Rational Power Functions

One of the topics our PreCalculus syllabus (Math 111) covers is “Rational Power Functions.” Since functions like  f(x)=x^n are called power functions, the rational power functions would be those of the form  f(x)= x^{p/q} (where  p/q \in \mathbb{Q} ). Unfortunately, this topic isn’t covered in our textbook (Zill’s “Essentials of Precalculus with Calculus Previews“).

Tom Kunkle, on his Math111 Homepage, provides his students a nice summary of how these functions behave: See ratpowfunc.pdf. When thinking about this week’s Lab Assignment, my goal was to give students practice on drawing graphs of functions like

 y = -(x-5)^{4/3}+1
Lab Description
I created nine cards, which I’m calling “Function Cards.” Each card is a half-page (printed on heavy-duty card stock). On one side, the card has a Graph of some translated rational power function. On the other side, the card has an Equation of a translated rational power function. However, the equation and the graph are not the same function!

Here are the directions I provided to students:

  1. When you get your Function Card, flip it to the Equation side. Make a note of the card number. Write down the equation.
  2. As a group, work together to sketch a graph of the equation. You may use the Rational Power Function handout (if you like). You may NOT use a calculator. Work together! Your sketch should clearly display & label all intercepts, any cusp points, any vertical asymptotes, any locations where the tangent line is vertical, and the parent function.
  3. Once your group has agreed upon what you think the graph looks like, draw your sketch on the Answer Sheet, along with the information listed in Step 2. Then go around to the other groups. Find the Function Card with the graph that matches your sketch. Ask nicely, then take it from that group.
  4. Return to Step 1.

The students will have the entire class period to create graphs of each of the nine Functions. Since they will track down the graph most closely matching theirs, they will have a chance to check their answers. Even once they see the answer (i.e., the graph), they are still asked to do some thinking since they have to provide information about the graph’s important features.

Hopefully this process will work. The only thing that concerns me is how long it will take them. I am really terrible at gauging how long it takes students to think about things. My best guess was that it would take about 4 minutes to sketch a quick graph, and then another 3 minutes to write down its important features. So, (7 minutes)x(9 Function Cards) should be a little over an hour.

I picked the nine functions from Tom’s “Homework Handout” on this Section, available here. If the weather complies, maybe we can do this activity outside. The only thing better than math class is math class in the sunshine.

Statistics Group Projects: In Progress

The Elementary Statistics students in my course are wandering the streets of downtown Charleston gathering data for their group projects. There are five groups and here are the questions they are asking (along with their best guess as to what they will find):

  • Do you own a bike? Best guess: At least 30% “Yes” response rate
  • Do you have a passport? At most 40%
  • Are you on vacation? About 30%
  • Do you consume alcohol? At least 80%
  • Have you ever had a fake ID? Around 50%

I look forward to seeing their data!

Statistics Group Project

Project Motivation
There are two class meetings left in my “Elementary Statistics” summer course. This class time will be devoted to students working together on a group project. Last semester when I taught this course for the first time I really wanted to implement some type of end-of-term project. I wanted the project to be collaborative in nature since both my own experiences and recent research in education have shown that students explaining concepts to each other is as important to their learning process as hearing their instructor’s explanations. I also wanted the project to be somewhat self-designed by the groups themselves. It was my hope that giving them some freedom in their projects would increase their interest level in what they were doing.

The topics we finished covering at the end of the course were about creating confidence intervals and performing hypothesis tests (sometimes called tests of significance). Because we discussed this material so recently, it seemed appropriate to have this be the jumping-off point for the projects.

Project Introduction
I wanted the students to have experience going out into the “real world” to gather data, so the project asks them to conduct interviews with people they find around campus. Since it’s only a week-long project (instead of over an entire semester), to make things easier each group has to agree on a single”Yes” or “No” question to ask their random sample. There are three rules for the question.

  • First, each member of the group must agree with the group’s decision on the question. They have to discuss different ideas, vote on them, and eventually reach consensus.
  • Second, the question must be “interesting.” This is hard to define, but basically I want them to avoid boring questions like “Are you a human being?” or “Have you ever been to Mars?” that will result in boring data.
  • Third, the question must be “appropriate” — it has to be something each group member would feel comfortable asking a perfect stranger or their grandmother or their kid brother. (Hopefully they would know to avoid offensive or disrespectful or inappropriately personal questions, but who knows?)

Once they have chosen their question, each individual is asked to guess (to the nearest 10%) what proportion of interviewees will answer “Yes” to the question. After reaching an individual conclusion, the groups discuss what they expect as a group. I wrote a handout describing the “What” and “How” of prior probability distributions and each group works on creating [a very basic] one before they are allowed to leave to gather data.

Project Report
The groups have the rest of the class time to gather data together. I tried to avoid giving them much direction on who they should interview, or where they should find the people, or what types of people to ask. (For instance, do they want to focus on College of Charleston undergrads, or are tourists okay too?) I suggested to them that they need to keep in mind a lot of the ideas we discussed in the class, like:

  • What’s an appropriate sample size?
  • What sampling method should we use? (Convenience, cluster, stratified, systematic, etc.)
  • Should we expect bias in our data? If so, what types? (Sampling bias, response bias, nonresponse bias, etc.)
  • Can we do anything to eliminate bias?

Eventually the groups must produce a typed project report, outlining their process from how they decided on a question and constructed their prior to where they conducted their interviews. They must use the methods of inferential statistics that we learned in our class to create a confidence interval for the proportion of subjects who said “Yes” and give a correct interpretation of the confidence interval. They also have to perform a one-proportion hypothesis test. They are expected to use their prior probability distribution to formulate a claim to test. They are graded on both their data analysis and interpretation of results.

Project Grading
I created a grading rubric for the project. It’s available as a public Dropbox file: 104-project-rubric.pdf A colleague looked over it in the copy room and commented, “You sure are overly detailed with that thing!” This is probably a fair criticism, but mostly I was trying to avoid hearing lots of student questions that boiled down to, “What is the least I have to do in order to get an A?

Something that comes up a lot in discussions about graded collaborative assignments is the “slacker problem”, i.e., How do you keep students from getting by doing zero work? I don’t have a good answer for this. I know that when I was a student, I was annoyed by free-loaders, so I have empathy for students who feel the same way. One of the categories on my grading rubric is “Teamwork Assessment.” Each student must individually send me a confidential e-mail discussing how their group functioned as a team and how they contributed to the overall project. They are asked to give their team a grade of how well they worked together. It was my hope that telling them on the project rubric that (a) they are responsible for their group functioning as a team and (b) they are also responsible for ensuring they contribute to the group that it would create a cultural pressure toward equal collaboration. I can’t say for sure how successful this was last semester, however I was happy that out of nearly 100 students, I only had one or two complain about slackers in their groups.

Looking Forward
This summer’s class has a strong group mentality, I think partly because we have been spending ten hours a week together in class. I hope that this will contribute to great collaborative effort toward these projects. I am also excited to see what questions they will ask and what their data will show. I’ll end this post with a few questions I remember from the class projects last spring:

  • Do you have a fake ID?
  • Do you own an iPhone?
  • Will you vote to re-elect Barack Obama?
  • Did you drink alcohol last weekend?
  • Do you have blue eyes?
  • Do you have a car on campus?