Objective/Rationale
My students regularly ask "Why do I need to know this?" and "When am I ever going to use this in life?" I think the most pertinent answer to these questions is that the content in my classes teaches them to be logical and critical thinkers. However, I think it is also important to acknowledge that a lot of my students are asking these questions because they do not understand the math. If my students are getting the mathematics education they deserve, they should have an in-depth understanding of the topics involved in algebra and students should be able to justify why the mathematics they did worked and how to interpret what the mathematics means. By frequently incorporating word problems in my classroom, I want to help my students start to see situations in which basic algebra is used all the time. If the word problems are accessible, my students will be less discouraged and they will be acquiring more in depth knowledge of the content.
It has been my experience that, when students enter my Algebra I class, either as 9 th graders or 10th graders repeating Algebra, they seem to have been programmed to think that, in every problem, they need to find an answer or to solve for x. They usually approach a problem by solving for x (or any available variable) regardless of what the question directs them to do. I believe that this is due to the fact that they do not have a clear idea of what a variable is or the ways variables are used. In addition, they do not have an adequate understanding of the equals sign. For so many years, the work needed to develop the understanding has been ignored and step-by-step processes have been emphasized. Most of the time I find that my students have memorized "shortcuts" that were given to them without justification or explanation. A lot of these shortcuts come directly from the bold or boxed terms in the textbook and students think they are important to memorize because they have been emphasized in this way. My students are memorizing the bold without understanding why it is bold. This leads to my students guessing what they should be doing without thinking why. Students learn all these ways to compress and make things more concise and simpler without having meaning attached, so that when they are asked to decompress or "unpack" what they did, they are clueless and left behind. My goal in this unit is to give my students the foundation and tools they need to be critical thinkers and be successful at justifying their process. The main tools I will focus on in this unit are:
a) a sense of what variables are and ways in which they are used;
b) an understanding of how to work with expressions, including
i) what expressions are,
ii) what they are used for,
iii) how expressions are formed, and the grammar of expressions, including reading, interpreting and writing expressions
iv) what it means for expressions to be equivalent, and how to manipulate expressions to produce equivalent ones, especially to simplify them;
c) what equations are and how they are used, including
i) manipulating equations,
ii) solving equations, and
iii) writing equations that represent situations presented verbally (i.e., translating word problems into equations).
I also want them to understand that solving an equation is a process of logical reasoning.
As their teacher I intend to emphasize that shortcuts, that they may have seen before entering my class, are a privilege for the experts and that they will be permitted to use the shortcuts once they have proven to me that they are truly experts. I believe once a student has been asked and expected to justify his steps throughout a process of simplifying or solving by decomposing/justifying, and has shown that he is clearly an expert, he should no longer be required to show all the steps. Students should be encouraged to use a "shortcut" once they have an understanding of why it works. Not only will this help them as they go through high school, but my students will become more familiar with the two column proof context. In Geometry my students are presented with the two-column proof and asked to explain their reasoning process, which has never been expected before. Therefore, requiring my students in Algebra to follow the justification process will be serving a dual purpose: teaching them to always think critically and ask why they are doing what they are doing; and preparing them for a format for later by making connections.
When my students are presented with a problem set, they rarely read what the problem is asking them to do, nor do they understand why they are doing the processes they are doing. Specifically, I believe that my students do not have an understanding of variables and the equals sign. Without such basic understandings, they cannot know how to approach a problem or explain it clearly. This superficial understanding of the mathematics involved and the inability to "unpack" carries throughout Algebra 1 and into all of my higher-level classes. At my school this poses a huge problem as they progress through the math courses, but are unable to build on understanding because they had no foundation to begin with. The majority of my students have to retake a math class sometime in their high school career due to the fact that they are lacking the fundamental knowledge. In the state of California, students now need three years of different math, and at my school that means they must complete Algebra 2. Failing a class puts a student at serious risk of not graduating from high school on time. Therefore it is imperative that a student is engaged in the content and can demonstrate her understanding clearly or her success in high school and her ability to graduate is at stake.
The major problem for my Algebra 1 students is their confusion between expressions (recipes for computation) and equations (answers to particular questions). I chose to focus on expressions and equations because this is where I first start to see my students struggling with the concepts. I think this is because it is when they are asked to use the tools they have previously been taught. I see a fundamental misunderstanding between what an expression is and what an equation is. I think this is because we often take an expression and when we simplify an expression we use an equals sign to create an equivalent expression. Because there is no concrete understanding of what an equals sign represents, my students see an equals sign and immediately want to solve for the variable or find an answer. This different contexts of the equals sign causes a lot of confusion. Often times my students "do the problem correctly" except at the end add an equals sign and solve an equation that never existed. When I say, "do the problem correctly" I mean they took the correct steps to simplify an expression, but misinterpreted what the question was initially asking or what the expression represents. Therefore it is my specific objective in this unit to decompose and break down the fundamentals of an expression, equivalent expressions and the differences among as well as connections between expressions and equations.
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