TEK 6.3 A Learning Goal: I can recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values.

TEK 6.3 B Learning Goal: I can determine, with and without computation, whether a quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one.

Vocabulary

Reciprocals: Two numbers that, when multiplied together, have a product of 1.

Rational Number: Any number that can be represented by a fraction.

Order of Operations: The rules that tell you which operations to do first within a mathematical expression. PEMDAS is a common way to remember these rules. The P stands for parenthesis, so anything inside parenthesis is done first. Next is E for exponents, so a number with an exponent gets raised to that power. The M and D stand for multiplication and division, which are on the same level. Lastly, the A and S stand for addition and subtraction, which also share a level.

Commutative Property: An expression is commutative when changing the order of operands does not change the result. Multiplying numbers together is a commutative process. 3*5 = 5*3. It doesn't matter whether the 5 or 3 comes first - the answer is still 15. This also works with addition - try adding 3 and 5 in different ways to see! You may have heard the word commute before, which means to travel. This definition is why the commutative property is named so - it doesn't matter what route you travel on to get to the answer because you end up arriving at the same place.

Numerator: The part of the fraction that is on top of the line.

Denominator: The part of the fraction that is underneath the line.

Product: The result of multiplying numbers together.

Multiplying Fractions Using Properties

You have probably heard that when multiplying two fractions you multiply across the numerators and denominators of the two fractions. But have you ever wondered why that works?

Fractions are basically a division problem, so multiplying fractions is problem that requires division and multiplication. Multiplication and division are opposites, but they actually go well together - so well that a division problem can be written as a multiplication problem, as we will see later.

Consider this problem:

You can probably tell that the answer to this problem is 4 pretty quick. 8 divided by 4 is 2 and 6 divided by 3 is also 2. Multiplying 2 by 2 gives us 4. However, did you know there is another way to solve it? Since fractions are just division, the problem can be rewritten as:

Multiplication and division are on the same level of the order of operation, so you do the problem from left to right. 8 divided by 4 is 2. 2 times 6 is 12. 12 divided by 3 is 4, which is our answer. You may have noticed that we did different operations during the problem, but we arrived at the same answer as last time. The Commutative Property (see vocabulary above) applies to many problems with only multiplication and division because, as we said before, division can also be rewritten as multiplication. So we can move the numbers of this problem around and multiply/divide them in various orders and still have the same result. Try this one with a calculator:

It comes out to 4 too.

The final answer of this problem is the result of multiplying 6 and 8 at some point, and dividing by 4 and 3 at some point. What if we moved 6 and 8 to the front:

6 times 8 is 48. 48 divided by 4 is 12, and 12 divided by 3 is 4. Now, here is the key: dividing a number by two other numbers in a row is the same as dividing by the product of those two other numbers. Imagine that you have a pizza, but you only want to eat one fourth of the pizza right now. First you would cut the pizza in half. Then you could cut one of the halves in half. Cutting something in half is the same as dividing it by two, so you have divided the pizza by 2 twice. You have also divided the pizza by 4 to get the fourth of the pizza you want to eat. Guess what? 2 times 2 is equal to 4, so dividing by two numbers and dividing by their product is the same thing. With this in mind, we can now write:

This is also written:

Do the problems in the parenthesis first because of order of operations. 48 divided by 12 is 4. If we look back at our original problem, this expression is the same as if we had multiplied across the numerators and denominators of the original fractions.

Reciprocals

As you know from looking at the vocabulary above, a pair of reciprocals multiply together to produce 1.

2 and 1/2 are examples of reciprocals. 2 can be rewritten as a fraction, and these two numbers can be multiplied together using fraction multiplication (and then simplified):

Notice how 2/1 and 1/2 are similar numbers. The only difference between them is that their numerators and denominators are flipped. This is how you find the reciprocal of a number: you switch the numerator and denominator. For integers, all you have to do is put the number under one. For fractions, you will have to switch whatever is on top of the fraction with whatever is on the bottom. Here is another example of reciprocals:

Earlier we mentioned that division problems can be written as multiplication problems. This is because dividing by a number is the same thing as multiplying by its reciprocal. Let's say you want to divide 8 by 4. This is the same thing as multiplying 8 by the reciprocal of 4, which is 1/4 (4 is an integer, so you put it under 1.) How can this be the same? Well, let's multiply these two numbers, with 8 turned into the fraction 8/1:

Since a fraction is just a division problem, the fraction we end up with is the same as the original problem.

Multiplying Fractions - Increase or Decrease?

If you see two reciprocals multiplied together, you know the answer will be 1. But for other fraction multiplication problems, is there a quick way to see whether multiplying a number by a fraction increases or decreases the value?

Let's look at a couple of examples:

1/2 is the same as 4/8, so the end product (3/8) is less than the first factor (1/2).

8/3 is more than 2/3 since you have 8 thirds instead of only 2.

After simplification, the first factor (3/7) is unchanged.

10 is less than 12, so the first factor decreased.

Notice the trends:

• If a number (any kind of number - fraction, integer, etc) is multiplied by a fraction that is less than 1, it decreases.
• If a number is multiplied by a fraction that is greater than 1, it increases.
• If a number is multiplied by a fraction that reduces to 1, the number is unchanged.

When you think about it, this kind of makes sense. A fraction that is less than 1 is a portion of 1, so when it is multiplied by the first factor, it will result in a portion of that number. A fraction that is greater than one simplifies to a mixed fraction, where you have at least one full copy of the first factor, and then an additional portion of it as well.

The way to tell if a fraction is less than or greater than 1 is by comparing the numerator and denominator:

• If the denominator is greater than the numerator, the fraction is less than 1
• If the numerator is greater than the denominator, the fraction is greater than 1
• If the numerator and denominator are equal, the fraction is equal to 1

Now you can just look at any problem and tell right away whether or not the first factor decreases or increases without doing any math.

For practicing vocabulary with flashcards, go to:

https://quizlet.com/_6rt3mc

Use these quizzes to make sure you fully understand what you have just learned. Click "view score" to see explanations for any wrong answers. Click "submit another response" to try again.