## Kiss those Math Headaches GOODBYE!

### How to Convert Percents to Decimals — Flawlessly

My last post showed how you convert decimals to percents.

Now I’ll dare to do the obvious with a post on how to convert percents to decimals.

Since converting percents to decimals is the opposite of converting decimals to percents, it makes sense — does it not? — that we’d use the opposite procedure. And that is the case.

Not only that, but we do the opposite procedure in the opposite order, too. How’s that for going totally opposite?

Since the final step in converting decimals to percents is tacking on a percent symbol (%), the very first step in converting percents to decimals is taking off that symbol.

And since the first step in converting decimals to percents is nudging the decimal point two places to the right, the last step in converting percents to decimals is pushing the decimal point two places to the left.

Let’s take a look at the process with this example.

Problem:  convert 73.2% to a decimal.

Step 1:  Take off the percent symbol. 73.2% changes to just 73.2

Step 2:  Move the decimal point two places to the left. 73.2 changes to .732 That’s all there is to it. This tells us that 73.2% is the same as .732, or 7 hundred thirty-two thousandths.

If you don’t recall the steps that you’re turning around, here’s a quick way to remember the process of converting percents to decimals. As I said in my last post, we can make use of alphabetical order, setting up the words for decimal and percent, in order, like this:

D-Decimal                              P-Percent

Then we draw an arrow showing that we’re converting from percent form to decimal form. The arrow shows the direction of the conversion: percent to decimal.

D-Decimal    <————–    P-Percent

This arrow points to the left, and that tells us that we move the decimal point to the left when we convert a percent to a decimal.

Let’s look at the process again, this time focusing now on how we use the arrow’s direction to help us.

Problem:  Convert  4.782% to a decimal.

Step 1:  Rip off the percent symbol. 4.782% changes to 4.782

Step 2:  Give the decimal point two shoves in the arrow’s direction. Since a percent to decimal conversion makes the arrow point left, we shove the decimal point two spaces to the left.  4.782 changes to .04782

This tells us that .04782 is the same as 4.782%

Note: if there are no digits showing to the left, we’re free to add 0s on the left side of the leftmost digit to create a place where the decimal point lands, after being shoved to the left.

In the last example, we had to tack a 0 on the left of 4.782 — making it 04.782, to get a digit (0) to the left of which we placed the final decimal point. Be confident that you can write as many 0 digits as you need to the left of a number’s leftmost digit. For example, it is just fine (though admittedly strange) to write 4.3 as 0004.3. You’d do this weird maneuver if you need that many zeros to the left of the 4. This occurs in converting numbers to scientific notation, for example.

So, now that you know the process, try your hand at converting the following percents to decimals: (Answers at the bottom of this post.)

a)  38%                                                                                                                           b)  19.3%                                                                                                                         c)   4.2%                                                                                                                         d)  175%                                                                                                                         e)  398.6%
f)  2,400%

Answers to the problems in the last post, converting decimals to percents:

a)  8590%                                                                                                                     b)  416.2%                                                                                                                     c)  20873.5%
d)  4.7%
e)  207,465%
f)  28.3%
g)  .569%

Answers to problems in this post:

a)  38%        =  .38
b)  19.3%     =  .193
c)   4.2%      =  .042
d)  175%      =  1.75
e)  398.6%   =  3.986
f)  2,400%    =  24.0, or just 24

### “Simple” equations? Not always so simple.

Have you ever wondered how students can make mistakes with equations that appear extremely simple to solve, equations as basic as: 9 – x = 11, or: – 9 + x = 11?

To me this was rather baffling until I started to see these equations through students’ eyes, thanks to some tutorees who told me why they were making some mistakes here.

Through this experience, I’ve come to realize that I should be careful before I label any equations “simple equations.”

Here’s a case in point, which I’ll call Example #1:

9 – x = 11

If you ask students how the 9 in this problem is connected to the left side of this equation, a distressingly LARGE NUMBER will say that the 9 is connected by SUBTRACTION. If you just think about it, it’s a very understandable mistake. There’s a big glaring negative sign to the the right of the 9. When students look at that negative sign, many think it tells them that the 9 is connected by SUBTRACTION. Using this analysis, and armed with the knowledge that you do “the opposite operation,” these students will blithely ADD 9 to both sides of the equation, but they’ve just gotten off on the wrong track, getting: 18 – x = 20.

Similarly, given an equation as “simple” as this, Example #2:

– 9 + x = 11

many students will tell you that the – 9 is connected by addition. Why? Because of the big glaring + sign to the right of the – 9. . Reasoning thus, these students will gleefully do the opposite operation and subtract 9 from both sides of the equation, landing in an equally sticky puddle of wrong thinking, getting: – 18 + x = 2

How can we help students avoid these common algebraic pitfalls? Here’s what I’ve found helps.

Tell students that if a number is connected to a variable by an ADDITION or SUBTRACTION sign, you have to look to the LEFT of the number — not to the right — to determine how it is connected to that side of the equation.

To see how this works, let’s look back at Example #1: 9 – x = 11

A student who understands the correct process knows she must look to the LEFT of the 9 to see how it is connected to this side of the equation. Since there’s NO VISIBLE SIGN to the LEFT of the 9, that means that there’s an invisible + sign, so the problem can be re-written, for sake of clarity as: + 9 – x = 11

Now the student is getting somewhere. Using the idea of looking to the LEFT, she looks to the left of the 9 and sees this + sign. This tells her that the 9 is connected by ADDITION, not by subtraction. Following the rule of inverse operations, she now knows to SUBTRACT 9 from both sides of the equation, to get: – x = 2

From there it’s just a matter of multiplying both sides by (– 1) to get the variable alone in its positive form, obtaining the answer: x = – 2

Similarly, Example 2 can be solved using the Look to the LEFT rule of thumb.

This problem is: – 9 + x = 11

The student must ignore the + sign to the right of the 9, and instead look to the LEFT of the 9, finding a – sign. That – sign tells the student that the 9 is connected to the left side of the equation by SUBTRACTION. Using the rule of inverse operations, he knows to add 9 to both sides of the equation, getting: x = 20 And that is the final answer.

As I think about this area of confusion, I think it occurs because we teachers sometimes tell kids that they must “get rid of the number that is connected to the variable.” I know I’ve been guilty of talking this way (but I am trying to reform myself).

When a number is multiplying or dividing a variable, it is perfectly appropriate to tell kids to “get rid of the number next to the variable.” For example, in: 3x = 21, students need to get rid of the 3 which is connected to x through multiplication. Likewise, in x/5 = 6, students must “get rid” of the 5 that is connected through division.

But my recent algebra epiphany is that when numbers are connected through addition or subtraction, as in these problems:

10 – x = 23

x + 7 = – 5

– 8 + x = 13

– 2 – x = 8

we need to be more subtle in our instructions. We must tell students to Look to the LEFT of the number on the same side of the variable. And then tell students to let the sign they see to the LEFT guide them as to how to get rid of the number term that is in the way.

Just a little self-confession that I thought I’d pass along.

Good news: I’ve already started using this “Look to the Left” approach, and it is clearing up lots of confusion for students.

Happy teaching!

— Josh