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Redefining the Term 'Term'

Lesson Plans > Mathematics > Algebra
 

Redefining the Term 'Term'

[This document is a work in progress, and is subject to change. The ideas within this document have not yet been implemented and tested in an algebra class. During the 2017-18 school year, I will be giving these concepts a test-run with some of my upper-level math classes; updates and clarifications will be added to the document as needed.]

Many algebra curricula go to great lengths to define what a term is, and often the definition given results in confusion at some point in the algebra curriculum.

If you search math websites and algebra textbooks, you'll usually find that a term is defined as "a number, a variable, or a  combination of numbers, variables, and exponents, combined using only multiplication and division as operators." This is really not a complete definition, and is more cumbersome than it needs to be. Why is it not complete? Here are a couple examples, both of which involve the distributive property. We tell students that when we have an expression like 3(2x + 6), we multiply the 3 times every term inside the parentheses:

3(2x + 6) = 3(2x) + 3(6) = 6x + 18

But consider these examples:

Example One: 3[2 + x(y + 1) + x]

Students who have the above definition of a term ingrained in their minds have no idea what to do with the expression above. What do they do with the x(y + 1) in the middle? Do they multiply the x by 3?  What about the y? And the 1? Maybe all three of them? They're all terms, right?

Example Two: 3(2 +
y + 1
x
+x)

The same bafflement occurs here. What do we do with that thing in the middle? multiply all the terms of the numerator by 3? What about the denominator? How many terms is that, really?

Students repeatedly get these expressions wrong, and the fault is not (entirely) theirs; it's the manner in which they've been taught what a term is.

What's at stake here is not how we implement the distributive property, but rather, what our understanding of a term is. And everyone who has been doing algebra for a long time implicitly understands (even if you didn't realize that you were doing this) that you were treating those compound expressions as a single term.

x(y + 1) is a term. So is 
y + 1
x
. Even though neither one of them passes the standard definition of a term.

To see that this is how we really think of terms, ask yourself how many terms are in the following expression? xyz + y(x + 1) + 4x. If you said "three", then you think of y(x + 1) as a term.*

At some point, every algebra teacher, when confronted with expressions like these, ends up telling students, "You treat that as though it's a single term," or they end up saying, "Well, that really is a single term, because the definition of term is actually broader than what we told you." Why do we tell them to treat it like a term, instead of defining a term in such a way that it is a term? Perhaps it's because we think defining it that way from the beginning would be too complex for an Algebra One student. Perhaps it's because we think it's not going to matter in the short term for Algebra One students who won't be facing complex expressions like this until later in the year.  But this lack of clarity in our thinking/explaining does cause problems for students. And even if it doesn't cause problem-solving problems, it does cause problems in students' understanding of the nature of mathematics, since mathematics depends on logic, precision, and clarity in its development and presentation.

So how do we define a term in such a way that it includes compound expressions like the ones above? I'd like to suggest that we begin by making sure we get our dependencies in our definitions of expression and term in the right place.

Some texts define "term" first, and then define an expression as follows: "One or more terms combined with addition and subtraction." This is flat out wrong, because according to this reasoning, if
y + 1
x
is not a term, then 2x +
y + 1
x
is not an expression. 

I'm going to define expression first, and then follow that up with a definition of term which references expressions. 

Expressions: An expression is a number, a variable, or a combination thereof, joined using any of the mathematical operators and/or grouping symbols.

I'll spend some time getting students to give me examples of expressions, such as:

8 - x,
7
3
x3 - y2 + 8, 7, x+1, z, 2xy2, 12x3(3y - 11) +
23

Now we'll use the following as our definition of a term:

Terms: A term is any expression in which all addition and subtraction operations are contained within grouping symbols.

The big changes in the definition are:

  1. We're now defining a term as a specific type of expression.
  2. We've skipped the "containing numbers and variables" phrase, because that phrase is in my definition of an expression, and is therefore implicit in the term definition.
  3. We've introduced the idea that a term may involve grouping. This is important, because it helps us to think of a term as a "unit" - something that should be treated as a group. This is the goal of our thinking when it comes to terms.

Let's see how this definition serves us.

First, we can see that of the examples above, the following are all considered to be terms:

7, z, 2xy2

Now let's consider the next part of our definition, which talks about "grouping." Our revised definition suggests that even though x + 1 is not a term, we can make it into a term simply by putting parentheses around it: (x + 1). That's a novel idea if you're used to the original definition of a term, but if you think about it, once you put parentheses around something, you're treating it as a single, contained unit. And that is, conceptually, what a term is. Furthermore, calling it a term reinforces the idea that it has a coefficient, and if no coefficient is written, it's one. When students see 3x - (x + 1), they are already trained to ask the question, "What's the coefficient of the term (x + 1)?" And the answer is: negative one. This way of processing a parenthetical expression may reduce the number of sign errors resulting from careless manipulation.

According to our revised definition, this is also a term: x(x + 1); even though it has addition, the addition is enclosed in a grouping symbol.

But what about
x + 1
y
?

Students will have no trouble with this if you do what I do: from the very beginning of algebra I drill into them that a fraction bar is a grouping symbol.

