Problems with Defining the Variable (Part 1)

Defining your variable is one of the easiest ways to earn points on tests/quizzes, and you can get those points even if you don't know how to solve the word problem. Here are a few examples from previous courses of mistakes that sometime happen. Remember, defining a variable is to pick a letter and tell me what it represents in the word problem.

•  Problem 1: Just picking the letter, but not telling me what it represents. Here are a couple of examples.

 

and

• Problem 2: Defining the variable as something that is already known. The variable always represents something that you want to know, but don't. Here is a good example of the student defining his/her variable to be the length of the pipe, which is already known to be 120 inches.

•  Problem 3: Defining the variable to be more than one quantity. A variable can only represent a single number/quantity. One common mistake in problems with more than one unknown is to define the variable to be both unknowns. Here the student has defined the variable to represent both the length and the width.

• Problem 4: The variable as defined is not consistent with the equation. This could be a simple typo, like the first example problem. The student's symbol in part 1 is not the same as the symbol he/she uses in the inequality of part 2. It could also be that the student

• Problem 5: The Variable's meaning is inconsistent with it's use in the equation. I have to get a good example of this from student work. Here is a type of word problem I often see this mistake made. 
• Anna, Lea, and Simon have ages that add up to be 78. If they were all born in consecutive years, how old is each one?
• Sometimes I'll get answer like this.
• Part 1, define your variable-- x: the youngest person's age
• Part 2, write your equation-- (x-1) + x + (x+1) = 78
• Both the variable and the equation are okay by themselves, but are not good together. The reason is that when you write an equation (x-1) + x + (x+1) = 78, you are letting "x" be the middle person's age. The youngest persons age is "x-1" and the oldest is "x+1". To rectify the situation the student can either change the way they define the variable or their equation.
• Changing the definition of the variable, the correct answer would be...
• Part 1, define your variable-- x: the middle person's age
• Part 2, write your equation-- (x-1) + x + (x+1) = 78
• Changing the equation the correct answer would be...
• Part 1, define your variable-- x: the youngest person's age
• Part 2, write your equation-- x + (x+1) + (x+2) = 78

• Problem 6: Make sure to state the obvious. A lot of times we use letters that help remind us of what a variable means, i.e. it reminds of us of its definition. Take this problem for example.
• The perimeter of is 108. If the length of a rectangle is twice the width, what are the dimensions?

Most people let the variable's definition be the width. When choosing a symbol, the smart choice is to choose w, because it reminds us of the definition even when we are working with the equation. (x, y, m, l...are all fine too.) Even though it is obvious when I see w in your equation, you still have to define the variable. I know this is a little persnickety, but indulge me :). Don't just write w in part 1, or worst of all leave it blank. A nice w: the width of the rectangle will do nicely.