The definition of "term" and its use in teaching algebra especially distribution

Question

Context: While teaching algebra I taught the distributive property of numbers. For any number A, B, and C, the following holds.

$$A(B+C)=AB+AC$$

(Recall juxtaposition symbolizes multiplication.) Read from left to right it means, "We can multiply A over the grouped sum of B and C." Read from right to left it means, "we can factor an A from the sum of AB+AC." Giving plenty of examples students still want to apply the property like so:

$$3[xy-2(x-y) +\frac x y]=3x3y-6(3x-3y) +\frac {3x} {3y}$$

Later I showed step by step how we got the generalized distribution property, which states for k well-formed expressions, $p(a+b+...+j+k)=pa+pb+...+pj+pk$.

But my first thought was to explain that distribution only applies to the terms inside the brackets. That we do not have 7 terms but 3 terms in $[xy-2(x-y)+\frac x y]$.

However, when looking at most books definition for terms, I find many options, none of which work.

My Questions: What is the definition for term? What good references can you give? Is this definition complete and correct? "Terms in a mathematical expression are separated from each other by addition or subtraction, just so long as the addition or subtraction is not itself inside a group (for instance inside a different operation or parentheses)."

This would correctly separate this expression into 6 terms. Slashes, /, showing where each new term begins. $$/3x(2-4) +/ log(x+3) -/ \frac {2-x} x +/ f(x-1) -/ 7^{2-x} +/ (3+3x)$$

Here are some other definitions for term.

Textbook Elementary Algebra, "A term is either a single number (called a constant term) or the product of a number and one or more variables." Thus $2(x-y)$ is not a term.

Most textbooks give statements like "To subtract a sum of terms, change the sign of each term and add the results." Before defining term or never defining it.

Wikipedia, "In elementary mathematics, a term is either a single number or variable, or the product of several numbers or variables. Terms are separated by a + or - sign in an overall expression." Thus $2(x$ and $-y)$ are two term unless you can rigorously define what "overall expression" means.

mathwords.com "Term: Parts of an expression or series separated by + or – signs, or the parts of a sequence separated by commas." And gives example, Expression $$\frac {p-2q} {a^2 +b}$$ has Terms $p,2q,a^2,b$.

Answer 1

A Friendly Introduction to Mathematical Logic by Leary and Kristiansen has a nice definition. I'll paraphrase:

If we have a language of mathematical symbols, a term of that language is a nonempty finite string $t$ of symbols such that either:

• $t$ is a variable
• $t$ is a constant
• $t \equiv f(t_1,t_2,...,t_n)$ is the result of a function of terms $t_i$, a recursive definition.

Going over the exact formalism presented in that text may be personally satisfying, but it'll be overkill for your students.

However, emphasizing that we are dealing with symbols is a good first step in abstraction. Wiki's definition is taking the next step. While working with real or complex numbers, a term is, pragmatically speaking, that symbol or almagamation of symbols separated from one another by the language's operational symbols $+$ and $-$. A separation can be achieved using grouping symbols (i.e. parenthesis and brackets) that act as organizational meta-symbols not within the language.

Personally, teaching students at that level, I emphasize terms by drawing colored boxed, demonstrating that context matters by enforcing precise language, and making this process a call and response discussion so they feel comfortable exploring the organization of the algebraic expression.

e.g. We can consider the following expression to be mde of two terms when written as

$$ \boxed{x} + \boxed{\frac{a + b^3 + bc}{cd}}.$$

We could also rewrite the expression such that it were four terms. Any idea on how?

$$ \boxed{x} + \boxed{\frac{a}{cd}} + \boxed{\frac{b^3}{cd}} + \boxed{\frac{bc}{cd}}$$

What if we're talking about the original second term. Note the terms in the numerator of that fraction. How many terms? Three.

$$ x + \frac{\boxed{a} + \boxed{b^3} + \boxed{bc}}{cd}$$

I cannot attest to the merits of this approach, but I hope it guides the students to look at their symbols a bit more abstractly. Some students, especially those who are already behind on material, are afraid to deviate from a specific algorithm for an algebraic scenario, leaving them hung up on more challenging problems. So, I hope that such exposition encourages students to play around with the algebra and discover how to appropriately identify the objects they see.

Answer 2

First and foremost a term is a SINGLE VALUE, a number or variable, or a product, anything that is enclosed in parenthesis. As - ab=(a)b=a(b)=(ab)=(a*b) That says implied multiplications are enclosed in parenthesis. So the simplest way to define a term is: A single value, a number or variable or an implied multiplication, anything in parenthesis.

Terms like [ 3,x,xy,2(x) ] are in general separated by operators. Here they are separated by commas, as they are in some forms of multiplication. The + or - operators have no special powers to separate terms. Any operator separates terms. Owing to the fact that [+ and -] operations are performed after [* and /] they naturally group [* or /] operations.
In a generalized equation, [one with + or - operators] you can go merrily along and place parenthesis between the [+ and -] thus grouping the other terms. Then the [+ or -] cut the equation into segments. However within each group you still have terms separated by operators.
3 + 4 is two terms, and 3 * 4 is also two terms. (2 * 3) is one term. Stating only + or - separate terms confuses people, and is untrue. Terms are routinely separated by [, : ;] in many equations also.

If you teach it this way, I believe you can avoid confusion. BTW a(b+c)=(ab+ac) You failed to maintain your parenthesis. Anyway this is how it was taught in the 1950's.