The associative and commutative rules are at the heart of many if not most of the operations we do in math. In the previous lesson, students simplified linear expressions such as 3 + 2 -9 +8 -5 and 2x + 3y –x + 2y. The rules of associativity and commutativity have allowed them to group the positive and negative numbers together and then proceed. Similarly with algebraic expressions, they learned to combine 2x with –x, and 3y with 2y. Now we focus on the formal rules that allowed them to do what they did.
From what I’ve seen when teaching this to seventh graders, they view these rules as one more set of formal rules which are regarded as something they will never use. I continue teaching it, not because it is called for in the applicable math standards, but because of its value to the subject and because I think it is well-served to try to produce educated people.
The topic isn’t so abstract that it is above the learning capabilities of seventh graders, nor even sixth graders. I can’t tell which students are going to end up becoming mathematicians and which ones are not. Because of this general ignorance of where my students will end up, I keep in mind the preface to the classic mathematics text, Naïve Set Theory by the late mathematician Paul Halmos (1955). In it he makes a statement about set theory that I find applicable here: “General set theory is pretty trivial stuff really, but if you want to be a mathematician you need some, and here it is; read it, absorb it, and forget it.”
Warm Ups. Continuing on with the review and preview style of warm-ups, we want to use problems that make use of the associative and commutative rules. Here are some suggestions:
1. 5 – 7 + 8 – 2 = ? (Answer: 4)
2. Combine like terms and simplify. 6y +3x – 2y + 7x + 9 (Answer: 4y+10x+9)
3. 5x + 5x + 5x = ? (Answer: 15x)
4. 3(5x) – 4x +3y +3(2y) (Answer: 15x - 4x + 3y + 6y= 11x + 9y)
Associativity of addition. I will ask students to add 252 + 60 + 40 being careful to remind them not to shout out their answers. I will ask how many students added straight across; that is 252 + 60, and then adding that sum to 40. I then ask how many added 252 to the sum of 60 and 40 first. I will then ask which method they thought was an easier way.
I write on the board: (252 + 60) + 40 = 252 + (60 + 40), and explain that the parentheses indicate the grouping of an operation. The right hand side illustrates adding the 60 and 40 separately, and then adding the total to 252. The left hand side illustrates adding the 252 and 60 first, and then adding that result to 40.
Does it matter how we group the numbers? I’ll ask. In my experience, the general consensus has always been “no” which is when I’ll state that this property is called the Associative Property, and then write it formally:
For every number a, b and c: a + (b+c) = (a + b) + c
I then give examples of associative groupings that do in fact matter, but the groupings are taken for granted. If I wrote 14 + 6 + x, we can combine the 14 and 6, but not the 6 and x. Therefore, we are using the associative property as (14 + 6) + x in order to combine the like terms—in this case, the numbers 14 and 6.
What does the grouping look like for 7 + 9 + x? (Answer: (7+9) + x)
Similarly when we combine like terms such as x + 2x + y, we are applying a grouping of (x+2x) + y to obtain 3x + y.
Associativity of multiplication. The same rule for groupings also holds for multiplication. In fact, multiplication is usually done concurrently with the presentation for associativity of addition.
I'll ask whether (4 x 5) x 2 = 4 x (5 x 2). And in fact, it does not matter as they’ll quickly say. I’ll point out that just as we found an easier way to add, we do the same type of thing with multiplication.
5 x (2 x 9) = (5 x 2) x 9. Is it easier to multiply 5 x 18, or 10 x 9?
The formal rule for associativity of multiplication is then given:
For every number a, b and c, a(bc) = (ab)c
3 x (7 x 9) = (3 x 7) x 9
Subtraction and Division. Next, is whether subtraction and division are associative?
I will ask whether these two expressions are equal 24 – (6 -2) =? (24 -6) -2
The quick answer is “no”; most students find this fairly obvious. I once had a student point out that if the expression were written as 24 + (-6) + (-2), then (24+ (-6) ) + (-2) = 24 + ( (-6) + (-2) ), then the expressions would be equal. In this student’s case, I discussed and explored it in private with him since his level of understanding was high. Trying to explain his reasoning to the class which did not yet have that level of understanding would result in confusion, I felt.
