In keeping with my namesake, “Eat, Pray, Solve for X,” it only makes sense for me to make my first *real* post about solving for x! My thoughts on Combining Like Terms were just a stepping stone in the process…

In the good ol’ days, I used to solve equations traditionally just like my teachers taught me: +x. +x below the equation on both sides, – 7 -7, etc., etc. I was great at it! I aced just about every math test and quiz as a kid, and so I walked into my first day of Algebra 1 with the **“THIS IS SO EASY!” mentality that a lot of new teachers fall victim to**. Only it wasn’t. I began to dread the days of solving equations because kids just didn’t get it. Oh, they could complete each step individually… but pulling everything together was next to impossible.

- Students regularly
**confused the steps**of combining like terms with moving all like terms to one side of the equation. - Students really did not understand that -2x meant -2*x, so they would
**add 2 instead of dividing by -2**in an attempt to eliminate the coefficient. -
**Students get lazy**really quickly and start writing +x on one side of the equation but not the other. The inconsistency drives me nuts! It’s not mathematical! - FORGET literal equations or manipulating linear equations into y = mx + b form…

I was at my wit’s end when BVD, a special ed team-teacher and dear friend of mine, introduced something she had learned: the river method. **With** **set steps that worked for any equation. Mind. Blown.** I was absolutely skeptical, but dove right in because I had been failing miserably up to that point. **And let me tell you… IT WORKED! ** We led the class through the unit with a foldable, notes, and activities, and consistently practiced with the students for the remainder of the year. **Two years later**, we got feedback from the Algebra 2 team that our kids were able to solve equations like never before. *(This method also contributed to our department’s Alg 1 state test scores rising from a 52% pass rate to 74% in a single year!)* **And if there’s one thing a kid should be able to do when he walks out of my Algebra 1 doors at the end of the year, it’s solve an equation. **

So… **(1)** If you, like me, **are at your wit’s end** and desperately need a change, THIS METHOD’S FOR YOU.

**(2)** If you love the good ol’ days and **don’t see anything wrong** with traditional solving, THIS METHOD’S FOR YOU.

**(3)** If you’re **totally skeptical** because there are literally infinite ways to solve equations, THIS METHOD’S FOR YOU. Basically, these step-by-step “rules” help students to create a framework and develop mathematical understanding of terms, balancing equations, etc. Then, once they’ve mastered the rules, they can venture outside the box because they *get* it. We, as math teachers, are mathematically inclined and it comes naturally to us. Many students just don’t see the world in the same way we do… yet.

**(4)** If nothing else has worked, consider this: one of my challenging kiddos in Algebra Part 1 was failing first quarter with a 22.9%, due to attendance and lack of understanding. He earned a B+ on our Solving Equations Quiz using the river method! Here’s proof:

****Now that you’re convinced… THE RIVER METHOD!****

When I teach this unit, I first **talk with students about inverse operations**. (The inverse of adding 6 is subtracting 6. The inverse of dividing by -4 is multiplying by -4.) We even do a quick worksheet practicing this concept. Then **I dive right in to solving the most difficult equations** – basically my philosophy in math (where I can apply it), and in life as a whole. If students can complete the most difficult kinds of problems up front, the rest are a cinch!

**Here are the steps for solving any equation:**

**(1) DISTRIBUTE**: Get rid of parentheses. Multiply the number in front of parentheses with each term inside:

**(2) GET RID OF FRACTIONS:** Box all terms. Include the sign in front. Fractions cannot be split up by boxes. Multiply each term by the denominator.

**(3) BOX TERMS: **Here, I use circles for number terms and boxes for x terms. Create a “river” down the equal sign to show the balance.

**(4) SORT TERMS:** Label a side for variables (*x* in this case) and the other for numbers (#). Here I put x on the left, but I could have chosen right instead. (*I love having that conversation where one kid tries the variable on the other side and ends up with the same. exact. answer.*)* * Also, when I teach, I work my way down the line and ask about each term “Does it stay or does it go? If it **changes sides**, it **changes signs**.”

**(5)** **COMBINE LIKE TERMS: **I let students use their calculators for basic operations as needed. Doing this step after sorting terms prevents students from doing things like subtracting 12*x* from 8*x* and 12*x* on the same side of the equation.

**(6) GET RID OF COEFFICIENTS:** Divide by the coefficient of x.

Aaand we’re done!

Now would be a good time to have your students check their answers, check for correct units, make sure they’re answering the question, etc.

**I am SO passionate about this method of solving equations and the success I’ve had that I presented on it with BVD at our county’s math development.** **Is it perfect? Practically. **** **I like that the river method reinforces the balance of equations. It’s also clean! Students aren’t writing* + 5* randomly below one side of an equation. This method gives them *permission* to write the change only once. And, **as students get really good at this method, I find that they no longer need to box their terms or actually label the sides**. The scaffolds that were so essential to developing student understanding slowly drift away naturally.

The skeptics out there are probably asking, “but **what about justification with the properties of equality**?” Honestly, the addition and subtraction properties of equality are a little tricky, but they tend to trip students up regardless of solving method. I teach my students to ask “What changed in this step? Well, if the term became a positive, it means I ADDED, so it’s addition property of equality.” You can also go so far as to copy the problem and write -5x on both sides of the equation, IN LINE with the rest of it. This demonstrates that when 5x and -5x cancel you’re left with only -5x on the left of the equation.

Pictures of the free (!) Solving Equations Foldable on Teachers Pay Teachers:

Best of luck!

*– Miss Elsie*

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