Before beginning the lesson, I divide the classroom boards into 3 sections. I title each section with a sign (or write in capital letters):
2. LINEAR COMBINATION
I then project the following system on the board:
2x - 5y = -9
3x + y = -5
I divide the class in 3 parts by assigning a number from 1 to 3, to each student. Students must solve the system using the method corresponding to his or her group number. I hand out graph paper to those students who are solving by graphing.
The main goal of this opener is to access prior learning and encourage a brief discussion among students on which method was best for solving this particular system and why. When the class is done solving the system I randomly choose 3 students, one for each method, to go to the board and solve the system. Graphing the system on the board may be difficult. I may give one or more students a sheet of graph chart paper to work on.
Since we are reviewing methods today, all three students should have the same solution (-2, 1). The student solving by graphing should have this solution or something really close, if they graphed correctly. I conclude the opener by asking the class which method they think is the better route to solving this problem. I expect that most of my students will say Multiplication (Linear Combination), but some may say substitution. It is always interesting to hear students debate over this. Mathematicians do have their favorite methods.
To describe a word problem using a system of equations, we need to figure out what the two unknown quantities are and give them names, usually x and y in my classroom. Next, we need to use the information we're given about those quantities to write two equations. But how can students know when a word problem requires them to write a system of linear equations?
I tell the class that once they know that the problem requires a system, when writing the equations, usually, one equation will relate to quantities (total number of coins, number of tickets sold), and the other equation will relate the values (price of tickets, weight, or number of wheels). The Word Problems Systems Slip sums up these ideas. I share a copy of this with each of my students and I have them paste it in their notebooks for reference.
The application section of today's lesson consist of word problems in which students must write a system of equations and solve by any method.
I hand each student the Pirate Problems handout and divide the class into working groups of twos or threes. Students should read the Pirate Problems silently within their groups before working together to solve them. As they work, I try my utmost not to let students lure me into giving too much information. I really want them to struggle with writing the system of equations and using it to find a solution (MP1, MP4).
Teacher's Note: When my students get "stuck" solving a word problem, they often start asking questions ceaselessly until they get what they want. I resist this "pressure" and encourage them to discuss ideas with their partners, and try different routes toward solving the problem. In preparing to do this, it is important for me to keep the goals of the lesson in mind; if I offer a tip it will be on something outside of the main learning goals for today's lesson.
As I walk around assessing students, I listen to their conversations. I will ask questions (carefully) to help guide a student's thinking.
Note: In part A of the Pirate handout, 4 of the coins were counterfeit and 8 were real gold. In part B, there were 17 bracelets and 35 necklaces in the chest.
Students love to go to the board when they've solved a problem like this one. I call on volunteers to write all their work on the board to parts A and B, and ask the rest of the class if they solved the problems using a different approach. Students may have multiplied instead of using substitution. Or may have found different variables first. I make sure all the possible routes are discussed before going on to closing the lesson. Comparing solutions deepens understanding of methods and processes (MP3).
To close the lesson, I formally assess students' ability to write a system of equations from a situation in a word problem. I give each student an exit ticket and have them complete it independently. I inform the students that they must define two variables in each and write two equations, but they do not need to solve the system.
My students always need more practice handling word problems. The more of these they see, the better they become at writing the system and solving. They soon see the similar patterns among problems involving two equations in two variables. Tonight's homework, Counterfeit Coins, offers some practice with problems that have a similar structure to the Pirate Problems in today's lesson. Students should show all their math work on the paper to be given full credit for it. It is also the only way we can see their work and give proper feedback.