# An Open-ended Function Problem

So in Precalculus today, I decided to start with something (after the warm-up) I’ve wanted to do for a little while.  It was good that I waited until now because there were things that we’ve learned in class that they were able to use.  I am certainly going to do this again, although I need to think about how to scaffold so that they do not take up the entire hour and a half of the block period.

I told them “I’ll give you five minutes to start on this” and an hour and a half later they were still working.  Here’s the problem I gave them:

Stove Heating Element Problem: When you turn on the heating element of an electric stove, the temperature increases rapidly at first, then levels off. Sketch a reasonable graph showing temperature as a function of time. Show the horizontal asymptote. Indicate on the graph the domain and range.

I embedded the problem into a link sheet, as shown below, so they had to come up with an equation, graph, and table, not necessarily in that order (I warned them to try it in an order that made the most sense).  Here’s the LINK sheet:

I handed out the worksheets, divided them into groups of 3, gave each group a whiteboard and a few markers (I didn’t have enough for every student to have one), and offered laptops from the laptop cart to them.  There were enough laptops for every group of 3 to have at least 2 laptops, but very few of the students went for the laptops first.  They had learned how to use Geogebra the previous week using this worksheet on sliding/shifting functions that I created last year.  Being able to slide/shift functions up/down and left/right really came in handy for many of them.  About half of the groups realized that Geogebra would be much, much easier than typing in equation after equation into the “y=” of their graphing calculator while the other half of the groups needed my prompting to grab a laptop.

I was excited by how much they discussed and the kinds of discussions they were having.  One by one the groups figured out various little important things by studying the graph.  Things such as (1) “hey, that looks like a square root graph, but can’t be because square root graphs don’t have asymptotes!” (2) when time is zero, temperature shouldn’t be zero, (3) which units they should use (seconds or minutes; Celsius, Fahrenheit, or Kelvin), (4) how “quickly” the graph should get close to the asymptote, (5) where to put the sliders into the function when using Geogebra, (6) which parent function to use, and (7) how to move a graph up/down and left/right.  Most of these topics were covered in most groups without prompting, which was wonderful!

The students really struggled, and I absolutely refused to help most of them–especially when I was the first one they turned to in order to see if something was “correct” or not.  Another great thing about this exercise was the number of good answers: there were at least 3 different types of functions used, all of which could have modeled the situation correctly.  I told the students “there is no one correct answer–there are several right answers!”  I also warned them that there were even more wrong answers, so they couldn’t just put down whatever they wanted and call it “right”–they had to justify their answer.

One of the downsides to this activity was the amount of time it took.  I didn’t realize that authentic discussion took so long, or that they would be so painstakingly slow at figuring out a good-looking solution to an open-ended problem.  That probably means that they have been “force-fed” too often in their math careers, although I know I did that last year and there will be times I’ll have to do that this year, too.  My hope is that the more they do this, the more they’ll appreciate it and the faster they’ll get.  I perhaps should have done this Geogebra activity on stretching functions before because that would have also come in handy, but I didn’t want them to have forgotten the shifting/sliding function activity.

Another cool tool that I showed them and a few groups were brave enough to use was the Desmos Graphing Calculator.  My students had not used this yet, but I think that making sliders and editing the function is a little easier to do there than Geogebra (just a little easier).  One problem was that we were using the school’s laptops, and they only had IE 8 (no Chrome, no Firefox, no other good browser…) so the latest Desmos did not work well at all for the students.  Geogebra is a little more versatile and I think I’ll keep teaching the students how to use it because of that.

Overall, I think this was a good exercise for the students and I’ll certainly try this again, hopefully with more scaffolding and more direction.  The students did not appreciate the fact that I would not “help” them early on, but they felt a sense of accomplishment at the end of the class period as their group began to get closer and closer to a good-looking graph.  Many of the students left class still feeling frustrated even though they were successful in the end.  I suppose they thought that they wasted an hour of their life because they were just frustrated, but I really hope they learned something from it.

Questions I have for you, if you care to respond:

1. What can I do to shorten the amount of time it takes them to solve this?  I gave them the “parent” functions a week ago, so they had the hyperbola, inverse square, exponential, and power function, and of which would have worked (with tweaking).
2. How would you change the problem?  Along with that, what extension questions can you think of?
3. Are there any other tools (program online) that might help students use this?  I’m looking for free thing–yes, I know Geometer’s Sketchpad is a wonderful tool, but when all these other tools are free, I find it difficult to request money for programs that are so similar to free programs.