Multiplication Notation

Many people perceive


as being something different than


Yet, when asked “what operation connects the “3” to the “x”, most 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. Most often, people 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

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

=~(3)(x) + (6)

people 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


people 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 people are usually confident in stating that the answer is


It is also worth noting that expanding multiplication into a sum of terms also works 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 a person become skilled at visualizing


as either

1x~~~or ~~~(1)(x)



as either


without conscious effort, mistakes in these situations usually tend to begin happening less frequently.

By Whit Ford

Math tutor since 1992. Former math teacher, product manager, software developer, research analyst, etc.

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