Educational Portfolio
Sayreville High School
Student Teaching
Pre-Calculus
Unit Aims
Generally, this unit will serve to teach students how to graph the various
trigonometric functions by differentiating instruction through the use of groups
and connecting the material to real-world applications.
When
developing this unit, there was a lot beyond the content that I wanted to
include, and a lot of different methods that I wanted to incorporate to make
this happen. Pre-Calculus is an
elective class at Sayreville High School and therefore the students choose to
take the course. Because of this
the classes and the students within the classes, come into the course at very
different levels. You first have
the junior students that have been very strong in math courses throughout their
high school careers and have been on an accelerated math program, taking them
into pre-calculus their junior year.
Seniors come into the course from either taking Algebra II or Algebra III
as juniors. The students are at
very different levels and therefore differentiating is very important.
Mathematics is a very traditional subject in which people generally expect to
see a teacher lecturing and giving notes to students.
Although lecturing can be unavoidable sometimes, this unit allows a lot
of different ways of presenting the material given the nature of the topic.
Graphing trigonometric functions is a very visual subject.
Transformations and shifts of the graphs are typically best described
through visual representation.
PowerPoint presentations of the topic greatly helped the students with this
visual aspect. I think that it is
also a very good tool, not just for the visual learners, but all the students,
as it allows them to understand not just what the numbers are and how to attain
them, but how the numbers impact the graphs of the functions.
It is important that the students have a full conceptual understanding of
the topic, which includes such a discussion.
The students will be able to use graphing calculators on their own,
allowing for more of a hands-on activity for more of a kinesthetic learner.
The
other elements of my educational philosophy support differentiated instruction
as well, such as cooperative learning or group work.
It allows students a different means of grasping the material: through
their peers. Cooperative learning
was very varied throughout the unit.
Groups allow students to ask their peers questions and get clarification
on material that for whatever reason, they did not get from the lesson.
It provides the students with a check being able to check their answers
with their peers. It also allows
the students to talk through solutions and difficulties with peers, enhancing
their understanding and can also bring to light for the teacher common
misconceptions that she can then address.
In this unit groups were able to be included easily because the
difficulty of the material dictated that the most useful activity that the
students could engage in was practicing completing problems.
Completing a good amount of problems is necessary for the students to
grasp and understand the concept and be able to do a variety of problems
themselves. Cooperative learning
activities kept the students interested and focused while completing these
problems. Having such activities as
seen in second lesson assures that the students are working together and
discussing the material, as they are all responsible for giving the solution to
one of the problems. The groups
will be designed and students will be placed in groups in such a way to maximize
the learning experience for each of the students.
The
focus groups will pull in not only cooperative learning, but differentiated
instruction. This unit contains
several different aspects, each of which is difficult in its own way.
Therefore, as a review activity, the students are given the ability to
choose what they feel is the most difficult aspect of the material.
The focus group materials allow for student’s different areas of
difficulty and different student ability levels as well.
The students are given the opportunity to say that they are completely
comfortable with the material, and are then offered the option of playing a
review game instead of focusing on one specific aspect they are having trouble
with. The other students will be
split into groups based upon what they said were their weaknesses, and also
taking into account the make-up of each group to ensure that each group will
succeed in their tasks. The focus
groups allow for peer cooperation which is very helpful, while focusing on
individual weaknesses and difficulties.
The
fourth lesson presented here introduces the unit project which ties together
differentiated instruction as well as real world applications of the material.
The project is a graded assessment for the third marking period, allowing
the students to demonstrate the knowledge that they have gained in a different
manner. The unit project included
several different parts. The first
part consisted of graphing several different functions, comparing and
contrasting different equations, and then specifying the reasons for these
similarities or differences. This
part of the project is very similar to the work that they have been completing
in class, in groups and on their homework.
Asking the students for more of a written explanation of the concepts
brings in some of the goals of the previous unit where the students worked on
verbalizing their mathematical knowledge and ideas.
The
second part of the project consisted of the students completing their own
biorhythms and finding an additional real-world application of the graphs of the
sine and cosine functions.
Biorhythms are based on a sine function curve, but they are also controversial
and something that is a little different for the students to complete.
It appeals to the creative sides of the students as I gave them a good
deal of leeway into how they displayed the graphs, as well as the whole concept
of biorhythms themselves. The
additional real-world application could be one impacting the students directly
or that is a part of a career that uses the graphs for some reason.
Being that the class is an elective class, it can be assumed that many of
the students will be going into mathematics related fields.
Being juniors and seniors, many of the students are trying to decide
exactly what the major is going to be.
This project also allows the students to become familiar with different
mathematical careers that they may be interested in.
In higher level mathematics classes a main difficulty is connecting the
ideas to the student’s lives so they are able to explain to themselves the
purpose of understanding and learning the concepts.
Students who are able to justify the time and energy they spend in
learning the topic are therefore more inclined to give themselves that extra
push to fully grasp the material.
Unit Lesson Plans
The first lesson that was
included below is the introduction lesson to graphing sine and cosine unit.
