# It’s True! 3x = (3)(x)

Many students I work with perceive

$3x$

as being something different than

$(3)(x)$

Yet, if I ask “what operation connects the “3” to the “x”, most students will think a second and respond “multiplication”. So, they can figure out what it stands for – but they do not perceive it that way initially.

This mis-perception contributes to a number of algebra errors on homework, quizzes, and tests. Most often, student do not seem to see that

$3x+6$

can be factored into

$3(x+2)$

or that

$3x+x$

is the same as

$4x$

I recommend three steps to resolve this situation:
– rewrite terms with coefficients explicitly as products (either by putting a dot between coefficient and variable, or putting each in parentheses)
– expand the product into a sum of like terms (a version of repeated addition)
– put a coefficient of “1” in front of any variables without coefficients. It is usually worth revisiting the notion that multiplying by one does not change anything, so students are always welcome to do this if it helps.

Faced with

$3x + 6$

students often fail to notice the common factor. However, if it is rewritten as

$(3)(x) + (6)$

students often spot that three is also a factor of six more quickly and are confident rewriting the expression as

$(3)(x)+(3)(2)$

before finally factoring to produce

$(3)(x + 2)$

Faced with

$3x+x$

students are often uncertain about what they can, or cannot do. The lack of a coefficient in front on the $x$ leads them to either doubt they can “combine like terms” because there is no coefficient for them to add, or to factor out the $x$ while forgetting to leave a 1 behind from the second $x$ term. If the expression is rewritten as

$3x+1x$

or perhaps even

$(3)(x)+(1)(x)$

this will often resolve uncertainty.

However if a student consistently does not recognize $x$ as the same thing as $1x$, something that seems common, the expression can also be rewritten as

$(3)(x)+(x)$

then expanded into

$(x+x+x)+(x)$

At this point students are usually confident in stating that the answer is

$4x$

It is also worth noting to students that expanding multiplication into a sum of terms will also work with decimal coefficients, but the terms will not all be identical and thus is not the usual “repeated addition”. For example,

$3.4x\\*~\\*=3x + 0.4x\\*~\\*=1x+1x+1x+0.4x$

Once students become skilled at visualizing

$x$

as either

$1x~~~or ~~~(1)(x)$

and

$3x$

as either

$(3)(x)~~~or~~~x+x+x~~~or~~~1x+1x+1x$

without conscious effort, mistakes in these situations usually begin to recede into their past.