Similarly, I do the same with division:
24 divided by (6 divided by 2) does not equal (24 divided by 6) divided by 2
Commutativity of addition. I start with a simple example of commutativity and ask whether 6 + 7 = 7 + 6. This is admittedly a no-brainer, and they have learned long ago that the order of addends does not matter. I then write the formal rule:
For every number a and b, a + b = b + a
I now go back to the first example: 252 + 60 + 40 and ask whether it would make a difference if we added the 60 and 40 first or later. The consensus is “no” and I’ll ask why. What I’m looking for is “the order doesn’t matter” but I’ll take anything that resembles it. If I hear something like 252 + 100 = 100 + 252, I will write that on the board immediately. If I don’t hear that, I’ll write it on the board anyway.
Since all agree that 252 + 100 = 100 + 252, I will rewrite it as:
252 + (60 + 40) = (60 + 40) + 252
Is it still true? If there is any doubt, I will remind them that we said earlier it doesn’t matter whether we add the 60 and 40 first or last—it will still equal 100.
With this established, I will ask whether 7 + (9 + 2) = (9 + 2) + 7
I give a few more examples, and now mix up associativity and commutativity and ask them to tell me which property is at play. Examples:
(7 + 3) +2 = 7 + (3 + 2) (Associative)
56 + (85 + 34) = (85 +34) + 56 (Commutative)
I ramp up the difficulty and will throw in problems like:
45 + (3 +56) = 45 + (56 + 3) (Commutative)
Some students will say associativity, because 45 stays in the same position. I’ll ask why it is commutative and then give another problem similar.
If you recall, in the previous section I said problems like these seem to feed in to the notion that math class often gives problems that are nothing like the examples they’ve seen in class. This is why I’ve presented the material in this fashion. I have given them examples of the problems they will be seeing, so they have had experience with the simple versions and more complex ones.
Commutativity of multiplication. Multiplication is shown the same way as above, and is usually done concurrently with addition, just as associativity of multiplication is done. The formal rule for commutativity of multiplication is:
For every number a and b, ab = ba
Subtraction and Division. As with associativity, the operations of subtraction and division are not commutative. This is easily seen with numerical examples: 6 – 3 does not equal 3 -6. And 25 divided by 5 does not equal 5 divided by 25. This last example usually requires some explanation, since students tend to forget that fractions are division. The fraction 5 divided by 25 is 5/25 or 1/5.
One More Type of Problem before Homework Assignment. To wrap up I provide one more type of problem that combines everything covered in the lesson.
I point out that some problems like 7 – 2 + 3 can be simplified by grouping addends that add up to ten. For example, the above expression can be reordered to 7 +3 – 2 . For such problems, I ask that each step be identified by providing parentheses and naming the property involved: commutative or associative. The first step would be to switch the –2 + 3 around, so the expression becomes 7 + (3 – 2). The next step is to write it as (7+3) – 2. The last step is simply 10 – 2, which is referred to as “simplifying”. For these problems, students may find it easier to write subtractions as adding the opposite; the above problem can be rewritten as 7 + 3 + (–2). Then it is easier to see that the commutative step is (–2 + 3).
These type of problems extend to multiplication as well, in which students find ways to make products of 10, 100 and so forth.
Students should be given some worked examples and then a few independent before starting them on the homework assignment. Examples of this last type of problem with steps identified follow:
-17 + (38 +3) = 17 + (3 + 38) (Commutative)
= (17 + 3) + 38 (Associative)
= 20 + 38 (Simplifying)
= 58 (Simplifying)
4 x (59 x 25) = 4 x (25 x 59) (Commutative)
= (4 x 25) x 59 (Associative)
= 100 x 59 (Simplifying)
= 5900 (Simplifying)
For each m,
5 + (m + 3) = 5 + (3 + m) (Commutative)
= (5 +3) + m (Associative)
= 8 + m (Simplifying)
In this case they are adding the like terms (5 and 3)