Smaller Values
Numbers farther to the left have smaller values.
0 < 1
Free Math Guide
Fractions on a Number Line Explained
Learn how the denominator creates equal intervals, how the numerator counts those intervals from zero, and how to locate proper and improper fractions on a number line.
A number line shows numbers in order from left to right. Fractions belong on the number line just like whole numbers and decimals. To locate a fraction, divide each whole into equal intervals according to the denominator and then count those intervals from zero according to the numerator.
On This Page
- What a number line represents
- Why the intervals must be equal
- How the denominator creates the intervals
- How the numerator locates the point
- Locating proper fractions
- Locating improper fractions
- Connecting improper fractions and mixed numbers
- Equivalent fractions on a number line
- Comparing fractions with benchmarks
- Common number-line mistakes
- Number-line checklist
- Interactive practice
What Does a Number Line Represent?
A number line places numbers in order. Values increase as you move to the right and decrease as you move to the left.
Numbers Increase From Left to Right
0
1
2
Greater Values
Numbers farther to the right have greater values.
2 > 1
Fractions Are Numbers
A fraction is not only part of a shape. It is a number with a specific location on the number line.
Why Number-Line Intervals Must Be Equal
Each space between neighboring tick marks must represent the same amount. Unequal spaces would make the number line inaccurate.
Correct Partition
0 —|—|—|— 1
The whole is divided into equal-size intervals.
Each interval has the same length
Incorrect Partition
0 —|———|—|— 1
The spaces have different lengths, so they cannot represent equal fractional parts.
Unequal spaces create incorrect values
Count Intervals, Not Only Tick Marks
A number line divided into four equal parts has four spaces between zero and one, but it has five tick marks when both endpoints are included.
The Denominator Creates the Equal Intervals
The denominator tells how many equal fractional intervals make one whole.
3
5
→
5 equal intervals per whole
Find the Denominator
Look at the bottom number of the fraction.
Divide Every Whole
Partition every whole-number interval into that many equal spaces.
Fifths Between 0 and 1
0
1
One Interval Is a Unit Fraction
When each whole is divided into five equal intervals, one interval represents 1/5.
The Numerator Counts the Intervals From Zero
After the number line has been partitioned, the numerator tells how many fractional intervals to move to the right from zero.
3
5
→
Count 3 fifth-size intervals
Location of 3/5
3/5
0
1
How to Locate 3/5
- The denominator 5 divides the whole into five equal intervals.
- Each interval represents 1/5.
- Begin at zero.
- Count three intervals to the right.
- Place the point at 3/5.
Locating Proper Fractions
A proper fraction has a numerator smaller than its denominator. Its value is greater than zero and less than one.
Proper-Fraction Location
Positive proper fractions are located between zero and one.
Example: Locate 2/3
3 equal intervals
- Divide the space from 0 to 1 into thirds.
- Each interval represents 1/3.
- Count two intervals from zero.
2/3 is the second tick after 0
Example: Locate 5/8
8 equal intervals
- Divide the whole into eighths.
- Each interval represents 1/8.
- Count five intervals from zero.
5/8 is greater than 1/2
Near Zero
A fraction with a small numerator compared with its denominator lies closer to zero.
1/8 is close to 0
Near One
A proper fraction whose numerator is nearly equal to its denominator lies closer to one.
7/8 is close to 1
Locating Improper Fractions
An improper fraction has a numerator greater than or equal to its denominator. It may be equal to one whole or located to the right of one.
Partition Every Whole
Use the denominator to divide each whole-number interval into equal parts.
Keep Counting Past One
Count the numerator's total number of intervals from zero, even after reaching one whole.
Location of 7/4
7/4
0
1
2
Example: Locate 7/4
- Divide each whole-number interval into four equal parts.
- Four fourths reach one whole.
- Continue three more fourth-size intervals.
- The seventh interval from zero is 7/4.
7/4 is between 1 and 2
Connecting Improper Fractions and Mixed Numbers
An improper fraction and its equivalent mixed number share exactly the same location on the number line.
7
4
=
1 3/4
Find the Whole Number
7 ÷ 4 = 1 remainder 3
The quotient 1 is the whole-number part.
Use the Remainder
Remainder = 3
The remainder becomes the numerator of the fractional part, while the denominator remains 4.
Same Value, Same Point
Because 7/4 and 1 3/4 are equivalent, they cannot be placed at different locations on the number line.
Equivalent Fractions on a Number Line
Equivalent fractions name the same number, so they appear at the same point on a number line.
1
2
=
2
4
=
3
6
Equivalent Fractions at One-Half
1/2 = 2/4 = 3/6
0
1
Different Partitions Can Name the Same Location
Halves, fourths, and sixths use different interval sizes, but 1/2, 2/4, and 3/6 all end at the same point.
Comparing Fractions With Number-Line Benchmarks
The benchmarks zero, one-half, one, and nearby whole numbers can help you interpret a fraction's value.
Compare With One-Half
3/8 < 1/2
Three-eighths is located to the left of one-half.
Compare With One
7/6 > 1
Seven-sixths is located one sixth to the right of one whole.
Closer to Zero
1/5
One-fifth is less than one-half, so it is closer to zero than to one.
Closer to One
4/5
Four-fifths is greater than one-half, so it is closer to one than to zero.
Number-Line Position Shows Comparison
When comparing positive numbers, the point farther to the right represents the greater value.
Common Fractions-on-a-Number-Line Mistakes
Counting Tick Marks Instead of Intervals
Four parts require only four total tick marks
Four equal intervals from zero to one require five tick marks when both endpoints are included.
Using the Numerator to Divide the Whole
For 3/5, divide the whole into 3 parts
The denominator creates the intervals. For 3/5, divide the whole into five equal intervals.
Using the Denominator to Count From Zero
For 3/5, count 5 intervals
The numerator tells how many intervals to count. Count three intervals for 3/5.
Starting at the First Tick Instead of Zero
Zero is the starting point. The first interval ends at 1 divided by the denominator.
Making Unequal Intervals
Every fractional interval must have the same length throughout the number line.
Stopping Improper Fractions at One
7/4 cannot be placed because it is greater than 1
Extend the number line past one and continue counting fourth-size intervals.
Placing Equivalent Fractions at Different Points
1/2 and 2/4 have different locations
Equivalent fractions represent the same value and therefore share one location.
Fractions on a Number Line Checklist
Use these questions before choosing the fraction's location.
Correct Denominator?
Use the bottom number to determine how many equal
intervals make one whole.
Equal Intervals?
Confirm that every space between neighboring tick
marks has the same length.
Counting From Zero?
Begin at zero before counting the numerator's
intervals.
Correct Numerator?
Count the number of intervals shown by the top
number.
Proper Fraction?
A positive proper fraction should be located between
zero and one.
Improper Fraction?
Extend the number line past one and continue
counting.
Equivalent Value?
Equivalent fractions must share the same point.
Position Reasonable?
Compare the point with zero, one-half, one, and
nearby whole numbers.
Ready to Practice?
Try the Interactive Fractions on a Number Line Tool
Practice identifying the denominator, counting the numerator's intervals, selecting the correct tick mark, locating improper fractions, and interpreting each fraction's position.
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