Lesson 4
Ordering Rational Numbers
Let’s order rational numbers.
4.1: How Do They Compare?
Use the symbols >, <, or = to compare each pair of numbers. Be prepared to explain your reasoning.
 12 _____ 19
 212 _____ 190
 15 _____ 1.5
 9.02 _____ 9.2
 6.050 _____ 6.05
 0.4 _____ \(\frac{9}{40}\)
 \(\frac{19}{24}\) _____ \(\frac{19}{21}\)
 \(\frac{16}{17}\) _____ \(\frac{11}{12}\)
4.2: Ordering Rational Number Cards
Your teacher will give you a set of number cards. Order them from least to greatest.
Your teacher will give you a second set of number cards. Add these to the correct places in the ordered set.
4.3: Comparing Points on A Line

Use each of the following terms at least once to describe or compare the values of points \(M\), \(N\), \(P\), \(R\).
 greater than
 less than
 opposite of (or opposites)
 negative number

Tell what the value of each point would be if:
 \(P\) is \(2\frac12\)
 \(N\) is 0.4
 \(R\) is 200
 \(M\) is 15
The list of fractions between 0 and 1 with denominators between 1 and 3 looks like this:
\( \frac{0}{1}, \, \frac{1}{1},\, \frac{1}{2},\, \frac{1}{3},\, \frac{2}{3}\)
We can put them in order like this: \( \frac{0}{1} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{1}{1}\)
Now let’s expand the list to include fractions with denominators of 4. We won’t include \(\frac{2}{4}\), because \(\frac{1}{2}\) is already on the list.
\( \frac{0}{1} <\frac{1}{4} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{1}{1}\)
 Expand the list again to include fractions that have denominators of 5.
 Expand the list you made to include fractions have have denominators of 6.
 When you add a new fraction to the list, you put it in between two “neighbors.” Go back and look at your work. Do you see a relationship between a new fraction and its two neighbors?
Summary
To order rational numbers from least to greatest, we list them in the order they appear on the number line from left to right. For example, we can see that the numbers
2.7, 1.3, 0.8
are listed from least to greatest because of the order they appear on the number line.
Glossary Entries

negative number
A negative number is a number that is less than zero. On a horizontal number line, negative numbers are usually shown to the left of 0.

opposite
Two numbers are opposites if they are the same distance from 0 and on different sides of the number line.
For example, 4 is the opposite of 4, and 4 is the opposite of 4. They are both the same distance from 0. One is negative, and the other is positive.

positive number
A positive number is a number that is greater than zero. On a horizontal number line, positive numbers are usually shown to the right of 0.

rational number
A rational number is a fraction or the opposite of a fraction.
For example, 8 and 8 are rational numbers because they can be written as \(\frac81\) and \(\text\frac81\).
Also, 0.75 and 0.75 are rational numbers because they can be written as \(\frac{75}{100}\) and \(\text\frac{75}{100}\).

sign
The sign of any number other than 0 is either positive or negative.
For example, the sign of 6 is positive. The sign of 6 is negative. Zero does not have a sign, because it is not positive or negative.