How many of you gagged a little bit when you saw “Literal Equations” in the title? Oops, there you go again. I know, it’s not your fault. Anyone who has taught Algebra 1 has been slightly traumatized by the process. Not by the process of solving them yourself, but by the process of helping students to understand them. If that’s how you feel, imagine what your students must be thinking. Also, if solving equations was even a tiny bit difficult for any of your kids, literal equations is like pulling teeth.
The good news is that by Solving Equations with the River Method, you truly set yourself up for success! By now, your students have spent a few weeks combining like terms and tackling some pretty intense fractions and distributive properties. If you’re lucky, they may have even taken a break to find slope and graph lines like we do in the Virginia Standards of Learning (Yes… SOLs. *The poor kids feel like they’re already out of luck before they even begin*). These other “filler” (but very important) topics buy time so your students can continue to practice solving equations on warm-ups and homework assignments.** So by now, they’re practically experts at equations!
**My friend CarFax used to say that sometimes students focus so hard on what they’re learning that they can’t quite grasp it in the moment. Then, when the class moves on to a new topic, and students begin to focus on another concept, suddenly something clicks. Their brains release that block and incredibly, it all makes sense! While I don’t advocate rushing, there’s something to be said for continuing the momentum of a class, because eventually students will “get” it
For those of you who are afraid to ask, a literal equation is any equation that has more than one variable (or letter) in it. Usually literal equation questions ask you to “isolate the y” or “solve for W,” etc., so your final answer contains both variables and numbers.
Alright. I’m going to walk you through 4 practice questions so you can see the magic at work for yourself. Then maybe you’ll reconsider adopting this method if the 4 reasons from my other solving equations post didn’t suffice… 😉
(1) Solve for y: AKA “rewrite the equation in Slope-Intercept Form” from Standard Form.
(a) No parentheses, no fractions. So box your terms…. maybe. To be honest, by this point my scaffolds have fallen away, and most students don’t even need the boxes to know that the negative sign goes with the y.
(b) Choose a side for “y” and the other for “others”–any term that’s not a y. Label your categories and sort terms. We still use a coefficient of -1 with the y! It’s important!
(c) Get rid of coefficients. Divide each individual term by -1 and simplify. *We mostly do this when talking about linear equations so we can obtain the slope and the y-intercept.*
I always say in a “British” accent: “A negative divided by a negative is a….” and the kids jump in with “POSITIVE!” My goal in this life is to be obnoxious enough to get those basics stuck in their brains forever.
(2) Solve for y: AKA “rewrite the equation in Slope-Intercept Form” from Point-Slope Form
(a) Parentheses? Distribute. Use those calculators as needed for -1/2*-8. My Algebra Part 1 kids and I had to practice some serious fraction simplification before this came up.
(b) Fractions? IGNORE THEM. “…Whaaaaat?! But Ms. Elsie, you always told us to get rid of our fractions next!” Yes, and now the rules are a *little* different when we’re isolating the y. If we multiply by a -2, suddenly we have a -2y. We can actually work with that and divide by -2 in the end. But if we’re smart about it, we recognize that we ONLY need to move the -5 in order to solve for y. Let’s be smart. Context, context, context…
And to be honest, in my 9 years of teaching, I don’t think I’ve ONCE seen my students try to get rid of the fraction like that. I think the kids are so stunned by all the letters that they don’t go there. Were they really ever listening?? Did I teach them ANYTHING at all?! Maybe we need to have a conversation…
(3) Solving for other letters. Gasp!
(a) No Parentheses? GET RID OF FRACTIONS. That means we need to box all of our terms. So the V gets one box, and the right side of the equation gets a single BIG box. Multiply EACH BOX by the denominator.
(c) Get rid of coefficients. Have a conversation with your students about the invisible operation that’s happening between each factor on the right side. Multiplication. The opposite of multiplication is??? Division. By everything we don’t want.
(4) The most complex non-factoring Literal Equation you can find… except maybe some physics formulas with subscripts…
(a) No parentheses. No fractions. Choose a side for “h” and a side for “others.” (I always encourage students to leave the variable where it is if it already has a positive coefficient. This helps, especially with distinguishing between multiple choice answers on a standardized test.) Then sort terms. Here’s where some boxes *may* come in handy.
(b) Divide by the coefficient(s) of h. Again, talk about the invisible multiplication that’s happening on the right side. What do you need to get rid of? (2πr) What is the inverse of multiplication? (Division) Here we also talk about the fact that there’s no *need* to divide each term separately. We can leave it as one big fraction.
Well, that’s it. I can’t promise I won’t write about solving equations and rivers again, but that’s the majority of what I have to say. I hope you find equally as much success with it as I did! The key is to keep practicing alllll year long.
And I told myself this was going to be a short post tonight…