Math Background

Lesson: Comparing and Ordering Fractions
Developing the Concept

Students should be comfortable using models to compare and order fractions before you ask them to use benchmarks and equivalent fractions to perform the same task.

Materials: none

Preparation: none

  • Take notes on the board or overhead projector as students devise rules for benchmarks for 0, one-half, and 1. Some students may be able to devise the rules without a number line; others may need to see how the fractions fall visually.
  • Ask: How can you tell when a fraction is close to 1?
    Discuss this until students agree that, since the denominator tells how many equal parts in 1, the closer the numerator is to the denominator, the closer the value of the fraction is to 1.
  • Ask: Is seven-eighths close to 1?
    By their definition, seven-eighths is close to 1.
  • Ask: Is one-half close to 1?
    Even though 1 is close to 2, you can't really say that one-half is close to 1.
  • Ask: How can you tell when a fraction is close to one-half?
    This question should also lead to lively discussion. Encourage students to come to the conclusion that a fraction is close to one-half if the numerator is about one half as large as the denominator.
  • Ask: Is five-eighths close to one-half?
    By their definition, five-eighths is close to one-half.
  • Ask: How do you think you can tell when a fraction is close to 0?
    Given the discussions so far, this one should easily lead to the conclusion that a fraction is close to zero when the numerator is very small compared to the denominator. This gets more obvious as the denominator gets larger (1 in the numerator makes a fraction seem close to 0 unless the denominator is 2).
  • Say: Use the benchmarks you just made up to compare these numbers to 0, one-half, or 1: seven-eighths, one-fifth, and three-eighths.
    seven-eighths is close to 1; one-fifth is close to 0; three-eighths is close to one-half but less than one-half.
  • Ask: Can you use benchmarks to order seven-eighths, one-fifth, and three-eighths from least to greatest?
    Given the previous exercise, students should easily order these numbers: one-fifth, three-eighths, seven-eighths.

    Try several more simple benchmark-ordering exercises.

  • Ask: What can you do if you're not sure about how two fractions compare?
    Help students to arrive at the conclusion that they can write equivalent fractions with like denominators if they're not sure of a comparison.

    Review writing equivalent fractions.
    Write the fractions three-fifths, two-tenths, and seven-tenths on the board.

  • Say: Suppose you want to order three-fifths, two-tenths, and seven-tenths from least to greatest. You can see that if all the fractions had the same denominator, you would only need to compare the numerators.
  • Ask: How can we find an equivalent fraction for three-fifths with a denominator of 10?
    Guide students to see that to find an equivalent fraction, you need to multiply both the numerator and denominator by the same number.
  • Ask: What number times 5 will give you 10? (2)
    Then show the multiplication.
    three-fifths = three times two over five times two = six-tenths
    Rewrite the fractions: six-tenths, two-tenths, seven-tenths
  • Say: Now order the fractions from least to greatest. (two-tenths, six-tenths, seven-tenths)

Wrap-Up and Assessment Hints
After students have worked comfortably with manipulatives, benchmarks, and equivalent fractions as they compare and order fractions, ask them to make up fractions to compare and work with a partner. You'll know they understand the concepts when they ask each other to compare fractions with greater denominators whose numerators are close to each other. When assessing students, check that they have a clear understanding of fraction concepts, such as the meaning of numerator, denominator, and improper fraction. Determine whether students can write improper fractions as mixed numbers, write fractions in simplest form, and find equivalent fractions. These skills must be mastered before students progress to computing with fractions.


Houghton Mifflin Math Grade 4