This idea is important, but often ignored. None of the algebra texts I have in my posession explicitly state this fact, which means it gets overlooked in a lot of algebra classes. The expression
x + 1
x - 1
is functionally equivalent to the expression
(x + 1)
(x - 1)
. If you were going to write the fraction as a division expression, you would not write x + 1 ÷ x - 1; you would write (x + 1) ÷ (x - 1).  Unless you point this fact out to students, they will repeatedly make careless mistakes with algebraic fractions.

If you have already taught your students that a fraction bar is a grouping symbol, then they will have no trouble recognizing that the following are terms:

x
y
;
x + 1
z
; 3(
x + 1
z - 1
)

In fact, the following is also a term, made up of terms nested two layers deep:

1 +
a
3
1 -
b
5

This way of understanding a term is somewhat different, but it more precisely meets the intention - that a term is something that should be treated as a single unit.

There are other things besides fraction bars and parentheses that are grouping symbols. A square root is a grouping symbol. To see this, consider that
x - 1
is equivalent to
(x - 1)
, and if we wanted to write it without a square root symbol, we'd have to write it as (x - 1)
1
2
, rather than x- 1
1
2
.

Thus,
x - 1
is a term. And, at an instinctive level, you know that it's a term, because if your student sees this:

5
x -1
+ 2
x -1
and asks you what to do, you'll tell them to "combine like terms."

5
x - 1
+ 2
x -1
= 7
x - 1

If you've taught your students from the very beginning that anything in a grouping symbol is a term, then this step becomes almost instinctive, rather than a struggle. It even carries through into more advanced mathematics, as we recognize that sigma notation, function notation, and even integral notation are grouping symbols, which makes the following simply a matter of combining like terms:

5
10
Σ(3n+4)
n=1
 +  7
10
Σ(3n+4)
n=1
 =  12
10
Σ(3n+4)
n=1

Returning to the idea of fractions, one common mistake students make is trying to cancel terms out of the numerator and the denominator like this example:

 x2 + 5x + 6
 x2 + 3x + 2

A proper understanding is that cancellation can only happen if the numerator and denominator are written as a single term, and then you can only cancel the entire term, or factors of that term (this understanding is in keeping with how we deal with arithmetic fractions). Thus, you must factor the numerator and denominator. And if you can't factor them to turn them each into a single term, the only way to turn them into a single term is to wrap them with parentheses:

(x2 + 5x + 6)
(x2 + 3x + 2)

Those parentheses are a visual cue that they can't cancel anything across groups. I make my students begin by putting parentheses around any multi-term numerators and denominators. The only way they're allowed to remove those parentheses is if they can find another way to write the expressions as single terms:

(x + 2)(x + 3)
(x + 2)(x + 1)

Cautions

Caution #1

This way of defining "terms" might change the way we define "like terms" - some books explain that two terms are like terms if they have the same variables, and the same exponents for each variable. We need to be more "general" in our definition: two terms are like terms if they are identical except for their coefficients.

Caution #2
If you choose to redefine "term" in such a way that it accurately reflects how we actually think of them, you'll need to be careful about how you use the word "term" in connection with monomials and polynomials.

A monomial is (according to one curriculum) "a number, a variable, or a product of a number and one or more variables." In an Algebra One curriculum, we rarely see problems in which there is a significant distinction between a "term" and a "monomial", so it's tempting to use them interchangeably. They're not interchangeable, because the definition of a monomial only allows for positive, integral exponents, but by definition, a variable in a term can have any exponent.

Even using the standard definition of a term, it's never a good idea to use the words interchangeably. Some texts, for example, say that the degree of a term is the sum of the exponents of its variables, and this is not correct, because - to an Algebra One student - that would make the degree of
x2
y3
five, because 2 + 3 = 5. An Algebra Two student might recognize that this is equivalent to x2y-3, which gives it a degree of negative one. So you get two different answers depending on how you write the term. We've always risked confusion with such a definition. We should more appropriately be talking about the degree of a monomial, and avoid using the phrase "degree of a term" (unless that term happens to be a monomial, and even then, we should avoid that terminology, since it might lead to the impression that all terms have a degree).

This distinction becomes even greater with our alternate way of describing a term; your students will have fun watching you squirm when they ask you to tell them the degree of the term
x(x + 1)
x - 2
x - 17
. Be precise in your language, and avoid squirmage! Monomials have degrees; terms do only if they are also monomials.

Caution #3
In implementing this idea, it's important to remember that students who come in from other schools, or leave for other schools, will be facing a different definition of term. Students coming in should be given a "crash course" on how we define the word, and all students should be reminded that different teachers will define the word in different ways.


As one final thought, this way of looking at terms allows us to give an alternate way of describing the distributive property: The distributive property is a property which allows us to combine multiple terms into a single term, or to break up a single term into multiple terms, and - in some circumstances - to break it up into monomials.

* This example was pulled directly from Khan Academy. Interestingly, their written definition of a term is identical to the standard definition of a monomial, and yet, in their video in the same unit of study, they state that y(x + 1) is a term, even though that doesn't match either their own written definition or the one that is typically used.

Lesson by Mr. Twitchell

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