Q.1 Identify smaller and larger fractions
What is a fraction?
A fraction represents a part of a whole. It is written with two numbers: a numerator and a denominator. The denominator tells you how many equal parts the whole is split into, and the numerator tells you how many of those parts you are considering.
- A pizza cut into 8 equal slices: one slice is 18 of the pizza.
- A chocolate bar broken into 12 equal squares: three squares are 312 of the bar.
- A gallon of milk divided into 4 equal quarts: one quart is 14 of the gallon.
The denominator is the “name” of the parts (halves, thirds, fourths), and the numerator tells you the “count” of those parts you have.
Understanding the denominator
The denominator is the bottom number in a fraction. It tells you the size of each equal part. The larger the denominator, the smaller each part is, because the whole is being divided into more pieces.
- In 12, the denominator 2 means the whole is split into 2 equal parts. Each part is one-half.
- In 18, the denominator 8 means the whole is split into 8 equal parts. Each part is one-eighth.
- One-eighth (18) is smaller than one-half (12) because the whole is cut into more pieces.
Think of sharing a cookie with a friend (2 parts) versus sharing it with 10 friends (10 parts). When you share with more people, each person’s piece is smaller. The denominator works the same way.
Comparing fractions with the same denominator
When two fractions have the same denominator, the parts are the same size. To compare them, look only at the numerators. The fraction with the larger numerator is the larger fraction because it has more of those equal parts.
- Compare 38 and 58. Since the denominators are the same (eighths), compare 3 and 5. 5 is greater, so 58 > 38.
- Compare 25 and 45. Both are fifths. 4 is greater than 2, so 45 > 25.
- If you have 16 of a pie and a friend has 46 of the same pie, your friend has more because 4 is larger than 1.
This is the simplest way to compare fractions. The denominator tells you the size of the piece, and the numerator tells you how many pieces. With the same denominator, it is a simple “more pieces means more of the whole.”
Comparing fractions with the same numerator
When two fractions have the same numerator, the fraction with the smaller denominator is the larger fraction. Why? Because a smaller denominator means the whole is divided into fewer, larger pieces, and you have the same number of those larger pieces.
- Compare 13 and 16. Both have a numerator of 1. Thirds are larger than sixths, so 13 > 16.
- Compare 34 and 38. Both have 3 pieces. Fourths are larger than eighths. Therefore, 34 is greater than 38.
- Imagine you have two identical sandwiches. You cut one into 2 equal halves and the other into 4 equal quarters. If you take 1 piece from each sandwich, you get half (12) from the first and a quarter (14) from the second. The half is clearly larger.
This rule can be tricky at first. Remember: with the same numerator, the fraction with the smaller denominator has the larger pieces, so it is the greater fraction.
Using benchmark fractions to compare
Benchmark fractions are common fractions that you know well, like 12, 14, and 34. You can compare other fractions to these benchmarks to decide if they are smaller or larger.
- Is 25 greater than or less than 12? Think: 12 is the same as 2.55. Since 2 is less than 2.5, 25 < 12.
- Compare 78 to 34. Since 34 is the same as 68, and 7 is greater than 6, then 78 > 34.
- If a fraction’s numerator is less than half of its denominator, it is less than 12. If the numerator is more than half of the denominator, it is greater than 12.
Using benchmarks like 0, 12, and 1 is a powerful mental math strategy. It helps you quickly estimate the size of a fraction without finding a common denominator.
Comparing fractions using a common denominator
When fractions have different numerators and different denominators, you can find a common denominator. This means rewriting both fractions as equivalent fractions with the same denominator. Then, compare the numerators.
- Compare 23 and 34. A common denominator is 12. 23 = 812 and 34 = 912. Since 9 > 8, 34 > 23.
- Compare 56 and 79. Use a common denominator of 18. 56 = 1518 and 79 = 1418. Since 15 > 14, 56 > 79.
- This method always works. You can find a common denominator by multiplying the two denominators together or by finding their least common multiple.
Finding a common denominator does not change the value of the fractions. You are simply renaming them in a way that makes them easier to compare.
Comparing fractions using number lines
A number line is a visual tool that helps you compare fractions. When you plot two fractions on a number line, the fraction that is farther to the right is the larger fraction.
- Plot 14 and 34 on a number line from 0 to 1. 14 is closer to 0, and 34 is closer to 1. Therefore, 34 > 14.
- Draw a number line divided into eighths. Mark 58 and 24. Since 24 is the same as 48, 58 is to the right of 48. So, 58 > 24.
- Number lines are especially helpful when fractions have different denominators because you can see the exact position of each fraction.
When using a number line, always make sure the intervals (the spaces between 0 and 1, 1 and 2, etc.) are equal. This ensures your comparison is accurate.
Identifying fractions greater than one
A fraction is greater than 1 when its numerator is larger than its denominator. These are called improper fractions. They represent more than one whole. For example, 54 means you have five quarters, which is more than one whole (since four quarters make a whole).
- 32 is greater than 1 because 3 > 2. It is equal to 1 and one-half.
- 74 is greater than 1 because 7 > 4. It is equal to 1 and three-quarters.
- When comparing an improper fraction to a proper fraction (numerator less than denominator), the improper fraction is always larger because it is greater than 1, while the proper fraction is less than 1.
Any fraction with a numerator that is equal to the denominator is equal to 1, such as 44 or 77.
Common Core alignment: CCSS.MATH.CONTENT.4.NF.A.2 – Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.4.NF.A.2. All content is 100% free, use it for whole-class instruction, independent study and practice, or homework.