The most recent Algebra Problem of the Week, Count Your Change, contained the following challenge:

[S]how that your answer is true for any case where there are three times as many nickels as dimes.

About half of the submissions that we received showed that the answer was true for any case, while the other half showed one or more examples but did not show that their answer was true for any case.

Showing your answer is true for any case means that you somehow have to represent the “any-ness” on one or more quantities in the problem. Then you have to look at the relationships in the problem and see if you can draw conclusions even with the “any-ness” represented.

What intrigued me reading the submissions is how students who worked mostly with special cases tried to “variablize” those cases without really connecting to the idea of a variable as representing an unknown (or varying or all possible) value(s) of a quantity.

Here’s an example from Sin Y. at Sekolah Ciputra:

n=5 cents
d=10 cents

number of n = 3x number of d
if number of d = 1
I’ve got 3n and 1d, which is
=(3×5)+(1×10) = 25 cents

reversed = 1n and 3d
so, I’ve got (1×5)+(3×10) = 35 cents

There’s so much interesting thinking going on here! First of all, I’m really excited about how organized and patterned the calculations are. I wonder how Sin would write out the calculations for beginning with two dimes. If they looked like this: (6×5)+(2×10) and (2×5)+(6×10) then I think Sin would be well set up to represent the work as (3*_*5)+(_*10) and (_*5)+(3*_*10).

Another thing that stands out for me about Sin’s work is the use of the letters n and d. I wouldn’t call them variables because they seem to represent the value of a nickel and a dime (which don’t vary, at least in our culture). I’ve noticed that students often use variables in the place of labels or words that they say. So Sin may have written 3n and 1d because out loud they would have said 3 nickels and 1 dime. Many students aren’t thinking of the letters they use in equations as varying quantities, they think of them as labels or abbreviations for words.

Let’s take a look at another example of that from Sin’s classmate Charina.

n = 3d = for every 10c (dime) you have 15c (nickels).
the total = 10 +15 = 25
when d = n, for every 5c (nickel) you have 30c (dimes)
The total = 5 +30 = 35

Again, there’s great thinking here. I love the idea of for every __ you have __. I think Charina has almost created a general solution here, because they recognized that for every 10 cents in dimes you have 15 cents in nickels and so Charina might be thinking that you’ll have groups of 25 cents and if you switch the number of dimes and the number of nickels you will have the same number of groups of 35 cents.

n groups of 25 cents converted to n groups of 35 cents is n increases of 10 cents per n groups of 35 cents. Proportionally, n 10′s out of n 35′s is the same as 10/35, or a 40% increase.

However, I wonder if Charina’s use of “c” which could be thought of as a variable, is actually a unit, “cents”? Charina’s work could be interpreted where c represents the number of dimes, and the value of dimes is 10c and nickels is 15c. But I think it’s more like that Charina is thinking about one dime and three nickels, and is using “c” for the units of cents.

Therefore, it’s hard to tell how much Charina has generalized the “for every 10c there are 15c” and if she is really on her way to a general solution.

Here’s one more example from Matthew at Sacajawea Middle School:

1 dime + 3 nickels = \$0.25.   3 dimes + 1 nickel = \$0.35

to represent this equation the algorithm d + d (3n) = T in other words dimes plus 3 nickels per dime equals total.

Here Justin has said in words the important relationship he is trying to represent: “dimes plus 3 nickels per dime equals total.” I have some questions about that relationship, especially “Total what? Total value? Total number of coins?”

Then I have some questions about his expression. He has d + d (3n) = T. Again I’m wondering if he is using the “d” and “n” partly as labels. His algorithm says to me, “write the number of times, then add that to that number times three which will be the number of nickels.”

Another possibility is that like Sin, Justin is thinking about d = 10, n = 5, with d and n representing the value of a nickel and a dime. In that case, the equation isn’t quite right, but 10 cents + 3 * number of dimes * 5 cents is an expression for the total value that’s not far from what Justin has…

In the Make a Mathematical Model activity from the Problem Solving and Communication Activity Series, we encourage students to name quantities and write relationships among them in words. It’s possible that may have helped Sin, Charina, and Justin translate their work, but I think all three students could articulate their thinking it words. The struggle seems to be figuring out what the heck a letter in an equation means. Is it a label? An abbreviation for a word?

In fact, it stands for a value of a quantity and so it works like a number, but a weird kind of number that can have any value. That’s a hard concept to grasp and I’m not sure what experiences our students need to have to grasp it. What do you think?