Instruction of the lesson
was completed
with the aid of a PowerPoint.
The second lesson is also on graphing sine and cosine with
varying period and amplitude. The
students were placed into pre-assigned groups of three to four people to
complete
on a worksheet graphing sine and
cosine functions. Each group was responsible for presenting the solution
and graph
of one problem to the class.
The third lesson that I have included is a lesson in which students were placed
in focus groups to review for a
quiz. The students were given exit cards the day before on which they were
asked to write what they felt were their
weaknesses in the given topic. Given the exit cards I put the students in
focus groups where they were given
packets of information with completed examples and problems for them to complete
during class. The first
group was a
review group in which the students could play a game to review for the quiz.
The second
group focused on graphing sine and
cosine while the
third group worked on cycles of sine and cosine graphs.
The fourth lesson included was used to introduce one of the major assessments of the unit which was a project. The Biorhythms and Sine Curve Project had many different aspects, one of which served to introduce more about the applications of sine and cosine graphs to the students. One such application is biorhythms curves, as described in the project packet. Each student had to construct their own biorhythms based around the idea that biorhythms are basic sine curves with varying periods. They then also had to complete some research and find other applications of sine and cosine graphs. Please take a look at some of the student projects which include graphs and also several different applications that the students found that use the graphs of sine and cosine including weather, music, beef prices, and tide applications.
Unit Reflection
Overall I was very happy with the unit, its presentation and the progress of the students throughout the unit, but there are definitely aspects that I would change. I received a lot of positive feedback from the students in the lessons such as lesson one is which I used PowerPoint to enhance instruction. The PowerPoint gave the more visual students a better grasp of the content. I also found the use of the PowerPoint very helpful as it allowed me to focus more on the explanation of the concepts rather then writing all the information on the board for the students to copy. Yet, I also think that this was a disadvantage because it allowed me a quicker pace then that which some of the students were prepared for. This would be something that I would be able to work on however for future lessons involving such technology. Giving the students abbreviated notes on the material helped but, they noted that it was difficult for them take notes and listen to explanations at the same time. I think that in the future, one of the goals I would add to the unit would be to develop the student’s abilities to multi-task. This is an important skill to have, not in just mathematics courses, but throughout the student’s academic careers.
I also received a lot of positive feedback from the students about the focus groups in the third lesson. I felt that the students that used their time effectively were able to get a lot out of the focus groups, however there were the groups of students that did not focus on what they were supposed to be working on. I reminded these students several times throughout the class that they had a task that they were responsible for working on, but with five to six other groups needing help and guidance, it was difficult to ensure that all groups were on task at all times. The next time that I have students complete focus groups I would change the assignment slightly so that the students would be responsible for handing in work that they completed during the groups so that I would be able to keep a better handle on students that were not staying on task.
One typical mistake that I made throughout the lessons in which I incorporated group work or cooperative learning was underestimating the amount of time that it would take the students to complete the assignment. As soon as you put the students into group you loose the time that they socialize, which I did not account for when organizing many of my lessons. Although I found myself improving on timing and scheduling, I found that timing was a general issue that I had throughout the unit. In lesson two I did not anticipate the activity taking as long as it did and therefore the students had not completed the first eight problems of the worksheet by the end of class. I still had the groups present their problems to the class, but because the majority of the students had not completed all of the problems, many of them did not pay attention to the presentations of the problems they themselves had not finished. I think that the activity was very good in it allowed the students to complete problems and receive help and guidance from their peers while focusing on individual weak areas, but at the same time, I would change it slightly the next time I completed it in class. I would have the students complete the problems that their group was presenting in class, along with any additional problems they were able to finish in the class period. Then I would have them complete the assignment for homework. The groups could then present the solutions to their problems during class the next day and everyone would have completed all the problems and would be more apt to pay attention to the presentations to check their answers.
I was extremely happy with the way that the students completed the Biorhythms Project. Many of them were not excited about the project when they first received it, but the final products that they handed in were very well done and beyond my expectations. I thought that the project really pulled in the idea of real-world applications of what we were learning and more a creative aspect that students don’t always expect to see in a math class. I would have changed slightly the fourth lesson in which I introduce the project to the students. The students had a lot more questions and concerns then I had anticipated and I was not prepared for this time wise. I had anticipated completing my own biorhythms with the students as an example but did not have enough time in the period to complete the example entirely. Although I thought the materials that I gave them with the project, outlined in a very detailed manner how to finish the project, many of the students had difficulty with how to carry out the formulation of their own biorhythms. I would definitely like to use the project again, but the next time I introduced the task I would give the students a period in which they would be able to work on the assignment in class to ensure that they knew how to complete it. I think this would also help the students to understand the full extent of the endeavor. Many of the students underestimated the breadth of the project and tried to complete the assignment the night before it was due and it was clear during grading which students did this. I think starting the project in class would allow the students to see the many different parts that needed to be included in the project and how much time it would take them.
Although I was happy with the unit overall, there are several things I would tweak, so that they would run a little more smoothly for both myself and my students. This tweaking process is one that I think will be a normal experience for me throughout my teaching career.
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