make this less painful for my students, I asked students to write why the sheet had them examine a 4 mile trip.  Once they saw a point to re-working the problem, they went back to the graph and measures of central tendency.

I plan on hitting students with this same type of problem next week, cutting out what I don’t need, and see how students progress.