5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

When your students were little children, they were probably adorable—until they started questioning everything around them. One moment, they barely know how to string a word together, and the next, they bombard you with questions like "Why is the sky blue? Why is water wet? If Fluffy can pee and poop in the backyard, why can't I?"

Yes, they were just eager to know more about the world around them. Yes, this curiosity was a wonderful thing and should have been rewarded. But now, it's payback time.

Students should think of solving equations and inequalities as answering questions—like those same questions they asked when they were younger. Only they'll be less about poop and pee and more about finding unknown values. Potato, potahto.

Students should know that valid equations are true equations—but just because we see an equation written down doesn't mean it's true or valid. For all we know, the equation could be lying through its little, mathematical teeth. It's up to students to determine whether or not the equation is valid or invalid (i.e., true or false).

You can start with simple equations like 3 = 3 and 1 = 2. Since 3 = 3 is true, it's a valid equation; since 1 = 2 isn't true, it's an invalid equation (but still an equation!). Same deal goes for inequalities, obviously.

Once they can do that, introduce them to equations with variables in them, like x + 1 = 2, for instance. Students should realize that solving this equation is the same as answering the question, "What value of x makes the equation x + 1 = 2 true?"

We don't expect students to use the properties of operations to solve equations for variables just yet. Instead, we can start by giving students values to plug into the equations and inequalities to determine whether said equation or inequality is true at that value. In other words, they should be able to answer the question, "Is x = 4 a solution to the inequality 2x – 1 ≤ 2?"

You might also find that students struggle more with inequalities than equations, so if needed, spend a little extra time making sure they know exactly what to do when an inequality rears its ugly head.