The Whole is Finished

Not sure anyone will see this, but I hope you all did well on your finals (WSU Math 251 students) and can't wait to see you in 252!

Fractions: The Last Piece of a Whole

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Percents and decimals. These are often confusing when looked at in a fractional light.

How many of the following columns and rows can you fill out?

If you can't do any of them, that's okay. 

Let's start with decimals to fractions. 

.1 = 1/10

.01 = 1/100

.001 = 1/1000

See the pattern? When we see .1 we should say it as one tenth. Let's use our 6.28 example. It doesn't show it above, because 1 = 1/1, but that means we have 6 wholes. A whole being 1.000. Then we have .2 which looks like 2/10 if we use the examples above. Next, .08, looks like the second example, so we get 8/100. If we add the two fractions together we get 28/100. Hence, 6.28% is also 6 and 28/100. If your teacher asks you to reduce, you can change 28/100 into 7/25.

Can you do it the other way? Let's try 1/4. Will 4 go into 10? No. Will it go into 100? Yes. 4x25=100. Hence, we need to multiply 1 by 25 as well. 25/100. From this point, you can probably write out the decimal. .25, right? Yup!

Here are the relationships in simpler forms.

6.28% is not the same at what we figured up top though. To turn a percent into a true decimal for extended equations, we have to; divide by 100, move the decimal to the left two spots, or some other trick. So, 6.28% is actually .0628. Why do you divide by 100 though? Truth be told, 100% is your whole. Instead of dividing by 100, you're actually dividing by 100%.

The image below shows you these relationships.

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Fractions: The Third Piece of a Whole

I like chocolate cheese cake, with lots of graham crackers in the crust. Anyone else? Anyone? Let's say you take 2/6 of a chocolate cheese cake. I want 3/4 of what's left (I don't know why I wouldn't just take 4/4). How much of the original whole pan do I get? Well, there's 4 pieces left (since you took 2), and I want 3 of them. But how many pieces am I taking in reference to the original 6? 3/6 is the same as 1/2. Yum! So, how do we write that formula? 6/6 - 2/6 - 3/4? Using common denominators, that gives us 24/24 - 8/24 - 18/24 = -2/24? That's not right... Why? The "whole" that I'm using for 6ths and 4ths are different. I'm really taking 3/4 OF 4/6. OF means multiply. 

The image above shows a simple way of showing what we do when multiplying fractions. Multiply across the top (1x1=1), and then across the bottom (2x2=4). The circles show the equivalent of half OF a half.

Another way to multiply fractions is by using the area model of multiplication. For example,

I think the image explains it better than I could. 

There is yet another way to figure out some problems. Swapping the top numbers, or the bottom numbers - but not from top to bottom or bottom to top. In the last problem, we could have had 3/3 x 2/4. That's actually 1 x 1/2, which is equivalent to 6/12.

This is what swapping looks like when put into action. This is the Commutative Property at work as well.

Next Step

Dividing Fractions! So.... Y'all probably know about flipping the second fraction and multiplying, and ya gotta use improper fractions... yada yada. Why in the world DO we flip a fraction? And why not the first fraction? Let's take a look.

4/5 divided by 3/8

If we take the typical route, we get 8x4 and 3x5. 32/15 which reduces to 2 2/15.

We'll come back.

What if I divide 3/5 by 1/5? You know it's 3. How do the 5's disappear? They don't. You're just dividing 3 by 1 and 5 by 5. 5 by 5 is 1, which can be ignored at the bottom of a fraction. Heeeyyyyy! Those were common denominators!

So what if I found a common denominator between 5 and 8 from the previous problem? 40. If I multiply 5 by 8, then I have to do the same to the 4. If I multiply 8 by 5, then I have to do the same to the 3. We get 32/40 divided by 15/40. If the 40's go to 1, then I'm left with 32/15. That's the same thing as we got above.

When you divide, you're asking how many groups of this number are in the other number. You may not know off the top of your head, but let's see an example.

Oh! They also divided across the top and bottom! That's another way to find easier answers. That all depends on both numbers being easily divided into one another though.

Think that's all... Hope you learned something!

Fractions: The Second Piece of a Whole

          Ever heard of a benchmark partial clock ? Me neither... But it's a great way to remember some good ways to look at fractions!

          Benchmarks really help with comparing fractions without finding common denominators. Who wants to find a common denominator between 23rds, 14ths, and 127ths? Definitely not me. Here's where benchmarks come in handy. They're typically 0, 1/2, 1, 1 1/2, 2, etc. If we have 7/14  6/23, and 125/127 to compare, I can tell you very quickly which is biggest and which is smallest based on how close it is to a benchmark. For example, 125/127 is very close to a whole, so it must be a large fraction. 7/14 is half, so it isn't as big as 125/127. 6/23 isn't even close to a half (11.5/23), so it must be the smallest one. No common denominators needed.

The image above may help you understand this concept. Two 1/3 pieces is more than half, but four 1/9 pieces is not. Four may be more than two, but 9ths are smaller pieces.

          

          Parts. There are different ways of adding whacky fractions. Traditional is very confusing. What if you could separate each of the fractions into nicer pieces? You can! See?

While adding to zero is very easy, your starting point could be at any other fraction you want to add. For example, if you were adding 2 2/3 to 3 2/3, you can split up one of the 2/3s and add a single third to the other one to make a whole. 2 2/3 can become 2 1/3 and the other 1/3 makes 3 2/3 into 4. Now you're adding 4 and 2 1/3. Much easier. No? Any fraction can be broken into smaller parts to make them easier to add. You can even add 5/6 and 1/2 without finding a common denominator. If you know that 3/6 is equal to 1/2 you can take 3/6 from 5/6 and add it to the half. Then you're adding 1 and 2/6.

          What about time? We have minutes and hours that make up our time system. 24 hours in a day and 60 minutes in an hour. How do you add 2 5/6 hours to 1 3/4 hours if you want the answer in minutes and hours?

If you fill in a few of the missing lines, you can see that the clock makes these fractions (5/6 and 3/4) for us. In terms of 6ths, you reach the 5th piece at 50 minutes. The 3rd piece of 4 is at 45 minutes. So I'm adding 2 hours 50 minutes to 1 hour 45 minutes. If minutes start over at 60, then you have to add 1 whole hour to your total. The end result is 4 hours 35 minutes.

These were the things that really helped me this week. Hope they help you too!