The basic steps for solving a system of equations by substitution, by addition and multiplication, or by graphing are relatively simple; what students often struggle with is the meaning of what they are doing. Students need to master three big ideas about systems of equations: an equation and a line on the graph mean the same thing; there are several different ways to a system of equations, but all give the same answer; and, systems of equations have real-world applications.
The equation and the line mean the same thing. Many students struggle with the basic concept that the line on the graph and the equation are two different ways of representing the same thing, and that the solution to a system of equations is where two lines cross. Play a game to help students master this idea.
Turn your classroom into a graph, and let your students practicing graphing systems of equations by going on a treasure hunt. Use rows of desks, or simply signs along the walls of the room, to lay our your classroom like a Cartesian plane. The only clue students get is a system of equations to solve by graphing. The "treasure" (a candy bar, goody bag, or other prize) will be hidden at the point where the two lines cross on your life-sized "graph."
No calculators for this game, since students will "check" their work by finding out if the treasure is where they expect it to be. For added excitement, divide the students into teams who can race to find the prize.
There are several ways to get the same solution. Many students are confused by the need to learn several different ways of solving the same system of equations. Students will gain confidence when you use the Sinister Imposters game to help them practice different methods for solving the same system of equations.
Announce to your students that a team of numbers has been handpicked for a top-secret mission. The problem? Your team has been infiltrated by one or more enemy agents. Headquarters has devised a test to find which numbers are the bad guys -- you guessed it, the secret test is a system of equations to solve! Numbers that are solutions to the system are good guys; numbers that aren't, are enemy operatives. It's up to the students to find out who belongs on the team, and who doesn't.
Provide students with several values for x, and several values for y. In each list, some of the numbers should be real solutions, and others not. Because this is a top-secret mission, your students must use all the different methods they know to keep the enemy from finding out what they are up to. Divide the students into groups, and have some use graphing, others substitution, others addition with subtraction and others tables to test the numbers on your list.
Systems of equations are useful in real life. Mixture problems are a staple of this part of the curriculum. Motivate your students by simply taking mixture problems out of the textbook and into the real world. Use a jar filled with different kinds of candy or fruit (that you have already counted, of course), and let your students use mixture problems to find out how many gummy bears or how many Snickers bars there are. Divide students into teams, each looking for a different number; or, have the teams race -- the first team to come up with all the numbers gets to choose one item from the jar.
Work problems are another important application. Help students relax with work problems by choosing volunteer applications for imaginary jobs. You can even bring uniforms or tools to go with the story. Have students design their equations and work their problems. If you want even more motivation, announce that those who get the correct answers will be paid for their jobs, or have students guess ahead of time whether something seems like a fair deal or not, then explain why they were right or wrong.