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.

Project Based Learning

Our classes for the Fall 2012 semester start today. Thankfully, my teaching schedule doesn’t include Tuesdays, so I don’t start until tomorrow! I’m hoping to use Tuesdays this semester to work on several other projects, including adding more blog postings. Wish me luck.

This semester I’ll be teaching two sections (Section 05 and 17) of our Pre-Calculus class (Math 111) and one section (Section 05) of our Calculus I class (Math 120). Each class meets for 50-minutes per day on Mondays, Wednesdays, and Fridays, and an additional 75-minutes on Thursdays. The longer meetings on Thursdays will be useful in my current quest to incorporate Project Based Learning (“PBL”) into my classes.

I’ve begun the task of designing “Lab assignments” for students to work on, in small groups, during our Thursday meetings. Ideally they would be assignments that require no pre-lecture and ask the students to draw from their course content knowledge to form connections between ideas. By working together in a group, the students could collaborate (hopefully allowing for some peer instruction), ask questions, have a discussion, and digest what we’ve talked about during our other class meetings. According to my calendar, the students will have ten lab assignments over the course of the semester.

Yesterday I began working on the third lab assignment for Calculus. The topics covered earlier that week will be limits at infinity; asymptotic behavior; and continuity. I found an activity called “Carousel Game” from the NCTM‘s Illuminations series and modified it for my class. Here’s a brief overview of this lab:

  • Topic: Graphing rational functions
  • Goal: To correctly determine the equation that corresponds to the problem situation or graph
  • Technology Required: None allowed!
  • Warm-up: Vocabulary assessment, including: asymptote,  rational function, exponential function, end behavior, domain, range
  • Activity: Students will use a description or a graph to find the equation for twelve functions
  • Assessment: After finding the functions, students will find domain, range, vertical asymptotes, horizontal asymptotes, and all intercepts. This will be turned in and graded.

I also uploaded a copy of the lab instructions to my public Dropbox. If you are interested in seeing the entire lab, check it out here: http://dl.dropbox.com/u/59433434/120-Lab2.pdf. (Notice that it’s 120-Lab2, even though I mentioned before it is really our third lab — I start numbering things with zero.)

I’m hoping to reuse this activity in Pre-Calculus later in the semester, once we cover material about rational functions.

Peer Instruction and IF-AT

Whys and Hows of Peer Instruction
I’ve been reading a lot recently about the “whys” and “hows” of peer instruction. Robert Talbert, a regular blogger for the Chronicle of Higher Education, has several blog posts about peer instruction with great information on why you might be interested in implementing it in the classroom. This month I’ve been teaching a four-week summer course in Elementary Statistics. This gave me a great opportunity to try to implement some peer instruction in my classroom.

A lot closer to home is John Peters, a colleague over in the College’s Biology Department. A few months ago he told me about IF-AT testing system. IF-AT stands for Immediate Feedback Assessment Technique. The basic idea of an IF-AT test form is a cross between an old style scantron form and a scratch-off lottery ticket. They have a cool demo you can click through to see how they work.

Basics of IF-AT Testing System
Basically, the students are given a multiple choice question. They answer the question and then scratch off their answer choice on the IF-AT form. If the answer is correct, a star will appear under their answer choice. If they are incorrect, they can re-attempt the question, choose a different answer, and try to find the star.

Now, as an instructor, you have to be a little clever when you write an IF-AT quiz. You have to make sure that the answer to Question 3 is where it is supposed to be, since you can’t move the stars around. There are many different IF-AT forms and they are all coded in a way that the instructor can figure out where the stars are when writing her quiz. The IF-AT folks have created test generation software to help (but I haven’t used it yet).

Why bother with the IF-AT?

  1. Students know immediately if their answer was correct.
  2. Students can get a second chance at a question, allowing for “partial credit” on a multiple choice test
  3. Instructors can also get immediate feedback about how students are doing

Peer Instruction with the IF-AT
Here’s how I introduced my statistics students to the IF-AT system: Two weeks ago, we had a “review day” before our first exam. I wrote three different 5-question quizzes covering various problems and topics related to the test material. I started by distributing the first 5-question quiz to the students. Each student finished the quiz individually. Once everyone was done, they were asked to find a partner and compare answers. If they agreed on an answer choice, great. If they found different answers for a question, they were asked to discuss and debate their answer choices until they found consensus. Once a  pair of students agreed on all five answers, they raised their hands and I gave them an appropriate IF-AT form. They worked together to scratch off their answers. If they found a star (i.e., got the correct answer) with their first try, they earned 5 points; on the second try, 2 points; on the third try, 1 point; and on the fourth try, 0 points. Scores were tallied.

After that, we continued with Rounds Two and Three: We went through the same process (individual / pair / consensus / IFAT) for the next two quizzes, but they were told they had to work with a different partner each time. At the end of the activity, each student had worked with three of their peers and had three scores for their quizzes. The top performing student earned 75 out of 75 points — impressive! As a reward, the student got a “late homework pass” entitling them to turn in homework 24 hours late (without the usual late penalty).

Cost-Benefit Analysis
The IF-AT forms cost about 25-cents each. I purchased 500 forms that are 25 questions with four answer choices (A,B,C,D) per question. Since I needed three 5-question quizzes, I took a few forms and then used the office paper-cutter to slice them into Questions 1-5, Questions 6-10, and Questions 11-15. Since the students were working in pairs, I only needed half as many forms (for a cost of $0.12/student). This is fairly inexpensive.

The students really loved this review activity. I don’t know much (any?) cognitive neuroscience, but I’m pretty sure that finding the star hit their brains’ dopamine reward center. After the first few minutes of the activity, when a pair of students would find a star, a very audible exclamation of joy usually followed. After we were done, we still had a bit of class time remaining and knowing which problems were the tough ones helped focus our time on reviewing material they struggled with the most.

I still had a bunch of “tail end” IF-AT forms (with questions 16-25) so on their test the next day, they found ten multiple choice questions. When I first started talking with colleagues about the IF-AT forms, someone mentioned they had used them in a testing situation and that it had gone poorly. They had given the IF-AT form along with the test sheet and some students would proceed like this:

Answer Question 1. Scratch off answer. Get it wrong the first time. Try again until correct.
Answer Question 2. Scratch off answer. Get it wrong the first time. Try again until correct.
Answer Question 3. Scratch off answer. Get it wrong the first time. Try again until correct.

But, as you might expect, having missed a few problems at the start of the exam, the students ended up feeling fairly crummy about their chances for the rest of the test and some “gave up”. Overall class performance was lower than expected.

To combat this issue, I made sure that the students had answered all questions before I gave them an IF-AT form. That way they wouldn’t become discouraged early on. After the test, all of the students reported that they were happy about the use of IF-AT on their test. They liked leaving knowing how they had performed on that portion of the test. They appreciated being able to have a second (or third!) try on some of the problems. I liked that half the test was graded before I left the classroom!

Future IF-AT Use
I’m hoping to use the IF-AT review strategy again before our last test, which happens next week. The IF-AT forms seem great for running a review session that is constructive. In the fall, I will be teaching both Pre-Calculus and Calculus I courses and I’m planning on using the IF-AT forms in both as part of an early diagnostic to catch any students who were placed into the wrong course.