Unit Reflection:
1. To what extend were unit goals achieved?
As a whole, I feel like the unit goals were achieved by most of the students. I believe this to be true based on the student’s scores on their study guides as well as their scores on the unit test. There were some lessons the students understood more than others, but I am happy with the results of this unit. First, the students were able to solve a system of equations using the methods from the unit. The method the students understood the most was the elimination and substitution method that we learned first. This is a method that relates most to what students have done before in previous math classes. Also, the students as a class were able to use Cramer’s Rule pretty successfully. This method is one that does not involve much decision-making, simply following a set procedure. Thus, I think students felt pretty comfortable performing this method. The two methods that some of the students had more difficulty with were Row Reduced Echelon Form and the Inverse Method. These two methods are incredibly intricate and involve a lot on the students’ part to make logical and systematic decisions. While many of the students were able to do these methods, others had trouble knowing exactly how to go about the process. Thus, I think my first goal was achieved, but I would still want to reinforce my students’ ability in these last two methods.
Another part of my first goal was to get students to organize their work and work in a systematic way. I think this is a skill that the students will continue to work on the rest of their career as a student, but I saw a great improvement in all of my students’ work throughout this unit. The students work became more clear and easy to follow. Also, after much practice with reducing matrices, it became apparent that students were working in a more systematic way. They had a set procedure that they used each time they solved a problem, and this helped them actually be able to find a solution to the problem.
Concerning Goal #2, I really believe the students now understand how to set up, reduce, and perform operations on matrices. From their pre-test to the study guide and unit test, the students actually knew what a matrix was and how to correctly use them to solve a problem. The students know how to create a matrix given a system of equations as well as perform operations. The only part of this goal that students can continue practicing is reducing a matrix and justifying their reasoning behind each step. I think the students generally understand the process, but I want them to be able to know why they’re performing each step.
Overall, I am very pleased with the outcome of this unit. I think the students do understand how to solve systems of equations as well as how to use matrices.
2. What changes, omissions or additions to the unit would you make if you were to teach again?
For this unit, I would add an extra day in order for students to practice reduced row echelon form. I essentially ended up doing this by having extra time for the students to work on their homework problems concerning reduced row echelon form. The students really need that extra practice time with their peers. It is easy for them to get lost on their own, but when they are initially learning the process, I think it is vital that they work together and know how to go through each step. Thus, I would add an extra day for 10.2, and take away a day from 10.3. 10.3 is not too difficult for the students to understand, so I do not think they would be at a disadvantage if they lost a day.
An addition I would make to the unit would be to have a class set of white boards in order for the students to be able to work on them during class. If they can work out the problems on the whiteboards, and not have to erase small mistakes and start over, I think the students would be more open to the complicated and intricate processes of some of the methods.
Furthermore, I would like to have the students work on even more real world applications in order for them to see how helpful matrices are to solving complex problems that occur. We got some good practice on a few examples, but I would like to get more of a variety of examples so that students can really get a well-rounded view of what matrices can do. I was pleased with their performance on the test of the word problems, and I think spending a day just working on real-world applications would be incredibly helpful for them to practice as well as spark their interest in learning about solving systems of equations.
3. What do you envision for the next unit?
For the next unit, students will be learning more about systems of equations as well as inequalities. The students will be completing problems that can only be solved in the end using one of the methods the students learned in this unit. Thus, rather than having the problem be solving systems of equations, that is only a component of an even larger problem that the students are trying to solve. Even more, we expand on the ideas of systems of equations to systems of inequalities. I am really excited for this section of the course because the students get to do a lot of fun real-world examples with systems of inequalities called linear programming. This will be a great part of the next unit because the students can get some cool projects that relate to other subjects they may be interested in, like history, and really see how useful linear programming is. Thus, as a whole, the next unit really expands on this unit, so I can have students recall everything they learned. It will be great to really reinforce these ideas and help them get even more practice with systems of equations. I plan to have a lot of projects and student learning in this next unit, which I think the students will really benefit from.