Combining Like Terms

So let's talk about combining like terms for a minute. This is a phrase that comes up a lot in algebra, as the teacher rattles off a whole list of things you are supposed to do. "Now you just combine the like terms, bring the x's over here, divide, and voile, here is your answer: x = 2!" Yeah, not only do you have no idea what he just said, you are not sure that x = 2 could ever be the answer to anything in your life!

So I guess we should first address the word "term". In an algebraic expression (click here for a short discussion on what that is) there can be one or more terms. Terms are the parts of the expression that are separated by + or - signs. Let's look at some examples:

x + 2y - 4 => There are 3 terms here: x, 2y, and -4
(Yeah, I know, you thought the minus sign was just there to separate the
2y and the 4, but it also belongs
to the 4)

3x² - 5x + 6 => There are 3 terms here as well: 3x², -5x, and 6

8y + 3 - 6y - 1 => There are 4 terms in this one: 8y, 3, -6y, and -1

Got it?

Now, like terms are terms that are enough the same that they can be combined. (I am indebted to this website for this explanation) Let's go grocery shopping again. (can you tell where the bulk of our money goes?!) Suppose you buy a bunch of gala apples, a few granny smith apples, some pears, and some cheerios. When the checker rings it up, sure they can just keep scanning individual apples, one by one -- or they can group all of the same type of apple together, as well as all of the pears. When it comes to the Cheerios, if there is more than one box, they will scan one and then hit the "mulitple" key on their register. Right?

So it's the same thing with algebra. Take the last example above. Sure, you can keep it at 8y + 3 - 6y - 1. There's nothing wrong with it! It's correct, but it is just more complicated than it needs to be. Why not combine the two terms with y's? Then the whole thing will be shorter. So we think to ourselves, 8y - 6y -- that must be 2y! (if you have trouble with this, think: 8 gala apples - 6 gala apples = 2 gala apples!)
Now instead of having to write (or say) 8y + 3 - 6y - 1,
you will write 2y + 3 - 1.

So now what do you think about the 3 and the -1? Can we combine them?? Sure!! That will make it:
2y + 2

See how easy this is? Does it matter that I kind of had to mix up the order to get this to work? Nope. It doesn't matter which order the checker rings your stuff up in, and it doesn't matter which order you write algebraic stuff in either. Just keep minus/negative signs with the numbers they "belong" to -- i.e. the number that follows them (so in our example here, the negative sign belongs to the 1)

Want to try a few? Have at it!

1) 8x - 3 + 4y - 2x
2) 2y² + 3y - y + 4
(hint: the 2y², the 3y, and the -y are all y's, but they are not all the same type -- like gala apples and granny smith. Only the y²'s will combine with each other, and the single y's (not squared) will combine together.
3) 97 - 3z + 25

Scroll down for answers:




























1) 6x - 3 + 4y
6x + 4y - 3 would also be correct.
2) 2y² + 2y + 4
3) 122 - 3z

Woo-hoo! Look how cool you are!!

Constants, Variables, and Algebraic Expressions

It’s almost back-to-school time! Perish the thought, I know . . . at our house, we will be in denial until the big day, if not longer. The kids at school will all be starting off in their math classes with some basic terms and definitions, so I thought I’d get us off to a start with some basic algebra.

The words “algebraic expression” may sound really complicated, but really it is just a group of numbers and letters, (constants and variables) that represent (express!) something. Everything in an algebraic expression is either a constant or a variable. The constants always remain the same (thus the term constant). They are numbers, like the 2 and 5 in 2x + 5, or the price of a loaf of bread at the grocery store, and their values obviously don’t change (unless there is a sale!)

Variables are represented by letters in an expression, and they can change. In our algebraic example above (2x + 5), the variable is x. In a real-life situation, a variable might be the number of loaves of bread you buy at the store. It can change with every trip to the store.

Now, the magic of algebra is that you can write one equation, but use it for a number of different scenarios. In our grocery example, you might say that one week you bought 2 loaves of bread. If bread costs $2.19/loaf, you will have spent 2 x $2.19, which equals $4.38. But what if you had bought 3 loaves? Or only 1? That’s where algebra comes in handy. Instead of writing 2 x $2.19, 4 x $2.19, or 1 x $2.19, let’s just write the “ x $2.19” part once, recognizing that no matter how many loaves we buy, we will multiply that number by $2.19.

______ x $2.19 = the amount you spent on bread

Now, let’s use the variable x for the number of loaves you bought. The only thing is, if I write x times $2.19, it will look like this: x x $2.19. That’s kind of confusing, isn’t it? So to keep us from confusing x and times, there are some new ways to represent multiplication:

1)instead of an x, in algebra, we use a dot between the two factors. I don’t know how to type this, but imagine the following without the bottom dot of the colon
4 : 3 = 12
2)we also use parenthesis to mean the same thing:
(4)(3) = 12
3)finally, whenever there is a variable and a constant (a letter and a number), we just write them together whenever we mean multiplication:
4x = 12

Ok, back to our example. Using an x for the number of loaves you bought, we have:

(x)(2.19) = the amount you spent on bread, or
2.19x = the amount you spent on bread.

Now you just use whatever number you want for x. It could be a 2, or a 10. You do the problem the same way!

Yeah, it’s basic, but that is the magic of algebra!!

I'm just wondering. . .

Ok, so one of the reasons I'm trying out this blog is to see if the functionality of blogger is enough for what I want to do here. So I'm just checking to see if this video embeds.

Beep beep!

Math!? You want to talk about MATH!?

Ok, so I know I'm a nerd. A math nerd anyway. And I know you probably didn't love math as much as I did in high school. You may have even hated it.

But here's the thing. I'll bet you remember math being really hard. I mean, who understood 2-column proofs, factoring, or trigonometry anyway? Well, I'm here to tell you a secret that I've learned through 17 years of tutoring and several years teaching math. . . Guess what? It's not really all that hard!

The first time you learned algebra, you were probably 12 or 13. You were all caught up in the drama of junior high, and you didn't have the mental capability to really fully wrap your mind around the abstractions of algebra anyway. And let's not even talk about geometry and proofs. But now you are an adult. We use logical thinking and abstract thinking every day! Did you ever have to explain anything to your boss (or your two-year-old?) Then you probably could handle some simple geometrical proofs. Do you have to think through how you will spend your money each month, even though some expenses are variable and others are fixed? Well, then you have the general idea of constants and variables down!

I am going to be covering random math topics here, in no particular order, for the use of adults who may want to brush up for one reason or another. Be patient with me, as I am in school and parenting 4 kids. It would take an eternity to cover absolutely everything, so keep checking back if you are interested. If you want to see something specific, send me a comment and I'll do my best to make a post about it. Hopefully this is useful to you. Let me know what you think! I welcome comments and suggestions.

How exciting! I'm going to have my own blog about math!