As you’re probably aware, I’m a big believer in using stories to bring math to life. Especially when you’re teaching tricky concepts, using a story can be the “magic switch” that flicks on the light of understanding. Armed with story-based understanding, students can recall how to perform difficult math processes. And since people naturally like stories and tend to recall them, skills based on story-based understanding really stick in the mind. I’ve seen this over and over in my tutoring.
The kind of story I’m talking about uses an extended-metaphor, and this way of teaching is particularly helpful when you’re teaching algebra. Ask yourself: what would you rather have? Students scratching their heads (or tearing out their hair) to grasp a process taught as a collection of abstract steps? Or students grasping a story and quickly seeing how it guides them in doing the math? I think the answer is probably pretty clear. So with this benefit in mind, let’s explore another story that teaches a critical algebraic skill: the skill of “unpacking” terms locked inside parentheses.
To get the picture, first imagine that each set of parentheses, weirdly or not, represents a corrugated cardboard box, the kind that moving companies use to pack up your possessions. Extending this concept, the terms inside parentheses represent the items you pack when you move your goodies from one house to another. Finally, for every set of parentheses (the box), imagine that you’ve hired either a good moving company or a bad moving company. (You can use a good company for one box and a bad company for a different “box” — it changes.) How can you tell whether the moving company is good or bad? Just look at the sign to the left of the parentheses. If the moving company is GOOD, you’ll see a positive sign to the left of the parentheses. If the moving company is BAD, you’ll spot a negative sign there.
Here’s how this idea looks:
+ ( ) The + sign here means you’ve hired a GOOD moving company for this box of stuff.
– ( ) This – sign means that you’ve hired a BAD moving company to pack up this box of things.
Now let’s put a few “possessions” inside the boxes.
+ (2x – 4) This means a GOOD moving company has packed up your treasured items: the 2x and the – 4.
– (2x – 4) Au contraire! This means that a BAD moving company has packed up the 2x and the – 4.
[Remember, of course, that the term 2x is actually a + 2x. No sign visible means there's an invisible + sign before the term.]
What difference does it make if the moving company is GOOD or BAD? A big difference! If it’s a GOOD company, it packs your things up WELL. Result: when you unpack your items, they come out exactly the same way in which they went into the box. So since a good moving company packed up your things in the expression: + (2x – 4), when you go to unpack your things, everything will come out exactly as it went in. Here’s a representation of this unpacking process:
+ (2x – 4)
= + 2x – 4
Note that when we take terms out of parentheses, we call this “unpacking” the terms. This works because algebra teachers fairly often describe the process of taking terms out of ( ) as “unpacking” the terms. So here’s a story whose rhetoric matches the rhetoric of the algebraic process. Convenient, is it not?
Now let’s take a look at the opposite situation — what happens when you work with a BAD (boo, hiss!) moving company. In this case, the company does such a bad job that when you unpack your items, each and every item comes out “broken.” In math, we indicate that terms are “broken” by showing that when they come out of the ( ), their signs, + or – signs, are the EXACT OPPOSITE of what they should be. So if a term was packed up as a + term, it would come out as a – term. Vice-versa, if it was packed up as a – term, it would come out as a + term. We show the process of unpacking terms packed by a BAD moving company, as follows:
– (2x – 4)
= – 2x + 4
And that pretty much sums up the entire process. Understanding this story, students will be able to “unpack” terms from parentheses, over and over, with accuracy and understanding.
But since Practice Makes Perfect, here are a few problems to help your kiddos perfect this skill.
“Unpack” these terms by removing the parentheses and writing the terms’ signs correctly:
a) – (5a + 3)
b) + (5a – 3)
c) – (– 3a + 2b – 7)
d) + (– 3a + 2b – 7)
e) 6 + (3a – 2)
f) 6 – (3a – 2)
g) 4a + 6 + (– 9a – 5)
h) 4a + 6 – (– 9a – 5)
a) – (5a + 3) = – 5a – 3
b) + (5a – 3) = + 5a – 3
c) – (– 3a + 2b – 7) = + 3a – 2b + 7
d) + (– 3a + 2b – 7) = – 3a + 2b – 7
e) 6 + (3a – 2) = + 3a + 4
f) 6 – (3a – 2) = – 3a + 8
g) 4a + 6 + (– 9a – 5) = – 5a + 1
h) 4a + 6 – (– 9a – 5) = + 13a + 11
Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com Just click the links in the sidebar for more information!