Developing Fraction Number Sense Through Part/Whole Thinking

Written by Donna Boucher

Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.

It’s no secret that fractions are a very difficult concept for students to understand. But why is that? Fractions are an extremely abstract concept, and without adequate concrete and representational experiences that build fraction number sense, students do not develop mental images of what the abstract symbols mean.  

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In Texas, students begin formal fraction instruction in 2nd grade under our TEKS, but the symbolic notation for fractions (1/4, 2/3, etc.) is not taught until 3rd grade. The 2nd grade standards include:

  • partition objects into equal parts and name the parts, including halves, fourths, and eighths, using words
  • explain that the more fractional parts used to make a whole, the smaller the part; and the fewer the fractional parts, the larger the part
  • use concrete models to count fractional parts beyond one whole using words and recognize how many parts it takes to equal one whole

Let’s take a closer look at each.

Partition into equal parts and name the parts

The most fundamental understanding about fractions is that the parts must be equal. Children have a basic understanding about equal parts prior to formal instruction on fractions. They know that when they are sharing a sandwich, for instance, the parts should be the same size for it to be fair.

The next stage of understanding is that there are words that describe the number of equal parts. Two equal parts are called halves, three equal parts are called thirds, four equal parts are called fourths, and so on. The denominator, then, is actually a noun—just like apple or orange

Count fractional parts using concrete models

We can count the equal parts, so we can count fourths just like we can count apples. For example, we can have 1, 2, 3, or 4 fourths. We can even have 5, 6, or 7 fourths, although that would be more than a whole. So the numerator is a counting number.
 
While it may seem obvious to us, students need to understand how many parts it takes to make a whole. For example, if the whole is divided into fourths, it takes 4 fourths to make a whole. This leads students to understand that 3 fourths is less than a whole, while 5 fourths is greater than a whole. Notice that in 2nd grade, students are actually experiencing improper fractions and mixed numbers, although they are not formally identified as such.

Students should also see a variety of representations for fractions. In the pictures above, I’ve used foam fraction circles, but you can also use Cuisenaire rodsfraction tiles or fractions strips. Click here to download both color and B & W fraction strips.

More parts = smaller pieces

Did you groan when you looked at the graphic above because you’ve had so many students tell you 1/8? That is such a common misconception. Without concrete and pictorial experiences seeing pieces of both sizes, students revert to what they know about whole numbers—a bigger number is bigger. Using that reasoning, 1/8 must be bigger, because 8 is greater than 4.

 
If your major takeaway from this post is that students need lots of concrete and pictorial experiences with fractions, then I’ve done my job. This is just so critical.
 

Decomposing fractions

I’m a big fan of number bonds. The part/whole understanding students develop when they work with number bonds is an essential component of being able to work flexibly with numbers. Some time back, I wrote a blog post about how number bonds evolve through more complicated math concepts, and fractions was one of the examples I used. In the example shown below, if students know they need 2 more fifths to make a whole and that they can get those 2 fifths by decomposing the 4 fifths, it sure makes improper fractions and mixed numbers a whole lot simpler. Typically, we teach procedures for adding fractions and converting the sum to a mixed number, and students often follow the steps with no real understanding. Worse yet, they add the denominators and end up with 7 tenths!

Students would, of course, experience this concept first with concrete materials, and it might look and sound something like this:

Then, I thought, why not develop number bond cards for fractions? And that’s exactly what you see pictured. The traditional number bond card is on the left with the fraction number bond card on the right.

Here’s the prep work…

f you know me, you know these fraction number bond cards are going to be hole-punched and put on O-rings! Don’t hole punch them with a single hole punch, though—use your 3-hole punch! It’s so much easier, and your holes line up nicely.

The finished product! I actually put the halves, thirds, and fourths on one O-ring and the sixths and eighths on another. Much more manageable that way.

Meaningful Practice

Students need lots of practice with this concept. And what makes practice engaging and meaningful? GAMES! The games shown here are from Fishing “Whole”: Composing and Decomposing Fractions, which also includes the fraction number bond cards.

There are 24 cards like the ones shown that can be used to play either Memory or Go Fish. The goal is to find two cards that make a whole.

The card shows 1 eighth.  I would ask my partner, “Do you have 7 eighths?”, because 1 eighth and 7 eighths make a whole. If he does, he hands it over and I lay down the pair. If he doesn’t, he says, “Go fish!”.

The other game is a Scoot game. Card 19 shows 4 sixths. It takes 2 more sixths to make a whole, so I’d write 2 sixths in space 19. It’s typically played whole class, but it’s also a great workstation.

I hope this post has provided you with some useful information! I’d love to hear your comments.

  

7 Comments

  1. TheElementary MathManiac

    Another classic post! Since I stopped introducing the symbolic notation of fractions to second graders they have developed fewer misconceptions and are much less likely to use improper whole number reasoning as a third or fourth grader. I like to talk with older kids about the numerator being an adjective and the denominator being a noun.

    Tara
    The Math Maniac

    Reply
    • Donna Boucher

      Thanks, Tara! Yeap Ban Har talked about the “noun” and “adjective” thing when I saw him speak this summer. Such a great analogy! The standards writers definitely got this progression of skills right, IMO, and it’s a big improvement over how we have traditionally taught fractions.

      Reply
  2. Ms F

    Wow this is really interesting and will be really helpful even with the students who have been struggling with fractions. Thank you for posting!

    Reply
  3. Ellen

    Fantastic article, I found this so helpful. I currently teach first grade and they adore playing “Go Fish” to practise combinations of 10. The idea of adapting this game for fractions in later grades is something I will definitely be sharing at my team meeting. Thank you!

    Reply
  4. Linda Dacey

    Love seeing number bonds applied to fractions! I’ve been playing with linear models of fractions as thinking of fractions on number lines are often challenging. I think I noticed on Twitter that you did some work for Shell Education. Great folks there! @Linda_Dacey

    Reply
  5. ali

    Hi Donna,

    In the US, are you required to familiarise your students with unit fractions only, or do you have to show them fractions of a collection of objects too?
    Here in Aus we have to do both from the beginning, but starting with halves, then quarters etc… The kids find unit fractions easy, but the fractions of collections hard to work out independently.
    Do you think students should be well acquainted with unit fractions before going on to fractions of collections?

    Ali

    Reply
    • Donna Boucher

      If I’m not mistaken, Common Core only mentions parts of a whole and number lines but, of course, there are other standards used throughout the US as well. In Texas, our standards start with parts of a whole in 2nd grade, and really hit fractions hard in 3rd. Fractions of sets are not mentioned at all, but the number line is much more prominently featured than it used to be.

      Reply

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