For today's Warm Up assignment, I have provided students one pair of linear equations that Oni has solved using elimination. I ask students to decide whether they agree with her or not. Since there are a variety of strategies they could use to decide, I want to see which students tackle this problem easily as well as those who struggle. I make note of each on my observation clipboard which I carry during Warm Up and Work Time.
Once the timer sounds after 5 minutes, I poll the class to see who agrees and who disagrees. I then ask for volunteers to share how they know. While some students may actually solve the system, others may simply substitute the values into the equations to see if they work. I want to continue to model that I value a variety of approaches, so I ask for other students to volunteer their ideas if they differ.
Once we come to consensus as a class, we move directly to the day's learning objective, which I display on the Smartboard.
After Warm Up, I introduce today's Learning Objective and ask the students to solve the problem displayed. I then point out that this system had one solution. I reveal the next slide and ask students to solve the left problem with me. After multiplying both sides of the first equation by 6 so that we can eliminate a variable, both variables add to zero on the left side of the equation. I then record the right side of the equation. I explain that just like when we solved one-step linear equations and when we graphed solutions to systems, solving by elimination can have three outcomes: One solution, No solutions, and Infinitely many solutions.
I ask which of these options does 0 = -48 match. I then poll the class and ask individuals to explain their votes.
I then proceed to the third example system and ask the students how they would like to solve it. Once we have multiplied the bottom equation by 2, I ask students to notice what is happening. I then poll the class about the solution when we get 0 = 0. I then suggest we put both equations in slope-intercept form to compare. When the equations are equivalent, I ask what that means as far as solutions. I want the students to begin processing the characteristics of the three types of solutions.
I then reveal a graphic organizer for a Solution Analysis. I direct students to turn and talk with their group about what specific characteristics they notice about each of the three example problems we just solved. I set the timer for 8 minutes and ask students to record their ideas in their journals.
When the timer sounds, I ask for volunteers to share their group's ideas. As a class, we come to consensus about whether the characteristic is something that is always true or sometimes true. I record only those characteristics that are always true.
After coming to consensus about characteristics of special case systems, I introduce today's Work Time assignment: Solving Systems by Elimination- Special Cases. It is made up of 10 systems problems that students must first classify by solution. If they decide the system has only one solution, I ask them to find that solution.
I set the timer for 15 minutes. When time is up, I call the students' attention to the Smartboard for consensus building.
For Building Consensus, I once again reveal the graphic organizer used previously to document the characteristics of the different systems. I ask one group to list the numbers of the problems they found to have one solution. I then ask another group to give me the numbers of the problems with no solutions. Finally, I write the remaining numbers under the 'infinitely many solutions' column. I ask for consensus. If all students agree, we move back to the first column and record the solutions.
If we do not have consensus, I ask students to refer back to their chart and site specific evidence that helped them to decide how many solutions the system would have. We continue until all students agree with our classification of today's problems.