Again, we’re going to apply what we’ve already learned to add and subtract mixed numbers. It’s just like adding and subtracting fractions, but you have to remember to change the mixed numbers into improper fractions, and then find common denominators.
An example of adding mixed numbers (with different denominators)
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An example of subtracting mixed numbers (with common denominators)
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As you might imagine, multiplying and dividing mixed numbers is as simple as applying what you learned about changing mixed numbers into improper fractions, and then applying what you learned about multiplying and dividing fractions to get the answer. You can even simplify the answer by changing the resulting improper fraction (if it is improper) back into a mixed number.
Let’s start with a multiplication example.
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Dividing with mixed numbers is just as simple.
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One of the trickier parts of working with fractions is dealing with mixed numbers. A mixed number is exactly what is sounds like- one part whole number, one part fraction. It looks like this:
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To work with a mixed number, it’s actually easier to convert it to an improper fraction, or a fraction where the numerator is larger than the denominator. To change a mixed number into an improper fraction, change the whole number into a fraction that has a common denominator with the fraction, and then add the two together.
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We now have an improper fraction that we can add, subtract, multiply, or divide with other fractions.
If you get an improper fraction as an answer, you’ll need to simplify it to a mixed number. The easiest way to do this is a little division.
These two skills will open a whole new level of fractions to you. During the rest of the week, we’ll revisit working with fractions, only this time we’ll look at working with mixed numbers and improper fractions.
The last post this week takes a look at the last operation: subtraction. Fortunately, subtracting fractions is a lot like adding them. The denominator is still nothing more than a unit. We’re just manipulating the numerators.
For example:
This makes sense. If I have two thirds and you take one away, I’m left with only one third. Again, we’re making sure to call these fractions by their proper names so we remember those thirds are units, not numbers we have to work with.
Subtracting fractions with denominators also works the same way. You start by finding what both numbers go into and change both fractions accordingly. If I have to solve the problem 
Now that we have the same unit, we can do the subtraction:
Again, the denominator doesn’t change because it’s a unit instead of a number.
So, this concludes the look at simple fractions. Next week, we’re going to go back through the fraction skills with improper fractions and mixed numbers.
All right, now that we have the easy skills out of the way, we’re going to tackle one of the harder fraction skills. Amazingly, adding fractions is very confusing for many people, and it really doesn’t need to be.
To really understand what we’re doing, I need you to think of the denominator as a unit instead of a number. You know how you combine inches with other inches and still get inches. Adding fractions if the same idea. You add fifths and still get fifths. It will help if you get into the habit of calling a fraction by its proper name.
Let’s try this. We’re going to add one fifth to two fifths. Note that we’re calling them by their actual names. The problem looks like this mathematically:
Your first instinct might be to add the two together and come up with 
We added one fifth to our two fifths, and found out we have three fifths all together.
But what happens if we’re trying to add one fifth to three tenths. Suddenly, we don’t have the same units! When we see problems like this that involve feet and inches, we just convert the feet to inches and add all of the inches together. It’s the same thing with fractions. We change the fractions until the have the same unit (you can change just one fraction or both fractions, depending on the problem).
In our case, we have fifths and tenths. Well, we know the five is a factor of ten, so we can change that one fifth into tenths. Five goes into ten twice, so we’re going to create an equivalent fraction of two tenths. We now have two tenths and three tenths to add together. The problem goes fairly simply at this point.
When we add those two tenths to our three tenths, we end up with five tenths. (If you want, you can practice simplifying that answer.)
The next topic we’re going to tackle is one I’ve been trying to work with because it affects so many different math skills: creating equivalent fractions.We sometimes need a fraction to be cut into more or fewer pieces to make it simpler to work with. To do this, we need to change the numbers in the fraction without changing the value of the number.
Let’s take
You’ll note that I have created the fraction 
We can now multiply straight across the numerators and the denominators.
That’s great, but what if I have something like 
We’ll be using this skill over the next couple of days as we look at adding and subtracting simple fractions.
Today, we’ll focus on the second easiest fraction skill. We’re going to divide fractions. As you might expect, it also involves multiplying fractions.
To remember how to divide fractions, remember this rhyme a fellow math teacher taught me a couple of years ago:
Dividing fractions, don’t ask why.
Flip-flop the second and multiply.
In practice, this little rhyme looks like this:
If you think about it, this makes sense. Draw a picture of
The important bit to remember here is that you only flip over the second fraction.
All right, it’s time for a series based out of some of my earlier work. I hope you guys are finding these useful. Leave a comment and let me know.
This week’s series is on the one thing that makes every math student panic: fractions. Honestly, fractions aren’t meant to be scary. They just parts of numbers. We use them all the time. Cooking relies on fractional parts of ingredients. Money relies on fractional parts of dollars. We talk about time in half-hours. Even music, with its half-notes, quarter notes, and eighth notes, involves the beauty of fractions.
Our first fraction skill is going to seem a little weird. When you first learned the operations, you learned addition and subtraction first. I’m going to shake things up and teach you multiplication first because it is honestly the easiest fraction skill, and you need to have it in hand to work with adding and subtracting fractions, as well as creating equivalent and simplified fractions.
To multiply fractions, you multiply the numerators together, and then you multiply the denominators together. For example:
Yes, it’s really that simple. Get the hang of it now, becasue you’re going to find us relying on this skill for the rest of the week.
Because they’re brief, I’m covering the last two rules together.
The first one makes sense.
Rule #6- a1 = a
Practically: a = a
One occurrence of a number can only equal itself.
The next one requires a little bit of an explanation.
Rule #7- a0 = 1
Practically: Think back to the multiplication and division posts. The only way either makes sense if one of the bases is raised to the zero power is if the base and exponent together equal one. One is the only number you can multiply or divide something by without changing its value. For example, x3x0 = x3+0 = x3
That wraps up all the laws of exponents. If you can, put them all on a sticky note or a set of flashcards to help yourself learn them.
The next rule is another one that stumps people. The rule itself is simple, but the application can be a little tricky.
Rule #5- a-x =
Put simply: x-3 =
Why this rule is tricky: 5x-3y2 =
Note that it’s only the term with the negative exponent that goes into the denominator. That’s the trick. Master that, and this rule will be a piece of cake!
