Multiplication Notation

Many students I work with perceive


as being something different than


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\\*~\\* =~3(x+2)

or that

3x+x\\*~\\* =~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


before finally factoring to produce

=~(3)(x + 2)

Faced with


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


or perhaps even


this will often resolve uncertainty, as they might see that it can be factored


and then simplified


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


then expanded into


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


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


as either

1x~~~or ~~~(1)(x)



as either


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

Published by

Whit Ford

Math teacher, substitute teacher, and tutor (along with other avocations)

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.