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So, I'm a mathematics tutor, and I always run into this mental block that students have: they don't seem to understand what "in terms of" means. Also, the phrase "in terms of" is so useful, and it kind of frustrates me when student don't understand it.

EDIT: Let me explain why I think it is so useful. Sometimes, when I am working with a word problem, we will define a variable to be something like "x=number of apples" or "y=cost of toaster oven". Then to help set up an equation to work through the word problem, I will ask them to tell me what the price of 5 toaster ovens in terms of y. Sometimes I can work around this phrase, but sometimes I just really wish I could use it. (I guess I could just say "the answer depends on x", but that kind of has the same set of problems.)

Here's two examples:

Bobby buys 20 apples for \$15 and eats x apples. He sells the rest for \$2 each. (1) How much money does he make total? Write your answer in terms of x. (2) How many apples can Bobby eat but still make a profit?

Solve the following equation for y in terms of x: 2y+3x=1.

Now while it is fairly easy to get a student to solve such problems, so many students just don't seem to understand the concept of "in terms of". Since it is such an important concept (it is essentially precursor to functions) I really want to convey some kind of understanding.

Does anyone have any good ideas and advice for explaining this or working around the use of the phrase?

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I do avoid using this phrase in all of my math classes. Not that I've ever thought of it as a particular goal, but I would want to reserve the word "term" to specify an addend.

In your first example, I would specify, "Write a formula using the variable $x$". In your second example, I specify, "Solve the following equation for $y$" (end of direction), having previously defined "solving" as isolating the indicated variable on one side.

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  • $\begingroup$ I like your instructions for the first example. For the second example, I would say solve the following equation for y. Then I would add, it's ok for your solution to contain other variables that are already in the equation (because usually the students are solving for a number). I would also clarify afterwards they solved in terms of x because x is included in their solution. $\endgroup$ – Amy B Oct 19 '15 at 0:49
  • $\begingroup$ I'm actually more confident about my response to the second example, because that direction matches what I see in any of the algebra books on my shelf, e.g.: Martin-Gay Introductory Algebra, Bittinger Intermediate Algebra, Sullivan Algebra & Trigonometry, etc. $\endgroup$ – Daniel R. Collins Oct 19 '15 at 1:34
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    $\begingroup$ @Jay: Actually I wouldn't want to use the word "solution" for that, either (not quite the definition). These days I'm calling that task "transforming formulas" in my classes. $\endgroup$ – Daniel R. Collins Oct 19 '15 at 3:29
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    $\begingroup$ Interesting. I like that. I'll probably use that. (However, most teachers still use "solution", so I am stuck with that to a certain extent.) $\endgroup$ – Jay Oct 19 '15 at 3:33
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    $\begingroup$ "In terms of" is standard terminology. You're harming your students by not exposing them to it. $\endgroup$ – Ben Crowell Oct 20 '15 at 1:51
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Your students may be wondering "What is the value added of the expression 'in terms of'."

In the example you gave: Solve the following equation for y in terms of x: 2y+3x=1, you could just as easily say "Solve for y." Since there are only two variables in the expression "solve for y" and "solve for y in terms of x" are the same (in this particular example). Adding the expression "in terms of y" is redundant. If the students are used to seeing just "solve for x," they may be trying to figure out how adding the phrase "in terms of y" is changing what they were doing before.

To get over that, you may want to try systems of equations...

x + 2 = y

y + 5 = z

...and then you can demonstrate the difference between "solve for x in terms of y" vs. "solve for x in terms of z."

Back to word problems: Word problems are supposed to reflect real-world situations for students. In a word problem, that deals with real objects and real things, I would never say, "express your answer in terms of ex." I would use the actual objects, "express your answer in terms of apples" or "what is the cost of 5 toasters in terms of dollars?" Mixing variable names with real-world objects in word problems sort of defeats the purpose of word problems and could be confusing to students.

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I had similar problems with the phrase "such that" as taught in set-builder notation. Still--40 years later--I don't like the expression "such that." It's just not a phrase I would ever use naturally. I learned to read (in my head) the vertical bar in set builder notation as "where."

I haven't experienced the problem you described with any students I've taught. But I might apply the same solution: give your students a number of equivalent expression they can substitute in their heads. For example, "in terms of x" = "using units of x" = "where the variable x appears in the answer." For the students that do seem to grasp concept, ask them to suggest equivalent phrases.

Three other observations: 1) The word "term" has at least two meanings. One meaning is a part of a polynomial that is expressed without addition or subtraction operators. But a second meaning is "a specific word or expression," as in "vocabulary terms." The phrase "in terms of" uses the 2nd meaning. If you have emphasized the first meaning with your students, it may be helpful to state explicitly that you're using the 2nd meaning.

2) I understand this may be an artifact of typing your question, but I found your example "x apples" a little confusing myself. In this case, x is a unitless number, and the "units" are apples. So asking for the answer "in terms of x" seems like you're asking for an answer "in terms of a unitless number." I don't think you mean "in terms of x." In that particular example, I think you mean "in terms of apples" = "using units of 'apples'."

3) Have your students been exposed to dimensional analysis (a.k.a. unit analysis) in your course or previous ones? The concept of a conversion factor is closely related to this question. Your students are probably used to converting between inches and centimeters or other units of measure. Perhaps you can give them a serious of word problems involving unit conversion, and the word problems can use the phrase "in terms of." This may help get them accustomed to the meaning of the phrase. For example, "Express the capacity of a 10 gallon gas tank in terms of liters."

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What's important is what came before. Early on, algebra students should be comfortable with slope-intercept form, y=mx+b. They may also have seen the notation f(x)=something. In both cases, all x's, powers of x with their coefficients are on the right of the equal sign. These equations are present "in terms of x" or "as a function of x."

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