Unit 2, Part 1: Identifing fractions

The main difference between fractions and whole numbers, of course, is that fractions are parts of a whole. They have a numerator, a word that means "enumerate," or "count," and a denominator. The word denominator is related to denominate, which means "to name." The denominator names the total number of parts in the whole, and the numerator tells how many of the parts are in the fraction. So you can say that each fraction has a name; students need to pay attention to that name when they compare, order, or compute with the fraction. Just as you wouldn't say that 7 inches are greater than 6 feet because 7 > 6, you wouldn't say that 7 eighths are greater than 6 fourths by comparing only the 7 and the 6. Fractions come in three basic forms.

• Proper fractions have a numerator that is always less than the denominator. In arithmetic, using positive numbers, proper fractions represent the numbers between 0 and 1. , , and are all proper fractions.

   

• Improper fractions have a numerator that is greater than the denominator. Improper fractions can be rewritten as fractions or mixed numbers. There is nothing "wrong" with improper fractions; in fact they're sometimes easier to compute with than mixed numbers. , , and are improper fractions.

   

• Mixed numbers have a whole-number part and a fractional part (usually proper). Mixed numbers can be rewritten as improper fractions. 1 and 8 are mixed numbers.

You may have learned how to write numbers in different forms, and you need to do the same with fractions. An important rule about numbers is that if you multiply or divide a number by 1, you don't change the value of the number. This is the Property of One. You should also know the division rule that states when you divide a number by itself, the quotient is 1. So any number divided by itself equals 1. One nice thing about fractions is that they provide you with an infinite number of forms of the number 1: , , , , and so on, or even a fraction over a fraction, such as .
So if you want to write a number with a denominator other than the one it came with, you can use this rule.

In order to work with fractions, you need to find equivalent fractions—that is, fractions with different numbers but the same value. If you're working with and you'd really rather be working with eighths, multiply by . Since 1 =, then has the same value as even though it's written as .

You also need to understand how to write fractions in the simplest form. When you can find a common factor other than 1 for both the numerator and denominator of a fraction, the fraction is not in simplest form. Use that common factor to find a simpler form of your fraction. You can divide the numerator and denominator by the same number.

Equivalent fractions are important when comparing fractions. When denominators are the same (like or common denominators), they can just compare the numerators: 7 eighths > 3 eighths.

When denominators are different, you may use benchmarks to compare them. A very easy benchmark is . If the numerator of a fraction is less than half of the denominator, the fraction's value is less than . Similarly, if the numerator is more than half of the denominator, the fraction's value is greater than . For example: > and <, so must be greater than .

When denominators are different, you may also model with a diagram or manipulative.

1. a ruler marked in eighths or sixteenths of an inch

    

   

    

    

2. same-length number lines, each dedicated to all the fractions between 0 and 1 with the same denominator

    

   

     

3. a collection of wax-paper fraction squares

    

     
4. coins and dollar bills

When denominators are different and benchmarks don't help, you should find equivalent fractions with like denominators.

Look for a denominator that can be used to name both fractions. Then multiply the fraction you wish to change by a form of one—such as , , , and so on—that produces the denominator you want. Just as inches can name measurements given in feet and yards, sixteenths can name fractions given in eighths and fourths.

can be written as by multiplying by .
can be written as by multiplying by .

Since and are both forms of 1, you haven't changed the value of either fraction, just the form.

With an understanding of how to compare fractions, you can start ordering fractions, or placing them in order.

For example, to order , , and from least to greatest, first find an equivalent fraction to that has a denominator of 8.

= = is equivalent to .

Then compare the numerators.

< < , so < < .

You can also use a number line to determine where the fractions fall in relation to each other.

 

is closest to the left.
is closest to the right.
So < < .

See Khan Academy on comparing fractions Links to an external site.

Once you understand the meaning of each part of a fraction, you will find it easy to write an improper fraction as a proper fraction. The denominator tells how many equal parts are in one whole unit. If a fraction represents more than one unit, the numerator will be greater than the denominator. Dividing out all of the whole units will give you the whole-number part of the number with the same value as the fraction. If there is a remainder from this division, it is the numerator of the proper-fraction part of a mixed number. You will already know the denominator: It's the denominator of your original improper fraction. For example: is an improper fraction; 25 ÷ 3 has 8 as its whole-number part and 1 as its remainder. So = 8 .

If you use this technique to write an improper fraction as a mixed number, you may end up with a fraction that is not in simplest form. In this case, you will need to take another step and write the fraction in simplest form.

To better understand how to operate with fractions you need to understand a few basic concepts.

 

Prime factorization

"Prime Factorization" is finding which prime numbers multiply together to make the original number. Prime factorization or integer factorization of a number is the determination of the set of prime numbers which multiply together to give the original integer. It is also known as prime decomposition.

