Have you ever noticed how much trouble students have figuring out the difference between these two kinds of expressions:

The key, I’ve found, is to use the Order of Operations to decode the meaning of each expression.

Ask your students to think through how many Order of Operation steps there are in each expression.

If they’re on the right track, they’ll see that the top expression has just one operation: working out the exponent by multiplying (– 3) by itself twice. This looks like this:

As for the bottom expression, this is more difficult because of that “loose” negative sign, a negative sign that is not enclosed inside parentheses. Just like a “loose canon” can be a danger in the military, a “loose” negative sign can spell trouble on the algebra playing-field. Students will make all kinds of weird mistakes in trying to figure out what to do with this little varmint.

But if you tell students to view that “loose” negative sign as meaning (– 1) times the expression that follows it, they can use the Order of Operations again to do the right thing first.

Specifically, they notice that there are two operations to be performed: multiplication and exponents, and they remember that they do exponents before multiplication, following the Order of Operations.

Following these steps, the simplification looks like this:

And while this is fine and correct, notice that you can “kick it up a notch” by using color to separate out the two operations. I use blue for decoding the negative sign and working it out; I use red for the 3-squared term. Check it out:

I’ve yet to meet a student who cannot follow this procedure, when color is used.

Try it out and see what kind of results you get.

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