Elimination to Solving Systems (1)
Lesson 6 of 13
Objective: SWBAT solve systems of equations by Elimination
To begin this lesson on Solving Systems of equations by Elimination, I pair students up and hand each a Launch Slip. One student in the pair will solve the system by addition and the other by substitution. After solving, each pair of students should compare the answers and methods used with each other.
Question 2 asks to state which method was easier to use and why. In this case, addition is probably the easier way to the solution, but it's always interesting to hear what students that chose substitution have to say. The idea is to teach students how the way a system is written can lend itself to one solving method better than another. Yet, students sometimes may prefer a method simply because they feel more comfortable with it. This is fine, as long as they demonstrate that they know how to solve the system correctly and understand the underlying concepts.
I ask volunteers to solve the system using both methods on the board. Then, I ask students to share their responses to Question 2. When I do get someone that prefers substitution over addition, I encourage discussion and debate over the latter. Many times, the student(s) will see why addition can be easier here, after other students offer their explanations.
The short Pencil Pierre PowerPoint shows our cartoon character, “Pencil Pierre” trying to make a system problem easier to solve by changing the sign of one of the coefficients. But does Pierre really help?
The purpose of this part of the lesson is to tap into the notion of equivalent equations. I want to lead students to see that Pencil Pierre needs to change ALL the signs in the equation, and not just one sign. In other words he needs to multiply the entire equation by -1 and obtain an equivalent equation, hence, equivalent systems.
I leave the last slide on the screen as students answer the questions on the New Info Q Pencil Pierre. Based on their background knowledge, students should be able to respond correctly with little intervention on my part. Later in this lesson, they should be able to figure out how to "eliminate” variables with coefficients other than one, in order to solve the system.
I ask that students use the back of the slip to write their responses as I walk around answering any questions. I want students to understand that the idea here is to use equivalent equations that allow solving the system by linear combination. I then call on volunteers to share answers with the entire class. (In the reflection in the previous lesson, I explain why I prefer students multiplying by negative one, instead of using subtraction.)
Expected answers for the questions are:
1) Pierre believes that changing the sign of y will eliminate the y variables by Addition, resulting in one equation in terms of one variable x. This is actually true.
2) Pierre is wrong despite having eliminated the y variables because he changed the second equation, resulting in a different system, that has a different solution. What he needed to do was multiply the entire equation by -1 (Multiplication Property of Equality), resulting in an equivalent equation and therefore, equivalent systems. Then proceed.
The Video Narrative explains the purpose of this Application section. The handout Skills Worksheet contains four questions in which student pairs have to assess a system and answer questions in reference to the elimination of a variable, then solve the system.
In Question 3 I've shown two very common mistakes students make when solving systems. Being able to locate these and realize what has been done will certainly help students avoid such mistakes.
In Question 4, students should figure that multiplying the second equation by the reciprocal of 10/3, keeping the coefficient negative, will lead to cancelling the y variable. I ask students not to convert any fraction to a decimal.
As I walk around the class assessing students, I make sure partners in each group share the work equally in Questions 1 and 2. I encourage partners to look over each other's work and find mistakes themselves, if any, instead of me pointing these out. Making sure students discuss the problems among each other saves time for the closure section, in which they write the problems on the board.
To close the lesson I call on volunteers to write their responses on the board. I tell students to only write the answers, to save time. Students should hand in their work on their way out of class. The alternative skills worksheet Alternate Skills Worksheet is for students who I find are struggling with the work. I like to make sure these students are able to eliminate a variable by multiplying one equation in the system, before they go on to the next part of the lesson.
The Homework is designed so that these struggling students can handle the first teo questions. This would be enough to be given credit for. Most students will be able to do the entire homework sheet.