Comparing and       Ordering Fractions   Introducing the Concept   Developing the Concept       Adding and       Subtracting Fractions       With Like       Denominators   Introducing the Concept   Developing the Concept

## Comparing and Ordering Fractions

Your students may need to be reassured that fractions are just numbers and, like other numbers, they can be compared, ordered, and used in computation. 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.
Students have learned how to write numbers in different forms, and they 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. Students 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, students often 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 .

Students will 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 tell students to divide the numerator and denominator by the same number.

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

When denominators are different, students 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. fraction strips (See page 328 in the text.)

5. coins and dollar bills

6. two-color counters and yarn loops
When denominators are different and benchmarks don't help, students 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 introduce the idea of ordering fractions. Suggest that students use the same logic to order fractions as they do to order whole numbers. Have them compare pairs of fractions and be sure every pair is related in the same way.

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 < < .

Students 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 < < .

Once students understand the meaning of each part of a fraction, they 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. Explain to students that they 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 students use this technique to write an improper fraction as a mixed number, they may end up with a fraction that is not in simplest form. In this case, you may want them to take another step and write the fraction in simplest form.

You can introduce the addition and subtraction of fractions with like denominators by explaining to students that it is similar to adding and subtracting with whole numbers. However, the sum or difference is written over the denominator. When the denominators of fractions are the same, you can say the fractions have the same name. When this is true, you can add or subtract the numerators without changing the denominator. So 3 fifths plus 3 fifths equals 6 fifths. This way of thinking only works for addition and subtraction.