Understanding Rational Numbers

 

Remember when you first learned to divide?

5 went into 10 exactly 2 times....no remainder...a real neat package.

The quotient, 2, is a whole number...but wait...

It is a rational number as well.

 

 

Then, your teacher thought you were so smart, so she gave you problems that were not so neat...they had remainders!

 

 

12 divided by 5 = 2 with 2 out of 5 left over...or 2 and 2/5ths!! OOOO, ick....2 and 2/5ths is not a whole number, but it is a rational number because it is the quotient of two integers.

Hey....a number is a number...right!

So, what is this about a rational number?

A rational number can be written as the quotient of two integers ( but not if the divisor is a 0).

 

Can you think of a few examples of rational numbers?

3 2/3rds, 4.025, .07, 456, 234 47/48, 787/21

432/0 is not a rational number...Can you guess why?

 

 

Communicating information sometimes requires that we be very specific about the amount of something.

This quantity may not be exactly expressed as a whole number...this is why rational numbers are soooo important.

Can you think of a situation when you would need to use rational numbers?

 

Is a whole number also a rational number?

 

 

***REMEMBER***

A FRACTION IS PART OF A WHOLE

OR A SET

 

 

WHEN YOU DIVIDE A WHOLE OR A SET

INTO EQUAL PIECES...YOU CREATE FRACTIONS!!!!

THUS....BITS AND PIECES!

 

Some examples might be...

"I want to eat 1/2 (one half) of this pizza." Meaning....if I cut the pizza into 2 equal size pieces, then I want 1 out of the 2 halves.

Another way to read this fraction is 1 out of two pieces of the pizza...1/2...one half.

 

Well, I don't feel really hungry today and I only want a couple of slices. If the pizza was already cut into 7 slices, what fraction of the pizza did I eat if I had 2 slices?

REMEMBER TO ASK YOURSELF...

WHAT NUMBER DID I EAT OUT OF HOW MANY SLICES.

 

 

 

The trumpet player is having trouble with the 1/8th notes...they are to be held only a small part of one beat. Meaning... if one beat is divided up into 8 equal size pieces of time, then the trumpet player should hold the note only 1 out of the 8 pieces of a beat...1/8...one eighth.

 

Jane bought one candy bar made up of 6 equal squares. She wanted to share it with 3 of her friends.

She quickly figured out that they could have 2 pieces each. Meaning they each got 2 out of 6 equal size pieces...2/6ths of the candy bar...two sixths of the bar.

 

So, what have we learned?

A FRACTION IS A PART OF A WHOLE!!!

 

OK, I know this isn't science but let's dissect what makes up a fraction.

 

2/5...two fifths....2 out of 5 equal pieces

2/...the top number, called a numerator, must be a whole number...it expresses "how many" out of the parts the whole has been divided.

/5... the bottom number, called a denominator, must be a whole number.... it shows how many equal parts the whole has been divided.

 

***REMEMBER***

FRACTIONS AS A RATIO

 

There were not enough computers in the classroom for all of the students. They had to share...2 computers for every 5 students.

Another way to say this is 2:5, or 2/5 or two to five.

 

***REMEMBER***

 

FRACTIONS ARE A DIVISION PROBLEM

 

If a fraction bar could speak it would say, "DIVIDED BY."

 

1/2 is simply 1 "divided by" 2

3/4 is simply 3 "divided by" 4

 

We will solve these division problems in the decimal portion of this unit.

 

 


EQUAL FRACTIONS

 

Two or more fractions that name the same quantity are called

equivalent fractions.

 

1/2 is equivalent to 2/4

2/4 is equivalent to 3/6

3/6 is equivalent to 4/8

4/8 is equivalent to 5/10

are you beginning to see a pattern?

If you take 1/2 and multiply the numerator

and the denominator by 2, your answer is 2/4.

What happens if you take 1/2 and multiply the numerator and the denominator by 3?

What would be the next 5 equal fractions to 1/2?

GREAT JOB!

 

So, we know that as long a we multiply both the numerator and denominator by the same number, we will get an equal fraction.

 

Well, what if we divided the numerator and the denominator of a fraction by the same number...would we get an equal fraction?

Take a guess!

Let's test this to be sure.

8/16ths

8 divided by 2 = 4

16 divided by 2 = 8

4/8ths is equal to 8/16ths

 

If you take 4/8ths and divide both the numerator and the denominator by 2 what is the equal fraction that you get?

And if you divide both by 2 again...what equal fraction do you get?

Name at least five fractions that are equivalent to 1/8.....2/5.....1/10?

***REMEMBER***

Equivalent fractions are equal in value

but have different numerators and denominators.

 


 

MIXED NUMBERS

A mixed number is a mix of a whole number and a fraction.

4 1/3 is an example of a mixed number.

4 is the whole number and 1/3 is the fraction.

 

You can turn this mixed number into an improper fraction.

HOW ?

IT'S EASY!

Multiply the denominator of the fraction by the whole number and then add the numerator to your answer. That number becomes the numerator of the fraction and the denominator stays the same.

To change 4 1/3 into a fraction

Multiply the denominator by the whole number...

3 x 4 = 12

Add the denominator to your answer...

12 + 1 = 13, this is the numerator of the fraction

13 = the numerator

3 = the denominator

The improper fraction 13/3 = 4 1/3

 

You try it!

Can you change the following mixed numbers into improper fractions?

4 2/5

6 4/7

10 4/5

 

Have you figured out that if you divide an improper fraction, you get a mixed number?

10 4/5 = 54/5

This explains that there are 54 5ths in 10 4/5.

 

and

 

54 divided by 5 = 10 and 4/5 left over

 

How many 4ths are there in 5 and 3/4ths?

HINT: Just convert 5 and 3/4ths to an improper fraction.

 

 

 


 

COMPARING FRACTIONS

 

To compare fractions you must first find their common denominator.

Convert each fraction into an equal fraction using the common denominator.

 

For example...compare the following fractions

 

Which is larger 2/5ths or 3/4ths

The least common denominator is 20...(that is the smallest number that both 4 and 5 are factors)

 

2/5 = 8/20 because 5x4 = 20, and, 2x4 = 8.

 

3/4 = 15/20 because 4x5 = 20, and, 3x5 = 15

 

Which is greater...8/20ths or 15/20ths?

Once you have converted the fractions you want to compare to equal fractions with the same denominator

....IT'S A NO BRAINER!!!

COMPARE THE FOLLOWING...

1/2 and 1/3rd?

Why was 6 the least common denominator?

Please note that it is easier to use the least common denominator,

but you may use any number which has both 2 and 3 as factors.

You are getting good!!!

Compare the following for practice...

Which is greater...

a. 1/5th or 1/3?

b. 2/7ths or 1/4

c. 5/6ths or 9/10ths

GOOD JOB!

 

 


 

NOW YOU ARE READY FOR MAGIC!

 

CHANGE FRACTIONS TO DECIMALS

 

Remember you learned earlier that if a fraction bar could speak...it would say "divided by"

For example...1/2 is... 1 divided by 2

the answer to this division problem is .5

 

Try another one....4/5ths or... 4 divided by 5

the answer to this division problem is .8

 

"The trick" to changing fractions to decimals

is to divide the numerator by the denominator.

 

 

 

If you want to convert a fraction with a denominator of 10 to a decimal you simply write the number in the proper place on the place value chart.

 

For example...7/10 would be written as .7

What would 3/10ths be as a decimal?

 

What would 7/100ths be as a decimal?

THINK: Where is the hundredths place on the place value chart?

The answer is .07

 

What would 17/100ths be?

.17

 

What would 43/100ths be?