Communicating mathematical reasoning is difficult if students don't know how to write out their thoughts using symbols. For instance, the symbol (A)c can mean the area of a circle which is a dependent variable of the radius. So, the entire equation can read (A)c = pi r^2. Well, suppose we have 2 circles we are considering, each with a different radius and therefore a different area. And suppose that one circle fits directly inside of the other.
How are we going to distinguish between these two circles given that both are variable sizes with one being larger than the other? Simple, we use a symbol to represent the variable areas of the circles. Simple as that. So start off by labeling one of the circles L for larger circle and S for smaller circle. We can then set up an equality such as follows:
(A)Lc > (A)Sc
The area of the large circle is greater than the Area of the small circle. See how the letters line up with the statement? Let's make it more clear.
The (A)rea of the (L)arge (C)ircle is greater than (>) the (A)rea of the (S)mall (C)ircle
Both symbols are variables and tell us that we are talking about something specific about each circle, which circle we are talking about and that we are talking about a circle. This means that we can talk about the perimeter of these two circles without ever seeing them or drawing them. Since perimeter starts with P let's use that to draw another relationship that makes since.
Since the perimeter of a circle = 2 pi r and the large circle Lc is larger than the small circle Sc we can assume that the perimeter of the large circle Lc is greater than the perimeter of the small circle Sc. So, let's draw another conclusion.
(P)Lc > (P)Sc
The (P)erimeter of the (L)arge (C)ircle is greater than (>) the (P)erimeter of the (S)mall (C)ircle
Again, these are variable amounts, but by developing the symbols that we are using, we can actually refer to specific items without ever seeing.
Now, remember that the small circle (Sc) is inside the large circle (Lc). Can you draw an equation that represents the space between the two circles? It's actually quite simple. The area of Lc minus area of Sc represents the area between the circles. Let's use some symbols that represent our variables to denote what we just said.
(A)Lc - (A)Sc = (A)Bc
Again, the (A)rea of the (L)arge (C)ircle minus the (A)rea of the (S)mall (C)ircle = the (A)rea (B)etween the (C)ircles
Learning how to communicate with variables requires us to define them in a way that makes sense to us. We can go further and define the areas of the circles in terms of their radius' and without ever being given a value, still make logical equations, however, we will have to denote the different radius with a 1 or a 2 or a s and an l. Probably an S and L would be the best choice since we are dealing with a small circle and a large circle.
So, without further adieu, we can summarize the areas of the circles as follows:
(A)Lc = pi Lr^2 Area of the large circle equals pi times the (l)arge (r)adius squared
(A)Sc = pi Sr^2 Area of the small circle equals pi times the (s)mall (r)adius squared
Substituting these values back into our original expression we get
pi Lr^2 - pi Sr^2 = (A)Bc
Following along with what we just went through, can you write this out by hand like the equations above?
The reason it is important to understand Variables as Symbols is because many state tests will offer credit for showing work! Even if you cannot solve the problem, if you can use your symbols right, you can set up the problem so that someone can read what you are thinking and assign you credit for your efforts.
Hope this helps!