This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational. (Readers are encouraged to provide examples of mistakes that students will make for the particular topic being discussed. They will be incorporated into the ever-evolving text, so you can be a part of this next book!)
Normally, before introducing the topic of combining like terms, comes a lesson on the foundational axioms of math: The associative and commutative rules, and the distributive property. I reverse the order because I have found that after working with combining like terms, the rules make more sense when it is revealed that these rules are what they have been using when simplifying expressions.
There’s no harm in teaching the rules prior to this lesson; I just find it easier to explain. Also, textbooks give problems that have students identify the principal (i.e., associative or commutative or both) at work with statements such as (x+y) + 7 = 7 + (x+y). Students find these confusing without having some direct experience first. Such problems sometimes feed into their perception of math class as being taught using a simple example, and then given problems that don’t look like anything that they’ve learned. More on this in the next section.
Warm-Ups. Warm-Ups for today’s lesson should include some of the basics of algebraic symbols as well as working with negative numbers.
1. Simplify the expression x + x (Answer: 2x)
2. -5 + 10 – 2 = ? (Answer: 3)
3. Evaluate ab + a where a = -2 and b = -5 (Answer: (-2)(-5) + (-2) = 10-2 = 8)
4. 5(3 – 7) = ? (Answer: 5(-4) = -20)
Terms. When addition or subtraction signs separate an algebraic expression into parts, each part is a term. For example a + b consists of two terms, a and b. Suppose I had 2a + 3b. Then 2a and 3b are terms.
Ramp it up a bit: What about 2ab + c?
The numerical part of a term that contains a variable is called the numerical coefficient of the variable. I will ask what the numerical coefficient is of 5x, of -24y, of 3xy.
At this point, I like to point out that variables with no coefficients, like a, b, n, and so on, have an unwritten coefficient of 1. Some students have a difficult time remembering this. The expression 5x + x is often stared at. I tell them to put a 1 in front of the x and then tell me the answer. While this is a good stop-gap technique to get them over the initial stare-down, it is important to get them in the habit of knowing that such variables have an unwritten 1 in front of it. Confusion over this can lead to errors later. Specifically, when they get into algebra and have to simplify expressions such as a/2a, some will write it as 0/2. The expression x/x2 I’ve seen written as 0/x. It is therefore important to keep reminding them about the “1” coefficient.
Like Terms. If two terms are the same variable, or the same combination of variables, they are called “like terms”. Examples such as 2x and 34x, 8ab and 5xyz, and so on. Because the terms are the same, they can be added, just like we added x + x previously. This tends to be confusing at first, so I start with single variables:
The expression x + x + y + y + y can be written as (x+x) + (y + y + y), which is 2x + 3y. Since the coefficients are 1 for each of the variables in the example, what we have done is add the coefficients: 1 + 1, and 1 + 1 + 1.
Taking it to the next step, 3x – 2x + 5x, is the same as adding the numerical coefficients—that is 3 – 2 + 5—and adding the variable afterward: 10x. The summary statement I give them is:
To combine like terms, add or subtract the coefficients.
We can only combine like terms. It is important, therefore, to make it clear that although 2x + 3x can be combined to form 5x, 2x + 3y cannot and must be left that way, although students will try their best to combine anything that gets in their way. It is helpful to emphasize the difference between like and unlike terms.
I write on the board examples and ask if they are like or unlike; if like, I want the combined result:
6a – 4a: Like terms, result is 2a.
6ab – 4b. Unlike.
6c + c. Like terms; results is 7c.
When terms are combined this is also known as simplifying the expression. The word “simplify” can mean many things in math, so it is important to explain what we mean for specific contexts. For example, later in algebra, saying “simplify (x+y)^2 means to expand the binomial. (Which is the term I use, though textbooks will say “simplify”). On the other hand, “simplify x^2 +5x+6” means to factor it: (x+2)(x+3).
Writing statements in the form of questions that students need to complete is how I will sometimes summarize this particular lesson:
1. In the expression 5x, the coefficient of x is ___.
2. In the expression 4a + b, the coefficient of b is ___.
3. Simplify if possible 2a +6a + 3. If not, say “not possible” (Answer 8a + 3)
4. Simplify is possible 5a + 3xy – xy + 4a. (Answer 9a + 2xy)
5. Simplify if possible. 8n +2. (Answer: Not possible)
6. Simplify if possible. 2m + 6 + 5m – 2n – 4. (Answer: 7m – 2n + 2)
7. Traditional Math: Simplifying Expressions: Combining Like Terms
Thank you Barry for yet another valuable article. I am an old chalkie and hope to pen you a couple of views of my own along the lines of "traditional" education. You may delete them if they offend.
Well done Barry.
A tangential thought: while you can't use associativity with subtraction as ordinarily expressed the rule works perfectly fine when combined in the right way with the distributive law and regarding subtraction as a combination of multiplication and addition:
9-5-3 = ?
On the one hand (9-5)-3 = 4-3=1 whereas 9-(5-3) = 9-2 = 7.
But (9-5)-3 = (9+(-1)5)+(-1)3 = 9+((-1)5+(-1)3) = 9+(-1)(5+3) = 9-(5+3) = 1
whereas 9-(5-3) = 9+(-1)(5+(-1)3) = 9+(-1)5+(-1)(-1)3 = (9+(-1)5)+3 = 7
Probably a TERRIBLE thing to get middle-schoolers to muddle through though while they're first mastering the rules. It has its place when simplifying algebra, though. Eventually one learns to fluently convert to sums as needed. And in compound quotients sometimes to convert to products.