How to explain to a child what a variable is..

Yesterday, my child was given a series of problems when she got a graph of the line (an image) and she should choose a corresponding equation of that line.

I got the impression that nevertheless she can write down or sound out the line equation without any problems. She can even graph it. But I got the impression that she does not really understand what the variables are. By some reason she did not feel confident to play with numbers, to think about the “function” of one or another term in the equation rather than to deal with it formally. So I decided that probably the source of that is that the variable concept might be too abstract for a child.

In elementary mathematics, a variable is an alphabetic character representing a number, called the value of the variable, which is either arbitrary or not fully specified or unknown.

I.e. in the particular case of line equation y=mx+c, we have two coefficients m and c, so if we are discussing some line in particular, m and c have some values. ( Let’s, for example, take m=2, c=1, so we have y=2x+1, but let’s still write the equation y=mx+c, but keeping in mind that m=2, c=1)

So we have 2 variables x and y, i.e. we have infinite number of values of x and corresponding values of y. Luckily to graph this line we need only two pairs, but usually teachers advise to add a couple more. But let’s think whether the m and c coefficients are just arbitrary numbers or they might mean something… The odd question for a child, is’n it?

What if the value of x=0?

Exactly! If we will pick zero value for x we will have that y is equal c.

I.e. if x=0, then y=c.

Let’s imagine what does it mean, if x=0, then the point (x=0, y=c) is located at the y axis. So the point (x=0, y=c) belongs to both lines x=0 (or y axis) and our line y=mx+c, i.e. our line crosses y axis at the point (x=0, y=c). In other word c is the value of y coordinate at the point of intersection between our line and y-axis.

We still need to imagine what m might illustrate or which geometrical meaning m coefficient might have…

But I am afraid it is already too long post so I will try to write another summary post about line equations and the mathematical meanings of coefficients there. :)