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Q&A #2048


Terms in an algebraic expressions

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From: Pat Ballew (for Teacher2Teacher Service)
Date: Sep 13, 1999 at 05:01:29
Subject: Re: Terms in an algebraic expressions

I am going to avoid an easy method and work on an effective one.  There are at
least three mathematics ideas here that students need to understand.

The first is the concept of a term.  Students are often confused about why
x + y is two terms and xy is one.  Give them a good clear foundation in what a
term is.  One workable definition is that a term is any number of constants
and variables multiplied together. This is the mathematical 'multiply' and
includes division. "x" is a term, as is "3/4", and so "3/4 x" is a term and
"x^2" is a term, etc.

The second has to do a little with bookkeeping to make a term prettier.  It is
hard even for us to quickly see that xyxxxyxx is the same as xxxyyxxx or
xxxxxxyy in this form, so they need to know that x^2 y and yx^2 are the same
but it is easier on everyone if we agree to use the form "coefficient, first
variable alphabetically, power, second variable, power, etc."  We have to
get to the idea that x^2y^3 is sort of a name for the term, and 5 is how many
of "those" there are. y^3x^2 means the same but none of them would yell
across the playground and say, "Hey, Wilson Billy, wait up!".

Just like in 5/8, the eighths (i.e. 1/8 in 5/8) tells us what kind of fraction
we are adding (Denominate literally means "to name"; the "nom" is the same
root for nominate, name a candidate; and nomenclature, a system of names.  The
time spent helping them to understand that these math words have meaning will
pay dividends in learning.) The five (i.e. 5 in the fraction 5/8) tells us how
many pieces there are (Numerator is shortened from enumerate which means "to
count").  Spend some time asking them to find the quantity and the name of
each term, written out in English.

At this point they should be able to do exercises where they can identify the
number of terms in an expression, simplify the terms to conventional style,
then tell what kind of term it is (x^2y) and how many (x-squared y)'s there
are.  Here is an example if that is not clear.

  7x+3+5x^2y-9yz^3

There are four terms. The first term is an x variable term and there are
seven of them.  The second term is a constant term and there are three of
them(units).  The third term is an x^2y variable term and there are five of
them.  When they can do this they know enough about terms to know which are
alike, and then they can understand the BIG math idea, what terms can be
combined.

The third is the really big abstract idea at work here, but they should have
learned it in first grade and it lasts forever.  It is the simple rule that
"two of these plus three of these is five of these; whatever these are".

I would actually walk them through a drill where I hold up two fingers on one
hand.

"What is this?"

"Two." is NOT an acceptable answer.  "Two What?" Eventually they hit upon
"two fingers" and I praise them lavishly. Now three fingers on the other
hand; "And what is this?" If "Three Fingers!" is the answer I hear from them
they are ready.

And if I add them together????  FIVE FINGERS (they are into it now). And we
summarize, so two fingers plus three fingers is five fingers.

Now I hold up two pencils.

"What is this?"  and then three pencils.  Then
I put them together in one hand.

Make them say it, the whole group chanting the whole sentence.  "Two
pencils plus three pencils is five pencils".

From here on out it goes fast.

What is two trucks plus three trucks?  (they shout back the answer)

What is two dogs plus three dogs?

What is two eighths plus three eighths?  (they may pause here, its starting to
sound like mathematics, but it still seems to work out.)

Has is two feet plus three feet?

What is 2x plus 3x ?

I actually come back to this time after time in higher math.  With radicals 2
sqrt(7) pleats 3 sqrt(7).  In complex numbers 2i+3i.  And in calculus,
2f'(x) + 3 f'(x) is still true.

This is all actually easier to do than to write out, but I believe that
language is an important part of understanding anything, especially
mathematics.  If they can learn to use everyday language to express
mathematical ideas, they are starting to develop a true cognitive picture of
the ideas.  I really believe that you have not understand anything you can not
find language to describe; or as I say it to the kids, "If you cannot say the
names, you cannot play the game".

Sorry, that takes a while to read, but I think each part is important.  I
hope it works for you and your students.  Good luck

-Pat Ballew, for the Teacher2Teacher service

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