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Comparing Fractions: Using a Benchmark of 1/2

“Understanding why a fraction is close to 0, 1/2, or 1 is a good beginning for fraction number sense. It begins to focus on the size of fractions in an important yet simple manner.”
(Van de Walle/ Lovin, 2006)

In this series of blog posts, I am exploring 5 different strategies for comparing fractions. The first post highlighted comparing fractions with like numerators or denominators, while the second post introduced a strategy for comparing fractions one unit fraction from a whole. Another useful strategy is to compare fractions to a benchmark of 1/2.

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Students should have many opportunities to explore fractions equivalent to 1/2 using hands-on materials. My favorite manipulative for this exploration is fraction tiles, because you can easily connect them to fractions on a number line. If you don’t have access to fraction tiles, you can download printable ones here. When the students are using the tiles, I like them to draw a number line above the 1 whole tile (see picture) so they have a visual representation of the fact that proper fractions fall between 0 and 1 on the number line. Why is that important? Check out this post!

Fractions Equivalent to One Half

Challenge students to use their tiles to find fractions that are equivalent to 1/2. Be sure to have the students work in pairs to facilitate mathematical discourse. It’s always interesting to watch this process–some kiddos are very methodical, trying each of the pieces in order (eg., thirds, fourths, fifths, etc.), while others employ a more ‘helter skelter’ approach. I like to highlight the students who use a more organized method and remind students that being organized often helps mathematicians do their work more efficiently.

After the students have found all of the fractions, write them on your whiteboard or an anchor chart. I like to use the order shown below. By not putting the fractions in order (2/4, 3/6, 4/8, etc.), students are more likely to focus on the relationship between the numerator and denominator, which is what you want.

Equivalent to One Half

Now ask students a simple question: What do you notice? Be prepared for many different responses, but these are a few I usually hear:

  • The numerators are “in order” or count by ones (2, 3, 4, 5, 6)
  • The denominators skip count by twos (4, 6, 8, 10, 12)
    Note: These are the relationships students usually see first. When students make them, I certainly record them, but I also challenge students to look for relationships between the numerator and denominator in each fraction
  • If you add the numerator again, you get the denominator (4 + 4 = 8)
  • Two times the numerator is the denominator (4 x 2 = 8)

Oddly enough, students rarely say that the numerator is half the denominator. Once they see it, though, the light bulbs go off! Next, I give them other denominators and ask what the numerator would need to be for the fraction to be equivalent to 1/2.

Equivalent to One Half Missing Numerator

Now is a good time to go back and ask students which tiles they could not use to make a fraction equivalent to 1/2 (thirds and fifths). This is a GREAT discussion! Using the tiles, some students will totally get that half of 5 is 2 1/2, so 2/5 is less than 1/2 and 3/5 is greater than 1/2.

The next step is to have students determine if fractions are less than, equal to, or greater than 1/2. Even though students might see the pattern (numerator is half the denominator), they will still need lots of practice and hands-on learning. To practice, download these fraction cards, show students a card, have them build it with tiles, and then explain why it is less than, equal to, or greater than 1/2 (7/12 is greater than 1/2, because 6/12 is equivalent to 1/2, and 7/12 is more than 6/12). Verbalizing their thinking is really critical because that helps them internalize the relationship between the numerator and denominator in fractions equivalent to 1/2.

comparing fractions

Because multiple representations are important, you can also use number lines for the same type of practice. Download a sheet with the number lines here, cut them into strips, and use them as described above.

comparing fractions

One more strategy to go! Read about the last strategy here.

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  1. I totally agree that the concrete should be infused into this lesson. I use the fraction strips daily when teaching 3rd grade fractions.

  2. I am a Differentiation Instructor (Interventionist) in my school so need to use concrete, hands-on activities. Thank you for offering the materials for free.

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