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
can be factored into
is the same as
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.
students often fail to notice the common factor. However, if it is rewritten as
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
students 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.
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 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,
Once students become skilled at visualizing
without conscious effort, mistakes in these situations usually begin to recede into their past.