The distributive property is frequently used in algebra and is an important skill for students to master in preparation for algebra. Students tend to find it fun when distributing a number across addition of two positive numbers; but confusing when distributing a positive number across subtraction, and even more so when distributing a negative number across subtraction. When negative numbers are involved, either outside or inside the parentheses, it’s helpful to represent expressions such as 5 – 6 as 5 + (–6). The warm-ups for the days lesson therefore include problems in this form to initiate discussion prior to the lesson.
Warm-Ups. Warm-Ups should include the following types of problems.
1. 4 + (–5) =? (Answer: 4 – 5 = –1 )
2. –5 + (–7) (Answer: –5 – 7 = –12
3. Tell what property is illustrated—associative or commutative: 3 – (9 –8) = – (9–8) +3 (Commutative)
4. Write as an algebraic expression. “The product of 3 and some number.” (Answer: 3x)
5. Dave works 3 hours Friday evening and 7 hours Saturday at $8 an hour. Find his total earnings for those two days. (Answer: 8 x 10 = $80—can be solved also by 8x3 + 8x7 = 24 + 56 = $80.)
Preliminaries. Problems 1 and 2 of the warm-ups provide a chance to review that subtraction is adding the opposite. Therefore 4 – 5 can be written as 4 + (-5). Provide some other examples, going back and forth; e.g., 5 + (-7) = 5 – 7; -7 – 8 = -7 + (-8).
Distributive Property of Multiplication Over Addition. Problem 5 of the warm-up provides an introduction to the distributive rule. In going over the warm-up answer, I like to ask how people solved it. Some will say they added the 7 and 3 for total hours and multiplied by the $8/hr rate. Others multiply the individual hours worked by $8. I then write on the board 8 x 7 + 8 x 3 = 80, and 8 x *(7+3) = 80.
Since both of these methods end up with the same result, we can write the following:
8(7 + 3) = 8 x 7 + 8 x 3
This is an illustration of the distributive property: the 8 can be multiplied by the each of the numbers inside the parentheses as illustrated. 6(5 + 4) can be solved as 6(9). It can also be solved by multiplying the 6 by each of the numbers inside the parentheses: 6 x 5 + 6 x 4.
After doing two or three more, I then write the formal rule on the board:
For any numbers a, b and c, a(b + c) = ab + ac.
Someone at this point may ask why bother using distribution when you can add the numbers in the parentheses and then be done with it in few steps? I explain that we don’t always have numbers inside the parentheses. For example, we might have an expression such as: 4(5 + x). Using the distributive property, this becomes 4(5) + 4(x), or 20 + 4x.
I inevitably have to remind students that when we multiply a number by a variable, such as 4(x), it becomes 4x. I typically include a problem like that in the warm-ups if I know it will come up in the lesson.
More examples for them to do: 3(x + 2); 2(7 + 2x); 5(a + b).
Distributive Property of Multiplication Over Subtraction. It doesn’t hurt to remind students one more time that subtraction is “adding the opposite” (or “adding the additive inverse”).
To introduce the shift from addition to subtraction, I write a problem on the board in this format:
2(3 - 5)
I immediately ask how I can rewrite the numbers in the parentheses, given that subtraction means “adding the opposite”. Upon hearing 3 + (-5), I then rewrite the problem:
2 (3 + (-5)
I work through this example so they can see its similarity to what we have done with addition:
2(3 + (-5) = 2(3) + 2(-5), and ask them to write in their notebooks or mini-whiteboards the multiplication and simplification, hoping to see 6 – 10 = -4.
We then try a few more, advising them before we begin to first rewrite the expression inside the parentheses before proceeding with the distribution step:
5 (x – 5) (Answer: 5(x + (-5)) = 5x – 25
6 (-2 – 7) (Answer: 6(-2 +(-7) ) = 6(-12) +6(-7) = -72 -42 = -114.
3 (-x – 2) (Answer: 3 (-x + (-2) ) = 3(-x) + 3(-2) = -3x -6
I will ask if anyone can do the distribution without rewriting the expression in the parentheses as adding the opposite. If someone can, I have them show the class, and re-demonstrate it in case the explanation wasn’t clear. (Middle schoolers do not always describe their method articulately.)
Taking one of the problems I posted, say 5 (x-5), one can see that the first term will be 5x, and the second term will be 5(-5), to obtain 5x – 25. I do not insist that students do this; if they are comfortable with rewriting the expression as adding the opposite, there is no harm done. My experience is that as students become more comfortable, they move towards the short-cut.
Now we move on to problems like -2 ( 3 – 5).
I will state that we will not do a short-cut just yet, but rewrite the expression in the parentheses as before, that is: -2 (3 + (-5)
Distributing the -2, we obtain: -2(3) + (-2)(-5) = -6 + 10 = 4
And, as before, they are left to try some examples:
-5(-2 – x) Answer; -5 (-2 + (-x) ) = (-5)(-2) + (-5)(-x) = 10 + 5x
This last may cause some confusion. There will likely need to be explanation that if the variable has a negative sign, then it is treated like a negative number for purposes of multiplication.
After a few more examples, I ask again if anyone has found a shortcut. Usually, there are more students who see that the subtracted value is going to be positive. So a distribution like -2(3 – 5) is seen s as -2(3) + 10, knowing that -2(-5) will yield positive 10. Again, I do not insist that they use the shortcut, but like before, the more problems students do, they will be inclined to use the shortcut.
Finally, I will end with a problem of this form: -1(5 – 7), which yields -5 + 7 or 2. I point out that sometimes they will see a problem written as –(5-7), which means there’s a -1 outside the parentheses. I will do maybe one or two more and ask if they see a pattern regarding the signs of the numbers inside after multiplying by negative 1. Someone will see that the signs are reversed.
This suggests yet another short-cut. One can consider a problem such as -2 (3 – 7) as 2(-1)(3 – 7). We start with distributing the -1, which changes the signs inside: 2(-3 + 7). Now they multiply to obtain -6 + 14 = 8. The simple rule is: when the outside number is negative, change it to positive, and change the signs of the numbers inside, and then distribute.
This particular short-cut is one which I reserve more for accelerated classes at this stage. For other classes, I will introduce it when we reach the unit on subtracting linear expressions since they will have had more experience and can more easily follow what is going on with the procedure.