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Rewriting the Substitution Property

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Rewriting the Substitution Property

[Note: the content of this article has been superceded because of this article: Redefining Terms. Rather than specifically talking about inserting parentheses, I use the definition of term given in the article above, and tell students, "If the expression you are substituting is not a single term (i.e. a fully grouped expression), you must turn it into a term before you substitute." Thus, if it's not already a term (group), you turn it into a term by putting parentheses around it. This is a cleaner and more helpful way of explaining it, but it depends on our alternate definition of "term."]

When I am teaching the substitution property to my Algebra One students, I don't teach it the way it appears in the textbook. In the textbook, it appears as follows: 

If x = y, then x can be substituted in for y in any equation.

I tell my students to put this in their notes as follows:

If x = y, then (x) can be substituted in for y in any equation.

I tell my students that any time they perform a substitution, they should enclose the substituted quantity in parentheses. There are two reasons I tell them this.

The first reason is that, not surprisingly, there is an occasional student who will take the expression 2x, where x = 3, and rewrite it as 23. Requiring them to put parentheses in transforms this into 2(3), and they've been taught that parentheses juxtaposed against a number or variable represents multiplication.

The second reason for this is, of course, that x may be a multi-term algebraic expression, rather than just a number. In that case, we're substituting the entire expression into another expression, and the parentheses may be needed in order to keep order of operations intact.

To help illustrate this, I give my students a problem like this: My second number is four more than twice my first number, and my two numbers add to 34. What are the two numbers?

We can solve this by saying that x is the first number, and y is the second number. Then we have:

y = 2x + 4
x + y = 34

I can replace y (in the second equation) with the quantity (2x + 4). Thus, the second equation becomes:

x + (2x + 4) = 34

This gives me an opportunity to show my students how the distributive property can be used to get rid of parentheses. "Would you agree," I say, "that (2x + 4) and 1(2x + 4) are the same thing?" My students agree. So I show them that I can do this:

Step 1: x + 1(2x + 4) = 34
Step 2: x + 1(2x) + 1(4) = 34
Step 3: x + 2x + 4 = 34
Step 4: 3x + 4 = 34
Step 5: 3x = 30
Step 6: x = 10
Step 7: y = 2(10) + 4 = 24.

One of my students will invariably notice that the parentheses "didn't matter" - if I'd left the parentheses out when I did the substitution, I would have arrived at Step 3 immediately.

"Right," I say, "But now look at this problem: My second number is twelve less than twice my first number, and the difference between my first number and my second number is 4. What are the two numbers?"

This time the equations look like this:

y = 2x - 12 and x - y = 4

Now when we perform the substitution, we obtain x - (2x - 12) = 4. How do we get rid of the parentheses? By putting a 1 in front of them and distributing again. "Except," I point out, "there's a negative symbol in front of the parentheses, so really, that makes it negative one outside the parentheses, right?"

Step 1: x - 1(2x - 12) = 4
Step 2: x - 1(2x) - 1(-12) = 4
Step 3: x - 2x + 12 = 4
Step 4: -x = -8
Step 5: x = 8
Step 6: y = 4

Now the same students who noticed I didn't need the parentheses in the first example, can clearly see that I would have obtained an incorrect equation if I'd skipped the parentheses in the second example.

As time goes on, my students recognize that there are circumstances under which the parentheses are not actually necessary, and they are able to correctly identify those circumstances. However, at the beginning, I find it helpful to force my students to include parentheses when doing substitutions, even if they aren't, technically, necessary!

Lesson by Mr. Twitchell

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