Variables
If you have ever gazed upon a sheet of math work and shudder to yourself, “alphabet soup!” you may relate to my 7th graders acclimating to prealgebra. All of a sudden, we’re not just encoding information in numerals (0 through 9), symbols (+, -, =), and position (ones place vs, tens place, superscript powers, numerator vs, denominator). Now we throw LETTERS into the mix!
Variables come on the scene much earlier than prealgebra. In fact, I just gave my first grade homeschool kids some fun little secret-code type problems where symbols stand in for numbers (like lucky-charms math) but by 7th grade, students aren’t just playing around. They have to develop fluency with these additional layers of abstraction. It can be very intimidating.
I developed this one metaphor to help my students wrap their heads around the idea of a variable:
A variable is a letter-box with number inside it. You can’t see the number, but you know it’s in the box, and so you can do ANYTHING to the letter box, and the number inside it will follow the rules of math.
This metaphor works great for “solving for x” type problems where the variable is truly standing for one real, concrete, numeric answer. It already exists, we just have to solve the puzzle and deduce what’s in the box. (NB: Variables do not always stand in for a specific solution like this, but we will deal with that in another post.) Even seniors in high school calculus can benefit from this simple box metaphor. When life gets overwhelming, bring it back to the basics. We have a box. Let’s figure out what number is inside the box.
For an ultra simple example, I made you a gif! You know I love my manipulatives!
1) I name my box “x” by drawing a big old X on it.
2) I give you all a clue about my number. My number plus five is fourteen.
3) I remind myself that the equal sign is THE LAW OF MATH, and so whatever happens on one side, I must balance it on the other to keep the LAW OF EQUAL true.
4) I separate my 14 into 5 plus the rest, and I pull the extra 5 off BOTH SIDES of my equation. This leaves my box alone on the left, and a countable number of beads on the right. I have successfully ISOLATED my variable.
5) I count up my beads and make the big reveal! My box indeed had 9 beads inside of it. I have solved for x.
So, this is a very simplistic problem, I know, I could have multiplied, divided, taken a square root. I could have done any number of fancy difficult things to my variable. The point is that the variable will follow the exact same rules as the number inside the box. Early mathematicians tend to use logic to reason their way through a problem. The further they go in math, the more they will rely on systematically unwrapping their number with algebra. But however you approach it, this box model can make it feel less intimidating.
One quick note:
The most common variable we use in math is the letter “x” I have no clue why, but if you know, do tell in the comments! It’s handy to have a default, but sometimes kids get TOO comfortable with x and freak out whenever they see another letter. No need to freak. It’s just a different box with a different letter drawn on it. It follows all the same rules. I could name my variable anything. I could make it a heart. I could call it “Bruno” after my neighbor’s pug. Eventually, we add GREEK letters to the mix! It’s a lot. Different letters end up taking on different special roles. For instance, i has a very special meaning in math, as do e and π. So, maybe avoid using the special letters for your number box. I steer clear of s, l, and o, which look too much like 5, 1, and 0 for my taste. If I must use them, I draw them in script. The letters t and z drive my students nuts on their homework, but I have learned to draw t with a tail and cross my z, like a seraph font, so I don’t confuse them with + and 2. These little adaptations can make the difference between getting problems right and tripping over your own handwriting. Even the most confident math scholars lose a lot of points to “stupid mistakes” — the kind of mistakes that happen despite good understanding. One of my greatest shocks moving from high school teaching down to middle school was that middle school students are still very much IN PROCESS on their fine motor skill development. I had just assumed that handwriting would have come together by middle school, but no. Give them time and grace when they’re working through a page of alphabet soup. They’ll have it down by high school.