For today's Warm Up assignment, I provide two systems of linear equations that ask students to apply the strategy of elimination in order to solve. This strategy was introduced in the previous day's lesson, so students are familiar, but may not be wholly confident. As students work, I move through the room watching students' approaches. I make note of anyone struggling on my observation clip board, so I can bring them to my table to work during Work Time, part 2, as we work on scaffolded problems.
Once the timer sounds indicating the end of Warm Up, I select students at random (from the cup of name sticks) to share their solutions. I then pose a question to the class: How could we prove that this is the correct solution? I want to see if students realize they can use previously learned strategies to check their work. Typically, at least one student remembers that we can substitute our answer back into the equations to test them. As students coach me in this process, I follow their directions at the board. Once we have solved the two systems to everyone's satisfaction, we move to today's learning objective.
After coming to consensus about the solutions for the two Warm Up questions, I introduce today's Learning Objective. I remind students that the objective has not changed from the previous day and that we are going to continue practicing the strategy of substitution but with systems in different forms. I then remind students of the forms we saw previously and ask how we would use substitution to solve it. With the class coaching me, I solve the system. To reiterate the idea of checking our work, I then ask the class to help me prove that this is the solution, which we do by substituting our values back into the system of equations.
I ask students to give me a quick learning scale regarding this strategy by asking the to hold one to five fingers to their chest (1 = I'm lost and 5 = I could teach others this strategy). I want to make sure the vast majority of students are ready to solve a system in which neither equation has been solved in terms of one of the variables. Although this adds just one step from the previous day's lesson, I have found that this is the step with which students seem to struggle most.
If the majority of the class indicates 3's or lower, I change the day's plan and provide more practice over the previous day's lesson. If, on the other hand, only a handful of students indicate 3's or lower, I move on to the second example to show students what type of problems they will learn to solve in today's lesson. I reveal the second example and ask students to tell me what they notice about this example problem. Students are usually quick to recognize that there is no "y=" or "x=" to substitute. I then ask them to turn and talk to their neighbor about what we might do about that if we want to solve the system. I then eavesdrop as students talk about it.
Invariably, one student realizes we can solve one of the equations in terms of a variable, so I ask him/her to share their thinking. I then model while recoding what was said so that students see me solving one of the equations in terms of a variable. I explain this is a strategy that is very helpful in algebra and that I want them to practice using it during Work Time, Part 1.
To give students some practice with solving for one variable and to help them see how this assists in finding the system's solution, I provide two quick practice questions for Work Time, Part 1. I set the timer for six minutes and ask them to solve the two systems.
Once the timer sounds, I select students at random to share first how they were able to solve one of the equations in terms of a variable. I select another student to explain how the newly rearranged equation can be used to solve the system. Once the class is done Building Consensus about the solutions, I explain that next, they will have an opportunity to do additional practice with the new skill.
For Work Time, Part 2, I have created an assignment of scaffolded systems problems called Green-Yellow-Blue Substitution that students will work on. I explain that they must solve at least five systems during work time and no more than three of the five can be green. Students can then decide if they would like to tackle two yellow, one yellow and one blue, or two blue cards to complete the assignment. For students who enjoy a challenge, I encourage them to complete all ten cards for a bonus (in our class token economy).
I ask students to complete all their work in their journals so that I can give them credit. Once the timer sounds, I ask students to prepare for a ticket out the door.
Although students have provided me feedback throughout class about their level of learning how to apply substitution in a system of linear equations, I would like to see their independent work to know where each student might be struggling. For this feedback, I employ a Ticket Out the Door
which is one system of equations for students to solve. As students leave class, they hand me their index card with their work so that can then analyze their responses and provide additional help as needed for those students who may need additional time or practice to develop deep understanding.