On the following pages are my lessons for my unit plan as well as my unit reflection. This unit covered Chapter 10, Sections 1-4 of the Precalculus Book. The main topic of this unit was Systems of Equations. Here is an outline of how the 10 days were used to teach the material:
Lesson 1 and Lesson 2: Systems of Linear Equations: Substitution and Elimination
Lesson 3 and Lesson 4: Systems of Linear Equations: Matrices
Lesson 5 and Lesson 6: Systems of Linear Equations: Determinants
Lesson 7 and Lesson 8: Matrix Algebra
Lesson 9: Review Day
Lesson 10: Test Day
Literacy, Technology, and Numeracy:
I was able to incorporate literacy into my unit by having students explain the processes we were completing. Rather than just going through them, the students needed to be able to summarize the general methods as well as justify their steps as they went along the way. Furthermore, they needed to be able to read word problems and interpret their meaning throughout the unit and on the assessment.
I incorporated technology by using the graphing calculator numerous times. The students are getting more and more familiar with how to use it, so it was great to be able to show them ways they could use the calculator to solve their problems. I also incorporated Google Forms into my unit. The students used them in class, and they could go back to the form whenever they wanted in order to study.
Numeracy is involved in math class every day. Students need to have an understanding of the underlying mathematics in every problem they do. They have to be able to know if an answer makes sense and is reasonable. Furthermore, the are expanding on their mathematical experiences every day. They need to use math they learned in elementary school, like fractions, all the way up to everything we learned the previous day.
Instructional Strategies:
1) Lecture
2) Guided Practice
3) Cooperative Learning in Groups or with Partners
4) Individual Practice
5) Discovery Learning through Gallery Walks and Complex Applications
Description of Topic:
Systems of equations are a set of mathematical equations that are solved together in order to the find values of the variables that satisfy each of the equations. When solving a system of two equations, the result is simply a point where the two lines would cross in a coordinate plane. When solving a system of three equations, the result can be a line or a point where the three planes intersect in a 3-dimensional space. By conceptualizing what is occurring when solving the system, we are able to actually visualize what the result actually means.
There are many methods for which to solve a system of equations. In this unit, we cover four of the methods. The first involves substituting and eliminating the variables in the equations. This process works well with 2 or 3 equations, but would be too difficult to complete once more equations were involved. Thus, we need to introduce the idea of a matrix to students. Matrices are essentially a shorthand way of writing a system of equations. There are rules we must follow when reducing a matrix to find our solutions, but this method is much more efficient once we start to have 3 or more equations in our system.
Another method that we learn in this unit is Cramer's Rule. It is a process that involves solving for the determinant of a matrix and comparing the values found to find the variables. This is a great method to use when the matrix has a lot of 0s and 1s as entries. This method is also efficient to use, yet if there are not a lot of 0s and 1s, and there are many rows to the matrix, the process can get very tedious and intricate. It would be very easy to make a simply mistake with Cramer's Rule, thus it is incredibly important to stay organized and be systematic with one's approach.
Finally, the students will learn how to find an inverse of a matrix. This will be a topic that relates greatly to the mathematics they have known since elementary school involving the real number system. This is also a great review topic to do at the end of the unit because it ties in many of the methods we have done before. The students are extending the idea of row reducing a matrix, now the matrix is simply augmented with the identity matrix. This method tends to involve a lot of fractions and other numbers that are more difficult to work with. Thus, this is a process that is usually not used as much when solving matrices by hand.
Relevance of the Topic:
Although not each topic is covered in the Common Core State Standards since this is an advanced math class covering college level material, there are many ideas that are represented in the standards. The elimination and substitution method is represented in the standards as well as creating a matrix, performing matrix algebra, and understanding the identity and inverse matrix. The more complicated methods of reducing a matrix into reduced row echelon form, Cramer's Rule, and the inverse method are not directly stated as standards in the Common Core since they are advanced topics beyond the high school level, yet they are still really beneficial methods for the students to learn. Thus, this unit is not fully represented in the standards, but there are the most basic aspects of the unit that are incorporated into the standards.
