The Whole is Greater than the Sum of Its Parts
Lesson 3 of 12
Objective: SWBAT prove that replacing one equation in a system by the sum of that equation and a multiple of a second equation produces a system with the same solution. SWBAT apply the combination method to solve systems of equations.
To open class, students independently work on the day's Entry Ticket for 5-10 minutes and get prepared for learning. Today's Entry Ticket focuses on solving systems of equations using substitution. I have been having students practice creating equations right from the beginning of this unit, but I provide equations for this entry ticket because I want to zoom in on the skill of solving the equations.
Problem 1 is a system that lends itself to the substitution method (because the second equation is already written with y in terms of x. Problem 2 can be completed using substitution by rearranging one of the equations to isolate x or y. However, this system lends itself to combining the two equations with addition to eliminate the y. When reviewing Problem 2 with the class I look for students to identify this strategy/pattern and if it does not come up I show students how to solve the system using the new method. This is one way to show the utility of the process/tool that we will be learning about in class today.
Students take Two-Column Notes during this section of class on applying the linear combination method to solve systems of equations. I model the notes on the whiteboard in two column form with the PowerPoint Slides being projected on the screen as well. At this point in the year some students still struggle with translating a PowerPoint slide into two-column note form and benefit from a model set of notes on the whiteboard.
One way to Differentiate Instruction in this section is to provide students with a typed set of notes, allow students to use a word processor to type notes, and/or assign 2 or 3 students as the class note-takers and make copies of those students notes for the class.
For this lesson I stress the goal of rewriting equations in equivalent forms as an important step in the solution process. I remind students that we are looking to get to the point where we have a single equation with one variable. From that point, we can use the tools and strategies from our prior work on solving equations with one variable.
I teach from this perspective in order to help my students make connections between units. I want them to see solving systems as one or two additional steps, rather a brand new set of procedures that they have to learn from scratch. The more I can support students in realizing they already possess many of the tools to solve these problems, the more engaged and motivated my students act.
After reviewing the class notes for the day, students work in collaborative groups to complete 1-2 problems on solving systems of equations (see Collaborative Work: Word Problems and the Combination Method to Solve Systems). These problems are revisions of a set of problems the class worked on for the lesson on Creating and Interpreting Equations. I like to use similar contexts for word problems from time to time so students have some familiarity with the scenario/problem.
I tend to assign students a small number of problems with high expectations for completeness. I find that this strategy helps students work together to understand concepts at a deeper level. When I assign lengthier problem sets, students rush, rather than reflect, as they complete tasks.
For this activity, each group is asked to document their work on one of the white boards in the room. I let the class know that we will be celebrating each other's work with a Gallery Walk after the group work.
Differentiation: As an alternative task, if I think that my class will benefit from having the equations already written and setup, I might use problems from from the Kuta Software Worksheet - Solving by Elimination (Combination) or the class text.
When each group has completed their work on the white boards, I have the class rotate between the different problems in a Gallery Walk. The intent of the activity is for students to receive feedback on their work and self-check their level of understanding. Students can also compare and contrast worked out solutions to their group work as an alternative to the gallery walk. I manage the time for the Gallery Walk as follows - if there are 6 problems to present, give each group 3 minutes to review and provide feedback to one problem and then rotate.
One way to quickly generate constructive feedback from classmates is to have each group put their feedback on sticky notes:
I like to give different groups different colored sticky notes so I can assess the detail and level of feedback each group is giving.
The Gallery Walk gives students an opportunity to practice Math Practice 3 and Math Practice 6. Presenting work on the white board for others to review pushes students to construct an argument in writing using precise language and notation. The sticky note feedback gives students peer feedback that helps the problem solvers understand how others interpreted and understood their work.
To conclude the lesson I have students complete an Exit Ticket. The activity can be used in a variety of ways. I like to utilize the ticket as a temperature check or quick formative assessment to gauge student understanding of the day's learning objective. The work I get back also informs my instruction and level of review problems for upcoming lessons.
In this particular Exit Ticket I give students two systems to solve - both that lend themselves to solutions with the combination method. I developed a problem where students have to subtract the two equations to eliminate one of the variables because forgetting to switch the sign for all terms is a common mistake I see students make.
For homework I assign a problem set on Delta Math. I typically have students complete ten problems in a row, with a penalty of one for an incorrect response. In other words, if students complete the first 8 problems correctly, but then get the 9th incorrect, then they have credit for 7 in a row and need to complete 3 more problems in a row to get full credit for the assignment.
For a great overview video on Delta Math and how to set up a free account, click here:
If students have trouble with the homework I also direct them to review Khan Academy videos on solving systems using the combination/elimination method - here is a Khan Academy Video using a Potato Chip problem to Solve Systems video. In this video SK walks students through the steps of defining variables and creating equations to model the potato chip scenario in addition to solving the system.