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Fractions as Part of a Set

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.

Fractions are a very abstract concept for students, and it’s important that we provide many different models and representations to help students develop a deep understanding of fraction concepts.

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Van de Walle describes three different models for fractions:

  • AREA MODELS These models involve partitioning something into smaller equal parts. We often use manipulatives such as circular “pie” pieces, geoboards, pattern blocks, and paper folding to represent area models.
  • LENGTH MODELS These models focus on length, rather than area. Materials such as Cuisenaire rods, number lines, and fraction tiles are useful for representing length models.
  • SET MODELS With set models, the whole is a group of objects and subsets of the objects are the fractional parts.

Introducing set models

If students have some familiarity with fractions from working with area or length models, you can introduce the idea of set models with a little problem-solving. Provide each student or pair of students with 12 teddy bear counters, then tell the following story:

Twelve teddy bears were having a picnic. One-half of the bears ate hot dogs and one-half ate hamburgers. How many bears ate hot dogs and how many ate hamburgers?

Give the students time to work on the problem using their counters. By starting with one-half, there is a high probability that most of the students will be able to solve the problem. Ask students to share their solutions, explaining how they arrived at the answer of 6 and 6. You want to draw out that they divided the bears into two equal groups, and each group had six bears. Summarize their thinking by writing the following on the board:

 1/2 of 12 is 6

Pose additional problems:

One-third of the bears drank lemonade. How many bears drank lemonade?

One-fourth of the bears did not have dessert. How many bears didn’t have dessert?

Notice that all of these problems involve a unit fraction, a fraction with a numerator of one. Next, try a problem with a fraction that is not a unit fraction.

Two-thirds of the bears wore shorts. How many bears wore shorts?

This involves determining how many bears are in each equal group and then how many bears are in two of the groups.

You can repeat these activities with different numbers of bears for your set. Consider allowing students to create their own stories!

I created a resource, with both print and digital versions, for exploring set models using the teddy bear theme. You can grab it here.

Here’s a little video of the digital version in action.

How do you teach set models in your classroom? Add a comment to share!

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