I wonder if the holidays leave
dogs rather perplexed. They watch us do things like hide painted eggs in the
yard in the springtime, watch fireworks in July, stick candles in food only to
light them and blow them out, dress up in costumes in October, and bring a tree
inside to decorate. (They totally get Thanksgiving though).
As if this
weren't confusing enough, Sparky watches us bring presents home and wrap them
up, only to
unwrap them again, save some of the silky
ribbons and sparkly bows, and do it all over again at the next event. What's
with the wrapping and unwrapping? I'm guessing he'd just as soon get the toy and
be done with it.
If you're like me, shiny wrapping paper and bows are one
of the best parts of holidays. What is a
nother benefit of wrapping and
unwrapping presents? It
makes understanding inverse operations so much
easier.
What's It
Called?
Inverse
Operations
Inverse operations are operations that undo
each other. For example: opening a box and closing a box are inverse operations.
They undo each other. Addition and subtraction are inverse operations. Addition
undoes subtraction, and subtraction undoes addition.
Below is a list of
operations and the operations that
undo them; in other words, their
inverse
operations!
Operation
How to Undo It (its inverse
operation)
Addition .....
Subtraction
Subtraction ....
Addition
Multiplication ......
Division
Division ......
Multiplication
Squaring ....... Taking
the Square Root
Taking the Square Root........Squaring
So if we start with the plain ol' x and we add 2, we get x+2.
Undoing that action would mean subtract 2, right? That looks
like this: x+2 -2 = x. We end up where we started, because we
undid adding of 2.
How about if we start with x and
multiply it by 3? So we'd do x--> 3x. To
undo that action and
get back to where we started, we'd divide by 3, right? So, 3x -->
3x =
x. Obviously, this
3
works for numbers
too. If we start with 5 and multiply it by 3, we get 5(3)=15. Then, to undo what
we've just done, we divide 15 by 3, and we're right back to where we started.
15 = 5.
3
Gift Wrapping 101
We've talked about undoing single operations
so far; it's just a one-step "undoing" process. The process
of undoing more than one step (like in sovling equations) is just
like unwrapping a gift.....or taking off boots!
Let's say you put on some ankle socks and then some cute
boots. To undo this, you'd have to first take off the boots and
then take of your ankle socks, obviously. So if you did A and then
did B, to UNDO what you've done, first you'd undo B and then you'd undo
A. Makes sense, right? Now let's see how it works when there are three
steps.
If you wanted to wrap a beautiful pink sweater for your
sister, first you'd put it in a box. Then you'd wrap the box in wrapping paper,
and then you'd stick a sparkly bow on it. When your sister unwraps it, she would
do the inverse of each action you did, in the reverse order.
Wrapping:
Unwrapping:
1. put in box
1. unstick sparkly bow
2. wrap with
paper 2. unwrap paper
3. stick on sparkly
bow 3. take out of box
See how the first thing she does to unwrap the
gift undoes the last thing you did when you wrapped it? And how the last
she did to unwrap it undoes the first thing you did to wrap
it?
And believe it or not, when we isolate x by undoing a
series of operations, it works just the same way!
Isolating
X
Let's take a look at
this: 2(x + 3) - 8. How did this come
to be?
Once upon a time, x was all
by himself. Then someone wrapped him up! Here's what happened to him:
First, 3 was added to him: x +3. Then the
whole thing was multiplied by 2, and here's how he looked at that point: 2(x
+3). THEN, 8 was subtracted from this whole thing, so this is how he looks now:
2(x +3) - 8.
He can hardly recognize himself. Let's get him out of
there! In order to unwrap x, first we'd need to add 8, and we'd get 2(x +3).
Then we could divide the whole thing by 2 and get x +3. And then, subtractiong
3, we'd finally get x back to his normal self again, totally unwrapped.
NICE.
See how isolating x has more to do with the holidays
than you might have thought?
Watch OUT!
Notice how I keep saying "the whole thing was
multiplied by 2" or "then we subtract 8 from the whole thing." When
we're wrapping x or unwrapping x, it's important that we always "do" things to
the entire expression, not just one part of the expression. So if we had 2x +1
and we wanted to wrap it up more by dividing by 2, we could NOT just divide the
2x by 2; we'd have to divide the whole thing by
2:
2x + 1.
2
This is because soon we'll be using these techniques to
solve for x, and when you do things to both sides of the equation, you need to
do things to each entire side of the equation in order to keep the
scales balanced.
Quick Note: I keep saying x, but of course this works
for any variable you want to isolate: a, b, c, n, w, x, y, z, :) , etc.
Doing the Math
I'll describe the steps that were
used to build an expression, starting with x (or some other variable). Your job
is to actually build it! Remember at each step to do "things" to the
entire expression. Then, list the unwrapping steps. I'll do the
first one for you.
1. Start with y. Divide by 8, then subtract 4, and
then multiply by 3.
Working out the solution: Okay, we start with
y, and we divide by 8. That can be written like this
y, right?
8
Next, we're supposed to subtract
4. Remember this would be wrong
y-4 .
8
We have to subtract 4 from the whole thing, so we
would have to write: y/8 - 4 .
So far, so good? BTW, you would
build y-4 by first subtracting 4 from y and then dividing by 8.
8
Do you see the difference?
Now we're ready for the next
instruction: "Multiply by 3." We know we must multiply the whole thing by 3, so
that means 3(y/8 -4). Okay, we're done with that part! The unwrapping steps
would be the inverse of the instructions we got, just like unwrapping a gift:
divide by 3, add 4, and multiply by 8. Done!
Answer: 3(y/8 -4). And to
unwrap it, we'd first divide by 3, then add 4, and then multiply by
8.
2. Start with x. Add 3, and then multiply by
4.
3. Start with y. Multiply by 4, and then add
3.
4. Start with z. Add 3, and then divide by
4.
5. Start with w. Divide by 3, then subtract 1, and
then multiply by 5.
6. Start with n. Multiply by 6, then subtract 5, and
then divide by 7.
Answers coming
soon.
Solving for x
Now let's apply
our unwrapping knowledge to solving for x. Before we do, let's review what it
means to solve equations. Remember, when you get an equation to solve like
5(2x+1)-6=29, you're being asked to find out what number x has to be in order
for the equation to indeed be a true statement. That's our job-to discover x's
value! Sure, we could just stick a bunch of values in until one of them works,
but x is often a fraction. Are you really going to guess every fraction you can
think of too? There's got to be a better way to find out x's value, and there
is!
