The Value of N

I've been thinking a lot lately about the way math is visually presented to young students. Looking at equations and the meaning of equal signs, as written about in this recent post, I also began to wonder about the use of answer blanks, such as can be found in questions like this one:

5-3=___

There are reasons to write it like this, most obviously to give the student a place to put the "answer" or, better put, to complete the equation. However, following the equal sign with a blank might also be the cause of some confusion when students reach introductory algebra. They may have learned to associate the = sign, and/or the blank with "this is where the answer goes" rather than understand that this is an equation in which the value of each side must balance.

Consider the following way of representing the same question:

5-3= n

which could be followed with:

n = ___

or simply the words, "What does n equal?" or perhaps better still, "What is the value of n?".
(I chose the letter "n" because it can stand for the word "number", but any letter would work as well)

While there is still that problem of the equal sign followed by the blank, the way the first part is represented manages to help convey some information that is missing in the first example, such as "what are we looking for?" and "how can we represent the unknown number that will balance the equation?".

Maybe a picture would help get the idea across better (please forgive my crude drawings!):


Of course, if you have a balance and unit weights handy, you could always use those to help solve the problem.

I wonder if presenting simple arithmetic with a variable rather than a blank from the outset would help students better understand the concept of equation and equality better, and also predispose them to accept variable notation when it becomes more crucial in algebra.

If you choose to use this idea with your students, I'd be very grateful if you would let me know how it goes.

On the Concept of Balancing Equations

So often in the early grades kids become accustomed to seeing problems written out as below:

3 + 4 = ________

When the answer blank appears in different places in the equation, such as on the left-hand side, it can help, as can the creative use of manipulatives to represent the symmetry of equations. However, the connection between the equal sign and the demand for an answer may continue to confuse some students. Some students learn to think of the = sign as meaning "insert answer here" rather than as the fulcrum of the equation.

What do I mean by this?

Consider the term "balancing an equation".
If you envision an equation as a balance scale, you can put the = sign at the centre, or fulcrum of the scale. In this way, the equation is balanced when both sides are equal to each other. There is a symmetry in the weight on each side.

This can be used to demonstrate the mathematical meaning of the equal sign in a hands-on concrete way.

For students who have difficulty with the concept, consider having them use the balance with weight manipulatives. x might be the name of the 1 gram weights, y the two gram weights etc. Let them play around.

What if they put 2x on one side and y on the other? When the balance is level, the sides are equal.
What if you put 2y on one side and x on the other? When they balance is tilted one way or the other, the sides are unequal. Instead of an equation, you have an inequality ≠.

You can take this a step further if your scale is the kind that has an arrow on the fulcrum. Label the point of balance with an equal sign =, and the space on either side with an inequality sign ≠.

Encourage students to write the equations as they work with the balance to solve problems and also to make predictions of what expressions will be equations and which will be inequalities, as well as determine what is needed to turn an inequality into a balanced equation.

More of my math activities can be found here.