Variables made easy

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 6-year-old Autumn explains that a variable is a slot that you can put a number into.  The slot is usually represented on paper as a letter (such as x) or a mark (such as ___).  Here’s the definition of a variable symbol:

VARIABLE. A variable symbol is a symbol that is a placeholder for a number from a specified set of numbers. The set of numbers is called the domain of the variable.

A variable is a placeholder for “a” number; this number does not “vary.” An (unfortunately) common Textbook School Mathematics description of variable in the U.S. textbooks is, “A variable is a quantity that varies.” How does “a quantity” vary? (No, really, explain how a quantity like the length of a football field varies!)  It is no surprise to me that students in the U.S. don’t understand descriptions that don’t make any sense.   However, the description, “a placeholder for a number,” is about a single, non-varying number: “A thing” is much more concrete to students than, “A thing that could be this thing or that thing or maybe that thing over here; it varies.”

The beauty of the correct description of variable (and a point that needs to be made over and over to our students) is that it is the person who is using the variable who has ultimate control over what number they wish to insert into the placeholder.   The power to choose the number they insert into the placeholder rests in the will of the student, not in the variable itself!  The power to choose (and possibly start over and choose again) is what “vary” in “variable” means.

Okay, here are some notables about the video above that I like:

• Around the 1 minute mark, I say “are going to be numbers,” and Autumn follows up with a slightly strange sounding addendum remark, “or a number.”  Autumn is actually clarifying my statement here: We don’t insert more than one number into a slot at a time.  Instead, we insert only one number into a slot and that same number is inserted into every instance of the slot.
• The unit language that we have been using throughout this video series (and which runs all throughout Eureka Math) is present in the way we speak of arithmetic with variables as well.  So, for example, “3 tens plus 2 tens is 5 tens” that students say in early grades is the same as “3 slots plus 2 slots is 5 slots” that Autumn says in this video.  This makes the link between arithmetic and algebra much more obvious—in fact the two statements are exactly the same if we insert the number 10 into the slot!
• The algebraic expressions discussed in this video are:  $3x+2x$, $11x+6x$, $15x-8x$, $3(7x)$, $x^3$, $x^5$, $x^{10}$ (and Autumn mentions $x^{100}$).
• Autumn explains that $3(7x) = 21x$, and shows that if you insert 2 for the slot,  then the expression becomes a numerical expression $21 \cdot 2$, or $42$.  Of course, we have the power to vary:  we could start over and insert $5$ into the slot instead, and then $3(7x)=21x$ becomes $3(7\cdot 5) = 21\cdot 5$ or $105$.  (Once we insert a number in for $x$, the number goes into all instances of $x$ at the same time.)

Finally, we hope you enjoy laughing (with us) at the blooper that runs after the credits.  We cut this segment from the main video because it was a rare “mental typo” that didn’t contribute to understanding the main point of the video.   In general, in the “Growing up with Eureka” videos, we actually make a point of showing the real errors because Autumn and my discussions of those errors can lead to a better understanding of the concept for the viewer (cf. the discussion of “42 slots” in this video).

As always, comments are welcomed.

CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Partially supported by NSF CAREER grant DMS-0748636

About Scott Baldridge

Distinguished Professor of Mathematics, LSU. Geometric topologist: gauge theory, exotic 4-manifolds, knot theory. Author: Elementary Mathematics for Teachers.

1 Response to Variables made easy

1. Josh says:

Wonderful. Simply wonderful. See the following article discussing how 5-year-olds can learn calculus. http://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/

I’m convinced that the games kids play are based rudimentary organizations and processes (in math we call these “algorithms”). If we can constructively inject math/logic ideas and algorithms into play, or make play into learning directly about procedures (as in this case), we will be doing a massive favor to our kids: they will be able to understand more quickly, and create and imagine with more self-made structure.

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