January 17, 2016

Solving One Variable Equations

I teach ninth graders who arrive to high school without a solid foundation in all the prerequisite math skills. Sometimes I have to correct firmly held misconceptions - students are sure their teacher last year did it that way. Other times students are equally sure they've never seen a topic before. So when I teach a concept that students may have seen before I aim to be as clear as possible. Many teachers think that the best way to be clear is to provide very detailed step by step directions on how to complete a process. But my version of clear is providing the bare minimum information for students to be successful. After we played with pennies, I boiled that experience down to two pieces of information to solve one variable equations:
  • Combine like terms
  • Use opposite operations

As I continued, I recognized that students needed a bit more clarity on when to use opposite operations (they were continuing to subtract from the same expression twice) so I modified it to:
  • Combine like terms
  • Use opposite operations when terms are in different expressions

Now when I encounter a student who is stuck when solving an equation, I ask them "where are your like terms?" If they're in the same expression - combine. If they're in different expressions - combine using opposite operations. If they're inside parentheses or absolute value - uh oh, they're trapped! Then students can either choose a different pair of like terms or apply knowledge of the distributive property, definition of absolute value, etc. 

Limiting my instructions to two items means students have a chance of remembering them. It means students learn to make choices: some kids like have the variable on the left side - go for it! Some kids want to combine all like terms within an expression first - awesome! Some kids don't think ahead and move terms back and forth across the equal sign several times - you're making progress! It's really important to me to value all my students ideas. Later in the unit I might stop a kid and say - you're moving all the numbers to the same expression as the variable, it's legal algebra but it's not the fastest way to get there. But at the beginning? Yes! That's a great idea! Well done recognizing like terms!

When we get to inequalities - same rules and one additional note on how to shade. When we get to equations in more than one variable - same rules and a note that an expression is a valid answer. By giving students the bare necessities I'm making it easier to see the connections. It still takes an awful lot of practice and some students struggle at first with paralysis of choice - it really doesn't matter whether I combine the numbers or variables first?? - but it's important for them to start making some decisions as early as possible so they aren't entirely paralyzed when they reach trig identities and the only way to solve them is just trying a substitution to see what works.

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