1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Who was it that said, "A journey of a thousand miles begins with a single step"? Was it Socrates? Confucius? Bugs Bunny?

Whoever it was, they probably weren't thinking about algebra when they coined that gem. To many students, solving an algebraic equation may feel like a journey of a thousand miles. Before students begin with their single step, they should probably get a compass or something.

Students should be able to figure out the logical next step in solving an equation using the previous step. Sounds simple, but it takes more than just putting one foot in front of the other. Knowing that whatever is done to one side of the equation must be done to the other is a good start, but that doesn't tell them what to do.

Ideally, students should not be memorizing a set of rules and procedures and using them to solve equations. They should understand how the next step in solving an equation can be logically derived from the previous step, but there is a general format for how to get started on their algebraic journey.

When we solve an equation, it's usually a good idea to get the variable we want on one side of the equal sign. Then, we do what we can to simplify it as much as we can. 

For example, let's solve the equation x² – 3x – 7 = 2x + 17. Subtracting 2x + 17 from both sides gets all our x's on one side of the equation. That's a good place to start.

x² – 3x – 7 – (2x + 17) = 2x + 17 – (2x + 17)

x² – 5x – 24 = 0

We also took the opportunity to simplify by combining like terms. Since the equation we have is a quadratic equation, we can factor it into the product of two linear terms.

(x – 8)(x + 3) = 0

This product will equal 0 when either x – 8 or x + 3 is equal to 0, so we can solve it by setting each of them to 0. As a result, x = 8 and -3.

If students are really struggling, we suggest starting simple. Give them easier linear equations and slowly work your way to more complex ones. Point out patterns in equations, make sure they know the quadratic formula, and remind them of helpful factoring tricks. 

Most of all, get them to practice. It's difficult to take even the first step of a journey if you can't walk. After enough exercises, your students should be able to tackle any thousand-mile journey faster than the Road Runner.