1. To what extent were lesson objectives achieved?
Students will know:
· How to use and understand the Remainder Theorem
· The ideas the make up the Bounds on Zeros Theorem
Students will be able to:
· Use the Remainder Theorem to find the Remainder of Polynomial Division
· Use the Bounds on Zeros Theorem to find the bounds and create a graph of the polynomial
I believe the lesson objectives were achieved by the majority of my students. I think all of the students fully understood the Remainder Theorem, although they might not understand why it works. By looking over the formative assessments, the students understood the general process and were able to arrive at the correct solution. While I was teaching, many students had an idea of what was going on in the theorem, and they saw the pattern right away. Since they had just learned how to use synthetic division to find the zeros of the polynomial, they all seemed to understand why the Remainder Theorem is so useful. A problem that many students have is deciding the c-value to use as the x-value when evaluating. For example, if dividing by x+2, students sometimes forget to use x= -2 rather than a positive 2. I tried to emphasize that we were dividing by x-c, therefore we have x+2=x-(-2). While I think many students understand this idea, it might take some more time for the idea to really sink in. They are all capable of understanding it, but since it’s against our intuition, it is a little unnatural.
I also think the majority of the students understand the ideas of the Theorem of Bounds on Zeros. At first, I think the notation did not make sense to the students. There is a lot of letters and numbers involved, and I think the students were intimidated by what it all meant. After I went through one example, I think about half the class understood what was going on. After the second example, I could see that most of the students felt like they would be able to do a similar problem on their own, or at least with a partner. After looking through the worksheets/formative assessments, I believe most students fully understand the process.
There were two main points of confusion during the lesson concerning this theorem. First, students would forget, especially during the worksheet, that they needed to factor out the coefficient in front of the largest exponent term if the number was not 1. Students seemed to catch this mistake fairly quickly and would simply start over. A more difficult concept for students to grasp was the lack of including an =1 in the list when finding maximums. Many students wanted to include this term in these formulas and would get confused why it did not work. While I did not have the best explanation, I think I helped clarify by saying that the an =1 is accounted for in the setup of the theorem.
Overall, the students seem to understand the theorems and the processes that go along with them.
2. What changes, omissions, or additions to the lesson would you make if you were to teach again?
If I could reteach this lesson, I would try to challenge the students a little more. While many students thought the theorems were confusing, there were students who understood them very quickly. It would have been great to have a challenging version of the problems so that the students would have been more engaged and thinking deeper. I was concerned with the students who did not understand while I should have given just as much attention to the students who needed more difficulty. Even more, by having more student involvement at the board, I think the students would have been more engaged. I started with bringing one student to the front of the room but did not continue this practice throughout the class period. I think it would have helped the students who understood the material to explain their thinking to the rest of the class as well as helping the students who still had some confusion gain a new perspective from a peer.
Furthermore, I wish I would have shown the whole class the proof of the Remainder Theorem. While I had a print-out of the proof and showed it to some students during their work time, I think a whole lass demonstration of the proof would have been very beneficial Not only would it have challenged those students who understood the material right away, it would have provided more of a basis for how the theorem works. The proof is not too advanced, and with enough explanation, I think the majority of the class would have understood. Also, by learning how the theorem works, the students are more likely to remember it. The proof is also interesting for those students who want to go above and beyond the computational components of math. Even more, I could have used a demonstration of this proof to add more instruction and fill the 90 minute period better. There were about 10 minutes at the end of class where most students were done with their work, and if I had a little more material to cover, I think I would have had better timing throughout the whole period.
Finally, I need to remember the level of mathematics that the students are at. While I am writing on the board, I am talking and explaining aspects of the equations that the students have known for years. For example, when walking through a problem of finding the bounds, I said “the absolute value of -9 is 9”. Mr. Mierzwa pointed out that they all know that and do not need to be reminded of these basics. As a teacher, I want to project the idea that I believe in my students’ ability and that I know they understand the basics. The last thing I want them to think is that I think they do not know anything. Also, by cutting down on all these clarifications, I am talking less which may allow the students to think more and access their prior knowledge.
3. What do you envision for the next lesson?
For the next lesson, I think it would be a great idea for the students to gain more experience learning how to graph the polynomials in the correct window and find the zeros. Since they now know the Theorem for the Bounds on Zeros, they will know how to choose a good viewing window for their graph. Since graphing and interpreting the graphs of polynomials is an important skill, I think spending some time on this topic in the next lesson would be very beneficial and worthwhile.
Overall, I had a great experience creating and teaching this lesson. After teaching a few lessons, I am discovering that I am fairly comfortable and relaxed with students. Now I know these students well and feel comfortable being myself around them. I think that I need to get better at getting students on task and stopping them from getting distracted by their friends or their phones. I am too lax at the moment and need to remember that I am the authority figure in the room.
I also think I am getting better, and quicker, at planning lessons. Now that I know my class, I understand, for the most part, what they need from me in order to learn new material. This was a good lesson to teach because I had to reteach myself the material, a task I know I will have to do when I am teaching a variety of classes. It was a good experience to have to relearn the material as well as figure out the best way to teach it to the students. Finally, I think I still need to practice more in teaching to different styles of learners. I should try to do more visually and kinesthetically. While I wrote examples on the board, I think I should do more to help a variety of learners learn in the way that is best for them.