Figure 1; prime factoring methods:

 

Prime Numbers

A prime number is a whole number that only has two factors which are itself and one. A composite number has factors in addition to one and itself.

The numbers 0 and 1 are neither prime nor composite.

All even numbers are divisible by two and so all even numbers greater than two are composite numbers.

All numbers that end in five are divisible by five. Therefore all numbers that end with five and are greater than five are composite numbers.

The prime numbers between 2 and 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97. It's good to know these, it helps when it comes to Prime Factorization.

See the Khan Academy tutorial on Prime Factoring Links to an external site.

The Greatest Common Factor

GCF is the largest number that is a common factor of two or more numbers.

How to find the greatest common factor:

• Determine if there is a common factor of the numbers. A common factor is a number that will divide into both numbers evenly. Two is a common factor of 4 and 14.
• Divide all of the numbers by this common factor.
• Repeat this process with the resulting numbers until there are no more common factors.
• Multiply all of the common factors together to find the Greatest Common Factor

See the Khan Academy on GCF Links to an external site.

Simplifying Fractions 

Fractions may have numerators and denominators that are composite numbers (numbers that has more factors than 1 and itself).

How to simplify a fraction:

• Find a common factor of the numerator and denominator. A common factor is a number that will divide into both numbers evenly. Two is a common factor of 4 and 14.
• Divide both the numerator and denominator by the common factor.
• Repeat this process until there are no more common factors.
• The fraction is simplified when no more common factors exist.

Another method to simplify a fraction

• Find the Greatest Common Factor (GCF) of the numerator and denominator
• Divide the numerator and the denominator by the GCF

See Khan Academy on simplifying fractions  Links to an external site.

Lets Practice Prime Factorization.

Example 1: Find the Prime Factors to the number 12

It is best to start working from with a prime number you know will "go into" 12:

12$\div$3 = 4 - Three goes into 12 four times. Four is not a prime number.

4$\div$2 = 2 - Two goes into Four 2 times. Two is a prime number. So, now I have all prime numbers.

It is best to use this method, the prime "factoring tree". 

 

 

The prime factors for 12 are: 3 x 2 x 2

Example 2: Find the Prime Factors to the number 18

 

The prime factors for 18 are: 2 x 3 x 3

2.1.A Practice Prime Factorization

Directions: Using the prime factoring "tree" example from above, conduct prime decomposition of the following numbers. Show all work.

1. 58

2. 16

3. 45

4. 88

5. 105

Lets practice comparing fractions.

Example 3: Place the following fractions in order from smallest to largest:

$\frac{1}{9},\:\frac{7}{12},\:\frac{4}{15},\:\frac{3}{5}$

 

First we need to find a common denominator. We need the smallest positive number that is a multiple of two or more numbers, this is the Least Common Multiple.

First factor all of the numerators.

9 = 3 x 3

12 = 3 x 2 x 2

15 = 3 x 5

5 = This is a prime number

The Least Common Multiple is: 3 x 3 x 2 x 2 x 5 = 180

We now need all denominator to be "180"

To change 9 into 180, we need to multiply 9 by 20, anything we do to the denominator, we must do to the numerator. So, we will also multiply 1 by 20. Our new first fraction looks like this:

$\frac{1}{9}=$$\frac{20}{180}$

 

To change 12 into 180 we need to multiply 12 by 15, and we need to change our numerator as well. 

$\frac{7}{12}=\frac{105}{180}$

 

To change 15 into 180 we need to multiply 15 by 12, and we need to change our numerator as well. 

$\frac{4}{15}=\frac{48}{180}$

 

To change 5 into 180 we need to multiply 5 by 36, and we need to change our numerator as well. 

$\frac{3}{5}=\frac{108}{180}$

 

Now we compare the fractions. Place in order from smallest to largest looking at the numerators: 20, 48, 105 and 108 - so the order is:

$\frac{1}{9},\:\frac{4}{15},\:\frac{7}{12},\:\frac{3}{5}$

2.1.B Practice Comparing Fractions

Directions: Place the following fractions in order from smallest to largest

1. $\frac{1}{12},\frac{1}{4},\frac{1}{3},\frac{3}{5}$

 

2. $\frac{6}{7},\:\frac{7}{22},\:\frac{4}{15},\:\frac{5}{19}$

 

3. $\frac{5}{9},\:\frac{4}{13},\:\frac{9}{22},\:\frac{12}{17}$

 

4. $\frac{6}{10},\:\frac{5}{15},\:\frac{7}{12},\:\frac{9}{25}$

 

5. $\frac{4}{24},\:\frac{6}{15},\:\frac{5}{22},\:\frac{7}{17}$

 

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