This unit fits nicely into the scope of the semester. Students, in the previous unit, gained an understanding of functions and how to solve for variables. Thus, this unit is a great extension into working with more than one equation. They can use the notation and ideas they learned in the previous unit and now apply them in more complex problems. Thinking about the unit that will occur directly after this unit, they are also directly tied together. The next unit will be the second half of Chapter 10. Students will learn how to graph systems of inequalities in order to see the values that satisfy the inequalities. Rather than simply having one solution, there are an infinite amount of solutions that make the inequalities true. Even more, students also need to use the information from this unit when they complete partial fraction decomposition, another major component of the next unit. Without knowing how to solve the systems at the end of the procedure, students would not be able to find the answers to the problems. Thus, this unit ties nicely into what the students learned previously as well as preparing them for the content that is to come.
Rationale for the Topic:
Systems of equations are a very important topic for students to gain experience with in order to help prepare them for their future math courses. Not only does it introduce the idea that we can look at 2-dimensional and 3-dimensional figures, which play a big importance in advanced math classes, but it also helps students gain the important skill of critical thinking and logic. Since we are learning four methods, and within each method there are countless options for how to go about the problem, students have to be smart about which step to do next. They have a lot of control over the problem, and I think this is incredibly beneficial as students grow as math students but also people in the real world. If we can think wisely and choose a course of action that makes the most sense and makes our lives easier or more efficient. Having those skills is imperative to doing well in any math course as well in the 21st century world.
Furthermore, students really learn the skill of being organized and moving through a problem in a systematic way. Without this skill, the problems are nearly impossible. They are very intricate involving many steps, so if students want to be successful, they need to learn how to stay organized and be systematic with their thinking. I think it is really important that they learn these skills for any class and for any aspect of their life. Students need to keep track of all of their classes, their lives outside of school, and plan ahead for their future. In all of these scenarios, if students can keep their lives organized so they know what needs to be completed, where to put important information, and so on, they will be that much more prepared to take on everything that is going on in their lives. Thus, learning to be organized is a crucial skill. Also, learning how to think systematically is incredibly useful. Especially if any student is considering a major or a profession that involves calculus or higher, students are going to need to know how to logically work through a problem. With a college-level math class, one math problem could take multiple pages of space as well as multiple days of work. Thus, it is critical that you know what you are doing and are able to follow your thinking. Just like with anything in life, if you think about something logically, you are ready for anything that can happen in the future.
Goals:
Goal #1: I want students to be able to solve systems of equations using all four methods in this unit.
Sub-goal 1a: Students should be able to organize their work in a systematic way.
Sub-goal 1b: Students should be able to support their decisions in the steps they choose to take and recommend a course of action in completing a problem.
Sub-goal 1c: Students should be able to summarize each of the four methods and their procedures.
Sub-goal 1d: Students use a graphing calculator, when appropriate, to assist in solving equations.
Sub-goal 1e: Students can compose a system of equations, and evaluate, given a real-world application.
Goal #2: I want students to be able to set up, reduce, and perform operations on matrices.
Sub-goal 2a: Students should be able to compose a matrix given a system of linear equations and vice versa.
Sub-goal 2b: Students should be able to systematically reduce a matrix and justify their reasoning for their procedure.
Sub-goal 2c: Students should be able to express how to add, subtract, and multiply matrices.
Objectives:
Students will be able to:
Goal #1:
Perform the substitution, elimination, and calculator methods to solve systems of linear equations
Analyze and interpret the solution of a system of linear equations
Construct a system of linear equations that describes a real-world application
Organize their steps in solving systems of three linear equations
Solve systems of three linear equations using elimination and substitution
Evaluate the determinant of a matrix
Solve a system of equations using Cramer’s Rule
Evaluate the determinant of a 3x3 matrix
Develop the determinants to solve a system of equations using Cramer’s Rule
Develop the inverse method to solve a system of equations
Summarize why we multiply the inverse by the solution matrix to solve for the variables
Goal #2:
Compose a matrix given a system of linear equations
Reduce a matrix into row echelon form and reduced row echelon form
Compose the row operations to reduce a matrix
Solve a matrix using row operations
Add, scalar multiply, and multiply matrices
Judge when addition and multiplication are appropriate