One of our customer’s contacted us asking for some teacher support.  They wanted to know how to arrange mixed numbers (fractions) and put them in order starting with the smallest.

We offered the following advice:

When ordering fractions, use 0, 1/2, and 1 as benchmarks for comparison. Drawing a long straight line and marking it with 0, 1/2 & 1 will help you to place the other fractions along the line. First work out whether the fraction is more or less than 1 (if the numerator (top) is bigger than the denominator (bottom) it is more than one if not it is less than one). If it is less than 1, check to see if it is more or less than 1/2 (if the numerator is worth more than half of the denominator then it is more than half and vice versa). Then further refine your comparisons to see if the fraction is closer to 0, 1/2, or 1.

After you organize fractions by benchmarks, you can use these methods:
  • Same denominators: If the denominators of two fractions are the same, just compare the numerators. The fractions will be in the same order as the numerators. For example, 5/7 is less than 6/7.
  • Same numerators: If the numerators of two fractions are the same, just compare the denominators. The fractions should be in the reverse order of the denominators. For example, 3/4 is larger than 3/5, because fourths are larger than fifths.
  • Compare numerators and denominators: You can easily compare fractions whose numerators are both one less than their denominators. The fractions will be in the same order as the denominators. (Think of each as being a pie with one piece missing: The greater the denominator, the smaller the missing piece, thus, the greater the amount remaining.) For example, 6/7 is less than 10/11, because both are missing one piece, and 1/11 is a smaller missing piece than 1/7.
  • Further compare numerators and denominators: You can compare fractions whose numerators are both the same amount less than their denominators. The fractions will again be in the same order as the denominators. (Think of each as being a pie with x pieces missing: The greater the denominator, the smaller the missing piece; thus, the greater the amount remaining.) For example, 3/7 is less than 7/11, because both are missing four pieces, and the 11ths are smaller than the sevenths.
  • Equivalent fractions: Find an equivalent fraction that lets you compare numerators or denominators, and then use one of the above rules.
  • Same denominators: If the denominators of two fractions are the same, just compare the numerators. The fractions will be in the same order as the numerators. For example, 5/7 is less than 6/7.
  • Same numerators: If the numerators of two fractions are the same, just compare the denominators. The fractions should be in the reverse order of the denominators. For example, 3/4 is larger than 3/5, because fourths are larger than fifths.
  • Compare numerators and denominators: You can compare fractions whose numerators (tops) are both one less than their denominators (bottoms). The fractions will be in the same order as the denominators. (Think of each as being a pie with one piece missing: The greater the denominator, the smaller the missing piece, thus, the greater the amount remaining.) For example, 6/7 is less than 10/11, because both are missing one piece, and 1/11 is a smaller missing piece than 1/7.
You can also compare fractions whose numerators are both the same amount less than their denominators. The fractions will again be in the same order as the denominators. (Think of each as being a pie with x pieces missing: The greater the denominator, the smaller the missing piece; therefore, the greater the amount remaining.) For example, 3/7 is less than 7/11, because both are missing four pieces, and the 11ths are smaller than the sevenths.  Equivalent fractions: For example 1/2 is the same as 2/4 or 4/8. Find an equivalent fraction that lets you compare numerators or denominators, and then use one of the above rules.