One way I’ve found to be sure students actually understand a process is to turn it around on them. My precalc classes have been graphing rational functions and seem to have the process down. But, I wonder, do they really, REALLY get it? So I do a switcher-roo. I give them a graph and they have to write the equation. Or I give them the special characteristics of the function and they have to build the equation.
We did a pass the pen today where each student using a color pen added a part of the equation and noted why that part was necessary. They all wanted to add to the equation…I guess they get it!! Woot-woot!!
Haven’t used my new whiteboards as much as I’d like but I sure love the kinesthetic aspects of them…and my kiddos love them as well. Today, my precalculus students worked on determining the end behavior asymptote for rational functions. In the past, we’ve practiced pencil and paper style, but the retention seemed poor. So I had a blinding brainstorm just before class to use the whiteboards to practice – novel, tactical, and very visual for me and the students. And I think it will do the trick! 100% engagement in otherwise dreary practice.
This was a red-letter day in precalculus…at least for the teacher. As an opening question, I asked my class this simple question:
Is x = 3 the location of an infinite discontinuity or a removable discontinuity? Explain how you know.
As I was walking into class to set up, I decided to change it up. I asked the students to answer the question silently and individually. Then, once everyone had an answer, I asked them to move to one side of the room or the other depending on their answer of vertical asymptote versus removable discontinuity (hole). Then as a group, they needed to come up with one convincing mathematical reason their choice was correct. You can see the two huddles below.
Then I was the scribe as each group gave a reason. After each group commented, the other group needed to give a convincing statement to either support their stance OR refute the other group’s statement…thus the different colors. After each group gave a statement, they had time to re-huddle and/or change sides. And then we repeated the process. I was after student voice and academic discourse targeted at each other and not me as the conduit. They desperately wanted the answer, but I didn’t give it. So they had to resort to the definitions and theorems to add sophistication and depth to their arguments.
Although this took longer that I had originally planned for the opener (and I had to adjust the subsequent lesson) the end result was that the students, through disciplined discourse, were able to convince themselves of the answer AND had a much deeper understanding of the underlying concepts. It was so exhilarating to see the metamorphosis of their graph of the ideas behind vertical asymptotes and point discontinuities. They were so proud, and somewhat surprised that they could actually reason through to the correct answer. I never did tell them 😛
Another great day in Precalculus. I am so blessed with students who are really interested in understanding the mathematics rather than just memorizing for the test! I am in “teacher heaven.” Today I gave my students the conditions for a rational function: Given the following information, write the rational function that has zeros at x = 2, x = 3, and vertical asymptote at x = 5. It has a removable discontinuity at x = -1. It has a horizontal asymptote of y = -3. I then had them submit their answers via a QuickPoll with the TI Navigator system. Then we compared and listed all the answers that were unique. And then they were to look at each one and talk about what they thought need to be “fixed.” Great discourse and they didn’t even need me to call on them…lots of cross-class discussion. Learn so much from “mistakes.”
Today in Precalc we looked at various ways rational functions can appear in real world situations. One of the more common ways is mixture problems which we talked about yesterday, you know give the problem and look at all of the different approaches students take and then help them compare strategies to determine the most efficient approach. So as our opener problem today, the kids did this problem: How many liters of a 70% alcohol solution must be added to 50 liters of a 40% alcohol solution to produce a 50% alcohol solution? One student presented his solution on the board, but it was apparent by the student discussion that many students weren’t getting why the 0.7x was added to the numerator and a 1x (instead of a .3x) to the denominator. So we made a model of the situation and “experimented” with a couple of friendly values to see what happened, followed by relating it to the problem. I tried to use color to enhance the connections, but am not sure if it helped. I do love, Love, LOVE my laminated CCSS magnet signs for the 8 practices….so convenient for highlighting how students used the practices!! How do you help students understand the concept underlying the idea of mixture problems?
Okay, I admit, this is a challenging Opener problem…and I suggested to my students that they should be able to finish the problem in approx. 7 minutes. Aren’t I demanding?! They thought so. But I used the problem to talk about efficiency when approaching an intense problem like this as well as the need for a clearly defined approach that always works. It was a wonderful opportunity to help students improve their mathematical habits of mind as well as clarify and deepen their understanding of the various attributes of the rational function. All in all, a great day in Precalculus.
We are studying rational functions in my precalculus class. I never can understand the depth of the difficulties students have with graphing these functions, although I guess the number of different tasks, concepts and procedures required to determine the overall shape can be daunting for them. So I’m trying the acronym, that “Fear of Rational Functions DIES with this procedure.” D = Discontinuities (infinite and removable), I = Intercepts (x and y), E = End Behavior (model and asymptote), S = Shape (use a table to fill out the shape if needed). Kids seem more successful this year, they seem to grasp the ideas of discontinuities better because of the work (and foldable) we did earlier in the year and they are working hard at the idea/procedure of determining vertical versus end behavior asymptotes from the function. They are completing an exploration of discontinuities today and beginning an exploration of end behavior using their Nspires. It will be fun to see how well the ideas coalesce in the next couple of days.