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

or that

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

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

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

Faced with

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

Once a person become skilled at visualizing

as either

and

as either

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

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