In ordinary English, ‘to evaluate’ means to determine something’s worth or value. Or it can mean weigh the parameters as they apply to a situation and determine what course to take. You want to visit your friend, but he leaves in a rough part of town. You weigh the pleasure of visiting your friend vs the dangers to you. Given your circumstances, you may decide one way or another.

You might think, ‘Well, I haven’t seen him in a while, and I’m really rough and tough, am licensed to carry a gun and look like a sasquatch. I think I might go see him” Or you might say, “I’m a short, thin person, can’t defend myself, and there’s just been a rash of mugging where my friend lives. Maybe it’s not worth it.”

In algebra, the term evaluate is similar but has a more specific application. You are determining the value of an expression. Value is like worth, or weighing risk vs. benefits, but it has a specific numerical value. As mathematicians are, ironically, fond of simplifying, evaluation means to simplify an abstract expression down to a single set of numerical values. This is done in the application of formulas all the time.

D = M/V

Density equals mass divided by volume.  So if you have an object that weighs 14 kg and its volume is 30 liters, it’s density will be .47 kg/cm^3

Thus, the abstract formula is “simplified” to a specific concrete example.

The practical value of this is obvious, as most of the time, whether in electrical work, determining tire pressure, or how long a trip will take, we put specific values into an abstract equation. This is known as “plugging in” a value.

Plugging in a value and then you chug along to find the answer. That was fairly straightforward in the example I gave.

However, if you have minus signs, uh, oh.

Evaluate: -a^2b – c + d for a = -2, b = 3, c = -5, and d =8

There are minuses in the specific numbers to plug in and minuses in the expression. To keep track, I put everything in parentheses.

-(-2)^2)(3) –(-5) + (8)

Then I follow PEMDAS (Parentheses, exponents, multiplication/division, addition/subtraction.)

So, we get -2 squared, which is 4, then we minus that, which is -4; times that by 3, which gives us -12. Then we subtract –(-5), doing the multiplication first which makes it +5. Think of a minus sign as multiplying by -1.

Thus we have -12+5+8.

Going from left to right, we get 1. (Applause!).

Let’s suppose you have an example like this:

Evaluate (a +b) squared for a = -5, b = 3, c= -8, and d = 13

Remember PEMDAS!  You do parentheses before you do exponents. So you first add -5 to 3 and get -2, which you then square to get 4.  You don’t go -5 squared plus 3 squared, and get 25 + 9, which gives you 34, the wrong answer. Never drive left of center.

Also, note that there was no c or d in the expression. They often give you more information than you need to see if you sort it out to get the information you do need.

I hope you had fun with this. Those who don’t need any more help can stay after class and clean the erasers.

Dr. Fred Young


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