How to Add and Multiply Monomials

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Before I was an accountant who quit her job to stay at home with her kids, I was the daughter of a math teacher.  You could just imagine my younger days, as a humble math prodigy, with a busy schedule of winning (and occasionally losing) math competitions, discussing the latest math tests with my fellow geeky friends, arguing math hypothesis with my dad in the study den….

Too bad it never happened.

My childhood was more about cool ways to make Lego people than solving equations for fun.  (I had no idea Lego mini figures existed!)

I was forced to do calculus in college and was put to shame by a young prodigy as he sailed through the exams….I got back at him in Spanish class though.  It was awesome to see him sweat as he tried to conjugate a verb!  Take that, you over-achiever!

Both my mother-in-law, Rita, and my father were high school math teachers for many years.  They were strict teachers who exhausted themselves in their craft.  Numerous times, their former students, now older and wiser, have found them on social media or in random public places and thanked them for their math & life lessons.

I can’t be the first teacher’s daughter to wrestle with higher level math…and lose.  Sorry, dad, it’s not in the genes.  I avoid my daughter’s homework time by planning my dinner prep at that very moment each day.  Oops, I’m sautéing the chicken, can’t leave the stove….

A few weeks ago, after my gardening post, Rita wrote up a random instructional sheet on how to add monomials versus how to multiply them.  She patiently explained it to myself and my daughter while I tried to follow along.

I am not ashamed to admit it – I didn’t know why adding/multiplying algebra would be such a big deal.

Until this summer.  Maybe Rita was working her magic.

This past July, I finally bought these popular Summer Bridge activity books I’d heard of.  They promised a good refresher of the past year and to also slowly introduce your child to new concepts.  It was an inexpensive purchase but you have to do all the grading yourself.

I was relieved to find an answer key.

Each week, I was like the all-knowing wizard, raising my wand over my daughter’s work and putting proud little checks on her correct answers.

One day, when I realized that my daughter had the wrong answers for an entire page, my wand quickly vanished.  My free ride was over – I had to get involved and teach some math.

I sat down with her and we attempted to work it out together.  Surprisingly, I found her mistakes were partly from her misunderstanding of working with monomials.

I remembered suddenly what my mother-in-law had tried to explain to me and then my daughter was back on track.  Rita’s worksheet glared at me from the corner of my desk.

After reading this, you might never know what a monomial is either or need to look for it hiding in your kitchen cupboards.  (Don’t worry, it’s not.)

BUT you are going to help your child clear up an often confused math concept and they’ll thank you for it….maybe before they graduate from high school.  Hang in there.


Have a look at these equations – which of the following are correct?

 (The multiplication sign is replaced with a middle dot · so you can tell the difference between multiplication sign and the x.  It’s genius but I didn’t learn math that way so it looks like a different language!).

Hey, guess what?  They’re all wrong.

And why should you care?

Because it’s an easy fix and you can help make the rest of your middle schooler’s math journey less formidable in one lesson.


To add monomials, think about how we calculate the perimeter of something – we add the lengths of the sides.

1 meter + 1 meter + 1 meter = 3 meters

Let’s get fancy:

1m + 1m + 1m = 3m


m + m + m = 3m

Same thing, right?  Just shorter and to the point.

And since math teachers love to use ‘x’ as the variable, let’s do the same:

1x + 1x + 1x = 3x


x + x + x = 3x

Remember that when you’re ADDING, you’re grouping all the variables (x) by their whole numbers or co-efficient (1x + 1x + 1x) and ending up with one nice, neat combination (3x).

Here’s a slightly more difficult one:

x2 + x2 = 2x2

Don’t let that little mini ‘2’ throw you off – that’s the exponent and nothing to be afraid of.  When you add monomials, you add the whole numbers and keep the exponents just the way they are.


To multiply monomials, you will need to work with exponents, such as  x2 above, the 2 being the exponent.

Think about how we calculate the AREA of something – we multiply the lengths of two sides, right?

1 meter  ·  1 meter = 1 square meter

Or written more concisely:

m  ·  m = m2

Wait!  Shouldn’t it be  m · m = 2m?  Nope.

When you multiply, you express the answer as the base variable (m) and write the sum of occurrences as the exponent.  

Here’s another example:

m3 · m3 = m6

Students would look at this and think the answer is 2m3 or even m9 but they would be wrong.

It’s like saying m ·  m ·  m ·  m ·  m ·  m = m6   but all you had to do is keep one ‘m’ and add the exponents.  The base variable stays the same – only the exponents change.

Another thing to repeat to yourself: when we multiply monomials, we ADD the exponents, not multiply them.

Now, if you see a problem with whole numbers in the mix, don’t worry – here’s how you handle it:

3a ·  2a = 6a2

The answer is not 5a.  Neither is it 6a.

You think of it as:

 3  ·   2   ·   a  ·  a =  6a2

Multiply the whole numbers, write the base variable ONCE and list the number of times it occurs as the exponent.

When in doubt, stretch it out and see what you have.


3a + 2a = 5a

3a ·  2a = 6a2

These are both correct.

And if you’re curious, here are the correct answers from our trick question:

Cute graphics courtesy of Scrapbook Gems.